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abstract: 'We present a scheme to establish non-classical correlations in the motion of two macroscopically separated massive particles without resorting to entanglement in their internal degrees of freedom. It is based on the dissociation of a diatomic molecule with two temporally separated Feshbach pulses generating a motional state of two counter-propagating atoms that is capable of violating a Bell inequality by means of correlated single particle interferometry. We evaluate the influence of dispersion on the Bell correlation, showing it to be important but manageable in a proposed experimental setup. The latter employs a molecular BEC of fermionic Lithium atoms, uses laser-guided atom interferometry, and seems to be within the reach of present-day technology.'
author:
- Clemens Gneiting
- Klaus Hornberger
title: Bell test for the free motion of material particles
---
[*Introduction.—*]{}The possibility of entangling macroscopically separate, non-interacting particles challenges our classical view of the world by putting into question the concepts of realism and locality [@Bell1987a]. Nowadays entangled states are routinely established with photons [@Brendel1999a]. Also the internal states of material particles have been entangled, e.g., using non-classical light as a carrier for the quantum correlations [@Julsgaard2001a; @Sherson2006a; @Matsukevich2006a; @Moehring2007a; @Rosenfeld2007akurz], or their Coulomb interaction in an ion trap [@Leibfried2005a; @Haffner2005a]. However, the original discussion of entanglement focused on the *motional* state of massive particles, whose spatial separation is a dynamic feature of the entangled two-particle wave function [@Einstein1935a; @Schrodinger1935a]. The latter is spatially extended, unlike with internal entanglement, where the positions only play the passive role of separating the parties. Since the positions and momenta are quantum observables with a direct classical analog, an observation of non-classical correlations arising from macroscopically distinct phase space regions would therefore be a striking demonstration of the failure of classical mechanics.
A convincing demonstration of non-classical correlations between two parties requires the experimental violation of a Bell-type inequality [@Clauser1969a]. In the simplest case, it involves detecting dichotomic properties on each side, such as the polarization of spin-1/2 systems, where a maximal violation is obtained if the spins are in a Bell state, say ${|\Phi\rangle} = ({{|\uparrow\rangle}_1} {{|\uparrow\rangle}_2} + {\mathrm{e}}^{{\mathrm{i}}\phi}{{|\downarrow\rangle}_1} {{|\downarrow\rangle}_2})/\sqrt{2}$. Such a dichotomic property is not readily available in the motional state of free, structureless particles. One possibility is to consider observables which have no classical analogue, such as the displaced parity or pseudospin operators used to discuss entanglement in the ‘original EPR state’ [@Banaszek1998a; @Chen2002akurz]. While they are expedient for characterizing the quadratures of light fields, their experimental implementation seems exceedingly difficult in the case of free, macroscopically separated material particles, where only position measurements are easily realized.
![\[BellTest\] (a) The two particles in a DTE state are each characterized by an early and a late wave packet component resulting from two different dissociation times. (b) Each particle is passed through an unbalanced Mach-Zehnder interferometer with a switch deflecting the early wave component into the long arm, while conducting the late component through the short arm. (c) The path difference is chosen such that the early and the late wave packets overlap. Detecting the particle in an output port, at a given phase $\varphi$ and splitting ratio $\vartheta$, amounts to a dichotomic measurement analogous to a spin $1/2$ detection in an arbitrary orientation. The existence of DTE is established, implying a macroscopically delocalized two-particle state, if the correlations at the single-particle interferometers violate a Bell inequality. ](fig1_300dpi.eps){width="0.95\columnwidth"}
Here we aim instead at producing a motional analogue of the Bell state ${|\Phi\rangle}$ and implementing a Bell test which requires a simple position measurement in the end. The idea is to expose a diatomic molecule to a sequence of two temporally separated dissociation pulses. Each of the two counter-propagating dissociated atoms then has an early and a late wave packet component corresponding to the two possible dissociation times, and their correlation may be called “dissociation time entanglement” (DTE), see Fig. \[BellTest\] (a). If one regards these components as spin state analogues in the motion (say, early corresponds to spin up, late to spin down), the DTE state shows the same structure as the Bell state ${|\Phi\rangle}$. Switched, unbalanced Mach-Zehnder interferometers on each side then serve to mimic arbitrary spin rotations, and the detection of each particle in one of the interferometer output ports completes the Bell test, see Fig. \[BellTest\] (b,c).
The DTE state is a variant of the “energy-time entangled” state introduced in [@Franson1989a], and it is closely related to the “time-bin entanglement” of photons [@Brendel1999a; @Tittel2000a; @Simon2005a], which has been used e.g. for establishing non-local correlations over fiber distances of more than 50km [@Marcikic2004a] in a similar setup as in Fig. \[BellTest\]. As a main difference, the DTE state is not composed of two single-particle product states, but it superposes their *relative* coordinate, while the center-of-mass state remains separable during the two-pulse dissociation process.
Our use of DTE reflects the necessity to come up with a state generation scheme appropriate to *material* particles. Their finite mass and internal structure entail substantial complications which require a careful investigation of whether a Bell violation can be expected at all. Most prominently, one must account for the unavoidable wave packet dispersion, and only the recent progress in manipulating ultra-cold molecules (such as their condensation [@Kohler2006a] and controlled dissociation [@MukaiyamaDurrGreiner]) suggests the possibility to generate motional states that allow one to keep the detrimental effect of dispersion under control. We note that other ways to demonstrate nonlocal correlations of molecular dissociation products have been proposed in [@Opatrny2001a; @Kheruntsyan2005a].
We will show below that, by appropriately choosing a sequence of magnetic field pulses, the Feshbach-induced dissociation of an ultracold ${}^6$Li$_2$ molecule within a guiding laser beam can generate a motional state of macroscopically entangled atoms that is capable of violating a Bell inequality. But first, to clarify the implications of dispersion, we determine the correlation function for a generic DTE state subject to correlated single-particle interferometry.
[*The general DTE Bell test.—*]{}It is natural to take the bound molecular two-particle state to be separable in the center-of-mass (${\text{c.m.}}$) and the relative (${\text{rel}}$) coordinate. Denoting by $\tau$ the period between the two dissociation pulses and assuming the transverse motion to be frozen in the ground state of the guiding laser beam, the longitudinal part of a pure DTE state then takes the form ${|\Psi_{\text{DTE}}\rangle} = ({\hat{\mathrm{U}}}^{\text{(0)}}_{z,\tau} {|\Psi_0\rangle} + {\mathrm{e}}^{{\mathrm{i}}\phi_{\tau}} {|\Psi_0\rangle})/\sqrt{2}$, where ${|\Psi_0\rangle} = {|\psi_0^{{\text{c.m.}}}\rangle} ({|\psi_0^{{\text{rel}}}\rangle}+{\hat{\mathrm{P}}}{|\psi_0^{{\text{rel}}}\rangle})/\sqrt{2}$. It involves the free time-evolution operator ${\hat{\mathrm{U}}}^{\text{(0)}}_{z,t}$, the parity operator ${\hat{\mathrm{P}}}$, and a relative phase $\phi_{\tau}$ determined by the details of the two-pulse dissociation process. The center-of-mass state of the original molecule ${|\psi_0^{{\text{c.m.}}}\rangle}$ is taken to rest in a wave guide, while ${|\psi_0^{{\text{rel}}}\rangle}$ propagates into positive direction.
Indicating the output ports of the two interferometers $j=1,2$ by $\sigma_j=\pm 1$, the immediate experimental outcome is characterized by the two-time probability density $\mathrm{pr}(\sigma_1,\sigma_2;t_1,t_2)$ for detecting particles behind the respective ports at the times $t_1,t_2$. It depends on the phase settings $\varphi_j$ and mirror transmission angles $\vartheta_j$ of the interferometers, and the dispersive evolution of the DTE state leads to a complicated fringe pattern as a function of $t_1$, $t_2$, and $\tau$. However, as suggested by the above analogy with a discrete Bell test, a robust quantity characterizing entanglement is obtained by integrating the port-specific probabilities over all times, ${P}_{\sigma_1,\sigma_2}=\int\mathrm{pr}(\sigma_1,\sigma_2;t_1,t_2){\mathrm{d}}t_1{\mathrm{d}}t_2$. For any reasonable model of the time-of-arrival detection, the correlation function can be equally evaluated by means of the projections ${\hat{\Pi}}_{\sigma_1}\otimes {\hat{\Pi}}_{\sigma_2}\equiv {\hat{\Pi}}^{\sigma_1}_{\sigma_2}$ onto the (unbounded) spatial regions behind the respective ports, ${P}_{\sigma_1,\sigma_2}= \lim_{t\to\infty} ||{\hat{\Pi}}^{\sigma_1}_{\sigma_2}{\hat{\mathrm{U}}}_{z,t}{|\Psi_{\text{DTE}}\rangle}||^2$.
Since we are only interested in the interferometer output, the effect of dispersion is best incorporated by time-dependent scattering theory, which separates the ‘raw action’ of the interferometers, described by the S-matrices ${\hat{\mathrm{S}}}_j$, from the free dispersive time evolution ${\hat{\mathrm{U}}}_{z,t}^{\text{(0)}}$. The projection of the DTE state component ${|\psi^{\text{c.m.}}_0\rangle}{|\psi^{\text{rel}}_0\rangle}$ onto a particular output-port combination $\sigma_1$, $\sigma_2$ then takes the form $$\label{eq:1}
{\hat{\Pi}}^{\sigma_1}_{\sigma_2} {\hat{\mathrm{U}}}_{z,t}^\text{(on)}
{|\psi^{\text{c.m.}}_0\rangle}{|\psi^{\text{rel}}_0\rangle} = {\hat{\mathrm{U}}}_{z,t}^{\text{(0)}}[{\hat{\mathrm{S}}}_{\sigma_1}^\text{(on)}\otimes {\hat{\mathrm{S}}}_{\sigma_2}^\text{(on)}]
{|\psi^{\text{c.m.}}_0\rangle}{|\psi^{\text{rel}}_0\rangle} \,.$$ Here we assume $t$ to be sufficiently large, so that the wave packets have passed the interferometers, and we take the entrance switches to be in the “on” configuration, i.e., routing towards the long arms. Implementing the phase shifts by the arm-length variations $\ell_j$, the projected S-matrices ${\hat{\mathrm{S}}}_{\sigma_j}^\text{(on)}={\hat{\Pi}}_{\sigma_j}{\hat{\mathrm{S}}}_j^\text{(on)}$ are given by ${\hat{\mathrm{S}}}^\textrm{(on)}_{\sigma_j = +1} = \exp({\mathrm{i}}\hat{p}_j \ell_j/\hbar) \cos \vartheta_j$ and ${\hat{\mathrm{S}}}^\textrm{(on)}_{\sigma_j = -1} = \exp({\mathrm{i}}\hat{p}_j \ell_j/\hbar) \sin \vartheta_j$. For the “off” configuration we have correspondingly ${\hat{\mathrm{S}}}^\textrm{(off)}_{\sigma_j = +1} = \sin \vartheta_j$ and ${\hat{\mathrm{S}}}^\textrm{(off)}_{\sigma_j = -1} = -\cos \vartheta_j$. Analogous relations hold for ${|\psi_0^{\text{c.m.}}\rangle}{\hat{\mathrm{P}}}{|\psi_0^{\text{rel}}\rangle}$. The setup requires the dispersion-induced broadening of the wave packets to remain much smaller than the separation between the early and the late components, so that the switches can be changed in between. In this case the correlation function ${P}_{\sigma_1,\sigma_2}$ can be evaluated by using Eq. (\[eq:1\]) and its variants even for non-pure and non-separable initial two-particle states, ${|\Psi_0\rangle}{\langle \Psi_0|}\to\varrho_0$. One obtains $$\begin{aligned}
\label{eq:2}
{P}_{\sigma_1,\sigma_2}(\ell_1,\ell_2) =&
\frac{1}{4} \Big\{ 1 + \sigma_1 \sigma_2 \; \text{Re} \Big[ {\mathrm{e}}^{- {\mathrm{i}}\phi_{\tau}}
\int \!{\mathrm{d}}p_1 \int \!{\mathrm{d}}p_2
\nonumber\\
& \times
\exp\Big(
{\mathrm{i}}\frac{\vec{p} \cdot \vec{\ell}}{\hbar}-{\mathrm{i}}\frac{\vec{p}^{\,2}\tau}{2 m \hbar}
\Big)\mathrm{pr}(p_1,p_2) \Big] \Big\},\end{aligned}$$ with $\vec{p} = (p_1,p_2)^{\text{T}}$, $\vec{\ell} = (\ell_1,\ell_2)^{\text{T}}$, and $\mathrm{pr}(p_1,p_2)={\langle p_1,p_2|}\varrho_0{|p_1,p_2\rangle}$ the momentum distribution function. For simplicity we take here the beam splitters to be symmetric ($\vartheta_j= \pi/4$) and the particles to have equal mass $m$. Note that (\[eq:2\]) is independent of the total time of flight and invariant under momentum phase transformations ${\langle p_1,p_2|}\Psi_0\rangle\to \exp[{\mathrm{i}}\xi(p_1,p_2)]{\langle p_1,p_2|}\Psi_0\rangle$, which includes spatial translations, thus rendering the correlation ${P}_{\sigma_1,\sigma_2}(\ell_1,\ell_2)$ a robust signal.
For generic Gaussian states in the center-of-mass and relative motion Eq. (\[eq:2\]) can be evaluated in closed form. The variances $\sigma^2_{p,{\text{c.m.}}}$ and $\sigma^2_{p,{\text{rel}}}$ then determine characteristic dispersion times, $T_{{\text{c.m.}}} = 2m \hbar/\sigma_{p,{\text{c.m.}}}^2$ and $T_{{\text{rel}}} = m \hbar/2\sigma_{p,{\text{rel}}}^2$, indicating the transition to a dispersion-dominated spatial extension of the wave packets. The expectation value of the relative momentum $p_{0,{\text{rel}}}= m v_{\text{rel}}/2 $ defines the reduced wave length $\lambdabar_{{\text{rel}}} = \hbar/p_{0,{\text{rel}}}$, which sets the scale for the non-local interference fringes in the explicit correlation function,
$$\begin{aligned}
\label{eq:3}
{P}_{\sigma_1,\sigma_2}( \ell_1,\ell_2) = \frac{1}{4} \Bigg\{& 1 + \sigma_1 \sigma_2
\frac{\big( 1+{\tau^2}/{T_{{\text{c.m.}}}^2} \big)^{-{1}/{4}}}{ \big( 1+{\tau^2}/{T_{{\text{rel}}}^2} \big)^{{1}/{4}}}
\exp\bigg[
- \frac{T_{{\text{rel}}}}{T_{{\text{rel}}}^2+\tau^2} \frac{(\ell_1-\ell_2-\tau v_{{\text{rel}}})^2}{2 v_{{\text{rel}}}\lambdabar_{{\text{rel}}}}
- \frac{T_{{\text{c.m.}}}}{T_{{\text{c.m.}}}^2+\tau^2}\frac{(\ell_1+\ell_2)^2}{2 v_{{\text{rel}}}\lambdabar_{{\text{rel}}}}\bigg]
\nonumber \\
& \times \cos \bigg[ \frac{\ell_1-\ell_2}{\lambdabar_{{\text{rel}}}}
+ \frac{\tau}{T_{{\text{rel}}}^2+\tau^2} \frac{(\ell_1-\ell_2-\tau v_{{\text{rel}}})^2}{2 v_{{\text{rel}}}\lambdabar_{{\text{rel}}}}
+ \frac{\tau}{T_{{\text{c.m.}}}^2+\tau^2} \frac{(\ell_1+\ell_2)^2}{2 v_{{\text{rel}}}\lambdabar_{{\text{rel}}}}
- \frac{\varphi_0}{2} \bigg] \Bigg\} ,\end{aligned}$$
with $\varphi_0 = \tau v_{{\text{rel}}}/ \lambdabar_{{\text{rel}}}+\arctan(\tau/T_{{\text{c.m.}}})+\arctan(\tau/T_{{\text{rel}}})+2\phi_{\tau}$.
These DTE correlations can violate a Bell inequality. This is seen from the structural similarity of (\[eq:3\]) to the correlation function of the standard spin-1/2-based setup, ${P}^{\text{spin}}_{\sigma_1,\sigma_2}(\varphi_1,\varphi_2) = \left\{ 1+\sigma_1 \sigma_2 \cos(\varphi_1-\varphi_2) \right\}/4$, where the $\sigma_j=\pm 1$ denote the spin measurement outcomes for analyzers tilted by the angles $\varphi_j$ with respect to the Bell state quantization axis. It follows from this analogy that an unambiguous demonstration of entanglement requires the fringe visibility of the correlation signal to exceed $1/\sqrt{2}$ over at least a few periods.
The dispersive suppression of this fringe visibility is described by those terms in (\[eq:3\]) which depend on the characteristic times $T_{{\text{c.m.}}}$ and $T_{{\text{rel}}}$. Specifically, the dispersion-induced distortion between the early and the late wave packet components is reflected in the Lorentzian reduction factor and in the quadratic compression of the fringe pattern, while their envelope mismatch causes the Gaussian suppression. Based on this result one finds that nonlocal correlations can be observed, for $\lambdabar_{{\text{rel}}}/(\tau v_{{\text{rel}}}) \ll 1$, provided $(1+\tau^2/T_{\text{c.m.}}^2)(1+\tau^2/T_{\text{rel}}^2)<4$. In the following, we present a conceivable scenario for the generation of a DTE state, which meets these conditions.
[*Experimental scenario for a DTE Bell test.—*]{}We suggest to use a dilute molecular Bose-Einstein condensate (BEC) produced from a 50:50 spin mixture of fermionic ${}^6\text{Li}$ as a starting point. It can be prepared efficiently and with near-perfect purity, displaying a huge lifetime of more than 10s due to the Pauli blocking of detrimental 3-body collisions [@Jochim2003a]. A truly macroscopic time separation between the two pulses, say $\tau=1\,$s, is thus conceivable, and for a realistic dissociation velocity of $v_{{\text{rel}}}=$1cm/s the de Broglie wave length $\lambda_{{\text{rel}}}=13.3\mu\text{m}$ is large enough to pose viable stability requirements for the interferometers.
The BEC is prepared in a red-detuned, far off-resonant laser beam (transverse trap frequency $\omega_\text{G}/2\pi= 300$Hz), strong enough to guide the dissociated atoms towards the single particle interferometers. The intersection with a second, weak laser beam creates an elongated dipole trap for the BEC within the laser guide [@Fuchs2007a]. At the end of the preparation steps it is very shallow (depth $U_\text{T}/k_B=100\,\text{nK}$, $\omega_\text{T}/2\pi= 0.5$Hz) and only a small number of molecules (on the order of $10^{2}$) remains in the BEC. These can be taken to be non-interacting, so that the initial longitudinal center of mass state ${|\psi_\text{T}\rangle}$ of the molecules is straightforwardly defined by the trap parameters.
Each interferometer (with path length difference $\tau v_{\text{rel}}/2\simeq 5\,$mm) is implemented by two more red-detuned laser beams crossing the guide in a triangular arrangement at small angles. While the crossings act as beam splitters, the required atom mirror may be realized using an evanescent light field or a blue-detuned laser beam perpendicular to the interferometer plane [@Adams1994a; @Kreutzmann2004a]. The time controlled appliance of such perpendicular blocking beams could also implement the switch. However, a simplified setup could replace the switch by an ordinary beam splitter, at the cost of 50% post-selection.
The fluorescence detection of the slow, strongly confined atoms at the two output guides can be done with near unit efficiency and with single particle resolution, since no particular spatial or temporal accuracy is needed. This single-particle resolution is crucial since events where more than one molecule gets dissociated are disregarded in the present scenario, relying on post-selection. In a more refined setup, it is conceivable to use a specially prepared optical lattice where each site is occupied by at most one molecule [@Volz2006a].
All this implies that the molecular dissociation in presence of the wave guide must meet a number of rather restrictive criteria to render a demonstration of macroscopic entanglement possible. Only the transverse ground state of the guide may be populated to admit the above quasi-one-dimensional description of the interferometers, while both the momentum spread of the wave packets and the dissociation probability must be sufficiently small. In order to judge the feasibility of our experimental scenario, we now discuss how the dissociated part of the state, ${|\Phi_{{\text{bg}}}(t)\rangle}$, depends on the magnetic field pulse sequence and the resonance parameters.
A Green function analysis within the two-channel single-resonance approach shows that after an arbitrary magnetic field pulse sequence (close to an isolated resonance) ${|\Phi_{{\text{bg}}}(t)\rangle}$ is described, for low energies, at positions far from the dissociation center, and at large times, by the asymptotic form ${|\Phi_{{\text{bg}}}(t)\rangle} \sim C_{{\text{bg}}} {|\varphi_{0,0}^{\text{c.m.}}\rangle}{|\varphi_{0,0}^{\text{rel}}\rangle}{\hat{\mathrm{U}}}^{\text{(0)}}_{{z},t}{|\Psi_{{z}}\rangle}$, where ${\hat{\mathrm{U}}}^{\text{(0)}}_{{z},t}$ is the free propagator in the longitudinal direction and where the transverse motion is frozen in the harmonic ground state, ${|\varphi_{0,0}^{\text{c.m.}}\rangle}$ and ${|\varphi_{0,0}^{\text{rel}}\rangle}$, respectively, of the guiding laser beam. The longitudinal state is determined by ${\langle p_{\text{c.m.}},p_{\text{rel}}|\Psi_{{z}}\rangle} = \tilde{C}(p_{\text{c.m.}}^2/4m\hbar+p_{\text{rel}}^2/m\hbar+2\omega_\text{G}){\langle p_{\text{c.m.}}|}\psi_\text{T}\rangle/||\tilde{C}||$, where $\tilde{C}(\omega)$ is the Fourier transform of the closed channel probability amplitude $C(t)$ and $||\tilde{C}||^2 = \int {\mathrm{d}}p_{\text{c.m.}}{\mathrm{d}}p_{\text{rel}}|\tilde{C}(p_{\text{c.m.}}^2/4m\hbar+p_{\text{rel}}^2/m\hbar+2\omega_\text{G})|^2 |{\langle p_{\text{c.m.}}|}\psi_\text{T}\rangle|^2$. ${C}(t)$, in turn, is determined by the coupled channel dynamics as induced by the externally controlled magnetic field $B(t)$. The dissociation probability is given by $|C_{{\text{bg}}}|^2 = \omega_{\text{G}} a_{{\text{bg}}} \mu_{{\text{res}}} \Delta B_{{\text{res}}} ||\tilde{C}||^2 / \pi \hbar^2$. It involves the background scattering length $a_{{\text{bg}}}$, the resonance width $\Delta B_{{\text{res}}}$, and $\mu_{{\text{res}}}$, the difference between the magnetic moments of the resonance state and the open channel.
While the association of the molecules is best done at a broad resonance [@Jochim2003a], shifting to a narrow resonance (e.g., $\Delta B_{{\text{res}}}=1$mG, $\mu_{{\text{res}}}=0.01\,\mu_\text{B}$, $a_{{\text{bg}}}=100 a_0$) one can ensure by choosing field pulses $B(t)$ with short duration (e.g. $T=60\,$ms) that about a single molecule dissociates on average. A sequence of two square pulses with base value $B_0$ and height $\Delta B$ sweeping over the resonance position $B_{{\text{res}}}$ then generates a DTE longitudinal wave packet of the required form, ${|\Psi_{{z}}\rangle} = [{\hat{\mathrm{U}}}^{\text{(0)}}_{{z},\tau} + {\mathrm{e}}^{{\mathrm{i}}\phi_\tau}] {|\Psi_0\rangle}/\sqrt{2}$. The momentum distribution is given by $|{\langle p_{\text{c.m.}},p_{\text{rel}}|}\Psi_0\rangle|^2=p_0\bar{p}^4\text{sinc}^2[(p_{\text{c.m.}}^2/4+p_{\text{rel}}^2-p_0^2)/\Delta p^2]|{\langle p_{\text{c.m.}}|}\psi_\text{T}\rangle|^2/\pi[(p_{\text{c.m.}}^2/4+p_{\text{rel}}^2-p_0^2+\bar{p}^2)\Delta p]^2$, where we define $p_0^2/m=\mu_\text{res}(B_0+\Delta B-B_\text{res})-2U_\text{T}-\hbar\omega_\text{G}$, as well as $\bar{p}^2/m=\mu_{\text{res}}\Delta B$ and $\Delta p^2=2m\hbar/T$ [^1]. Instead of evaluating Eq. (\[eq:2\]) directly with this momentum distribution, it is more transparent to approximate the latter by Gaussians, which allows one to apply the analysis following Eq. (\[eq:3\]). For reasonable pulses $B_0+\Delta B-B_\text{res}=350$mG these Gaussians are centered at $p_{0,{\text{rel}}}=\pm m v_{\text{rel}}/2$ and $p_{0,{\text{c.m.}}}=0$, with spreads of $\sigma_{p,{\text{rel}}}=1.196 m \hbar/(p_0 T)$ and $\sigma_{p,{\text{c.m.}}}=\sqrt{\hbar\omega_\text{T}m/2}$, respectively. This yields $T_{{\text{rel}}} = 3.4$s and $T_{\text{c.m.}}=0.64\,$s, implying a visibility of about 72% in the correlation signal, which exceeds the threshold value of $1/\sqrt{2}$.
Our analysis thus shows that the observation of a macroscopic DTE state is feasible with material particles, even though dispersion poses tight constraints. The technological challenge is substantial, but not insurmountable. Stable lasers are required and the magnetic pulse sequence must be reproducible with a relative accuracy of $10^{-5}$ from shot to shot, so that the relative phase between the early and late components, given by $\phi_\tau \simeq [2 U_\text{T}\tau-\mu_{{\text{res}}} \Delta B \, T + \mu_{{\text{res}}} (B_{{\text{res}}}-B_0) \tau]/\hbar+\omega_\text{G}\tau$, varies less than 50mrad. Realistic choices of the laser wave length ($1\,\mu$m) and the vacuum pressure ($10^{-8}\,$mbar) suffice to suppress decoherence due to scattering of off-resonant photons or background gas particles.
A great advantage of this setup is that no interferometric stability is required between the two interferometers, so that truly macroscopic separations are feasible. Moreover, the DTE state reveals its entanglement robustly since neither a spatial nor temporal resolution is required in the detection. This work was supported by the DFG Emmy Noether program.
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|
---
abstract: 'We give a short overview of the proof of Shelah’s eventual categoricity conjecture in universal classes with amalgamation [@ap-universal-v9].'
address: 'Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA'
author:
- Sebastien Vasey
bibliography:
- 'uc-categ-overview.bib'
date: |
\
AMS 2010 Subject Classification: Primary 03C48. Secondary: 03C45, 03C52, 03C55, 03C75, 03E55.
title: 'The lazy model theoretician’s guide to [S]{}helah’s eventual categoricity conjecture in universal classes'
---
Introduction
============
We sketch a proof of:
\[main-thm\] Let ${K}$ be a universal class with amalgamation. If ${K}$ is categorical in[^1] *some* $\lambda > H_2$, then ${K}$ is categorical in *all* $\lambda' \ge H_2$.
The reader should see the introduction of [@ap-universal-v9] for motivation and history. Note that (as stated there) the amalgamation hypothesis can be removed assuming categoricity in cardinals of arbitrarily high cofinality. However this relies on hard arguments of Shelah [@shelahaecbook Chapter IV], so we do not discuss it. There are plans for a sequel where the amalgamation hypothesis will be removed under categoricity in a single cardinal of arbitrary cofinality (earlier versions actually claimed it but the argument contained a mistake).
Note that this is not a self-contained argument, we simply attempt to outline the proof and quote extensively from elsewhere. For another exposition, see the upcoming [@bv-survey].
We attempt to use as few prerequisites as possible and make what we use explicit. We do not discuss generalizations to tame AECs with primes [@categ-primes-v3], although we end up using part of the proof there.
We assume familiarity with a basic text on AECs such as [@baldwinbook09] or the upcoming [@grossbergbook]. We also assume the reader is familiar with the definition of a good ${\mathcal{F}}$-frame (see [@shelahaecbook Chapter II] for the original definition of a good $\lambda$-frame and [@ss-tame-toappear-v3 Definition 2.21] for good ${\mathcal{F}}$-frames), and the definition of superstability (implicit in [@shvi635], but we use the definition in [@indep-aec-v5 Definition 10.1]). All the good frames we will use are *type-full*, i.e. their basic types are the nonalgebraic types, and we will omit the “type-full”.
This note was written while working on a Ph.D. thesis under the direction of Rami Grossberg at Carnegie Mellon University and I would like to thank Professor Grossberg for his guidance and assistance in my research in general and in this work specifically. I thank John Baldwin for early feedback on this note.
The proof
=========
The argument depends on [@sh394], on the construction of a good frame and related results in [@ss-tame-toappear-v3], on Boney’s theorem on extending good frames using tameness [@ext-frame-jml] (the subsequent paper [@tame-frames-revisited-v4] is not needed here), and on the Grossberg-VanDieren categoricity transfer [@tamenesstwo]. The argument also depends on some results about unidimensionality in III.2 of [@shelahaecbook] (these results have short full proofs, and have appeared in other forms elsewhere, most notably in [@tamenesstwo; @tamenessthree]).
There is a dependency on the Shelah-Villaveces theorem ([@shvi635 Theorem 2.2.1]), which can be removed in case one is willing to assume that ${\text{cf} (\lambda)} > {\text{LS}}({K})$. This is reasonable if one is willing to assume that $K$ is categorical in unboundedly many cardinals: then by amalgamation, the categoricity spectrum will contain a club, hence cardinals of arbitrarily high cofinality.
We proceed in several steps.
1. Without loss of generality, ${K}$ has joint embedding and no maximal models.
\[Why? Let us define a relation $\sim$ on ${K}$ by $M \sim N$ if and only if $M$ and $N$ embed into a common extension. Using amalgamation, one can see that $\sim$ is an equivalence relation. Now the equivalence classes ${\langle {K}_i : i \in I \rangle}$ of $\sim$ form disjoint AECs with amalgamation and joint embedding, and by the categoricity assumption (recalling that the Hanf number for existence is bounded by $H_1$) there is a unique $i \in I$ such that ${K}_i$ has arbitrarily large models. Moreover $({K}_i)_{\ge H_1} = {K}_{\ge H_1}$ so it is enough to work inside ${K}_i$.\]
2. ${K}$ is ${\text{LS}}({K})$-superstable.
\[Why? By [@shvi635 Theorem 2.2.1], or really the variation using amalgamation stated explicitly in [@gv-superstability-v2 Theorem 6.3]. Alternatively, if one is willing to assume that ${\text{cf} (\lambda)} > {\text{LS}}({K})$, one can directly apply [@sh394 Lemma 6.3].\]
3. ${K}$ is $(<\aleph_0)$-tame.
\[Why? See [@ap-universal-v9 Section 3][^2] (this does not use the categoricity hypothesis).\]
4. ${K}$ is stable in $\lambda$.
\[Why? By [@ss-tame-toappear-v3 Theorem 5.6], ${\text{LS}}({K})$-superstability and ${\text{LS}}({K})$-tameness imply stability everywhere above ${\text{LS}}({K})$.\]
5. \[sat-step\] The model of size $\lambda$ is saturated.
\[Why? Use stability to build a $\mu^+$-saturated model of size $\lambda$ for each $\mu < \lambda$. Now apply categoricity.\]
6. ${K}$ is categorical in $H_2$.
\[Why? By the proof of [@sh394 II.1.6], or see [@baldwinbook09 14.8].\]
7. ${K}$ has a good $H_2$-frame.
\[Why? By [@ss-tame-toappear-v3 Theorem 7.3] which tells us how to construct a good frame at a categoricity cardinal assuming tameness and superstability below it.\]
8. For $M \in {K}_{H_2}$, $p \in {\text{gS}}(M)$, let ${K}_{\neg^\ast p}$ be defined as in [@ap-universal-v9 Definition 5.7]: roughly, it is the class of $N$ so that $p$ has a unique extension to ${\text{gS}}(N)$ (so in particular $p$ is omitted in $N$), but we add constant symbols for $M$ to the language to make it closed under isomorphisms. Then ${K}_{\neg^\ast p}$ is a universal class.
\[Why? That it is closed under substructure is clear. That it is closed under unions of chains is because universal classes are $(<\aleph_0)$-tame, so if a type has two distinct extensions over the union of a chain, it must have two distinct extension over an element of the chain. Here is an alternate, more general, argument: ${K}_{H_2}$ is $\aleph_0$-local (by the existence of the good frame), so using tameness it is not hard to see that ${K}_{\ge H_2}$ is $\aleph_0$-local. Now proceed as before.\]
9. If $K$ is not categorical in $H_2^+$, then there exists $M \in {K}_{H_2}$ and $p \in {\text{gS}}(M)$ so that ${K}_{\neg^\ast p}$ has a good $H_2$-frame.
\[Why? See [@categ-primes-v3 Theorem 2.15][^3]: it shows that if $K_{H_2}$ is weakly unidimensional (a property that Shelah introduces in III.2 of [@shelahaecbook] and shows is equivalent to categoricity in $H_2^+$), then the good $H_2$-frame that $K$ has, restricted to ${K}_{\neg^\ast p}$ (for a suitable $p$) is a good $H_2$-frame. The definition of weak unidimensionality is essentially the negation of the fact that there exists two types $p \perp q$ (for a notion of orthogonality defined using prime models).\]
10. If $K$ is not categorical in $H_2^+$, $K_{\neg^\ast p}$ above has arbitrarily large models.
\[Why? By Theorem \[step-3\] below (recalling that ${K}_{\neg^\ast p}$ is a universal class), ${K}_{\neg^\ast p}$ has a good $(\ge H_2)$-frame. Part of the definition of such a frame requires existence of a model in every cardinal $\mu \ge H_2$.
11. If $K$ is not categorical in $H_2^+$, the model of size $\lambda$ is not saturated. This contradicts (\[sat-step\]) above, therefore $K$ is categorical in $H_2^+$.
\[Why? Take $M \in {K}_{\neg^\ast p}$ of size $\lambda$ (exists by the previous step). Then $M$ omits $p$ and the domain of $p$ has size $H_2 < \lambda$.\]
12. $K$ is categorical in all $\lambda' \ge H_2$.
\[Why? We know that $K$ is categorical in $H_2$ and $H_2^+$, so apply the upward transfer of Grossberg and VanDieren [@tamenesstwo Theorem 0.1].
To complete the proof, we need the following:
\[step-3\] Let $K$ be a universal class. Let $\lambda \ge {\text{LS}}({K})$. If ${K}$ has a good $\lambda$-frame, then ${K}$ has a good $(\ge \lambda)$-frame.
1. $K$ is $\lambda$-tame for types of length two.
\[Why? See [@ap-universal-v9 Section 3].\]
2. $K$ has weak amalgamation: if[^4] ${\text{gtp}}(a_1 / M; N_1) = {\text{gtp}}(a_2 / M; N_2)$, there exists $N_1' \lea N_1$ containing $a_1$ and $M$ and $N \gea N_1'$, $f: N_2 \xrightarrow[M]{} N$ so that $f (a_2) = a_1$.
\[Why? By the isomorphism characterization of Galois types in AECs which admit intersections, see [@non-locality Lemma 2.6] or [@ap-universal-v9 Proposition 2.17]. More explicitly, set $N_1' := {\text{cl}}^{N_1} (a_1 M)$, where ${\text{cl}}^{N_1}$ denotes closure under the functions of $N_1$. Then chase the definition of equality of Galois types.\]
3. $K$ has amalgamation.
\[Why? By [@ap-universal-v9 Theorem 4.15].\]
4. $K$ has a good $(\ge \lambda)$-frame.
\[Why? By Boney’s upward frame transfer [@ext-frame-jml] which tells us that amalgamation, $\lambda$-tameness for types of length two, and a good $\lambda$-frame imply that the frame can be extended to a good $(\ge \lambda)$-frame.\]
[^1]: Here and below, we write ${h (\theta)} := \beth_{(2^{\theta})^+}$. We see universal classes as AECs so that for $K$ a universal class, ${\text{LS}}({K}) = |L ({K})| + \aleph_0$. For ${K}$ a fixed AEC, we write $H_1 := {h ({\text{LS}}({K}))}$ and $H_2 := {h (H_1)}$.
[^2]: The main idea there is due to Will Boney, see [@tameness-groups].
[^3]: The original argument in [@ap-universal-v9] is harder, as it requires building a global independence relation.
[^4]: Since we do not assume amalgamation, Galois types are defined using the transitive closure of atomic equivalence, see e.g. [@shelahaecbook Definition II.1.9].
|
---
abstract: 'Coherent nonlinear multi-pulse processes, nonlinear waves and echo effects in resonant media are the topical problems of modern optics and important tools of coherent spectroscopy and quantum information science. We generalize the McCall-Hahn area theorem to the formation of an arbitrary photon echo generated during the multi-pulse excitation of the optically dense resonant media. The derived theorem made it possible to reveal the nonlinear mechanism of generation and evolution of the photon echo signals inside the media after a two-pulse excitation. We find that a series of self-reviving echo signals with total area of $2\pi$ or $0\pi$ is excited and propagates in the media depth, with each pulse having an individual area less than $\pi$. The resulting echo pulse train is a new alternative to the well-known soliton or breather. The developed pulse-area approach paves the way for more precise coherent spectroscopy, studies of different photon echo signals and quantum control of light pulses in the optically dense media.'
author:
- 'Sergey A. Moiseev$^{1}$'
- 'Mahmood Sabooni$^{2,3}$'
- 'Ravil V. Urmancheev$^1$'
bibliography:
- 'main.bib'
title: Photon echoes in optically dense media
---
Studies of coherent multi-pulse nonlinear effects like photon echo and four-wave mixing open wide opportunities for understating of light-atom interactions, fundamental processes of nonlinear and quantum optics, provide powerful techniques for spectroscopic investigation of atoms and molecules and are considered as a principal tool for implementation of basic processes in practical quantum information science [@Yetzbacher2007; @Christensson2008; @Dorfman2016; @Pezz2018; @Mourou2019]. Herein, photon echo technique [@KopvillemPhEcho1963; @Kurnit1964] attracts an especial everlasting attention in coherent spectroscopy [@Kurnit1964] and light pulse storage [@HEER197749; @Samartsev1980; @1981JEPTStyrkov; @Mossberg:82; @Carlson:831]. Recently, the photon echo in optically dense media opened promising opportunities for quantum storage of a large number of light pulses [@MoiseevKroll2001; @Tittel2009; @Lvovsky2009; @SparkesNatComm2011; @Usmani2010] and quantum processing [@OpticalQMapplicationReview] that determined a steady interest and elaboration of numerous protocols of photon echo based quantum memory [@Tittel2009; @Hosseini2011; @Rani2017; @Minnegaliev_2018; @Saglamyurek2018; @Guo2019; @Mazelanik2019], which are important for the creation of quantum repeater [@RevModPhys.83.33], microwave quantum memory [@PhysRevLett.105.140503; @Moiseev2018], etc.
The study of the properties of a two- and three-pulse photon echoes in optically dense media is the main task in the development of the multi-pulse spectroscopy and photon echo quantum memory schemes in such media. The most general theoretical description of the coherent resonant interaction of multi-pulse light fields with resonant atoms can be provided by the pulse area theorem [@McCallHahn1969; @Lamb1971; @allen1975optical; @Eberly:98; @PhysRevLett.88.243604; @Chaneliere:14; @PhysRevA.92.063815; @ThreePulseAreaTheorem]. In early works on the two-pulse (primary) photon echo, it was found that the initial excitation could result in the generation of multiple echo signals [@HAHN1971265; @allen1975optical] followed by a long-term investigation of the underlying mechanism [@FRIEDBERG1971285; @Lamb1971; @allen1975optical; @Moiseev1987; @1998-Azadeh-PRA; @1998-Wang-OC; @1999-Wang-PRA; @PhysRevA.79.053851; @Li2010; @Tsang:03]. Quite early an analytic solution for total area of all the echoes was obtained [@HAHN1971265; @allen1975optical; @1998-Azadeh-PRA], that proved that the total pulse area can tend asymptotically towards $2\pi$ in the media depth if the initial pulse area of two exciting laser pulses exceeds $\pi$. However, this solution does not allow one to describe the behavior of each individual echo pulse.
Previously acquired solution for the primary echo pulse area predicted that the echo pulse area never exceeds $\pi$ and generally decays in the depth of the media [@Moiseev1987] . This finding again stressed the ambiguity of the known physical picture behind the formation of the total nonlinear response to the multi-pulse excitation. In the recent years the stakes were raised by the demand for an efficient optical solid-state quantum memory and the noted interest in coherent multi-pulse interactions in the optically dense media.
In this Rapid Communication we find an analytical solution of the photon echo pulse area theorem posed in [@HAHN1971265; @FRIEDBERG1971285; @Lamb1971] in 1971. By analysing the solution we for the first time discover the mechanism of self-induced transparency [@McCallHahn1969] for two- and many-pulse excitation of the atomic media leading to the formation of many echo pulses. To do that we find the general analytic solution for the pulse area of an arbitrary secondary photon echo signal. The found solutions show that the echo signals are excited coherently one after another in a certain area of the medium and then disappear, generating new echo signals and creating a self-reviving echo sequence. We show that depending on the input pulse areas this echo pulse train forms a multi-pulse analogue to the well-known single pulse $2\pi$ optical soliton or a $0\pi$ optical breather despite each individual echo pulse area never exceeding $\pi$. Herein, by using the highly non-linear nature of the light-atom interaction we can control the total response of the media. Being near the thresh-old, when the incoming area of the second pulse is close to $\pi$, and by slightly changing it to being $<\pi$ or $> \pi$ one can initiate a huge change in the outcome from an optical soliton to an optical breather, respectively. This also demonstrates the potential of the pulse area approach for coherent spectroscopy of the optically dense media.
First we reproduce the McCall-Hahn area theorem and derive the general equation for the pulse area of an arbitrary echo pulse starting with the usual reduced set of Maxwell-Bloch equations [@allen1975optical] for the light field and atomic system: $$\begin{aligned}
\begin{split}
[ \partial_z + c^{-1}\partial_t ] \Omega & = i \frac{\mu}{2} \langle P\rangle,
\\
\partial_t u & = - \Delta v - \gamma u,
\\
\partial_t v & = \Delta u - \gamma v + \Omega w,
\\
\partial_t w & = -\Omega v,\\
\end{split}
\label{eq:mb_set}
\end{aligned}$$ where $\vec{r} = \vec{r}(t,z,\Delta) = (u,v,w)^T$ is the Bloch vector, each component depending on time $t$, spatial coordinate $z$ and atomic detuning $\Delta$; $P = u-iv$ - atomic polarization, electric field $E(t,z) = \varepsilon(t,z) \exp[i(kz-\omega t)] + c.c.$ is described by a complex light field envelope $\varepsilon(t,z)$ with corresponding Rabi frequency $\Omega (t,z) = (2 d/\hbar) \varepsilon (t,z)$; $\mu = 4\pi N d^2\omega/\hbar c$, $\gamma = 1/T_2$, $T_2$ is the coherence lifetime of the atomic transition and $\langle...\rangle\equiv \int_{-\infty}^{\infty} G(\Delta) ...d\Delta$ is the averaging over the inhomogeneous broadening. From now on for simplicity, we do not denote the existing dependence on $z$ in atomic and field variables $\vec{r}$ and $\Omega$.
By transferring to the pulse area $\theta = \int_{-\infty}^\infty dt \Omega (t)$ and follow [@McCallHahn1969; @Eberly:98] to find that incoming pulse areas $\theta_1, \theta_2$ satisfy the well-known pulse area theorem: $$\partial_z \theta = \tfrac{1}{2}\alpha w_0(z)\, \sin \theta(z),
\label{eq:area1}$$ where $w_0$ is the initial inversion of the atomic system, $\alpha$ is the resonant absorption coefficient [@allen1975optical]. The first pulse propagates in the undisturbed media, with $w_0=-1$ and partially inverts for the second pulse, so $w_0= -\cos\theta_1$. Substituting $w_0$ into Eq. we get the well-known solutions [@allen1975optical]: $$\begin{aligned}
\begin{split}
\theta_1(z) & = 2 \arctan \left[e^{-\alpha z/2} \tan \dfrac{\theta_1(0)}{2}\right],
\\
\theta_2(z) & = 2\arctan \left[ \kappa ~\mathrm{sech} \left( \beta - \frac{\alpha}{2}z \right)\right],
\end{split}
\label{eq:th_two_solution}
\end{aligned}$$ where $\beta=\ln\{\tan[\frac{\theta_1(0)}{2}]\}$ and $\kappa=\tan[\frac{\theta_2(0)}{2}]/\sin[\theta_1(0)]$.
Eqs. and can be used to find the total area of all excited photon echoes [@HAHN1971265; @FRIEDBERG1971285; @allen1975optical; @1998-Azadeh-PRA]: $$\theta_{\Sigma e}(z) = 2 \arctan \left[e^{-\alpha z/2} \tan \tfrac{\theta_{1}(0)+\theta_{2}(0)}{2}\right]-\theta_2(z)-\theta_1(z).
\label{eq:sum_area}$$
This solution predicts that if $\theta_2(0)<\pi, \theta_1(0)+\theta_2(0) > \pi$, the total area of all echo pulses asymptotically tends to $2\pi$ [@HAHN1971265]. It leaves however a lot of uncertainty about the mechanism and physics of the photon echo generation, since any information about the particular photon echo signals remains hidden. How exactly different echoes combine into $2\pi$ pulse area? What is the contribution of an individual echo? Moreover, if input pulse areas $\theta_1(0)<\pi/2, \theta_2(0) >\pi$, Eq. predicts the sum of all echoes to be $0$. What happens with the different echo signals in this case and does that mean that there will be no echoes? To answer all these questions, we have to analyze the generation of each echo signal individually.
To find the area theorem for an arbitrary individual photon echo signal we integrate the first of Eqs. over time around the time of echo emission $t_e$, from $t_0 = t_e-\tau/2$ to $t_1=t_e+\tau/2$, where $\tau$ is the delay between the pulses. We should also clarify the time scales assumed for the following derivation. Firstly, we assume non-overlapping pulses $\tau \gg \delta t_{1,2}$ with pulse duration being much smaller than coherence time $\delta t_i \ll T_2$, $i=1,2,e1,...$, to neglect the relaxation during the pulses. Secondly, inhomogeneous broadening of the atomic system is much larger that the pulse spectrum $\Delta_{in} > 1/\delta t_{1,2}$. Thirdly, for simplicity we consider a solid state system, meaning $T_1\gg T_2$ and thus we can neglect the population decay between the pulses. In short, $1/\Delta_{in} < \delta t_{1,2} \ll \tau \lesssim T_2$.
The expressions under the integrals, $P_0(z,\Delta)$ and $w(t,z,\Delta)$ are complex expressions consisting of several oscillating components. However most of these components will give $0$ after averaging over $\Delta$ in Eq. . To find the proper expression for the echo area we need to only take into account the phasing components of polarization and inversion that contribute to the echo formation. The details of the integration and equation handling can be found in the Supplemental material.
As a result we obtain the general equation for an arbitrary echo pulse area: $$\partial_z \theta (z)= \frac{1}{2}\alpha [ 2 v_0(z) \cos^2\frac{\theta(z)}{2} + w_0(z)\sin\theta(z)],
\label{eq:area_general}$$ where $w_0(z)$ and $v_0(z)$ are the initial values ($t=t_e-\tau/2$) of the Bloch vector resonance components with $\Delta = 0$ which only give nonzero response in the field equation in (1). After transition to $\eta = \tan\tfrac{\theta(z)}{2}$ we get a linear equation $\partial_z \eta(z) = \frac{\alpha}{2}[v_0(z)+w_0(z) \eta(z) ]$ with clear solution.
Equation describes the pulse area of a chosen echo signal given the phasing coherence $v_0$ in a presence of spectral uniform inversion $w_0$ and Eq. comes down to finding $v_0(z)$ and $w_0(z)$ for each echo signal. In Supplemental material we describe the algorithm that allows to find the $v_0, w_0$ for an arbitrary echo. But whatever they may be, we note that $|\theta|$ never exceeds $\pi$. Below we investigate the analytic solutions for the pulse areas of all the echo signals.
For primary echo we have $\vec{r}(t)=U(t-\tau)T(\theta_2)U(\tau)T(\theta_1)\vec{r}(0)$, $t_0 = 3\tau/2$ and the correct phasing components of $\tilde{v}_{0}(3\tau/2), \tilde{w}_{0}(3\tau/2)$ [@Moiseev1987; @2019-OptExpress]: $$\begin{aligned}
\begin{split}
v_{0}(3\tau/2,z) & = \Gamma_{\tau}^2 \sin\theta_1(z)\sin^2\tfrac{\theta_2(z)}{2},
\\
\tilde{w}_{0}(3\tau/2,z) & = -\cos\theta_1(z)\cos\theta_2(z),
\end{split}
\label{eq:primary_vw}
\end{aligned}$$ where $\Gamma_{\tau} = e^{-\gamma\tau}$ is the relaxation term. Corresponding Eq. gives primary photon echo pulse area: $$\theta_{e1}(z) =2\arctan\left[\Gamma_{\tau}^2 \sin\theta_1(0) \sin^2 {\tfrac{\theta_2(z)}{2}}\sinh\tfrac{\alpha z}{2} \right].
\label{eq:echo_area_solution}$$
After the incoming pulses and the primary echo pulse we have $\vec{r}(t)=U(t-2\tau)T(\theta_{e1})U(\tau)T(\theta_2)U(\tau)T(\theta_1) \vec{r}(0),~ t_0=5\tau/2$ and the phasing components $v_{0}(5\tau/2,z), w_{0}(5\tau/2,z)$ are: $$\begin{aligned}
\begin{split}
v_{0} = v_{01} & + v_{02} =
\tfrac{1}{2} \Gamma_\tau^2 \sin\theta_1(z) \sin\theta_{e1}(z) \sin\theta_2(z)
\\
+ & \Gamma_\tau^2 \cos\theta_1(z)\sin^2\tfrac{\theta_{e1}(z)}{2} \sin\theta_2(z),
\\
w_{0} = w_{01} & + w_{02} = -\Gamma_\tau^2 \sin\theta_1(z)\sin^2\tfrac{\theta_2(z)}{2} \sin\theta_{e1}(z)
\\
- & \cos\theta_1(z) \cos\theta_2(z)\cos\theta_{e1}(z).
\end{split}
\label{eq:2nd_echo_v_w}\end{aligned}$$
The first terms in both equations $v_{01} (z)= \tfrac{1}{2}\Gamma_\tau^2 \sin\theta_1\sin\theta_2\sin\theta_{e1}$ and $w_{01}(z) = - \Gamma^2_\tau \sin\theta_1\sin^2\tfrac{\theta_2}{2}\sin\theta_{e1}$ are proportional to $\sin\theta_1(z)$ and vanish when the first pulse is absorbed. They are responsible for stimulated photon echo generated by incoming pulses and the primary echo pulse. The other two components $v_{02}(z) = \Gamma_\tau^2\cos\theta_1\sin\theta_2\sin^2\tfrac{\theta_{e1}}{2}$ and $w_{02}(z) = -\cos\theta_1\cos\theta_2\cos\theta_{e1}$ are proportional to $\cos\theta_1$ are correspond to the secondary two-pulse photon echo created by the second pulse and the primary echo pulse.
Analysis of the successive echoes follows the same procedure but requires more calculations since $v_{0}$ and $w_{0}$ have more terms with each step. In the Supplemental material we introduce the phasing polarization and inversion components for the third and the fourth echoes and discuss the physical meaning of different contributions. It is obvious that the described procedure can be applied for the case with comparable transverse and longitudinal relaxations and for other light-atom equations.
We will now proceed to clarify the mechanism of the total $2\pi$ pulse area formation when $\theta_1(0)+\theta_2(0)>\pi$. Figure \[fig:echo\_areas\_999\] shows the spatial behavior of the area of incoming pulses, echo pulses and the total area depending on the optical density of the medium for $\theta_1(0)=0.1\pi, \theta_2(0)= 0.999\pi$. We see that incoming pulses excite primary and secondary echoes that in turn excite subsequent echos. Each echo pulse is born, propagates and eventually dies out within a finite spatial interval. However the total area of all existing pulses behaves strictly in accordance with McCall-Hahn area theorem Eq. and remains close to $2\pi$. This is realized due to the precise spatial consistency of all the echoes involved.
The case of $\theta_2(0)>\pi$ really helps to highlight the benefits of looking at an individual echo signal rather than at the sum of all echo signals. The second incoming pulse is big enough to form a $2\pi$-soliton on its own, and McCall-Hahn area theorem predicts that the sum of all echoes will equal $0\pi$. The impression could be that after some point in the medium there are no echoes at all. The real picture however is much more vivid, there are many hidden echoes with nontrivial areas working together to comply with the McCall-Hahn area theorem. Figure \[fig:echo\_areas\_1001\] showcases this echo pulses’ behavior for $\theta_1(0)=0.1\pi, \theta_2(0)=1.001\pi$. Each two of the subsequent echoes have opposite phases, so they are canceling each other in a dynamical equilibrium, resulting in $0\pi$ total pulse area at any point of the medium. Figure \[fig:echo\_areas\_1001\] also shows that the primary echo assists the formation of the $2\pi$ total area, which would otherwise happen much further into the medium.
We note that the echo areas in Figs. \[fig:echo\_areas\_999\],\[fig:echo\_areas\_1001\] behave very similar, differing only in their spatial delays. This is the case, when we can neglect the stimulated echo terms in Eqs. and find a highly accurate approximate analytic solution for each pulse area. For example, we write for the secondary echo area ($z>z_1$): $$\tan \frac{\theta_{e2}}{2} =\Gamma_{\tau} \sin\theta_2(z_1) \sin^2 {\frac{\theta_{e1}(z)}{2}}\sinh\tfrac{\alpha}{2} (z-z_1),
\label{eq:approx_sol}$$ where $\theta_{e1}$ is given in Eq. with the initial pulse areas taken at the transition point $z_1$: $(\theta_1(0),\theta_2(0)) \rightarrow (\theta_2(z_1),\theta_{e1}(z_1))$. By doing so we assume that at $z=z_1$ the first pulse was successfully absorbed by the media and neglect polarization and inversion components acquired at $z<z_1$. solution for $\theta_{e2}$ is shown with dashed lines in Figs. \[fig:echo\_areas\_999\],\[fig:echo\_areas\_1001\].
Equations and describe the pulse area at the output of the optically dense media. Moreover, given $\delta t_1 > \delta t_2$ they can also accurately describe the peak energy of the echo pulse [@1999-Wang-PRA; @2019-OptExpress]. This easy to measure quantity can be used for coherent multi-pulse spectroscopy of the optically dense media, where usual spectroscopy is complicated due to strong nonlinear light-atoms interaction. In this highly nonlinear regime the conventional Beer law $I_{echo} = I_0 \Gamma_\tau^2$ is not valid while Eq. can be used to measure $\Gamma_\tau$ dependence.
It also is interesting to discuss the experimental detection of photon echo train generation and what it can lead to. As it is seen in Figs. \[fig:echo\_areas\_999\],\[fig:echo\_areas\_1001\], one can experimentally observe only $2$ or $3$ light pulses at the output of the optical density medium, while other pulses will be highly suppressed. Herein in media with higher optical densities, we will see only higher order echo pulses, characterized experimentally by later arrival times. The photon echo experiments in such media are quite typical for many quantum memory protocols. In particular, interesting opportunity is to try detecting the spatial evolution of photon echo inside such media, for example in the rare-earth ions doped crystals [@Tittel2009; @CHANELIERE201877; @Hua_2018].
One possible candidate for high optical density and large Rabi frequency is [$^4\!I_{9/2}-\,^4F_{3/2}\,$]{}transition of [Nd$^{3+}$:YVO$_4$]{}$\:$ at $897.705\: \text{nm}$ with dipole moment $d = 9.16\times 10^{-32}\: \text{C.m}$. Considering $P=100\: mW$ and beam radius of $r=1\mu$m one could reach up to $\Omega \sim 250\: \text{MHz}$. The $\pi$-pulses can be as brief as several nanoseconds which is much shorter than $T_2$. These pulses are spatially squeezed in the medium up to 4 orders of magnitude by the group velocity reduction in the presence of a spectral hole in the optical transition [@Sabooni2013b], this would allow to observe spatial evolution of the solitons and echo pulses inside the medium.
It is worth noting that only soliton-like pulses can propagate through the medium without changing their temporal form and transferring atoms to their initial state. Accordingly, the photon echo pulses in the generated train will be stretch in time and ultimately overlap with each other deep in the medium forming a single $2\pi$-soliton in case of Fig. \[fig:echo\_areas\_999\]. Similarly the stretching echo pulses will asymptotically form a $0\pi$-breather, for the case of Fig. \[fig:echo\_areas\_1001\]. In the core of these transformations lies conservation laws of Maxwell-Bloch equations [@Lamb1972].
Finally, we summarize and conclude the long-lasting derivation of the two-pulse photon echo area theorem started over 45 years ago in [@HAHN1971265; @FRIEDBERG1971285; @Lamb1971], providing an analytic solution for the pulse area of any desired photon echo signal. We showcase the power of the pulse area approach by exploring the rich physics behind the two-pulse echo excitation of an optically dense medium in two previously understudied cases: $\theta_1(0) < \pi, \theta_2(0) \lesssim(\gtrsim) \, \pi$. For the first time we demonstrate that in both these cases a self-reviving echo train is excited deep in the medium with total pulse area $2\pi$ in the first case and $0\pi$ in the second previously unknown case. Thus a slight change in the second pulse area can lead to the dramatic change in the nonlinear multi-pulse media response: an optical soliton in one case or a soliton followed by a breather in the other case. At the same time the complex spatial dynamic of the total nonlinear media response after the two-pulse excitation is precisely aligned with the general McCall-Hahn area theorem prediction.
The developed approach of photon echo pulse area theorem can provide new insights in general analysis of coherent multi-pulse interactions with various photon echo experiments. Although the two-pulse photon echo itself cannot be used for quantum storage [@PhysRevA.79.053851], the developed pulse area approach provides intensity independent universal tool for deeper studies of quantum memory (especially for intensive light pulses and cavity assisted storage), coherent spectroscopy and generation of nonlinear waves in optically dense media. It could also be used in both optical and microwave wavelength regions, for two- and three-level atomic ensembles with arbitrary transverse and longitudinal relaxation times, etc. Next important analytic step could be to generalize and extend the results acquired here for multi-pulse excitation using inverse scattering transform, as was done in [@Kaup1977] for McCall-Hahn area theorem.
The reported study was funded by Russian Foundation for Basic Research, research project no.17-52-560009.
Supplemental material {#supplemental-material .unnumbered}
=====================
Arbitrary echo pulse area
-------------------------
Here we derive the general equation for an arbitrary individual echo pulse area. To do so we integrate the first of Eqs. (1) over time around the time of echo emission $t_e$, from $t_0 = t_e-\tau/2$ to $t_1=t_e+\tau/2$, where $\tau$ is the delay between the pulses. By assuming that $\tau \gg \delta t$, $\delta t$ being the pulse duration, we arrive to the equation for pulse area where we substitute the formal solution for $P$ from Eqs. (1): $$\begin{gathered}
\partial_z \theta = i\frac{\mu}{2}\langle \int_{t_0}^{t_1} dt \Big[ P_0(\Delta) e^{-\gamma t_e-i \Delta (t-t_e)}
\\
-i \int_{t_0}^{t} dt' \Omega (t') w(t',\Delta) e^{-(i \Delta + \gamma) (t-t')} \Big] \rangle,
\label{eq:area_1}\end{gathered}$$ where we introduced $P_0(\Delta)e^{-\gamma t_0} = P(t_0,\Delta)e^{-i \Delta \tau/2}$.
The key to finding the correct solution is proper handling of the integrals over $t$ in these two terms. One can show that $P_0(\Delta)$ and $w(t,\Delta)$ can be presented as a sum of several components $P_0 = P_0^{(0)}+P_0^{(1)}+...$ and $w = w^{(0)}+w^{(1)}+w^{(2)}+...$ with the total number of the components depending on the echo signal of interest (see Eqs. and and the following discussion). These components have a from $P_0^{(n)} \sim \exp[-i\Delta (t-t_e) -i n\Delta\tau + \varphi_n]$, $w^{(n)} \sim \cos[n\Delta\tau+\varphi_n], \text{ where } n\in \mathbb{Z},$ the phase $\varphi_n$ is either $0$ or $\pi/2$.
For $n \neq 0$, $P_0^{(n)}$ and $w^{(n)}$ are rapidly oscillating functions of $\Delta$ near the echo pulse emission time $t_e$ since $\tau\gg \delta t$. Averaging over $\Delta$ leads to that only the slowly varying terms $P^{(0)}_0$ and $w^{(0)}$ contribute to the echo pulse area in Eq. . After using $P_0(\Delta) = P_0^{(0)}$, we simply integrate the first term by taking into account: $\int_{t_0}^{t_1} dt e^{-i \Delta (t-t_e)} \rightarrow 2\pi \delta(\Delta)$ (this limit is valid assuming no temporal overlapping between the light pulses). In the second term we switch the order of temporal integrals, similar to [@allen1975optical; @Eberly:98], and arrive to the integral: $$\begin{aligned}
\begin{split}
& \langle \int_{t_0}^{t_1} dt' \Omega (t') w(t',\Delta) \int_{t'}^{t_1} dt e^{-(i \Delta + \gamma) (t-t')} \rangle =
\\
& \langle \int_{t_0}^{t_1} dt \Omega (t)\tfrac{ w^{(0)}(t,\Delta)}{\gamma+i \Delta } \rangle = \pi G(0)\int_{t_0}^{t_1} dt\Omega (t)w^{(0)}(t,0),
\end{split}\end{aligned}$$ where we have also taken into account that $w^{(0)}(t',\Delta)$ and $G(\Delta)$ are even functions of $\Delta$. Thus Eq. comes to: $$\partial_z \theta = \frac{\alpha}{2} \left[ 2 \tilde{v}_0
+\int_{t_0}^{t_1} dt \Omega(t) \tilde{w}(t) \right],
\label{eq:area_3}$$ where $\alpha = \mu\pi G(0)$ is the resonant absorption coefficient, $ \tilde{v}_0=iP_0^{(0)}(0) e^{-\frac{1}{2}\gamma\tau}$ is the resonant component of the phased coherence, $\tilde{w}(t) = w^{(0)}(t,0)$ is the resonant component of the atomic inversion. To find $\tilde{w}(t)$ and to integrate Eq. , we write the Bloch equation set for the case $\Delta = 0$, ignoring relaxation during the pulses, since $\gamma \delta t\ll 1,$: $$\begin{aligned}
\begin{split}
& \tilde{v}(t) = \tilde{v}_0 \cos \theta(t) + \tilde{w}_0 \sin \theta (t),
\\
& \tilde{w}(t) = \tilde{w}_0 \cos \theta(t) - \tilde{v}_0 \sin \theta (t),
\end{split}
\label{eq:delta_sol}\end{aligned}$$ where $\theta(t) = \int_{t_0}^t \Omega(t) dt$, and $\tilde{v}(t)$ is a resonant part of the phased coherence, $\tilde{w}_0 = \tilde{w}(t_0)$.
Equation can now be integrated, and after reassigning $\tilde{v}_0 \rightarrow v_0,~\tilde{w}_0 \rightarrow w_0 $we obtain Eq. (5): $$\partial_z \theta (z)= \frac{1}{2}\alpha [ 2 v_0(z) \cos^2\frac{\theta(z)}{2} + w_0(z)\sin\theta(z)],
\label{eq:area_general}$$
Phasing components of polarization and inversion
------------------------------------------------
Here we show in detail the calculation of $v_0$ and $w_0$ for the secondary echo and give the expressions for the third and the fourth echoes. We assume that the medium is excited by two incoming pulses having pulse areas $\theta_1, \theta_2$, that give rise to multiple photon echoes having pulse areas $\theta_{ei}$.
Under a multi-pulse excitation a two level system engages in two processes: it is either interacting with the electric field of the applied pulse, or it is left to its own devices and experiences free oscillations decaying as $e^{-\gamma t}$. In the assumed timescales of these processes the influence of the pulse with area $\theta$ can be written as a rotation of the Bloch vector around $u$-axis: $$T(\theta)\vec{r}=
\begin{pmatrix}
1 & 0 & 0 \\
0 & \cos \theta & \sin\theta \\
0 & -\sin\theta & \cos\theta
\end{pmatrix}
\begin{pmatrix}
u \\
v \\
w
\end{pmatrix}.$$ And free nutation is described with another rotation matrix, this time around $w$-axis: $$U(t)\vec{r}=
\begin{pmatrix}
e^{-\gamma t} \cos \Delta t & - e^{-\gamma t} \sin \Delta t & 0 \\
e^{-\gamma t} \sin \Delta t & e^{-\gamma t} \cos \Delta t & 0 \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
u \\
v \\
w
\end{pmatrix},$$
The secondary echo is emitted at the time $t=3\tau$, and to find the phasing parts of the coherence and inversion we write the Bloch vector $\vec{r}(t)=U(t-2\tau)T(\theta_{e1})U(\tau)T(\theta_2)U(\tau)T(\theta_1) \vec{r}(0), t_0=5\tau/2$. The calculation gives: $$\begin{aligned}
\begin{split}
v(t) = & - \Gamma_\tau c_1 c_2 s_{e1} c_t
\\
& - \Gamma_\tau^2 [c_1 s_2 c_{e1} c_\tau c_t + c_1 s_2 s_\tau s_t - s_1 s_2 s_{e1} c_t c_\tau]
\\
+ \Gamma^3_\tau s_1 [ & c_\tau s_\tau s_t + c_2 c_\tau s_\tau s_t - c_2 c_{e1} c_\tau^2 c_t + c_{e1} s_\tau^2 s_t],
\end{split}
\label{eq:App_v_All}
\\
\begin{split}
w(t) = & - c_1 c_2 c_{e1} + \Gamma_\tau [s_1 s_2 c_{e1} c_\tau+c_1 s_2 s_{e1} c_\tau ]
\\
& -\Gamma_\tau^2 [s_1 c_2 s_{e1} c^2_\tau - s_1 s_{e1} s_\tau^2]
\end{split}
\label{eq:App_w_All}
\end{aligned}$$ here we use a short notation for trigonometric functions: $s_i = \sin \theta_i, c_i = \cos \theta_i, i=1,2,e1,$ $s_\tau = \sin \Delta\tau,$ $c_\tau = \cos \Delta\tau, s_t = \sin \Delta(t-\tau), c_t = \cos \Delta(t-\tau)$.
This includes the phasing components, responsible for the echo generation and that are proportional to $ \cos [\Delta (t-2\tau)]$ and non phasing components. For example the first term of $v(t)$ contains only $c_t = \cos \Delta (t-\tau)$ and is non phasing, while the second term contains $c_\tau c_t = \cos \Delta\tau \cos \Delta(t-\tau) = \frac{1}{2}[\cos\Delta(t-2\tau)+\cos\Delta t] = \frac{1}{2}\cos\Delta(t-t_e)+\frac{1}{2}\cos(\Delta (t-t_e)+2\Delta\tau)$, so we get a phasing term $-\frac{1}{2}\Gamma_\tau^2 c_1 s_2 c_{e1} \cos\Delta(t-2\tau)$ that contributes to $P^0$. For $w(t)$ it is similar, except we are now interested in the time independent terms, like the first term in Eq. . The terms with $c_\tau^2$ or $s_\tau^2$ also contribute since $c_\tau^2 (s_\tau^2) = \frac{1}{2}(1 \pm \cos 2\Delta\tau),$ where the second term will vanish after averaging over $\Delta$.
We now leave only the terms that contribute to the echo: $$\begin{aligned}
v(t) = & \frac{1}{2}\Gamma_\tau^2 [-(c_{e1} +1)c_1 s_2 + s_1 s_2 s_{e1}] \cos \Delta (t-2\tau),
\\
w(t) = & - c_1 c_2 c_{e1} + \frac{1}{2}\Gamma_\tau^2 [1-c_2 ] s_1 s_{e1},
\end{aligned}$$ and we get for $\tilde{v}_0(3/2\tau)$ and $\tilde{w}_0(3/2\tau)$: $$\begin{aligned}
\begin{split}
v_0(3/2\tau,z) = & -\Gamma^2_\tau \cos\theta_1 \sin\theta_2 \cos^2\tfrac{\theta_{e1}}{2}
\\
& + \tfrac{1}{2} \Gamma^2_\tau \sin\theta_1 \sin\theta_2 \sin\theta_{e1},
\end{split}
\label{eq:App_Sec_V}
\\
\begin{split}
w_0(3/2\tau,z) = & -\cos\theta_1 \cos\theta_2 \cos\theta_{eq}
\\
& + \Gamma^2_\tau \sin\theta_1 \cos^2\tfrac{\theta_2}{2} \sin\theta_{e1}.
\end{split}
\label{eq:App_Sec_W}
\end{aligned}$$ The first terms in Eqs. and are very similar to those of primary echo pulse and correspond to the two-pulse echo generation by the $\theta_2(z), \theta_{e1}(z)$. This contribution to the secondary echo is presented as color yellow in Fig. \[fig:App\_Pulse\_Sequences\]. The second terms in Eqs. , correspond to the stimulated echo generation and are presented by the color blue in Fig. \[fig:App\_Pulse\_Sequences\].
In the same fashion we can write the phasing coherence and inversion after four pulses, two incoming pulses and two echo pulses: $$\begin{aligned}
\begin{split}
v_0 (7\tau/2,z) = & \tfrac{1}{2} \Gamma_\tau^2 \times
\\
& \big[\sin\theta_1\sin\theta_2\cos\theta_{e1}\sin\theta_{e2} \\
& +\cos\theta_1\sin\theta_2\sin\theta_{e1}\sin\theta_{e2} \\
& +2\cos\theta_1\cos\theta_2\sin\theta_{e1}\sin^2\tfrac{\theta_{e2}}{2}\big]
\\
& + \Gamma_\tau^4 \big[\sin\theta_1\cos^2\tfrac{\theta_2}{2}\sin^2\tfrac{\theta_{e1}}{2}\cos^2\tfrac{\theta_{e2}}{2} \\
& -\sin\theta_1 \sin^2\tfrac{\theta_2}{2}\cos^2\tfrac{\theta_{e1}}{2}\sin^2\tfrac{\theta_{e2}}{2} \big],
\end{split}
\label{eq:App_3rd_Echo_V}
\\
\begin{split}
w_0(7\tau/2,z) = & -\cos\theta_1\cos\theta_2\cos\theta_{e1}\cos\theta_{e2} \\
& - \Gamma_\tau^2 \times [ \sin\theta_1\sin^2\frac{\theta_2}{2}\sin\theta_{e1}\cos\theta_{e2} \\
& + \cos\theta_1\sin\theta_2\sin^2\frac{\theta_{e1}}{2}\sin\theta_{e2} \\
& + \frac{1}{2}\sin\theta_1\sin\theta_2\sin\theta_{e1}\sin\theta_{e2} ] = \\
& -v_4 \sin\theta_{e2} + w_4 \cos\theta_{e2}.
\end{split}
\label{eq:App_3rd_Echo_W}
\end{aligned}$$
The first term $\sim \sin\theta_1\sin\theta_2\cos\theta_{e1}\sin\theta_{e2}$ in $ v_0 (7\tau/2,z)$ is the stimulated echo from the two incoming and the second echo pulses naturally proportional to $\Gamma_\tau^2$. The second term $\sim \cos\theta_1\sin\theta_2\sin\theta_{e1}\sin\theta_{e2}$ is another stimulated echo generated by the second incoming and the first two echo pulses. The third term $\sim\cos\theta_1\cos\theta_2\sin\theta_{e1}\sin^2\tfrac{\theta_{e2}}{2}$ represents the contribution of the two-pulse echo from the two echo pulses. The next term $\sim \sin\theta_1\cos^2\tfrac{\theta_2}{2}\sin^2\tfrac{\theta_{e1}}{2}\cos^2\tfrac{\theta_{e2}}{2}$ is the two-pulse echo generated by the first incoming and primary echo pulses. The last term, $\sim \sin\theta_1 \sin^2\tfrac{\theta_2}{2}\cos^2\tfrac{\theta_{e1}}{2}\sin^2\tfrac{\theta_{e2}}{2}$ is the revived primary echo, generated by the first two incoming pulses and recovered by the second echo pulse (first echo pulse just suppresses its amplitude by the factor $\cos^2\tfrac{\theta_{e1}}{2}$).
|
---
abstract: 'Theoretical and interpretative study on the subject of photodetachment of H$^{-}$ near a partial reflecting surface is presented, and the absorption effect of the surface is investigated on the total and differential cross sections using a theoretical imaging method. To understand the absorption effect, a reflection parameter $K$ is introduced as a multiplicative factor to the outgoing detached-electron wave of H$^-$ propagating toward the wall. The reflection parameter measures, how much electron wave would reflect from the surface; $K=0$ corresponds to no reflection and $K=1$ corresponds to the total reflection.'
author:
- |
A. Afaq[^1]\
*Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China.*
title: '****'
---
[**PACS**]{} number: 32.80.Gc\
It has been observed both theoretically as well as experimentally that the photodetachment cross section of H$^-$ shows a smooth behavior in the free space [@Smith1959; @Ohmura1960]; while in the presence of a reflecting surface it displays oscillations [@YZC1; @YZC2]. These oscillations are similar as if they are in the presence of a static electric field [@Fabrikant1980; @Bryant1987; @Stewart1988; @Rau1988; @Greene1988; @Du1988; @Kondratovich1990].
Quite recently, Afaq and Du [@Afaq2007] have argued that this oscillatory effect in the photodetachment cross section of H$^-$ is because of two-path interference of the detached-electron wave from the negative ion. To the observation point, one path comes directly from the source H$^-$ , while the second path appears to be coming from an image of the source behind the wall. In reference [@Afaq2007], the idea has been discussed without considering the absorption effect of a wall. What would happen on the photodetachment cross section of negative ion when the wall in use will be absorbing? This problem is still interesting and has to be discussed. I use a simple model for H$^-$ and provide quantitative answer to the problem.
Near an absorbing wall the physical picture of the photodetachment process may be described as: When the detached-electron wave is made incident on the wall, a part of it is absorbed by the wall and the other part is reflected with low intensity. This low intensity electron wave propagates away from the system and appears to be coming from an image behind the wall, and at a very large distance it interferes with the direct outgoing detached-electron wave. Consequently, we obtain an outgoing electron flux interference pattern on the screen. The photodetachment cross section is proportional to the integrated outgoing electron flux across a large enclosure in which the source H$^-$ sits.
A partial reflecting wall $(0\leq K\leq1)$ is used for the electron scattering and it is placed from H$^-$ at a distance more than $50$ Bohr radii, so that the asymptotic approximations can be valid. The assumptions about the partial reflecting wall are the same as in reference [@Afaq2007]. For an observer at large distance from H$^{-}$, there are two components of detached-electron wave going from H$^{-}$ to the observer. The first component propagates directly from H$^{-}$ to the observer as if there is no wall; the second component first propagates toward the wall, after being partially reflected by the wall, it then propagates from the wall to the observer. In the theoretical imaging method, the first component comes from H$^{-}$ directly, the second component appears coming from an image of the H$^{-}$ behind the wall. The detached-electron flux can be calculated from the above two component outgoing waves. By integrating the detached-electron flux for all angles, we are able to derive analytic formula for the total photodetachment cross section of H$^{-}$. Atomic units are used unless otherwise noted.
A schematic diagram for the photodetachment of H$^{-}$ near a partial reflecting wall is shown in Fig. 1. A hydrogen negative ion H$^{-}$ acting as a source (S) of detached electron wave is on z-axis, its distance from the wall (W) is d. The reflecting surface of the wall coincides with the x-y plane. A laser polarized in the z direction is applied for the photodetachment of H$^{-}$. Three components of detached electron wave $\Psi_{1}$, $\Psi_{2}$ and $\Psi_{K}$ are also shown. $\Psi_{1}$ is the direct component, $\Psi_{2}$ is the reflected component and $\Psi_{K}$ is the absorbed components by the wall. The reflected component appears as if it is from an image (I) behind the wall. The $\pm$ symbols indicate the sign of two lobes of P-orbital wave function.
The photodetachment process can be regarded as a two step process[@Du1987; @du1988; @dU1988; @Bracher]: in the first step, the negative ion absorbs one photon energy $E_{ph}$ and generates an outgoing electron wave; in the second step, this outgoing wave propagates to large distances. Let $\Psi_{1}$ be the direct outgoing electron wave after the photodetachment of H$^{-}$ in the absence of the wall and let $\theta_1$ be the angle between the detached electron and the z-axis. Half of this wave with $\theta_1$ smaller than $\pi/2$ propagates away from the negative ion to large distance. The other half of the wave with $\theta_1$ larger than $\pi/2$ first propagates to the wall, after being partially reflected by the wall, it then propagates to large distance. We call this reflected wave $\Psi_{2}$. A part of this incoming wave towards the wall is absorbed by the surface $\Psi_{K}$ which can be measured by introducing reflection parameter $K$ as a multiplicative factor to the wave moving toward the wall. The total outgoing electron wave $\Psi^{+}$ at large distance from the system is given by $\Psi^{+}=\Psi_{1}+\Psi_{2}.$
The expression for the direct wave $\Psi_{1}$ has been derived before[@Du3]. Using $(r_1,\theta_1,\phi_1)$ as the spherical coordinates of the electron with respect to the source (S) and $(r_2,\theta_2,\phi_2)$ as the spherical coordinates of the electron with respect to image (I), we have $$\begin{aligned}
\Psi_{1}(r_{1},\theta_{1},\phi_{1}) &=&
U(k,\theta_{1},\phi_{1})\frac{\exp(ikr_{1})}{k r_{1}}, \nonumber \\
\Psi_{2}(r_{2},\pi-\theta_{2},\phi_{2}) &=&
U(k,\pi-\theta_{2},\phi_{2})K\frac{\exp i(kr_{2}-\mu\pi/2)}{k
r_{2}}.\end{aligned}$$ Where $k=\sqrt{2E}$, and E is the detached-electron energy, $k_{b}$ is related to the binding energy $E_{b}$ of H$^-$ by $E_{b}=\frac{k^2_{b}}{2}$, B is a normalization constant and is equal to $0.31552$, and $K$ is the reflection parameter that accounts how much electron wave reflects from the wall. $U(k,\theta,\phi)$ is an angular factor, and for laser polarization parallel to z-axis it can be written as $U(k,\theta_{1},\phi_{1})=\frac{4k^2Bi}{(k_{b}^{2}+k^{2})^{2}}\cos\theta_{1}$, $U(k,\theta_{2},\phi_{2})=U(k,\pi-\theta_{2})=-\frac{4k^2Bi}{(k_{b}^{2}+k^{2})^{2}}\cos\theta_{2}.$
Eqs.(1) becomes $$\begin{aligned}
\Psi_{1}(r_{1},\theta_{1},\phi_{1})&=&\frac{4k^2Bi}{(k_{b}^{2}+k^{2})^{2}}\cos(\theta_{1})\frac{\exp(ikr_{1})}{kr_{1}},\nonumber
\\ \Psi_{2}(r_{2},\theta_{2},\phi_{2})&=&-\frac{4k^2Bi}{(k_{b}^{2}+k^{2})^{2}}K\cos(\theta_{2})\frac{\exp
i(kr_{2}-\mu\pi/2)}{kr_{2}}.\end{aligned}$$
Since $r_{1}$ and $r_{2}$ are large compared to the distance between H$^-$ and the wall $d$, we can simplify further. Let $(r,\theta,\phi)$ be the spherical coordinates of the detached-electron relative to the origin. For phase terms, we approximate as $ r_{1}\approx r-d\cos\theta$, $ r_{2}\approx
r+d\cos\theta$, and in all other places in Eqs. (2), we use $r_{1}\approx r_{2}\approx r$, and $\theta_{1}\approx
\theta_{2}\approx \theta$. With these approximations, the outgoing detached electron wave $\Psi^{+}$ from the system is given by $$\begin{aligned}
\Psi^{+}(r,\theta,\phi)=\frac{4k^2Bi}{(k_{b}^{2}+k^{2})^{2}}
\cos(\theta) \left[e^{-ikd\cos\theta}-K e^{
i(kd\cos\theta-\mu\pi/2)}\right]\frac{\exp( i k r)}{k r}.\end{aligned}$$ Eq. (3) represents the outgoing electron wave produced in the detachment of H$^-$ near a partial reflecting wall. We now find electron flux distribution on a screen at large distance and then total photodetachment cross section. The electron flux is defined as [@Afaq2007] $$\vec{j}(r,\theta,\phi)=\frac{i}{2}(\Psi^{+}\vec{\bigtriangledown}\Psi^{+\ast}
-\Psi^{+\ast}\vec{\bigtriangledown}\Psi^{+}).$$ Using the expression for $\Psi^{+}(r,\theta,\phi)$ in Eq. (3) and flux formula in Eq. (4), we obtain the electron flux distribution along the radial direction $${j_{r}}(r,\theta,\phi)=\frac{16k^3B^2}{(k_{b}^{2}+k^{2})^{4}}\cos^{2}(\theta)\left[\frac{1+K^{2}+2K\cos(2kd\cos\theta+\pi-\mu\pi/2)}{r^{2}}\right].$$
We now calculate the total photodetachment cross section of negative ion near a partial reflecting wall. Imagine a large surface $\Gamma$ such as the surface of a semi-sphere enclosing the source region, a generalized differential cross section $\frac{d\sigma(q)}{ds}$ may be defined on the surface from the electron flux crossing the surface [@Afaq2007], $\frac{d\sigma(q)}{ds}=\frac{2\pi
E_{ph}}{c}\vec{{j_{r}}}\cdot\hat{n}$, where c is the speed of light approximately equal to 137 a. u., q is the coordinate on the surface $\Gamma$, $\hat{n}$ is the exterior norm vector at q, $ds=r^{2}\sin\theta d\theta d\phi$ is the differential area on the spherical surface. The total cross section may then be obtained by integrating the differential cross section over the surface, $\sigma(q)=\int_{\Gamma}\frac{d\sigma(q)}{ds}ds.$ Therefore, the first part of total photodetachment cross section of negative hydrogen ion near a partial reflecting wall is $$\sigma_1(E)=\frac{\sigma_{0}(E)}{2}\left[1+K^2-6K
A_1(2d\sqrt{2E})\right].$$ with $$A_1(u)=\left[\frac{\sin(u-\mu\pi/2)}{u}+2\frac{\cos(u-\mu\pi/2)}{u^{2}}-2\frac{\sin(u-\mu\pi/2)}{u^{3}}-2\frac{\sin(\mu\pi/2)}{u^3}\right]$$
Where, $\sigma_{0}(E)=\frac{16\sqrt{2} \pi^{2}B^2
E^{3/2}}{3c(E_b+E)^3}$, is the photodetachment cross section of $H^{-}$ in the absence of reflecting wall, the argument $u=2d\sqrt{2E}$ of $A_1$ in Eq. (6) is equal to the action of the detached-electron going from the negative ion to the partial reflecting wall and back to the negative ion.
When the electron wave incidents on the surface of a wall, a part of it is absorbed. Let we denote this part be $\Psi_K$ such that the sum of the reflected part and the absorbed part would be equal to the incoming electron wave to the wall. We introduce an absorption parameter $T$ that measures how much of the electron wave is absorbed by the surface of the wall such that $T^2+K^2=1$. The absorbed part of the detached-electron wave is then given by $\Psi_K(r,\theta,\phi)=T\Psi(r,\theta,\phi)$, where $\Psi(r,\theta,\phi)$ is an electron wave from the source H$^{-}$ toward the wall. To calculate total cross section for the absorbed part of the detached-electron wave, we performed similar steps as for $\Psi^{+}(r,\theta,\phi)$ but integration limits for $\theta$ would be from $\pi/2$ to $\pi$. Hence, the second part of the total cross section comes out $$\sigma_2(E)=\sigma_{0}(E)\left[\frac{K^2-1}{2}\right].$$ After adding the Eq. (6) and the Eq. (7), the total photodetachment cross section for H$^{-}$ near a partial reflecting surface becomes
$$\sigma(E,K)=\sigma_{0}(E)A(2d\sqrt{2E})a^2_0.$$
Where $A(u)$ is the modulation function and is defined by
$$A(u)=1-3K\left[\frac{\sin(u-\mu\pi/2)}{u}+2\frac{\cos(u-\mu\pi/2)}{u^{2}}-2\frac{\sin(u-\mu\pi/2)}{u^{3}}-2\frac{\sin(\mu\pi/2)}{u^3}\right].$$
It is clear that Eq. (8) reduces to the case as there is no wall for $K=0$. Hence for this particular condition, the wall acts like a transparent medium for electron waves. For $K=1$ Eq. (8) reduces to the results by Afaq and Du [@Afaq2007].
Fig. 2 using Eq. (9) shows the behavior of the modulation function $A(u)$ for different values of $K=1,~0.7,~0.4$ and $\mu=1,~1.5,~2$. In Fig. 1(a), the soft wall case [@Afaq2007] is represented by thick solid line and the hard wall case [@Afaq2007] is represented by solid line, doted lines represent for the intermediatory case. In Fig. 2 (a)-(c), we observe that the amplitude of oscillation decreases and phase changes due to change in values of $K$ and $\mu$. Fig. 3 using Eq. (8) with the exact value of modulation function in Eq. (9) shows total photodetachment cross section for the same values of $K$ and $\mu$ as in Fig. 2. For $K=1$, the oscillation amplitude is large Fig. 2(a) and for $K=0.4$, it becomes very small Fig. 2(c). It shows that oscillations in the photodetachment cross section can effectively be controlled by reflection parameter $K$.
The reason is, the electron wave that initially propagates toward wall, a part of it is absorbed. Due to this absorption, the reflected part will possess low intensity electron wave. This reflected part appears coming from the image behind the wall. Two path interference occurs on a screen placed at very large distance from the system. Consequently, we observe a decrease in the oscillation amplitude of photodetachment cross section.
Assuming a screen perpendicular to z-axis is placed at a distance L from the wall, L is much greater than d and usually equal to thousands atomic units in the experiments[@A11; @A12]. The flux distributions on the screen is cylindrically symmetric, it depends only on the distance between any point $(x,y)$ on the screen and the z-axis $\rho=\sqrt{x^2+y^2}$. By projecting the radial flux in Eq. (5) along z-direction and then adding flux due to absorbed electron wave function , the flux crossing the screen is $$j_z(\rho)=\frac{32k^3B^2}{(k_{b}^{2}+k^{2})^{4}}\frac{L^{3}}{(\rho^{2}+L^{2})^{5/2}}\left[1+K\cos(\frac{2kdL}{\sqrt{\rho^{2}+L^{2}}}+\pi-\mu\pi/2)\right].$$
Fig. (4) using Eq. (10) represents the differential cross section across z-axis for different values of $K=1,~0.5,~0.1$ and $\mu=1,~2$. We have fixed photon energy $E_{ph}=1eV$ and the distance between the wall and the screen $L=10000$ Bohr radii. Fig. (4) represents the wall-induce interference for the differential cross section, which may be observable in the photodetachment microscopy experiments [@A11; @A12].
In summary, the photodetachment of $H^{-}$ near a partial reflecting surface was studied using the theoretical imaging method. This method made possible the derivation of analytical formulas of total and differential cross sections in a straightforward manner. It is concluded that there is a strong relation between the reflection parameter $K$ and the oscillation of the cross sections. The reflection parameter measures, how much of the electron wave would reflect from the wall. For $K=0$ we get no oscillation, and for $K=1$ we get maximum oscillation in the cross sections. On analyzing the photodetachment spectra for different values of $K$, it may be possible to characterize surfaces using an electron beam as a probe just as various imaging and diffraction techniques, which have been developed for surface study [@Yagi; @Minoda]. The detached-electron flux distributions on a screen placed at large distance from the negative ion is also obtained. The distributions displayed strong interference patterns. These patterns may be observable just as in the photodetachment microscopy experiments for negative ions in the presence of a static electric field [@A11; @A12; @Du]. I hope that this theoretical study of photodetachment of H$^-$ near a partial reflecting surface will stimulate experiments, and may be useful in studying the surfaces.
I would like to thank Prof. M. L. Du for his useful discussion and comments.
[^1]: e mail: afaq@itp.ac.cn
|
---
abstract: 'We develop a Hamilton-Jacobi theory for singular lagrangian systems using the Gotay-Nester-Hinds constraint algorithm. The procedure works even if the system has secondary constraints.'
address:
- 'Manuel de León: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), c$\backslash$ Nicolás Cabrera,n. 13-15, Campus Cantoblanco, UAM 28049 Madrid, Spain'
- 'Juan C. Marrero: Unidad asociada ULL-CSIC “Geometría Diferencial y Mecánica Geométrica", Departamento de Matemática Fundamental, Universidad de La Laguna, Tenerife, Canary Islands, Spain'
- 'David Martín de Diego: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), c$\backslash$ Nicolás Cabrera, n. 13-15, Campus Cantoblanco, UAM 28049 Madrid, Spain'
- 'Miguel Vaquero: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), c$\backslash$ Nicolás Cabrera, n. 13-15, Campus Cantoblanco, UAM 28049 Madrid, Spain'
author:
- Manuel de León
- Juan Carlos Marrero
- David Martín de Diego
- Miguel Vaquero
title: 'On the Hamilton-Jacobi theory for singular lagrangian systems'
---
[^1]
Introduction
============
One of the most classical problems of theoretical mechanics is the study of constrained systems. Essentially, there are two different meanings to understand constrained systems. One refers to systems where we externally impose constraints allowing some particular motions (external constraints). The second case is when the degeneracy of a lagrangian function imposes constraints on the phase space of the system (internal constraints). In this paper, we will restrict ourselves to this last situation.
At a first step, when the lagrangian is singular, there appear constraints restricting the admissible positions and velocities. Later on, the evolution of these initial constraints may produce new constraints.
The theory of degenerate (or singular) lagrangian systems is relevant in Field theory, and just the quantization of these systems led to Dirac [@dirac] to develop a wonderful theory of constraints, later geometrized by Gotay, Nester and Hinds [@gotaythesis; @gotay1; @gotay0; @gotay5]. Recently, M. Leok and collaborators [@sosa] have studied degenerate lagrangians arising from truly mechanical systems, even in presence of additional nonholonomic constraints (see also the paper by de León and Mart[í]{}n de Diego [@lmm5]).
Another important topic in theoretical mechanics is the Hamilton-Jacobi theory which allows us to find solutions of a hamiltonian systems by means of solutions of a partial differential equation, the Hamilton-Jacobi equation. Conversely, we can treat to solve a PDE using the characteristic curves of a Hamiltonian system (see these two standard books [@AM; @Arnold] for a general view of the theory and some modern approaches in terms of lagrangian submanifolds; see also [@rund] for a more classical view). In [@lmm1; @lmm3], we have successfully extended the classical Hamilton-Jacobi theory for nonholonomic systems, and in [@lmm2; @lmv] for classical field theories. Therefore, it seems quite relevant to extend the Hamilton-Jacobi theory also for degenerate lagrangian systems, and this is just the goal of the present paper.
Briefly, the standard formulation of the Hamilton-Jacobi problem is to find a function $S(t,q^A)$ (called the principal function) such that $$\label{hamiltonjacobi1}
\frac{\partial S}{\partial t}+H(q^A,\frac{\partial S}{\partial q^A})=0$$
If we put $S(t,q^A)=W(q^A)-tE$, where $E$ is a constant, then $W$ satisfies $$\label{hamiltonjacobi2}
H(q^A,\frac{\partial W}{\partial q^A})=E;$$ $W$ is called the characteristic function. Equations and are indistinctly referred as the Hamilton-Jacobi equations.
There have been several attempts to develop a Hamilton-Jacobi theory for degenerate lagrangian system ([@gomis1; @gomis2; @Rothe]). These procedures were based on the homogeneization of the given lagrangian, which leads to a new lagrangian system with null energy; then, it is possible to discuss the Hamilton-Jacobi equation for the constraints themselves. The main problem is that, due to the integrability condition for the resultant partial differential equation, one can only consider first class constraints. Therefore, the treatment of the cases when second class constraints appear should be developed by [*ad hoc*]{} arguments (as in [@Rothe], for instance). Thus, in [@gomis1] and [@gomis2] the authors only discuss the case of primary constraints.
Therefore, the Hamilton-Jacobi problem for degenerate lagrangian is far to be solved.
Our procedure to develop a geometric Hamilton-Jacobi theory is strongly inspired in two main issues. The first one in the recent approach to the Hamilton-Jacobi theory developed by Cariñena [*et al*]{} [@cari] (see also [@cari2; @cari3; @marmo2], and [@lmm3] for the applications to nonholonomic mechanics and field theory); and the second one, is the geometric theory of constraints due to Gotay and Nester [@gotay0].
Let us recall that given an almost regular lagrangian $L:TQ\rightarrow \mathbb{R}$ one can define a presymplectic system on $M_1=FL(TQ)\subset T^*Q$, the primary constraint submanifold where $\omega_1$ is the restriction of the canonical symplectic form on $T^*Q$ to $M_1$, and $FL: {TQ}\rightarrow T^*Q$ is the Legendre transformation defined by $L$. The dynamics is obtained from the equation $$i_X\omega_1=dh_1,$$ where $h_1\in C^{\infty}(M_1)$ is the projection of the energy $E_L\in C^{\infty}(TQ)$.
The above equation produces a sequence of submanifolds $$\cdots M_k \hookrightarrow \cdots \hookrightarrow M_2 \hookrightarrow M_ 1
\hookrightarrow T^*Q$$ and, eventualy, a final constraint submanifold $M_f$ if the algorithm stabilize at some step.
The strategy is to consider the projection of the constraint submanifolds provided by the constraint algorithm, so that we obtain new surjective submersions onto submanifolds of the given configuration manifold. This fact permits to connect a given solution of the final constraint submanifolds $M_f$, with its projection onto $Q_f$ ($\pi_f:M_f\rightarrow Q_f$ is the surjective submersion) using a section of $\pi_f$.
The SODE problem is also discussed such that one can obtain the corresponding lagrangian picture.
We also discuss the relation of the geometric Hamilton-Jacobi problem with the Hamilton-Jacobi problem (in a traditional sense) for arbitrary extensions of $h_1$, in terms of first and second class primary and secondary constraints. Therefore, this work can be considered as the natural extension to the Hamilton-Jacobi problem of the geometrization by Gotay and Nester of the Dirac constraint algorithm.
Several examples are discussed along the paper in order to illustrate the theory.
Classical Hamilton-Jacobi theory (geometric version)
====================================================
The standard formulation of the Hamilton-Jacobi problem is to find a function $S(t, q^A)$ (called the [**principal function**]{}) such that $$\label{hjj1}
\frac{\partial S}{\partial t} + h(q^A, \frac{\partial S}{\partial
q^A}) = 0,$$ where $h=h(q^A, p_A)$ is the hamiltonian function of the system. If we put $S(t, q^A) = W(q^A) - t E$, where $E$ is a constant, then $W$ satisfies $$\label{hjj2}
h(q^A, \frac{\partial W}{\partial q^A}) = E;$$ $W$ is called the [**characteristic function**]{}.
Equations (\[hjj1\]) and (\[hjj2\]) are indistinctly referred as the [**Hamilton-Jacobi equation**]{}.
Let $Q$ be the configuration manifold, and $T^*Q$ its cotangent bundle equipped with the canonical symplectic form $$\omega_Q = dq^A \wedge dp_A$$ where $(q^A)$ are coordinates in $Q$ and $(q^A, p_A)$ are the induced ones in $T^*Q$. In what follows, $\pi_Q : T^*Q \longrightarrow Q$ will denote the canonical projection.
Let $h : T^*Q \longrightarrow {\mathbb{R}}$ a hamiltonian function and $X_h$ the corresponding hamiltonian vector field, say $$i_{X_h} \, \omega_Q = dh.$$ Therefore, the integral curves $(q^A(t), p_A(t))$ of $X_h$ satisfy the Hamilton equations: $$\frac{dq^A}{dt} = \frac{\partial h}{\partial p_A} \, , \; \frac{dp_
A}{dt} = - \frac{\partial h}{\partial q^A}.$$ We can define also the Poisson bracket ot two functions. Given $f$ and $g$ real functions on $T^*Q$, we define a new function $\{f,g\}$ by $$\{f,g\}=\omega_Q(X_f,X_g)\; ,$$ where $X_f$ and $X_g$ are the corresponding hamiltonian vector fields.
The Poisson bracket gives us the evolution of observables, since given the hamiltonian $h$ we have $$\dot{f}=X_h(f)=i_{X_h}(i_{X_f}\omega_Q)=\omega_Q(X_f,X_h)=\{f,h\},$$ and then we can rewrite the Hamilton equations as $$\begin{array}{l}
\dot{q}^A=\{q^A,h\}\\ \noalign{\medskip}
\dot{p}_A=\{p_A,h\}.
\end{array}$$ Let $\lambda$ be a closed 1-form on $Q$, say $d\lambda=0$; (then, locally $\lambda = dW$).
The following theorem gives us the relation of the Hamilton-Jacobi equation and the solutions of the Hamilton equations (see [@AM; @Arnold]).
\[HJTh\]
The following conditions are equivalent:
1. If $\sigma: I\to Q$ satisfies the equation $$\frac{dq^A}{dt} = \frac{\partial h}{\partial p_A},$$ then $\lambda\circ \sigma$ is a solution of the Hamilton equations;
2. $d (h\circ \lambda)=0$
We can reinterpret Theorem \[HJTh\] as follows (see [@cari; @lmm1; @lmm3]).
Define a vector field on $Q$: $$X_h^{\lambda}=T\pi_Q\circ X_h\circ \lambda\;$$ The following diagram illustrates the construction of the vector field $X_h^\lambda$: $$\xymatrix{ T^*Q
\ar[dd]^{\pi_Q} \ar[rrr]^{X_h}& & &T(T^*Q)\ar[dd]^{T\pi_Q}\\
& & &\\
Q\ar@/^2pc/[uu]^{\lambda}\ar[rrr]^{X_h^{\lambda}}& & & TQ }$$
Then the following conditions are equivalent:
1. If $\sigma: I\to Q$ satisfies the equation $$\frac{dq^A}{dt} = \frac{\partial h}{\partial p_A},$$ then $\lambda\circ \sigma$ is a solution of the Hamilton equations;
2. If $\sigma: I\to Q$ is an integral curve of $X_h^{\lambda}$, then $\lambda\circ \sigma$ is an integral curve of $X_h$;
3. $X_h$ and $X_h^{\lambda}$ are $\lambda$-related, i.e. $$T\lambda(X_h^{\lambda})=X_h \circ \lambda$$
Next, we have the following intrinsic version of Theorem \[HJTh\].
\[HJTh2\] Let $\lambda$ be a closed $1$-form on $Q$. Then the following conditions are equivalent:
1. $X_h^\lambda$ and $X_h$ are $\lambda$-related;
2. $d (h\circ \lambda)=0$
[**Proof:**]{} In local coordinates, we have that $$X_h = \frac{\partial h}{\partial p_A} \frac{\partial}{\partial q^A} -
\frac{\partial h}{\partial q^A} \frac{\partial}{\partial p_A}$$ and $$\lambda = \lambda_A(q) \, dq^A$$ Then, $$\begin{aligned}
X_h^\lambda &=& \frac{\partial h}{\partial p_A}(\lambda(q)) \, \frac{\partial}{\partial q^A}\; , \\
T\lambda(X_h^\lambda) &=& \frac{\partial h}{\partial p_A} \, \frac{\partial}{\partial q^A} +
\frac{\partial h}{\partial p_A} \frac{\partial \lambda_B}{\partial q^A} \, \frac{\partial}{\partial p_B}\; ,\\
d(h \circ \lambda) &=& (\frac{\partial h}{\partial q^A} + \frac{\partial h}{\partial p_B} \frac{\partial \lambda_B}{\partial q^A} \,) dq^A\; .\end{aligned}$$ Since $d\lambda=0$ if and only if $$\frac{\partial \lambda_A}{\partial q^B} =
\frac{\partial \lambda_B}{\partial q^A},$$ we have the equivalences between (i) and (ii). $\Box$
If $$\lambda = \lambda_A(q) \, dq^A$$ then the Hamilton-Jacobi equation becomes $$h(q^A, \lambda_A (q^B)) = const.$$ If $\lambda = dW$ then we recover the classical formulation $$h(q^A, \frac{\partial W}{\partial q^A}) = const.$$ since $$\lambda_A = \frac{\partial W}{\partial q^A}\; .$$
The Hamilton-Jacobi theory in the lagrangian setting {#lagrangian}
====================================================
Let $L : TQ \longrightarrow {\mathbb{R}}$ be a lagrangian function, that is, $$L=L(q^A, \dot{q}^A)$$ where $(q^A, \dot{q}^A)$ denotes the induced coordinates on the tangent bundle $TQ$ of the configuration manifold $Q$. In what follows, $\tau_Q : TQ \longrightarrow Q$ will denote the canonical projection.
Let us denote by $$S = dq^A \otimes \frac{\partial}{\partial \dot{q}^A}$$ and $$\Delta = \dot{q}^A \, \frac{\partial}{\partial \dot{q}^A}$$ the vertical endomorphism and the Liouville vector field on $TQ$ (see [@Leonrodrigues] for intrinsic definitions).
The Poincaré-Cartan 2-form is defined by $$\omega_L = - d\alpha_L \; , \, \alpha_L = S^*(dL)$$ and the energy function $$E_L = \Delta(L)-L$$ which in local coordinates read as $$\begin{aligned}
\alpha_L &=& \hat{p}_A \, dq^A \\
\omega_L &=& dq^A \wedge d\hat{p}_A\\
E_L &=& \dot{q}^A \hat{p}_A - L (q, \dot{q})\end{aligned}$$ where $\displaystyle{\hat{p}_A = \frac{\partial L}{\partial \dot{q}^A}}$ stand for the generalized momenta. Here $S^*$ denotes the adjoint operator of $S$.
The lagrangian $L$ is said to be regular if the Hessian matrix $$\left( W_{AB} = \frac{\partial^2 L}{\partial \dot{q}^A \partial
\dot{q}^B} \right)$$ is regular, and in this case, $\omega_L$ is a symplectic form on $TQ$.
We define the Legendre transformation as a fibred mapping $FL : TQ \longrightarrow T^*Q$ such that $$\langle FL(v_q), \beta_q \rangle = \langle \tilde{X}_{v_q}, \alpha_L(v_q) \rangle$$ where $T\tau_Q(\tilde{X}_{v_q}) = v_q \in T_qQ$ and $\beta_q \in T_q^*Q$. In local coordinates we get $$FL(q^A, \dot{q}^A) = (q^A, \hat{p}_A),$$ and $L$ is regular if and only if $FL$ is a local diffeomorphism.
If $L$ is regular, then there exist a unique vector field $\xi_L$ on $TQ$ satisfying the symplectic equation $$i_{\xi_L} \, \omega_L = dE_L$$ and moreover, it automatically satisfies the second order differential equation (SODE) condition, i.e. $$S \, \xi_L=\Delta.$$
If, in addition, we assume that $L$ is hyperregular, that is, $FL:TQ\rightarrow T^*Q$ is a global diffeomorphism; then we can define a (global) hamiltonian function $h:T^*Q \rightarrow \mathbb{R}$ by $h=E_L\circ FL^{-1}$. It is easy to show that $FL^*\omega_Q=\omega_L$ and that $\xi_L$ and, then $X_h$ are $FL$-related. So, the solutions of the Euler-Lagrange equations transform by $FL$ into solutions of the Hamilton equations and viceversa. Given a vector field $Z$ on $Q$ we define a new vector field on $Q$ by $$\xi_L^Z = T\tau_Q \circ \xi_L \circ Z \; ,$$ that is, we have the following commutative diagram
$$\xymatrix{ TQ
\ar[dd]^{\tau_Q} \ar[rrr]^{\xi_L}& & &T(TQ)\ar[dd]^{T\tau_Q}\\
& & &\\
Q\ar@/^2pc/[uu]^{Z}\ar[rrr]^{\xi_L^{Z}}& & & TQ }$$
Now, Theorem \[HJTh2\] can be reformulated as follows.
\[HJTh2-bis\] Let $Z$ be a vector field on $Q$ such that $FL \circ Z$ is a closed 1-form. Then the following conditions are equivalent:
1. $\xi_L^Z$ and $\xi_L$ are $Z$-related;
2. $d (E_L \circ Z)=0$
[**Proof:**]{} The result follows as a direct consequence of Theorem \[HJTh2\] and the fact that $\xi_L$ and $X_H$ are $FL$-related. $\Box$
The Hamilton-Jacobi theory for singular lagrangian systems {#sec}
==========================================================
In this section we shall give a geometric approach to the Hamilton-Jacobi theory in terms of the Gotay-Nester-Hinds constraint algorithm [@gotay1; @gotay2].
Let $L : TQ \longrightarrow {\mathbb{R}}$ be a singular lagrangian, that is, the Hessian matrix $$\left( W_{AB} = \frac{\partial^2 L}{\partial \dot{q}^A \partial
\dot{q}^B} \right)$$ is not regular, or, equivalently, the closed 2-form $\omega_L$ is not symplectic.
Therefore, the equation $$\label{lagrangian}
i_{\xi} \, \omega_L = dE_L$$ has no solution in general, or the solutions are not defined everywhere. Moreover the solutions do not necessarily satisfy the SODE condition. Recall that SODE condition is $$\label{soe}
S \, \xi=\Delta$$ or, equivalently, $$T\tau_Q(X)=\tau_{TQ}(X),$$ where $\tau_Q:TQ\rightarrow Q$ and $\tau_{TQ}:TTQ\rightarrow TQ$ are the canonical projections.
Singular lagrangian system have been extensively studied by P.A.M. Dirac and P. Bergmann (see[dirac]{}), in order to obtain a procedure for canonical quantization of local gauge theories. They developed an algorithm (called Dirac-Bergmann theory of constraints) that has been later geometrized by M.J. Gotay and J. Nester [@dirac; @gotay1; @gotay2].
In the sequel, we will assume that $L$ is almost regular, which means that:
- $M_1= FL(TQ)$ is a submanifold of $T^*Q$;
- The restriction of the Legendre mapping $FL_1 : TQ \longrightarrow M_1$ is a submersion with connected fibers.
In this case, $M_1$ is called the submanifold of primary constraints.
If $L$ is almost regular, since $\ker( TFL)=\ker( \omega_L)\cap V(TQ)$, where $V(TQ)$ denotes the vertical bundle, and the fibers are connected then a direct computation shows that $E_L$ projects onto a function $$h_1 : M_1 \longrightarrow {\mathbb{R}}\; .$$
Denote by $j_1 : M_1 \longrightarrow T^*Q$ the natural inclusion and define $$\omega_1 = j_1^* (\omega_Q) \; .$$
Consider now the equation $$\label{singular}
i_X \, \omega_1 = dh_1\; .$$
There are two possibilities:
- There is a solution $X$ defined at all the points of $M_1$; such $X$ is called a global dynamics and it is a solution (modulo $\ker \, \omega_1$). In other words, there are only primary constraints.
- Otherwise, we select the submanifold $M_2$ formed by those points of $M_1$ where a solution exists. But such a solution $X$ is not necessarily tangent to $M_2$, so we have to impose an additional tangency condition, and we obtain a new submanifold $M_3$ along which there exists a solution. Continuing this process, we obtain a sequence of submanifolds $$\cdots M_k \hookrightarrow \cdots \hookrightarrow M_2 \hookrightarrow M_ 1
\hookrightarrow T^*Q$$ where the general description of $M_{l+1}$ is $$M_{l+1}:=\{p\in M_{l} \textrm{ such that there exists } X_p\in T_pM_l \\
\textrm{ satisfying } i_X\omega_1=dh_1 \}.$$ If the algorithm stabilizes at some $k$, say $M_{k+1}=M_k$, then we say that $M_k$ is the final constraint submanifold which is denoted by $M_f$, and then there exists a well-defined solution $X$ of (\[singular\]) along $M_f$.
There is another characterization of the submanifolds $M_l$ that we will describe now. If $N$ is a submanifold of $M_1$ then we define $$TN^{\perp}=\{Z\in T_p(M_1), \ p\in N \textrm{ such that } \omega_1(X,Z)=0 \textrm{ for all } X\in T_pN\}.$$
Then, at any point $p\in M_l$ there exists $X_p\in T_pM_l$ verifying $i_X\omega_1=dh_1$ if and only if $\langle TM_k^{\perp},dh_1 \rangle=0$, (see [@gotay0]).
Hence, we can define the $l+1$ step of the constraint algorithm as $$M_{l+1}:=\{p\in M_l\textrm{ such that }\langle TM_l^{\perp},dh_1 \rangle(p)=0\} \; ,$$ where ${TM_l}^{\perp}$ is defined as above.
Case I: There is a global dynamics
----------------------------------
In this case there exists a vector field $X$ on $M_1$ such that $$(i_X \, \omega_1 = dh_1)|_{M_1}.$$
Moreover, we have $\pi_1(M_1)=Q$, where $\pi_1$ is the restriction to $M_1$ of the canonical projection $\pi_Q : T^*Q \longrightarrow Q$.
Next, assume that $\gamma$ is a closed 1-form on $Q$ such that $\gamma(Q)
\subset M_1$. Define now a vector field $X^\gamma$ on $Q$ by putting $$X^\gamma = T\pi_1 \circ X \circ \gamma\; .$$ The following diagram summarizes the above construction: $$\xymatrix{
M_1\ar[dd]^{\pi _1} \ar[rrr]^{X}& & &TM_1\ar[dd]^{T\pi _1}\\
& & &\\
Q\ar@/^2pc/[uu]^{\gamma_1}\ar[rrr]^{X^\gamma}& & & TQ }$$
Here $\gamma_1$ denotes the restriction of $\gamma$.
We have $$\begin{aligned}
\gamma_1^*(i_{X-T\gamma_1(X^\gamma)} \, \omega_1) &=& \gamma_1^*(i_X \,
\omega_1) - \gamma_1^*(i_{T\gamma_1(X^\gamma)} \, \omega_1) \\
&=& \gamma_1^* d(h_1) - \gamma_1^*(i_{T\gamma_1(X^\gamma)} \, \omega_1)\\
&=&d(h_1 \circ \gamma_1)\end{aligned}$$ since $\gamma^*(i_{T\gamma_1 X^\gamma} \, \omega_1) = i_{X^\gamma} \,
(\gamma_1^* \omega_1) = 0$, because $$\gamma_1^* \omega_1 = \gamma_1^* j_1^* \omega_Q = (j_1 \circ \gamma_1)^* \omega_Q = \gamma^* \omega_Q = - d\gamma = 0$$
Therefore, taking into account that $V\pi_1\oplus T\gamma_1(TQ)= TM_1$ and $\omega_1$ (as it happens with $\omega_Q$) vanishes acting on two vertical tangent vectors with respect to the canonical projection $\pi_1:M_1
\rightarrow Q$, we deduce the following: $$\label{condition-1}
X-T\gamma_1(X^\gamma) \in \ker (\omega_1) \Leftrightarrow d(h_1 \circ
\gamma_1) = 0$$
Moreover, we will show that it is possible to refine condition (\[condition-1\]) and to prove that $X$ and $X^{\gamma}$ are $\gamma_1$ related.
First at all, it is clear that for any point $p$ of $M_1$ $$T_p(T^*Q)=T_pM_1+V_p(T^*Q)$$ where $V(T^*Q)$ denotes the space of vertical tangent vectors at $p$.
In addition, $X-T\gamma_1(X^{\gamma})$ is vertical at the points of $\textrm{Im}(\gamma_1)$, so given any $Z\in V_p(T^*Q)$, $p\in \textrm{Im}(\gamma_1)$, we deduce $$\omega_Q(X-T\gamma_1(X^{\gamma}),Z)=0 \hbox{ along $\textrm{Im} ( \gamma_1)$}$$ since $\omega_Q$ vanishes on two vertical tangent vectors.
Now, given $Z\in T_pM_1$ we have $$\omega_Q(X-T\gamma_1(X^{\gamma}),Z)=\omega_1(X-T\gamma_1(X^{\gamma}),Z)=0$$ because $X-T\gamma_1(X^{\gamma})\in \ker( \omega_1)$, and we obtain that $\omega_Q(X-T\gamma_1(X^{\gamma}),Z)$ $=0$ for any tangent vector $Z\in T_p(T^*Q)$ on any point of $\textrm{Im} ( \gamma_1)$. Since $\omega_Q$ is non-degenerate we deduce that $X=T\gamma_1(X^{\gamma})$ on $\textrm{Im} ( \gamma_1)$.
In conclusion, we have the following result
\[hj111\] $$X \textrm{ and }T\gamma_1(X^\gamma) \textrm{ are $\gamma_1$ related } \Leftrightarrow d(h_1 \circ
\gamma_1) = 0$$
[As a consequence of the above result, if $h_1$ is constant along $\gamma_1(Q)$ then $\gamma_1$ maps the integral curves of $X^\gamma$ on integral curves of $X$. So $d(h_1\circ \gamma_1)=0$ can be considered as the Hamilton-Jacobi equation in this case. ]{}
Case II: There are secondary constraints
----------------------------------------
In this case, the algorithm produces a sequence of submanifolds as follows $$\cdots M_k \hookrightarrow \cdots\hookrightarrow M_2 \hookrightarrow M_ 1
\hookrightarrow T^*Q$$
We assume that the projections $Q_r:=\pi_Q(M_r)$ are submanifolds, and that the corresponding projections $\pi_r:M_r\rightarrow Q_r$ are fibrations, where $\pi_r$ is the restriction of $\pi_Q$ to $M_r$.
The constraint algorithm produces a solution $X$ of the equation $$(i_X \, \omega_1 = dh_1)_{|M_f} \; ,$$ where $X$ is a vector field on $M_f$.
Coming back to the Gotay-Nester-Dirac algorithm we can summarize the situation in the following diagram: $$\xymatrix{
TQ \ar@{->}[r]^{FL} \ar[dr]_{FL_1} & T^*Q \ar[dr]^{\pi _Q} & \\
& M_1 \ar@{_{(}->}[u] \ar[r]^{\pi _1} & Q \\
& M_2 \ar@{_{(}->}[u] \ar[r]^{\pi _2 }&\ar@{_{(}->}[u] Q_2 \\
& \ar@{_{(}->}[u] & \ar@{_{(}->}[u] \\
& \ar@{.}[u] &\ar@{.}[u] \\
& M_f \ar@{_{(}->}[u] \ar[r]^{\pi _f } & \ar@{_{(}->}[u] Q_f
}$$
Assume now that $\gamma$ is a closed 1-form on $Q$ such that
- $\gamma(Q) \subset M_1$.
- $\gamma(Q_f) \subset M_f$.
As in Case I, $\gamma$ allows us to define a vector field $X^\gamma\in \mathfrak X (Q_f)$ by $$X^{\gamma} = T\pi_f \circ X \circ \gamma_f\; .$$ $$\xymatrix{
M_f\ar[dd]^{\pi _1} \ar[rrr]^{X}& & &TM_f\ar[dd]^{T\pi _f}\\
& & &\\
Q_f\ar@/^2pc/[uu]^{\gamma_f}\ar[rrr]^{X^\gamma}& & & TQ_f
}$$ Here $\gamma_f$ is the restriction of $\gamma$ to $Q_f$.
Now, given $q\in Q_f$, we have $$\begin{array}{l}
i_{\left(X(\gamma_1(q))-T_{q}\gamma_f(X^{\gamma}(q))\right)}\, \omega_1\circ T_{q}\gamma_1= i_{X(\gamma(q))}\, \omega_1\circ T_{q}\gamma_1-i_{T_{q}\gamma_f(X^{\gamma}(q))}\, \omega_1\circ T_{q}\gamma_1\\ \noalign{\medskip}
=dh_1({\gamma_f(q)})\circ T_{q}\gamma=d(h_1\circ \gamma_1)(q)\; .
\end{array}$$ Observe that since $\gamma_f$ is the restriction of $\gamma_1$ we have $T_{q}\gamma_f(X^{\gamma}(q))=T_{q}\gamma(X^{\gamma}(q))$. Therefore, given $Y_{q}\in T_{q}Q$ then $T_{q}\gamma_1(Y(q))=T_{q}\gamma(Y(q))$, and we deduce that $$\begin{array}{l}
i_{T_{q}\gamma_f(X^{\gamma}(q))}\, \omega_1\circ T_{q}\gamma(Y(q))= \omega_1(T_{q}\gamma(X^{\gamma}(q)), T_{q}\gamma(Y(q)))\\ \noalign{\medskip}
=(\gamma_1^*\omega_1)(X^{\gamma}(q),Y(q))=\gamma_1^*j_1^*\omega_Q(X^{\gamma}(q),Y(q))=d\gamma(X^{\gamma}(q),Y(q))=0\; .
\end{array}$$ The previous discussion can be applied to every point $q\in Q_f$; therefore, taking into account that $\omega_1$ vanishes acting on two vertical tangent vectors and $V\pi_1\oplus T\gamma_1(TQ)= TM_1$, we can deduce the following $$X-T\gamma_f(X^\gamma) \mathop{\in}_{M_f} \ker( \omega_1) \Leftrightarrow d(h_1 \circ
\gamma_1)_{|Q_f} = 0$$
Usin g a similar argument that in Case I, it is possible to deduce that $X$ and $X^{\gamma}$ are $\gamma_f$ related since we have $$T_p(T^*Q)=T_pM_1+V_p(T^*Q)$$ for all $p\in M_f$.
Therefore, we deduce the following.
\[hj111\] $$X \textrm{ and }T\gamma_1(X^\gamma) \textrm{ are $\gamma_f$ related } \Leftrightarrow d(h_1 \circ
\gamma_1)_{|Q_f} = 0$$
Notice that the condition $X-T\gamma_f(X^\gamma)\in\textrm{Ker}(\omega_f)$ along $\gamma_f(Q_f)$ implies that, if $\sigma : {\mathbb{R}}\longrightarrow Q_f$ is an integral curve of $X^\gamma$, then $\sigma_\gamma = \gamma \circ \sigma : {\mathbb{R}}\longrightarrow M_f$ is an integral curve of $X$
Therefore, the condition $$d(h_1 \circ \gamma_1)_{|Q_f} = 0$$ could be still considered as the Hamilton-Jacobi equation in this context.
Hamilton-Jacobi theory for further geometric constraint equations
-----------------------------------------------------------------
Besides of the equation $i_X \, \omega=dh_1$ introduced in the previous section, other equations have been studied in the literature starting with the same data, that is, a singular lagrangian. For completeness, in this section we will discuss some of these equation of motions.
A good reference for these topics is [@gotaythesis].
### Extended equation of Motion and the Dirac conjeture {#DIRAC-a}
A constraint is called first class provided its Poisson bracket with every other constraint weakly vanishes, and second class otherwise (see Section \[dirac\] for more details). Dirac [@dirac] conjectured that all first-class secondary constraints generate ‘gauge transformations’ which leave the physical state invariant. See, for instance [@gotay4] and references therein for the discussion about the avaibility of the Dirac conjeture. Moreover, the motivation of our study will be more clear in section \[dirac\].
Withour entering in physical discussions, we will analyze if it is possible to extend our Hamilton-Jacobi formalism for the equations derived assuming Dirac conjeture. Therefore, we need first to discuss the geometry of this ‘extended equation’ for singular lagrangians.
Suppose that we are in the conditions of the previous section. We have $j_1:M_1\rightarrow T^*Q$ where $M_1$ is a submanifold and $j_1$ the inclusion, and a constrained hamiltonian $h_1:M_1\rightarrow \mathbb{R}$. As before, we study the presymplectic system $(M_1, \omega_1, dh_1)$ and apply the Gotay-Nester-Hinds algorithm, assuming that we reach to a final constraint manifold $M_f$. Denote by $j_f:M_f\rightarrow M_1$ the inclusion. Now we say that a vector field $X$ on $M_f$ is a solution of the extended equations of motion if $X$ can be writen $$\label{extended}
X=Y+Z$$ where $Y$ and $Z$ are vector fields on $M_f$, such that $i_Y \, \omega_1=dh_1$ and $Z\in\ker(\omega_f)$ where $\omega_f:=j_f^*(\omega_1)$.
We can now obtain a less restrictive version of the previous Hamilton-Jacobi theory, which gives solutions of the extended equations of motion.
Assume again that $\gamma$ is a $1$-form on $Q$ such that
1. Im$(\gamma)\subset M_1$
2. Im$(\gamma_f)\subset M_f$
3. $d\gamma=0$
From a fixed solution $X$ of the extended equation, we can define $X^{\gamma} = T\pi_f \circ X \circ \gamma_f$.
Proceeding as in the previous section, we have $$\begin{aligned}
\gamma_f^*(i_{X-T\gamma_f(X^\gamma)} \, \omega_f) &=& \gamma_f^*(i_X
\, \omega_f) - \gamma_f^*(i_{T\gamma_f(X^\gamma)} \, \omega_f) \\
&=& \gamma_f^* dh_1 - \gamma_f^*(i_{T\gamma (X^\gamma)} \, \omega_f)\\
&=&d(h_1 \circ \gamma_f)\; .\end{aligned}$$ since $\gamma$ is closed.
Using similar arguments that in the previous section, we deduce the following
\[112\] Under the above conditions, we have$$X-T\gamma_f(X^\gamma)\mathop{\in}_{M_f} \ker(\omega_f)\Leftrightarrow d(h_1\circ\gamma_f)=0.$$
If $d(h_1\circ\gamma_f)=0$, then $X-T\gamma_f(X^\gamma)\in\ker (\omega_f)$.
[**Proof:**]{} It follows the same lines of the proofs of Proposition \[hj111\] but now observing that $$TM_f=T\gamma_f(TQ_f)\oplus V\pi_f$$ and $V\pi_f\subset V_{M_f}\pi_Q$. $\Box$.
By the last proposition $T\gamma_f(X^\gamma)=X+\tilde{Z}$, with $\tilde{Z}\in \ker \omega_f$. Then, form (\[extended\]) we have that $T\gamma_f(X^\gamma)=Y+(Z+\tilde{Z})$. So, $T\gamma_f(X^\gamma)$ is a solution of the extended equations of motion.
Therefore, the condition $$d(h_1 \circ \gamma_f) = 0$$ could be still considered as the Hamilton-Jacobi equation in this context.
### Hinds algorithm {#HINDS-a}
Besides of the Gotay-Nester-Hinds algorithm, other approaches have been discussed in the literature. In particular we briefly recall the algorithm introduced by Hinds (see Gotay [@gotay0] for a detailed discussion). Hinds algorithm also start considering the equation $i_X\omega_1=dh_1$ as the Gotay-Nester-Hinds algorithm. The algorithm generates a descending sequence of constraint submanifolds. In the favorable case, the algorithm stabilizes at a final constraint submanifold which we will denote again by $N_f$ (see discussion below). It is important to point out that, in general, this constraint submanifold $N_f$ will be different from the final constraint submanifold obtained by the Gotay-Nester-Hinds algorithm, that is $N_f\neq M_f$. In principle, both algorithms start to diverge from each other after the second step.
In more geometric terms, assume that we are in the conditions of the previous section. Define $N_1:=M_1$ as we did before and denotes $N_{l+1}$ for $l>1$ the following subset $$N_{l+1}:=\{p\in N_l \textrm{ such that exists } X\in T_pN_l \textrm{ verifying }i_{X}\omega_l=dh_l\},$$ where, if we call $k_l:N_l\rightarrow N_1$ the natural inclusion, then $\omega_l:=k_l^*\omega_1$ and $h_l:=k_l^*h_1$. We obtain the sequence of submanifolds $$\cdots N_k \hookrightarrow \cdots \hookrightarrow N_2 \hookrightarrow N_ 1=M_1
\hookrightarrow T^*Q.$$ Again if the algorithm stabilizes, i.e. $N_k=N_{k+1}$, then we say that $N_k$ is the final constraint manifold, $N_f$. In this case, the Hinds algorithm produces a solution $X\in {\mathfrak X}(N_f)$ of the equation $$i_{X}\omega_f=dh_f.$$ This equation is less restrictive than , and so the two algorithms diverge for $l\geq 2 $. We will come back later to the above equation.
Now, we can develop a Hamilton-Jacobi theory in this setting.
Assume that there exists a $1$-form $\gamma$ on $Q$ satisfying
1. Im$(\gamma_f)\subset N_f$ ,
2. $d\gamma=0$ along $N_f$.
Then we can define $X^{\gamma} = T\pi_f \circ X \circ \gamma_f$ and state the equivalent Hamilton-Jacobi theory. The proof follws the same lines that in proposition \[112\].
\[113\] $$X-T\gamma_f(X^\gamma)\in \ker(\omega_f)\Leftrightarrow d(h_1\circ\gamma_f)=0.$$
Relation to the Dirac-Bergmann theory of constraints {#dirac}
----------------------------------------------------
In this section we will discuss the relation of the Gotay and Nester theory with the original Dirac-Bergmann theory of constraints.
Assume that we begin with an almost regular lagrangian $L:TQ\rightarrow \mathbb{R}$. Then there exists an open neighbourhood, $U\subset T^*Q$ where in canonical coordinates $(q^A,p_A)$, $M_1\cap U$ is given by the vanishing of functions $\Phi^i(q^A,p_A)$ defined on $U$. The functions $\Phi^i$ are called primary constraints.
Remember that we can project $E_L$ to $h_1:M_1\rightarrow \mathbb{R}$, and any extension of $h_1$ to $U$ should be of the form $$H=h+u_i\Phi^i \; ,$$ where $h$ is an arbitrary extension to $U$ of $h_1$. The functions $u_i$, $1\leq i\leq 2\dim Q-\dim M_1$ are Lagrange multipliers to be determined.
Acording to Dirac the equations of motion are $$\begin{array}{l}
\dot{q}^A=\frac{\partial H}{\partial p_A}+u_i\frac{\partial \Phi^i}{\partial p_A} \\ \noalign{\medskip}
\dot{p}_A=-\frac{\partial H}{\partial q^A}-u_i\frac{\partial \Phi^i}{\partial q^A}
\end{array}$$ which must hold over $U_1:=M_1\cap U$. If we denote $j_1:U_1\rightarrow U$ the inclusion, and $\omega_1=j_1^*\omega_Q$, the preceding equations can be equivalen rewritten as $$\label{ecu}\begin{array}{lcr}
i_X\omega_1=dh_1 &\textrm{or}& (i_X\omega_Q=dh+u_id\Phi^i)_{|U_1}
\end{array}$$ which are the equations that we have considered in the Gotay-Nester-Hinds algorithm.
Since $X$ must be tangent to $U_1$ we should have $$\begin{array}{lll}
0&=&(X(\Phi^i))_{|U_1}=\{\Phi^i,H\}_{|U_1}=\{\Phi^i,h+u_j\Phi^j\}_{|U_1} \\ \noalign{\medskip}
&=&(\{\Phi^i,h\}+u_j\{\Phi^i,\Phi^j\})_{|U_1}.
\end{array}$$
These equations can be trivially satified, determine some Lagrange multipliers or add new constraints on the variables $q^A$, $p_A$ over $U_1$. These new constraints, if any, are called secondary constraints. Suppose that we have obtained the secondary constraints $\xi^{\alpha}$. So, we have to restrict the dynamics to $U_2:=U_1\cap (\xi^{\alpha})^{-1}\{0\}$.
Again, the solution must be tangent to $U_2$ and it requires that $$\begin{array}{lll}
0&=&(X(\xi^{\alpha}))_{|U_2}=\{\xi^{\alpha},H\}_{|U_2}=\{\xi^{\alpha},h+u_i\Phi^i\}_{|U_2} \\ \noalign{\medskip}
&=&(\{\xi^{\alpha},h\}+u_i\{\xi^{\alpha},\Phi^i\})_{|U_2}.
\end{array}$$ As before, these equations may determine more Lagrange multipliers or add new constraints to the picture, that is, new secondary constraints. Iterating this procedure, if the algorithm stabilizes, we arrive to a set $U_f$ which is an open subset of the final constraint manifold $M_f$ obtained by the Gotay-Nester-Hinds algorithm (see [@gotay0] for a proof ).
It is necesary to introduce some definitions. We say that a function defined on $U$ is **first class** if its Poisson bracket with every constraint (primary and secondary) vanishes. Otherwise, it is said to be of **second class**.
We can reorder constraints into first class or second class. We will denote by $\chi^a$ and $\xi^b$, the primary first and second class constraints, respectively; and by $\psi^c$ and $\theta^d$, the secondary first and second class constraints, respectively. We will also denote by $\mu_a,\lambda_b$ the corresponding Lagrange multipliers for the primary first and second class constraints, respectively.
So, if the problem has a solution, we must obtain a vector field $X$ over $U_f$, which satisfies the equations $$(i_X \, \omega_Q = dh+\mu_ad\chi^a+\lambda_bd\xi^b)_{|M_f}.$$ The $\lambda_b$’s are determined functions and the $\mu_a$’s can be varied to obtain other admissible solutions. In consequence, it is also clear that primary first class constraints correspond to gauge transformations which leave the physical state invariant. As we have discussed before, Dirac conjectured that the first class secondary constraints may also generate gauge transformations, therefore, the generalized equations of motion discussed in Subsection \[DIRAC-a\] are locally rewritten as $$\label{dirac1}
(i_X \, \omega_Q = dh+\mu_ad\chi^a+\lambda_bd\xi^b+v_cd\psi^c)_{|M_f}.$$ where $\lambda_b$ are still determined functions and $\mu_a$ and $v_c$ can be varied arbitrarily. The hamiltonian $h+\mu_a\chi^a+\lambda_b\xi^b+v_c\psi^c$ is called the extended hamiltonian, and equation the extended equation of motion following the notation of [@gotaythesis]. Geometrically the solutions of are just $$X=Y+Z,$$ where $Y$ is a vector field on $M_f$ solution of the equations of motion, \[ecu\], and $Z\in \ker(\omega_f)$ where $\omega_f$ is the restriction of $\omega_1$ to $M_f$.
If we proceed in the same way with the Hinds algorithm developed in Subsection \[HINDS-a\], we will arrive to solutions $X$ satisfying $$(i_X \, \omega_Q = dh+\mu_ad\chi^a+\lambda_bd\xi^b +\overline{v}_{\overline{c}}\overline{\psi}^{\overline{c}}+\overline{w}_{\overline{d}}\overline{\theta}^{\overline{d}})_{|N_f},$$ where $\overline{v}_{\overline{c}}$, $\overline{w}_{\overline{d}}$ are the Lagrange multipliers corresponding to the constraints $\overline{\psi}^{\overline{c}}$ and $\overline{\theta}^{\overline{d}}$. Note that $\overline{\psi}^{\overline{c}}$ and $\overline{\theta}^{\overline{d}}$ now correspond to the secondary constraints of the final constraint manifold $N_f$ in the Hinds algorithm.
Examples
--------
Now we illustrate the previous propositions with several examples.
### There are only primary constraints
This example is discussed by O. Krupkova in [@Krup1]. Let $L$ be the Lagrangian $L:T{\mathbb{R}}^3\rightarrow \mathbb{R} $ given by $$L(q^1,q^2,q^3,\dot{q}^1,\dot{q}^2,\dot{q}^3)=\frac{1}{2}(\dot{q}^1+\dot{q}^2)^2.$$ Then $FL$ is given by $FL:T{\mathbb{R}}^3\rightarrow T^*{\mathbb{R}}^3$ $$FL(q^1,q^2,q^3,\dot{q}^1,\dot{q}^2,\dot{q}^3)=(q^1,q^2,q^3,\dot{q}^1+\dot{q}^2,\dot{q}^1+\dot{q}^2,0),$$ and the primary constraints are $$\begin{array}{lr}
\Phi^1(q^A,p_A)=p_1-\,p_2 & \Phi^2(q^A,p_A)=p_3.
\end{array}$$ So $$M_1=\{(q^1,q^2,q^3,p_1,p_2,p_3)\in\mathbb{R}^6 \textrm{ such that } p_1=p_2, \ p_3=0\}$$ and we can use $(q^1,q^2,q^3,p_1)$ as coordinates on $M_1$.
It follows that $$\begin{array}{l}
E_L=(\dot{q}^1+\dot{q}^2)\dot{q}^1+(\dot{q}^1+\dot{q}^2)\dot{q}^2-L=L \\ \noalign{\medskip}
h_1=\frac{1}{2}({p_1})^2 \\ \noalign{\medskip}
\omega_1=dq^1\wedge dp_1+dq^2\wedge dp_1 \\ \noalign{\medskip}
\textrm{Ker}(\omega_1)=\displaystyle{\left\{\frac{\partial}{\partial q^3},\frac{\partial}{\partial q^1}-\frac{\partial}{\partial q^2} \right\}} \\
\end{array}$$ and a particular extension of the hamiltonian is $$h(q^A,p_A)=\frac{1}{2}({p_1})^2$$
It is easy to see that at the points of $M_1:=\textrm{Im}(FL)$ $$\{\Phi^i, h +u^1\Phi^1+u^2\Phi^2\}=0, \ i=1,2.$$
So we have global dynamics on $M_1$ and it holds that $$\{\Phi^1,\Phi^2\}=0$$ and we conclude that there are only first class constraints. The solutions of $ (i_X\omega_1=dh_1)_{|M_1}$ on $M_1$ are given by $$X=p_1\frac{\partial}{\partial q^1}+f_1\frac{\partial}{\partial q^3}+f_2(\frac{\partial}{\partial q^1}-\frac{\partial}{\partial q^2}),$$ where $f_1$ and $f_2$ are functions on $M_1$.
We now look for $\gamma\in \Lambda^1(Q)$ such that
1. $\gamma(Q)\subset M_1$
2. $d(h_1\circ \gamma)=0$
3. $d\gamma=0$
Suppose $$\gamma(q^1,q^2,q^3)=(q^1,q^2,q^3,\gamma_1(q^1,q^2,q^3),\gamma_2(q^1,q^2,q^3),\gamma_3(q^1,q^2,q^3)),$$ then, $\gamma(Q)\subset M_1$ implies that $\gamma_1=\gamma_2$ and $\gamma_3=0$.
The condition $d(h_1\circ \gamma)=0$ implies that $\frac{1}{2}({\gamma_1})^2=\textrm{constant}$, and because of that $\gamma_1=c$ where $c$ is a constant and so $\gamma_2=c$.
Now $\gamma(q^1,q^2,q^3)=(q^1,q^2,q^3,c,c,0)$ and $d\gamma=0$ is trivially satisfied.
If we take the general solution $p_1\frac{\partial}{\partial q^1}+f_1\frac{\partial}{\partial q^3}+f_2(\frac{\partial}{\partial q^1}-\frac{\partial}{\partial q^2})$ then we obtain $X^{\gamma}=T\pi_q\circ X\circ \gamma=c\frac{\partial}{\partial q^1}+(f_1\circ \gamma)\frac{\partial}{\partial q^3}+(f_2\circ\gamma)(\frac{\partial}{\partial q^1}-\frac{\partial}{\partial q^2})$. If we apply $T\gamma(q^A)(X^{\gamma}(q^A))=X(\gamma(q^A))$, then we recover the solution $X$ over the points of $\gamma$. It is clear, that integral curves of $X^{\gamma}$ are applied by $\gamma$ into integral curves of $X$ along ${\hbox{Im }\gamma}$.
This example has been discussed by J. Barcelos-Neto and N.R.F. Braga [@barcelos]. Let $L$ be the Lagrangian $L:T{\mathbb{R}}^4\rightarrow \mathbb{R} $ given by $$L(q^1,q^2,q^3,q^4,\dot{q}^1,\dot{q}^2,\dot{q}^3,\dot{q}^4)=(q^2+q^3)\dot{q}^1 +q^4\dot{q}^3+\frac{1}{2}\left((q^4)^2-2q^2q^3-(q^3)^2\right).$$
Then $FL$ is given by $FL:T{\mathbb{R}}^4\rightarrow T^*{\mathbb{R}}^4$ $$FL(q^1,q^2,q^3,q^4,\dot{q}^1,\dot{q}^2,\dot{q}^3,\dot{q}^4)=(q^1,q^2,q^3,q^4,q^2+q^3,0,q^4,0)$$ and the primary constraints are $$\begin{array}{lr}
\Phi^1(q^A,p_A)=p_1-q^2-q^3 , & \Phi^2(q^A,p_A)=p_2 , \\ \noalign{\medskip} \Phi^3(q^A,p_A)=p_3-q^4 , & \Phi^4(q^A,p_A)=p_4.
\end{array}$$ So $$\begin{array}{l}
M_1=\{(q^1,q^2,q^3,q^4,p_1,p_2,p_3,p_4)\in\mathbb{R}^8 \textrm{ such that }\\ \noalign{\medskip}\quad \quad \quad \quad p_1=q^2+q^3, \ p_2=0, \ p_3=q^4, \ p_4=0\}.
\end{array}$$ and we can use $(q^1,q^2,q^3,q^4)$ as coordinates on $M_1$.
It follows that $$\begin{array}{l}
E_L=(q^2+q^3)\dot{q}^1+q^4\dot{q}^3-L=-\frac{1}{2}((q^4)^2-2q^2q^3-(q^3)^2) \\ \noalign{\medskip}
h_1=-\frac{1}{2}((q^4)^2-2q^2q^3-(q^3)^2) \\ \noalign{\medskip}
\omega_1=dq^1\wedge dq^2+dq^1\wedge dq^3+ dq^3\wedge dq^4 \\ \noalign{\medskip}
\textrm{Ker}(\omega_1)=\{0\} \\
\end{array}$$ so $(M_1, \omega_1)$ is a symplectic manifold.
It we prefer to follow the Dirac-Bergmann algorithm, then one should take an extension $
h(q^A,p_A)=-\frac{1}{2}((q^4)^2-2q^2q^3-(q^3)^2)
$ of $h_1$. It is easy to see that at the points of $M_1:=\textrm{Im}(FL)$ $$\begin{array}{l}
\{\Phi^1, h +u^1\Phi^1+u^2\Phi^2+u^3\Phi^3+u^4\Phi^4\}=-u^2-u^3\\\noalign{\medskip}
\{\Phi^2, h +u^1\Phi^1+u^2\Phi^2+u^3\Phi^3+u^4\Phi^4\}=-q^3+u^1\\\noalign{\medskip}
\{\Phi^3, h +u^1\Phi^1+u^2\Phi^2+u^3\Phi^3+u^4\Phi^4\}=-q^2-q^3+u^1-u^4 \\\noalign{\medskip}
\{\Phi^4, h +u^1\Phi^1+u^2\Phi^2+u^3\Phi^3+u^4\Phi^4\}= u^3+u^4,
\end{array}$$ which determine completely the Lagrange multipliers: $$u^1=q^3\, , u^2=q^4\, ,
u^3=-q^4\, ,
u^4=-q^2,$$ and then all the constraints are of second class.
The solution of the equation $(i_X\omega_1=dh_1)_{|M_1}$ is given by $$X=q^3\frac{\partial}{\partial q^1}+q^4\frac{\partial}{\partial q^2}-q^4\frac{\partial}{\partial q^3}+(2q^3-q^2)\frac{\partial}{\partial q^4}-q^2\frac{\partial}{\partial p_3}$$
We will study now the solutions of the Hamilton-Jacobi equation. So, we look for $\gamma\in \Lambda^1(\mathbb{R}^4)$ such that
1. $\gamma(\mathbb{R}^4)\subset M_1$
2. $d(h_1\circ \gamma)=0$
3. $d\gamma=0$
If $\gamma(q^A)=(q^A,\gamma_1(q^A),\gamma_2(q^A),\gamma_3(q^A),\gamma_4(q^A))$, then the condition $\gamma(Q)\subset M_1$ gives $$\begin{array}{l}
\gamma_1(q^1,q^2,q^3,,q^4)=q^2+q^3 \\ \noalign{\medskip}
\gamma_2(q^1,q^2,q^3,,q^4)=0 \\ \noalign{\medskip}
\gamma_3(q^1,q^2,q^3,,q^4)=q^4 \\ \noalign{\medskip}
\gamma_4(q^1,q^2,q^3,,q^4)=0 \\ \noalign{\medskip}
\end{array}$$ But $h_1\circ \gamma=-\frac{1}{2}((q^4)^2-2q^2q^3-(q^3)^2)$, so the equation $d(h_1\circ \gamma)=0$ if and only if $\gamma(q)=(q^A,0,0,0,0)$.
\[exe\]
This example has been discussed by K. Sundermeyer [@sundermeyer]. Let $L$ be the Lagrangian $L:T{\mathbb{R}}^2\rightarrow \mathbb{R} $ given by $$L(q^1,q^2,\dot{q}^1,\dot{q}^2)=\frac{1}{2}(\dot{q}^1)^2+ \dot{q}^2\, q^1+\dot{q}^2\, q^1.$$
Then $FL$ is given by $FL:T{\mathbb{R}}^4\rightarrow T^*{\mathbb{R}}^4$ $$FL(q^1,q^2,\dot{q}^1,\dot{q}^2)=(q^1,q^2,\dot{q}^1+q^2,q^1)$$ and the primary constraints are $$\Phi^1(q^A,p_A)=p_2-q^1$$ So $$\begin{array}{l}
M_1=\{(q^1,q^2,p_1,p_2)\in\mathbb{R}^4 \textrm{ such that } \ p_2=q^1\},
\end{array}$$ and we can use $(q^1,q^2,p_1)$ as coordinates on $M_1$.
It follows that $$\begin{array}{l}
E_L=\frac{1}{2}\dot{q}^1 \\ \noalign{\medskip}
h_1=\frac{1}{2}(p_1-q^2) \\ \noalign{\medskip}
\omega_1=dq^1\wedge dp^1+dq^2\wedge dq^1 \\ \noalign{\medskip}
\textrm{Ker}(\omega_1)=\displaystyle{\left<\frac{\partial }{\partial p_1}-\frac{\partial }{\partial q^2}\right>} \\
\end{array}$$ Let $$h(q^A,p_A)=\frac{1}{2}(p_1-q^2)$$ be an extension of the hamiltonian.
It is easy to see that at the points of $M_1:=\textrm{Im}(FL)$ $$\{\Phi^1, h +u\Phi^1\}=0$$ and therefore we have global dynamics.
The solution of the equation $(i_X\omega_1=dh_1)_{|M_1}$ is given by $$X=(p_1-q^2)\frac{\partial}{\partial q^1}+f\frac{\partial}{\partial q^2}+f\frac{\partial}{\partial p_1}+(p_1-q^2)\frac{\partial}{\partial p_2},$$ where $f\in C^{\infty}(M_1)$
If we now look for $\gamma\in \Lambda^1(\mathbb{R}^4)$ such that
1. $\gamma(\mathbb{R}^4)\subset M_1$
2. $d(h_1\circ \gamma)=0$
3. $d\gamma=0$
then $\gamma(q^1,q^2)=(q^1,q^2,\gamma_1(q^1,q^2),\gamma_2(q^1,q^2))$ given by $$\gamma(q^1,q^2)=(q^1,q^2,q^2,q^1)$$ satisfies all the requiered conditions, because $p_1(\gamma(q^1,q^2)=q^2$, $\gamma=d(q^1\cdot q^2)$ and $h_1\circ \gamma(q^1,q^2)=\frac{1}{2}(q^2-q^2)=0$. Given an arbitrary solution $X=(p_1-q^2)\frac{\partial}{\partial q^1}+f\frac{\partial}{\partial q^2}+f\frac{\partial}{\partial p_1}+(p_1-q^2)\frac{\partial}{\partial p_2}$ of the constarined dynamics, we have that $$X^{\gamma}=(f\circ \gamma)\frac{\partial}{\partial q^2}$$ and also $$T\gamma(X^{\gamma})=(f\circ \gamma)\frac{\partial}{\partial q^2}+(f\circ \gamma)\frac{\partial}{\partial p^1}$$ which is precisely $X$ along $\textrm{Im}(\gamma)$.
### There are secondary constraints
Next, we are going to describe several examples where secondary constraints appear.
This example has been discussed by M.J. Gotay and J.M. Nester [@gotay3]. Let $L$ be the Lagrangian $L:T\mathbb{R}^2\rightarrow \mathbb{R} $ given by
$$L(q^1,q^2,\dot{q}^1,\dot{q}^2)=\frac{1}{2}(\dot{q}^1)^2+q^2(q^1)^2.$$
Then $FL$ is given by $FL:T\mathbb{R}^2\rightarrow T^*\mathbb{R}^2$ $$FL(q^1,q^2,\dot{q}^1,\dot{q}^2)=(q^1,q^2,\dot{q}^1,0)$$ and the primary constraints are $$\Phi^1(q^A,p_A)=p_2.$$
So $$M_1=\{(q^1,q^2,p_1,p_2)\in\mathbb{R}^4 \textrm{ such that } p_2=0\}$$ and we can use $(q^1,q^2,p_1)$ as coordinates on $M_1$.
It follows that $$\begin{array}{l}
E_L=\frac{1}{2}(\dot{q}^1)^2-q^2(q^1)^2 \\ \noalign{\medskip}
h_1=\frac{1}{2}(p_1)^2-q^2(q^1)^2\\ \noalign{\medskip}
\omega_1=dq^1\wedge dp_1 \\ \noalign{\medskip}
\textrm{Ker}(\omega_1)=\left<\frac{\partial}{\partial q^2} \right> \\
\end{array}$$ Let $$h(q^A,p_A)=\frac{1}{2}(p_1)^2-q^2(q^1)^2$$ be an arbitrary extension of the constarined hamiltonian $h_1$ to $T^*{\mathbb{R}}^2$.
It is easy to see that at the points of $M_1:=\textrm{Im}(FL)$ we have $$\begin{array}{l}
\{\Phi^1, h +u^1\Phi^1\}=-(q^1)^2\\
\end{array}$$ and therefore we need to restrict the dynamics adding a new constraint $$\Phi^2(q^A,p_A)=q^1.$$ Now $M_2:=\{(q^1,q^2,p_1,p_2)\in\mathbb{R}^4 \textrm{ such that } p_2=0, \ q^1=0\}$ and $Q_2:=\pi_Q(M_2)=\{(q^1,q^2)\in\mathbb{R}^2\ \textrm{ such that } q^1=0\} $. We have on $M_2$ $$\begin{array}{l}
\{\Phi^1, h +u^1\Phi^1\}=0\\
\{\Phi^2, h +u^1\Phi^1\}=p_1,
\end{array}$$ and we need to restrict again the dynamics, adding the constraint $$\Phi^3(q^A,p_A)=p_1$$
Now $M_3:=\{(q^1,q^2,p_1,p_2)\in\mathbb{R}^4 \textrm{ such that } p_2=0, \ q^1=0, \ p^1=0\}$ and $Q_3=Q_2=\{(q^1,q^2)\in\mathbb{R}^2\ \textrm{ such that } q^1=0\} $. Along $M_3$ we have $$\begin{array}{l}
\{\Phi^1, h +u^1\Phi^1\}=0\\
\{\Phi^2, h +u^1\Phi^1\}=0\\
\{\Phi^3, h +u^1\Phi^1\}=0,
\end{array}$$ and $M_3$ is the final constraint submanifold, $M_f$. We can easily check that $\Phi^1$ is a first class constraint and $\Phi^2$, $\Phi^3$ are second class.
The solutions of the equation $(i_X\omega_1=dh_1)_{|M_3}$ are of the form $$X=f\frac{\partial}{\partial q^2},$$ where $f\in C^{\infty}(M_3)$.
A solution of the Hamilton-Jacobi equation, should be $\gamma(q^1,q^2)=(q^1,q^2,\gamma_1(q^1,q^2),\gamma_2(q^1,q^2))$, such that
1. $\gamma(Q)\subset M_1$ and $\gamma_f(Q_f)\subset M_f$
2. $d(h_1\circ \gamma)_{|Q_f}=0$
3. $d\gamma=0$
The condition $\gamma(Q)\subset M_1$ implies $\gamma_2=0$. Next we compute $d\gamma$, $$d\gamma=\frac{\partial \gamma_1}{\partial q^2}dq^2\wedge dq^1+ \frac{\partial \gamma_2}{\partial q^1}dq^1\wedge dq^2=\frac{\partial \gamma_1}{\partial q^2}dq^2\wedge dq^1=0$$ and we deduce that $\frac{\partial \gamma_1}{\partial q^2}$ must vanish and $\gamma_1$ must be a function of $q^1$.
The condition $d(h_1\circ \gamma)_{|Q_f}=0$ can also be easily computed. We have $$d(h_1\circ \gamma)=d(\frac{1}{2}(\gamma_1)^2-q^2(q^1)^2)=(\gamma_1\frac{\partial \gamma_1}{\partial q^1}-2q^2q^1)dq^1 + (q^1)^2dq^2$$ and, along $Q_f=\{(q^1,q^2)\in\mathbb{R}^2\ \textrm{ such that } q^1=0\}$, we deduce $$d(h_1\circ \gamma)_{|Q_f}=\gamma_1(0)\frac{\partial \gamma_1}{\partial q^1}(0) \, dq^1$$
For example, if we take $\gamma_1=q^1$, $\gamma_2=0$, all the above conditions are satisfied, and $\gamma_f(Q_f)\subset M_f$.
Now, take a solution $X=f\frac{\partial}{\partial q^2}$; at the points of $Q_f$ we get $$X^{\gamma}(0,q^2)=(\pi_f)_*(f(0,q^2,0,0)\frac{\partial}{\partial q^2})= f(0,q^2,0,0)\frac{\partial}{\partial q^2}$$ and so $$T\gamma_f(X^{\gamma}(0,q^2))=f(0,q^2,0,0)\frac{\partial}{\partial q^2},$$ and we obtain the solution $X$ along $\textrm{Im}(\gamma_f)$.
This example has been discussed by M.J. Gotay [@gotay4]. Let $Q:=\{(q^1,q^2)\in \mathbb{R}^2 \textrm{ such that } q^1\neq 0 \}$ and $L$ be the Lagrangian $L:TQ\rightarrow \mathbb{R} $ given by $$L(q^1,q^2,\dot{q}^1,\dot{q}^2)=\frac{1}{2q^1}(\dot{q}^2)^2.$$
Then $FL$ is given by $FL:TQ\rightarrow T^*Q$ $$FL(q^1,q^2,\dot{q}^1,\dot{q}^2)=(q^1,q^2,0,\dot{q}^2/q^1)$$ and the primary constraints are $$\Phi^1(q^A,p_A)=p_1.$$
So $$M_1=\{(q^1,q^2,p_1,p_2)\in TQ \textrm{ such that } p_1=0\}.$$ and we can use $(q^1,q^2,p_2)$ as coordinates on $M_1$.
It follows that $$\begin{array}{l}
E_L=L \\ \noalign{\medskip}
h_1(q^A,p_A)=\frac{q^1}{2}(p_2)^2 \\ \noalign{\medskip}
\omega_1=dq^2\wedge dp_2 \\ \noalign{\medskip}
\textrm{Ker}(\omega_1)=\left<\frac{\partial}{\partial q_1} \right> \\
\end{array}$$ Let $h(q^A,p_A)=\frac{q^1}{2}(p_2)^2$ be an extension of the hamiltonian.
It is easy to see that at the points of $M_1:=\textrm{Im}(FL)$ we get $$\begin{array}{l}
\{\Phi^1, h +u^1\Phi^1\}=-\frac{(p_2)^2}{2}\\
\end{array}$$ and therefore we need to restrict the dynamics adding a new constraint $$\Phi^2(q^A,p_A)=p_2.$$ Now $M_2:=\{(q^1,q^2,p_1,p_2)\in TQ \textrm{ such that } p_1=0, \ p_2=0\}$ and $Q_2:=\pi_Q(M_2)=Q $. At the points of $M_2$ we have $$\begin{array}{l}
\{\Phi^1, h +u^1\Phi^1\}=0\\
\{\Phi^2, h +u^1\Phi^1\}=0,
\end{array}$$ and $M_2$ is the final contraint manifold. From $\{\Phi^1, \Phi^2\}=0$ we deduce that the constraints are all first class.
The solutions are of the form $X=f\frac{\partial}{\partial q^1}$ where $f\in C^{\infty}(M_2)$.
If we look for a solution of our Hamilton-Jacobi equation, $\gamma$, such that $\gamma(q^1,q^2)=(q^1,q^2,\gamma_1(q^1,q^2),\gamma_2(q^1,q^2))$, then the condition $\gamma(Q)\subset M_1$ implies $\gamma=0$ and $\gamma_f=0$. All conditions are verified and, given a solution $X$, we obtain that $X^{\gamma}$ and $X$ are trivially $\gamma_f$-related.
This example has been discussed by R. Skinner and R. Rusk [@ski]. Let $L$ be the Lagrangian $L:T\mathbb{R}^3\rightarrow \mathbb{R} $ given by $$L(q^1,q^2,q^3,\dot{q}^1,\dot{q}^2,\dot{q}^3)=\frac{1}{2}q^2(q^3)^2+\dot{q}^1\dot{q}^3.$$
Then $FL:T\mathbb{R}^3\rightarrow T^*\mathbb{R}^3$ is given by $$FL(q^1,q^2,q^3,\dot{q}^1,\dot{q}^2,\dot{q}^3)=(q^1,q^2,q^3,\dot{q}^3,0,\dot{q}^1) \;,$$ so that we have a primary constraint $
\Phi^1(q^A,p_A)=p_2.
$ This means that the primary constraint submanifold is $$M_1=\{(q^1,q^2,q^3,p_1,p_2,p_3)\in\mathbb{R}^6 \textrm{ such that } p_2=0\},$$ and then we can use $(q^1,q^2,q^3,p_1,p_3)$ as coordinates on $M_1$.
It follows that $$\begin{array}{l}
E_L=\dot{q}^3\dot{q}^1+\dot{q}^1\dot{q}^3-L=-\frac{1}{2}q^2(q^3)^2+\dot{q}^1\dot{q}^3 \\ \noalign{\medskip}
h_1(q^A,p_A)=p_1p_3-\frac{1}{2}q^2(q^3)^2 \\ \noalign{\medskip}
\omega_1=dq^1\wedge dp_1+dq^3\wedge dp_3 \\ \noalign{\medskip}
\textrm{Ker}(\omega_1)=\left<\frac{\partial}{\partial q^2} \right> .
\end{array}$$ As in the previous cases, take an arbitrary extension of the hamiltonian $h_1$, for instance$$h(q^A,p_A)=p_1p_3-\frac{1}{2}q^2(q^3)^2.$$
It is easy to see that at the points of $M_1:=\textrm{Im}(FL)$ $$\begin{array}{l}
\{\Phi^1, h +u^1\Phi^1\}=\frac{1}{2}(q^3)^2\\
\end{array}$$ and therefore we should restrict the dynamics adding a secondary constraint $$\Phi^2(q^A,p_A)=q^3.$$ Now $M_2:=\{(q^1,q^2,q^3,p_1,p_2,p_3)\in\mathbb{R}^6 \textrm{ such that } p_2=0, \ q^3=0\}$ and $Q_2:=\pi_Q(M_2)=\{(q^1,q^2,q^3)\in\mathbb{R}^3 \textrm{ such that } \ q^3=0\} $. Along $M_2$, we have $$\begin{array}{l}
\{\Phi^1, h +u^1\Phi^1\}=0\\
\{\Phi^2, h +u^1\Phi^1\}=p_1.
\end{array}$$ Therefore, we need again to restrict the dynamics, adding the constraint $
\Phi^3(q^A,p_A)=p_1\; .
$ Now $M_3:=\{(q^1,q^2,q^3,p_1,p_2,p_3)\in\mathbb{R}^6 \textrm{ such that } p_2=0, \ q^3=0, \ p_1=0\}$ and $Q_3=Q_2=\{(q^1,q^2,q^3)\in\mathbb{R}^3 \linebreak \textrm{such that} \ q^3=0\} $. Along $M_3$ we have $$\begin{array}{l}
\{\Phi^1, h +u^1\Phi^1\}=0\\
\{\Phi^2, h +u^1\Phi^1\}=0\\
\{\Phi^3, h +u^1\Phi^1\}=0
\end{array}$$ and then $M_3$ is the final contraint manifold, denoted by $M_f$; therefore, $Q_f=Q_3$. We deduce that the constraints are all first class.
The solutions of the equation $(i_X\omega_1=dh_1)_{|M_3}$ are of the form $$X=p_3\frac{\partial}{\partial q^1}+f\frac{\partial}{\partial q^2},$$ where $f\in C^{\infty}(M_3)$.
Now we look for a solution of our Hamilton-Jacobi equation, that is $\gamma(q^1,q^2,q^3)=(q^1,q^2,q^3,\gamma_1(q^1,q^2,q^3),\gamma_2(q^1,q^2,q^3),\gamma_3(q^1,q^2,q^3))$, such that
1. $\gamma(Q)\subset M_1$ and $\gamma_f(Q_f)\subset M_f$
2. $d(h_1\circ \gamma)_{|Q_f}=0$
3. $d\gamma=0$
The condition $\gamma(Q)\subset M_1$ implies $\gamma_2=0$; the condition $\gamma_f(Q_f)\subset M_f$ implies $(\gamma_f)_i
=0$ for $i=1,2$ and, the condition $d(h_1\circ \gamma)=0$ is $$\begin{array}{l}
d(h_1\circ \gamma)=\gamma_1\gamma_3-\frac{1}{2}q^2(q^3)^2\\
\noalign{\medskip}
=\left(\frac{\partial \gamma_1}{\partial q^1}\gamma_3 +\frac{\partial \gamma_3}{\partial q^1 }\gamma_1 \right)dq^1 +\left( \frac{\partial \gamma_1}{\partial q^2}\gamma_3 +\frac{\partial \gamma_3}{\partial q^2 }\gamma_1 + \frac{1}{2}(q^3)^2\right)dq^2 \\ \noalign{\medskip}+\left(\frac{\partial \gamma_1}{\partial q^3}\gamma_3 +\frac{\partial \gamma_3}{\partial q^3 }\gamma_1 +q^2q^3 \right)dq^3
\end{array}$$ Hence, $$\begin{array}{l}
d(h_1\circ \gamma)_{|Q_f}=\left(\frac{\partial \gamma_1}{\partial q^1}\gamma_3 +\frac{\partial \gamma_3}{\partial q^1 }\gamma_1\right)dq^1+ \left(\frac{\partial \gamma_1}{\partial q^2}\gamma_3 +\frac{\partial \gamma_3}{\partial q^2 }\gamma_1\right)dq^2 \\ \noalign{\medskip}+\left(\frac{\partial \gamma_1}{\partial q^3}\gamma_3 +\frac{\partial \gamma_3}{\partial q^3 }\gamma_1 \right)dq^3
\end{array}$$ The condition $d\gamma=0$ implies $$\begin{array}{ll}
d\gamma= &\frac{\partial \gamma _1}{\partial q^2}dq^2\wedge dq^1 +\frac{\partial \gamma_1}{\partial q^3}dq^3\wedge dq^1 \\ \noalign{\medskip}
&+\frac{\partial \gamma_3}{\partial q^1}dq^1\wedge dq^3+\frac{\partial \gamma_3}{\partial q^2}dq^2\wedge dq^3=0
\end{array}$$ taking into account that $\gamma_2=0$, and therefore $$\begin{array}{l}
\frac{\partial \gamma_1}{\partial q^2}=0 \\ \noalign{\medskip}
\frac{\partial \gamma_1}{\partial q^3}=\frac{\partial \gamma_3}{\partial q^1}\\ \noalign{\medskip}
\frac{\partial \gamma_3}{\partial q^2}=0
\end{array}$$
A particular solution is obtained putting $\gamma_1=\gamma_2=0$, and $\gamma_3$ an arbitrary function of $q^3$, for example $\gamma_3=q^3$.
For instance, take $X=p_3\frac{\partial}{\partial q^1}+f\frac{\partial}{\partial q^2}$ and $S=\frac{1}{2}(q^3)^2$, then $$\gamma(q^1,q^2,q^3)=(q^1,q^2,q^3,0,0,q^3)$$ and at the points of $Q_f$ we obtain $$X^{\gamma}(q^1,q^2,0)=T\pi_f(0\frac{\partial}{\partial q^1}+f(q^1,q^2,0,0,0,0)\frac{\partial}{\partial q^2})=f(q^1,q^2,0,0,0,0)\frac{\partial}{\partial q^2},$$ so that $$T\gamma_f( X^{\gamma}(q^1,q^2,0))=f(q^1,q^2,0,0,0,0)\frac{\partial}{\partial q^2}$$
We can also apply proposition \[112\] to the latter example and obtain solutions of the extended equation.
For instance, consider $\gamma(q^1,q^2,q^3)=(q^1,q^2,q^3,\gamma_1(q^A),\gamma_2(q^A),\gamma_3(q^A))$ given by $$\gamma(q^1,q^2,q^3)=(q^1,q^2,q^3,q^3,0,q^1)$$
We have
1. $\gamma_f(Q_f)\subset M_f$ because $$\gamma_f(Q_f)=\{(q^1,q^2,0,0,0,q^1)\in\mathbb{R}^6 \textrm{ such that } q^1, \, q^2 \in \mathbb{R}\}.$$
2. If we take coordinates $(q^1,q^2)$ in $Q_f$, then $d(h_1\circ \gamma_f)=d(0\cdot 0 -q^2 \cdot 0)=0$.
3. $d\gamma=0$, in fact, $\gamma=d(q^1 \cdot q^3)$.
If we consider a solution $X=p_3\frac{\partial}{\partial q^1}+f\frac{\partial}{\partial q^2}$, we can compute $$X^{\gamma}(q^1,q^2,0)=T\tau_Q\circ X\circ \gamma_f(q^1,q^2,0)=q^1\frac{\partial}{\partial q^1}+f(q^1,q^2,0)\frac{\partial}{\partial q^2}$$ and also $$T\gamma_f (X^{\gamma}(q^1,q^2,0))=q^1\frac{\partial}{\partial q^1}+f(q^1,q^2,0)\frac{\partial}{\partial q^2}+q^1\frac{\partial}{\partial p_ 3}$$ which is a solution of the equation $i_X \, \omega_3=dh_3$ where, if $i_3:M_3\rightarrow T^*Q$ is the inclusion on $T^*Q$ and $j_3 :M_3\rightarrow M_1$, then $\omega_3=i_3^*(\omega_Q)$ and $h_3=j_ 3^*(h_1)$.
Note that $\gamma$ in this case is not a solution of our Hamilton-Jacobi problem because $d(h\circ \gamma)(q^1,q^2,0)=q^1dq^3\neq 0$
Relation to classical Hamilton-Jacobi Theory
--------------------------------------------
In this section we will connect the Hamilton-Jacobi theory developed in the previous sections with the classical Hamilton Jacobi theory on $T^*Q$ using an appropriate extended hamiltonian.
We will use the same notation that in section \[dirac\]. We start with an almost regular lagrangian $L:TQ\rightarrow \mathbb{R}$, and then $\textrm{Im}(FL)=M_1$ is a differentiable submanifold of $T^* Q$ and, in addition, we can define $h_1$ implicitly by $h_1\circ FL=E_L$. We denote $\omega_1=j_1^*\omega_Q$, where $j_1:M_1\rightarrow T^*Q$ is the inclusion and $\omega_Q$ is the canonical symplectic form of the cotangent bundle. We take local coordinates $(q^A,p^A)$ in an open set $U\subset T^*Q$, such that $M_1$ is given locally by the vanishing of independent functions $\Phi^i(q^A,p^A)$, called primary constraints.
Remember that the equations of motion have the form $(i_X\omega_1=dh_1)_{|U_1}$, where $U_1:=M_1\cap U$. This equations are equivalent to $(i_X\omega_Q=dh+\mu_i d\Phi^i)_{|U_1}$ where $h$ is any extension of $h_1$ to $U$ defined on $M_1$ and $\mu_i$ are Lagrange multipliers.
### Case I: There are only primary constraints
First, we suppose that there exist a global solution $X$, i.e. $X$ is a vector field on $M_1$ that satisfies the equations of motion. We reorder constraint functions in two classes: first class constraints denoted by $\chi^a$ and second class constraints denoted by $\xi^b$. We also denote by $ u_a$ and $\lambda_b$ the corresponding Lagrange multipliers. Then the equations of motion are $$(i_X \, \omega_Q = dh+u_ad\chi^a+\lambda_b d\xi^b)_{|M_1}.$$ Now, suppose that $u_a$ and $\lambda_b$ are functions defined on $U$. It is clear that $X$ is the restriction to $M_1$ of the hamiltonian vector field corresponding to a hamiltonian of the form $h+u_a\chi^a+\lambda_b\xi^b$. In fact, all the solutions of the equations of motion are obtained in this way varying the functions $u_a$ arbitrarily and with prescribed values of $\lambda_b$. Next, we are looking for a solution of our Hamilton-Jacobi problem, that is, a 1-form $\gamma$ satisfying
1. $d\gamma=0$
2. $\textrm{Im}(\gamma)\subset M_1$
3. $d(h_1\circ \gamma)=0$
Condition (iii) can be easily checked that it is equivalent to $d((h+u_a\chi^a+\lambda_b\xi^b)\circ \gamma)=0$ because $(h+u^a\chi^a+\lambda^b\xi^b)_{|M_1}=h_1$. So, it is evident that the solutions of the classical Hamilton-Jacobi equation for the hamiltonians $h+u^a\chi^a+\lambda^b\xi^b$ (where $u^a$ are arbitrary functions and the rest are fixed) inside $M_1$ and the solutions for our Hamilton-Jacobi problem coincide.
### Case II: The general case
Suppose now that the algorithm do not stop at $M_1$, then we obtain the sequence of manifolds $$\cdots M_k \hookrightarrow \cdots\hookrightarrow M_2 \hookrightarrow M_ 1
\hookrightarrow T^*Q.$$ and we suppose that the algorithm stabilizes in a manifold $M_f$ of dimension$>0$.
We can reorder the constraints in first and second class (maybe changing the independent set of constrainsts). We will denote $\chi^a$ and $\xi^b$ the primary first and second class constraints and by $\psi^c$ and $\theta^d$ the secondary first and second class constraints. We will also denote by $u_a,\lambda_b,v_c$ and $w_d$ the corresponding Lagrange multipliers. Again a solution $X$ of the equations of motion verifies $$(i_X \, \omega_Q = dh+u_ad\chi^a+\lambda_bd\xi^b)_{|M_f}.$$ As above, $X$ is the restriction to $M_f$ of the hamiltonian vector field given by the hamiltonian $h+u_a\chi^a+\lambda_b\xi^b$ where some multipliers are determined applying the constraint algorithm.
We are looking for $\gamma\in \Lambda^1(Q)$ satisfying
1. $d\gamma=0$
2. $\textrm{Im}(\gamma)\subset M_1$
3. $\gamma(Q_f)\subset M_f$
4. $d(h_1\circ \gamma_1)=0$
Note that (iv) is equivalent to the equation $d((h+u_a\chi^a+\lambda_b\xi^b)\circ \gamma)=0$ because $(h+u_a\chi^a+\lambda_b\xi^b)_{|M_1}=h_1$, and so, the solutions of the classical Hamilton-Jacobi theory contained in $M_f$ for the hamiltonians $h+u_a\chi^a+\lambda_b\xi^b$ are just the solutions of our Hamilton-Jacobi problem.
Relation to other theories
--------------------------
The Hamilton-Jacobi theory for degenerate lagrangians have been discussed by several authors in the last 20 years. Let us recall some previous attempts.
1. In the papers by Longhi [*et al.*]{} [@gomis1; @gomis2] it is discussed the case of a time independent lagrangian which is homogeneous in the velocities. It is shown that we can substitute an arbitrary lagrangian by an homogeneous one using the traditional procedure by adding new variables and, then, this new homogeneous lagrangian has zero energy. The authors show that the hamiltonian can be added as a new constraint and, in consequence, they restrict themselves to the case when the hamiltonian is identically zero. The integrability condition for the resultant Hamilton-Jacobi equations implies that they can only consider first class constraints. On the other hand, in the paper by Rothe and F. G. Scholtz [@Rothe] an almost-regular lagrangian $L(t,q^A,\dot{q}^A)$ is considered. If the Hessian ($\frac{\partial^2L}{\partial \dot{q}^A \partial \dot{q}^B}$), has rank $n-m_1$ then, the constraint submanifold $M_1$ is locally described by coordinates $(q^A, p_a)$, where only , $a=m_1+1,\ldots,n$. The remaining momenta $p_\alpha$; $\alpha=1,\ldots,m_1$ are functions of $t$, $q^A$, $p_a$, that is, $p_{\alpha}=-f_{\alpha}(t,q^A,p_a)$ and represent the primary constraints $\phi_{\alpha}(t,q^A,p_A)=p_{\alpha}+f_{\alpha}(t, q^A,p_a)$. Then they consider the system of partial differential equations $$\begin{array}{l}
\frac{\partial S}{\partial t}+h_1(t,q^A,\frac{\partial S}{\partial q_a})=0 \\ \noalign{\medskip}
\frac{\partial S}{\partial q^{\alpha}}+ f_{\alpha}(t,q^A,\frac {\partial S}{\partial q_a})=0 \quad b=1,\ldots,m_1
\end{array}$$ where $h_1$ is the hamiltonian defined on the primary constraint manifold by the projection of the lagrangian energy.
2. The theory discussed in [@cari] is similar to our theory in the case of global dynamics, but they do not take into account secondary constraints. The authors also use the lagrangian homogeneous formalism to obtain the standard Hamilton-Jacobi theory for time dependent systems.
3. M. Leok and collaborators [@sosa] use the Dirac structures setting, and secondary constraints are not considered.
Lagrangian setting
------------------
The equations of motion are globally expressed by the presymplectic equation $$\label{equ}
i_ {\xi}\,\omega_L=dE_L,$$ where a possible solution $\xi$ is not in principle a SODE.
Therefore, in addition to the problem of finding solutions for , we must study the second order differential problem, that is, we shall obtain a solution of satisfying the additional condition $S\xi=\Delta$.
If we apply the constraint algorithm to the presymplectic system $(TQ,\omega_L,dE_L)$ we obtain a sequence of submanifolds. $$\cdots P_k \hookrightarrow \cdots \hookrightarrow P_2 \hookrightarrow P_ 1:= TQ$$ Assume that the algorithm stabilizes at some $P_{k+1}=P_k=P_f$, which is the final constraint submanifold.
If we consider, as above, the presymplectic system $(M_1,\omega_1,dh_1)$, and apply the constraint algorithm to the equation $$\label{equ2}
i_{X}\, \omega_1=dh_1$$ we obtain a sequence of submanifolds $$\cdots M_k \hookrightarrow \cdots \hookrightarrow M_2 \hookrightarrow M_ 1
\hookrightarrow T^*Q.$$ such that $$FL(P_i)=M_i \textrm{, for any $i$, }$$ and $$FL_i:=FL_{|P_i}:P_i\rightarrow M_i$$ are surjective submersions.
As a consequence, both algorithms stabilizes at the same step, say $k$, and then $$FL(P_f)=M_f$$ and $$FL_f:P_f\rightarrow M_f$$ is a surjective submersion. Moreover, we have the following results.
If $\xi$ is a $FL_f$-projectable solution of , then its projection $TFL_f(\xi)$ is a solution of .
Conversely, if $X$ is a solution of , then any $FL_f$ projectable vector field on $P_f$ which projects on $X$, is a solution of .
Next, we shall discuss the SODE problem as it was stated by M. J. Gotay and J. Nester [@gotaythesis; @gotay2] (see [@cari0] for an alternative description).
The results in [@gotaythesis; @gotay2] can be summarized in the following result.
\[SODE\]
1. If $\xi$ is a $FL_f$-projectable vector field on $P_f$ then for any $p\in M_f$ there exists a unique point in each fiber $FL_f^{-1}(p)$, denoted by $\eta_{\xi}(p)$ at which $\xi$ is a SODE. The point $\eta_{\xi}(p)$ is given by $$\eta_{\xi}(p):=T\tau_Q(\xi(p))$$
2. The map $$\begin{array}{rccl}
\beta_{\xi}:& M_f &\longrightarrow & P_f \\ \noalign{\medskip}
& p&\rightarrow & \beta_{\xi}(p):= \eta_{\xi}(p)
\end{array}$$ is a section of $FL_f:P_f\rightarrow M_f$ and on $\textrm{Im}(\beta_{\xi})$ there exists a unique vector field, denoted by $X_{\xi}$, which simultaneously satisfies the equations $$\begin{array}{l}
i_{X_{\xi}} \, \omega_L=dE_L\\ \noalign{\medskip}
SX_{\xi}=\Delta
\end{array}$$
We will now recall the construction of a solution of the dynamical equation which simultaneously satisfies the SODE condition. If $X:=(FL_f)_{*}(\xi)$, then $X$ is a vector field on $M_f$ satisfying $i_{X} \, \omega_1=dh_1$. The vector field $X_{\xi}$ described in is given by $$X_{\xi}(\beta_{\xi}(p))=T\beta_{\xi}(X(p))$$
A detailed proof can be seen in [@gotaythesis; @gotay2], but for the sake of completness, we recall here the way to choose the points on the fibers as it is stated in the Theorem \[SODE\] (i).
In the last part of this section we come back to the Hamilton-Jacobi problem, but now in the lagrangian setting.
The application of the constraint algorithm is summarized in the following diagram $$\xymatrix{
P_1=TQ\ar[rr]^{FL} \ar[rrd]^{FL_1} && T^*Q\\
P_2 \ar[u]^{g_2} \ar[rrd]^{FL_2}&& M_1 \ar[u]^{j_1}\\
\vdots & & M_2\ar[u]^{j_2} \\
P_f\ar[u]^{g_f}\ar[rrd]^{FL_f} && \vdots\\
&& M_f\ar[u]^{j_f}
}$$
Assume, as before, that $Q_i=\pi_Q(M_i)$ are submanifolds and $\pi_i={\pi_Q}_{|M_i}:M_i\rightarrow Q_i$ are surjective submersions. Since $\tau_Q=\pi_Q\circ FL$, then $\tau_Q(P_i)=\pi_Q(M_i)=Q_i$, and $P_i$ also projects onto $Q_i$. We denote $\tau_f={\tau_Q}_{|P_f}:P_f\rightarrow Q_f$.
In consequence, the following diagram is commutative. $$\xymatrix{
P_f \ar[ddr]^{\tau_f}\ar[rrd]^{FL_f} && \\
&& M_f \ar[dl]^{\pi_f}\\
&Q_f&
}$$
Now, if $X$ is a solution of $i_{X}\,\omega_1=dh_1$ on $M_f$ and $\gamma$ is a $1$-form which is a solution of the Hamilton-Jacobi problem, that is,
1. $\gamma(Q)\subset M_1$ and $\gamma_f(Q_f)\subset M_f$
2. $d(h_1\circ \gamma_1)_{|Q_f}=0$
3. $d\gamma=0$
then we can define $X^{\gamma}=T\pi_f\circ X\circ \gamma_f$. From Proposition \[112\] we deduce that $X$ and $X^{\gamma}$ are $\gamma_f$-related.
On the other hand we can construct a $FL_f$-projectable vector field $\xi$ on $P_f$ which projects on $X$. Next we can apply Proposition \[SODE\] and obtain the section $\beta_{\xi}:M_f\rightarrow P_f$. Recall that $X_{\xi}(\beta_{\xi}(p))=T\beta_{\xi}(X(p))$ is the unique vector field on $\textrm{Im}(\beta_{\xi})$ which satisfies the SODE condition and the equation $i_{X_{\xi}} \, \omega_L=dE_L$. The following lemma gives the relation between $\textrm{Im}(\beta_{\xi})$ and $Q_f$.
$\textrm{Im}(\beta_{\xi})$ is a submanifold of $ TQ_f$.
[**Proof:**]{} Since $X_{\xi}$ verifies the SODE condition, then $$T\tau_Q(X_{\xi}(p))=\tau_{TQ}(X_{\xi}(p))$$ for any $p\in \textrm{Im}(\beta_{\xi})$.
Since $X_{\xi}$ is tangent to $\textrm{Im}(\beta_{\xi})$, and since $\textrm{Im}(\beta_{\xi})$ is a submanifold of $ P_f$ and $\tau_Q(P_f)=Q_f$, then $T\tau_Q(X_{\xi}(p))\in TQ_f$.
On the other hand $\tau_{TQ}(X_{\xi}(p))=p\in \textrm{Im}(\beta_{\xi})$, and using the SODE condition we deduce that $p\in TQ_f$.$\Box$
Remember that $X_{\xi}$ and $X$ are $\beta_{\xi}$-related and $X$ and $X^{\gamma}$ are $\gamma_f$-related, so we deduce that $X_{\xi}$ and $X^{\gamma}$ are $\beta_{\xi}\circ \gamma_f$-related too. Moreover, since $X_{\xi}$ satisfies the SODE condition, we can find a better description of $\beta_{\xi}\circ \gamma_f$.
We have $$\beta_{\xi}\circ \gamma_f=X^{\gamma}.$$
[**Proof**]{} Since $X_{\xi}$ verifies the SODE condition, then given $q\in Q_f$ we obtain $$T\tau_q(X_{\xi}(\beta_{\xi}\circ\gamma_f(q)))=\tau_{TQ}(X_{\xi}(\beta_{\xi}\circ\gamma_f(q))).$$ Therefore, $$T\tau_Q(X_{\xi}(\beta_{\xi}\circ\gamma_f(q)))=T\tau_Q\circ T\beta_{\xi}(X(q))=T\pi(X(\gamma(q)))=X^{\gamma}(q)$$ where we have used that $\tau_Q=\pi\circ FL$ and $FL\circ \beta_{\xi}=id_{M_f}$.
On the other hand, $$\tau_{TQ}(X_{\xi}(\beta_{\xi}\circ\gamma_f(q)))=(\beta_{\xi}\circ\gamma_f(q))$$ Then, using the SODE condition we get $X^{\gamma}=\beta_{\xi}\circ\gamma_f$. $\Box$
The following corollary is immediate.
The vector fields $X_{\xi}$ and $X^{\gamma}$ are $X^{\gamma}$-related, i.e. $$X_{\xi}(\beta_{\xi}\circ \gamma_f(q))=TX^{\gamma}(X^{\gamma}(q))$$ or equivalently $$X_{\xi}(\beta_{\xi}\circ \gamma_f(q))=(X^{\gamma})^{C}(X^{\gamma}(q)),$$ where $(X^{\gamma})^C$ denotes the complete lift of the vector field $X^{\gamma}$.
Example: Lagrangian setting
---------------------------
We will revisite example \[exe\] and discuss the Hamilton-Jacobi problem for the Euler-Lagrange equation. The lagrangian function is $$L(q^1,q^2,\dot{q}^1,\dot{q}^2)=\frac{1}{2}(\dot{q}^1)^2+ \dot{q^2}\, q^1+\dot{q^1}\, q^1.$$
Then $FL$ was given $$FL(q^1,q^2,\dot{q}^1,\dot{q}^2)=(q^1,q^2,\dot{q}^1+q^2,q^1)$$ and the primary constraints are $$\Phi^1(q^A,p_A)=p_2-q^1$$ So $$\begin{array}{l}
M_1=\{(q^1,q^2,p_1,p_2)\in\mathbb{R}^4 \textrm{ such that } \ p_2=q^1\}.
\end{array}$$ and we can use $(q^1,q^2,p_1)$ as coordinates on $M_1$.
It follows that $$\begin{array}{l}
E_L=\frac{1}{2}\dot{q}^1 \\ \noalign{\medskip}
h_1=\frac{1}{2}(p_1-q^2) \\ \noalign{\medskip}
\omega_1=dq^1\wedge dp^1+dq^2\wedge dq^1 \\ \noalign{\medskip}
\textrm{Ker}(\omega_1)=\left<\frac{\partial }{\partial p_1}-\frac{\partial }{\partial q^2}\right> \\
\end{array}$$ Let $$h(q^A,p_A)=\frac{1}{2}(p_1-q^2)$$ be an extension of the hamiltonian $h_1$.
It is easy to see that, at the points of $M_1:=\textrm{Im}(FL)$, we have $$\{\Phi^1, h +u\Phi^1\}=0$$ and therefore we are in presence of global dynamics
The solution of the equation $(i_X\omega_1=dh_1)_{|M_1}$ is given by $$X=(p_1-q^2)\frac{\partial}{\partial q^1}+f\frac{\partial}{\partial q^2}+f\frac{\partial}{\partial p1}+(p_1-q^2)\frac{\partial}{\partial p^2},$$ where $f\in C^{\infty}(M_1)$
Recall also, that a solution of our Hamilton-Jacobi problem, $\gamma(q^1,q^2)$ $=(q^1,q^2,\gamma_1(q^1,q^2),\gamma_2(q^1,q^2))$, was given by $$\gamma(q^1,q^2)=(q^1,q^2,q^2,q^1)$$ If we take a solution $X=(p_1-q^2)\frac{\partial}{\partial q^1}+f\frac{\partial}{\partial q^2}+f\frac{\partial}{\partial p_1}+(p_1-q^2)\frac{\partial}{\partial p_2}$ we can compute $$X^{\gamma}=0\frac{\partial}{\partial q^1}+f\frac{\partial}{\partial q^2}$$ and also $$(X^{\gamma})^C(q^1,q^2,\dot{q^1},\dot{q^2})=(f\circ\gamma)\frac{\partial}{\partial q_1}+\left((\frac{\partial (f\circ \gamma)}{\partial q^1})\dot{q}^1+(\frac{\partial (f\circ \gamma)}{\partial q^2})\dot{q}^2\right)\frac{\partial}{\partial \dot{q}^2},$$ then $$(X^{\gamma})^C(X^{\gamma})=(X^{\gamma})^C(q^1,q^2,0,(f\circ \gamma)(q^1,q^2))=(f\circ\gamma)\frac{\partial}{\partial q_1}+(\frac{\partial (f\circ \gamma)}{\partial q^2})\dot{q}^2\frac{\partial}{\partial \dot{q}^2}.$$
This vector field along $X^{\gamma}$ satisfies the SODE condition. We can consider now the equation $i_ {\xi}\, \omega_L=dE_L$ $$\begin{array}{ll}
\omega_L&=d(\frac{\partial L}{\partial \dot{q}^1}dq^1+\frac{\partial L}{\partial \dot{q}^2}dq^2)=d((\dot{q}^1+q^2)dq^1+q^1dq^2)\\ \noalign{\medskip} &=d\dot{q}^1\wedge dq^1+dq^2\wedge dq^1+dq^1\wedge dq^2=d\dot{q}^1\wedge dq^1.
\end{array}$$ So, $i_{(X^{\gamma})^C(X^{\gamma})}\, \omega_L=0$ and $dE_L(X^{\gamma})=\dot{q}^1d\dot{q^1}(X^{\gamma})=0$ and thus $$i_{(X^{\gamma})^C(X^{\gamma})}\, \omega_L=dE_L(X^{\gamma}).$$Therefore $(X^{\gamma})^C(X^{\gamma})$ satisfies Euler-Lagrange equations and the SODE condition.
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[^1]: This work has been partially supported by MICINN (Spain) Grants MTM2009-13383, MTM2010-21186-C02-01 and MTM2009-08166-E, the ICMAT Severo Ochoa project SEV-2011-0087 and the European project IRSES-project “Geomech-246981”. M. Vaquero wishes to thank MICINN for a FPI-PhD Position.
|
---
abstract: 'Radioactive decays contribute significantly to the re-heating of supernova ejecta. Previous works mainly considered the energy deposited by $\gamma$-rays and positrons produced by [$\mathrm{^{56}Ni}$]{}, [$\mathrm{^{56}Co}$]{}, [$\mathrm{^{57}Ni}$]{}, [$\mathrm{^{57}Co}$]{}, [$\mathrm{^{44}Ti}$]{}, and [$\mathrm{^{44}Sc}$]{}. We point out that Auger and internal conversion electrons constitute an additional heat source. At late times, these electrons can contribute significantly to supernova light curves for reasonable nucleosynthetic yields. In particular, the internal conversion electrons emitted in the decay of [$\mathrm{^{57}Co}$]{} are an important heating channel for supernovae that have become largely transparent to $\gamma$-rays. We show that when the heating by these electrons is accounted for, the slowing down of the light curves of SN 1998bw and SN 2003hv is naturally obtained for typical nucleosynthetic yields. Additionally, we show that for SN 1987A the effects of internal conversion electrons are likely significant for the derivation of [$\mathrm{^{44}Ti}$]{} yields from its late time bolometric light curve.'
author:
- |
I. R. Seitenzahl$^{1}$\
$^1$Max-Planck-Institut für Astrophysik, Garching, Germany
title: Internal conversion electrons and supernova light curves
---
Introduction
============
A great success of nuclear astrophysics was the demonstration that radioactivity powers the light curves of Type Ia supernovae (SN Ia, most likely thermonuclear) and, at least for some events at late times, Type Ib/c and Type II (most likely core collapse) supernovae. Shortly after its explosion a supernova enters a state of homologous expansion. The temperature drops during the expansion until nuclear fusion reactions cease; radioactive decays, however, still take place, since the matter is only slightly ionized. It is now widely accepted that the energy liberated in the decay chain of radioactive [$\mathrm{^{56}Ni}$]{} is the most important nuclear source for re-heating the supernova ejecta to temperatures high enough for the spectrum to peak at optical wavelengths [@truran1967a; @colgate1969a]. At first, the bulk of the heating is produced by the energetic $\gamma$-rays which thermalize and deposit their energy via Compton scattering and photoelectric absorption. In the homologous expansion, the column density (and therefore also approximately the Compton opacity) decreases with time as $t^{-2}$, and the ejecta become more and more transparent to these high energy photons. Once $\gamma$-rays escape, the positrons produced in the decay of [$\mathrm{^{56}Co}$]{} and [$\mathrm{^{44}Sc}$]{} were thought to be the main heating sources. Here, we draw attention to often overlooked additional leptonic heating channels: Auger and internal conversion electrons. In section \[sec:decay\] we review the physics of nuclear decays relevant for supernova light curves. In section \[sec:lc\] we demonstrate the impact of internal conversion electrons from the decay of [$\mathrm{^{57}Co}$]{} on different supernova light curves. We conclude in section \[sec:conc\] with an outlook how this effect could be used to constrain supernova explosion models in the future. For the published refereed journal article that first pointed out the significance of internal conversion electrons on supernova light curves, please see [@seitenzahl2009d].
Nuclear decays {#sec:decay}
==============
The time dependence of $n$ nuclide abundances $N_i$ in a ecay chain is governed by the Bateman equations: $$\begin{aligned}
\label{eq:bateman}
\frac{dN_1}{dt}&=&-\lambda_1 N_1\\
\frac{dN_i}{dt}&=& \lambda_{i-1} N_{i-1}-\lambda_{i} N_{i} \hspace{8pt}.\end{aligned}$$ For $n=2$ and initial abundances $N_1(0)$ and $N_2(0)$ we get the solution $$\begin{aligned}
\label{eq:sol1}
N_1(t)&=& N_1(0)\exp(-\lambda_1 t) \\
\label{eq:sol2}
N_2(t)&=& N_1(0) \frac{\lambda_1}{\lambda_2-\lambda_1} [
\exp(-\lambda_1 t)-\exp(-\lambda_2 t) ] + N_2(0)\exp(-\lambda_2 t) \hspace{8pt}.\end{aligned}$$ The decay constants $\lambda_i$ are related to the half-lives $t^{1/2}_i$ and the mean life-time $\tau_i$ via $$\lambda_i = \frac{1}{\tau_i} = \frac{\ln(2)}{t^{1/2}_i} \hspace{8pt}.$$ The rate of energy deposition by decays of nucleus $i$ is given by the activity multiplied by the energy deposited per decay: $$\epsilon_i = \lambda_i N_i(t) q_i(t) \hspace{8pt},$$ where the number $N_i$ is given by eq. \[eq:sol1\] or eq. \[eq:sol2\] and the energy deposited, $q_i$, is a function of time due to the increasing escape fraction of $\gamma$-rays and possible late time escape of positrons.
[9cm]{}
[Nucleus]{} [Auger $e^-$ ]{} [IC $e^-$]{} [$e^+$]{} [X-ray ]{}
------------------------ ------------------ -------------- ----------- ------------
[$\mathrm{^{57}Co}$]{} 7.594 10.22 0.000 3.598
[$\mathrm{^{56}Co}$]{} 3.355 0.374 115.7 1.588
[$\mathrm{^{55}Fe}$]{} 3.973 0.000 0.000 1.635
[$\mathrm{^{44}Ti}$]{} 3.519 7.064 0.000 0.768
[$\mathrm{^{44}Sc}$]{} 0.163 0.074 595.8 0.030
Most of the nuclei synthesized in supernova explosions have atomic masses in the range $A\approx 12-70$. The isotopes are generally either stable or on the proton-rich side of the valley of stability. For these unstable nuclei, radioactive decay occurs along an isobar towards neutron richness, either by electron capture or positron emission. Electron capture proceeds via the capture of an atomic (typically K or L shell) electron by a proton in the nucleus and corresponding emission of an electron neutrino. Positron emission proceeds via the decay of a proton in the nucleus into a neutron and the corresponding emission of a positron and an electron neutrino. For both processes, the transition to the daughter nucleus is a statistical process to a distribution of (excited) nuclear states. Following electron capture, the daughter is formed with a hole in its atomic electron cloud. Higher lying atomic electrons transition to fill the gaps in the lower lying atomic shells, which results in characteristic X-rays being emitted from the cascade. For every such electron transition, there is also a probability that instead of an X-ray one or more higher lying atomic electrons are ejected. These electrons are known as Auger electrons.
[R]{}[2.5in]{} 
Analogous to the X-ray cascade of the atom, the excited daughter nucleus typically undergoes a series of transitions towards the ground state accompanied by emission of characteristic $\gamma$-ray photons. If the nucleus is surrounded by an electron cloud, then there is again a probability that instead of a $\gamma$-ray one or more atomic electrons are ejected. This process, in which the energy difference of the nuclear levels is carried away by the ejection of an inner atomic electron and there is no $\gamma$-ray photon emitted, is called internal conversion. The probability for internal conversion to occur for a given level is measured by the internal conversion coefficient $\alpha = \frac{\# \mathrm{of} \; e^- \; \mathrm{de-excitations}}{\#
\mathrm{of} \; \gamma \; \mathrm{de-excitations}}$. $\alpha$ is normally small for nuclei with low atomic number, but increases for levels close to the ground state. For the decay of [$\mathrm{^{57}Co}$]{} the probability for production of internal conversion electrons is very large, due to the existence of a low-lying nuclear level in the daughter nucleus [$\mathrm{^{57}Fe}$]{} (see Fig. \[fig1\]). For the first exited state (14.4 keV $3/2^{-}$) of [$\mathrm{^{57}Fe}$]{}, the internal conversion coefficient $\alpha = 8.58$. This decay is 100% electron capture, and over 99.8% all decays are into the 136 keV level. Captures to the higher lying 366.74 and 706.42 keV levels contribute only marginally. For ground-state to ground-state electron capture transitions, such as in the decay of [$\mathrm{^{55}Fe}$]{}, no $\gamma$-rays or positrons are emitted. Assuming that the neutrino escapes without interactions, in such transitions the Auger electrons and the X-rays constitute the only sources of radioactive heating.
The following four decay chains contribute most to bolometric supernova light curves: $$\begin{aligned}
& &^{56}\mathrm{Ni} \;\stackrel{t_{1/2} = \; 6.08d}{\hbox to
60pt{\rightarrowfill}} \; ^{56}\mathrm{Co} \;
\stackrel{t_{1/2} = \; 77.2d}{\hbox to 60pt{\rightarrowfill}} \; ^{56}\mathrm{Fe} \\
& &^{57}\mathrm{Ni} \;\stackrel{t_{1/2} = \; 35.60 h}{\hbox to
60pt{\rightarrowfill}}\; ^{57}\mathrm{Co} \;
\stackrel{t_{1/2} = \; 271.79d}{\hbox to 60pt{\rightarrowfill}} \; ^{57}\mathrm{Fe}\\
& &^{55}\mathrm{Co} \;\stackrel{t_{1/2} = \; 17.53 h}{\hbox to
60pt{\rightarrowfill}}\; ^{55}\mathrm{Fe} \;
\stackrel{t_{1/2} = \; 999.67 d}{\hbox to 60pt{\rightarrowfill}} \; ^{55}\mathrm{Mn}\\
& &^{44}\mathrm{Ti} \;\stackrel{t_{1/2} = \; 58.9 y}{\hbox to
60pt{\rightarrowfill}}\; ^{44}\mathrm{Sc} \; \stackrel{t_{1/2} =
\; 3.97 h}{\hbox to 60pt{\rightarrowfill}} \; ^{44}\mathrm{Ca}\end{aligned}$$ Here we do not model the radiative transport, we only compare leptonic (and X-ray) energy injection rates. We do not include the energy produced by the pair annihilation and further assume that the kinetic energy of the leptons is completely and locally deposited and thermalized. The energy generation rates presented here do not include any heating due to $\gamma$-rays and therefore are not predictions for bolometric light curves. This approach, however, does allow for a direct comparison of the relative importance of the different leptonic heating channels (positrons from the decays of [$\mathrm{^{56}Co}$]{} and [$\mathrm{^{44}Sc}$]{} and electrons from the decays of [$\mathrm{^{57}Co}$]{} and [$\mathrm{^{55}Fe}$]{}.) The relevant energies of the different decay channels are listed in table \[tab1\]. These data are taken from the National Nuclear Data Center[^1].
Late-time bolometric light curves {#sec:lc}
=================================
Bolometric light curves have to be reconstructed from multi-band photometry. For reliable reconstruction, the contribution of the UV/optical ($UBV\!RI$) and near-infrared ($JHK$) bands have to be included (UVOIR light curve). However, near-IR observations are rare at very late epochs, and sometimes only $B$-through-$I$ band observations with a near-IR correction extrapolated from earlier epochs are used [@sollerman2002a]. Below, we discuss some aspects and examples of different supernova light curves.
*Type Ia supernovae \[sec:snia\]*
---------------------------------
{width="8cm"}
SNe Ia are thought to be thermonuclear disruptions of white dwarf stars and as such do not have an extended envelope. Consequently, their ejecta become transparent to $\gamma$-rays relatively soon and the positron-dominated phase generally starts $150$–$300$ days after the explosion [@milne2001a; @sollerman2004a]. Between $300$ and $600$ days, the UVOIR light curves fall exponentially with the [$\mathrm{^{56}Co}$]{} decay half-life [@stritzinger2007a], a clear sign that a constant fraction of positrons is trapped. Unfortunately, only few SN Ia were observed at even later times, but there are indications of a slow-down in the light curves of some optical bands after $\sim$$600$ days [@sollerman2004a; @lair2006a]. A particularly interesting case is SN 2003hv, which shows a slow-down in the [ *bolometric*]{} light curve 786 days after B-band maximum light [@leloudas2009a]. It has been suggested that additional (possibly radioactive) heating sources may be needed to explain this observed effect. We point out that the magnitude and time of occurrence of the observed slow down of the light curve is expected and a natural consequence of the leptonic energy injection from the decay of [$\mathrm{^{57}Co}$]{}. To illustrate this, consider the leptonic energy generation rates of important long lived isotopes between 500 and 2000 days for W7 [@nomoto1984a], a common reference SN Ia model (see Fig. \[fig2\]). For this particular choice of yields [@iwamoto1999a], the light curve is significantly higher from assuming only heating from [$\mathrm{^{56}Co}$]{} positrons after about 750 days. 1000 days after the explosion, the heating from [$\mathrm{^{57}Co}$]{} electrons equals the heating from [$\mathrm{^{56}Co}$]{} positrons.
*Core collapse supernovae\[sec:ccsn\]*
--------------------------------------
The progenitors of core collapse SNe are massive stars, which have an extended, massive envelope at the time of explosion, which provides opacity for the emerging $\gamma$-rays. Compared to SNe Ia, the nuclear burning also takes place at higher entropy, which means that core collapse supernovae synthesize more [$\mathrm{^{44}Ti}$]{}. These differences imply that the moment when internal conversion electrons from [$\mathrm{^{57}Co}$]{} significantly contribute to the light curves is delayed. Nevertheless, two observed examples of core collapse SN where internal conversion electrons do significantly alter the shape of the light curve are given below.
### SN 1998bw
{width="8cm"}
The well observed SN 1998bw, which was the first case of a $\gamma$-ray burst associated with a SN[@galama1998a], was classified as a SN Ic. The explosion was a very energetic , asymmetric disruption of a massive, stripped stellar core ($\geq 10$ [$M_\odot$]{}; [@maeda2002a]). The high ejecta velocities and the lack of an extended envelope reduce the $\gamma$-ray opacity such that by $1000$ days almost all $\gamma$-rays are freely streaming. A nucleosynthetic yield calculation for hypernovae like 1998bw [@nakamura2001b] predicts a [$\mathrm{^{57}Ni}$]{} to [$\mathrm{^{56}Ni}$]{} mass ratio $\mathcal{R}^{57/56} \approx 0.0366$, $\sim 1.5$ times the solar value for the [$\mathrm{^{57}Fe}$]{} to [$\mathrm{^{56}Fe}$]{} ratio (which is $0.0234$; [@lodders2003a]). After modeling the UVOIR light curve to $\sim$$1000$ days after the explosion, [@sollerman2002a] showed that a simple model without contributions from freeze-out effects, circumstellar interaction, accretion onto a central compact object or light echoes requires a [$\mathrm{^{57}Ni}$]{} to [$\mathrm{^{56}Ni}$]{} ratio $\sim 13.5$ times greater than solar. This large discrepancy between the value of $\mathcal{R}^{57/56}$ predicted from explosive nucleosynthesis calculations and the one derived from light curve modeling reduces if the effects of internal conversion and Auger electrons are included in the light-curve calculations. In Fig. \[fig3\] we show the combined leptonic and X-ray luminosity corresponding to the nucleosynthetic yield calculations of [@nakamura2001b]. The non-$\gamma$-ray heating due to [$\mathrm{^{57}Co}$]{} is the dominant contribution between $\sim1000$ and 1600 days. The observed slow down of the light curve of 1998bw at $\sim900$ days (see Fig. \[fig3\]) is thereby naturally obtained without the need for strongly super solar $\mathcal{R}^{57/56}$.
### SN 1987A
{width="8cm"}
SN 1987A has demonstrated that light curves can be constructed for nearby supernovae extending for several years after the explosion. This has led to the consideration of longer lived radioactive species. The most important of these is [$\mathrm{^{57}Co}$]{}, which is expected to be produced in significant amounts as the decay product of the short lived [$\mathrm{^{57}Ni}$]{}. Due to its relatively longer half-life (271.79 days) and the high opacity of the ejecta to the emitted $\gamma$-rays, its $\gamma$-rays alone may dominate the bolometric light curve, especially of core collapse supernovae, at late times [@pinto1988a; @woosley1989a]. In fact, due to its extended envelope, the opacity to $\gamma$-rays in the remnant of 1987A remained high for such a long time that it never entered the [$\mathrm{^{56}Co}$]{} positron dominated phase [@fransson2002a]. It is argued, that the only other radionuclide which noticeably contributes to the bolometric light curve is [$\mathrm{^{44}Ti}$]{} [@kumagai1989a]. [$\mathrm{^{44}Ti}$]{} has an even longer half-life of 59.8 years, and the short lived daughter [$\mathrm{^{44}Sc}$]{} has a strong positron channel, which means that the [$\mathrm{^{44}Ti}$]{} decay chain is the dominant energy source at very late times. Photometric data of SN 1987A lead to the conclusion that the $\mathcal{R}^{57/56}$ was rather large, about five times the corresponding solar value [@suntzeff1992a; @dwek1992a]. Due to the proximity of SN 1987A, the presence of [$\mathrm{^{56}Co}$]{} and [$\mathrm{^{57}Co}$]{} was inferred not only photometrically from their imprint on the light curve, but also through a direct detection of escaping $\gamma$-rays from their decays. The observed gamma ray flux from [$\mathrm{^{57}Co}$]{} with the Oriented Scintillation Spectrometer Experiment (OSSE) on board the [*Compton Gamma Ray Observatory*]{} [@kurfess1992a] favored at most moderate enhancement of [$\mathrm{^{57}Ni}$]{} (factor 1.5), in good agreement with nuclear reaction network calculations [@nomoto1988a]. Similarly, based on the light curve, a rather high [$\mathrm{^{44}Ti}$]{} abundance was claimed in several studies [@dwek1992a; @woosley1991a; @timmes1996a]. The great difficulty of explaining the light curve of 1987A between 1000 and 1500 days without strong overproduction of [$\mathrm{^{57}Ni}$]{} can be ameliorated by including “freeze-out effects” [@fransson1993a] or by conveniently re-arranging the spatial distribution of the radionuclides [@clayton1992a], but some disagreement of data and models persists. By comparing the various leptonic energy deposition rates for a nucleosynthesis model of 1987A [@nomoto1988a], we show that the leptonic channels of [$\mathrm{^{57}Co}$]{} decay are significantly contributing between 1000 and 2000 days (see Fig. \[fig1987a\]).
Conclusion and Outlook {#sec:conc}
======================
Fitting models to observed late time light curves provides a unique and independent method to directly measure the [*isotopic*]{} yields of prominent radioactive nuclei synthesized in the explosion (in particular [$\mathrm{^{56}Ni}$]{}, [$\mathrm{^{57}Ni}$]{}, and [$\mathrm{^{44}Ti}$]{}). We have shown that at late times, when the ejecta have become largely transparent to $\gamma$-rays, the energy carried by Auger and internal conversion electrons may constitute a significant source of heating. These additional decay channels have to be considered for reliable isotopic abundance determinations from light curves. In particular, we have shown that a re-analysis of the bolometric light curve of 1987A (taking the hitherto unconsidered effect of internal conversion electrons into account) would likely yield significantly different (smaller) [$\mathrm{^{44}Ti}$]{} and [$\mathrm{^{57}Ni}$]{} masses. The new derived values for [$\mathrm{^{57}Ni}$]{} and [$\mathrm{^{44}Ti}$]{} would allow us to gain more insight into the explosion mechanism of core collapse supernovae. In particular, observationally driven inferences about the location of the mass cut can be made, which would give us valuable information about the physical processes separating neutron star and black hole formation from massive stars. Last but not least, SNe Ia are considered as the source of the positrons required to explain the Galactic 511 keV annihilation line observed by Integral/SPI [@knoedlseder2005a]. The question whether enough positrons can escape the remnant remains unanswered. An understanding of the physics that underlies the light curves of SNe is a crucial step in order to constrain the escape fraction of positrons.
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[^1]: http://www.nndc.bnl.gov/
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---
abstract: 'We demonstrate an exceptionally bright photon source based on a single nitrogen-vacancy center (NV-center) in a nanodiamond (ND), placed in the nanoscale gap between two monocrystalline silver cubes in a dimer configuration. The system is operated near saturation at a stable photon rate of 850kcps, while we further achieve strongly polarized emission and high single photon purity, evident by the measured auto-correlation with a g$^{(2)}$(0)-value of 0.08. These photon source features are key parameters for quantum technological applications, such as secure communication based on quantum key distribution. The cube antenna is assembled with an atomic force microscope, which allows us to predetermine the dipole orientation of the NV-center and optimize cube positioning accordingly, while also tracking the evolution of emission parameters from isolated ND to the 1 and 2 cube configuration. The experiment is well described by finite element modelling, assuming an instrinsic quantum efficiency of 0.35. We attribute the large photon rate of the assembled photon source, to increased quantum efficiency of the NV-center and high antenna efficiency.'
author:
- 'Sebastian K.H. Andersen'
- Shailesh Kumar
- 'Sergey I. Bozhevolnyi'
title: Ultrabright Linearly Polarized Photon Generation from a Nitrogen Vacancy Center in a Nanocube Dimer Antenna
---
Introduction
============
The single photon source is a fundamental component in the quantum technological evolution[@QuantTechReview], enabling secure communication based on single photon quantum key distribution [@QuantumKey; @NVCryptography], optical quantum computing[@LinearQuantumComputing] or light-matter interfaces for quantum infomation storage[@QuantumRegister]. For practical applications the single photon source should efficiently deliver a stable stream of indistinguisble, strongly polarized single photons at high rate and high purity, preferably under room temperature operation. Single photon emission may be achieved by spontaneous emission from a wealth of quantum emitters (QE) such as a molecule, quantum dot, defect in diamond, boron nitride or carbon nanotubes etc. [@SolidStateEmitterRev]. However the particular single photon emission properties are a function of both the intrinsic characteristics of the emitter and the surrounding dielectric enviroment. The desirable single photon source properties are hence achieved by proper choice of QE and engineering of the photonic enviroment, i.e. by coupling the QE to an antenna[@SpecModulation; @PolarizationModulation; @EmissionPattern], waveguide[@VGroove-NV; @NDonSilverWire] or directing the emission with a solid immersion lens. For photostable operation at room-temperature the nitrogen-vacancy center in diamond (NV-center) is an excellent emitter[@NVPhotoStable], while the emission is poorly polarized given phonon promoted population averaging of the doublet excited state[@ExcitedStateTimeAvg1; @ExcitedStateTimeAvg2]. Several works have previously focused on improving the efficiency and photon rate of the NV-center either by top-down fabrication in bulk diamond[@DiamondPillar; @DiamondAgAperture] or by incorporating a nanodiamond (ND) containing an NV-center into a plasmonic antenna[@GoldSphereDimer]. Though often, engineering the photonic enviroment comes at the cost of reducing the purity of the source, as background fluorescence from impurities in the diamond or materials introduced during fabrication limit the single photon quantum character of the source. The degree to which background compromises the single photon character, may be measured by the dip of the auto-correlation function at time zero (g$^{(2)}(0)$), indicating the probability of detecting 2 photons at the same time. The measured g$^{(2)}(0)$-value of previously demonstrated NV-center based photon sources have typically been $\sim$ 0.3[@DiamondAgAperture; @DiamondMembraneBullseye; @AgGratingAperture; @GoldSphereDimer] Schietinger *et. al.* assembled a dimer antenna consisting of two gold spheres around a ND containing an NV-center and demonstrated photon rates up to $\sim$420kcps with g$^{(2)}(0)=$0.3[@GoldSphereDimer]. Choy *et. al.* realized an NV-center in a diamond post surrounded by a silver film with grating corrugations and reported an asymptotic photon rate limit up to 704kcps, achievable at infinite laser power, with g$^{(2)}(0)\sim0.2$[@AgGratingAperture]. A record asymptotic limit of 2.7Mcps from an NV-center in a diamond membrane with an etched in grating was reported by *Li et. al.* with g$^{(2)}(0)=0.28$[@DiamondMembraneBullseye]\
\
In this work, we demonstrate an exeptionally pure photon source, consisting of a nanodiamond containing a single NV-center situated in the gap between two monocrystalline silver nanocubes. We measure a g$^{2}(0)$-value of 0.08 for the total system. The configuration is stable under large pump powers, as we operate the source near saturation, at a detected photon rate of 850kcps, similar to state of the art NV-center sources based on solid immersion lenses in diamond ($\sim$ 1Mcps)[@DiamondSolidImmersionLens] or ZrO$_2$ ($\sim$ 850kcps, stable emission)[@ZrOSolidImmersionLens]. Though unlike such sources, we further demonstrate strongly linearly polarized emission from our NV-center with a polarization ratio of 9, between the power detected along the major and minor axis. The system is assembled with an atomic force microscope (AFM), which allows us to directly probe the evolution of the NV-center emission properties from the isolated ND to the 1 cube- and 2 cube configuration. The freedom to precharacterize our NV-center before system assembly allows us to determine the orientation of the NV-center dipole axes and optimize the position of our nanocubes accordingly, for optimal NV-to-antenna coupling. Such optimization improves both the excitation efficiency and photon rate of the NV-center, as the excitation rate is improved by a factor of 5.86, while the detected photon rate at saturation increases by a factor 6.6, relative to the isolated ND. Finite element modelling agree well with experimental observations assuming an intrinsic quantum efficiency of 0.35. The enhanced photon rate, detected from the assembled photon source, is attributed to increased quantum efficiency and high radiation efficiency of the antenna.
Results and Discussion
======================
Figure \[fig:Figure1\]a illustrates the physics of our experiment. A single NV-center in a ND, decays from the excited electronic state by excitation of plasmonic charge density oscillations in a single- (figure \[fig:Figure1\]c) or 2 cube configuration (figure \[fig:Figure1\]d). The excited antenna mode subsequently decays either by ohmic loss or scattering of single photons, polarized along the axis of charge oscillation, hence imposing the radiation properties of the antenna on the NV-center. Further, the photon rate of the system is increased, as the NV-center spends less time in the excited state between emission events, when allowed to efficently dissipate its energy into the antenna mode. Clearly, these desirable single photon source features rely on achieving a large NV-to-antenna coupling rate, compared to direct photon emission or metal quenching. Optimizing the NV-to-antenna coupling rate is typically done by maximizing the quality factor to mode volume ratio of the antenna, spectrally tuned for the optical transition. The NV-center should further be positioned at the point of maximum mode amplitude, with the NV-dipole axis coaligned with the eletric field of the antenna mode [@OptimizingCouplingRate]. The near-field interaction of two cubes hybridize the dipolar cube modes (figure \[fig:Figure1\]c) into an “antibonding” and a “bonding” mode (figure \[fig:Figure1\]d) shifted to respectively higher and lower energies (figure \[fig:Figure1\]e). The “bonding” mode is particular well-suited for coupling to a QE, as the mode is strongly confined to the nano scale gap[@DimerFieldEnhancement], while the superradiant damping of the in-phase cube oscillations, ensure high antenna radiation efficiency[@DimerEfficiency]. The capacitor-like field distribution in the gap requires the cube facet to be aligned normal to the dipole axes of the NV-center for optimal coupling. We realize the optimal coupling configuration by precharacterization of the NV-center dipole orientation and deterministic nanoassembly of the cube dimer antenna with an AFM.\
The NV-center is pumped with a 532nm linearly polarized continous wave or pulsed laser. The pump light is focused onto the ND by an oil immersion objective (NA 1.4), situated below the quartz glass sample. Photons spontanously emitted into the same objective are filtered by a dichroic mirror (cut-off 550nm) and detected by two avalanche photo diodes (APD) in a Hanbury Brown-Twiss configuration, a grating spectrometer or a CCD camera. The pump polarization is controlled by a halfwave plate in the excitation light path, while an analyzer introduced in the detection path probe the polarization of emitted photons. Nano particle manipulation is performed by an AFM positioned above the sample (see supporting information (SI) for a schematic). Pumping a $\sim$ 35nm ND (figure \[fig:Figure2\]b), we observe single photon emission from an NV-center, identified by a measured g$^{(2)}(0)$-value of 0.14 (Figure \[fig:Figure3\]c) and the zero phonon line fingerprint for the neutral(NV$^{0}$, 575nm) and negative charge state(NV$^{-}$, 637nm)(figure \[fig:Figure1\]e). The NV-center continuously flip-flop between the charge states by photoionization[@PhotoIonization], however photon emission is dominated by the NV$^{-}$-state, populated $\sim$ 75% of the time. Our experiment is hence well described by considering purely the NV$^{-}$-state (see SI for cube coupled NV-center spectra). Excitation and emission from the excited doublet state of NV$^{-}$-center is facilitated by two orthogonal dipole axes (**p**$_x$, **p**$_y$) lying in the plane normal to the nitrogen atom - vacancy axis [@DipoleOrientation] (figure \[fig:Figure2\]a). We probe the projection of **p**$_x$, **p**$_y$ on the sample plane by rotating the pump polarization, while operating the NV-center in the non-saturated regime (105$\mu$W pump power) (figure \[fig:Figure2\]e). The detected photon rate then follow $R(\varphi)=\eta{q_e}\gamma_{ex}(\varphi)$. $\eta$ being the collection effiency of the objective and q$_e$ the quantum efficiency of the emitter. The excitation rate $\gamma_{ex}(\varphi) \propto \sum_{i=x,y} |\textbf{p}_i\cdot{\textbf{E}}(\varphi)|^2$ depends on the electric field of the pump laser $\textbf {E}=\textbf{E}_{pump}$, at the emitter position, polarized along the sample plane at the azimuth angle $\varphi$. We fit $R(\varphi)$ for the dipole plane containing $\textbf{p}_x$, $\textbf{p}_y$ and determine the largest dipole projection on the sample plane by the pump angle $\varphi^{NV}_p$ resulting in maximum photon rate (figure \[fig:Figure2\]e). $\varphi^{NV}_p$ is indicated by the blue arrow, wrt. the system configurations in figure \[fig:Figure2\]b-d. Two 80nm chemically synthesized silver cubes are subsequently positioned along $\varphi^{NV}_p$ in a dimer configuration, for optimal NV-to-antenna coupling (figure \[fig:Figure2\]c, d). $R(\varphi)$ for the cube-coupled system is well described by adding an additional term for the electric field generated by the cube(s) $\textbf{E}_{cube}=f_e\overset{\text{\scriptsize$\leftrightarrow$}}{\alpha}\textbf{E}_{pump}$, such that $\textbf{E}=\textbf{E}_{pump}+\textbf{E}_{cube}$. $f_e$+1 being the electric field enhancement and $\overset{\text{\scriptsize$\leftrightarrow$}}{\alpha}(\varphi^{cube}_p)$ the polarizability tensor of the cube, defined by the orientation of the cube dipole moment ($\varphi^{cube}_p$), which is coaligned with the electric field of the cube mode. We account for the enhancement factor $\eta{q}_e/\eta_0{q}_{e,0}$ of 1.52 (1 cube) and 2 (2 cubes), relative to the isolated ND indexed 0, and fit f$_e$, $\varphi^{cube}_p$ to experiment (figure \[fig:Figure2\]f, g)(see SI for details). For both cube configurations we find $\varphi^{cube}_p$-$\varphi^{NV}_p$=15$^o$, confirming near optimal alignment of the NV dipole axes with the electric field of the cube(s). The orientation of $\varphi^{cube}_p$ wrt. cube configurations is given by respectively a red or green arrow in figure \[fig:Figure2\]c, d. The enhancement of excitation rate is determined by $\gamma_{ex}/\gamma_{ex,0}=Rq_{e0}\eta_0/R_{0}q_{e}\eta$ for which we find values of 5.1 (1 cube) and 5.86 (2 cubes) at pump orientation $\varphi^{cube}_p$. Having established the dipole orientation of the NV-center and optimized the antenna configuration accordingly, we turn to the emission properties of the photon source. Rotating an analyzer in front of our detector, we find weakly polarized photons emitted from the isolated ND, with a polarization ratio of r$_{pol}$=2.1, between the photon rate detected along the major and minor axis (figure \[fig:Figure2\]h). The major axis of polarization is coaligned with $\varphi^{NV}_p$, as photons are generally polarized along the dipole axis of emission, while the two dipole configuration of the NV$^{-}$-state result in an overall weak photon polarization. The photon emission becomes increasingly polarized throughout assembly of the cube antenna, as we find r$_{pol}=$6.9 for a single cube and r $_{pol}$=9 for two cubes (figure \[fig:Figure2\]i, j). We note a slight shift of the major axis of polarization toward $\varphi^{cube}_p$, in good agreement with the polarization of emission, being the result of polarized photon scattering from plasmonic charge oscillations along the antenna dipole axis. The increase of photon polarization is naturally accompanied by an increase of the excited state decay rate ($\gamma$), as the degree of photon polarization scales with the increasing rate at which the NV-center decays from the excited state, by driving charge oscillations in the cube antenna. We recorded the temporal response to a sharp excitation pulse and found an enhancement of the excited state decay rate of respectively $\gamma{/}\gamma_0=2.25$ for 1 cube and $\gamma{/}\gamma_0=3.28$ for 2 cubes (figure \[fig:Figure3\]a). Unfortunately, the faster decay rate does not translate directly to an increase in photon rate, as the decay rate enhancement may partially result from metal quenching, or emission coupled to the antenna may be lost ohmically. The brightness of the source is hence determined by tracing out the saturation curve in terms of detected photon rate as a function of pump power (figure \[fig:Figure3\]b). After substraction of background from the plain sample surface, the detected photon rate is fitted to the conventional model:
$$\begin{aligned}
R(P)=R_{\infty}\frac{P}{P+P_{sat}}\end{aligned}$$
P being the pump power, P$_{sat}$ the saturation power and $R_{\infty}$ the asymptotic photon rate limit, detected at infinite pump power. Curiously, the R$_{\infty}$/R$_{\infty{0}}$ enhancement is significantly larger than the decay rate enhancement, as we find R$_{\infty}$/R$_{\infty{0}}$=3.41 for 1 cube and R$_{\infty}$/R$_{\infty{0}}$=6.58 for two cubes, with R$_{\infty}$=914kcps. The system is stable under large pump power as we operate the source at a photon rate of 850kcps, close to the photon rate limit. An AFM scan after $\sim$10min of operation indicates no morphological changes of the antenna. We attribute such power stability to the low ohmic heating losses of pristine monocrystalline silver, impeding thermal deformation even at large pump powers, while the strong radiative damping of the dimer mode may also be a contributing factor. Further, the large interband transision energy of silver $\sim$3.4eV[@SilverBand], prevents background photoluminescence from the metal, thereby ensuring high single photon purity. The photon purity is examined by histogramming the time interval ($\tau$) between photon detection events, yielding the 2. order correlation function g$^{(2)}(\tau)$. g$^{(2)}(0)$-events hence correspond to simultaneous detection of 2 photons, only possible in the presence of background as the NV-center may only emit 1 photon at a time. The g$^{(2)}(0)$-value is determined by fitting a 3-level rate model including a background term, normalized for $\tau\rightarrow\infty$[@HechtNovotny] (figure \[fig:Figure3\]c-e). The g$^{(2)}(0)$-value slightly improves by the addition of cubes to 0.08 from 0.14 measured for the isolated ND. The improved g$^{(2)}(0)$-value may be a result of the increased brightness of the NV-center effectively improving the signal-to-background ratio.\
Concluding the description of photon source emission properties, we now numerically examine the relation of the experimentally observed enhancement factors $\gamma/\gamma_0$ and R$_{\infty}$/R$_{\infty{0}}$. We write the decay rate of the isolated NV-center in terms of a radiative rate ($\gamma_{r0}$) and an intrinsic non-radiative rate ($\gamma_{nr0}$), such that $\gamma_0=\gamma_{r0}+\gamma_{nr0}$ with a corresponding intrinsic quantum efficiency $q_{e0}=\gamma_{r0}/\gamma_{0}$. Introducing a silver cube accelerates the radiative decay rate as additional decay channels, such as the plasmonic cube mode or metal quenching, are available for emission, while $\gamma_{nr0}$ is unaffected by the enviroment. The changed radiative decay is written $\gamma_r=\Gamma\gamma_{r0}$ such that the decay rate for the one or two cube configuration is given by $\gamma=\Gamma\gamma_{r0}+\gamma_{nr0}$. The decay rate enhancement in this case takes the form:
$$\begin{aligned}
\frac{\gamma}{\gamma_0}=\Gamma{q_{e0}}+(1-q_{e0})
\label{eq:DecayRateEnh}\end{aligned}$$
The photon rate limit is given by $R_{\infty}=\eta\gamma_r$. We define collection efficiency as the probability of a radiative decay event, resulting in a photon being emitted into the objective. Losses due to metal quenching or ohmic dissipation of antenna excitations are thereby included in $\eta$ and the photon rate enhancement can be written:
$$\begin{aligned}
\frac{R_\infty}{R_{\infty{0}}}=\frac{\eta}{\eta_0}\Gamma
\label{eq:PhotonRateEnh}\end{aligned}$$
The NV-center emission is modelled by 3D finite element simulations, as the power dissipated at 680nm wavelength by two orthogonal electric dipoles, oriented along the experimentally determined dipole plane and positioned in the center of a 35nm ND. The ND is modelled as a 4-sided truncated pyramid. $\Gamma$ is then obtained as the total power dissipated in the one or two cube configuration, relative to the ND situated on a glass substrate. Decay rate enhancement is calculated by equation \[eq:DecayRateEnh\] for various q$_{e0}$ values (figure \[fig:Figure4\]a). We find good agreement of modelled and experimental values for realistic gap sizes 40-45nm, for an intrinsic quantum efficiency of q$_{e0}\sim{0.35}$, in good agreement with previous experimental studies of the ND product[@NDQuantumEff]. Setting q$_{e0}=0.35$ and inserting experimental values $\gamma/\gamma_0$, $R_\infty/R_{\infty{0}}$ in equation \[eq:DecayRateEnh\] and \[eq:PhotonRateEnh\], we confirm the consistency of experiment and model by predicting $\eta/\eta_0$ in similar good agreement with modelling (figure \[fig:Figure4\]b (inset)), while $R_{\infty}/R_{\infty{0}}$ may be directly modelled for similar gap sizes (figure \[fig:Figure4\]b). The disparity of $\gamma/\gamma_0$ and $R_\infty/R_{\infty{0}}$ is thereby well explained by the non-unity instrinsic quantum efficiency of the NV-center. For q$_{e0}$=0.35 we find $\Gamma=4.57$ for one cube and $\Gamma=7.51$ for 2 cubes using equation 2. The corresponding quantum efficiency is calculated to respectively 0.71 and 0.80 by $q_e=\Gamma{q_{e0}}/(1+q_{e0}(\Gamma-1))$. The modelled and experimentally predicted increase of collection efficiency, going from one to two cubes (figure \[fig:Figure4\]b (inset)), is attributed to an increase of antenna efficiency as the numerical model finds respectively $\sim$ 63$\%$ and $\sim$ 97$\%$ of dipole emission reaching the far-field, while the fraction of far field emssion collected by the objective, is nearly unchanged. The increase in antenna efficiency should be expected given the superradiative damping of the dimer mode, compared to the plasmonic mode of a single nano particle. The large photon enhancement is thereby a result of an increased quantum efficiency of the NV-center, while an improved antenna efficiency of the dimer configuration, is a contributing factor to the enhancement over the single cube.\
We conducted the experiment 4 times, labelled experiment A-D, A being the experiment presented up to this point(figure \[fig:Figure4\]c-f) (see SI for measurements). Consistently we find $R_{\infty}/R_{\infty{0}} > \gamma/\gamma_{0}$ suggestive of a non-unity intrinsic quantum efficiency. The photon rate limit improved for all assembled photon sources by the addition of a second cube, with values $R_{\infty}$=671 - 1460kcps and a consistently high photon purity of g$^{(2)}(0)$=0.08 - 0.26 for the dimer configuration (figure \[fig:Figure4\]c, d). Enhancement factors for a single cube are $\gamma/\gamma_0$=1.8 - 3.4 ; $R_{\infty}/R_{\infty{0}}$=2.1 - 4.0, while the spread is more significant for two cubes $\gamma/\gamma_0$=2.1 - 5.9 ; $R_{\infty}/R_{\infty{0}}$=2.7 - 18.0. The larger spread is expected, given the strong dependence on cube separation, which is limited to the size of the ND, varying between experiments $\sim$30-35nm. Asymmetry of cube configurations and varying dipole orientation, and position of NV-center in the ND, should further contribute to the spread in experimental results.
Conclusion and outlook
======================
In summary we have presented a remarkably pure photon source, with a g$^{(2)}(0)$-value of 0.08, based on an NV-center contained in a $\sim$ 35nm ND, placed in the gap between two monocrystalline silver nano cubes. The low ohmic heating losses of pristine monocrystalline silver, allowed for stable operation under large laser powers, at a detected photon rate of 850kcps near the saturation limit of 914kcps. We demonstrated how AFM assembly of the photon source allowed for near optimal alignment of nano cubes for the maximum inplane dipole moment of the NV-center, while futher tracking the photon polarization properties, from weak polarization for the bare ND to strong linear polarization for the assembled system. The experimental finds is consistent with modelling of an NV-center with an instrinsic quantum efficiency of $\sim$ 0.35. The presented results are quite encouraging as significant improvements is within reach, by going for smaller gap sizes by employing smaller ND’s down to 5nm in size, for stable NV-center emission[@NVND] or 1.6nm for the Silicon vacancy center[@SiVND]. A scalable approach to realizing the photon source, is also concievable with molecular self-assembly as controlled assembly of face-face or egde-edge cube dimer configurations have already been realized.[@CubeAssembly].
Methods
=======
Sample preparation
------------------
A 0.18mm thick fused quartz glass slide (SPI supplies) was cleaned by an RCA1 cleaning step. 5ml Mili-Q and 1ml 28-30$\%$ NH$_4$OH(aq) solution heated to 65C, were removed from heating plate and 1ml 30$\%$ H$_2$O$_2$(aq) and slide glass added for 10min, followed by 2 step submersion in Mili-Q baths, 5min each. The cleaning step removed organic residue and promoted surface hydrophilicity for subsequent spincoating of a ND solution <50nm (Microdiamant), 100nm mean width silver cubes (nanoComposix) coated in a <5nm polyvinylpyrrolidone (PVP) layer and finally $\sim$10$\mu$m long silver wires synthesized in house. The macroscopic silver wires were used as reference markers, during experiment.
Experimental characterization
-----------------------------
The sample was mounted on a piezo scan stage, which allowed for identification of single NV-centers by confocal mapping of fluorescence using a 532nm linear polarized continuous or pulsed laser with pulse width/period 50ps/400ns. The pump light was focused by a 1.4NA oil immersion objective, used for excitation and collection of photon, while a halfwave plate controlled the pump polarization. The laser light was filtered from fluorescent photons by dichroic mirrors (SEMROCK) (cut-off 550nm) before being detected by a CCD camera (Hamamatsu-Orca-Flash4LT), an EMCCD (Andor - iXon Ultra888) connected to a grating spectrometer (Andor - Shamrock 500i) or two APDs (Picoquant - $\tau$-SPAD) in a Hanbury Brown-Twiss configuration. Photon rate as a function of pump polarization or power, was obtained by the time average of 2s time traces accummulated from both APDs. Decay rate curve and g$^{(2)}$($\tau$) was obtained by histogramming the time interval from respectively a laser sync pulse or APD detection event, to a detection event on the other APD, using an electronic timing box (Picoquant - PicoHarp 300) in a start-stop configuration. The polarization of emission was probed by the accumulated CCD image count (4s int) for various analyzer orientations. All Experiments were completed within 5 days of spincoating nanocubes.
Dark field spectrum
-------------------
The dark field spectrum of a single silver cube was obtained in transmission mode, illumination through the glass slide with a 1.2NA DF oil condenser lens and collecting light above the sample with a x50 NA 0.75. objective. A 400$\mu$m pinhole positioned in image plane in front of a fiber coupled spectrometer, allowed for selection of a single cube. The background signal obtained in the absence of a sample was subtracted from cube- and reference spectrum, using the diffuse scattering from the glass substrate as reference resulting in the cube scattering spectrum.
Finite element modelling
------------------------
3D finite element modelling was conducted in the commercially available Comsol Multiphysics 5.1. The parameter $\Gamma$ is modelled classically using the relation: $\Gamma=\gamma_r/\gamma_{r0}=P^{NV}/P_0^{NV}$. P$^{NV}$ being the total power dissipated by two orthogonal classical electric dipoles, lying in the dipole plane experimentally determined by the model fit in figure \[fig:Figure2\]e. wrt. to the antenna axis $\varphi_p^{cube}$. Simulation performed at emission wavelength 680nm, were found to be independent of the dipoles orientation in the plane. Index 0 refer to the reference system of dipoles situated in the center of a 35nm tall, 4-sided truncated pyramid shaped, diamond on a semi-infinite glass substrate, bordered from above by an air hemisphere. The modelled domain is bounded by a perfectly match layer (PML). A 8nm corner/side rounding radius were used for the 80nm cubes introduced symmetrically around the diamond. Material parameters were based on interpolation of tabulated data[@JC; @OpticalDiamondProperties; @FusedQuartz]. The collection efficiency was modelled as $\eta=P_{obj}^{NV}/P^{NV}$. P$_{obj}^{NV}$ being the power integrated over a spherical surface in the glass substate, corresponding to the solid collection angle of the experimental 1.4NA objective.
![(a) Illustration of ND containing a single NV-center situated in the gap between two silver nano cubes. The NV-center emission is accelerated as it couples to the plasmonic cube mode.(b) Electron micrograph of silver cubes scalebar 200nm. (c, d) Charge distribution of the dipolar mode of a single and coupled cubes.(e) Fluorescence spectrum of single NV-center(blue) and measured scattering spectrum of single nanocube $\sim$100nm(red) with corresponding simulations for single (red dashed) and coupled cubes(green dashed) separated by a 35nm gap.[]{data-label="fig:Figure1"}](Figures/Figure1.pdf)
![(a) Sketch of the NV-center in diamond. The excitation rate of the NV$^{-}$ charge state scales with the pump field projection on the dipole axes **p**$_{x}$, **p**$_{y}$. Phonon promoted population averaging of the doublet excited state E$_x$, E$_y$ (inset) facilitate spontanoues emission, polarized along either dipole axis. (b-d) AFM assembly of photon source color coded to corresponding experimental measurements, for (b) isolated ND (blue), (c) single cube (red) and (d) 2 cube configuration(green) scale bar 200nm. Blue arrow represent the orientation of largest inplane dipole moment of the NV-center, while red and green arrow give the electric field orientation of the antenna mode. (e-g) Normalized photon rate vs pump polarization angle, measured (dot) and model fit (solid). Dash curves in (e) indicate potential linear contributions of **p**$_{x}$ and **p**$_{y}$ to the model. (h-j) Normalized photon rate vs analyzer angle, measured (dot) and interpolation (solid).[]{data-label="fig:Figure2"}](Figures/Figure2.pdf)
![Color coded model fits to experimental data(gray) for isolated ND(blue), single cube(red) and 2 cubes(green). (a) excited state decay curves, fitted to single exponentials. (b) Saturation curves, background corrected for signal from nearby plain surface and (c-e) 2. order correlation measurement (raw data).[]{data-label="fig:Figure3"}](Figures/Figure3.pdf)
![ (a,b) Modelling of experiment A(dashed) and experimental values (solid, horizontal) for a single (red) or two cube configuration (green). (a,inset) Modelled values are given for an NV-center situated in the center of a 35nm ND for realistic gap separation of nano cubes. (a) Decay rate enhancement for intrinsic quantum efficiency q$_{e0}$=0.2-0.5 labelled on the curve. (b) Enhancement of detected photon rate at saturation, inset gives the experimentally predicted ratio of collection efficiency for q$_{e0}=0.35$ (solid) together with modelling (dashed). (c-f) Data summary of experiment A-D, for isolated ND (blue), 1 cube (red) and 2 cubes (green). Errorbars indicate the 95% confidence interval, not shown for (c) as errors $\sim$ 0.01.[]{data-label="fig:Figure4"}](Figures/Figure4.pdf)
The authors gratefully acknowledge the financial support of the European Research Council (Grant 341054 (PLAQNAP)).
Section S1: Schematic of experimental setup. Section S2: Fluorescence spectra of respectively isolated ND, 1 cube and 2 cube configuration for experiment A. Section S3: Detailed explanation of the model, fitted to photon rate as function of pump polarization angle, for isolated ND and cube coupled configurations. Section S4: Experimental data plots for experiment B-D.
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abstract: 'We study the transport properties of an ultracold gas of Bose-Einstein condensate that is coupled from a magnetic trap into a one–dimensional waveguide. Our theoretical approach to tackle this problem is based on the truncated Wigner method for which we assume the system to consist of two semi-infinite non-interacting leads and a finite interacting scattering region with two constrictions modelling an atomic quantum dot. The transmission is computed in the steady-state regime and we find a good agreement between truncated Wigner and Matrix-Product State calculations. We also identify clear signatures of inelastic resonant scattering by analyzing the distribution of energy in the transmitted atomic matter wave beam.'
author:
- Julien Dujardin
- Arturo Argüelles
- Peter Schlagheck
bibliography:
- 'bibtex/BEC.bib'
- 'bibtex/AtomLaser.bib'
- 'bibtex/ComScal.bib'
- 'bibtex/Peter.bib'
- 'bibtex/AndLoc.bib'
- 'bibtex/TruncWig.bib'
- 'bibtex/Transport.bib'
- 'bibtex/CAP.bib'
- 'bibtex/MPS.bib'
- 'bibtex/Tronic.bib'
- 'bibtex/Julien.bib'
title: 'Elastic and inelastic transmission in guided atom lasers: a truncated Wigner approach'
---
Introduction
============
The progress of the last two decades in the field of ultracold atoms has opened the possibility of investigating mesoscopic transport properties of interacting matter waves. A very important step in this context is the realization of atom lasers [@Bloch1999PRL; @Cennini2003PRL; @Hagley1999S; @Mewes1997PRL] permitting to create a beam of atoms by coherently outcoupling a trapped Bose–Einstein condensate (BEC) into an optical waveguide at a well-defined energy and flux [@Guerin2006PRL; @Couvert2008EEL; @Riou2008PRA; @Gattobigio2009PRA; @Debs2010PRA; @KleineBuening2010APB; @Gattobigio2011PRL]. This research is particularly interesting in view of the perspective to realize bosonic atomtronic devices [@Micheli2004PRL; @Daley2005PRA; @Seaman2007PRA; @Pepino2009PRL] and to study analogies with their fermionic counterpart [@Brantut2012S; @Bruderer2012PRA; @Kristinsdottir2013PRL]. A typical configuration of such a device is an atomic quantum dot that features resonant transport [@Carusotto2001PRA] and atom blockade [@Schlagheck2010NJoP; @Kristinsdottir2013PRL]. These features can be used as building blocks for atomic transistors [@Micheli2004PRL; @Seaman2007PRA; @Pepino2009PRL].
A theoretical modeling of such scattering processes within guided atom lasers faces the challenge of dealing with interactions between atoms. A full many-body treatment of such an open system is very complex and impossible to solve exactly in practice. During last years, these scattering processes have been studied in the mean–field approximation described by a nonlinear Gross–Pitaevskii (GP) equation [@Paul2005PRL; @Paul2007PRA; @Ernst2010PRA]. While this description gives satisfactory results for a weak nonlinearity, the question of validity arises rapidly in the case of strong nonlinear dynamics [@Ernst2010PRA] where dynamical instabilities occur. It has also been pointed out that in the presence of disordered potentials, even a weak atom-atom interaction strength can lead to inelastic scattering processes [@Geiger2012PRL; @Geiger2013NJoP], which can not be accounted for in the framework of the mean–field GP approximation.
The main focus of this work is to study such inelastic scattering processes in an atom laser context. We employ the truncated Wigner method (tW) [@Wigner1931; @Wigner1932PR; @Moyal1949PCPS] for this purpose. The latter amounts to sampling the initial quantum state by classical fields and to propagating them according to a slightly modified GP equation. This method has been used to study the reflection of a BEC on abrupt potential barriers at zero temperature [@Scott2006PRA] and at finite temperature [@Scott2007LP]. It can also be used to study the dynamics of a trapped BEC [@Isella2006PRA] when an optical lattice is adiabatically superimposed to the trapping potential. It has also been used to study the many-body Landau-Zener effect [@Altland2009PRA] as well as far from equilibrium dynamics (in particular non-thermal fixed-points) of many-body systems [@Schmidt2012NJoP]. Finally, the tW method can also take into account a continuous measurement process [@Lee2014PRA] during the evolution of the system.
We specifically apply the tW method to study the transmission of a one-dimensional guided atom laser beam across a double barrier potential forming an atomic quantum dot as described in Sec. \[sec:scattconf\]. In this scenario, we are particularly interested in resonant transport. To this end, we suppose a finite extent of the interacting region and discretize the one-dimensional space according to a finite-difference scheme. We generalize, in Sec. \[sec:tWBH\], the tW method to open systems using Smooth Exterior Complex Scaling [@Balslev1971CMP; @Simon1973AM; @Simon1979PLA; @Kalita2011JCP; @Dujardin2014APB]. We then study numerically, in Sec. \[sec:TransQD\], the transmission properties through the quantum dot model described in Sec. \[sec:scattconf\]. The obtained results are then confronted, in Sec. \[subsec:transmQD\], to the predictions provided by the mean–field approximation and to Matrix-Product State (MPS) calculations. In Secs. \[sec:TransQD\] [B–D]{}, we analyze the energy distribution of the transmitted beam using the tW method and develop a Bogoliubov approach to understand the physical origin of the inelastic peaks that appear in the energy distribution.
Scattering configuration {#sec:scattconf}
========================
We consider a guided atom laser experiment such as the one represented in Ref. [@Guerin2006PRL], where a magnetically trapped BEC plays the role of a coherent source of atoms. In this particular experiment, the atoms are out-coupled by a rf-knife rendering the final state insensitive to the magnetic field, but sensitive to the optical potential formed by an elongated far off-resonance optical beam constituting an atomic waveguide. Ideally, the propagation of the atoms at well defined energy is quasi one–dimensional (1D) along the waveguide. It is then possible to engineer an atomic quantum dot geometry by focusing two far–detuned laser beams perpendicular to the waveguide. In this paper, we specifically consider a waveguide configuration in which spatial inhomogeneities and atom-atom interactions are non-vanishing only in a finite region of space. Such a system is represented in Fig. \[fig:qdot\_geom\](a).
In order to properly implement the tW method, we discretize the 1D space by a series of points or sites separated by a constant distance $\Delta$, thereby forming a spatial grid. The wavefunction is then defined on these points. The sites are labeled with an index $l\in\mathbb{Z}$. One additional site $S$ is introduced in order to represent the source of atoms. This additional site $S$ is connected to the waveguide at site $l_S$ as illustrated in Fig. \[fig:qdot\_geom\](b).
We treat the spatial derivatives with a finite-difference approximation. The Hamiltonian is then given by $$\begin{aligned}
\label{eq:BHham}
\hat{\mathcal{H}} &= \displaystyle\sum_{l=-\infty}^{+\infty}& \Bigg[
-J(\hat{a}^\dagger_{l+1}\hat{a}_{l} + \hat{a}^\dagger_{l}\hat{a}_{l+
1}) \nonumber \\
&&+ \frac{g_l}{2} \hat{n}_l(\hat{n}_l-1
) + V_l\hat{n}_l \Bigg] \nonumber \\
&&+ \kappa^*(t)\hat{b}^\dagger\hat{a}_{l_S} +
\kappa(t)\hat{a}^\dagger_{l_S}\hat{b} + \mu \hat{b}^\dagger \hat{b}.\end{aligned}$$ Here $\hat{b}$ and $\hat{b}^\dagger$ is the annihilation and creation operator of the reservoir, respectively, $\mu$ is its chemical potential defined relative to the center of the band, and $\hat{a}_l$ and $\hat{a}_l^\dagger$ are the annihilation and creation operators, respectively, on the site $l$ of the chain. The hopping strength to the nearest neighbors is given by $J=\hbar^2/2m\Delta^2$, the on-site interaction strength is $g_l$, and the on-site potential is $V_l$. The coupling strength $\kappa(t)$ is related to the out-coupling process of atoms from the reservoir and can be controlled in a time-dependent manner (*e.g.* through the variation of the intensity of a radio-frequency field in the case of Refs. [@Guerin2006PRL; @Riou2008PRA]). We suppose that the source is adiabatically switched on from zero to a maximal value of $\kappa$, *i.e.* $$\lim_{t\to\infty} \kappa(t) = \kappa.$$
This Hamiltonian is similar to a Bose-Hubbard (BH) system describing an optical lattice in which only the lowest band in the Brillouin zone is considered. For the case where the on-site potential and interaction strength vanish, the dispersion relation is identical to the one of the free lattice and is given by $$\label{eq:DispRel}
E(k) = -2J \cos(k),$$ with a wavenumber $k/\Delta$. In the limit $k\ll1$, we have $E(k) = -2J + J k^2$ which, apart from a constant shift, corresponds to the dispersion relation of a free atom.
The scattering configuration of an atomic quantum dot is modeled by two sites with non-zero on-site potential. Between these two sites, we allow atoms to interact as depicted in Fig. \[fig:qdot\_geom\](b). This can be justified if, for instance, the waist of the elongated optical beam is particularly narrow at the position where the quantum dot is located. The perpendicular confinement is then rather strong and it is likely that collisions occur between atoms. Formally, this model can be encoded as
\[eq:qdot\] $$\begin{aligned}
V_l &= V(\delta_{l,l_0} + \delta_{l,l_0+L_{\mathrm{D}}}), \label{eq:qdotV} \\
g_l &= g \sum_{j=1}^{L_{\mathrm{D}}-1}\delta_{l,l_0+j}, \label{eq:qdotg}
\end{aligned}$$
where $l_0\in\mathbb{Z}$ is arbitrary and $L_{\mathrm{D}}$ is the length of the quantum dot. We set $L_{\mathrm{D}}=6$ in the rest of the paper.
![(color online) (a) A trapped BEC depicted by a (green) circle is loaded into a waveguide with two constrictions modeling an atomic quantum dot. The bold (black) lines represent the isopotentials of the waveguide. (b) One dimensional infinite Bose-Hubbard (BH) chain for the quantum dot model (see Eqs. \[eq:qdot\]). The condensate is prepared within a trap represented by the green circle ($S$) and coupled to the infinite BH chain (dashed green line). The big (red) circles represent a non-vanishing on-site atom-atom interaction. The two displaced sites enclosing the interaction region represent two sites where the on-site potential is nonzero.[]{data-label="fig:qdot_geom"}](eps/qdot_al.eps){width="0.9\linewidth"}
Truncated Wigner Method for open BH systems {#sec:tWBH}
===========================================
Phase-space methods were introduced by Wigner [@Wigner1931; @Wigner1932PR] and Moyal [@Moyal1949PCPS] and their development started in the 60’s with successful applications in quantum optics by Glauber [@Glauber1963PR] and Sudarshan [@Sudarshan1963PRL]. These methods allow to go beyond the mean–field GP description by, essentially, sampling the initial quantum state by classical fields. The prescription to sample the initial state and the equation of motion are not unique. In this paper we choose the truncated Wigner method (tW). The evolution of the system is then given by a classical equation of motion similar to the GP equation. In particular, the tW method maps the density matrix of the system onto a quasi–distribution function fulfilling a Fokker–Planck equation. It is then possible to replace this equation with a system of Langevin equations that can be numerically solved by a Monte-Carlo method. This section is devoted to generalize the tW method to open systems.
Truncated Wigner Method for BH systems {#subsec:tWinfBH}
--------------------------------------
Let us consider a general Bose-Hubbard (BH) system with on-site two-body interaction. Denoting by $\mathcal{A}=\{S,0,\pm1,\pm2,\cdots\}$ the ensemble of sites of the BH system, the many-body Hamiltonian of the system can be written as $$\label{eq:MBHBH}
\mathcal{\hat{H}} = \sum_{\alpha\in \mathcal{A} } \Bigg[ \sum_{\alpha'\in \mathcal{A} }h_{\alpha\alpha'}\hat{a}^\dagger_{\alpha}\hat{a}_{\alpha'} + \frac{g_\alpha}{2} \hat{n}_\alpha(\hat{n}_\alpha-1)\Bigg],$$ where we defined by $\hat{a}_\alpha$ and $\hat{a}_\alpha^\dagger$ the annihilation and creation operators, respectively, on the site $\alpha$ of the chain, and by $\hat{n}_\alpha = \hat{a}^\dagger_\alpha\hat{a}_{\alpha}$ the corresponding number operator. The matrix elements $h_{\alpha\alpha'}$ represent on-site energies as well as possible hoppings between the sites. We impose $h_{\alpha\alpha'}=h_{\alpha'\alpha}^*$ to ensure that the Hamiltonian remains hermitian. This general form makes our description also valid for more involved connections between different sites of the grid, such as small-world networks.
The general idea of the Wigner approach is to map the evolution of the density matrix prescribed by the von Neumann equation $$i\hbar\frac{\partial\hat{\rho}(t)}{\partial t} = [\mathcal{\hat{H}}, \hat{\rho}(t)]$$ to the evolution of the Wigner function $\mathcal{W}~\equiv~\mathcal{W}(\{\psi_\alpha,\psi_\alpha^*\},t)$ that is defined in the phase space spanned by the classical amplitudes $\psi_\alpha$ associated with each site $\alpha$. The Wigner function represents a quantum quasi-probability distribution and is defined as $$\begin{aligned}
\mathcal{W}(\{\psi_\alpha,\psi_\alpha^*\},t) &=& \prod_{\alpha \in \mathcal{A}}\frac{1}{\pi^2} \iint d\lambda_\alpha d\lambda^*_\alpha e^{-\lambda_\alpha\psi^*_\alpha + \lambda_\alpha^*\psi_\alpha} \nonumber \\ && \times\,\,\, \chi_{\mathcal{W}}(\{\lambda_\alpha,\lambda_\alpha^*\},t),\end{aligned}$$ which is the Fourier transform of the characteristic function $\chi_{\mathcal{W}}$ $$\chi_{\mathcal{W}}(\{\lambda_\alpha,\lambda_\alpha^*\},t) = \operatorname{Tr}\left[\hat{\rho}(t) \prod_{\alpha \in \mathcal{A}} e^{\lambda_\alpha\hat{a}_\alpha^\dagger - \lambda_\alpha^*\hat{a}_\alpha} \right].$$ The classical amplitudes $\psi_\alpha$ and $\psi_\alpha^*$ are complex canonical variables representing coherent states in phase space. The evolution of the Wigner function is then given by $$\begin{aligned}
\label{eq:Wignerevol}
i\hbar\frac{\partial {\mathcal{W}}}{\partial t} &=& \sum_{\alpha\in\mathcal{A}} \Bigg[ -\sum_{\alpha'\in\mathcal{A}} \left(h_{\alpha\alpha'}\frac{\partial}{\partial \psi_\alpha}\psi_{\alpha'} - h_{\alpha'\alpha}^* \frac{\partial}{\partial \psi_\alpha^*}\psi_{\alpha'}^* \right) \nonumber\\
&-& g_\alpha \left(\frac{\partial}{\partial\psi_\alpha}\psi_\alpha - \frac{\partial}{\partial\psi_\alpha^*}\psi_\alpha^*\right)(|\psi_\alpha|^2-1) \nonumber \\
&+& \frac{g_\alpha}{4} \left(\frac{\partial^2}{\partial \psi_\alpha^2}\frac{\partial}{\partial \psi^*_\alpha}\psi_\alpha - \frac{\partial}{\partial \psi_\alpha}\frac{\partial^2}{\partial \psi^{*^2}_\alpha}\psi_\alpha^*\right) \Bigg ] {\mathcal{W}}. \end{aligned}$$ Numerical integration of this equation is practically impossible since the dimension of the phase space is very large.
The so–called *truncated Wigner approximation* consists in neglecting the third order derivatives in Eq. . The resulting equation is commonly called the *truncated Wigner equation* and corresponds to a Fokker–Planck equation with only a drift term. It can be shown [@Gardiner2004] that this approximation is valid if there is locally a large number of atoms in the waveguide. The evolution of the Wigner function can be mapped to a set of coupled Langevin equations where the canonical variables $\psi_\alpha\equiv\psi_\alpha(t)$ and $\psi^*_\alpha\equiv\psi^*_\alpha(t)$ are now time-dependent. They satisfy $$\label{eq:CanonicalEvolution}
i\hbar\frac{\partial \psi_\alpha}{\partial t} = \sum_{\alpha'\in\mathcal{A}} h_{\alpha\alpha'}\psi_{\alpha'} + g_\alpha(|\psi_\alpha|^2 -1) \psi_\alpha.$$ The mapping gives another set of equations for the evolution of $\psi_\alpha^*$ which correspond to the complex conjugate of Eq. .
For the specific case of our guided atom-laser configuration, we can now write the final set of equations of motion for the sites representing the waveguide and the site corresponding to the source as
\[eq:StochAll\] $$\begin{aligned}
i\hbar\frac{\partial \psi_l}{\partial t} &= (V_l-\mu)\psi_l -J\left(\psi_{l+1} + \psi_{l-1}\right) \nonumber\\
& + g_l(|\psi_l|^2 -1) \psi_l + \kappa(t)\psi_S\delta_{l,l_S}, \\
\label{eq:StochSource}
i\hbar\frac{\partial \psi_S}{\partial t} &= \kappa^*(t)\psi_{l_S}.
\end{aligned}$$
It is nearly identical to a discrete GP equation except for a slightly different interaction term.
The initial state {#subsec:tWinit}
-----------------
The initial Wigner function $\mathcal{W}(\{\psi_\alpha,\psi_\alpha^*\},t_0)$ represents the initial quantum state of the system and has to be sampled by the classical fields $\psi_\alpha$. The latter can, for instance, represent coherent, thermal, squeezed or Fock states [@Olsen2009OC] and its time evolution is governed by classical trajectories evolving according to Eqs. . We consider that initially, at $t=t_0$, the waveguide is empty and the ground state of the reservoir trap is macroscopically populated with a large number $N$ of atoms at zero temperature. The Wigner function can then be written as $${\mathcal{W}}= \mathcal{W}_G(\{\psi_l,\psi_l^*\},t_0) \, \times \, \mathcal{W}_S(\psi_S,\psi_S^*,t_0),$$ at time $t=t_0$, where $\mathcal{W}_G(\{\psi_l,\psi_l^*\},t_0)$ and $\mathcal{W}_G(\{\psi_l,\psi_l^*\},t_0)$ correspond to the Wigner function of the source of atoms and the waveguide, respectively.
Since the waveguide is initially empty, the corresponding Wigner function has the form [@Sinatra2002JPBAMOP] $$\mathcal{W}_G(\{\psi_l,\psi_l^*\},t_0) = \prod_{l \in \mathbb{Z}} \left(\frac{2}{\pi}\right) \exp(-2|\psi_l|^2).$$ We can therefore sample the initial state with complex Gaussian random variables. More precisely, the initial values of the amplitudes $\psi_l$ can be written as $$\label{eq:InitCond}
\psi_l(t=t_0) = \frac{1}{2}\left(A_l + i B_l\right),$$ where $A_l$ and $B_l$ are real, independent Gaussian random variables with unit variance and zero mean, *i.e.* for each $l,l' \in \mathbb{Z}$ we have
\[eq:initcond\] $$\begin{aligned}
\overline{A_l} &= \overline{B_l} = 0,\\
\overline{A_{l'}A_l} &= \overline{ B_{l'}B_l} = \delta_{l,l'}, \\
\overline{A_{l'}B_l} &= 0,
\end{aligned}$$
where the overline denotes the average of the random variables. As a consequence, each site $l$ of the grid representing the empty waveguide has the average atom density $\overline{|\psi_l(t_0)|^2}=1/2$.
We are now considering the source part which represent a BEC with a high number $N$ of atoms such that it can be safely described by a coherent state ${\ensuremath{|\psi_S^0\rangle}}$. The initial Wigner function $\mathcal{W}_S(\psi_S,\psi_S^*,t_0)$ therefore reads $$\mathcal{W}_S(\psi_S,\psi_S^*,t_0) = \left(\frac{2}{\pi}\right) \exp(-2|\psi_S - \psi_S^0|^2).$$ As $N$ is very large, the relative uncertainty of both the amplitude $|\psi_S^0| = \sqrt{N}$ and the associated phase of the source are negligibly small. We therefore treat the source term completely classically, *i.e.* we set $\psi_S^0~=~\sqrt{N}$.
Supposing, in addition, that the coupling $\kappa(t)$ tends to zero such that $N|\kappa(t)|^2$ remains finite, we can safely neglect the depletion of the source or any back-action of the waveguide to the source since $\psi_S(t) = \sqrt{N}(1+\mathcal{O}(|\kappa|^2)$ at any finite time $t>t_0$. This allows us to solely focus on the evolution of the field in the chain. The equation to solve reads $$\begin{aligned}
\label{eq:StochFinal}
i\hbar\frac{\partial \psi_l}{\partial t} &=& (V_l-\mu)\psi_l -J\left(\psi_{l+1} + \psi_{l-1}\right) \nonumber\\
&&+ g_l(|\psi_l|^2 -1) \psi_l + \kappa(t)\sqrt{N}\delta_{l,l_S}.\end{aligned}$$ One can notice that if $|\psi_l|^2$ is very large, we recover the discrete GP equation.
Observables {#subsec:Observables}
-----------
It can be shown [@Cahill1969PR] that the time-dependent expectation value of the symmetrically ordered product of the operator $\hat{a}_l$ and $\hat{a}^\dagger_l$ is of the form $$\begin{aligned}
\label{eq:obscomputation}
\left\langle\left\{\prod_{l \in \mathbb{Z}} (\hat{a}_l^\dagger)^{r_l}\hat{a}_l^{s_l}\right\}_{\text{sym}}\right\rangle_t &=& \prod_{l \in \mathbb{Z}}\int d\psi_l d\psi^*_l\, (\psi_l^*)^{r_l}\psi_l^{s_l} \nonumber \\
&& \times \mathcal{W}(\{\psi_l,\psi_l^*\},t),\end{aligned}$$ where $\{ (\hat{a}_l^\dagger)^{r_l}\hat{a}_l^{s_l} \}_{\text{sym}}$ denotes the symmetrically ordered product *i.e.* the average of $(r_l+s_l)!/(r_l!s_l!)$ possible orderings of $r_l$ creation operators and $s_l$ annihilation operators. For instance, setting $r_l=2$ and $s_l=1$ we have $$\left\{ (\hat{a}_l^\dagger)^2\hat{a}_l \right\}_{\text{sym}} = \frac{1}{3}\left[ (\hat{a}_l^\dagger)^2\hat{a}_l + \hat{a}_l^\dagger \hat{a}_l \hat{a}_l^\dagger + \hat{a}_l(\hat{a}_l^\dagger)^2 \right].$$
This equation allows us to calculate the expectation value of observables on a particular site. Specifically, the expectation value of the total density $n_l(t)$ and the total current $j_l(t)$ on a site $l$ are given by $$\begin{aligned}
n_l(t) &=& \langle\hat{n}_l(t)\rangle = \langle\hat{a}_l^\dagger(t)\hat{a}_l(t)\rangle = \overline{|\psi_l(t)|^2} - 0.5, \\
j_l(t) &=& \langle\hat{j}_l(t)\rangle \nonumber\\
&=& \frac{i\hbar}{2}\left( \langle\hat{a}_{l+1}^\dagger(t)\hat{a}_l(t) - \hat{a}_l^\dagger(t)\hat{a}_{l+1}(t)\rangle\right) \nonumber \\
&=& \frac{i\hbar}{2}\left( \overline{\psi^*_{l+1}(t)\psi_l(t) - \psi^*_l(t)\psi_{l+1}(t)}\right),\end{aligned}$$ where the overline denotes the statistical average over all classical initial states. In addition, we can determine the coherent part of the density $n_l^{\text{coh}}(t)$ as well as the coherent part of the current $j_l^{\text{coh}}(t)$ through $$\begin{aligned}
n_l^{\text{coh}}(t) &=& |\langle\hat{a}_l(t)\rangle|^2 = \left|\overline{\psi_l(t)}\right|^2,\\
j_l^{\text{coh}}(t) &=& \frac{i\hbar}{2}\left( \overline{\psi^*_{l+1}(t)}\,\,\overline{\psi_l(t)} - \overline{\psi^*_l(t)}\,\,\overline{\psi_{l+1}(t)}\right).\end{aligned}$$ In the mean–field limit, the coherent part of the density and the current correspond to the usual GP density and current. We can also identify the incoherent part as the difference of the total and coherent parts of the density and the current: $$\begin{aligned}
n_l^{\text{incoh}}(t) &=& n_l(t) - n_l^{\text{coh}}(t),\\
j_l^{\text{incoh}}(t) &=& j_l(t) - j_l^{\text{coh}}(t).\end{aligned}$$
The situation evidently simplifies in the special case of a waveguide without any on-site potential or interaction between the atoms. In this case, the GP as well as the tW evolution equations reduce to the standard one-body Schrödinger equation and hence the coherent and total densities in the waveguide are identical. The stationary density $n^{\varnothing}$ is given by $$\begin{aligned}
\label{eq:freen}
n^{\varnothing} &=& \lim_{t\to\infty} |\overline{\psi(t)}|^2 = \lim_{t\to\infty} \overline{|\psi(t)|^2} - 0.5 \nonumber \\
&=& \frac{N|\kappa|^2}{4J^2-\mu^2}\end{aligned}$$ and the stationary current $j^{\varnothing}$ is given by $$\label{eq:freej}
j^{\varnothing} = \frac{N|\kappa|^2}{\sqrt{2J(4J^2-\mu^2)}}.$$
Truncated Wigner for open systems {#subsec:tWSECS}
---------------------------------
We are able to represent the infinite chain in terms of a finite open system if we assume that the on-site potential and the contact interaction are non-vanishing only in a finite region of space. This finite region will be named the *scattering region* and the regions on the left and the right hand side of it are called the *left* and *right leads*, in close analogy to electronic mesoscopic physics. Without loss of generality, we shall assume that the scattering region is defined in the interval $l\in\{1,\cdots,L\}$ on the grid. The dynamics in the leads is linear and can therefore be solved analytically. We then find that the evolution equation can be written as [@Dujardin2014APB] $$\begin{aligned}
\label{eq:integro_evol}
i\hbar \frac{\partial \psi_l}{\partial t} &=& (V_l-\mu)\psi_l + g_l |\psi_l|^2\psi_l + \kappa(t)\sqrt{N}\delta_{l,l_S} \nonumber\\
& & - J\left[\psi_{l-1}(1-\delta_{l,1})+ \psi_{l+1}(1-\delta_{l,L})\right] \nonumber\\
& & - \frac{2i}{\hbar}(\delta_{l,1} + \delta_{l,L})J^2 \int_{t_0}^t dt'\, \mathcal{M}_{1}(t-t')\psi_l(t') \nonumber\\
& & +\delta_{l,1} \chi_1(t) + \delta_{l,L} \chi_L(t),\end{aligned}$$ for the site $l$ within the scattering region ($l=1,\cdots,L$) with
\[eq:classicalnoise\] $$\begin{aligned}
\chi_1(t) &= 2J \sum_{l'=-\infty}^0 \mathcal{M}_{l'-1}(t-t_0)\psi_{l'}(t_0), \\
\chi_L(t) &= -2J\sum_{l'=L+1}^\infty \mathcal{M}_{l'-L}(t-t_0)\psi_{l'}(t_0),
\end{aligned}$$
and $$\mathcal{M}_l(\tau) = \frac{i^l}{2} \left[ J_{l-1}
\left(\frac{2J\tau}{\hbar}\right)+J_{l+1}\left(\frac{2J\tau}{\hbar}\right)\right]e^{i\mu\tau/\hbar}$$ where $J_l$ is the Bessel functions of the first kind of the order $l$.
As no approximation has yet been made, Eq. reproduces the true evolution of the infinite nonlinear system under consideration described by Eq. . The integral term in the third line of Eq. exactly describes the decay into the left and right leads and therefore yields a perfectly transparent boundary condition that is defined on the first and last site of the scattering region. The terms $\chi_1(t)$ and $\chi_L(t)$ in Eq. account for the propagation of the initial quantum fluctuations that arise in the framework of the tW approximtion and that eventually, during the time propagation, enter in the scattering region. These terms $\chi_1(t)$ and $\chi_L(t)$, considering the initial emptiness of the leads in the tW prescription (see Eqs. ), take the form of *quantum noise* entering the system. The autocorrelation functions related to these noise terms read are given by $$\overline{\chi_1^*(t)\chi_1(t+\tau)} = \overline{ \chi_L^*(t)\chi_L(t+\tau)} = -i \mathcal{M}_1(\tau).$$
The integral term in Eq. renders the numerical simulation rapidly inefficient because the whole integral has to be recomputed at every time step. The most efficient way to avoid this problem in the numerical computations [@Dujardin2014APB] is to remove this integral term and replace it by *Smooth Exterior Complex Scaling* [@Balslev1971CMP; @Simon1973AM; @Simon1979PLA; @Junker1982AAMP; @Reinhardt1982ARPC; @Ho1983PR; @Loewdin1988AQC; @Moiseyev1998PR]. The evolution of the finite open system is now governed by the following equation $$\begin{aligned}
\label{eq:SECS}
i\hbar\frac{\partial \psi_l}{\partial t} &=& \left(V_l- \mu q_l\right)\psi_l + g_l(|\psi_l|^2-1) \psi_l \nonumber \\
& & + \kappa(t)\sqrt{N}\delta_{l,l_S} +2J( q_l + q_l^{-1}) \psi_l \nonumber \\
& & -J\left[\frac{1}{q_{l+1}} +\frac{1}{2}\frac{q'_{l+1}}{ q^2_{l+1}}\right] \psi_{l+1} \nonumber \\
& & -J \left[\frac{1}{q_{l-1}} -\frac{1}{2}\frac{q'_{l-1}}{ q^2_{l-1}}\right]\psi_{l-1} \nonumber \\
& & +\delta_{l,1}\chi_1(t) + \delta_{l,L}\chi_L(t),\end{aligned}$$ where $q_l$ is a smooth function of the site index $l$. In the scattering region ($1\leqslant l\leqslant L$) we impose $q_l=1$, while $q_l$ is smoothly ramped to $e^{i\theta}$ within the left ($l<1$) and the right ($l>L$) leads where $\theta$ is an arbitrary positive angle. The function $q'_l$ represents the discrete derivative of $q_l$ with respect to $l$. If $q_l\neq 1$, the Hamiltonian is not hermitian any longer and the outgoing atoms are absorbed without reflection, provided that the discrete function $q_l$ is sufficiently smooth (*i.e.* $q_{l+1}-q_l\simeq q'_l\ll q_l$). This approach was successfully tested in Ref. [@Dujardin2014APB] for the case of a linear and a nonlinear Schrödinger equation with or without quantum fluctuations as described in Eqs. and Eqs. .
Transmission across a quantum dot {#sec:TransQD}
=================================
Transmission spectrum {#subsec:transmQD}
---------------------
We now study transport across a symmetric double barrier potential that can be seen as a resonator. Hence, in absence of interaction, we know that the transmission spectrum will give rise to a series of Fabry-Pérot or Breit-Wigner peaks at resonances. As explained in Ref. [@Paul2005PRL], the presence of atom-atom contact interaction bends these peaks. Depending on the strength of the nonlinearity within the resonator, bistability can occur as seen in Fig. \[fig:V0.5g0.1\]. This bistability can be seen as an artifact of the mean–field approximation since many-body quantum scattering processes are linear from a microscopic point of view and, as a consequence, we expect a unique many-body scattering state to establish.
We now discuss the effects of the interaction on the Fabry-Pérot peaks beyond the mean–field GP description. We fix the maximal coupling strength between the source and the waveguide to $N|\kappa|^2 = J^2$. In Fig. \[fig:V0.5g0.1\], we plot the transmission across the quantum dot against the chemical potential with an interaction strength $g=0.2J$ and an on-site potential $V=J$. The total transmission $T$ is determined by comparing the total current in the downstream region to the stationary current obtained in the case of a perfectly homogeneous and interaction-free waveguide: $$\label{eq:totaltransmission}
T = \lim_{t\to\infty} j(t)/j^{\varnothing}.$$ It can be decomposed into its coherent $T^{\text{coh}}$ and incoherent $T^{\text{incoh}}$ part by respectively comparing the coherent and incoherent current to the free current :
$$\begin{aligned}
T^{\text{coh}} &= \lim_{t\to\infty} j^{\text{coh}}(t)/j^{\varnothing}, \\
T^{\text{incoh}} &= \lim_{t\to\infty} j^{\text{incoh}}(t)/j^{\varnothing}.
\end{aligned}$$
In the mean–field description, we observe that the GP curve is bent and features bistability as it was shown by Paul *et. al.* [@Paul2005PRL]. This curve has been obtained by solving the stationary GP equation in the same way as it was done in Ref. [@Paul2007PRA]. The dashed black curves correspond to solutions of the stationary GP equation that are unstable (middle branch of the resonance peak) or inaccessible through a time-dependent loading of the waveguide at constant chemical potential (upper branch of the resonance peak).
In order to benchmark our tW calculations, we compare the total transmission given by Eq. to the one obtained by a genuinely quantum simulation using Matrix-Product State (MPS) [@Vidal2003PRL; @Verstraete2004PRL; @Vidal2004PRL] calculations. This method is based on the Density-Matrix Renormalization Group [@White1992PRL] (DMRG) which uses renormalization techniques to express in an optimized way the density matrix of a block within the system under consideration. The states produced by this process belong to the class of matrix-product states [@Verstraete2004PRL; @Vidal2004PRL; @Vidal2003PRL], which offer a highly optimized way of treating the full problem as long as no highly entangled states are present. When the number of atoms is quite low and the system is very small, the full Hilbert space can be efficiently truncated by removing the degrees of freedom that are not involved in the dynamical evolution of the system. Such an optimized method enables us to numerically simulate the atomic quantum dot.
![(color online) Transmission across the quantum dot configuration versus $\mu/2J$ for $N|\kappa|^2 = J^2$, $g=0.2J$, $V=J$. The black curve corresponds to the mean–field (GP) calculation, the red curve to tW method and the green dots to the MPS method. We can see that for the GP method bistability occurs for the first resonance. The dashed black line depicts states that are not accessible during a time-dependent loading of the waveguide. The tW and MPS curves are in good agreement and exhibit an imperfect transmission at resonances. The incoherent part of the transmission is represented by an orange line. This curve shows that an appreciable amount of incoherent atoms are generated at the resonances, demonstrating a departure from of the GP model. The MPS method becomes numerically inefficient near the band edges, *i.e.* for $\mu\approx -2J$.[]{data-label="fig:V0.5g0.1"}](eps/V0.5g0.1.eps){width="\linewidth"}
The results which are displayed in Fig. \[fig:V0.5g0.1\], show a good agreement between the tW and MPS methods. Both methods clearly show that the transmission is not perfect at resonance, meaning that full resonant transmission is prohibited. The orange dotted curve in Fig. \[fig:V0.5g0.1\] displays the incoherent part of the transmission. We can see that about ten to twenty percent of the transmission comes from incoherent atoms at the resonances, which appears to be a consequence of the enhanced atomic density within the quantum dot at resonance. Indeed, in contrast to the coherent part of the transmitted beam, the incoherent atoms may exit the quantum dot to either one of the leads. They thereby inhibit perfect transmission of the atomic beam at resonance.
Energy distribution of the transmitted atoms
--------------------------------------------
We are now interested in signatures of inelastic scattering in the transmitted beam. To this end, we take a large but finite number of sites $L_{\textrm{ft}}=1000$ in the transmitted region and define $\hat{a}(k)$ as $$\hat{a}(k) = \frac{1}{\sqrt{L_{\textrm{ft}}}}\sum_{l=L_D+2}^{L_D+2+L_{\textrm{ft}}} \hat{a}_l e^{-ikl},$$ corresponding to the annihilation operator associated with the momentum eigenstate $e^{ikl}$ within the right lead. Noting that $[\hat{a}(k),\hat{a}^\dagger(k)]=1$ from this definition and following the procedure explained in the section \[subsec:Observables\], we can calculate the steady-state average total and coherent number of atoms moving with a wavenumber $k$ through
$$\begin{aligned}
n(k) &= \langle\hat{n}(k) \rangle = \langle \hat{a}^\dagger(k)\hat{a}(k)\rangle=\overline{|\psi(k)|^2} - 0.5,\\
n^\textrm{coh}(k) &= |\langle \hat{a}(k) \rangle|^2= |\overline{\psi(k)}|^2,
\end{aligned}$$
with $$\psi(k) = \frac{1}{\sqrt{L_{\textrm{ft}}}}\sum_{l=L_D+2}^{L_D+2+L_{\textrm{ft}}} \psi_l e^{-ikl}.$$ Since all the transmitted atoms have $k>0$, we define the total $n_E$ and coherent $n_E^\textrm{coh}$ average number of transmitted atoms moving with energy $E$ by
$$\begin{aligned}
n_E &\equiv n(k_E),\\
n_E^\textrm{coh} &\equiv n^\textrm{coh}(k_E),
\end{aligned}$$
where $k_E$ is obtained by inverting the dispersion relation : $$k_E = \arccos(-E/2J).$$
[eps/V2g0.01.eps]{} (-1,30) (-1,45)
In Fig. \[fig:V2g0.01\](a) we plot the transmission versus the normalized chemical potential of the incoming atoms with $V=4J$, $g=0.02J$ and $N|\kappa|^2 = \,J^2$. We can the see the appearance of well-resolved resonance peaks. Compared to Fig. \[fig:V0.5g0.1\] the visibility of the peaks is enhanced, which is expected as the enhancement of the potential barrier forming the quantum dot leads to a greater lifetime of the corresponding quasi-bound states. In Figs. \[fig:V2g0.01\](b–g), we plot the energy distribution of the transmitted atoms. We can see the appearance of additional peaks depending on the value of the chemical potential. For Fig. \[fig:V2g0.01\](d), where the chemical potential $\mu/2J=-0.37$ is far away from any resonance, we can only observe one peak corresponding to the coherent beam atoms coming from the source. In Fig. \[fig:V2g0.01\](b,c,e–g) we can identify the appearance of two types of peaks (designated by arrows of different colors). As first type, we have two side peaks on the left- and right-hand side of the main peak at the incident beam energy, as seen for example in Fig. \[fig:V2g0.01\](b,g) (black arrows). This will be further discussed in Sec. \[subsec:collosc\]. The second type of peaks correspond to inelastic scattering processes of atoms that thereby undergo a transition between different single-particle levels within the atomic quantum dot. They can be seen in Fig. \[fig:V2g0.01\](c–f) (blue arrows) and will be discussed in Sec. \[subsec:bogolqd\].
Collective oscillations {#subsec:collosc}
-----------------------
To understand the appearance of the two side peaks in the immediate vicinity of the incident beam energy, we study a leaky and driven single-level model with energy $E_0$ and two-body interaction between atoms with an interaction strength $g$. In the Heisenberg picture, the evolution equation of the field operator $\hat{a}\equiv\hat{a}(t)$ related to the level reads $$\label{eq:SingLevQuanEvolEq}
i\hbar \frac{\partial \hat{a}}{\partial t} = ( E_0 - i\gamma/2)\hat{a} + g \hat{a}^\dagger\hat{a}\hat{a} + [\kappa \hat{b} + \hat{\xi}(t)]e^{-i\mu t/\hbar},$$ where $\hat{b}$ correspond to the annihilation operator of the source which is coupled to the level with a coupling strength $\kappa$. The use of an imaginary leaky term $i\gamma/2$ implies that the losses are Markovian, which is justified in the limit of weak coupling between the single-level system and the leads. As was discussed in Sec. \[subsec:tWSECS\], reducing the infinite waveguide to a finite open system introduces additional noise terms emerging from the initial vacuum fluctuations outside the quantum dot. Theses noise terms are accounted for by a time-dependent noise operator $\hat{\xi}(t)$ satisfying $$[\hat{\xi}(t), \hat{\xi}^\dagger(t)] = \xi_0^2\, \delta(t-t'),$$ for some $\xi_0\in\mathbb{R}$. For the sake of simplicity we consider here a white noise. The commutation relations for the bosonic field operators are given by
$$\begin{aligned}
[\hat{a}(0), \hat{a}^\dagger(0)] &= 1, \\
[\hat{b}, \hat{b}^\dagger] &= 1.
\end{aligned}$$
This model has the same ingredients as the atomic quantum dot system but offers the advantages to allow for analytical results.
For this particular system, we are interested in the appearance of side peaks near the resonance for a weak atom-atom interaction and large population of the single-particle level. As a consequence, the truncated Wigner evolution equation of the wavefunction $\psi\equiv\psi(t)$ can be written as $$\label{eq:SingLevClassEvolEq}
i\hbar \frac{\partial \psi}{\partial t} = ( E_0 + g|\psi|^2 - i\gamma/2)\psi + [\kappa\sqrt{N}+\xi(t)]e^{-i\mu t/\hbar},$$ where $N$ is the number of atoms in the source and the term $|\psi|^2-1$ is well approximated by $|\psi|^2$. The classical equivalent $\xi(t)$ of the quantum noise $\hat{\xi}(t)$ has following properties
$$\begin{aligned}
\overline{\xi(t)} &= 0, \\
\overline{\xi^*(t)\xi(t')} &= \frac{\xi_0^2}{2}\, \delta(t-t'),
\end{aligned}$$
in perfect analogy with the truncated Wigner prescription to sample the initial quantum state with classical fields.
Instead of determining the number of atoms at energy $E$ by means of a spatial Fourier transform in the transmitted beam, we define it through a temporal Laplace transform of the amplitude on the level under consideration. We define the Laplace transform as $$\label{eq:LaplaceTransform}
\tilde{\psi}(E) = \frac{1}{\sqrt{\hbar T}}\int_0^\infty \, \psi(t) \exp\left[ \left( \frac{1}{T} - i \frac{E}{\hbar} \right) t \right] dt,$$ for a fixed (and ideally very large) observation time $T$. The number of atoms at energy $E$ is calculated according to Eq. and reads $$\label{eq:tWObsPresc}
\langle n_E\rangle = \overline{|\tilde{\psi}(E)|^2} - \frac{1}{2}[\tilde{a}(E),\tilde{a}^\dagger(E)].$$
We are interested in collective oscillations of the condensate. For that purpose, we assume that we are close to a stationary state $\phi_0$ defined as the solution of the stationary GP equation $$(E_0-\mu- i\gamma/2 + g|\phi_0|^2) \phi_0 + \kappa\sqrt{N} = 0.$$ We then decompose the wavefunction $\psi(t)$ as $\psi(t)=(\phi_0 + \delta\psi(t))e^{-i\mu t / \hbar}$ and linearize the resulting evolution equation for $\delta\psi(t)$. We thereby obtain the Bogoliubov equations associated with Eq. which read $$\label{eq:BogolClassicalSysEq}
\left(
\begin{array}{cc}
\Sigma-E & g\phi_0^2 \\
-g\phi_0^{*^2} & -(\Sigma^*+E)
\end{array}
\right)
\left(
\begin{array}{c}
\delta\tilde{\psi}(E) \\
\delta\tilde{\psi}^*(-E)
\end{array}
\right)
=
\left(
\begin{array}{c}
-\tilde{\xi}(E) \\
\tilde{\xi}^*(-E)
\end{array}
\right)$$ after applying a Laplace transform according to Eq. , with $$\Sigma=E_0-\mu+2g|\phi_0|^2 - i \left(\frac{\gamma}{2}-\frac{\hbar}{T}\right).$$ Solving the system of equations , we find $$\overline{|\delta\tilde{\psi}(E)|^2} = \frac{(|\Sigma+E|^2 + g^2|\phi_0|^4)\xi_0^2 / 4\hbar}{|(\Sigma-E)(\Sigma^*+E) - g^2|\phi_0|^4|^2},$$ which yields $$\overline{|\tilde{\psi}(E)|^2} = \frac{1}{\hbar T}\frac{|\phi_0|^2}{T^{-2} + (E-\mu)^2/\hbar^2} + \overline{|\delta\tilde{\psi}(E-\mu)|^2}.$$ Following the same steps as in the previous lines, and supposing that $N\to\infty$, $\kappa\to0$ in such a way that $N|\kappa|^2$ remains constant, we can compute the commutator of Eq. , which is given by $$[\tilde{a}(E),\tilde{a}^\dagger(E)] = \frac{(|\Sigma+E|^2 - g^2|\phi_0|^4)\xi_0^2 / 2\hbar}{|(\Sigma-E)(\Sigma^*+E) - g^2|\phi_0|^4|^2}.$$ The total number of atoms at energy $E$ finally reads $$\begin{aligned}
\label{eq:bogolne}
\langle n_E \rangle &=& \frac{1}{\hbar T}\frac{|\phi_0|^2}{T^{-2} + (E-\mu)^2/\hbar^2} \nonumber \\
& & + \frac{g^2|\phi_0|^4\xi_0^2 / 2\hbar}{|(\Sigma-E)(\Sigma^*+E) - g^2|\phi_0|^4|^2}.\end{aligned}$$
In Fig. \[fig:BdG\_1site\], we plot $\langle n_E \rangle$ for $\mu/E_0=1.08$ for an observation time $E_0T=500\hbar$ and $\xi_0/E_0=0.5$. The interaction strength is set to $g/E_0=0.02$, the leak rate to $\gamma/E_0=0.001$ and the source of atom to $\sqrt{N}\kappa/E_0=0.05$. We directly see the spectral signature of collective oscillations for $\mu/E_0=1.08$ which is close to the nonlinear resonance (*i.e.* the population of the single-level system is high). This is in accordance with our previous findings for the quantum dot where collective oscillations appear near the resonances (see Fig. \[fig:BdG\_qdot\](a)). The occurrence of these side-peaks is, furthermore, in perfect qualitative agreement with the atom blockade study of Carusotto in Ref. [@Carusotto2001PRA].
\[r\]\[r\][$10^{\textrm{-}3}$]{} \[r\]\[r\][$10^{\textrm{-}2}$]{} \[r\]\[r\][$10^{\textrm{-}1}$]{} \[r\]\[r\][$10^0$]{} \[r\]\[r\][$10^1$]{} \[r\]\[r\][$10^2$]{} \[r\]\[r\][$10^3$]{} \[r\]\[r\][$10^4$]{} \[r\]\[r\][$10^5$]{} ![(color online) Average number of atoms $\langle n_E \rangle$ at energy $E$ for $g/E_0=0.02$, $\gamma/E_0=0.001$, $\sqrt{N}\kappa/E_0=0.05$, $E_0T=500\hbar$ and $\xi_0/E_0=0.5$. The presence of collective oscillations is clearly manifested in form of two side peaks appearing at $E-\mu\approx\pm0.36 E_0$. The (red) dots show the results obtained by numerically integrating Eq. and applying a Laplace transform according to Eq. . They are in perfect agreement with the theoretical prediction (black line) of Eq. .[]{data-label="fig:BdG_1site"}](eps/BdG_1site.eps "fig:"){width="\linewidth"}
Bogoliubov excitations in the quantum dot {#subsec:bogolqd}
-----------------------------------------
We are now interested in the Bogoliubov modes within the multi-mode quantum dot configuration that we focus on in this paper. To this end, we numerically solve the Bogoliubov equations defined with respect to the stationary solution of the effective GP-like equation . The stationary wavefunction of Eq. defined on the grid is given by $\phi_{0,l}$ on site $l$. We can solve the Bogoliubov equations $$\mathcal{T} \mathbf{y}^{(n)} = \epsilon_n \mathbf{y}^{(n)},$$ where $\epsilon_n$ is the n$^\textrm{th}$ eigenvalue and $\mathbf{y}^{(n)}$ the related eigenvector. The matrix $\mathcal{T}$ is defined as $$\mathcal{T} =
\begin{pmatrix}
\mathcal{L} && \mathcal{C} \\
-\mathcal{C}^* && -\mathcal{L}^*
\end{pmatrix},$$ with the matrix elements of $\mathcal{L}$ and $\mathcal{C}$ defined by $$\begin{aligned}
\mathcal{L}_{ll'} &=& (V_l-\mu q_l+2g_l|\phi_{0,l}|^2)\delta_{l,l'} \\
& & - J_{l'}(\delta_{l+1,l'} + \delta_{l-1,l'}) , \\
\mathcal{C}_{ll'} &= &g\phi_{0,l}^2 \delta_{l,l'},\end{aligned}$$ with $l,l^{'}= 0,1,\cdots, L$ and $$J_l = J \left[\frac{1}{q_{l}} -\frac{1}{2}\frac{q'_{l}}{ q^2_{l}}\right],$$ in the presence of SECS, see Eq. . Clearly, $\mathcal{T}$ is not hermitian. Hence, the corresponding eigenvalues $\epsilon_n$ are complex, and their imaginary part is related to the width of the corresponding resonance peak.
The numerically computed results are plotted for two different values of $\mu$ in Fig. \[fig:BdG\_qdot\]. The vertical black lines correspond to the expected Bogoliubov eigenenergies Re$(\epsilon_n)$ and the grey zones correspond to the expected width of the peaks given by $2$Im($\epsilon_n$). The upper panel shows the results for $\mu/2J=-0.86$ and we can see that collective oscillations appear within the quantum dot in agreement with the Bogoliubov theory. The lower panel corresponds to a chemical potential $\mu/2J=-0.04$ that is close to the energy corresponding to the 3rd resonance. It shows a richer structure of peaks arising from the superposition of collective oscillations and inelastic scattering. Indeed, two colliding atoms at the incident energy $\mu/2J=-0.04$ can exchange energy through a collision process. After the collision, the first atom can end up on the 4th energy level and the second can end up on the 2nd energy level as depicted in Fig. \[fig:BdG\_qdot\]. The results given in the tW calculation are in very good agreement with the Bogoliubov calculation.
[eps/V2g0.01\_peaks.eps]{}
Conclusions
===========
In the present work, we studied one-dimensional resonant transport of Bose–Einstein condensates within a guided atom laser configuration. For this purpose, we introduced a generalization of the truncated Wigner method to open systems. The reduction from an infinite system to a finite scattering region introduces an additional term accounting for quantum fluctuation which takes the form of a quantum noise. We made use of smooth exterior complex scaling to absorb the outgoing flux of atoms. This allowed us to study resonant and non-resonant transport across a one-dimensional atomic quantum dot beyond the mean–field Gross–Pitaevskii description.
The truncated Wigner method was used to compute the transmission across a quantum dot configuration. We observed that perfect resonant transmission is inhibited due to incoherent atoms creating a transmission blockade. This effect is in quantitative agreement with a Matrix-Product State calculation. The incoherent atoms originate from two different physical process. The first one is the creation of collective oscillations on an individual single-particle level within the quantum dot leading to two side peaks in the direct vicinity of the incident beam energy. The second one is related to inelastic collisions of atoms where atoms are transfered to other energy levels within the quantum dot.
The truncated Wigner method appears to be a very convenient tool to study transport of interacting Bose–Einstein condensates across more involved scattering configurations such as one–dimensional disordered potentials. This shall be discussed in a forthcoming publication [@Dujardin]. The approach presented in this paper can, furthermore, be extended to account for a more realistic description of the experimental configurations at hand involving, for instance, two reservoirs of $N$ atoms at ultralow but finite temperatures. This extension will then allow to simulate source-drain transport processes across quantum dot like configurations, paving the way to a realistic theoretical study of atomtronics devices or atomic transistors.
The authors want to thank Boris Nowak for fruitful discussions. Computational resources have been provided by the Consortium des Equipements de Calcul Intensif (CECI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11.
|
---
author:
- 'P. Salas[^1][^2]'
- 'M. A. Brentjens'
- 'D. D. Bordenave'
- 'J. B. R. Oonk'
- 'H. J. A. Röttgering'
bibliography:
- 'lofar\_holog.bib'
title: 'Tied-array holography with LOFAR'
---
[A radio interferometer uses time delays to maximize its response to radiation coming from a particular direction. These time delays compensate for differences in the time of arrival of the wavefront at the different elements of the interferometer, and for delays in the instrument’s signal chain. If the radio interferometer is operated as a phased array (tied array), the time delays cannot be accounted for after an observation, so they must be determined in advance.]{} [Our aim is to characterize the time delays between the stations in the core of the LOw Frequency ARray (LOFAR).]{} [We used radio holography to determine the time delays for the core stations of LOFAR (innermost $3.5$ km). Using the multibeaming capability of LOFAR we map the voltage beam faster than with a raster scan, while simultaneously calibrating the observed beam continuously.]{} [For short radio holographic observations ($60$ s and $600$ s) of 3C196, 3C147, and 3C48 we are able to derive time delays with errors of less than one nanosecond. After applying the derived time delays to the beamformer, the beam shows residuals of less than $20\%$ with respect to the theoretical beam shape.]{} [Tied-array holography could be a way towards semi-real-time beam calibration for the Square Kilometer Array.]{}
Introduction
============
A radio telescope works by combining the signals received by the elements that constitute its aperture (a reflecting surface in the case of a dish, or an array of antennas in the case of a phased array). In order to maximize the sensitivity of the telescope towards a particular direction, the signals arriving from that direction must be combined in phase, i.e., the time difference between the signals received by different aperture elements must be zero. In the case of a dish this is accomplished by shaping the reflecting surface in such a way that all the signals arrive at the receiver at the same time; in a phased array it is done by introducing instrumental time delays between its elements to compensate for the time of arrival of the signal at the antennas [e.g., @Thompson2017].
Deviations from a perfect phase alignment when the signals are combined lead to a loss in the efficiency of the telescope [e.g., @Ruze1952; @Ruze1966; @D'Addario2008]. These phase misalignments can be caused by the telescope itself, or they can be produced in the path between the source of the signals and the telescope. An example of the former are phase differences caused by misaligned panels in a reflector [e.g., @Baars2007] or by uncorrected cable delays in a phased array.
Different methods to reduce phase misalignments between the elements of an aperture have been developed. These include photogrammetric measurements [e.g., @Wiktowy2003], direct measurement of the aperture distribution [e.g., @Chen1998; @Naruse2009], holographic measurements [e.g., @Napier1973; @Bennett1976; @Scott1977; @Baars2007; @Hunter2011], and calibration using astronomical sources [e.g., @Fomalont1999; @Intema2009; @Thompson2017; @Rioja2018]. This paper focuses on the holographic measurement of the aperture illumination of a large phased array telescope.
Since the work of @Scott1977, holographic measurements have been used to calibrate the dishes of the Very Large Array [VLA, e.g., @Kesteven1993; @Broilo1993], the Atacama Large Millimeter Array [ALMA, e.g., @Baars2007], and the Green Bank Telescope [GBT, e.g., @Hunter2011]; to study the primary beam response of the Westerbork radio telescope [WSRT, e.g., @Popping2008] and the Allen telescope array dishes [ATA, e.g., @Harp2011]; and to characterize the beam and aperture of the LOw Frequency ARray (LOFAR) stations \citep[Brentjens et al. in prep.][]{}. All these measurements have been restricted to the study of apertures $\lesssim100$ m in diameter.
In the regime of low frequencies and large apertures, holographic measurements are particularly challenging. At low frequencies the ionosphere will introduce additional time delays depending on its total electron content [TEC, e.g., @Intema2009]. To accurately measure the intrinsic phase errors between the elements of the phased array without ionospheric distortion, the phased array must be smaller than the diffractive scale of the ionosphere. Night time observations of the ionosphere at $150$ MHz show that its diffractive scale is between $30$ and $3$ km [@Mevius2016].
LOFAR operates at frequencies between $10$ MHz and $240$ MHz [@vanHaarlem2013]. This frequency range is covered by two different types of antennas: low band antennas (LBA, $10$–$90$ MHz) and high band antennas (HBA, $120$–$240$ MHz). The HBA antennas are combined in a $4\times4$ tile with an analog beamformer. The antennas and tiles are grouped into stations, and the stations are further combined to form an array. For the core stations of LOFAR, the LBA stations consist of $96$ antennas, while the HBA stations have $48$ tiles split into two fields. Of the $96$ antennas in a core LBA station the available electronics permits only $48$ to be actively beamformed. There are $24$ stations in the core of LOFAR. The core stations are connected via fiber to a central clock, thus their signals can be added coherently to form a telescope with a maximum baseline of $3.5$ km. The stations in the innermost $350$ m are known as the *Superterp*.
Each LOFAR LBA dipole observes the entire sky, while the HBA tiles have a field of view (FoV) of $30\degr$ at $150$ MHz. Since the signals from the antennas and tiles are combined digitally, the stations can simultaneously point in multiple directions within their FoV [e.g., @Barton1980; @Steyskal1987]. When the signals from different stations are added together coherently, a phased array (known as a tied array) is formed. This enables LOFAR to form multiple tied-array beams (TABs) that point in different directions.
Method {#sec:method}
======
We want to determine the time delays for the array formed by the stations in LOFAR’s core. We refer to the tied array formed by these stations as the array under test (AUT). In order to determine the time delays, we start from a map of its complex-valued beam $B$. The basic procedure used to measure $B$ is the same as that employed by @Scott1977, with a difference in its implementation. In their work, a raster scan was used to map the region around the bright unresolved source. Here, we take advantage of LOFAR’s multi-beaming capability to map the region around the bright unresolved source. Using multiple TABs the whole region is mapped simultaneously, and there is always a TAB pointing towards the bright unresolved source. In addition to speeding up the process by a factor equal to the number of simultaneous beams, this allows continuous calibration of the AUT and the reference stations by always having a TAB at the central calibrator source.
At the frequencies at which LOFAR operates, the Milky Way is bright and it will distort the observed map of $B$. To reduce the contribution from the Milky Way to the measurements, we use a reference station to produce a baseline that resolves out large-scale Galactic structure [e.g., @Colegate2015]. The contribution from smaller bright sources (e.g., Cassiopeia A or Cygnus A) cannot be completely resolved out, and is reduced by limiting the field of view (FoV) through time and frequency smearing [e.g., @Bridle1999]. Moreover, the AUT and the reference station “see” different portions of the ionosphere, which will introduce an additional time delay between them. The effects of the different ionosphere seen by the AUT and the reference station are calibrated using the bright point source.
---------------- --------- ------------ ------------------------------ ---------- --------- ------------- ----------- --------
Observation ID Antenna \# antenna Start time Duration \# TABs TAB spacing FoV Source
fields (s) (arcmin) (degrees)
L658168 HBA 46 June $14$ $13$:$40$:$00$ UT 60 $169$ $1.6$ $0.37$ 3C147
L658158 HBA 46 June $14$ $13$:$30$:$00$ UT 60 $169$ $1.6$ $0.37$ 3C196
L650445 LBA 24 April $19$ $09$:$20$:$00$ UT 600 $271$ $5$ $1.32$ 3C48
L645357 LBA 24 March $20$ $19$:$45$:$00$ UT 600 $271$ $5$ $1.32$ 3C196
---------------- --------- ------------ ------------------------------ ---------- --------- ------------- ----------- --------
Following the measurement equation formalism [@Hamaker2000], we obtain the visibility generated by cross-correlating the signals from the AUT and the reference station as $$V_{b}^{}=J_{\mathrm{AUT},b}^{}EJ_{\mathrm{ref}}^{\dagger}\delta_{b,\mathrm{ref}},$$ where $E$ represents the coherency matrix formed by the pure sky visibilities, $J_{\mathrm{AUT},b}$ and $J_{\mathrm{ref}}$ are respectively the Jones matrices [@Jones1941] of the AUT and the reference station, the subscript $b$ represents the TABs formed with the AUT, the $\dagger$ symbol denotes taking the conjugate transpose of the corresponding matrix, and $\delta_{b,\mathrm{ref}}$ is the Kroneker delta-function due to the spatial dependence of the product. The calibration consists of finding the inverse of the visibility of the central TAB, $V_{c}^{-1}$, and right multiplying all the visibilities with it. This is possible since $V_{c}$ is non-singular, as $E$ is non-singular by definition and the AUT measures two orthogonal polarizations. After this, for the central beam $\tilde{V}^{}_{b=c}=V^{}_{c}V^{-1}_{c}=1$, where $1$ represents the identity matrix. For the remaining directions $\tilde{V}^{}_{i}=V^{}_{i}V^{-1}_{c}=J^{}_{\mathrm{AUT},i}J^{-1}_{\mathrm{AUT},c}$. This means that the calibrated visibility for the $i$-th beam only depends on the Jones matrix of the AUT, and not on the sky brightness distribution. This relation holds if the sky coherency matrix is that of a single point-like source [e.g., @Smirnov2011]. The calibrated visibilities map $B$. The details behind the calibration method will be presented in Brentjens et al. (in prep.).
From the observed map of $B$ we determine the amplitude and phase over the aperture of the AUT, $A$. In the far-field approximation, and for a coplanar array, they are related by [e.g., @D'Addario1982; @Baars2007; @Thompson2017], $$B(l,m)\propto\iint A(p,q)e^{2\pi i(pl+qm)\frac{\nu}{c}}dpdq,
\label{eq:ft}$$ where $i$ denotes the imaginary unit, $c$ is the speed of light, $\nu$ is frequency, $p$ and $q$ are orthogonal coordinates in the aperture plane, and $l$ and $m$ are the direction cosines measured with respect to $p$ and $q$. For LOFAR, the $(p,q)$ coordinate system has its origin at the center of the aperture and it lies in the plane of the station, or in this case the plane of the *Superterp* stations. The phase of $A(p,q)$ is set, for example, by uncalibrated errors in the clock distribution, cable length, antenna position, and ionospheric phase variations across the aperture.
Observations {#sec:observations}
============
LOFAR holography observations {#ssec:lofobs}
-----------------------------
Table \[tab:obs\] summarizes the observations. Each TAB recorded complex voltages in two orthogonal polarizations (X and Y) at $5.12$ $\mu$s time resolution in ten spectral windows $195.3125$ kHz in width each. The data were subsequently ingested into the LOFAR long-term archive. The calibrator sources are selected to be small compared to the size of the TAB, and compared to the fringe spacing of the baselines between the AUT and the reference stations. The former prevents systematic distortions in the measured beam, while the latter guarantees high signal-to-noise ratios (S/N) on the baselines towards the reference stations.
The complex valued beam maps were measured on a regular hexagonal grid, $1.32$ and $0.37$ deg across for LBA and HBA, respectively. The map size is limited by the number of TABs, spectral windows and stations that the beamformer, COBALT [@Broekema2018], can process simultaneously. Per Fourier relation Equation \[eq:ft\] this implies a spatial resolution in the aperture plane of $270$ m (HBA at $174$ MHz) and $170$ m (LBA at $68$ MHz), comparable to the diameter of the *Superterp* ($350$ m). The separation between TABs was set at $\lambda/D$ at the highest frequency, and kept constant for lower frequencies. This maximizes the FoV while avoiding the overlap of aliasing artifacts with the AUT in the aperture plane, and enables simultaneous observations at different frequencies.
The required integration time is set by the error on the phase in the aperture plane, $\Delta\phi$, [@D'Addario1982] $$\Delta\phi\approx\dfrac{\pi D}{4\sqrt{2}d\mathrm{S/N}_{\mathrm{bm}}},
\label{eq:dphi}$$ where $D$ is the telescope diameter, $d$ the spatial resolution on the aperture plane, and $\mathrm{S/N}_{\mathrm{bm}}$ the signal-to-noise ratio in the complex-valued beam map; in other words, the ratio of the peak response of the array to the root mean square (rms) over the complex-valued beam map, $\mathrm{S/N}_{\mathrm{bm}}=I/\sigma$. Sigma can be estimated as [e.g., @Napier1982] $$\sigma=\sqrt{(\mbox{SEFD}_\mathrm{CS}/N_\mathrm{CS})(\mbox{SEFD}_\mathrm{RS})}/(\sqrt{\Delta\nu\Delta t}),$$ where $N_\mathrm{CS}$ is the number of stations in the AUT; $\mbox{SEFD}_\mathrm{CS}$ and $\mbox{SEFD}_\mathrm{RS}$ are the system equivalent flux density (SEFD) of a core station and a reference station, respectively; $\Delta\nu$ is the bandwdith; and $\Delta t$ the integration time. For the LBA the SEFD of each antenna field is $\approx30$ kJy at $60$ MHz and for the HBA $\approx3$ kJy at $150$ MHz [@vanHaarlem2013]. Thus, for an observation of 3C196 with integration times of $600$ s at $60$ MHz and $60$ s at $150$ MHz we can determine the time delays with errors of $1.8$ ns and $0.4$ ns, respectively.
From raw voltages to beam maps
------------------------------
To obtain a complex-valued map of the array beam we cross-correlate the voltage from the AUT with that of the reference station. This is done using an FX correlator [e.g., @Thompson2017] implemented in *python*. To channelize the time series data from each spectral window we use a polyphase filter bank [PFB, e.g., @Price2016] with a Hann window to alleviate spectral leakage and scalloping losses. We produce spectra of $64$ $3$ kHz channels, and a time resolution of $327.68$ $\mu$s. This enables us to, at a later stage, flag narrowband radio frequency interference (RFI) without flagging the entire time sample. In each spectral window we discard $25\%$ of the channels at the edges, leaving a bandwidth of $146.48$ kHz per spectral window. The two orthogonal polarizations are combined to produce four cross-correlation products, i.e., XX, XY, YX, and YY products.
Before proceeding, we check that the bright unresolved source in the map center dominates the signal. In this case, a time delay versus fringe rate plot will show a peaked response in the center of the diagram. An example of such a diagram is presented in Figure \[fig:waterfalls\].
After cross-correlation, the visibilities are time averaged to ensure that their S/N is high enough ($>3$) for calibration. For the HBA observations we average to a time resolution of $0.4$ s, which results in a S/N of $6$. For the LBA, which has a lower sensitivity and is more severely affected by the ionosphere (Figure \[fig:b0phase\]), the averaging times are longer. For L645357 we average to $20$ s and for L650445 — $5$ s.
After time averaging, we remove visibilities affected by RFI in the frequency-time domain. We use a SumThreshold method [AOFlagger, -@Offringa2012] on each TAB, polarization, and spectral window independently. For the LBA and HBA observations the fraction of flagged data is $\approx5\%$. After RFI flagging we average each spectral window in frequency to a single $146.48$ kHz channel.
The amount of time and bandwidth smearing on the visibility measured by a baseline can be approximated by [e.g., @Smirnov2011; @Thompson2017] $$\langle V\rangle=V\operatorname{sinc}(\Delta\Psi)\operatorname{sinc}(\Delta\Phi),$$ where $\Delta\Psi=\pi\theta_{\mathrm{s}}\Delta\nu/(\theta_{\mathrm{b}}\nu)$, $\Delta\Phi=\pi\theta_{\mathrm{s}}\omega_{\mathrm{e}}\Delta t/\theta_{\mathrm{b}}$, $\theta_{\mathrm{s}}$ is the distance from the array’s phase center, $\theta_{\mathrm{b}}$ is the size of the synthesized beam formed by the baseline, and $\omega_{\mathrm{e}}$ is the Earth’s rotational angular velocity ($7.2921159\times^{-5}$ radians s$^{-1}$). Then, for a baseline of $52$ km, a bandwidth of $146.48$ kHz, an integration time of $0.4$ s, and a frequency of $115$ MHz ($\theta_{\mathrm{b}}\approx10\arcsec$) time and bandwidth smearing reduce the amplitude of Taurus A (the closest A-team source to 3C147) by $1.7\times10^{-4}$. For an integration time of $5$ s, an observing frequency of $37$ MHz, and the same baseline and bandwidth the amplitude of Cassiopeia A (closest A-team source to 3C48) is smeared by $2.3\times10^{-4}$.
The flagged and averaged visibilities are then calibrated by multiplying by the inverse of the Jones matrix of the central beam. This has the effect of removing most of the undesired systematic effects present in the data, such as the dependence of the observed visibilities on the sky brightness distribution, beam pattern of the reference station, or ionospheric delays between reference station and AUT.
After calibration, we further average the visibilities in time to one time sample with a duration of one minute for the HBA and ten minutes for the LBA (Table \[tab:obs\]). After averaging in time, we are left with one calibrated complex visibility for each polarization (XX, XY, YX, and YY), spectral window, and TAB. These calibrated complex visibilities map the complex-valued beam. Finally, we compute the inverse-variance weighted mean beam maps, averaged over all reference stations.
Results {#sec:results}
=======
Beam and aperture maps {#ssec:bamaps}
----------------------
An example of the observed beam of the LBA is presented in the top left panel of Figure \[fig:bmap\_lba\]. There the main lobe of the beam is at the map center, and we can also see that there is a side lobe with a similar amplitude at $(l,m)=(-6,25)$. This is produced by improperly calibrated time delays between stations. For the HBA (top left panel of Figure \[fig:bmap\_hba\]) the side lobes have amplitudes $\approx30\%$ of the main lobe. The time delays between HBA stations are regularly calibrated using synthesis imaging observations.
The voltage beam is the Fourier transform of the aperture illumination (Eq. \[eq:ft\]), shown in the bottom left panel of Figures \[fig:bmap\_lba\] and \[fig:bmap\_hba\]. There we can see that the amplitudes are non-zero at the location of the stations in the AUT. The amplitudes are larger in the *Superterp* because there the stations are unresolved and their amplitudes, and phases, overlap.
Time delays and $0$ Hz phase offsets {#ssec:tau}
------------------------------------
To measure the phase of the stations in the AUT we pose Equation \[eq:ft\] as a linear problem, i.e., B=Ax with $$A=\begin{pmatrix}
\exp[2\pi i(p_{s}l_{j}+q_{s}m_{j})] & \dots & \exp[2\pi i(p_{N_\mathrm{CS}}l_{j}+q_{N_\mathrm{CS}}m_{j})]\\
\vdots & \ddots & \vdots \\
\exp[2\pi i(p_{s}l_{N}+q_{s}m_{N})] & \dots & \exp[2\pi i(p_{N_\mathrm{CS}}l_{N}+q_{N_\mathrm{CS}}m_{N})]\\
\end{pmatrix},$$where $N$ is the number of TABs and x is a complex vector whose argument is the phase of each station, $\phi$. The linear complex problem is recast to a real problem following @Militaru2012. Then, we use least squares parameter estimation to determine the amplitude and phase at the locations of the stations. The phases derived are not meaningful on their own, as an interferometer only measures relative phases [e.g., @Jennison1958]. To remove the arbitrary offset from the phases we reference them with respect to one of the stations in the AUT.
From the referenced phases we can recover the time delay, $\tau$, and the $0$ Hz phase offset, $\phi_{0}$, of each station. These are related to the phase by the linear relation $\phi=2\pi\nu\tau+\phi_{0}$. An example of the observed phases and their best fit linear relation are presented in Figure \[fig:phaseeg\]. There we can see that the phases show a linear relation with frequency and that the error bars on the phases become larger for stations closer to the array center.
Examples of the measured $\tau$ and $\phi_{0}$ for the HBA stations derived from the L658168 observations are shown in Figure \[fig:dgains\_hba\]. For the *Superterp* stations, CS$002$ to CS$007$, the error bars are a factor of three larger than for the rest of the stations. This is a consequence of the larger phase errors obtained for the *Superterp* stations (see Figure \[fig:phaseeg\]). This is also reflected in the larger aperture residuals at the *Superterp* (bottom right panel of Figure \[fig:bmap\_hba\]). For the stations outside the *Superterp*, the errors on $\tau$ have a mean value of $1.4\pm1.2$ ns and $1.2\pm0.9$ ns for the XX and YY polarizations, respectively. For the observation L658158 the same stations have errors on $\tau$ with a mean of $3.9\pm1.7$ ns and $3.7\pm1.7$ ns for the XX and YY polarizations, respectively. Since the flux density of 3C196 is a factor of $1.2$ higher than that of 3C147, the larger errors on $\tau$ for L658158 are produced by the larger phase fluctuations in this observation (Figure \[fig:b0phase\]).
The measured time delays and $0$ Hz phase offsets for the LBA stations derived from L645357 and L650445 are presented in Figure \[fig:dgains\_lba\]. In both observations the derived values of $\tau$ and $\phi_{0}$ agree to within $3\sigma$, even though in L645357 the S/N is higher by a factor of $12$; for the L645357 observations the ionosphere over the array produces a smooth slow time-varying phase rotation, while for L650445 the changes are faster and more pronounced (Figure \[fig:b0phase\]). The time delays for the *Superterp* stations have errors that are a factor of four larger than for the rest of the stations.
For both HBA and LBA (Figures \[fig:dgains\_hba\] and \[fig:dgains\_lba\]), the $0$ Hz phase offsets are consistent with being zero at the $5\sigma$ level. Motivated by this, we fit a linear relation to the phases with $\phi_{0}=0$. The values of $\tau$ for the LBA stations under this assumption are presented in Figure \[fig:tau\_lba\_fix\]. We can see that the derived time delays are consistent with those presented in Figure \[fig:dgains\_lba\], but in this case the error bars are smaller because there is one less free parameter and setting $\phi_{0}=0$ is a strong constraint. Using $\phi_{0}=0$ the mean value of the error of the derived time delays is $0.26\pm0.16$ ns and $0.17\pm0.10$ ns for the HBA and LBA, respectively.
In Figure \[fig:tau\_lba\_fix\] we also show the time delays for the LBA stations derived from imaging observations. In imaging observations the phases for each station are derived from observations of a bright calibrator source and a model of the sky brightness distribution [e.g., @Fomalont1999]. Then the contribution to the phase from the station delays and the ionosphere are separated [e.g., @vanWeeren2016; @deGasperin2018; @deGasperin2019]. We see that the time delays derived using holography and imaging observations agree to within $3\sigma$ for $20$ out of the $23$ stations present in both observations. This shows that the time delays derived here, where no model of the sky brightness distribution is used, are indistinguishable from those derived in imaging observations. The interferometric time delays have smaller error bars because they are obtained using $488$ $195.3125$ kHz spectral windows.
We check that the derived time delays and $0$ Hz phase offsets capture the status of the AUT by using them to simulate the array beam and comparing it with the observed beam. To simulate the array beam we use the time delays and $0$ Hz phase offsets shown in Figures \[fig:dgains\_hba\] and \[fig:dgains\_lba\] for the HBA and LBA stations, respectively. These are used to evaluate the phase of each station at a particular frequency, producing a complex valued map of the AUT. The Fourier transform of the AUT simulates the array beam. The residuals between the observed and simulated array beams are presented in the top right panel of Figures \[fig:bmap\_lba\] and \[fig:bmap\_hba\]. The residuals in the image plane have no obvious structure and show amplitudes of $\lesssim10\%$. This shows that we can reproduce the array beam using the derived time delays and $0$ Hz phase offsets. However, the Fourier transform of the beam residuals reveals that there is significant structure in the aperture plane (bottom right panel of Figures \[fig:bmap\_lba\] and \[fig:bmap\_hba\]). This can be seen as a larger amplitude ($11\%$ for the LBA) at the location of the *Superterp*, for which we are not capturing the phase behavior as accurately as for the stations away from it (see also Figure \[fig:phaseeg\]).
Discussion {#sec:discussion}
==========
Corrected time delays {#ssec:cortau}
---------------------
We use the derived time delays (Figure \[fig:tau\_lba\_fix\]) to update the instrumental time delays in LOFAR’s beamformer. To test the effect of updating the instrumental time delays in the beamformer we observed Cygnus A with the core stations of LOFAR. The observation was one minute long using the imaging mode, where the signals of different stations are cross-correlated instead of added. Since the signal path between stations and the beamformer is the same in tied-array and imaging modes, these observations have a beam equivalent to the one observed using holography. Cygnus A has a size of $\approx1\arcmin$ [e.g., @McKean2016], so it will be unresolved by the LOFAR core at LBA frequencies (at $90$ MHz the spatial resolution of the core is $3\farcm3$). Hence, a dirty image obtained from this observation will show the array beam.
A comparison between the LBA beam after the update and a model of the beam is presented in Figure \[fig:corrbeam\]. In the beam model the phase of each station is given by its location with no additional time delays. We see that the sidelobes in the observed beam are similar to those of the beam model. After subtracting the beam model the residuals are $\lesssim20\%$.
We compare the S/N of the observations of Cygnus A with the theoretical S/N in Figure \[fig:snr\_ratio\]. The ratio of the observed S/N to its theoretical value has a mean of $0.88\pm0.06$. These results show that after updating the time delays in the beamformer the array beam is close to the ideal case. The remaining differences are produced by propagation delays introduced by the ionosphere and any remaining errors in the instrumental time delays.
When using an unpolarized bright unresolved source to determine the time delays, there will be an arbitrary offset between the X and Y polarizations. When these time delays are applied to the beamformer, the offset will produce a rotation of Stokes $U$ into Stokes $V$. In order to find this offset we need to observe a linearly polarized source for which the sign of the rotation measure is known. From this observation the offset is determined from the angle between the apparent Stokes $U$ and $V$ [e.g., @Brentjens2008]. This step has not been performed yet.
Comparison to other methods
---------------------------
Previous to holography LOFAR used interferometric observations to derive time delays between its stations using the methods described by @Wijnholds2009. In these interferometric array-calibration observations pairs of antennas were cross-correlated using the station correlator. The antenna gains were derived by calibrating against multiple calibrator sources in their FoV. The observations lasted $6$ hours and $24$ hours for the HBA and LBA, respectively. Variations in the sky brightness distribution due to the ionosphere were partially averaged out during the observations [e.g., @Wijnholds2011]. By comparison, holography requires $1$ minute and $10$ minutes of observations for the HBA and LBA, respectively, and it does not require the use of a sky model.
Time delays between stations are also derived during imaging observations. In the case of Figure \[fig:tau\_lba\_fix\] the time delays derived from imaging observations have smaller error bars because they are derived from $488$ $195.3125$ kHz spectral windows. If we derive time delays from the imaging observations using the same ten spectral windows as for the holographic observations, then the errors on the time delays have a difference of less than $5\%$. This makes holography a competitive alternative, as its accuracy can be scaled up by adding more reference stations and more spectral windows. For the latter an increase in computing power is required.
The method presented here is also used to calibrate the antennas within a LOFAR station. For this calibration a station is under test (instead of the AUT). The station under test generates multiple station-beams to map its beam, while another station acts as reference. The complex voltages from the station under test and reference are then cross-correlated and calibrated following the same procedure as that outlined here. From the calibrated complex visibilities the complex gains for each antenna within a station are derived.
Improvements to holographic measurements with LOFAR
---------------------------------------------------
One of the main limitations of the holographic measurements presented is the spatial resolution over the telescope aperture. For a constant number of TABs this can be improved by observing a larger portion of the beam using a mosaic while keeping a TAB at the calibrator source. Additionally, the separation between TABs could be made smaller, reducing aliasing artifacts in the aperture plane.
For the experiments presented in this work we used only four reference stations. Outside its core, LOFAR has $14$ stations within the Netherlands and $13$ stations distributed all over Europe. Any of these stations can be used as reference station, as long as the baseline formed with the AUT does not resolve the source used to map the beam. This means that there can be an improvement in the S/N of the complex-valued beam map of up to a factor of four using the same sources. With this level of improvement in the maps of the complex-valued beam, the integration times could be made shorter or more precise time delays could be derived.
The time delays derived from holographic measurements can be used to update the instrumental time delays prior to an observation with the tied array (e.g., of a pulsar). Moreover, if the integration time required to reach nanosecond precision could be made shorter, and the post-processing of the holographic measurements could be done in real time, then it would be possible to interleave holographic observations during the tied-array observations. This could be a way towards semi-real-time beam calibration for the Square Kilometer Array (SKA). Implementing a dedicated holography mode in the supercomputer that processes the raw LOFAR data is one of the next steps towards this goal.
Summary {#sec:summary}
=======
In this work we used radio holography along with a new calibration method to characterize the time delays between LOFAR’s core stations. This new calibration method consists in calibrating the measured complex-valued beam map by right multiplying by the matrix inverse of the map center. This calibration makes the observed complex-valued reception pattern independent of the sky brightness distribution.
Four HBA and three LBA reference stations were used simultaneously to produce maps of the tied-array voltage beam. Using $60$ s (HBA) and $600$ s (LBA) long observations of 3C196, 3C147, and 3C48 we derived time delays with an error $<1$ ns. We find that the main limitations in reaching nanosecond precision in the measured time delays are the condition of the ionosphere over the array and the ability to spatially resolve the array elements. LOFAR now uses the derived time delays operationally.
P. S., J. B. R. O. and H. J. A. R. acknowledge financial support from the Dutch Science Organisation (NWO) through TOP grant 614.001.351. This research made use of Astropy, a community-developed core Python package for Astronomy [@Astropy2013], and matplotlib, a Python library for publication quality graphics [@Hunter2007].
[^1]: psalas@nrao.edu
[^2]: Current address: Green Bank Observatory, Green Bank, WV 24944, USA
|
---
abstract: |
We present a near-infrared cooled grating spectrometer that has been developed at the Arcetri Astrophysical Observatory for the 1.5 m Infrared Telescope at Gornergrat (TIRGO).
The spectrometer is equipped with cooled reflective optics and a grating in Littrow configuration. The detector is an engineering grade Rockwell NICMOS3 array (256$\times$256 pixels of $40 \mu$m). The scale on the focal plane is 1.73 arcsec/pixel and the field of view along the slit is 70 arcsec. The accessible spectral range is $0.95-2.5\mu$m with a dispersion, at first order, of about 11.5 [Å]{}/pixel. This paper presents a complete description of the instrument, including its optics and cryo-mechanical system, along with astronomical results from test observations, started in 1994. Since January 1996, LonGSp is offered to TIRGO users and employed in several Galactic and extragalactic programs.
author:
- 'L. Vanzi, M. Sozzi, G. Marcucci, A. Marconi, F. Mannucci, F. Lisi, L. Hunt, E. Giani, S. Gennari, V. Biliotti, and C. Baffa'
date: 'Received ...'
title: 'LonGSp: the Gornergrat Longslit Infrared Spectrometer '
---
2.0cm
Introduction
============
The development of the spectrometer LonGSp (Longslit Gornergrat Spectrometer) was part of a project aimed at providing the 1.5-m Infrared Telescope at Gornergrat (TIRGO) with a new series of instruments based on NICMOS3 detectors. The infrared (IR) camera, ARNICA, developed in the context of this project is described in Lisi et al. (1995). LonGSp is an upgrade of GoSpec (Lisi et al. 1990), the IR spectrometer operating at TIRGO since October 1988. Thanks to the NICMOS detector, the new spectrometer enables longslit spectroscopy with background limited performance (BLP). The GoSpec characteristics of compactness and simplicity are maintained in the new instrument. Only a subsection of the engineering grade array (40$\times$256 pixels) is used. A description of the optics, cryogenics, and mechanics is presented in Section \[opt\] and \[cry\]; the electronics, software and the performance of the detector are presented in Section \[elect\] and \[detec\]. Finally, in Section \[obs\] we present details regarding the observations and data reduction, and in Section \[astro\] the results of the first tests at the telescope.
Optical Design {#opt}
==============
The optical scheme of the instrument is sketched in Fig. 1; it is designed to match the f/20 focal ratio of the TIRGO telescope.
Following the optical path from the telescope, the beam encounters the window of the dewar, the order sorting filter, a field lens, and the slit; the latter resides at the focal plane of telescope. The window and field lens are composed of calcium fluoride. Filters and slits are respectively mounted on two wheels and can be quickly changed during the observations. The field lens images the pupil on the secondary mirror of an inverted cassegrain (with focal length of 1400 mm) that produces a parallel beam 70 mm in diameter. This beam is reflected onto the grating by a flat mirror tilted by $10^{\circ}$. The grating, arranged in Littrow configuration, has 150 grooves/mm and a blaze wavelength of $2 \mu$m at first order; rotation around the $10^{\circ}$ tilted axis allows the selection of wavelengths and orders. A modified Pfund camera (with focal length of 225 mm) following the grating, collects the dispersed beams on the detector. The sky-projected pixel size is 1.73 arcsec, and the total field covered along the slit direction is 70 arcsec.
The back face of the grating is a flat mirror so that, when the grating is rotated by 180 degrees, the instrument functions as a camera, in the band defined by the filters, with a field of view of about 1.5 arcmin square. This facility can be useful for tests, maintenance, and for centering weak sources on the slit.
All the mirrors are gold coated to provide good efficiency over a wide spectral range, and the optics are acromatic at least up to $5\mu$m. The optical components are cooled to about 80 K by means of thermal contact with a cryogenic vessel filled with liquid nitrogen at atmospheric pressure as described below. The mounting of optical elements is designed to take into account the dimensional changes between mirrors (in pyrex) and supports (in aluminium) generated by the cooling and the differences in thermal expansion coefficents.
The resolving power is (for first order) about 600 in the center of J band, and 950 in the center of the K band, using a slit of two pixels (3.46 arcsec).
Cryogenics and Mechanics {#cry}
========================
As can be seen in Fig. 2, where we show some parts of the mechanical structure of the instrument, the core of the instrument is the liquid nitrogen reservoir, which has a toroidal shape with rectangular cross section and a capacity of 3 liters. It provides support and cooling for two optical benches, which are located on opposite sides of the vessel. The central hole of the toroid allows the beam to pass from one optical bench to the other.
The grating motion is assured by an external stepper motor via a ferrofluidic feedthrough, and the position is controlled by an encoder connected to the motor axis outside the dewar. Two springs acting on the worm gear guarantee good stability of the grating position. Two internal stepper motors, modified to operate at cryogenic temperatures (Gennari et al. 1993), drive the filter and slit wheels.
Mechanics and optics are enclosed in a radiation shield. The internal cold structure is supported by nine low-thermal-conductivity rods, which are fixed between the internal liquid nitrogen reservoir and the external vacuum shield, and are rigidly linked to the focal plane adaptor of the telescope. Externally, the instrument has the form of a cylinder with a base of about 40 cm in diameter and length of about 60 cm.
A small amount of active charcoal is present to maintain the value of the pressure required (less than 10$^{-4}$ mb) for a sufficiently long time (more than 20 days). The charcoal is cooled by an independent cryogenic system; in the rear optical bench there is a smaller nitrogen vessel ($\sim$ 0.5 l) thermally insulated from the surrounding environment. The regeneration of the charcoal must be carried out once a month in order to maintain a sufficiently high absorption rate. This operation consists of heating the charcoal to 300 K, while the pressure inside the dewar is maintained below 10$^{-1}$ mb by means of a rotary vacuum pump. Because the charcoal is cooled by an independent cryogenic system, the heating of the optics and the main part of the mechanical structure is not necessary and the operation can be completed in about four hours.
To cool and warm the entire instrument reasonably quickly, the dewar is filled with gaseous nitrogen at a pressure of about 200 mb during the cooling and heating phases: in this way the thermal transients prove to be shorter than seven hours. The rate of evaporation of the nitrogen from the main reservoir allows about 16 hours of operation in working conditions, more than a winter night of observation.
Electronics and Software {#elect}
========================
The electronics of LonGSp comprise two main parts: “upper” electronics, that are situated near the instrument, and the “lower” electronics in the control room. The connection between the two parts is assured via a fiber-optics link. Two boards, close to the cryostat, house part of the interface electronics, that is a set of four preamplifiers and level shifters and an array of drivers and filters that feed the clock signals and the bias to the array multiplexer.
The “upper” electronics are composed of an intelligent multi–part sequence generator and a data acquisition section. The first is controlled by a microprocessor (a Rockwell 65C02), and the sequencer is capable of generating many different waveforms template (at 8 bits depth) stored in an array of 128 Kbytes of memory. The final waveform is generated by selecting, via software, the templates needed together with their repetitions.
The data acquisition segment consists of a bank of four analog-to-digital converters at 16 bits, and the logic for converting them to serial format. A transceiver sends data to the telescope control room through a fiber optics link. Data are sent as groups of four, one for each quadrant, and are presented together with the quadrant identifier (two bits) to the frame grabber. The fiber optics link is bidirectional, so that it is possible to send instructions to the control microprocessor in the “upper” electronics, and to communicate with the motor control through a serial connection (RS-232) encoded on the same fiber-optics link. The “upper” electronics are completed by the box which contains the power supply, the stepper motors controllers, and the temperature controller of the array.
The “lower” electronics implement the logic to decode the serial data protocol, in order to correctly reconstruct the frame coming from the array detector. Data are collected by the custom frame grabber (known as the “PingPong”) which is capable of acquiring up to four images in each of its two banks. When a bank is written, the other can be read, enabling continuous fast acquisitions. Also of note is the ability to re-synchronize data acquisition to the quadrant address, virtually eliminating mis–aligned frames.
The instrument is controlled by an MS-DOS PC equipped with a 80486 CPU (33 MHz clock), high-resolution monitor, and 600 Mbytes of hard disk space. At the end of a data acquisition sequence, each single frame or the stack average of a group of frames is stored on the PC hard disk, and are later transferred to optical disk (WORM) storage. In the near future, the WORM cartridges will be superseded by more standard writable CD-ROM cartridges. A local Ethernet network connects the control computer to the TIRGO Sun workstation, so that it is possible to transfer the data for preliminary reduction using standard packages.
The software developed for this instrument is “layer organized”, that is to say organized as a stack of many layers of subroutines of similar levels of complexity. To accomplish its task, each routine need rely only on the immediately adjacent level and on global utility packages. Such a structure greatly simplifies the development and maintenance of the software.
Our efforts were directed towards several different requirements. Our first priority was to have a flexible laboratory and telescope engine, capable of acquiring easily the large quantity of data a panoramic IR array can produce. The human interface is realized through a fast character-based menu interface. The operator is presented only with the options which are currently selectable, and the menu is rearranged on the basis of user choices or operations.
We have also stressed the auto–documentation of data. After the decision to store data in standard FITS format, it was deemed useful to fully exploit the header facility to label each frame with all relevant information, such as telescope status, instrument status, and user acquisition choices. Data are also labelled with the observer name in order to facilitate data retrieval from our permanent archive. In particular, the form of the FITS file is completely compatible with the context IRSPEC of the ESO package MIDAS.
Finally, one of our main goals was to produce an easy-to-use software and with the smallest “learning curve”. Our idea was that data acquisition must [*disappear*]{} from observer attention, giving him/her the possibility to concentrate on the details of the observations; in this way, observing efficiency is much higher. As a result, we have implemented automatic procedures such as multi–position (“mosaic”), and multi–exposure (stack of many frames).
Detector Performance {#detec}
====================
Although the spectrometer was initially designed to use a subsection 40x100 of a NICMOS3 detector, we found later that very good performance can be obtained on an even larger area. Using 256 pixels in the wavelength direction, we have a spectral coverage of almost 0.3 $\mu m$. This means that with a single grating setting we can measure a complete J spectrum and have good coverage in H and K.
The best 40x256 subsection was selected on the basis of good cosmetics (low percentage of bad pixels) and low dark current and readout noise. We measured the percentage of bad pixels, the dark current, and the readout noise via laboratory tests based on sets of images taken at a series of exposure times of a spatially uniformly illuminated scene, and without any illumination (by substituting the filter with a cold stop).
The readout noise is determined as the mean standard deviation of each pixel in the stack of short integration times where the dark current is negligible. The dark current and gain measurement are based on two linear regressions: values of dark frames as a function of exposure time in the first case, and spatial medians of the stack variance relative to the stack median in the second one. Details of these tests are presented in Vanzi et al. (1995). In Table 1, we present the results of further tests carried out in April 1995.
\[riv\]
---------------- ---------------
Bad pixels 2.9%
Dark current 0.9 $e^-$/sec
Read out noise 45 $e^-$
---------------- ---------------
: Measured parameters of the detector
Observations and Data Reduction {#obs}
===============================
The procedures for LonGSp observations are those commonly used in NIR spectroscopy, optimized for the characteristics of the instrument. For compact sources, observations consist of several groups of frames with the object placed at different positions along the slit. In the case of extended sources, observations consist of several pairs of object and sky frames. On-chip integration time is 60 sec or less for a background level of roughly 6000 counts/pixel because of ensuing problems with sky line subtraction (see below). At a given position along the slit, several frames can be coadded.
The main steps in the reduction of NIR spectroscopic data are flat–field correction, subtraction of sky emission, wavelength calibration, correction for telluric absorption, and correction for optical system $+$ detector efficency. Data reduction can be performed using the IRSPEC context in MIDAS, the ESO data analysis package, properly modified to take into account the LonGSp instrumental setup. We have found it useful to acquire dark and flat frames at the beginning and the end of the night; we obtain flat-field frames by illuminating the dome with a halogen lamp.
Observations of a reference star are taken for a fixed grating position. An early type, featureless star (preferably an O star) is needed to correct for telluric absorption and differential efficency of the system, and a photometric standard star is needed if one wants to flux calibrate the final spectrum (only one grating position in each band is required). An alternative technique, proposed by Maiolino et al (1996), consists of using a G star corrected through data of solar spectrum. Both methods have been succesfully tested.
Flat field frames are first corrected for bad pixels, then dark-current subtracted, and normalized. Dark current is subtracted from all raw frames, and then divided by the normalized flat field.
For compact sources (the frames taken at the different positions along the slit are denoted by A, B and C), the sky is subtracted by considering $A-B$, $B-(A+C)/2$, and $C-B$, and taking a median of the three differences. In case of extended objects, if A and B denote object and sky frames, the sky is subtracted by considering $(A1+A2)/2-(B1+B2)/2$ (the order of observations is $A1\,B1\,B2\,A2$). However, a simple sky subtraction is almost never sufficient to properly eliminate the bright OH lines whose intensity varies on time scales comparable with object and sky observations. Moreover, mechanical instabilities can produce movements of spectra (usually a few hundreds of a pixel) which are nevertheless enough to produce residuals which exceed the detector noise. To correct for these two effects, the sky frames are multiplied by a correcting factor and shifted along the dispersion direction by a given amount. These factors and shifts are chosen automatically by minimizing the standard deviation in selected detector areas where only sky emission is present. Because this effect increases with the integration time, it is advisable not to exceed 60 seconds for each single integration.
Slit images at various wavelengths are tilted as a consequence of the off-axis mount of the grating. Sky subtracted frames are corrected by computing analytically the tilt angle from the instrumental calibration parameters, or by directly measuring it from the data.
Wavelength calibration in LonGSp data is performed using OH sky emission lines. The wavelength dispersion on the array is linear to within a small fraction of the pixel size and is computed analytically once the central wavelength of the frame is known. At the beginning of the data reduction, the nominal central wavelength used in the observations is assigned to a properly chosen sky frame. Then the calibration is refined using the bright OH sky lines (precise wavelengths of OH lines as well as a discussion of their use as calibrators are given in Oliva & Origlia 1992).
The same procedures are applied to the reference stars frames to obtain the calibration spectra, and the spectrum of a photometric standard star can be used to flux calibrate the final frames.
Astronomical Results {#astro}
====================
The first tests at the telescope took place successfully in early 1994. From these observations we measured the efficiency of the instrument (through the observation of photometric standard stars) and its sensitivity (1$\sigma$ in 60 sec of integration time); these are reported in table 2.
\[efi\]
[cccc]{} Band(order) & Efficency & Line$^{1}$ & Continuum$^{2}$\
J (I) & 0.045 & 4$\times 10^{-14}$ & 2$\times 10^{-15}$\
H (I) & 0.10 & 2$\times 10^{-14}$ & 8$\times 10^{-16}$\
K (I) & 0.08 & 2$\times 10^{-14}$ & 8$\times 10^{-16}$\
\
Since January 1996, LonGSp is offered to TIRGO users and employed in several galactic and extragalactic programs. To give an impression of the capabilities of the instrument, we show (in Figs. 3,4 and 5) some acquired spectra of various type of sources: extended, compact and extragalactic, without comment as to their astrophysical significance.
Gennari S., Mannucci, F., Vanzi L., 1993, Cryogenic stepper motors for infrared astronomical instrumentation. In: A. M. Fowler (ed.) Proc. SPIE 1946, International Symposium on Optical Engigneering and Photonics in Aerospace and Remote Sensing, p. 610
Gennari S., Vanzi L, 1993, LonGSp: the new infrared spectrometer of TIRGO. In: R. Bandiera (ed.) Proc. XXXVII Annual Meeting of the S.A.It., p. 752
Gennari S., Vanzi L, 1994, LonGSp: a near infrared spectrometer. In: I. McLean (ed.) Infrared astronomy with arrays, Kluwer Academic Publishers, p. 351
Lisi F, Baffa C., Biliotti V., et al., 1995, PASP 108, 364
Maiolino R., Rieke G.H., Rieke M.J., 1996, AJ 111, 537
Oliva E., Origlia L. 1992, A&A 254, 466
Vanzi L., Gennari S., Marconi A., 1994, NICMOS3 detector for astronomical spectroscopy, IAU Symp. 167, Com. N. 167.Or.031
|
---
abstract: |
It is proved that for all but a finite set of square-free integers $d$ the value of transcendental function $\exp~(2\pi i ~x+\log\log y)$ is an algebraic number for the algebraic arguments $x$ and $y$ in the real quadratic field of discriminant $d$. Such a value generates the Hilbert class field of imaginary quadratic field of discriminant $-d$.
[*Key words and phrases: real multiplication; Sklyanin algebra*]{}
author:
- 'Igor Nikolaev [^1]'
title: 'On algebraic values of function $\exp ~(2\pi i ~x+\log\log y)$'
---
Introduction
============
It is an old problem to determine if given irrational value of a transcendental function is algebraic or transcendental for certain algebraic arguments; the algebraic values are particularly remarkable and worthy of thorough investigation, see \[Hilbert 1902\] [@Hil1], p. 456. Only few general results are known, see e.g. \[Baker 1975\] [@B]. We shall mention the famous Gelfond-Schneider Theorem saying that $e^{\beta\log\alpha}$ is a transcendental number, whenever $\alpha\not\in \{0, 1\}$ is an algebraic and $\beta$ an irrational algebraic number. In contrast, Klein’s invariant $j(\tau)$ is known to take algebraic values whenever $\tau\in {\Bbb H}:=\{x+iy\in {\Bbb C}~|~y>0\}$ is an imaginary quadratic number.
The aim of our note is a result on the algebraic values of transcendental function $${\cal J}(x,y):=\{e^{2\pi i ~x ~+ ~\log\log y} ~|-\infty<x<\infty, ~1<y<\infty\}$$ for the arguments $x$ and $y$ in a real quadratic field; the function ${\cal J}(x, y)$ can be viewed as an analog of Klein’s invariant $j(\tau)$, hence the notation. Namely, let ${\goth k}={\Bbb Q}(\sqrt{d})$ be a real quadratic field and ${\goth R}_{\goth f}=
{\Bbb Z}+{\goth f}O_{\goth k}$ be an order of conductor ${\goth f}\ge 1$ in the field ${\goth k}$; let $h=|Cl~({\goth R}_{\goth f})|$ be the class number of ${\goth R}_{\goth f}$ and denote by $\{{\Bbb Z}+{\Bbb Z}\theta_i ~|~ 1\le 1\le h\}$ the set of pairwise non-isomorphic pseudo-lattices in ${\goth k}$ having the same endomorphism ring ${\goth R}_{\goth f}$, see \[Manin 2004\] [@Man1], Lemma 1.1.1. Finally, let $\varepsilon$ be the fundamental unit of ${\goth R}_{\goth f}$ and let $f\ge 1$ be the least integer satisfying equation $|Cl~(R_f)|=|Cl~({\goth R}_{\goth f})|$, where $R_f={\Bbb Z}+fO_k$ is an order of conductor $f$ in the imaginary quadratic field $k={\Bbb Q}(\sqrt{-d})$. Our main result can be formulated as follows.
\[thm1\] For each square-free positive integer $d\not\in\{1,2,3,7,11,19,$ $43,67,163\}$ the values $\{{\cal J}(\theta_i,\varepsilon) ~|~ 1\le i\le h\}$ of transcendental function ${\cal J}(x,y)$ are algebraically conjugate numbers generating the Hilbert class field $H(k)$ of the imaginary quadratic field $k={\Bbb Q}(\sqrt{-d})$ modulo conductor $f$.
\[rk1\] [Since $H(k)\cong k(j(\tau))\cong {\Bbb Q}(f\sqrt{-d}, j(\tau))$ with $\tau\in R_f$, one gets an inclusion ${\cal J}(\theta_i,\varepsilon))\in {\Bbb Q}(f\sqrt{-d}, j(\tau))$.]{.nodecor}
[Note that even though the absolute value $|z|=\sqrt{z\bar z}$ of an algebraic $z$ is an algebraic number, the absolute value of ${\cal J}(\theta_i,\varepsilon)$ is transcendental. It happens because $|z|$ belongs to a quadratic extension of the real field ${\Bbb Q}(z\bar z)$ which may have no real embeddings at all. (Compare with the CM-field, i.e. a totally imaginary quadratic extension of the totally real number field.)]{.nodecor}
The structure of article is as follows. Some preliminary facts can be found in Section 2. Theorem \[thm1\] is proved in Section 3 and Section 4 contains an example illustrating the theorem.
Preliminaries
=============
The reader can find basics of the $C^*$-algebras in \[Murphy 1990\] [@M] and their $K$-theory in \[Blackadar 1986\] [@BL]. The noncommutative tori are covered in \[Rieffel 1990\] [@Rie1] and real multiplication in \[Manin 2004\] [@Man1]. For main ideas of non-commutative algebraic geometry, see the survey by \[Stafford & van den Bergh 2001\] [@StaVdb1].
Noncommutative tori
-------------------
By a [*noncommutative torus*]{} ${\cal A}_{\theta}$ one understands the universal [*$C^*$-algebra*]{} generated by the unitary operators $u$ and $v$ acting on a Hilbert space ${\cal H}$ and satisfying the commutation relation $vu=e^{2\pi i\theta}uv$, where $\theta$ is a real number.
\[rmk1\] [Note that ${\cal A}_{\theta}$ is isomorphic to a free ${\Bbb C}$-algebra on four generators $u,u^*,v,v^*$ and six quadratic relations: $$\label{eq2}
\left\{
\begin{array}{cc}
vu &= e^{2\pi i\theta} uv,\\
v^*u^* &= e^{2\pi i\theta}u^*v^*,\\
v^*u &= e^{-2\pi i\theta}uv^*,\\
vu^* &= e^{-2\pi i\theta}u^*v,\\
u^*u &= uu^*=e,\\
v^*v &= vv^*=e.
\end{array}
\right.$$ Indeed, the first and the last two relations in system (\[eq2\]) are obvious from the definition of ${\cal A}_{\theta}$. By way of example, let us demonstrate that relations $vu=e^{2\pi i\theta} uv$ and $u^*u=uu^*=v^*v=vv^*=e$ imply the relation $v^*u = e^{-2\pi i\theta}uv^*$ in system (\[eq2\]). Indeed, it follows from $uu^*=e$ and $vv^*=e$ that $uu^*vv^*=e$. Since $uu^*=u^*u$ we can bring the last equation to the form $u^*uvv^*=e$ and multiply the both sides by the constant $e^{2\pi i\theta}$; thus one gets the equation $u^*(e^{2\pi i\theta}uv)v^*=e^{2\pi i\theta}$. But $e^{2\pi i\theta}uv=vu$ and our main equation takes the form $u^*vuv^*= e^{2\pi i\theta}$. We can multiply on the left the both sides of the equation by the element $u$ and thus get the equation $uu^*vuv^*= e^{2\pi i\theta}u$; since $uu^*=e$ one arrives at the equation $vuv^*= e^{2\pi i\theta}u$. Again one can multiply on the left the both sides by the element $v^*$ and thus get the equation $v^*vuv^*= e^{2\pi i\theta}v^*u$; since $v^*v=e$ one gets $uv^*= e^{2\pi i\theta}v^*u$ and the required identity $v^*u = e^{-2\pi i\theta}uv^*$. The remaining two relations in (\[eq2\]) are proved likewise; we leave it to the reader as an exercise in non-commutative algebra.]{.nodecor}
Recall that the algebra ${\cal A}_{\theta}$ is said to be [*stably isomorphic*]{} (Morita equivalent) to ${\cal A}_{\theta'}$, whenever ${\cal A}_{\theta}\otimes {\cal K}\cong
{\cal A}_{\theta'}\otimes {\cal K}$, where ${\cal K}$ is the $C^*$-algebra of all compact operators on ${\cal H}$; the ${\cal A}_{\theta}$ is stably isomorphic to ${\cal A}_{\theta'}$ if and only if $$\label{eq3}
\theta'={a\theta +b\over c\theta+d}\quad
\hbox{for some matrix} \quad \left(\matrix{a & b\cr c & d}\right)\in SL_2({\Bbb Z}).$$ The $K$-theory of ${\cal A}_{\theta}$ is two-periodic and $K_0({\cal A}_{\theta})\cong K_1({\cal A}_{\theta})\cong {\Bbb Z}^2$ so that the Grothendieck semigroup $K_0^+({\cal A}_{\theta})$ corresponds to positive reals of the set ${\Bbb Z}+{\Bbb Z}\theta\subset {\Bbb R}$ called a [*pseudo-lattice*]{}. The torus ${\cal A}_{\theta}$ is said to have [*real multiplication*]{}, if $\theta$ is a quadratic irrationality, i.e. irrational root of a quadratic polynomial with integer coefficients. The real multiplication says that the endomorphism ring of pseudo-lattice ${\Bbb Z}+{\Bbb Z}\theta$ exceeds the ring ${\Bbb Z}$ corresponding to multiplication by $m$ endomorphisms; similar to complex multiplication, it means that the endomorphism ring is isomorphic to an order ${\goth R}_{\goth f}={\Bbb Z}+{\goth f}O_{\goth k}$ of conductor ${\goth f}\ge 1$ in the real quadratic field ${\goth k}={\Bbb Q}(\theta)$ – hence the name, see \[Manin 2004\] [@Man1]. If $d>0$ is the discriminant of ${\goth k}$, then by ${\cal A}_{RM}^{(d, {\goth f})}$ we denote a noncommutative torus with real multiplication by the order ${\goth R}_{\goth f}$.
Elliptic curves
---------------
For the sake of clarity, let us recall some well-known facts. An [*elliptic curve*]{} is the subset of the complex projective plane of the form ${\cal E}({\Bbb C})=\{(x,y,z)\in {\Bbb C}P^2 ~|~ y^2z=4x^3+axz^2+bz^3\}$, where $a$ and $b$ are some constant complex numbers. Recall that one can embed ${\cal E}({\Bbb C})$ into the complex projective space ${\Bbb C}P^3$ as the set of points of intersection of two [*quadric surfaces*]{} given by the system of homogeneous equations $$\label{eq4}
\left\{
\begin{array}{ccc}
u^2+v^2+w^2+z^2 &=& 0,\\
Av^2+Bw^2+z^2 &=& 0,
\end{array}
\right.$$ where $A$ and $B$ are some constant complex numbers and $(u,v,w,z)\in {\Bbb C}P^3$; the system (\[eq4\]) is called the [*Jacobi form*]{} of elliptic curve ${\cal E}({\Bbb C})$. Denote by ${\Bbb H}=\{x+iy\in {\Bbb C}~|~y>0\}$ the Lobachevsky half-plane; whenever $\tau\in {\Bbb H}$, one gets a complex torus ${\Bbb C}/({\Bbb Z}+{\Bbb Z}\tau)$. Each complex torus is isomorphic to a non-singular elliptic curve; the isomorphism is realized by the Weierstrass $\wp$ function and we shall write ${\cal E}_{\tau}$ to denote the corresponding elliptic curve. Two elliptic curves ${\cal E}_{\tau}$ and ${\cal E}_{\tau'}$ are isomorphic if and only if $$\label{eq5}
\tau'={a\tau +b\over c\tau+d}\quad
\hbox{for some matrix} \quad \left(\matrix{a & b\cr c & d}\right)\in SL_2({\Bbb Z}).$$ If $\tau$ is an imaginary quadratic number, elliptic curve ${\cal E}_{\tau}$ is said to have [*complex multiplication*]{}; in this case lattice ${\Bbb Z}+{\Bbb Z}\tau$ admits non-trivial endomorphisms realized as multiplication of points of the lattice by the imaginary quadratic numbers, hence the name. We shall write ${\cal E}_{CM}^{(-d,f)}$ to denote elliptic curve with complex multiplication by an order $R_f={\Bbb Z}+fO_k$ of conductor $f\ge 1$ in the imaginary quadratic field $k={\Bbb Q}(\sqrt{-d})$.
Sklyanin algebras
-----------------
By the [*Sklyanin algebra*]{} $S_{\alpha,\beta,\gamma}({\Bbb C})$ one understands a free ${\Bbb C}$-algebra on four generators and six relations: $$\left\{
\begin{array}{ccc}
x_1x_2-x_2x_1 &=& \alpha(x_3x_4+x_4x_3),\\
x_1x_2+x_2x_1 &=& x_3x_4-x_4x_3,\\
x_1x_3-x_3x_1 &=& \beta(x_4x_2+x_2x_4),\\
x_1x_3+x_3x_1 &=& x_4x_2-x_2x_4,\\
x_1x_4-x_4x_1 &=& \gamma(x_2x_3+x_3x_2),\\
x_1x_4+x_4x_1 &=& x_2x_3-x_3x_2,
\end{array}
\right.$$ where $\alpha+\beta+\gamma+\alpha\beta\gamma=0$. The algebra $S_{\alpha,\beta,\gamma}({\Bbb C})$ represents a twisted homogeneous [*coordinate ring*]{} of an elliptic curve ${\cal E}_{\alpha,\beta,\gamma}({\Bbb C})$ given in its Jacobi form $$\left\{
\begin{array}{ccc}
u^2+v^2+w^2+z^2 &=& 0,\\
{1-\alpha\over 1+\beta}v^2+
{1+\alpha\over 1-\gamma}w^2+z^2 &=& 0,
\end{array}
\right.$$ see \[Smith & Stafford 1993\] [@SmiSta1], p.267 and \[Stafford & van den Bergh 2001\] [@StaVdb1], Example 8.5. The latter means that algebra $S_{\alpha,\beta,\gamma}({\Bbb C})$ satisfies an isomorphism $\hbox{{\bf Mod}}~(S_{\alpha,\beta,\gamma}({\Bbb C}))/
\hbox{{\bf Tors}}\cong \hbox{{\bf Coh}}~({\cal E}_{\alpha,\beta,\gamma}({\Bbb C}))$, where [**Coh**]{} is the category of quasi-coherent sheaves on ${\cal E}_{\alpha,\beta,\gamma}({\Bbb C})$, [**Mod**]{} the category of graded left modules over the graded ring $S_{\alpha,\beta,\gamma}({\Bbb C})$ and [**Tors**]{} the full sub-category of [**Mod**]{} consisting of the torsion modules, see \[Stafford & van den Bergh 2001\] [@StaVdb1], p.173. The algebra $S_{\alpha,\beta,\gamma}({\Bbb C})$ defines a natural [*automorphism*]{} $\sigma$ of elliptic curve ${\cal E}_{\alpha,\beta,\gamma}({\Bbb C})$, [*ibid.*]{}
Proof of theorem \[thm1\]
=========================
For the sake of clarity, let us outline main ideas. The proof is based on a categorical correspondence (a covariant functor) between elliptic curves ${\cal E}_{\tau}$ and noncommutative tori ${\cal A}_{\theta}$ taken with their “scaled units” ${1\over\mu}e$. Namely, we prove that for $\sigma^4=Id$ the norm-closure of a self-adjoint representation of the Sklyanin algebra $S_{\alpha,\beta,\gamma}({\Bbb C})$ by the linear operators $u=x_1,u^*=x_2, v=x_3, v^*=x_4$ on a Hilbert space ${\cal H}$ is isomorphic to the $C^*$-algebra ${\cal A}_{\theta}$ so that its unit $e$ is scaled by a positive real $\mu$, see lemma \[lem2\]; because $S_{\alpha,\beta,\gamma}({\Bbb C})$ is a coordinate ring of elliptic curve ${\cal E}_{\alpha,\beta,\gamma}({\Bbb C})$ so will be the algebra ${\cal A}_{\theta}$ modulo the unit ${1\over\mu}e$. Moreover, our construction entails that a coefficient $q$ of elliptic curve ${\cal E}_{\alpha,\beta,\gamma}({\Bbb C})$ is linked to the constants $\theta$ and $\mu$ by the formula $q=\mu e^{2\pi i\theta}$, see lemma \[lem1\]. Suppose that our elliptic curve has complex multiplication, i.e. ${\cal E}_{\tau}\cong {\cal E}_{CM}^{(-d,f)}$; then its coordinate ring $({\cal A}_{\theta}, {1\over\mu}e)$ must have real multiplication, i.e. ${\cal A}_{\theta}\cong {\cal A}_{RM}^{(d, {\goth f})}$ and $\mu=\log\varepsilon$, where $|Cl~(R_f)|=|Cl~({\goth R}_{\goth f})|$ and $\varepsilon$ is the fundamental unit of order ${\goth R}_{\goth f}$, see lemma \[lem3\]. But elliptic curve ${\cal E}_{CM}^{(-d,f)}$ has coefficients in the Hilbert class field $H(k)$ over imaginary quadratic field $k={\Bbb Q}(\sqrt{-d})$ modulo conductor $f$; thus $q\in H(k)$ and therefore one gets an inclusion $$\mu e^{2\pi i\theta}\in H(k),$$ where $\theta\in {\goth k}= {\Bbb Q}(\sqrt{d})$ and $\mu=\log\varepsilon$. (Of course, our argument is valid only when $q\not\in {\Bbb R}$, i.e. when $|Cl~(R_f)|\ge 2$; but there are only a finite number of discriminants $d$ with $|Cl~(R_f)|=1$.) Let us pass to a detailed argument.
\[lem1\] If $\sigma^4=Id$, then the Sklyanin algebra $S_{\alpha,\beta,\gamma}({\Bbb C})$ endowed with the involution $x_1^*=x_2$ and $x_3^*=x_4$ is isomorphic to a free algebra ${\Bbb C}\langle x_1,x_2,x_3,x_4\rangle$ modulo an ideal generated by six quadratic relations $$\label{eq9}
\left\{
\begin{array}{cc}
x_3x_1 &= \mu e^{2\pi i\theta}x_1x_3,\\
x_4x_2 &= {1\over \mu} e^{2\pi i\theta}x_2x_4,\\
x_4x_1 &= \mu e^{-2\pi i\theta}x_1x_4,\\
x_3x_2 &= {1\over \mu} e^{-2\pi i\theta}x_2x_3,\\
x_2x_1 &= x_1x_2,\\
x_4x_3 &= x_3x_4,
\end{array}
\right.$$ where $\theta=Arg~(q)$ and $\mu=|q|$ for a complex number $q\in {\Bbb C}\setminus \{0\}$.
[*Proof.*]{} (i) Since $\sigma^4=Id$, the Sklyanin algebra $S_{\alpha, \beta, \gamma}({\Bbb C})$ is isomorphic to a free algebra ${\Bbb C}\langle x_1,x_2,x_3,x_4\rangle$ modulo an ideal generated by the skew-symmetric relations $$\label{eq10}
\left\{
\begin{array}{ccc}
x_3x_1 &=& q_{13} x_1x_3,\\
x_4x_2 &=& q_{24}x_2x_4,\\
x_4x_1 &=& q_{14}x_1x_4,\\
x_3x_2 &=& q_{23}x_2x_3,\\
x_2x_1&=& q_{12}x_1x_2,\\
x_4x_3&=& q_{34}x_3x_4,
\end{array}
\right.$$ where $q_{ij}\in {\Bbb C}\setminus\{0\}$, see \[Feigin & Odesskii 1989\] [@FeOd1], Remark 1.
\(ii) It is verified directly, that relations (\[eq10\]) are invariant of the involution $x_1^*=x_2$ and $x_3^*=x_4$, if and only if $$\label{eq11}
\left\{
\begin{array}{ccc}
q_{13} &=& (\bar q_{24})^{-1},\\
q_{24} &=& (\bar q_{13})^{-1},\\
q_{14} &= & (\bar q_{23})^{-1},\\
q_{23} &= & (\bar q_{14})^{-1},\\
q_{12} &= & \bar q_{12},\\
q_{34} &= & \bar q_{34},
\end{array}
\right.$$ where $\bar q_{ij}$ means the complex conjugate of $q_{ij}\in {\Bbb C}\setminus\{0\}$.
[The invariant relations (\[eq11\]) define an involution on the Sklyanin algebra; we shall refer to such as a [*Sklyanin $\ast$-algebra*]{}.]{.nodecor}
\(iii) Consider a one-parameter family $S(q_{13})$ of the Sklyanin $\ast$-algebras defined by the following additional constraints $$\left\{
\begin{array}{ccc}
q_{13} &=& \bar q_{14},\\
q_{12} &=& q_{34}=1.
\end{array}
\right.$$ It is not hard to see, that the $\ast$-algebras $S(q_{13})$ are pairwise non-isomorphic for different values of complex parameter $q_{13}$; therefore the family $S(q_{13})$ is a normal form of the Sklyanin $\ast$-algebra $S_{\alpha, \beta, \gamma}({\Bbb C})$ with $\sigma^4=Id$. It remains to notice, that one can write complex parameter $q:=q_{13}$ in the polar form $q=\mu e^{2\pi i\theta}$, where $\theta=Arg~(q)$ and $\mu=|q|$. Lemma \[lem1\] follows. $\square$
\[lem2\] [**(basic isomorphism)**]{} The system of relations (\[eq2\]) for noncommutative torus ${\cal A}_{\theta}$ with $u=x_1, u^*=x_2, v=x_3, v^*=x_4$, i.e. $$\label{eq13}
\left\{
\begin{array}{cc}
x_3x_1 &= e^{2\pi i\theta}x_1x_3,\\
x_4x_2 &= e^{2\pi i\theta}x_2x_4,\\
x_4x_1 &= e^{-2\pi i\theta}x_1x_4,\\
x_3x_2 &= e^{-2\pi i\theta}x_2x_3,\\
x_2x_1 &= x_1x_2=e,\\
x_4x_3 &= x_3x_4=e,
\end{array}
\right.$$ is equivalent to the system of relations (\[eq9\]) for the Sklyanin $\ast$-algebra, i.e. $$\label{eq14}
\left\{
\begin{array}{cc}
x_3x_1 &= \mu e^{2\pi i\theta}x_1x_3,\\
x_4x_2 &= {1\over \mu} e^{2\pi i\theta}x_2x_4,\\
x_4x_1 &= \mu e^{-2\pi i\theta}x_1x_4,\\
x_3x_2 &= {1\over \mu} e^{-2\pi i\theta}x_2x_3,\\
x_2x_1 &= x_1x_2,\\
x_4x_3 &= x_3x_4,
\end{array}
\right.$$ modulo the following “scaled unit relation” $$\label{eq15}
x_1x_2=x_3x_4={1\over\mu}e.$$
[*Proof.*]{} (i) Using the last two relations, one can bring the noncommutative torus relations (\[eq13\]) to the form $$\label{eq16}
\left\{
\begin{array}{ccc}
x_3x_1x_4 &=& e^{2\pi i\theta}x_1,\\
x_4 &= & e^{2\pi i\theta}x_2x_4x_1,\\
x_4x_1x_3 &=& e^{-2\pi i\theta}x_1,\\
x_2 &=& e^{-2\pi i\theta}x_4x_2x_3,\\
x_1x_2 &=& x_2x_1 =e,\\
x_3x_4 &=& x_4x_3 =e.
\end{array}
\right.$$
\(ii) The system of relations (\[eq14\]) for the Sklyanin $\ast$-algebra complemented by the scaled unit relation (\[eq15\]), i.e. $$\label{eq17}
\left\{
\begin{array}{cc}
x_3x_1 &= \mu e^{2\pi i\theta}x_1x_3,\\
x_4x_2 &= {1\over \mu} e^{2\pi i\theta}x_2x_4,\\
x_4x_1 &= \mu e^{-2\pi i\theta}x_1x_4,\\
x_3x_2 &= {1\over \mu} e^{-2\pi i\theta}x_2x_3,\\
x_2x_1 &= x_1x_2={1\over\mu}e,\\
x_4x_3 &= x_3x_4={1\over\mu}e,
\end{array}
\right.$$ is equivalent to the system $$\label{eq18}
\left\{
\begin{array}{cc}
x_3x_1x_4 &= e^{2\pi i\theta}x_1,\\
x_4 &= e^{2\pi i\theta}x_2x_4x_1,\\
x_4x_1x_3 &= e^{-2\pi i\theta}x_1,\\
x_2 &= e^{-2\pi i\theta}x_4x_2x_3,\\
x_2x_1 &= x_1x_2={1\over\mu}e,\\
x_4x_3 &= x_3x_4={1\over\mu}e
\end{array}
\right.$$ by using multiplication and cancellation involving the last two equations.
\(iii) For each $\mu\in (0,\infty)$ consider a [*scaled unit*]{} $e':={1\over\mu} e$ of the Sklyanin $\ast$-algebra $S(q)$ and the two-sided ideal $I_{\mu}\subset S(q)$ generated by the relations $x_1x_2=x_3x_4=e'$. Comparing the defining relations (\[eq14\]) for $S(q)$ with relation (\[eq13\]) for the noncommutative torus ${\cal A}_{\theta}$, one gets an isomorphism $$\label{eq19}
S(q)~/~I_{\mu}\cong {\cal A}_{\theta}.$$ The isomorphism maps generators $x_1,\dots,x_4$ of the Sklyanin $\ast$-algebra $S(q)$ to such of the $C^*$-algebra ${\cal A}_{\theta}$ and the [*scaled*]{} unit $e'\in S(q)$ to the [*ordinary*]{} unit $e\in {\cal A}_{\theta}$. Lemma \[lem2\] follows. $\square$
[It follows from (\[eq19\]) that noncommutative torus ${\cal A}_{\theta}$ with the unit ${1\over\mu}e$ is a coordinate ring of elliptic curve ${\cal E}_{\tau}$. Moreover, such a correspondence is a covariant functor which maps isomorphic elliptic curves to the stably isomorphic (Morita equivalent) noncommutative tori; the latter fact follows from an observation that isomorphisms in category [**Mod**]{} correspond to stable isomorphisms in the category of underlying algebras. Such a functor explains the same (modular) transformation law in formulas (\[eq3\]) and (\[eq5\]).]{.nodecor}
\[lem3\] The coordinate ring of elliptic curve ${\cal E}_{CM}^{(-d,f)}$ is isomorphic to the noncommutative torus ${\cal A}_{RM}^{(d, {\goth f})}$ with the unit ${1\over \log\varepsilon}e$, where ${\goth f}$ is the least integer satisfying equation $|Cl~({\goth R}_{\goth f})|=|Cl~(R_f)|$ and $\varepsilon$ is the fundamental unit of order ${\goth R}_{\goth f}$.
[*Proof.*]{} The fact that ${\cal A}_{RM}^{(d, {\goth f})}$ is a coordinate ring of elliptic curve ${\cal E}_{CM}^{(-d,f)}$ was proved in \[Nikolaev 2014\] [@Nik1]. We shall focus on the second part of lemma \[lem3\] saying that the scaling constant $\mu=\log\varepsilon$. To express $\mu$ in terms of intrinsic invariants of pseudo-lattice $K_0^+({\cal A}_{RM}^{(d, {\goth f})})\cong {\Bbb Z}+{\Bbb Z}\theta$, recall that ${\goth R}_{\goth f}$ is the ring of endomorphisms of ${\Bbb Z}+{\Bbb Z}\theta$; we shall write ${\goth R}_{\goth f}^{\times}$ to denote the multiplicative group of units (i.e. invertibe elements) of ${\goth R}_{\goth f}$. Since $\mu$ is an additive functional on the pseudo-lattice $\Lambda={\Bbb Z}+{\Bbb Z}\theta$, for each $\varepsilon, \varepsilon'\in {\goth R}_{\goth f}^{\times} $ it must hold $\mu(\varepsilon\varepsilon' \Lambda)=\mu(\varepsilon\varepsilon') \Lambda=
\mu(\varepsilon)\Lambda+\mu(\varepsilon')\Lambda$. Eliminating $\Lambda$ in the last equation, one gets $$\mu(\varepsilon\varepsilon')=\mu(\varepsilon)+\mu(\varepsilon'),
\qquad \forall \varepsilon, \varepsilon'\in {\goth R}_{\goth f}^{\times}.$$ The only real-valued function on ${\goth R}_{\goth f}^{\times}$ with such a property is the logarithmic function (a regulator of ${\goth R}_{\goth f}^{\times}$); thus $\mu(\varepsilon)=\log\varepsilon$, where $\varepsilon$ is the fundamental unit of ${\goth R}_{\goth f}$. Lemma \[lem3\] is proved. $\square$
[**(Second proof of lemma \[lem3\])**]{} [The formula $\mu=\log\varepsilon$ can be derived using a purely measure-theoretic argument. Indeed, if $h_x: {\Bbb R}\to {\Bbb R}$ is a “stretch-out” automorphism of real line ${\Bbb R}$ given by the formula $t\mapsto tx,~\forall t\in {\Bbb R}$, then the only $h_x$-invariant measure $\mu$ on ${\Bbb R}$ is the “scale-back” measure $d\mu={1\over t} dt$. Taking the antiderivative and integrating between $t_0=1$ and $t_1=x$, one gets $$\mu=\log x.$$ It remains to notice that for pseudo-lattice $K_0^+({\cal A}_{RM}^{(d,{\goth f})})\cong {\Bbb Z}+{\Bbb Z}\theta$, the automorphism $h_x$ corresponds to $x=\varepsilon$, where $\varepsilon>1$ is the fundamental unit of order ${\goth R}_{\goth f}$. Lemma \[lem3\] follows. $\square$.]{.nodecor}
One can prove theorem \[thm1\] in the following steps.
\(i) Let $d\not\in\{1,2,3,7,11,19,43, 67,163\}$ be a positive square-free integer. In this case $h=|Cl~(R_f)|\ge 2$ and ${\cal E}_{CM}^{(-d,f)}\not\cong {\cal E}({\Bbb Q})$.
\(ii) Let $\{{\cal E}_1,\dots, {\cal E}_h\}$ be pairwise non-isomorphic elliptic curves having the same endomorphism ring $R_f$. From $|Cl~(R_f)|=|Cl~({\goth R}_{\goth f})|$ and lemma \[lem3\], one gets $\{{\cal A}_1,\dots, {\cal A}_h\}$ pairwise stably non-isomorphic noncommutative tori; the corresponding pseudo-lattices $K_0^+({\cal A}_i)={\Bbb Z}+{\Bbb Z}\theta_i$ will have the same endomorphism ring ${\goth R}_{\goth f}$. Thus for each $1\le i\le h$ one gets an inclusion $$(\log\varepsilon) e^{2\pi i\theta_i}\in H(k),$$ where $H(k)$ is the Hilbert class field of quadratic field $k={\Bbb Q}(\sqrt{-d})$ modulo conductor $f$. Since $(\log\varepsilon)\exp (2\pi i\theta_i)=\exp (2\pi i\theta_i+\log\log\varepsilon):={\cal J}(\theta_i, \varepsilon)$, one concludes that ${\cal J}(\theta_i, \varepsilon)\in H(k)$.
\(iii) Finally, because $Gal~(H(k)|k)\cong Cl~(R_f)\cong Cl~({\goth R}_{\goth f})$, it is easy to see that the set $\{{\cal J}(\theta_i,\varepsilon) ~|~ 1\le i\le h\}$ is invariant of the action of group $Gal~(H(k)|k)$ on $H(k)$; in other words, numbers ${\cal J}(\theta_i, \varepsilon)$ are algebraically conjugate.
Theorem \[thm1\] is proved. $\square$
Example
=======
In this section we shall use remark \[rk1\] to estimate ${\cal J}(\theta, \varepsilon)$ for special values of the discriminant $d$; the reader is encouraged to construct examples of his own.
[Let $d=15$ and $f=1$. It is well known that the class number of order $R_{f=1}\cong O_k$ of the field $k={\Bbb Q}(\sqrt{-15})$ is equal to 2. Because the class number of the field ${\goth k}={\Bbb Q}(\sqrt{15})$ is also 2, one concludes from equation $|Cl~({\goth R}_{\goth f})|=|Cl~(R_f)|$ that conductor ${\goth f}=1$. Let $\tau\in O_k$; it is well known that in this case $j(\tau)\in {\Bbb Q}(\sqrt{5})$, see e.g. \[Silverman 1994\] [@S], Example 6.2.2. In view of remark \[rk1\], one gets an inclusion ${\cal J}(\theta_i,\varepsilon)\in {\Bbb Q}(\sqrt{-15}, \sqrt{5})$. Since one of $\theta_i$ is equal to $\sqrt{15}$ and the fundamental unit $\varepsilon$ of the field ${\goth k}={\Bbb Q}(\sqrt{15})$ is equal to $4+\sqrt{15}$, one gets the following inclusion $${\cal J}(\sqrt{15}, ~4+\sqrt{15}):=e^{2\pi i\sqrt{15}+\log\log (4+\sqrt{15})}\in {\Bbb Q}\left(\sqrt{-15}, \sqrt{5}\right).$$]{.nodecor}
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M. A. Rieffel, Non-commutative tori – a case study of non-commutative differentiable manifolds, Contemp. Math. 105 (1990), 191-211. Available [http://math.berkeley.edu/$\sim$rieffel/]{}
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<span style="font-variant:small-caps;">The Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada, E-mail:</span> [igor.v.nikolaev@gmail.com]{}
[^1]: Partially supported by NSERC.
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abstract: |
Textual queries are largely employed in information retrieval to let users specify search goals in a natural way. However, differences in user and system terminologies can challenge the identification of the user’s information needs, and thus the generation of relevant results. We argue that the explicit management of ontological knowledge, and of the meaning of concepts (by integrating linguistic and encyclopaedic knowledge in the system ontology), can improve the analysis of search queries, because it enables a flexible identification of the topics the user is searching for, regardless of the adopted vocabulary.
This paper proposes an information retrieval support model based on semantic concept identification. Starting from the recognition of the ontology concepts that the search query refers to, this model exploits the qualifiers specified in the query to select information items on the basis of possibly fine-grained features. Moreover, it supports query expansion and reformulation by suggesting the exploration of semantically similar concepts, as well as of concepts related to those referred in the query through thematic relations. A test on a data-set collected using the OnToMap Participatory GIS has shown that this approach provides accurate results.
author:
- Noemi Mauro
- Liliana Ardissono
- Adriano Savoca
title: 'Concept-aware Geographic Information Retrieval'
---
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[ **On the orthogonal component of BSDEs in a Markovian setting** ]{}
[**Abstract -** ]{} [ In this Note we consider a quadratic backward stochastic differential equation (BSDE) driven by a continuous martingale $M$ and whose generator is a deterministic function. We prove (in Theorem \[theorem:main\]) that if $M$ is a strong homogeneous Markov process and if the BSDE has the form then the unique solution $(Y,Z,N)$ of the BSDE is reduced to $(Y,Z)$, *i.e.* the orthogonal martingale $N$ is equal to zero showing that in a Markovian setting the “usual” solution $(Y,Z)$ has not to be completed by a strongly orthogonal even if $M$ does not enjoy the martingale representation property. ]{}
[**Sur la composante orthogonale d’une EDSR dans un contexte markovien** ]{}
[*[**Résumé -** ]{}*]{} [ Dans cette Note nous considérons une équation différentielle stochastique rétrograde (EDSR) de générateur déterministe et quadratique dirigée par une martingale continue $M$. Nous prouvons (dans le Théorème \[theorem:main\]) que si $M$ est un processus de Markov homogène fort et si l’EDSR est de la forme l’unique solution $(Y,Z,N)$ de l’EDSR se réduit à $(Y,Z)$, *i.e.* la martingale orthogonale $N$ vaut zéro. Cela prouve que dans un contexte markovien la solution “habituelle” $(Y,Z)$ n’a pas à être complétée par une martingale fortement orthogonale même si $M$ ne possède pas la propriété de représentation martingale. ]{}
0.5cm [**Version française abrégée**]{}\
Dans cette Note nous considérons une équation différentielle stochastique rétrograde (EDSR) dirigée par une martingale continue $M$, de générateur quadratique $f$ et admettant $F(X_T)$ pour condition terminale où $F:\real\to\real$ dénote une fonction déterministe suffisamment régulière et $X$ l’unique solution forte d’une équation différentielle stochastique (EDS) également dirigée par $M$. Dans ce contexte il a été démontré dans [@ElKarouiHuang] and [@Morlais] qu’il existe un unique triplet $(Y,Z,N)$ solution de l’EDSR considérée où $Y$ est un processus stochastique borné, $Z$ un processus prévisible de carré intégrable et $N$ une martingale fortement orthogonale à $M$. Puisque nous ne supposons pas que $M$ possède la propriété de représentation martingale, la solution habituelle $(Y,Z)$ doit *a priori* être complétée par une martingale $N$ fortement orthogonale à $M$. Si le générateur $f$ est supposé Lipschitz, les auteurs de [@ElKarouiHuang] obtiennent la solution de l’EDSR *via* une itération de Picard de la forme . Notons que la troisième compostante de la solution, la martingale orthogonale $N$ est “statique” lors de cette itération.\
\
L’objet de cette Note est de démontrer que dans un contexte markovien (*i.e.* avec une condition terminale comme exposée plus haut et un générateur déterministe dépendant uniquement de $y$ et $z$) la solution $(Y,Z,N)$ se réduit au couple $(Y,Z)$ autrement dit, la composante orthogonale $N$ est nulle même si la propriété de représentation martingale n’est pas vérifiée pour $M$. Afin de simplifier la preuve du résultat principal (Théorème \[theorem:main\]) nous considérons une diffusion $X$ de dérive nulle et toutes les équations mises en jeu sont uni-dimensionnelles (le cas d’un générateur dépendant de $(X,M)$ fera l’objet d’un travail futur). Ce résultat permettera (dans un travail en préparation) de simplifier l’étude des propriétés des EDSR quadratiques de la forme comme en particulier donner une preuve de différentiabilité par rapport aux paramètres initiaux $(x,m)$ (voir ) sans l’hypothèse additionnelle (MRP) (*c.f.* [@ImkellerReveillacRichter Section 4.2]) utilisée dans [@ImkellerReveillacRichter Theorem 4.6].
Preliminaries {#section1}
=============
Let $M:=(M_t)_{t\in [0,T]}$ be a real-valued continuous square integrable martingale with respect to a continuous filtration $(\mathcal{F}_t)_{t\in [0,T]}$ both defined on a probability space $(\Omega,\mathcal{F},\P)$. Assume that $M$ is an homogeneous strong Markov process with respect to $(\mathcal{F}_t)_{t\in [0,T]}$. For $(t,m)$ in $[0,T]\times \real$ we denote by $M^{t,m}$ the process defined as $ M_s^{t,m}:=m+M_s-M_t, \quad s \in [t,T].$ Let $C:=(C_t)_{t\in[0,T]}$ be the $(\mathcal{F}_t)_{t\in [0,T]}$-predictable, increasing process defined by $ C_t:=\arctan(\langle M, M \rangle_t), \; t \in [0,T].$ On this filtered probability space we also consider a stochastic process $X^{t,x,m}:=(X_s^{t,x,m})_{s\in [t,T]}$ defined as the unique strong solution of the following one-dimensional stochastic differential equation $$\label{SDE}
X_s^{t,x,m}=x+\int_t^s \sigma(X_r^{t,x,m},M_r^{t,m}) dM_r, \quad s \in[t,T], \; t \in [0,T]$$ where $\sigma:\real\times\real \to \real$ is deterministic, of class $C^2(\real\times \real)$ with locally Lipschitz partial derivatives and such that there exists a positive constant $k$ satisfying $ \vert \sigma(x_1,m_1)-\sigma(x_2,m_2) \vert \leq k \vert x_1-x_2 \vert, \quad \forall (x_1,x_2,m_1,m_2) \in \real^4.$ Let us finally introduce the object of interest of this Note that is the following backward stochastic differential equation (BSDE) coupled with the forward process $X^{t,x,m}$ as $$\begin{aligned}
\label{BSDE}
Y_s^{t,x,m}=&F(X_T^{t,x,m})-\int_t^T Z_r^{t,x,m} dM_r + \int_t^T f(r,Y_r^{t,x,m},Z_r^{t,x,m}) dC_r-\int_t^T dN_r^{t,x,m}\nonumber\\
&+ \frac{\kappa}{2} \int_t^T d\langle N^{t,x,m},N^{t,x,m}\rangle_r.\end{aligned}$$ where $F:\real\to\real$ is a bounded deterministic function of class $\mathcal{C}^2(\real)$ with bounded derivatives. The generator $f: [0,T] \times \real \times \real \to \real$ is assumed to be $\mathcal{B}([0,T]) \otimes \mathcal{B}(\real) \otimes \mathcal{B}(\real)$-measurable where $\mathcal{B}(\real)$ is for the Borel $\sigma$-filed on $\real$ (so that $f(r,x,m)$ is deterministic for non-random $(r,x,m)$ in $[0,T]\times\real^2$) and is such that there exists a deterministic constant $c$ satisfying $ \sup_{r \in [0,T]} \vert f(r,0,0) \vert \leq c.$ We assume in addition that the generator $f$ is quadratic in $z$ and Lipschitz in $y$. The typical example being when $f$ is of the form $f(s,y,z)=l(s,y)+\eta \vert z \vert^2$ where $\eta$ is a fixed constant and $l$ is Lipschitz in $y$ (the more general “quadratic” assumptions can be found for example in [@ImkellerReveillacRichter]). We recall that in this setting, it is shown in [@Morlais] that there exists a unique triple $(Y^{t,x,m}, Z^{t,x,m}, N^{t,x,m}) \in \mathcal{S}^\infty\times L^2(d \langle M, M \rangle \otimes d\P)\times \mathcal{M}^2$ where $\mathcal{S}^\infty$ is the space of bounded and continuous $(\mathcal{F}_t)_t$-adapted processes, $L^2(d \langle M, M \rangle \otimes d\P)$ denotes the space of square integrable $(\mathcal{F}_t)_t$-predictable processes and $\mathcal{M}^2$ the space of square integrable $(\mathcal{F}_t)_t$-martingales $N$ strongly orthogonal to $M$ (*i.e.* $\langle M, N \rangle=0$). We also mention that these processes are real-valued. We finally stress that all the conditions and assumptions previously mentioned will be assumed to hold in the rest of this Note and that $K$ denotes a constant which can differ from one line to another. We conclude this section by recalling some important facts. First let us mention that only the couple $(X,M)$ is an homogeneous strong Markov process.
([@CinlarJacodProtterSharpe Theorem (8.11)] or [@Protter V. Theorem 35]) The process $(X_s^{t,x,m},M_s^{t,m})_{s\in [t,T]}$ is an homogeneous strong Markov process for the filtration $(\mathcal{F}_t)_{t\in [0,T]}$. If in addition $M$ is assumed to enjoy the independent increments property then the stochastic process $(X_s^{t,x,m})_{s\in [t,T]}$ is a strong Markov process.
The Markov property of the couple $(X,M)$ transfers to the solution of and .
([@ImkellerReveillacRichter Proposition 3.2, Theorem 3.4]) \[prop:Markovprop\] There exist two deterministic functions $u,v:[0,T] \times \real^2 \to \real$, $\mathcal{B}([0,T]) \otimes \mathcal{B}(\real^2)$ such that $(Y^{t,x,m},Z^{t,x,m})$ in and satisfy $$Y_s^{t,x,m}=u(s,X_s^{t,x,m},M_s^{t,m}), \quad Z_s^{t,x,m}=v(s,X_s^{t,x,m},M_s^{t,m}) \sigma(s,X_s^{t,x,m},M_s^{t,m}), \quad s \in [t,T]$$ where $\mathcal{B}_e(\real^2)$ is the $\sigma$-field on $\real^2$ generated by functions $(x,m) \mapsto \E\L[\phi(s,X_s^{t,x,m},M_s^{t,m}) dC_s\R]$ with $\phi:\Omega \times [0,T] \times \real^2 \to \real$ a continuous bounded function.
Finally we will use the following [^1][property]{} for the solution of the BSDE .
(Particular case of [@ImkellerReveillacRichter Theorem 4.6]) \[prop:Diff\] The map $ (x,m)\mapsto Y^{1,t,x,m} $ is of class $\mathcal{C}^1(\real^2)$ $\P$-a.s. where $Y^{1,t,x,m}$ is as in below.
Main result
===========
We are now ready to state and prove the main result of this Note.
\[theorem:main\] Assume that assumptions of Section \[section1\] are in force then $N^{t,x,m}$ in is equal to zero and equation becomes $$Y_s^{t,x,m}=F(X_T^{t,x,m})-\int_t^T Z_r^{t,x,m} dM_r + \int_t^T f(r,Y_r^{t,x,m},Z_r^{t,x,m}) dC_r.$$
First note that it is enough to assume that the generator $f$ is Lipschitz in $(y,z)$. Indeed, in [@Morlais Theorems 2.5-2.6], the existence and uniqueness of the solution $(Y^{t,x,m},Z^{t,x,m},N^{t,m})$ of the BSDE is given as a limit of solutions of Lipschitz BSDEs. As a consequence, $N^{t,x,m}$ is equal to zero in if the orthogonal martingale parts $N$ in the approximating Lipschitz BSDEs vanish. So assume $f$ to be Lipschitz in $(y,z)$. In [@ElKarouiHuang] the authors show that the unique solution of is obtained as the limit of the following Picard iteration: $$\begin{aligned}
\label{eq.Picard}
Y_s^{0,t,x,m}&=Z_s^{0,t,x,m}=0,\nonumber\\
Y_s^{k+1,t,x,m}&=F(X_T^{t,x,m})- \int_s^T Z_r^{k,t,x,m} dM_r +\int_s^T f(r,Y_r^{k,t,x,m},Z_r^{k,t,x,m}) dC_r\nonumber\\
&- \int_s^T dN_r^{t,x,m} + \kappa \int_s^T d\langle N^{t,x,m}, N^{t,x,m}\rangle_r, \quad k\geq 0. \end{aligned}$$ Note that $N^{t,x,m}$ is not part of the iteration (we refer to [@ElKarouiHuang Proof of Theorem 6.1] for more details). This remark leads to the main idea of the proof. Indeed, consider the first iteration, *i.e.* $(Y^{1,t,x,m}, Z^{1,t,x,m}, N^{t,x,m})$ defined by $$\label{eq.first}
Y_s^{1,t,x,m}=F(X_T^{t,x,m})- \int_s^T Z_r^{1,t,x,m} dM_r +\int_s^T f(r,0,0) dC_r- \int_s^T dN_r^{t,x,m} + \kappa \int_s^T d\langle N^{t,x,m}, N^{t,x,m}\rangle_r.$$ By the a priori estimates obtained in [@ElKarouiHuang Proposition 6.3] the triplet $(Y^{1,t,x,m}, Z^{1,t,x,m}, N^{t,x,m})$ is unique. As a consequence if we show that $N^{t,x,m}=0$ in equation then the Theorem is proved since $(Y^{k,t,x,m}, Z^{k,t,x,m}, N^{t,x,m})$ converges to the unique solution of . The rest of the proof is devoted to this fact.\
\
Since $Y^{1,t,x,m}$ is $(\mathcal{F}_\cdot)$-adapted it holds by Markov property that $$Y_s^{1,t,x,m}=g(s,X_s^{t,x,m},M_s^{t,m}), \; \textrm{ with } g(s,x,m):=\E\left[F(X_{T-s}^{t,x,m}) -\int_s^T f(r,0,0) dC_r\right].$$ In addition, Proposition \[prop:Diff\] applied to gives that the application $(x,m) \mapsto g(t,x,m)$ is of class $\mathcal{C}^1(\real\times\real)$ for every $t$. We mimic a technique given in [@ImkellerReveillacRichter] and compute $\langle Y^{1,t,x,m}, N^{1,t,x,m} \rangle_s$ for $s\geq t$. Let $\pi^{(n)}:=\{t=t_0^{(n)} \leq t_1^{(n)} \leq \cdots \leq t_N^{(n)}=s\}$ be a family of subdivisions of $[t,s]$ whose mesh $\vert \pi^{(n)} \vert$ tends to zero as $n$ goes to the infinity. For sake of simplicity the superscript $(n)$ will be omitted in the following computations. $$\begin{aligned}
\label{bracket1}
\langle Y^{1,t,x,m}, N^{t,x,m} \rangle_s&=&\langle g(\cdot,X_\cdot^{t,x,m},M_\cdot^{t,x,m}), N^{t,x,m} \rangle_s\nonumber\\
&\overset{\P}{=}&\lim_{n \to \infty} \sum_{j=1}^r (g(t_{j+1},X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})-g(t_j,X_{t_j}^{t,x,m},M_{t_j}^{t,x,m})) \Delta_j N^{t,x,m}\nonumber\\
&\overset{\P}{=}& \lim_{n \to \infty} \sum_{j=1}^r\bigg[ (g(t_j,X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})-g(t_j,X_{t_j}^{t,x,m},M_{t_j}^{t,x,m})) \Delta_j N^{t,x,m}\nonumber\\
&&+(g(t_{j+1},X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})-g(t_j,X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})) \Delta_j N^{t,x,m} \bigg].\end{aligned}$$ We consider the two sumands above separately. For the first part we follow a technique used in [@ImkellerReveillacRichter] and apply the mean theorem. Let $\bar{M}_j$ (respectively $\bar{X}_j$) below a random point between $M_{t_j}^{t,x,m}$ and $M_{t_{j+1}}^{t,x,m}$ (resp. $X_{t_j}^{t,x,m}$ and $X_{t_{j+1}}^{t,x,m}$) in the computations below. We have $$\begin{aligned}
\label{bracket2}
&&\sum_{j=1}^r (g(t_j,X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})-g(t_j,X_{t_j}^{t,x,m},M_{t_j}^{t,x,m})) \Delta_j N^{t,x,m}\nonumber\\
&=&\sum_{j=1}^r (g(t_j,X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})-g(t_j,X_{t_j}^{t,x,m},M_{t_{j+1}}^{t,x,m})) \Delta_j N^{t,x,m}\nonumber\\
&&+\sum_{j=1}^r (g(t_j,X_{t_j}^{t,x,m},M_{t_{j+1}}^{t,x,m})-g(t_j,X_{t_j}^{t,x,m},M_{t_j}^{t,x,m})) \Delta_j N^{t,x,m}\nonumber\\
&=&\sum_{j=1}^r \bigg[\partial_2 g(t_j,X_{t_j}^{t,x,m},M_{t_j}^{t,x,m}) \Delta_j X \Delta_j N^{t,x,m}+ \partial_3 g(t_j,X_{t_j}^{t,x,m},M_{t_j}^{t,x,m}) \Delta_j M \Delta_j N^{t,x,m} +R_{j,r}\bigg] \quad\quad\quad{}\end{aligned}$$ where $R_{j,r}$ is defined as $$\begin{aligned}
R_{j,r}&:=& (\partial_2 g(t_j,\bar{X}_j,M_{t_{j+1}}^{t,x,m}-\partial_2 g(t_j,X_{t_j}^{t,x,m},M_{t_j}^{t,x,m})) \Delta_j X \Delta_j N^{t,x,m}\\
&&+ (\partial_3 g(t_j,X_{t_j}^{t,x,m},\bar{M}_j) -\partial_3 g(t_j,X_{t_j}^{t,x,m},M_{t_j}^{t,x,m})) \Delta_j M \Delta_j N^{t,x,m}.\end{aligned}$$ Since $(x,m) \mapsto g(s,x,m)$ is of class $\mathcal{C}^1$ for every $s$ in $[0,T]$ the remainder term $\sum_{j=0}^r R_{j,r}$ as $r$ goes to infinity (we refer to [@ImkellerReveillacRichter Proof of (5.13)] for the complete justifications). Then it follows using that $$\begin{aligned}
&&\lim_{r\to\infty} \sum_{j=1}^r (g(t_j,X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})-g(t_j,X_{t_j}^{t,x,m},M_{t_j}^{t,x,m})) \Delta_j N^{t,x,m}\\
&=&\langle \int_t^\cdot \partial_2 g(r,X_r^{t,x,m},M_r^{t,x,m}) \sigma(r,X_r^{t,x,m},M_r^{t,x,m})+ \partial_3 g(r,X_r^{t,x,m},M_r^{t,x,m}) dM_r, N_\cdot^{t,x,m} \rangle_s=0\end{aligned}$$ by strong orthogonality between $M$ and $N$. As a consequence, relation reduces to $$\label{bracket5}
\langle Y^{1,t,x,m}, N^{t,x,m} \rangle_s\overset{\P}{=} \lim_{n \to \infty} \sum_{j=1}^r (g(t_{j+1},X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})-g(t_j,X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})) \Delta_j N^{t,x,m}.$$ We have that $$\begin{aligned}
&&\L\vert \sum_{j=1}^r (g(t_{j+1},X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})-g(t_j,X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})) \Delta_j N^{t,x,m} \R\vert^2\\
&\leq&\sum_{j=1}^n \left\vert g(t_{j+1},X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})-g(t_j,X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,x,m})\right\vert^2 \times \sum_{j=1}^n \vert \Delta_j N \vert^2\\
&=&\sum_{j=1}^n \left\vert\E\left[F(X_{T-t_{j+1}}^{0,X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,m}})-F(X_{T-t_j}^{0,X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,m}})-\int_{t_j}^{t_{j+1}} f(r,0,0) dC_r\right]\right\vert^2 \times \sum_{j=1}^n \vert \Delta_j N \vert^2\\
&\leq& 2 \left[\sum_{j=1}^n \left\vert\E\left[F(\tilde{X}_{T-t_{j+1}})-F(\tilde{X}_{T-t_j}) \right]\right\vert^2 + \sum_{j=1}^n \left\vert\E\left[\int_{t_j}^{t_{j+1}} f(r,0,0) dC_r\right]\right\vert^2 \right]\times \sum_{j=1}^n \vert \Delta_j N \vert^2\\\end{aligned}$$ where for simplicity of notations we set $\tilde{X}_s:=X_s^{0,X_{t_{j+1}}^{t,x,m},M_{t_{j+1}}^{t,m}}$. Let $\bar{X}_j$ be a random point between $\tilde{X}_{T-t_{j+1}}$ and $\tilde{X}_{T-t_j}$. Writing $\E\L[F(\tilde{X}_{T-t_{j+1}})-F(\tilde{X}_{T-t_j})\R]$ as $$\begin{aligned}
&&\E\L[F(\tilde{X}_{T-t_{j+1}})-F(\tilde{X}_{T-t_j})\R]\\
&=& \E\L[F'(\tilde{X}_{T-t_{j+1}}) \L(\tilde{X}_{T-t_{j+1}}-\tilde{X}_{T-t_j}\R)\R] + \frac12 \E\L[F''(\bar{X}_j) \L\vert \tilde{X}_{T-t_{j+1}}-\tilde{X}_{T-t_j}\R\vert^2\R]\\
&=& \E\L[F'(\tilde{X}_{T-t_{j+1}}) \E\L[\tilde{X}_{T-t_{j+1}}-\tilde{X}_{T-t_j}\vert \mathcal{F}_{T-t_{j+1}}\R]\R] + \frac12 \E\L[F''(\bar{X}_j) \L\vert \tilde{X}_{T-t_{j+1}}-\tilde{X}_{T-t_j}\R\vert^2\R]\\
&\leq& K \E\L[\L\vert \tilde{X}_{T-t_{j+1}}-\tilde{X}_{T-t_j}\R\vert^2\R]\end{aligned}$$ and $f(r,0,0)$ as $ f(r,0,0)= \max\{f(r,0,0),0\}-\max\{-f(r,0,0),0\}$ it follows that $$\begin{aligned}
&&\left\vert\sum_{j=1}^n \left(g(t_{j+1},M_{t_{j+1}})-g(t_j,M_{t_{j+1}})\right) \Delta_j N \right\vert^2\\
&\leq& K \left[\sum_{j=1}^n \left\vert E\left[ \vert \tilde{X}_{T-t_{j+1}}-\tilde{X}_{T-t_j}\vert^2 \right] \right\vert^2+ \sup_{r\in[0,T]} \vert f(r,0,0)) \vert \sum_{j=1}^n \left\vert\E\left[C_{t_{j+1}}-C_{t_j}\right]\right\vert^2 \right] \times \sum_{j=1}^n \vert \Delta_j N \vert^2\\
&\leq& K \left[\sum_{j=1}^n E\left[ \vert \tilde{X}_{T-t_{j+1}}-\tilde{X}_{T-t_j}\vert^4 \right] + \sup_{r\in[0,T]} \vert f(r,0,0)) \vert \sum_{j=1}^n \E\left[\vert M_{t_{j+1}}-M_{t_j} \vert^4\right] \right] \times \sum_{j=1}^n \vert \Delta_j N \vert^2\\
&\leq& K \left(E\left[\sum_{j=1}^\infty \vert \tilde{X}_{T-t_{j+1}}-\tilde{X}_{T-t_j}\vert^4 \right] + \E\left[ \sum_{j=1}^\infty \vert M_{t_{j+1}}-M_{t_j} \vert^4\right] \right) \times \sum_{j=1}^\infty \vert \Delta_j N \vert^2\\
&=&0\end{aligned}$$ since the quartic variations of a martingale are zero. The previous computation and the equality entail that $$\label{bracket6}
\langle Y^{1,t,x,m}, N^{t,x,m} \rangle_s\overset{\P}{=} 0.$$ On the other hand, the covariation $\langle Y^{1,t,x,m}, N^{t,x,m} \rangle_s$ in the BSDE equals to $$\label{bracket7}
\langle Y^{1,t,x,m}, N^{t,x,m} \rangle_s\overset{\P}{=} \langle N^{t,x,m}, N^{t,x,m} \rangle_s.$$ Hence relations and give that $N_s^{t,x,m}=N_0^{t,x,m}$ for every $s$ in $[t,T]$.
[1]{}
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[^1]: Note that this result ([@ImkellerReveillacRichter Theorem 4.6]) has been proved under an additional technical assumption (MRP) with $f$ a quadratic generator. Since the generator in equation is very simple, using only an exponential change we can apply the computations realized in [@ImkellerReveillacRichter Theorem 4.6] without assuming the hypothesis (MRP). The full proof of this fact will be presented in a paper in preparation.
|
---
abstract: |
In this paper, some relations between the topological parameter $d$ and concurrences of the projective entangled states have been presented. It is shown that for the case with $d=n$, all the projective entangled states of two $n$-dimensional quantum systems are the maximally entangled states (i.e. $C=1$). And for another case with $d\neq n$, $C$ both approach $0$ when $d\rightarrow
+\infty$ for $n=2$ and $3$. Then we study the thermal entanglement and the entanglement sudden death (ESD) for a kind of Yang-Baxter Hamiltonian. It is found that the parameter $d$ not only influences the critical temperature $T_{c}$, but also can influence the maximum entanglement value at which the system can arrive at. And we also find that the parameter $d$ has a great influence on the ESD.
author:
- Chunfang Sun
- Kang Xue
- Gangcheng Wang
- Chunfeng Wu
title: A study on the relations between the topological parameter and entanglement
---
Introduction
============
Quantum Entanglement(QE)[@R.F.; @Werner], the most surprising nonclassical property of quantum system, provides a fundamental resource in realizing quantum information and quantum computers [@M.; @Nielsen] and is widely exploited in quantum cryptography[@A.K.; @Ekert], dense coding, teleportation [@C.H.; @Bennett]. It has been clarified that the entanglement of a quantum state is one of the most important properties not only in quantum information science but also in condensed matter physics. The thermal entanglement has been investigated in the system of the Heisenberg XXX [@XXX; @XXX1], XX [@XX], XXZ [@XXZ], and the Ising [@D.G] models. Recently in Ref.[@T.; @Yu1] it has been shown that there exists a certain class of two-qubit states which display a finite entanglement decay time. This phenomenon is aptly called ESD and cannot be predicted from quantum decoherence which is an asymptotic phenomenon. It has received a lot of attentions both theoretically and experimentally[@T.; @Yu1; @T.; @Yu2; @Z.F; @S.G; @M.; @P].
The Temperley-Lieb algebra(TLA) first appeared in statistical mechanics as a tool to analyze various interrelated lattice models[@TLA] and was related to link and knot invariants[@wda]. Either algebraically by generators and relations as in Jones¡¯ original presentation[@Jones], or as a diagram algebra modulo planar isotopy as in Kauffman¡¯s presentation[@Kauffman3], the TLA has always hitherto been presented as a quotient of some sort. Recently in Ref.[@zhang], the TLA is found to present a suitable mathematical framework for describing quantum teleportation, entangle swapping, universal quantum computation and quantum computation flow. In a very recent work[@Abramsky3], Abramsky traced some of the surprising and beautiful connections from knot theory to logic and computation via quantum mechanics. However, the physical meaning of the important topological parameter $d$ (describing the unknotted loop ¡°$
\bigcirc $¡± in topology) is still unclear. Motivated by this, in this paper we focus on studying the relations between the parameter $d$ and entanglement to explore what role do the parameter $d$ play in the entanglement.
The paper is organized as follows: In Sec.2, we study the relations between the topological parameter $d$ and concurrences of the projective entangled states. It is shown that for the case with $d=n$, all the projective entangled states of two $n$-dimensional quantum systems are the maximally entangled states (i.e. $C=1$). And for another case with $d\neq n$, $C$ both approach $0$ when $d\rightarrow\infty$ for $n=2$ and $3$. In Sec.3, the thermal entanglement for a kind of Yang-Baxter Hamiltonian related to the TLA is investigated. We find that the parameter $d$ has great influences on the thermal entanglement. It not only influences the critical temperature $T_{c}$, but also can influence the maximum entanglement value at which the system can arrive at. In Sec.4, the ESD for the same Yang-Baxter Hamiltonian is investigated. It is found that the parameter $d$ has a great influence on the ESD. A summary is given in the last section.
some relations between the parameter $d$ and $C$ {#sec2}
=================================================
In this section, we first obtain the projective entangled state $|\Psi_{i,i+1}\rangle$ of two $n$-dimensional quantum systems, which contains the topological parameter $d$. Then we come to investigate the concurrences $C$ of the states $|\Psi_{i,i+1}\rangle$ to explore some relations between the parameter $d$ and $C$.
In order to keep the paper self-contained, we first briefly review the theory of the TLA[@TLA]. It is a unital algebra generated by $U_{i}$( $i=1,2,...,N-1$) which satisfy the following relations, $$\begin{aligned}
\label{1}
U_{i}^{2}&=&dU_{i} \nonumber\\
U_{i}U_{j}U_{i}&=&U_{i}~~~~~~~~~~|i-j|=1 \nonumber\\
U_{i}U_{j}&=&U_{j}U_{i} ~~~~~~~|i-j|>1\end{aligned}$$ where $d$ ($ d\in\mathbb{C}$ and $ d\neq 0$) is the unknotted loop ¡°$ \bigcirc $¡± in the knot theory which does not depend on the sites of the lattices. The notation $U_{i}\equiv U_{i,i+1}$ is used, $U_{i,i+1}$ is short for $1_{1}\otimes \cdots \otimes 1_{i-1}\otimes
U_{i,i+1}\otimes 1_{i+2} \otimes \cdots \otimes1_{N}$, and $1_{j}$ represents the unit matrix of the $j$-th particle. These relations are diagrammatically represented in Fig. 1.
![Diagrammatic the TLA[]{data-label="fig1"}](TLA.eps)
In the following, we write the $n^{2}\times n^{2}$ $(n=2,3,...,n)$ matrix $U$ as a form of projectors in the tensor product of two nearest $n$-dimensional quantum spaces as follows, $$\label{U}
U_{i,i+1}=d|\Psi_{i,i+1}\rangle\langle\Psi_{i,i+1}|,$$ where $|d|^{\frac{1}{2}}|\Psi_{i,i+1}\rangle$ describes $\bigcup$ and $|d|^{\frac{1}{2}}\langle\Psi_{i,i+1}|$ describes $\bigcap$ in topology. Although the parameter $d$ can be arbitrary, in this paper we restrict ourselves on $d> 0$ for convenience. The projective entangled state $|\Psi_{i,i+1}\rangle$ of two $n$-dimensional quantum systems, which contains the topological parameter $d$, takes of the following form, $$\begin{aligned}
\label{state}
\begin{array}{ll}
|\Psi_{i,i+1}\rangle=\sum_{\lambda,\mu=0}^{n-1}\alpha_{\lambda\mu}|\lambda\rangle_{i}|\mu\rangle_{i+1}.
\end{array}\end{aligned}$$ where $|\lambda\rangle_{i}$ and $|\mu\rangle_{i+1}$ are the orthonormal bases of the Hilbert spaces $i$ and $i+1$ respectively, and $\alpha_{\lambda\mu}$’s are complex numbers satisfying the normalization condition $\sum_{\lambda,\mu=0}^{n-1}|\alpha_{\lambda\mu}|^{2}=1$. And we set in each row $\lambda$ and each column $\mu$ of the matrix $\alpha$ there is a single nonzero element. The generators can be written as, $$\label{5}
(U_{i,i+1})^{\lambda\mu}_{\lambda^{'}\mu^{'}}=d
\alpha_{\lambda\mu}\alpha_{\lambda^{'}\mu^{'}}^{\ast}~~~~~~
\lambda,\mu,\lambda^{'},\mu^{'}=0,1,2,...,n-1.$$ By calculation, it is easy to see that the first relation of Eq.(\[1\]) is automatically satisfied. In order to satisfy the second relation of Eq.(\[1\]), the fulfilled conditions read, $$\label{6}
\left\{
\begin{array}{ll}
d^{2}\sum_{\lambda,\nu,\sigma=0}^{n-1}\alpha_{\nu\lambda}^{\ast}\alpha_{\lambda\mu}\alpha_{\nu\sigma}\alpha_{\sigma\beta}^{\ast}=\delta_{\mu\beta},\\
&\\
d^{2}\sum_{\lambda,\nu,\sigma=0}^{n-1}\alpha_{\mu\lambda}\alpha_{\lambda\nu}^{\ast}\alpha_{\beta\sigma}^{\ast}\alpha_{\sigma\nu}=\delta_{\mu\beta},
\end{array}
\right.$$ where $\mu=0,1,2,...,n-1$. By this limited conditions Eq.(\[6\]), the projective entangled state $|\Psi_{i,i+1}\rangle$ of two $n$-dimensional quantum systems and the corresponding topological parameter $d$ can be determined. Via Eq.(\[U\]), the corresponding $n^{2}\times n^{2}$ matrix $U$ can also be obtained. Next via two classes of the parameter $d$, we come to study the relations between the parameter $d$ and the concurrences $C$ of the corresponding states $|\Psi_{i,i+1}\rangle$.
Example I: the case with the parameter $d=n$
--------------------------------------------
In example $I$, we will discuss a series of the generalized $n^{2}\times n^{2}$ ($n=2,3,...,n$) matrix $U$ with the topological parameter $d=n$.
For the case with $\lambda=\mu$ and $\lambda^{'}=\mu^{'}$($\lambda,\mu,\lambda^{'},\mu^{'}=0,1,2,...,n-1$) in the tensor product of two nearest $n$-dimensional quantum spaces, via Eq.(\[6\]) and Eq.(\[state\]), the corresponding state is $$\label{n}
|\Psi\rangle=\sum_{\lambda=0}^{n-1}\frac{1}{\sqrt{n}}e^{ik_{\lambda\lambda}}|\lambda\lambda\rangle,$$ where the topological parameter $d=n$ and the parameters $k_{\lambda\lambda}$ are arbitrary real. By means of concurrence, we study these entangled states. In Ref.[@Albeverio], the generalized concurrence (or the degree of entanglement[@Hill]) for two qudits is given by, $$\label{C}
C=\sqrt{\frac{n}{n-1}(1-I_{1})},$$ where $I_{1}=Tr[\rho_{A}^{2}]=Tr[\rho_{B}^{2}]=|\kappa_{0}|^{4}+|\kappa_{1}|^{4}+\cdots+|\kappa_{n-1}|^{4}$, with $\rho_{A}$ and $\rho_{B}$ are the reduced density matrices for the subsystems, and $\kappa_{j}$’s($j=0,1,\ldots,n-1$) are the Schmidt coefficients. Then we can obtain the generalized concurrence of the state $|\Psi\rangle$ (\[n\]) as follows, $$\label{20}
C=1.$$ It is interesting that for the series of $n^{2}\times n^{2}$ matrix $U$ with the topological parameter $d=n$, all the projective states $|\Psi\rangle$ have the maximum entanglement. The state $|\Psi\rangle$ in Eq. (\[n\]) can be considered as a straightforward generalization of the symmetric Bell state $|\Phi^{+}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ when $e^{ik_{\lambda\lambda}}=1$.
Example II: the case with the parameter $d\neq n$
--------------------------------------------------
In example $II$, we will discuss another class of the $n^{2}\times n^{2}$ matrix $U$ with the topological parameter $d\neq n$. Because in this case we can’t obtain the general generalized $n^{2}\times n^{2}$ matrix $U$, we will study the cases with $n=2$ and $n=3$.
For the case with $n=2$, via Eq.(\[6\]) and Eq.(\[state\]), the corresponding state is $$\label{state 1}
\begin{array}{ll}
|\Psi\rangle=\frac{1}{\sqrt{1+q^{2}}}(q
e^{ik_{01}}|01\rangle+e^{ik_{10}}|10\rangle),
\end{array}$$ where the topological parameter $d=q+q^{-1}$ and $k_{01},k_{10},q\in$ real. Hereafter, $q> 0$. The generalized concurrence of the state (\[state 1\]) is, $$\label{concurrence2}
C=\frac{2}{d}, ~~~~~~~~where ~~~d\geq 2.$$ For the case with $n=3$, via Eq.(\[6\]) and Eq.(\[state\]), there are three sets of solutions and the corresponding states are, $$\label{state 2}
\begin{array}{ll}
|\Psi\rangle^{(1)}=\frac{1
}{\sqrt{1+q+q^{2}}}(qe^{ik_{02}}|02\rangle+\sqrt{q}e^{ik_{11}}|11\rangle+e^{ik_{20}}|20\rangle),\\
&\\
|\Psi\rangle^{(2)}=\frac{1
}{\sqrt{1+q+q^{2}}}(qe^{ik_{01}}|01\rangle+e^{ik_{10}}|10\rangle+\sqrt{q}e^{ik_{22}}|22\rangle),\\
&\\
|\Psi\rangle^{(3)}=\frac{1
}{\sqrt{1+q+q^{2}}}(\sqrt{q}e^{ik_{00}}|00\rangle+qe^{ik_{12}}|12\rangle+e^{ik_{21}}|21\rangle),
\end{array}$$ where the parameter $d=q+q^{-1}+1$ and $k_{\lambda\mu}$ $\in$ real ($\lambda,\mu=0,1,2$). All their concurrences are the same as, $$\label{concurrence3}
C=\sqrt{\frac{3}{d}},~~~~~~~~where~~~ d\geq 3.$$
Via these two examples, it is shown that there are some relations between the topological parameter $d$ and concurrences $C$ of the entangled states. In other words, the parameter $d$ has great influences on the entanglement. Example $I$ and Example $II$(i.e., $q=1$) show that for the series of the parameter $d=n$, all the projective states $|\Psi\rangle$ of two $n$-dimensional quantum systems are the maximally entangled states (i.e., $C=1$). For another class of the parameter $d\neq n$ (i.e., $q\neq1$) in example $II$, via investigating the cases with $n=2$ and $n=3$, it is found that the concurrences $C$ both decrease when $d$ goes up, and it approaches $0$ when $d\rightarrow +\infty$, as shown in Fig. 2. We guess that for the generalized $n^{2}\times n^{2}$ matrix $U$ with the parameter $d\neq n$, the conclusion, which is that when the parameter $d\rightarrow +\infty$, $C$ approaches $0$, is also correct. Another fact in Fig. 2 is that for the same value of loop $d$, the concurrence of the entangled two-qutrit states (\[state 2\]) is always larger than the concurrence of the entangled two-qubit states (\[state 1\]). This means that the concurrence not only depends on the topological parameter $d$, but also depends on the dimension $n$.
![The concurrence is plotted versus the parameter $d$. The solid line corresponds to $C=\frac{2}{d}$ for $n=2$, and the dotted line corresponds to $C=\sqrt{\frac{3}{d}}$ for $n=3$[]{data-label="fig2"}](C23.eps)
Thermal entanglement in a YANG-BAXTER SYSTEM {#sec4}
============================================
In this section, we come to study the thermal entanglement for a kind of Yang-Baxter Hamiltonian, which is related to the TLA for $n=2$, to explore the influences of the parameter $d$ on the thermal entanglement.
By substituting Eq.(\[state 1\]) into Eq.(\[U\]) for $n=2$, in the standard basis $\{|00\rangle, |01\rangle, |10\rangle,
|11\rangle\}$, the $4\times 4$ matrix $U$ is, $$\begin{aligned}
U=\left(
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & q & e^{i\varphi} & 0 \\
0 & e^{-i\varphi} & q^{-1} & 0 \\
0 & 0 & 0 & 0 \\
\end{array}
\right),\end{aligned}$$ with the topological parameter $d=q+q^{-1}$ and $\varphi=k_{01}+k_{10} \in $ real.
As is known, the Yang-Baxter equation (YBE)[@yang; @baxter; @drin] is given by, $$\breve{R}_{i}(x)\breve{R}_{i+1}(xy)\breve{R}_{i}(y)=\breve{R}_{i+1}(y)\breve{%
R}_{i}(xy)\breve{R}_{i+1}(x) , \label{YBE}$$ where $x$ and $y$ are spectrum parameters. Via the trigonometric Yang-Baxterization approach[@ckg2], it gives
$$\begin{aligned}
\breve{R}(x)=[q^{2}+q^{-2}-(x^{2}+x^{-2})]^{-\frac{1}{2}}[(q
x-q^{-1}x^{-1})I
-(x-x^{-1})U], \nonumber\\
\breve{R}^{-1}(x)=[q^{2}+q^{-2}-(x^{2}+x^{-2})]^{-\frac{1}{2}}[(qx^{-1}-q^{-1}x)I
+(x-x^{-1})U].\end{aligned}$$
It is easy to check that $\breve{R}^{+}(x)=\breve{R}^{-1}(x)=\breve{R}(-x)$ for $
x=e^{i\theta}$, where $\theta\in$ real. It is worth to mention that in this paper, the real parameters $\theta$ and $\varphi$ are time-independent.
Here we study a original Hamiltonian describing two spin-1/2 particles (particle 1 and 2) interaction, $$H_{0}=\mu_{1}S_{1}^{z}+\mu_{2}S_{2}^{z}+gS_{1}^{z}S_{2}^{z},$$ where $\mu_{i}$ ($i=1,2$) represent external magnetic field and $g$ is the interaction of $z$-component of two-qubit spins. Taking the Schr$\ddot{o}$dinger equation $i\hbar
\partial|\Psi\rangle/\partial
t=H|\Psi\rangle$ into account, where $|\Psi\rangle=\check{R}(x)|\Psi_{0}\rangle$ and $|\Psi_{0}\rangle$ is the eigenstate of $H_{0}$, one can get a new Hamiltonian as $H(\theta,\varphi)=\check{R}(x)H_{0}\breve{R}^{-1}(x)$[@sun], where the real parameters $\theta$ and $\varphi$ are time-independent. For convenience, we let $x=i$ (i.e., $\theta=\frac{\pi}{2}$). Then we arrive at a new Hamiltonian, $$\begin{aligned}
\label{H}
H=(B+J(1-\frac{8}{d^{2}}))S^{z}_{1}+(B-J(1-\frac{8}{d^{2}}))S^{z}_{2}
+gS^{z}_{1}S^{z}_{2}
-\frac{4J\sqrt{d^{2}-4}}{d^{2}}(e^{i\varphi}S^{+}_{1}S^{-}_{2}+e^{-i\varphi}S^{-}_{1}S^{+}_{2})
,\end{aligned}$$ where $B=\frac{\mu_{1}+\mu_{2}}{2}$ and $J=\frac{\mu_{1}-\mu_{2}}{2}$, and $S^{\pm}_{i}=S^{x}_{i}\pm i
S^{y}_{i}$ are raising and lowering operators respectively for the $i$-th particle. Specifically, we find that when $\varphi=\pi$, this model is the two-qubit anisotropic Heisenberg $XXZ$ model under an inhomogeneous magnetic field. $B \geq 0$ is restricted, and the magnetic fields on the two spins have been so parameterized that $J(1-8/d^{2})$ controls the degree of inhomogeneity. For the system (\[H\]), its corresponding eigenstates read $|\Psi_{1}\rangle=|00\rangle$, $|\Psi_{2}\rangle=|11\rangle|$, $|\Psi_{3}\rangle=\frac{2}{d}(\frac{-\sqrt{d^{2}-4}e^{i\varphi}}{2}|01\rangle+|10\rangle)$, $|\Psi_{4}\rangle=\frac{2}{d}(|01\rangle+\frac{\sqrt{d^{2}-4}e^{-i\varphi}}{2}|10\rangle)$, with corresponding energies $E_{1}=B+\frac{g}{4}$, $E_{2}=-B+\frac{g}{4}$, $E_{3}=J-\frac{g}{4}$, $E_{4}=-J-\frac{g}{4}$.
Next to quantify the entanglement of formation of a mixed state $\rho$ of two qubits, we use the Wootters concurrence [@W.K] defined as, $$\begin{aligned}
\label{c}
C(t)
=\max\{0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\},\end{aligned}$$ where $\{\lambda_{i}\}$ are the eigenvalues of the matrix $\rho(\sigma_{y}^{A}\otimes
\sigma_{y}^{B})\rho^{\ast}(\sigma_{y}^{A}\otimes \sigma_{y}^{B})$, with $\rho^{\ast}$ denoting complex conjugation of the matrix $\rho$ and $\sigma_{y}^{A/B}$ are the Pauli matrices for atoms A and B. When spin chains are subjected to environmental disturbance, they inevitably become thermal equilibrium states. The thermal state at finite temperature $T$ is $\rho(T)=\frac{1}{Z}\exp(-\frac{H}{kT})$, where $Z={\rm Tr}[\exp(-\frac{H}{kT})]$ is the partition function and $k$ is the Boltzmann constant. For simplicity, we write $k=1$. By calculation, the density matrix $\rho(T)$ of the system (\[H\]) can be written as, $$\begin{aligned}
\rho(T)=\frac{1}{2(\cosh\frac{B}{T}+e^{\frac{g}{2T}}\cosh\frac{J}{T})}
\left(
\begin{array}{cccc}
e^{\frac{-B}{T}} & 0 & 0 & 0 \\
0 & e^{\frac{g}{2T}}(\cosh\frac{J}{T}-(1-\frac{8}{d^{2}})\sinh\frac{J}{T}) & \frac{4\sqrt{d^{2}-4}}{d^{2}}e^{\frac{g}{2T}}\sinh\frac{J}{T}e^{i\varphi} & 0 \\
0 & \frac{4\sqrt{d^{2}-4}}{d^{2}}e^{\frac{g}{2T}}\sinh\frac{J}{T}e^{-i\varphi} & e^{\frac{g}{2T}}(\cosh\frac{J}{T}+(1-\frac{8}{d^{2}})\sinh\frac{J}{T}) & 0 \\
0 & 0 & 0 & e^{\frac{B}{T}} \\
\end{array}
\right).\end{aligned}$$ The concurrence is calculated as, $$\begin{aligned}
\label{c2}
C = \max\left(
\frac{\frac{4\sqrt{d^{2}-4}}{d^{2}}e^{\frac{g}{2T}}\sinh\frac{|J|}{T}-1}{\cosh\frac{B}{T}+
e^{\frac{g}{2T}}\cosh\frac{J}{T}},0\right).\end{aligned}$$ Now we do the limit $T \rightarrow 0$ on the concurrence (\[c2\]), we obtain, $$\begin{aligned}
\label{maximum}
\lim_{T\rightarrow 0}C_1 &=& \frac{4\sqrt{d^{2}-4}}{d^{2}}~~~~~ for ~~~~~|B|>|J|+\frac{g}{2}, \nonumber \\
&=&\frac{2\sqrt{d^{2}-4}}{d^{2}} ~~~~~ for ~~~~~|B|=|J|+\frac{g}{2}, \nonumber \\
&=&0 ~~~~~ ~~~~~for ~~~~~~~~~~|B|<|J|+\frac{g}{2}.\end{aligned}$$ It is worth to mention that the influences of the parameters $g$ and $B$ on the thermal entanglement have been discussed in our paper[@sun], whose model corresponds to the topological parameter $d=2$ (i.e., $q=1$). Here we emphasize on exploring the parameter $d$’s influences on the thermal entanglement. From Eq. (\[maximum\]) we can see that at $T=0$, the entanglement vanishes as $|B|$ crosses the critical value $|J|+\frac{g}{2}$, which means that the critical magnetic field $B_{c}$ is independent on the parameter $d$. An important point revealed by Eq. (\[maximum\]) is that the maximum entanglement value at which the system can arrive at, which is $C_{max}=\frac{4\sqrt{d^{2}-4}}{d^{2}}$ for $|B|>|J|+\frac{g}{2}$ at $T=0$, is dependent on the parameter $d$. Fig. 3 shows that when the parameter $d=2\sqrt{2}$, the ground states $|\Psi_{3}\rangle=\frac{1}{\sqrt{2}}(-|01\rangle+|10\rangle)$ or $|\Psi_{4}\rangle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$ both become the maximally entangled states, so the maximum entanglement value $C_{max}=1$. When the parameter $d \rightarrow 2$ or $d
\rightarrow +\infty$, the ground states $|01\rangle$ or $|10\rangle$ both have no entanglement, then the maximum entanglement value $C_{max}=0$. Another important character revealed by Eq. (\[c2\]) is that the critical temperature $T_{c}$, which is determined by the nonlinear equation $\frac{4\sqrt{d^{2}-4}}{d^{2}}e^{\frac{g}{2T_{c}}}\sinh\frac{|J|}{T_{c}}=1$, is also dependent on the parameter $d$. From Fig. 4, it is shown that when $d=2\sqrt{2}$, $T_{c}$ arrive at the maximum value (i.e., $T_{c}$ is about 1.5). When $d \rightarrow 2$ or $d \rightarrow
+\infty$, all the four eigenstates are unentangled states, so the critical temperature $T_{c}=0$. Thus we can obtain a higher entanglement at a fixed temperature via changing the values of the parameter $d$.
![The maximum entanglement value $C_{max}$ is plotted versus the parameter $d$.[]{data-label="fig3"}](Cmax.eps)
![The critical temperature $T_{c}$ is plotted versus the parameter $d$. Coupling constant $J=1$ and the parameter $g=1$.[]{data-label="fig4"}](Tc.eps)
ESD in the same YANG-BAXTER SYSTEM
==================================
In this section, we study the ESD in the same Yang-Baxter system (\[H\]) to explore the influence of the parameter $d$ on the ESD.
The time evolution $U(t)=exp\{-iHt\}$ is written in the basis $\{|00\rangle,|01 \rangle,|10\rangle, |11\rangle\}$, $$\begin{aligned}
\label{uu}
\begin{array}{lll}
U_{11}=e^{-i(B+\frac{g}{4})t}\\
\\
U_{44}=e^{i(B-\frac{g}{4})t}\\
\\
U_{22}=e^{i\frac{g}{4}t}(\cos[J t]-i (1-\frac{8}{d^{2}})\sin[J t]),\\
U_{33}=e^{i\frac{g}{4}t}(\cos[J t]+i (1-\frac{8}{d^{2}})\sin[J t])\\
\\
U_{23}=e^{i\frac{g}{4}t}\frac{4i e^{i\varphi}\sqrt{d^{2}-4}\sin[J t]}{d^{2}}, ~~~~~ U_{32}=e^{i\frac{g}{4}t}\frac{4i e^{-i\varphi}\sqrt{d^{2}-4}\sin[J
t]}{d^{2}}.
\end{array}\end{aligned}$$ It is convenient to choose the initial state $\rho_{0}=\frac{1-\gamma}{4}+\gamma|\psi\rangle\langle\psi|$ $(0<\gamma\leq1)$ with $|\psi\rangle$=$sin\alpha|01\rangle
+cos\alpha|10\rangle$. It is worth to mention that in our paper[@sun1], it has been shown that in Yang-Baxter systems, the ESD is not only sensitive to the initial condition , but also has relations with the different Yang-Baxter systems. And it has been found that the meaningful parameter $\varphi$ has a great influence on the ESD. Here we emphasize on studying the influences of the unknotted loop $d$ on the ESD. For convenience, we let the parameters $\alpha=\frac{\pi}{4}$, $\gamma=0.5$, $J=\frac{1}{2}$ and $\varphi=\pi$. The system model (\[H\]) corresponds to the two-qubit anisotropic Heisenberg $XXZ$ model under an inhomogeneous magnetic field. Then the entanglement for $\rho(t)=U(t)\rho_{0}U^{+}(t)$ can be given easily, and according to Eq.(\[c\]), the concurrence can be obtained as follows, $$\begin{aligned}
\label{c1}
C=\frac{\sqrt{(16(d^{2}-4)+(d^{2}-8)^{2}\cos
t)^{2}+d^{4}(d^{2}-8)^{2}\sin^{2} t}}{2d^{4}}-\frac{1}{4}.\end{aligned}$$ In Fig. 5, we give a plot of the concurrence as a function of the time $t$ and the parameter $d$. It is clear that in our closed Yang-Baxter system, the ESD happens in some special times and then the entanglement revives after a while. One can note that the topological parameter $d$ has a great influence on the ESD when the initial condition is determinate. It is obvious that the ESD happens only when the parameter $d$ changes in a certain range. This means that in the Yang-Baxter system, one can realize the ESD via changing the values of the parameter $d$ when the initial condition is determinate.
Summary {#sec5}
=======
In this paper, we have presented some relations between the topological parameter $d$ and concurrences of the projective entangled states. Specifically, it is shown that for the case with the parameter $d=n$, all the projective entangled states of two $n$-dimensional quantum systems are the maximally entangled states (i.e. $C=1$). And for another case with the parameter $d\neq n$, via investigating the cases with $n=2$ and $n=3$, we find $C$ both approach $0$ when $d\rightarrow\infty$. Then we construct a kind of Yang-Baxter Hamiltonian related to the $4\times 4$ matrix $U$, with the topological parameter $d=q+q^{-1}$ for $n=2$. The thermal entanglement and the ESD for the Yang-Baxter system have been investigated. It is found that the parameter $d$ has great influences on the thermal entanglement. It not only influences the critical temperature $T_{c}$, but also can influence the maximum entanglement value at which the system can arrive at. Finally we find that the parameter $d$ also has a great influence on the ESD, and one can realize the ESD via changing the values of the parameter $d$ when the initial condition is determinate. It is worth to mention that via our paper, it is obvious that the topological parameter $d$ plays an important role in the entanglement.
Acknowledgments
===============
We would like to thank Chengcheng Zhou for his useful discussions. This work was supported by NSF of China (grants No. 10875026) and NUS research (grant No. WBS: R-710-000-008-271).
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abstract: 'Near the superconductor-insulator (S-I) transition, quench-condensed ultrathin Be films show a large magnetoconductance which is highly anisotropic in the direction of the applied field. Film conductance can drop as much as seven orders of magnitude in a weak perpendicular field ($\le$ 1 T), but is insensitive to a parallel field in the same field range. We believe that this negative magnetoconductance is due to the field de-phasing of the superconducting pair wavefunction. This idea enables us to extract the finite superconducting phase coherence length, L$_{\phi}$, in nearly superconducting films. Our data indicates that this local phase coherence persists even in highly insulating films in the vicinity of the S-I transition.'
address:
- 'Department of Physics and Astronomy, University of Rochester,'
- 'Rochester, New York 14627'
author:
- 'E. Bielejec, J. Ruan, and Wenhao Wu'
title: 'Anisotropic Magnetoconductance in Quench-Condensed Ultrathin Beryllium Films '
---
[2]{}
The superconductor-insulator (S-I) transition remains a controversial subject after nearly two decades of intense research. Experimental systems have generally been grouped into two seemingly different categories, displaying somewhat different features. These are granular versus uniform films. In apparently granular films [@Ekinci], the transition temperature, T$_{c}$, and the energy gap, $\Delta$, for the individual grains are essentially constant throughout the S-I transition [@White]. It is well established that superconductivity is destroyed by the breakdown of long-range phase coherence between the grains [@White; @Orr]. Granular films tend to lose their zero resistance state at a normal state sheet resistance, R$_{N}$, that is close to the quantum pair resistance, R$_{Q}$ = h/(2e)$^{2}$ $\approx$ 6.5 k$\Omega$. In addition, granular films display quasireentrance in films not far from being superconducting: film resistance initially drops as the temperature is cooled below T$_{c}$, but eventually increases at low temperatures. A truly insulating state is not seen in granular films until R$_{N}$ $\gg$ R$_{Q}$. Films considered to be uniformly disordered, such as quench-condensed Bi/Ge and Pb/Ge films [@Haviland], amorphous InO$_{x}$ films [@Hebard], and $\em{a}$-MoGe films [@Yazdani], undergo a much sharper S-I transition. On the superconducting side, T$_{c}$ decreases with decreasing film thickness, approaching zero at the S-I transition. Tunneling experiments have suggested [@Valles] that in Bi/Ge and Pb/Ge films the superconducting gap, $\Delta$, decreases with decreasing film thickness until the pair wavefunction, $\Delta^{1/2}$e$^{i\phi}$, with $\phi$ being the phase of the order parameter, vanishes and the film becomes insulating. These results suggest that the S-I transition is driven by the vanishing of the superconducting gap. In the alternative “dirty-boson” model [@Fisher], superconductivity is suppressed by $\em{phase}$ fluctuations, and the Cooper pairs persist even on the insulating side of the transition. In recent years, the “dirty-boson” model has been applied nearly exclusively to explain the scaling analyses of film resistance in the disorder-driven and field-driven S-I transitions in uniform films [@Hebard; @Yazdani; @Liu; @Markovic], although the existence of Cooper pairs in the insulating states of these films has yet to be demonstrated. The “dirty-boson” model is, in fact, expected to describe best the S-I transition in granular films, since the amplitude of the order parameter is well defined on both side of the S-I transition in granular films.
The key concept of the “dirty-boson” model is that phase fluctuations drive a continuous S-I transition. Thus, in the vicinity of the S-I transition, there should exist a finite superconducting phase coherence length, L$_{\phi}$, even on the insulating side of the transition. This L$_{\phi}$ should scale with the correlation length of the transition and should diverge approaching the transition. In this Letter, we report on magnetoconductance (MC) measurements in the vicinity of the S-I transition in quench-condensed ultrathin Be films. As we argue below, this MC study provides the first direct measurement of L$_{\phi}$ in insulating films. We have observed that, for a given insulating film, L$_{\phi}$ drops as temperature is lowered. This underscores the competition between localization and superconductivity. Approaching the superconducting state with increasing film thickness, we have found L$_{\phi}$ to grow drastically.
Our ultrathin Be films were quench-condensed onto bare glass substrates which were held near 20 K during the evaporations. We chose Be mainly for two reasons. First, earlier studies have suggested [@Yatsuk] that quench-condensed Be films are nearly amorphous. Scanning force microscopy studies of our Be films, after warming up to room temperature, have found no observable granular structure down to 1 nm. This indicates that the length scale of disorder in these Be films must be much smaller than the typical grain size in apparently granular films. Second, Be has a very weak spin-orbit coupling [@Tedrow]. As a result, a magnetic field applied parallel to the film plane, H$_{\parallel}$, couples to electron spin only and it does not couple to the orbital motion of the electrons. However, a perpendicular field, H$_{\perp}$, couples to both. Thus the MC
can be highly anisotropic in the direction of the applied field. In our Be films, the MC is negative and varies as much as seven orders of magnitude in weak H$_{\perp}$ up to 1 T, but it is insensitive to H$_{\parallel}$ in the same field regime. The low-field MC in our films is thus clearly an orbital effect. We, therefore, believe that this negative and highly aniostropic MC provides a direct measurement of the superconducting phase coherence length, L$_{\phi}$, as was suggested by Barber and Dynes [@Barber] in a MC study of superconducting granular Pb films. This method can eventually be used to measure the divergence of L$_{\phi}$ as films cross the S-I transition with varying film thickness.
It should be pointed out that the length scale of the disorder in films considered uniformly disordered is still not understood. Even if microscopy techniques fail to reveal any granular structure, there still can exist metallic clusters, which can support superconductivity and which are connected electrically by relatively narrow and insulating or metallic links. For example, the Ge underlayer in Bi/Ge and Pb/Ge films may produce tunneling channels connecting the superconducting clusters. Recently, Kapitulnik and collaborators [@Yazdani] have proposed that, near the S-I transition in $\em{a}$-MoGe, there exist both insulating and superconducting puddles, with transport being dominated by tunneling or hopping between them. Presumably, the typical size of the superconducting puddles grows approaching the S-I transition and eventually become the longest length scale of the system. Another example is the InO$_{x}$ films studied by Hebard $\em{et}$ $\em{al.}$ [@Hebard], which are believed to be amorphous, yet they display the quasireentrant behavior of granular films. Thus it is not clear as to what are the fundamental differences between the S-I transitions observed in uniform and granular films, other than that the different morphologies may lead to different universality classes of the transitions.
The details regarding our quench-condensation appa-
ratus, a rotating sample stage, as well as 4-terminal dc I-V measurements from which the film sheet resistance, R$_{\Box}$, was obtained, have been described elsewhere [@Bielejec]. In Fig. 1, we show the temperature dependence of R$_{\Box}$ for one film section deposited on a bare glass substrate following successive deposition steps to increase film thickness. The film changed its behavior from insulating to superconducting when R$_{\Box}$ at 20 K was reduced to below 10 k$\Omega$/$\Box$ with increasing thickness. Film $\#$10 in Fig. 1, which was superconducting with a T$_{c}$ $\sim$ 6 K, had a critical field H$_{c}$ above the 10-T field our magnet could reach at 4.2 K. Thus the H$_{c}$ is not far below the spin-paramagnetic limit [@Fulde], which we estimated [@Bielejec] to be $\sqrt{2}\Delta$/g$\mu_{B}$ $\approx$ 11.2 T, where g $\approx$ 2 is the Land$\acute{e}$ g-factor, $\mu_{B}$ is the Bohr magneton, and $\Delta$ = 0.92 mV is the superconducting gap for Film $\#$10. Early studies [@Lazarev] estimated that the critical field was 18 $\sim$ 20 T in quench-condensed Be films of T$_{c}$ = 8 $\sim$ 10 K, suggesting that these films were highly disordered with a very short penetration depth. The data in Fig. 1 do show the quasireentrant behavior, such as in Film $\#$7, which is typically seen in granular films. However, this quasireentrance is seen in a range of R$_{N}$ that is much narrower than in the case of typical granular films [@White; @Orr]. In addition, the T$_{c}$ of these Be films appears to increase significantly with increasing film thickness, which is typically seen in uniform films. Thus these Be films show certain properties of both uniform and granular films.
In Fig. 2(a), we show the MC measured in H$_{\perp}$ at a
number of temperatures. In the low-field regime below 1 T, the MC is $\em{negative}$ and varies as much as seven orders of magnitude. This can not be due to weak-localization [@Bergmann], which should lead to a $\em{positive}$ and relatively small MC in weak spin-orbit materials such as Be. In H$_{\parallel}$, film conductance was found to be insensitive to the field below 1 T, as shown in Fig. 2(b). Such highly anisotropic behavior indicates that the MC in H$_{\perp}$ is an orbital effect. We believe that this negative MC in the low H$_{\perp}$ regime arises as the superconducting phase coherence is suppressed when H$_{\perp}$ exceeds the crossover value H$^{\ast}$ that produces one flux quanta, $\Phi_{0}$ = h/2e, in a coherent area, or when L$_{\phi}^{2}$H$_{\perp}$ $\sim$ $\Phi_{0}$. Determining this crossover field H$^{\ast}$ when the conductance drops from its zero-field value, thus provides a measurement of L$_{\phi}$. A few years ago, Barber and Dynes [@Barber] made this argument to calculate L$_{\phi}$ in the descending resistance tail of superconducting granular Pb films, showing that for a superconducting film L$_{\phi}$ increases with decreasing temperature. In our non-superconducting films $\#$7 and $\#$8, the data plotted on a logarithmic field scale in Fig. 2(b) show that, at 100 mK, H$^{\ast}$ was near 0.002 T for Film $\#$8 and 0.1 T for Film $\#$7, as indicate by the arrows in Fig. 2(b). This translates into a coherence length, L$_{\phi}$, at 100 mK of about 1.0 $\mu$m for Film $\#$8 and 0.14 $\mu$m for Film $\#$7. We therefore see a growing L$_{\phi}$ as the films approach the superconducting state with increasing thickness. We note that the drop in conductance at high H$_{\parallel}$, seen in Fig. 2(b), is likely due to the suppression of the amplitude of the superconducting order parameter as the H$_{\parallel}$ approaches the spin-paramagnetic limit.
Not only did we observe L$_{\phi}$ to vary with film thickness, but it varied with temperature as well. In the temperature range in which R$_{\Box}$ decreases with decreasing temperature, we observed that the MC peak was sharper at lower temperatures, indicating an increasing L$_{\phi}$ with decreasing temperature. Such behavior is identical to that
observed by Barber and Dynes [@Barber]. It is due to the suppression of thermal fluctuations with lowering temperature. However, we have also observed, for the first time, that the MC peak is broader at lower temperatures in the quasireentrant regime where R$_{\Box}$ increases with decreasing temperature, as we show in Fig. 3 (a). Such behavior is seen in all quasireentrant films similar to Film $\#$ 7. Using the crossover field values, H$^{\ast}$, obtained from the data in Fig. 3 (a), we find a reduction of L$_{\phi}$ with decreasing temperature in this temperature range, as shown in Fig. 3 (b). We believe that this observation demonstrates the suppression of the superconducting phase coherence as localization effects are enhanced at lower temperatures.
The above proposal that the MC probes the superconducting phase coherence is further supported by the nonlinear I-V curves we have measured near the S-I transition. Insulating and nearly superconducting films near the S-I transition each has a distinct type of I-V curve [@Orr]. In the low bias regime, the I-V curves of nearly superconducting films show the supercurrent-type behavior: the I-V curves have a downward curvature; while the I-V curves of insulating films show the Coulomb-blockade-like behavior: the I-V curves have an upward curvature. The supercurrent-type behavior indicates the existence of a small supercurrent associated with local superconducting regions. There have been observations that the I-V curves evolve from the Coulomb-blockade-type to the supercurrent-type as the films cross the transition from the insulating side [@Orr]. We have seen the same type of behavior in our films. We have also observed that, in nearly superconducting films, the I-V curves changed from the supercurrent-type to the Coulomb-blockade-type as the conductance of the films is suppressed by a weak H$_{\perp}$, as shown in the inset in the top-left corner of Fig. 4. This is therefore additional evidence that the application of H$_{\perp}$ suppresses the superconducting fluctuations.
This negative MC and supercurrent-type I-V persisted even in much more insulating films such as Film $\#$6, which did not show any quasireentrant behavior. In this case, the supercurrent-type I-V could only be observed in a narrow temperature range between 0.8 $\sim$ 1.2 K. In the main part of Fig. 4, we plot the I-V curves measured at 1.05 K on Film $\#$6, for a number of perpendicular field values. Although the effect was much weaker in Film $\#$6 than in less insulating films $\#$7 and $\#$8, we can see clearly that, below a bias voltage of 15 mV, the curvature of the I-V curves changes from downward to upward with a increasing perpendicular field. However, as shown on a higher bias scale in the inset in the low-right corner of Fig. 4, the I-V curves always show an upward curvature regardless of the magnetic field, indicating the insulating nature of Film $\#$6. Thus although Films $\#$6 was very insulating, there still existed a finite L$_{\phi}$, which resulted in an observable supercurrent-type I-V in zero-field at temperatures not so low that the effect is completely suppressed by localization.
In conclusion, we have directly observed for the first time the finite superconducting phase coherence length L$_{\phi}$ on the insulating side of the S-I transition. Our quench-condensed Be films show both the quasireentrant behavior of granular films and the varying T$_{c}$ usually seen in uniformly disordered films. Scanning force microscopy studies have shown that the length scale of disorder is much shorter in these Be films than that of apparently granular films. The MC is negative, large, and highly anisotropic in the direction of the field. Our results demonstrate that this MC gives a direct probe of the length scale associated with the S-I transition. In nearly insulating films, L$_{\phi}$ is observed to $\em{decrease}$ with $\em{decreasing}$ temperature, highlighting the competition between localization and superconductivity. With increasing film thickness, we expect L$_{\phi}$ to grow and to diverge as the films eventually develop a global superconducting phase with zero resistance.
We gratefully acknowledge numerous invaluable discussions with S. Teitel, Y. Shapir, Y. Gao, and P. Adams. We thank S. Zorba and Y. Gao who performed scanning force microscopy studies of our quench-condensed Be films.
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abstract: 'Due to long chemical equilibration times within standard hadronic reactions during the hadron gas phase in relativistic heavy ion collisions it has been suggested that the hadrons are “born" into equilibrium after the quark gluon plasma phase. Here we develop a dynamical scheme in which possible Hagedorn states contribute to fast chemical equilibration times of baryon anti-baryon pairs (as well as kaon anti-kaon pairs) inside a hadron gas and just below the critical temperature. Within this scheme, we use master equations and derive various analytical estimates for the chemical equilibration times. Applying a Bjorken picture to the expanding fireball, the kaons and baryons as well as the bath of pions and Hagedorn resonances can indeed quickly chemically equilibrate for both an initial overpopulation or underpopulation of Hagedorn resonances. Moreover, a comparison of our results to $(B+\bar{B})/\pi^{+}$ and $K/\pi^{+}$ ratios at RHIC, indeed, shows a close match.'
author:
- 'J. Noronha-Hostler$^{1}$'
- 'C. Greiner$^{2}$'
- 'I. A. Shovkovy$^{3}$'
title: Fast Equilibration of Hadrons in an Expanding Fireball
---
(Anti-)strangeness enhancement was first observed at CERN-SPS energies by comparing anti-hyperons, multi-strange baryons, and kaons to $pp$-data. It was considered a signature for quark gluon plasma (QGP) because, using binary strangeness production and exchange reactions, chemical equilibrium could not be reached within the hadron gas phase [@Koch:1986ud]. It was then proposed that there exists a strong hint for QGP at SPS because strange quarks can be produced more abundantly by gluon fusion, which would account for strangeness enhancement following hadronization and rescattering of strange quarks. Later, multi-mesonic reactions were used to explain secondary production of anti-protons and anti-hyperons [@Rapp:2000gy; @Greiner]. At SPS they give a typical chemical equilibration time $\tau_{\bar{Y}}\approx 1-3\frac{\mathrm{fm}}{c}$ using an annihilation cross section of $\sigma_{\rho\bar{Y}}\approx\sigma_{\rho\bar{p}}\approx 50\mathrm{mb}$ and a baryon density of $\rho_{B}\approx \rho_{0}\;\mathrm{to}\;2\rho_{0}$, which is typical for SPS. Therefore, the time scale is short enough to account for chemical equilibration within a cooling hadronic fireball at SPS.
A problem arises when the same multi-mesonic reactions were employed in the hadron gas phase at RHIC temperatures where experiments show that the particle abundances reach chemical equilibration close to the phase transition [@Braun-Munzinger]. At RHIC at $T=170$ MeV, where $\sigma\approx
30\mathrm{mb}$ and $\rho_{B}^{eq}\approx\rho_{\bar{B}}^{eq}\approx0.04\mathrm{fm}^{-3}$, the equilibrium rate for (anti-)baryon production is $\tau\approx
10\frac{\mathrm{fm}}{\mathrm{c}}$, which is considerably longer than the fireball’s lifetime in the hadronic stage of $\tau<5\frac{\mathrm{fm}}{\mathrm{c}}$. Moreover, $\tau\approx
10\frac{\mathrm{fm}}{\mathrm{c}}$ was also obtained in Ref. [@Kapusta] using a fluctuation-dissipation theorem and a significant deviation was found in the population number of various (anti-)baryons from experimental data in the $5\%$ most central Au-Au collisions [@Huovinen:2003sa]. These discrepancies suggest that hadrons are “born" into equilibrium, i.e., the system is already in a chemically frozen out state at the end of the phase transition [@Stock:1999hm; @Heinz:2006ur].
In order to circumvent such long time scales it was suggested that near $T_{c}$ there exists an extra large particle density overpopulated with pions and kaons, which drive the baryons/anti-baryons into equilibrium [@BSW]. But it is not clear how this overpopulation should appear, and how the subsequent population of (anti-)baryons would follow. Moreover, the overpopulated (anti-)baryons do not later disappear [@Greiner:2004vm]. Therefore, it was conjectured that Hagedorn resonances (heavy resonances near $T_{c} $ with an exponential mass spectrum) could account for the extra (anti-)baryons [@Greiner:2004vm]. Baryon anti-baryon [@Greiner:2004vm; @Noronha-Hostler:2007fg] and kaon anti-kaon production develop according to $$\begin{aligned}
\label{eqn:decay}
n\pi&\leftrightarrow &HS\leftrightarrow n_{i,b}\pi+B\bar{B},\nonumber\\
n\pi&\leftrightarrow &HS\leftrightarrow n_{i,k}\pi+K\bar{K},\end{aligned}$$ which provide an efficient method for producing baryons and kaons because of the large decay widths of the Hagedorn states. In Eq. (\[eqn:decay\]), $n$ is the number of pions for the decay $n\pi\leftrightarrow HS$, which can vary, and $n_{i,b}$ ($n_{i,k}$) is the number of pions that a Hagedorn state will decay into when a baryon anti-baryon (kaon anti-kaon) pair is present. Since Hagedorn resonances are highly unstable, the phase space for multi-particle decays drastically increases when the mass is increased. Therefore, the resonances catalyze rapid equilibration of baryons and kaons near $T_{c} $ where the Hagedorn states show up. Here we use a Bjorken expansion within a cooling fireball in order to see at which temperature the chemical equilibrium values are reached or maintained. In this letter we also briefly discuss an analytical solution of the chemical equilibration time, which is valid at a constant temperature near $T_{c}$. Moreover, our numerical results for the baryon anti-baryon pairs and kaon anti-kaon pairs suggest that the hadrons can, indeed, be born *out* of equilibrium.
We use a truncated Hagedorn mass spectrum [@Hagedorn:1968jf] $$g(m)=\int_{M_{0}}^{M}\frac{A}{\left[m^2 +(m_{0})^2\right]^{\frac{5}{4}}}e^{\frac{m}{T_{H}}}dm$$ where the Hagedorn temperature is set to $T_{H}=180$MeV, which lies within the present range of Lattice QCD predictions [@Karsch], the normalization factor is $A=0.5\;\textrm{MeV}^{\frac{3}{2}}$, and $m_{0}=0.5$ GeV. We consider only mesonic, non-strange resonances and discretize the spectrum into mass bins of 100 MeV that range from the mass $M_{0}=2$ GeV to $M=7$ GeV. The effects of the truncation and Hagedorn temperature are further discussed in [@NHCGISbig]. However, the values we have chosen are acceptable.
The abundances’ evolution of the Hagedorn states, pions, and baryon anti-baryon pairs due to the reactions in Eq. (\[eqn:decay\]) are described by the following rate equations $$\begin{aligned}
\label{eqn:rate}
\dot{\lambda}_{i}&=&\Gamma_{i,\pi}\left(\sum_{n=2}^{\infty} B_{i, n}
\lambda_{\pi}^{n}-\lambda_{i}\right)+\Gamma_{i,B\bar{B}}\left(
\lambda_{\pi}^{\langle n_{i,b}\rangle} \lambda_{B\bar{B}}^2 -\lambda_{i}\right),\nonumber\\
\dot{\lambda}_{\pi }&=&\sum_{i} \Gamma_{i,\pi}\frac{N_{i}^{eq}}{N_{\pi}^{eq}} \left(\lambda_{i}\langle n_{i}\rangle-\sum_{n=2}^{\infty}
B_{i, n}n\lambda_{\pi}^{n} \right)\nonumber\\
&+&\sum_{i} \Gamma_{i,B\bar{B}} \langle n_{i,b}\rangle\frac{N_{i}^{eq}}{N_{\pi}^{eq}}\left(\lambda_{i}-
\lambda_{\pi}^{\langle n_{i,b}\rangle} \lambda_{B\bar{B}}^2\right), \nonumber\\
\dot{\lambda}_{B\bar{B}}&=&\sum_{i}\Gamma_{i,B\bar{B}}\frac{N_{i}^{eq}}{N_{B\bar{B}}^{eq}}\left( \lambda_{i}- \lambda_{\pi}^{\langle n_{i,b}\rangle} \lambda_{B\bar{B}}^2\right),\end{aligned}$$ where the fugacity is $\lambda=\frac{N}{N^{eq}}$, $N$ is the total number of each particle, and its respective equilibrium value is $N^{eq}$. The summation over $i$ represents the $i^{th}$ Hagedorn resonance bin. The structure of the rate equations for the kaon anti-kaon pairs is the same as in Eq. (\[eqn:rate\]), however, $K\bar{K}$ is substituted in for $B\bar{B}$.
The branching ratios for $HS\leftrightarrow n\pi$ are described by a Gaussian distribution $B_{i, n}\approx
\frac{1}{\sigma_{i}\sqrt{2\pi}}e^{-\frac{(n-\langle n_{i}\rangle)^{2}}{2\sigma_{i} ^{2}}}$ where $\langle n_{i}\rangle=0.9+1.2\frac{m_{i}}{m_{p}}$ is the average pion number each Hagedorn state decays into, found in a microcanonical model[@Liu], $\sigma_{i}^{2}=(0.5\frac{m_{i}}{m_{p}})^{2}$ is the chosen width of the distribution, and $n\geq 2$ is the cutoff for the pion number. Moreover, the branching ratios are normalized such that $\sum_{n=2}^{\infty} B_{i, n}=1$, which gives $\langle n_{i}\rangle\approx 2$ to $9$ and $\sigma_{i}^{2}\approx 0.8$ to $11$. The total decay width, $\Gamma_{i}\approx 0.17m_{i}-88$ MeV, which ranges from $\Gamma_{i}=250\;\mathrm{to}\;1100$ MeV, was found using the mass and decay widths in [@Eidelman:2004wy] and fitting them linearly similarly to what was done in Ref. [@LizziSenda]. The decay widths for the baryon anti-baryon decay are $\Gamma_{i,B\bar{B}}=\langle
B\rangle\Gamma_{i}$ and $\Gamma_{i,\pi}=\Gamma_{i}-\Gamma_{i,B\bar{B}}$. The average baryon number $\langle B\rangle $ per unit decay of Hagedorn resonances within a microcanonical model ranges from $0.06\mathrm{\;to}\;0.4$, so $\Gamma_{i,B\bar{B}}=15\;\mathrm{\;to}\;400$ MeV [@Greiner:2004vm]. We use only the average values in Eq. (\[eqn:decay\]) so that $\langle n_{i,b}\rangle=\langle n_{i,k}\rangle=3$ to 6 [@Liu; @Greiner:2004vm] is used for both the baryons and kaons. For the kaons $\Gamma_{i,K\bar{K}}=\langle K\rangle \Gamma_{i}$ where $\langle K\rangle=0.4$ to $0.5$ [@Liu; @Greiner:2004vm]. Thus, heavier resonances equilibrate more quickly because of large decay widths.
Using a Bjorken expansion, we find a relationship between the temperature and the time, $T(t)$, for which the total entropy is held constant $$\label{eqn:constrain}
\mathrm{const.}=s(T)V(t)=\frac{S_{\pi}}{N_{\pi}}\int \frac{dN_{\pi}}{dy} dy$$ where $\int \frac{dN_{\pi}}{dy} dy=874$ from Ref. [@Bearden:2004yx] for the $5\%$ most central collisions within one unit of rapidity and the entropy per pion $S_{\pi}/N_{\pi}=5.5$ is larger than that for a gas of massless pions according to the analysis in Ref. [@Greiner:1993jn]. The volume [@Greiner] is $$\label{eqn:bjorken}
V_{eff}(t\geq t_{0})=\pi\;ct\left[r_{0}(t_0)+v_{0}(t-t_{0})+\frac{a_{0}}{2}(t-t_{0})^2 \right]^2$$ where the initial radius is $r_{0}(t_0)=7.1$ fm, the average transversal velocity varies $v_{0}=0.3c,0.5c,$ and $0.7c$, and the corresponding accelerations are taken as $a_{0}=0.035,0.025,$ and $0.015$, respectively.
The equilibrium values of pions, $N_{\pi}^{eq}$, shown in Fig. \[fig:density\] are found using a statistical model [@StatModel].
[c]{}
Here we consider both the direct pions and the indirect pions, which come from resonances such as $\rho$, $\omega$ etc, and both the direct and indirect kaons. In Fig. \[fig:density\] we see that $N_{\pi}^{eq}$ increases with decreasing temperature. This occurs because the Hagedorn states dominate the entropy at high temperatures, which affects $N_{\pi}^{eq}$ due to the entropy constraint in Eq. (\[eqn:constrain\]). Therefore, we must consider the number of “effective pions" in the system, i.e., the total number of pions plus the potential number of pions from the Hagedorn resonances, defined as $$\begin{aligned}
\label{eqn:effpi}
\tilde{N}_{\pi,K\bar{K}}&=&N_{\pi}+\sum_{i}N_{i}\left(\frac{\Gamma_{i,\pi}}{\Gamma_{i}}\langle n_{i}\rangle +\frac{\Gamma_{i,K\bar{K}}}{\Gamma_{i}}\langle n_{i,k}\rangle\right)\nonumber\\
\tilde{N}_{\pi,B\bar{B}}&=&N_{\pi}+\sum_{i}N_{i}\left(\frac{\Gamma_{i,\pi}}{\Gamma_{i}}\langle n_{i}\rangle +\frac{\Gamma_{i,B\bar{B}}}{\Gamma_{i}}\langle n_{i,b}\rangle\right)\end{aligned}$$ for the kaons and baryons, respectively. In both cases $\tilde{N}_{\pi}^{eq}$ remain roughly constant throughout the Bjorken expansion. Additionally, throughout this paper our initial conditions are the various fugacities $\alpha\equiv\lambda_{\pi}(t_0)$, $\beta_{i}\equiv\lambda_{i}(t_0)$, and $\phi\equiv\lambda_{B\bar{B}}(t_0)$ or $\phi\equiv\lambda_{K\bar{K}}(t_0)$, which are chosen by holding the contribution to the total entropy from the Hagedorn states and pions constant i.e. $s_{Had}(T_{0},\alpha)V(t_{0})+s_{HS}(T_{0},\beta_{i})V(t_{0})=s_{Had+HS}(T_{0})V(t_{0})=const$.
The initial estimate for the Hagedorn state equilibration time is $\tau_{i}\equiv1/\Gamma_{i}$. In order to estimate the chemical equilibration time, we use Eq. (\[eqn:rate\]) in a static environment to find the equilibration time to be in the general ballpark [@Greiner; @Greiner:2004vm] of $$\begin{aligned}
\label{eqn:taubbkk}
\tau_{B\bar{B}}&\equiv&\frac{N_{B\bar{B}}^{eq}}{\sum_{i}\Gamma_{i,B\bar{B}}N_{i}^{eq}}=0.2-0.7\;\frac{\mathrm{fm}}{c}\nonumber\\
\tau_{K\bar{K}}&\equiv&\frac{N_{K\bar{K}}^{eq}}{\sum_{i}\Gamma_{i,K\bar{K}}N_{i}^{eq}}=0.1-0.3\;\frac{\mathrm{fm}}{c}\end{aligned}$$ between $T=180$ to $170$ MeV. As will be proven in detail in Ref. [@NHCGISbig], these time scales are only precise when the pions and Hagedorn states are held in equilibrium. In reality the chemical equilibration times are more complicated due to non-linear effects and the evolution of the equilibration must be divided into separate stages for a sufficient analysis.
Time Scale $T=180 - 170 $MeV
-------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------
$\lambda_{\pi}\approx 0$ $\tau_{\pi}^{0}\equiv\frac{N_{\pi}^{eq}}{\sum_{i} \Gamma_{i} N^{eq}_{i} \beta_{i}}$ $0.1-0.4\;\frac{\mathrm{fm}}{c}$
$\lambda_{\pi}\approx 1$ $\tau_{\pi}\equiv\frac{N_{\pi}^{eq}}{\sum_{i} \Gamma_{i} N^{eq}_{i} \langle n_{i}^2\rangle}$ $0.02-0.06\;\frac{\mathrm{fm}}{c}$
QE $\tau^{QE}_{\pi}\equiv\frac{N_{\pi}^{eq}}{\sum_{i} \Gamma_{i} N^{eq}_{i} \sigma_{i}^2}+\frac{\sum_{QE}N_{i}^{eq}\langle n_{i}^2\rangle}{\sum_{i}\Gamma_{i}N_{i}^{eq}\sigma_{i}^2}$ $2.7-3.7\;\frac{\mathrm{fm}}{c}$
TOT $\tau^{tot}\equiv\tau_{2GeV}+\tau^{QE}_{\pi}$ $3.5-4.5\;\frac{\mathrm{fm}}{c}$
: Equilibration times from analytical estimates where QE is quasi-equilibrium and TOT is total equilibrium[]{data-label="tab:tau"}
To find time scale estimates, we consider the more simplified case near $T_{c}$ excluding the baryons and kaons, i.e., Eq. (\[eqn:rate\]) without the baryonic terms. The evolution of the rate equations can be divided into three stages as shown in Tab. \[tab:tau\] and derived in [@NHCGISbig]. Initially, when the pions are far from equilibrium ($\lambda_{\pi}\approx 0$) the Hagedorn states can be held constant at a constant fugacity $\beta_{i}$. Substituting $\lambda_{\pi}\approx 0$ and $\lambda_{i}\approx\beta_{i}$ into Eq. (\[eqn:rate\]), we obtain $\tau_{\pi}^{0}$. As the pions near equilibrium, we can then use $\lambda_{\pi}\rightarrow 1$ to obtain $\tau_{\pi}$. Eventually, the right-hand sides of Eq. (\[eqn:rate\]) become roughly zero before full equilibration (known as quasi-equilibrium), which occurs once the lightest resonance reaches quasi-equilibrium $\tau_{2GeV}=0.8\frac{fm}{c}$. To obtain $\tau^{QE}_{\pi}$ we solved Eq. (\[eqn:effpi\]) without the baryonic term, assuming $\lambda_{\pi}\rightarrow 1$ and that the right-hand side of the pion rate equation equals zero. Then the total equilibration time $\tau^{tot}$ is just the addition of $\tau_{2GeV}$ and $\tau^{QE}_{\pi}$. Since $\tau^{tot}$ includes all the non-linear effects, which occur even after equilibrium is neared, the more appropriate time scale is on the order of $\tau_{\pi}^{0}$. The equilibration times increase directly with $N_{\pi}^{eq}$, $\langle n_{i}^2\rangle$ and are shortened by large $\Gamma_{i}$’s and wide branching ratio distributions $\sigma_{i}$’s. Because $N_{i}^{eq}$ decreases quickly as the system is cooled, the equilibration time is significantly longer at lower temperatures. In Fig. \[fig:pifree\] our analytical fit, which are exponential fits [@NHCGISbig] based on Tab. \[tab:tau\], match our numerical results well and nicely concur with the numerical results in Fig. \[fig:BBalso\]. Additionally, the baryons take slightly longer than predicted in Eq. (\[eqn:taubbkk\]), but they still equilibrate quickly (Fig. \[fig:BBalso\]).
-- --
-- --
\
In Fig. \[fig:expanshown\] the baryons and kaons are shown for an expanding system where we see that the baryons reach chemical equilibrium by $T=165$ MeV ($t-t_{0}\approx2-3\;\frac{fm}{c}$) and the kaons at $T=160-140$ MeV. As with the pions, we consider the effective number of baryons and kaons because of the effects of Hagedorn resonance on the entropy at high temperatures, so $$\begin{aligned}
\label{eqn:effbbkk}
\tilde{N}_{B\bar{B}}&=&N_{B\bar{B}}+\sum_{i}N_{i}\frac{\Gamma_{i,B\bar{B}}}{\Gamma_{i}}\nonumber\\
\tilde{N}_{K\bar{K}}&=&N_{K\bar{K}}+\sum_{i}N_{i}\frac{\Gamma_{i,K\bar{K}}}{\Gamma_{i}},\end{aligned}$$ which are shown in Fig. \[fig:expanshown\]. Not surprisingly, $\tilde{N}_{\pi,B\bar{B}}^{eq}$ and $\tilde{N}_{\pi,K\bar{K}}^{eq}$ remain almost constant due to the constraint set in Eq. (\[eqn:constrain\]). Moreover, our expansion is not strongly affected by $v_{0}$ and, therefore, in the following graphs it is set to $v_{0}=0.5c$.
In Fig. \[fig:expan\] we compare our total baryon to pion ratio $(B+\bar{B})/\pi^{+}$ to experimental data from PHENIX [@PHENIX] and STAR [@STAR]. $(B+\bar{B})/\pi^{+}$ is calculated by $B+\bar{B}=p+\bar{p}+n+\bar{n}\approx 2(p+\bar{p})$. It should be noted here that in our calculations we use both the effective number of baryons, in Eq. (\[eqn:effbbkk\]), and pions, in Eq. (\[eqn:effpi\]). We obtain $(B+\bar{B})/\pi^{+}\approx 0.3$, which matches the experimental data well. Moreover, our results are independent of the chosen initial conditions. Also, in Fig. \[fig:expan\] we compared the kaon to pion ratio to the data at PHENIX [@PHENIX] and STAR [@STAR] (both $K/\pi^{+}$ and $\bar{K}/\pi^{+}$ are shown). Again, we use the effective number of kaons (\[eqn:effbbkk\]) and pions (\[eqn:effpi\]). Our $K/\pi^{+}$ ratios compare to the experimental data very well and they level off between 0.16 to 0.17. As with the baryon anti-baryon pairs we do not see a very strong dependence on our initial conditions. In Fig. \[fig:expan\] both figures agree well with experimental data. Moreover, they remain roughly constant after $T=170-160$ MeV. This demonstrates that the potential Hagedorn states can be used to explain dynamically the build up of the known particle yields.
In future work we will consider strange baryonic degrees of freedom and thoroughly study the effects of our initial conditions and parameters. To conclude, we used Hagedorn resonances as a dynamical mechanism to quickly drive baryons and kaons into equilibrium between temperatures of $T=165-140$ MeV. Once a Bjorken expansion was employed, we found that our calculated $K/\pi^{+}$ and $(B+\bar{B})/\pi^{+}$ ratios matched experimental data well, which suggests that hadrons do not at all need to start in equilibrium at the onset of the hadron gas phase.
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|
**DECAYS OF CHARMED MESONS TO $PV$ FINAL STATES**
Bhubanjyoti Bhattacharya[^1] and Jonathan L. Rosner[^2]
*Enrico Fermi Institute and Department of Physics*
*University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637*
> New data on the decays of the charmed particles $D^0$, $D^+$, and $D_s$ to $PV$ final states consisting of a light pseudoscalar meson $P$ and a light vector meson $V$ are analyzed. Following the same methods as in a previous analysis of $D \to PP$ decays, one can test flavor symmetry, extract key key amplitudes, and obtain information on relative strong phases. Analyses are performed for Cabibbo-favored decays and then extended to predict properties of singly- and doubly-Cabibbo-suppressed processes.
INTRODUCTION
============
In the past few years rich data on charmed particle decays have been contributed by a variety of experiments. Among the decays studied are those involving $PV$ final states, where $P$ and $V$ denote light pseudoscalar and vector mesons, respectively. These decays obey an approximate flavor SU(3) symmetry [@Chau:1983; @Chau:1986; @Chiang:2002mr], allowing one to investigate such questions as the strong phases of amplitudes in these decays. These strong phases can be important when analyzing $D$ decay Dalitz plots in the context of studies of CP violation in $B \to D X$ decays. We have recently performed a similar analysis of $D \to PP$ decays [@Bhattacharya:2008ss].
The diagrammatic approach to flavor symmetry is reviewed briefly in Section II. Cabibbo-favored decays are discussed in Section III, singly-Cabibbo-suppressed decays in Section IV, and doubly-Cabibbo-suppressed decays in Section V. It is possible to obtain a few of the relevant amplitudes using factorization techniques. We discuss factorization calculations in Section VI and conclude in Section VII.
DIAGRAMMATIC AMPLITUDE EXPANSION
================================
A flavor-topology description of $D \to PV$ decays uses amplitudes defined as in Ref. [@Chiang:2002mr]. Cabibbo-favored (CF) amplitudes, proportional to the product $V_{ud} V^*_{cs}$ of Cabibbo-Kobayashi-Maskawa (CKM) factors, will be denoted by unprimed quantities; singly-Cabibbo-suppressed amplitudes proportional to $V_{us} V^*_{cs}$ or $V_{ud} V^*_{cd}$ will be denoted by primed quantities; and doubly-Cabibbo-suppressed quantities proportional to $V_{us} V^*_{cd}$ will be denoted by amplitudes with a tilde. These amplitudes are in the ratio $1:\lambda:-\lambda:-\lambda^2$, where $\lambda = \tan \theta_C =
0.2317$ [@Amsler:2008], with $\theta_C$ the Cabibbo angle.
The relevant amplitudes are labeled as $T$ (“tree”), $C$ (“color-suppressed”), $E$ (“exchange”), and (“A”) (annihilation). For $PV$ final states, a subscript on the amplitude denotes the meson ($P$ or $V$) containing the spectator quark.
The partial width $\Gamma(H \to PV)$ for the decay of a heavy meson $H$ may be expressed in terms of an invariant amplitude ${\cal A}$ as (H PV) = |[A]{}|\^2 , where $p^*$ is the center-of-mass (c.m.) 3-momentum of each final particle, and $M_H$ is the mass of the decaying particle.
CABIBBO-FAVORED DECAYS
======================
In Table \[tab:CFPV\] we summarize predicted and observed amplitudes for Cabibbo-favored decays of charmed mesons to $PV$. The experimental values are based on those in Ref. [@Amsler:2008] unless noted otherwise. Topological amplitudes are then obtained from these processes by algebraic solution. The values of $|T_V|$ and $|E_P|$ are uniquely given by the rates for $D_s \to \pi^+ \phi$ and $D^0 \to \ol{K}^{0} \phi$, respectively. A two-fold ambiguity then is found for the amplitude $|C_P|$ and phases of $C_P$ and $E_P$, as summarized in Table \[tab:cfampsa\].
As explained in Ref. [@Rosner:1999], the solution “B” with $|C_P| <
|T_V|$ is expected for a color suppressed amplitude. However, on the basis of fits to data from singly-Cabibbo-suppressed $D \to PV$ decays, it will turn out that we will prefer the solution “A” with $|C_P| > |T_V|$. In Fig.\[fig:TCEamps\] we plot these two solutions for amplitudes and relative phases of $T_V$, $C_P$ and $E_P$.
--------- ----------------------- ------------------------------------------ --------------------------- ------- -----------------
Meson Decay Representation ${\cal B}$ [@Amsler:2008] $p^*$ $|{\cal A}|$
mode ($\%$) (MeV) $(10^{-6})$
$D^0$ $K^{*-} \pi^+$ $T_V + E_P$ $5.91 \pm 0.39$ 710.9 $4.80 \pm 0.16$
$K^- \rho^+$ $T_P + E_V$ $10.8 \pm 0.7$ 675.4 $7.01 \pm 0.23$
$\ol{K}^{*0} \pi^0$ $\frac{1}{\sqrt{2}}(C_P - E_P)$ $2.82 \pm 0.35$ 709.3 $3.33 \pm 0.21$
$\ol{K}^0 \rho^0$ $\frac{1}{\sqrt{2}}(C_V - E_V)$ $1.54 \pm 0.12$ 673.7 $2.66 \pm 0.14$
$\ol{K}^{*0} \eta$ $\frac{1}{\sqrt{3}}(C_P + E_P - E_V)$ $0.96 \pm 0.3$ 579.9 $2.63 \pm 0.41$
$\ol{K}^{*0} \eta\,'$ $-\frac{1}{\sqrt{6}}(C_P + E_P + 2 E_V)$ $< 0.11$ 101.9
$\ol{K}^0 \omega$ $-\frac{1}{\sqrt{2}}(C_V + E_V)$ $2.26 \pm 0.4$ 670.0 $3.25 \pm 0.29$
$\ol{K}^0 \phi$ $-E_P$ $0.868 \pm 0.06$ 520.6 $2.94 \pm 0.10$
$D^+$ $\ol{K}^{*0} \pi^+$ $T_V + C_P$ $1.83 \pm 0.14$ 711.8 $1.68 \pm 0.06$
$\ol{K}^0 \rho^+$ $T_P + C_V$ $9.2 \pm 2.0$ 677.0 $4.06 \pm 0.44$
$D_s^+$ $\ol{K}^{*0} K^+$ $C_P + A_V$ $3.9 \pm 0.6$ 682.4 $3.97 \pm 0.31$
$\ol{K}^0 K^{*+}$ $C_V + A_P$ $5.3 \pm 1.2$ 683.2 $4.61 \pm 0.52$
$\rho^+ \pi^0$ $\frac{1}{\sqrt{2}}(A_P - A_V)$ 825.2
$\rho^+ \eta$ $\frac{1}{\sqrt{3}}(T_P - A_P - A_V)$ $13.0 \pm 2.2$ 723.8 $6.63 \pm 0.56$
$\rho^+ \eta\,'$ $\frac{1}{\sqrt{6}}(2T_P + A_P + A_V)$ $12.2 \pm 2.0$ 464.8 $12.5 \pm 1.0$
$\pi^+ \rho^0$ $\frac{1}{\sqrt{2}}(A_V - A_P)$ 824.7
$\pi^+ \omega$ $\frac{1}{\sqrt{2}}(A_V + A_P)$ $0.25 \pm 0.09$ 821.8 $0.76 \pm 0.14$
$\pi^+ \phi$ $T_V$ $4.38 \pm 0.35$ 711.7 $3.95 \pm 0.16$
--------- ----------------------- ------------------------------------------ --------------------------- ------- -----------------
: Branching ratios and invariant amplitudes for Cabibbo-favored decays of charmed mesons to one pseudoscalar and one vector meson. \[tab:CFPV\]
----------- ----------------- ---------------------------------------- ----------------- --------------------------------------------------
$PV$ Magnitude Relative Magnitude Relative
amplitude ($10^{-6}$) strong phase ($10^{-6}$) strong phase
$T_V$ $3.95 \pm 0.07$ — $3.95 \pm 0.07$ —
$C_P$ $4.88 \pm 0.15$ $\delta_{C_PT_V} = (-162 \pm 1)^\circ$ $2.84 \pm 0.09$ $\delta_{C_PT_V} = (-158.2^{+2.0}_{-1.9})^\circ$
$E_P$ $2.94 \pm 0.09$ $\delta_{E_PT_V} = (-93 \pm 3)^\circ$ $2.94 \pm 0.10$ $\delta_{E_PT_V} = (92.8^{+3.6}_{-3.7})^\circ$
----------- ----------------- ---------------------------------------- ----------------- --------------------------------------------------
: Solutions in Cabibbo-favored charmed meson decays to $PV$ final states. \[tab:cfampsa\]
Using the solutions for $T_V$, $C_P$ and $E_P$ as inputs, the other amplitudes $T_P$, $C_V$ and $E_V$ were obtained. The amplitude $T_P$ was assumed real relative to $T_V$, in accord with the expectation from factorization. Six sets of solutions were obtained for each of the two cases $|T_V| < |C_P|$ (“A”) and $|T_V| > |C_P|$ (“B”). These solutions are listed in Table \[tab:cfampsc\]. The solutions A1 and A2 are found to give the best fit to the data available for singly-Cabibbo-suppressed $D \to PV$ decays, and so will be singled out for special consideration. Note the identical magnitudes and phases of $T_P$, $C_V$ and $E_V$ in Solutions A1 and B1.
![Amplitudes $T_P$, $C_V$, and $E_V$ in solutions A1 (top) and A2 (bottom). \[fig:cftp\]](cftpa1.ps "fig:"){width="84.00000%"} ![Amplitudes $T_P$, $C_V$, and $E_V$ in solutions A1 (top) and A2 (bottom). \[fig:cftp\]](cftpa2.ps "fig:"){width="80.00000%"}
The magnitudes and phases of solutions A1 and A2 are illustrated in Fig. \[fig:cftp\]. The amplitudes $T_P + E_V = {\cal A}(D^0 \to K^-
\rho^+)$, $C_V - E_V = \sqrt{2}{\cal A}(D^0 \to \ok \rho^0)$, and $T_P + C_V =
{\cal A}(D^+ \to \ok \rho^+)$ form a triangle whose shape is specified by their magnitudes. The amplitudes $C_V$ and $E_V$ form the sides of a quadrangle whose diagonals are $C_V - E_V = \sqrt{2}{\cal A}(D^0 \to \ok \rho)$ and $C_V +
E_V = -\sqrt{2}{\cal A}(D^0 \to \ok \omega)$, and whose vertices lie on a circle with midpoint $M$. Two vertices are fixed, while the other two ($A$ and $B$ in Fig. \[fig:cftp\]) lie at any two opposite points on the circle. An additional constraint is the magnitude of $C_P + E_P - E_V = \sqrt{3}{\cal
A}(D^0 \to \overline{K}^{*0} \eta)$. A discrete ambiguity remains, corresponding to the solutions listed in Tables \[tab:cfampsa\] and \[tab:cfampsc\].
-------- ------- ---------------- -------------------------------------------- -------------------------------------------
No. $PV$ Magnitude Relative ${\cal{B}}(D^0 \to \ol{K^{*0}}\,\eta\,')$
ampl. ($10^{-6}$) phase ($10^{-4}$)
A1$^a$ $T_P$ 7.46$\pm$0.21 Assumed 0
$C_V$ 3.46$\pm$0.18 $\delta_{C_VT_V} = (172 \pm 3)^\circ$ $1.52 \pm 0.22$
$E_V$ 2.37$\pm$0.19 $\delta_{E_VT_V} = (-110 \pm 4)^\circ$
A2$^b$ $T_P$ 6.51$\pm$0.23 Assumed 0
$C_V$ 2.47$\pm$0.22 $\delta_{C_VT_P} = (-174 \pm 4)^\circ$ $1.96 \pm 0.23$
$E_V$ 3.39$\pm$0.16 $\delta_{E_VT_P} = (-96 \pm 3)^\circ$
A3 $T_P$ –5.67$\pm$0.22 Assumed 0
$C_V$ 3.64$\pm$0.27 $\delta_{C_VT_P} = (-46 \pm 4)^\circ$ $1.42 \pm 0.28$
$E_V$ 2.09$\pm$0.28 $\delta_{E_VT_P} = (-122^{+5}_{-6})^\circ$
A4 $T_P$ –5.60$\pm$0.24 Assumed 0
$C_V$ 1.68$\pm$0.24 $\delta_{C_VT_P} = (-20 \pm 6)^\circ$ $2.21 \pm 0.25$
$E_V$ 3.85$\pm$0.15 $\delta_{E_VT_P} = (-94 \pm 3)^\circ$
A5 $T_P$ –3.22$\pm$0.21 Assumed 0
$C_V$ 1.79$\pm$0.32 $\delta_{C_VT_P} = (-104 \pm 5)^\circ$ $2.18 \pm 0.25$
$E_V$ 3.79$\pm$0.13 $\delta_{E_VT_P} = (-180^{+4}_{-5})^\circ$
A6 $T_P$ 3.21$\pm$0.21 Assumed 0
$C_V$ 1.78$\pm$0.31 $\delta_{C_VT_P} = (105 \pm 5)^\circ$ $2.18 \pm 0.25$
$E_V$ 3.80$\pm$0.13 $\delta_{E_VT_P} = (-180^{+5}_{-4})^\circ$
B1 $T_P$ 7.46$\pm$0.21 Assumed 0
$C_V$ 3.46$\pm$0.17 $\delta_{C_VT_P} = (172 \pm 3)^\circ$ 0.33$\pm$0.05
$E_V$ 2.37$\pm$0.19 $\delta_{E_VT_P} = (-110 \pm 4)^\circ$
B2 $T_P$ 6.43$\pm$0.22 Assumed 0
$C_V$ 3.95$\pm$0.24 $\delta_{C_VT_P}=(-143 \pm 4)^\circ$ $0.052^{+0.020}_{-0.021}$
$E_V$ 1.40$\pm$0.32 $\delta_{E_VT_P}= (-71^{+6}_{-7})^\circ$
B3 $T_P$ 4.53$\pm$0.24 Assumed 0
$C_V$ 0.80$\pm$0.21 $\delta_{C_VT_P}=(130^{+16}_{-15})^\circ$ 1.18$\pm$0.10
$E_V$ 4.12$\pm$0.15 $\delta_{E_VT_P}=(72 \pm 3)^\circ$
B4 $T_P$ 4.97$\pm$0.22 Assumed 0
$C_V$ 3.28$\pm$0.29 $\delta_{C_VT_P}=(126 \pm 4)^\circ$ 0.42$\pm$0.10
$E_V$ 2.61$\pm$0.25 $\delta_{E_VT_P}=(47 \pm 5)^\circ$
B5 $T_P$ –3.33$\pm$0.22 Assumed 0
$C_V$ 0.75$\pm$0.19 $\delta_{C_VT_V}=(164^{+14}_{-15})^\circ$ 1.19$\pm$0.11
$E_V$ 4.13$\pm$0.17 $\delta_{E_VT_V}=(-140 \pm 2)^\circ$
B6 $T_P$ –7.70$\pm$0.21 Assumed 0
$C_V$ 4.01$\pm$0.17 $\delta_{C_VT_V}=(17^{+3}_{-4})^\circ$ 0.020$\pm$0.011
$E_V$ 1.24$\pm$0.22 $\delta_{E_VT_V}=(-52^{+9}_{-8})^\circ$
-------- ------- ---------------- -------------------------------------------- -------------------------------------------
: Alternative solutions for $T_P$, $C_V$, and $E_V$ amplitudes in Cabibbo-favored charmed meson decays to $PV$ final states. Solutions A1 – A6 correspond to $|T_V| < |C_P|$, while the solutions B1 – B6 correspond to $|T_V| > |C_P|$ \[tab:cfampsc\]
Predictions for the branching ratio for $D^0 \to \overline{K}^{*0} \eta'$, listed in the last column of Table \[tab:cfampsd\], in principle allow one to distinguish among various solutions. In addition, we shall see that only solutions A1 and A2 give rise to acceptable fits to singly-Cabibbo-suppressed decays.
We now state a relationship between $|T_P|$ and Cabibbo-favored $D_s$ decay amplitudes: |A(D\_s \^+ ’)|\^2 = |T\_P|\^2 + |A(D\_s \^+ )|\^2 - |A(D\_s \^+ )|\^2 Using the value of $|T_P|$ from solution A1 of Table \[tab:cfampsc\] and the decay amplitudes $(D_s \to \rho^+ \eta,\ \pi^+ \omega)$ from Table \[tab:CFPV\], we calculate the amplitude: $|A(D_s \to \rho^+ \eta')| =
(3.50 \pm 1.15) \times 10^{-6}$, which deviates from the experimental value (Table \[tab:CFPV\]) by a large amount. This problem with the quoted experimental rate for $D_s \to \rho^+ \eta'$ was already noted in Ref.[@Rosner:1999]. It indicates either the importance of neglected amplitudes involving the flavor-singlet component of $\eta'$, or an overestimate of the experimental decay rate in this mode.
The remaining parameters $A_P$ and $A_V$ were determined using the amplitudes of $D_s \to (\ol{K}^{*0} K^+,\ \ol{K}^0 K^{*+},\ \pi^+ \omega)$ and have been listed in Table \[tab:cfampsd\]. A direct calculation of the amplitudes for $D_s \to \rho^+ (\eta,\ \eta')$ is now possible using these amplitudes. For the amplitude solutions (A1, A2) preferred by fits to singly-Cabibbo-suppressed decays, we find ${\cal B}(D_s \to \rho^+ \eta) = (5.6 \pm 1.2,~5.55 \pm 0.60)\%$, to be compared with the experimental value of $(6.63 \pm 0.56)\%$, and ${\cal
B}(D_s \to \rho^+ \eta') = (2.9 \pm 0.3,~1.89 \pm 0.20)\%$, to be compared with the experimental value of $(12.5 \pm 1.0)\%$. The agreement between prediction and experiment for ${\cal B}(D_s \to \rho^+ \eta)$ is good for the solutions A1, A2, B1, and B2, while no solution gives agreement for ${\cal B}(D_s \to
\rho^+ \eta')$. We await forthcoming CLEO data on this mode.
-------- ----------- ------------------------ ------------------------------------------- ---------------------------------------- ---------------------------
No. $PV$ Magnitude Relative
amplitude ($10^{-6}$) phase
A1$^a$ $A_P$ $1.36_{-1.04}^{+1.16}$ $\delta_{A_P} = (-151^{+83}_{-74})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $5.6 \pm 1.2$
$A_V$ $1.25_{-0.31}^{+0.34}$ $\delta_{A_V} = (-19^{+10}_{-9})^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+)=$ $2.9 \pm 0.3$
A2$^b$ $A_P$ $2.15_{-0.18}^{+0.22}$ $\delta_{A_P} = (-179^{+32}_{-9})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $5.55 \pm 0.60$
$A_V$ $1.23_{-0.19}^{+0.31}$ $\delta_{A_V} = (-19^{+34}_{-14})^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+) =$ $1.89 \pm 0.20$
A3 $A_P$ $1.24_{-0.24}^{+0.34}$ $\delta_{A_P} = (-89^{+10}_{-14})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $4.20 \pm 0.81$
$A_V$ $0.96_{-0.22}^{+0.27}$ $\delta_{A_V} = ( 34^{+21}_{-14})^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+) =$ $1.45 \pm 0.28$
A4 $A_P$ $4.27_{-0.21}^{+0.42}$ $\delta_{A_P} = (-109^{+14}_{-5})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $2.77 \pm 0.27$
$A_V$ $3.20_{-0.19}^{+0.23}$ $\delta_{A_V} = (+72^{+6}_{-4})^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+) =$ $1.77 \pm 0.18$
A5 $A_P$ $2.88_{-0.24}^{+0.35}$ $\delta_{A_P} = (-123^{+6}_{-4})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $0.58 \pm 0.06$
$A_V$ $1.93_{-0.27}^{+1.21}$ $\delta_{A_V} = (69^{+15}_{-5})^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+) =$ $0.70 \pm 0.07$
A6 $A_P$ $2.88_{-0.31}^{+0.22}$ $\delta_{A_P} = (+122^{+5}_{-6})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $1.61 \pm 0.17$
$A_V$ $2.85_{-0.26}^{+0.21}$ $\delta_{A_V} = (-36 \pm 7)^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+) =$ $0.43 \pm 0.04$
B1 $A_P$ $1.57_{-0.32}^{+0.82}$ $\delta_{A_P} = (+121^{+19}_{-9})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $7.08 \pm 1.03$
$A_V$ $1.74_{-0.28}^{+0.44}$ $\delta_{A_V} =(-96^{+7}_{-6})^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+) =$ $2.53 \pm 0.37$
B2 $A_P$ $1.35_{-0.27}^{+0.51}$ $\delta_{A_P} = (- 74^{+12}_{-9})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $5.38_{-2.11}^{+2.03}$
$A_V$ $1.52_{-0.21}^{+0.70}$ $\delta_{A_V} = (+150^{+44}_{-10})^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+) =$ $1.86_{-0.73}^{+0.70}$
B3 $A_P$ $3.85_{-0.24}^{+0.39}$ $\delta_{A_P} = (+111^{+14}_{-5})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $2.42 \pm 0.16$
$A_V$ $2.78_{-0.22}^{+0.37}$ $\delta_{A_V} = (-68^{+17}_{-7})^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+) =$ $1.01 \pm 0.07$
B4 $A_P$ $1.74_{-0.23}^{+0.34}$ $\delta_{A_P} = (+ 77^{+41}_{-10})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $3.04 \pm 0.70$
$A_V$ $1.16_{-0.23}^{+0.27}$ $\delta_{A_V} = (-140 \pm 12)^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+) =$ $1.18 \pm 0.27$
B5 $A_P$ $4.12_{-0.31}^{+0.24}$ $\delta_{A_P} = (+111^{+6}_{- 9})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $1.30 \pm 0.10$
$A_V$ $3.22_{-0.38}^{+0.29}$ $\delta_{A_V} = (-60^{+8}_{-11})^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+) =$ $0.571_{-0.044}^{+0.045}$
B6 $A_P$ $0.67_{-0.29}^{+0.26}$ $\delta_{A_P} = (+45^{+22}_{-25})^\circ$ ${\cal{B}}(D_s^+ \to \eta\, \rho^+) =$ $4.80 \pm 2.54$
$A_V$ $1.28_{-0.20}^{+0.23}$ $\delta_{A_V} = (+168^{+11}_{-15})^\circ$ ${\cal{B}}(D_s^+ \to \eta'\,\rho^+)=$ $3.42 \pm 1.81$
-------- ----------- ------------------------ ------------------------------------------- ---------------------------------------- ---------------------------
: Solution for annihilation amplitudes in Cabibbo-favored charmed meson decays to $PV$ final states. \[tab:cfampsd\]
SINGLY-CABIBBO-SUPPRESSED DECAYS
================================
The topological amplitude decomposition of singly-Cabibbo-suppressed decays of $D^0 \to PV$ is listed in Table \[tab:SCSPVa\] along with the measured branching ratios and amplitudes for the decays. Unlike the $D \to PP$ case [@Bhattacharya:2008ss], here we have neglected the Okubo-Zweig-Iizuka (OZI) suppressed disconnected diagrams that form the Singlet-Exchange $(SE\,')$ and Singlet-Annihilation $(SA\,')$ amplitudes.
--------- ----------------------- ------------------------------------------------------------ --------------------------- ------- ---------------
Meson Decay Representation ${\cal B}$ [@Amsler:2008] $p^*$ $|{\cal A}|$
mode ($\%$) (MeV) $(10^{-6})$
$D^0$ $\pi^+\, \rho^-$ $-(T_V\,' + E_P\,')$ 0.497$\pm$0.023 763.8 1.25$\pm$0.03
$\pi^-\, \rho^+$ $-(T_P\,' + E_V\,')$ 0.980$\pm$0.040 763.8 1.76$\pm$0.04
$\pi^0\, \rho^0$ $\frac{1}{2}(E_P\,' + E_V\,' - C_P\,' - C_V\,')$ 0.373$\pm$0.022 764.2 1.08$\pm$0.03
$K^+\, K^{*-}$ $T_V\,' + E_P\,'$ 0.153$\pm$0.015 609.8 0.97$\pm$0.05
$K^-\, K^{*+}$ $T_P\,' + E_V\,'$ 0.441$\pm$0.021 609.8 1.65$\pm$0.04
$K^0\, \ol{K}^{*0}$ $E_V\,' - E_P\,'$ $< 0.18$ 605.3
$\ol{K}^{0}\, K^{*0}$ $E_P\,' - E_V\,'$ $< 0.09$ 605.3
$\pi^0\, \phi$ $\frac{1}{\sqrt{2}} C_P\,'$ 0.124$\pm$0.012 644.7 0.81$\pm$0.04
$\pi^0\, \omega$ $\frac{1}{2}(E_P\,' + E_V\,' - C_P\,' + C_V\,')$ 761.2
$\eta\, \rho^0$ $\frac{1}{\sqrt{6}}(2 C_V\,' - C_P\,' - E_P\,' - E_V\,')$ 652.0
$\eta\, \omega$ $- \frac{1}{\sqrt{6}}(2 C_V\,' + C_P\,' + E_P\,' +E_V\,')$ 488.8
$\eta\, \phi$ $\frac{1}{\sqrt{3}}(C_P\,' - E_P\,' - E_V\,')$ 648.1
$\eta\,' \rho^0$ $\frac{1}{2 \sqrt{3}}(E_P\,' + E_V\,' + C_P\,' +C_V\,')$ 342.5
$\eta\,' \omega$ $\frac{1}{2 \sqrt{3}}(E_P\,' + E_V\,' + C_P\,' - C_V\,')$ 333.5
$D^+$ $\rho^0\, \pi^+$ $\frac{1}{\s}(A_P\,'-A_V\,'-C_P\,'-T_V\,')$ 0.082$\pm$0.015 $767$ 0.32$\pm$0.03
$\omega\, \pi^+$ $-\frac{1}{\s}(A_P\,'+A_V\,'+C_P\,'+T_V\,')$ $<0.034$ $764$
$\phi\, \pi^+$ $C_P\,'$ 0.620$\pm$0.070 $647$ 1.13$\pm$0.06
$\ol{K}^{*0}\, K^+$ $(T_V\,'-A_V\,')$ 0.435$\pm$0.048 $611$ 1.03$\pm$0.06
$\pi^0\, \rho^+$ $\frac{1}{\s}(A_V\,'-A_P\,'-C_V\,'-T_P\,')$ $767$
$\eta\, \rho^+$ $\frac{1}{\sx}(A_V\,'+A_P\,'+2C_V\,'+T_P\,')$ $<0.7$ $656$
$\eta\,' \rho^+$ $\frac{1}{\sx}(C_V\,'-A_V\,'-A_P\,'-T_P\,')$ $<0.5$ $349$
$\ol{K}^0\, K^{*+}$ $(T_P\,'-A_P\,')$ 3.18$\pm$1.38 $612$ 2.78$\pm$0.60
$D_s^+$ $\pi^+\, K^{*0}$ $(A_V\,'-T_V\,')$ 0.225$\pm$0.039 $773$ 0.79$\pm$0.07
$\pi^0\, K^{*+}$ $-\frac{1}{\s}(C_V\,'+A_V\,')$ $775$
$\eta\, K^{*+}$ $\frac{1}{\st}(T_P\,'+2C_V\,'+A_P\,'-A_V\,')$ $661$
$\eta\,' K^{*+}$ $\frac{1}{\sx}(2T_P\,'+C_V\,'+2A_P\,'+A_V\,')$ $337$
$K^0\, \rho^+$ $(A_P\,'-T_P\,')$ $743$
$K^+\, \rho^0$ $-\frac{1}{\s}(C_P\,'+A_P\,')$ 0.27$\pm$0.05 $745$ 0.92$\pm$0.09
$K^+\, \omega$ $-\frac{1}{\s}(C_P\,'-A_P\,')$ $741$
$K^+\, \phi$ $T_V\,'+C_P\,'+A_V\,'$ $<0.057$ $607$
--------- ----------------------- ------------------------------------------------------------ --------------------------- ------- ---------------
: Branching ratios and invariant amplitudes for singly-Cabibbo-suppressed decays of charmed mesons to one pseudoscalar and one vector meson. \[tab:SCSPVa\]
-------- ---------- ---------------------------- ---------------------------- ------------------------ ----------------
No. Global
$\chi^2$ Decay Channel ${\cal{B}}_{th}(\%)$ ${\cal{B}}_{expt}(\%)$ $\Delta\chi^2$
A1$^a$ 61.8 $D^+ \to \ol{K}^{*0}\,K^+$ $0.17 \pm 0.04$ $0.435 \pm 0.048$ 16.1
$D^+ \to \omega\,\pi^+$ $0.16 \pm 0.04$ $<0.034$ 11.6
A2$^b$ 65.9 $D^+ \to \ol{K}^{*0}\,K^+$ $0.17 \pm 0.03$ $0.435 \pm 0.048$ 21.4
$D^0 \to \rho^0\,\pi^0$ $0.27 \pm 0.02$ $0.373\pm 10.1
0.022$
A3 341.4 $D^0 \to \rho^0\,\pi^0$ $(4.3\pm3.1)\times10^{-3}$ $0.373\pm0.022$ 275.2
$D^+ \to \rho^0\,\pi^+$ $(1.5\pm4.0)\times10^{-3}$ $0.082\pm0.015$ 25.1
A4 167.1 $D^0 \to \rho^0\,\pi^0$ $0.12 \pm 0.01$ $0.373 \pm 0.022$ 95.4
$D^+ \to \rho^0\,\pi^+$ $0.73 \pm 0.12$ $0.082 \pm 0.015$ 31.4
A5 324.1 $D^0 \to \rho^0\,\pi^0$ $(6.1\pm3.1)\times10^{-3}$ $0.373\pm0.022$ 272.6
$D^+ \to K^{*0}\,\ol{K}^0$ $0.19 \pm 0.02$ $<0.09$ 11.9
A6 149.8 $D^+ \to \rho^0\,\pi^+$ $0.91 \pm 0.09$ $0.082 \pm 0.015$ 51.1
$D^+ \to \ol{K}^{*0}\,K^+$ $0.12 \pm 0.03$ $0.435 \pm 0.048$ 32.1
B1 244.0 $D^0 \to \rho^0\,\pi^0$ $0.12 \pm 0.01$ $0.373 \pm 0.022$ 95.3
$D^0 \to \phi\,\pi^0$ $0.042\pm0.003$ $0.124 \pm 0.012$ 45.3
B2 155.7 $D^0 \to \phi\,\pi^0$ $0.042\pm0.003$ $0.124 \pm 0.012$ 45.3
$D^+ \to \phi\,\pi^+$ $0.21 \pm 0.01$ $0.62 \pm 0.07$ 32.9
B3 165.7 $D^0 \to \phi\,\pi^0$ $0.042\pm0.003$ $0.124 \pm 0.012$ 45.3
$D^+ \to \phi\,\pi^+$ $0.21 \pm 0.01$ $0.62 \pm 0.07$ 32.9
B4 151.7 $D^0 \to \phi\,\pi^0$ $0.042\pm0.002$ $0.124 \pm 0.012$ 45.3
$D^+ \to \rho^0\,\pi^+$ $1.44 \pm 0.23$ $0.082 \pm 0.015$ 34.4
B5 518.8 $D^0 \to \rho^0\,\pi^0$ $(5.4\pm2.8)\times10^{-3}$ $0.373\pm0.022$ 274.8
$D^+ \to \rho^0\,\pi^+$ $1.71 \pm 0.21$ $0.082 \pm 0.015$ 59.3
B6 401.3 $D^0 \to \rho^0\,\pi^0$ $0.015\pm0.006$ $0.373 \pm 0.022$ 245.9
$D^0 \to \phi\,\pi^0$ $0.042\pm0.003$ $0.124 \pm 0.012$ 45.3
-------- ---------- ---------------------------- ---------------------------- ------------------------ ----------------
: Global $\chi^2$ values for fits to singly-Cabibbo-suppressed $D \to
PV$ decays. Also included are the process that contribute the most to a high $\chi^2$ value.[]{data-label="tab:scschisq"}
We now make use of the amplitudes determined in Section III to predict the singly-Cabibbo-suppressed decay amplitudes. Here we assume the simple hierarchy of amplitudes explained in Section II. Based on the available data we calculated the global $\chi^2$ of singly-Cabibbo-suppressed $D \to PV$ decays for solutions A1–A6 and B1–B6. Solutions A1 and A2 have the two lowest values of $\chi^2$ and hence were chosen as the preferred and alternative solutions. Table \[tab:scschisq\] summarizes the global $\chi^2$ values for each of the twelve solutions. It also includes, for each solution, two processes that contribute the most towards a high value of $\chi^2$.
One notes in Table \[tab:scschisq\] that the main processes contributing to high global $\chi^2$ for all solutions are $D^0 \to \phi \pi^0$ and $D^0 \to \rho^0 \pi^0$. The solutions B1-6, which correspond to $|C_P| < |T_V|$, yield high $\chi^2$ for the process $D^0 \to \phi \pi^0$. The amplitude of this process depends only on $C_P\,'$. This shows that $|C_P| < |T_V|$ is not favored by the process $D^0 \to \phi \pi^0$. The processes $D^0 \to \rho^0 \pi^0$ and $D^+ \to \rho^0 \pi^+$ contribute to high $\chi^2$ for the solutions A3-6.
The predicted and experimental $D^0$ branching ratios are in qualitative agreement but with some notable exceptions. The predictions for $D^0 \to \pi
\rho$ fall slightly short of experiment for all charge states, most prominently for $\pi^0 \rho^0$. Recall that the predicted branching ratio for $D^0 \to
\pi^+ \pi^-$ lies significantly [*above*]{} the experimental value [@Bhattacharya:2008ss]. The predictions for $D^0 \to K^+ K^{*-}$ and $D^0 \to K^- K^{*+}$ are not badly obeyed, while those for $D^0 \to K^0
\overline{K}^{*0}$ and $D^0 \to \overline{K}^0 K^0$ are far below the current experimental upper limits. The predicted branching ratio for $D^0 \to \pi^0 \phi$ is approximately same as the observed value. The value of $\chi^2$ for solutions \# A1 and A2 are respectively 61.8 and 65.9 (Table \[tab:scschisq\]), where we have used the 18 data points for which the branching ratios are available.
In Table \[tab:SCSPVb\] we present our predictions for branching ratios of singly-Cabibbo-suppressed $D \to PV$ modes corresponding to the two solutions A1 and A2 having the lowest value of global $\chi^2$ for these modes. There is little one can do to distinguish between them given the available data on branching ratios. Both solutions yield fairly similar central values for most of the singly-Cabibbo-suppressed $D \to PV$ modes. A slight distinction may be made in a few cases. For example, the predicted central values of $\b(D^0 \to (K^0\,\ol{K}^{*0}, \ol{K}^0\,K^{*0}))$ are larger for solution A1 than for A2, though differing only by $1.5 \sigma$. Another example is the process $D^0 \to \pi^0 \omega$, for which the central value of the branching ratio in solution A2 is nearly three times its value in A1. Still another example is the process $D^+ \to \eta' \rho^+$, for which the predicted (very small) branching ratio in A1 is twice its value in A2. Measurements of the branching ratios for both Cabibbo-favored and singly-Cabibbo-suppressed decays with higher precision will be necessary in order to distinguish between the two solutions.
--------- --------------------- ------------------ ------------------- -------------------
PV Decay Experimental
Meson Mode ${\cal{B}}~(\%)$ Solution A1 Solution A2
$D^0$ $\pi^+\, \rho^-$ $0.497\pm0.023$ $0.39\pm0.03$ $0.39\pm0.03$
$\pi^-\, \rho^+$ $0.980\pm0.040$ $0.84\pm0.06$ $0.84\pm0.06$
$\pi^0\, \rho^0$ $0.373\pm0.022$ $0.29\pm0.02$ $0.27\pm0.02$
$K^+\, K^{*-}$ $0.153\pm0.015$ $0.20\pm0.01$ $0.20\pm0.01$
$K^-\, K^{*+}$ $0.441\pm0.021$ $0.43\pm0.03$ $0.43\pm0.03$
$K^0\, \ol{K}^{*0}$ $< 0.18$ $0.0080\pm0.0036$ $0.0020\pm0.0016$
$\ol{K}^0\, K^{*0}$ $< 0.09$ $0.0080\pm0.0036$ $0.0020\pm0.0016$
$\pi^0\, \phi$ $0.124\pm0.012$ $0.122\pm0.007$ $0.122\pm0.007$
$\pi^0\, \omega$ $0.043\pm0.008$ $0.119\pm0.012$
$\eta\, \rho^0$ $0.106\pm0.013$ $0.095\pm0.010$
$\eta\, \omega$ $0.140\pm0.009$ $0.127\pm0.009$
$\eta\, \phi$ $0.093\pm0.009$ $0.14\pm0.01$
$\eta\,' \rho^0$ $0.0154\pm0.0009$ $0.0158\pm0.0009$
$\eta\,' \omega$ $0.0066\pm0.0005$ $0.0077\pm0.0005$
$D^+$ $\rho^0\, \pi^+$ $0.082\pm0.015$ $0.097\pm0.048$ $0.23\pm0.12$
$\omega\, \pi^+$ $<0.034$ $0.15\pm0.04$ $0.14\pm0.12$
$\phi\, \pi^+$ $0.620\pm0.070$ $0.62\pm0.04$ $0.62\pm0.04$
$\ol{K}^{*0}\, K^+$ $0.435\pm0.048$ $0.17\pm0.04$ $0.17\pm0.03$
$\pi^0\, \rho^+$ $0.062\pm0.047$ $0.012\pm0.015$
$\eta\, \rho^+$ $<0.7$ $0.0017\pm0.0040$ $0.0057\pm0.013$
$\eta\,' \rho^+$ $<0.5$ $0.083\pm0.010$ $0.044\pm0.005$
$\ol{K}^0\, K^{*+}$ $3.18\pm1.38$ $1.66\pm0.20$ $1.66\pm0.12$
$D_s^+$ $\pi^+\, K^{*0}$ $0.225\pm0.039$ $0.15\pm0.04$ $0.15\pm0.03$
$\pi^0\, K^{*+}$ $0.049\pm0.012$ $0.020\pm0.008$
$\eta\, K^{*+}$ $0.014\pm0.011$ $0.012\pm0.008$
$\eta\,' K^{*+}$ $0.029\pm0.006$ $0.015\pm0.003$
$K^0\, \rho^+$ $1.29\pm0.15$ $1.29\pm0.09$
$K^+\, \rho^0$ $0.27\pm0.05$ $0.33\pm0.05$ $0.42\pm0.05$
$K^+\, \omega$ $0.108\pm0.029$ $0.072\pm0.033$
$K^+\, \phi$ $<0.057$ $0.038\pm0.009$ $0.037\pm0.028$
--------- --------------------- ------------------ ------------------- -------------------
: Comparison between predicted amplitudes based on Cabibbo-favored decays and the experimental values for singly-Cabibbo-suppressed decays of $D^0$ to a pseudoscalar and a vector meson. Predictions are listed for preferred (A1) and alternative (A2) solutions. \[tab:SCSPVb\]
DOUBLY-CABIBBO-SUPPRESSED DECAYS
================================
We now characterize the doubly-Cabibbo-suppressed or wrong-sign (WS) decays of $D \to PV$. A detailed list of possible decays and the corresponding topological amplitude decompositions are given in Table \[tab:WSPV\]. We used the Cabibbo-favored amplitudes calculated in section III to predict the WS amplitudes, using the simple hierarchy of amplitudes as explained in Section II. The predicted amplitudes have been included in Table \[tab:WSPV\] for the preferred (A1) and alternative (A2) solutions.
The experimental values for the following decays are available in the literature [@Amsler:2008]: (D\^0 K\^[\*+]{} \^-) &=& (3.0\^[+3.9]{}\_[-1.2]{}) 10\^[-4]{}\
(D\^+ K\^[\*0]{} \^+) &=& (4.35 0.9) 10\^[-4]{} The predicted values for these branching ratios (Table \[tab:WSPV\]) are in satisfactory agreement with the experimental values quoted above. An interesting point to note is that both solutions A1 and A2 give the same predicted central values for these branching ratios, but A2 has a larger error bar on both of them. Several other branching ratios in Table \[tab:WSPV\] predicted to exceed $10^{-4}$ may help to distinguish between solutions A1 and A2. These include $\b(D^0 \to K^{*0} \pi^0)$, $\b(D^+ \to K^{*+} \pi^0)$, and $\b(D^+ \to K^+ \rho^0)$. Reduction in errors on predictions will be needed in order that these distinctions exceed $2$–$2.5 \sigma$. Some of the doubly-Cabibbo-suppressed decays in Table \[tab:WSPV\] may be observable in Dalitz plots of $D$ decays to three pseudoscalars through interference with Cabibbo-favored $PV$ decays. For example, $D^0 \to K_S \pi^+ \pi^-$ might be able to provide new information about the decay process $D^0 \to K^{*+} \pi^-$, while $D^+ \to K_S \pi^+ \pi^0$ could provide information about $D^+ \to K^{*+}
\pi^0$.
[c l c c c c]{} Meson & Decay & Representation &$p^*$&\
& mode & &(MeV)& Solution A1 & Solution A2\
$D^0$ &$K^{*+}\, \pi^-$ &$T_P\, '' + E_V\,''$ &$711$&$3.63\pm0.26$ &$3.63\pm0.27$\
&$K^{*0}\, \pi^0$ &$(C_P\,'' - E_V\,'')/\s$ &$709$&$0.55\pm0.06$ &$0.80\pm0.08$\
&$K^{*0}\, \eta$ &$(C_P\,'' - E_P\,'' + E_V\,'')/\st$ &$580$&$0.38\pm0.04$ &$0.37\pm0.04$\
&$K^{*0}\, \eta\,'$&$-(C_P\,'' + 2E_P\,'' + E_V\,'')/\sx$&$102$&$0.0046\pm0.0004$&$0.0052\pm0.0004$\
&$K^+\, \rho^{-}$ &$T_V\,'' + E_P\,''$ &$675$&$1.46\pm0.10$ &$1.46\pm0.10$\
&$K^0\, \rho^{0}$ &$(C_V\,'' - E_P\,'')/\s$ &$674$&$0.70\pm0.07$ &$0.39\pm0.05$\
&$K^0\, \omega$ &$-(C_V\,'' + E_P\,'')/\s$ &$670$&$0.58\pm0.06$ &$0.52\pm0.06$\
&$K^0\, \phi$ &$- E_V\,''$ &$521$&$0.16\pm0.03$ &$0.33\pm0.03$\
$D^+$ &$K^{*0}\, \pi^+$ &$C_P\,'' + A_V\,''$ &$712$&$2.94\pm0.52$ &$2.94\pm0.65$\
&$K^{*+}\, \pi^0$ &$(T_P\,'' - A_V\,'')/\s$ &$714$&$3.74\pm0.49$ &$2.71\pm0.30$\
&$K^{*+}\, \eta$ &$-(T_P\,'' - A_P\,'' + A_V\,'')/\st$ &$586$&$3.37\pm0.43$ &$3.37\pm0.25$\
&$K^{*+}\, \eta\,'$&$(T_P\,'' + 2A_P\,'' + A_V\,'')/\sx$ &$137$&$0.0095\pm0.0029$&$0.0026\pm0.0010$\
&$K^0\, \rho^+$ &$C_V\,'' + A_P\,''$ &$677$&$3.43\pm0.75$ &$3.43\pm0.47$\
&$K^+\, \rho^0$ &$(T_V\,'' - A_P\,'')/\s$ &$679$&$2.17\pm0.40$ &$3.01\pm0.24$\
&$K^+\, \omega$ &$(T_V\,'' + A_P\,'')/\s$ &$675$&$0.64\pm0.21$ &$0.26\pm0.07$\
&$K^+\, \phi$ &$A_V\,''$ &$527$&$0.12\pm0.06$ &$0.12\pm0.06$\
$D_s^+$&$K^{*0}\, K^+$ &$T_V\,'' + C_P\,''$ &$682$&$0.20\pm0.02$ &$0.20\pm0.02$\
&$K^{*+}\, K^0$ &$T_P\,'' + C_V\,''$ &$683$&$1.18\pm0.16$ &$1.18\pm0.18$\
FACTORIZATION COMPARISONS
=========================
In the current section we compare our results for the amplitudes of $T_P$ and $T_V$ with the values extracted from explicit evaluation of the tree diagram assuming factorization [@Bjorken:1990]. In order to calculate $T_P$ we use the decay $D^0 \to K^-\, \rho^+$. In this scenario the spectator $\ol{u}$ quark goes from $D^0$ to the pseudoscalar $K^-$ and so we use the standard form of the $(D \to P)$ current [@Wisgur:1990]: H\_= f\_+(q\^2) (p\_D + p\_K)\_- f\_-(q\^2) (p\_D - p\_K)\_where $f_+$ and $f_-$ are the relevant form factors. The current we use for the vector meson is [@Rosner:1990]: \^ = \^ m\_ f\_ where $\epsilon^{\mu}$ represents the polarization of the vector meson, $m_{\rho}$ is its mass and $f_{\rho}$ is the associated decay constant. The invariant amplitude and the decay rate for the process $D^0 \to K^-\, \rho^+$ via the tree diagram may then be written as (D\^0 K\^-\^+)\_[T\_P]{}&=&-[*[i]{}*]{}V\_[cs]{} V\_[ud]{}\^[\*]{}H\_\^\
(D\^0 K\^-\^+)\_[T\_P]{}&=&\_[\_ q\^=0]{}|(D\^0 K\^- \^+)\_[T\_P]{}|\^2 After summing over the $\rho$ polarization and taking the modulus squared of the invariant amplitude one obtains the final form for $|T_P|$: |T\_P|&=&\
&&\
&=& (5.45 0.07) 10\^[-6]{} which is to be compared with the values quoted in Table \[tab:cfampsc\], and favors solution A2 over A1.
In obtaining the result stated above we used $|f_+(m_\rho^2)||V_{cs}| = 0.869
\pm 0.009$ [@Shipsey:2008]. The particle masses and the quantity $|V_{ud}|$ were taken from [@Amsler:2008]. $p^*$ is as quoted in Table \[tab:CFPV\]. We calculated the value of $f_\rho$ using the following formula: f\_&=&f\_\^\
&=&(209 1.6)[ MeV]{} where once again the particle masses and branching fractions were taken from [@Amsler:2008].
A similar approach may be taken in order to evaluate $|T_V|$ by looking at the decay $D^0 \to K^{*-}\, \pi^+$ via the tree diagram. In this case the spectator $\ol{u}$ quark goes from $D^0$ to the vector meson $K^{*-}$, so we use the standard forms of the $(D \to V)$ vector and axial-vector currents [@Wisgur:1990] and the pion current [@Rosner:1990]: V\_&=&*[i]{}g\_\^[\*]{}(p\_D + p\_[K\^\*]{})\^(p\_D - p\_[K\^\*]{})\^\
A\_&=&f\^\*\_ + a\_+(\^\*p\_D)(p\_D + p\_[K\^\*]{})\_ + a\_-(\^\*p\_D)(p\_D - p\_[K\^\*]{})\_\
\^&=&*[i]{}f\_q\^ We obtain for the amplitude $|T_V|$ the following expression: |T\_V|=|V\_[cs]{}||V\_[ud]{}|f\_ |f + a\_+ (m\_D\^2 - m\_[K\^\*]{}\^2) + a\_- m\_\^2| In principle this can be used to calculate $T_V$ once the form factors are given. However, we may adopt a simplification using a result from Ref. [@Rosner:1999], based on the earlier discussion in Ref.[@Rosner:1990]. In the heavy-quark limit one expects $\Gamma(D \to
\bar K^* \pi^+)_T = \Gamma(D \to\bar K \pi^+)_T$ and hence p\^\*\_[K ]{} |T|\^2\_[K ]{} = (p\^\*\_[K\^\* ]{})\^3 |T|\^2\_[K\^\* ]{} , where $T_{K^* \pi} = T_V$. In Ref. [@Bhattacharya:2008ss] we found in a fit to $D \to PP$ amplitudes that $|T|_{K \pi} = (2.78 \pm 0.13) \times
10^{-6}$ GeV. With $p^*_{K \pi} = 0.861$ GeV and $p^*_{K^* \pi} = 0.711$ GeV we then obtain the result T\_V = (4.3 0.2) 10\^[-6]{} , in reasonable agreement with the value of $(3.95 \pm 0.07) \times 10^{-6}$ quoted in Table \[tab:cfampsa\], especially considering the uncertainties associated with QCD corrections and with the use of the heavy-quark limit for the final strange quark.**
CONCLUSIONS
===========
We have used the flavor topology description to study the validity of flavor SU(3) for describing $D \to PV$ decays, to obtain relative phases and magnitudes of various contributing amplitudes, and to predict rates for as-yet-unseen singly- and doubly-Cabibbo-suppressed decays. We assumed flavor $SU(3)$ to be an exact symmetry for the tree level diagrams. We found that singly-Cabibbo-suppressed decays favor a ratio of color-suppressed to tree amplitudes $|C_P/T_V| > 1$, where the subscript denotes the meson ($P$ or $V$) containing the spectator quark. The present data for the Cabibbo-favored decays are compatible with twelve distinct sets of solutions for the amplitudes $T_P$, $C_P$, $E_P$, $A_P$, $T_V$, $C_V$, $E_V$, and $A_V$ (up to discrete ambiguities). However, on the basis of experimental branching ratios for singly-Cabibbo-suppressed decays we were able to choose two sets of solutions giving substantially lower values for $\chi^2$ than the other ten.
Our predictions of the branching ratios for singly-Cabibbo-suppressed decays deviate from the available experimental data in several cases, such as those in the first four lines of Table VI. This shows that flavor $SU(3)$ is not an exact symmetry. However flavor $SU(3)$ breaking, though present, is no worse in $D \to PV$ decays than in the $D \to PP$ decays discussed in Ref.[@Bhattacharya:2008ss].
Our prediction for the $D^+_s \to \eta\,'\rho^+$ branching ratio is much lower than the available experimental value. Either there are additional contributions to $\eta'$ production which we have neglected, or the experimental situation needs to be re-evaluated.
Our analysis of the singly-Cabibbo suppressed decays shows that processes such as $D^0 \to \pi^0 \omega$ can be used to distinguish between the two most likely amplitude solutions. The mean values predicted for the branching ratios of these processes differ by nearly a factor of three in the two solutions, but experimental data are not yet available to resolve this problem.
The branching ratios predicted for doubly-Cabibbo-suppressed decays are close to the experimental values in the two cases for which data are available. A precise measurement of a few of the other branching ratios may help select one of the two most-favored amplitude solutions.
Finally, factorization computations of the tree amplitudes agree with results obtained in direct analyses. However, a more precise calculation of the amplitudes using the factorization assumption could be done if data on the relevant form factors were available.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This work was supported in part by the United States Department of Energy through Grant No. DE FG02 90ER40560.
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[^1]: bhujyo@uchicago.edu
[^2]: rosner@hep.uchicago.edu
|
---
abstract: 'We investigate the minimum cases for realtime probabilistic machines that can define uncountably many languages with bounded error. We show that logarithmic space is enough for realtime PTMs on unary languages. On binary case, we follow the same result for double logarithmic space, which is tight. When replacing the worktape with some limited memories, we can follow uncountable results on unary languages for two counters.'
author:
- 'Maksims Dimitrijevs, Abuzer Yakaryilmaz'
bibliography:
- 'tcs.bib'
title: Uncountable realtime probabilistic classes
---
Introduction
============
When using uncountable transitions, bounded-error probabilistic and quantum models can recognize uncountably many languages [@ADH97; @SayY14C]. It is interesting to identify the minimum resources that are sufficient to follow this result. Some of the known results [@SayY14C; @DY16A] are as follows:
- Uncountably many unary languages can be defined by poly-time double log-space probabilistic Turing machines (PTMs) and linearithmic ($ O(n \log n) $) time log-space one-way PTMs.
- Uncountably many $k$-ary languages ($k>1$) can be defined by poly-time constant-space quantum Turing machines, linear-time linear-space two-way probabilistic counter machines, and arbitrarily small but non-constant-space PTMs.
In this paper, we investigate *realtime* probabilistic models that read the input in a streaming mode such that there is no pause on the input symbols. (This is also referred as strict realtime.) On general alphabets, it is known that bounded-error one-way PTMs cannot recognize any nonregular language in space $ o(\log \log n) $ [@Fr85]. Here we show that $ O(\log \log n) $-space is enough for realtime PTMs to define uncountably many languages. Therefore, this bound is tight for general alphabets. On unary alphabet, we follow the same result for $ O(\log n) $ space and we leave open whether realtime PTMs can recognize any unary nonregular languages in $ o(\log n) $ space. Lastly, we follow the same result for unary realtime probabilistic automata with counters and we show that two counters are sufficient. It is known that one counter is not enough since unary one-way probabilistic automata with one stack can recognize only regular languages with bounded error [@KGF97]. On the other hand, the case of two stacks is trivial since a work tape can be simulated by two stacks. We leave open to determine the minimum number of counters that use sublinear or sublogarithmic space on the counters.
In the next section, we present some background to follow the rest of the paper and then we present our results in Section \[sec:main-results\] under two subsections. We first present the results for unary languages (Section \[sec:unary\]), and then for general alphabet languages (Section \[sec:binary\]).
Background
==========
We assume the reader is familiar with the basics of complexity theory and automata theory. Throughout the paper, $ \Sigma $ not containing ${\mbox{\textcent}}$ (the left end-marker) and ${\$}$ (the right end-marker) denotes the input alphabet, $ {\tilde{\Sigma}}$ is the set $ \Sigma \cup \{ {\mbox{\textcent}},{\$}\} $, $ \Gamma $ not containing blank symbol denotes the work tape alphabet, $ {\tilde{\Gamma}}$ is the set $ \Gamma \cup \{ \mbox{blank symbol} \} $, and $ {\Sigma^{*}}$ is set of all strings obtained from the symbols in $\Sigma$ including the empty string.
Formally, a realtime PTM $ P $ is a 7-tuple $$P = (S,\Sigma,\Gamma,\delta,s_1,s_a,s_r ),$$ where $ S $ is the set of finite internal states, $ s_1 \in S $ is the initial state, $ s_a \in S $ and $ s_r \in S $ ($s_a \neq s_r$) are the accepting and rejecting states, respectively, and $ \delta $ is the transition function
$$\delta: S \times {\tilde{\Sigma}}\times {\tilde{\Gamma}}\times S \times {\tilde{\Gamma}}\times { \{ \leftarrow,\downarrow,\rightarrow \} }\rightarrow [0,1]$$ that governs the behaviour of $P $ as follows: When $ P $ is in state $ s \in S $, reads symbol $ \sigma \in {\tilde{\Sigma}}$ on the input tape, and reads symbol $ \gamma \in {\tilde{\Gamma}}$ on the work tape, it enters state $ s' \in S $, writes $ \gamma' \in {\tilde{\Gamma}}$ on the cell under the work tape head, and then the work tape head is updated with respect to $ d \in { \{ \leftarrow,\downarrow,\rightarrow \} }$ with probability $$\delta(s,\sigma,\gamma,s',\gamma',d),$$ where “$ \leftarrow $” (“$\downarrow$” and “$\rightarrow$”) means the head is moved one cell to the left (the head does not move and the head is moved one cell to the right). Note that input head can only perform “$\rightarrow$” moves. To be well-formed PTM, the following condition must be satisfied: for each triple $(s,\sigma,\gamma) \in S \times {\tilde{\Sigma}}\times {\tilde{\Gamma}}$, $$\sum_{s' \in S,\gamma' \in {\tilde{\Gamma}},d \in { \{ \leftarrow,\downarrow,\rightarrow \} }} \delta(s,\sigma,\gamma,s',\gamma',d) = 1.$$
The computation starts in state $ s_1 $, and any given input, say $ w \in \Sigma^* $, is read as $ {\mbox{\textcent}}w {\$}$ from the left to the right symbol by symbol, and the computation is terminated and the given input is accepted (rejected) if $ P $ enters $ s_a $ ($s_r$). It must be guaranteed that the machine enters a halting state after reading ${\$}$.
The space used by $ P $ on a given input is the number of all cells visited on the work tape during the computation with some non-zero probability. The machine $ P $ is called to be $ O(s(n)) $ space bounded machine if it always uses $ O(s(n)) $ on any input with length $n$.
If (realtime) $ P $ is allowed to spend more than one step on an input symbol, then it is called one-way. Formally, its transition function is extended by the move of the input head with $ \{\downarrow,\rightarrow\} $ in each transition, and then, the well-formed condition is updated accordingly.
Moreover, any PTM without work tape is called probabilistic finite automaton (PFA).
A counter is a special type of memory containing only the integers. Its value is set to zero at the beginning. During the computation, its status (whether its value is zero or not) can be read similar to reading blank symbol or non-blank symbol on the work tape, and then its value is incremented or decremented by 1 or not changed similar to the position update of the work head. (A counter can be seen as a unary stack.)
A realtime probabilistic automaton with $ k $ counters (P$k$CA) is a realtime PTM having $ k $ counters instead of a working tape. In each step, instead of reading the symbol under the work tape head, it checks the statuses of all counters; and then, it updates the value of each counter by a value from $ \{-1,0,1\} $ instead of updating the content of the work tape.
The language $L$ is said to be recognized by a PTM with error bound $\epsilon$ ($0 \leq \epsilon < 1/2$) if every member of $ L $ is accepted with probability at least $1-\epsilon$ and every non-member of $L$ ($w \notin L$) is accepted with probability not exceeding $\epsilon$.
We denote the set of integers $ \mathbb{Z} $ and the set of positive integers $ \mathbb{Z}^+ $. The set $ \mathcal{I} = \{ I \mid I \subseteq \mathbb{Z^+} \} $ is the set of all subsets of positive integers and so it is an uncountable set (the cardinality is $ \aleph_1 $) like the set of real numbers ($ \mathbb{R} $). The cardinality of $ \mathbb{Z} $ or $ \mathbb{Z^+} $ is $ \aleph_0 $ (countably many).
The membership of each positive integer in any $ I \in \mathcal{I} $ can be represented as a binary probability value: $$p_I = 0.x_1 0 1 x_2 0 1 x_3 0 1 \cdots x_i 0 1 \cdots,~~~~ x_i = 1 \leftrightarrow i \in I.$$
Our results {#sec:main-results}
===========
In our proof we use a fact presented in our previous paper [@DY16A].
\[fact:DY16A\] [@DY16A] Let $x=x_1 x_2 x_3 \cdots$ be an infinite binary sequence. If a biased coin lands on head with probability $p = 0. x_1 0 1 x_2 0 1 x_3 0 1 \cdots$, then the value $x_k$ can be determined with probability at least $\frac{3}{4}$ after $64^k$ coin tosses.
The proof of this fact involves the analysis of probabilistic distributions for the number of heads after tossing $64^k$ coins that land on the head with probability $p$. The $(3 \cdot k+3)$-th bit from the right in obtained number of heads is equal to $x_k$ with probability at least $\frac{3}{4}$.
Unary languages {#sec:unary}
---------------
In [@YS13B], it was shown that realtime deterministic Turing machines (DTMs) can recognize unary nonregular languages in $O(\log n)$ space. By adopting the technique given there, we can show that bounded-error realtime PTMs can recognize uncountably many unary languages.
\[thm:log-uPTM\] Bounded-error realtime unary PTMs can recognize uncountably many languages in $O(\log n)$ space.
We start with defining a unary nonregular language that can be recognized by bounded-error log-space realtime PTMs: $${\mathtt{ULOG}}= \{ 0^{k_i} \mid k_1=64 \cdot 28 \mbox{ and } k_i=k_{i-1} + 64^{i} \cdot (18i+10) \mbox{ for } i > 1 \},$$ where each member is defined recursively. Since it is not a periodic language, $ {\mathtt{ULOG}}$ is nonregular.
For any $ I \in \mathcal{I} $, we define the following language: $${\mathtt{ULOG(I)}}= \{ a^{k_i} \mid a^{k_i} \in {\mathtt{ULOG}}\mbox{ for } i \geq 1 \mbox{ and } i \in I \}.$$
We describe a bounded-error log-space PTM for $ {\mathtt{ULOG(I)}}$, say $P_I$. Then, we can follow the proof since there is a bijection (one-to-one and onto) between $ I \in \mathcal{I} $ and $ {\mathtt{ULOG(I)}}$ and $ \mathcal{I} $ is an uncountable set.
The PTM $ P_I $ uses a coin landing on head with probability $$p_I=0. x_1 0 1 x_2 0 1 x_3 0 1 \cdots x_i 0 1 \cdots,$$ where $ x_i = 1 $ if and only if $ i \in I $. The aim of $ P_I $ is iteratively finding the values of $ x_1, x_2, \ldots $ with high probability. If all input is read before reaching a decision on one of these values, then the input is always rejected.
During the computation, $ P_I $ uses two binary counters on the work tape. At the beginning, the iteration number is one, $ i = 1 $. The machine initializes the work tape as “\#000000\#000000\#” by reading 15 ($=9 \cdot 1 + 5 + 1$) symbols from the input (after 15-th symbol the working tape head is placed on the first zero to the left from the third $\#$). We name the separator symbols $ \# $s for the counters as the first, second, and third ones from left to the right. The first (second) counter is kept between the last (first) two $ \# $s.
By using the first counter, the machine counts up to $ 64^{i} $ and so meanwhile also tosses $ 64^{i} $ coins. By using the second counter, it counts the number of heads. The value of each counter can be easily increased by 1 when the working tape head passes on the counters from right to left once. Thus, when the working tape head is on the third $\#$, it goes to the first $\#$, and meanwhile increases the value of the first counter by 1, then tosses its coin, and, if it is a head, it also increases the value of the second counter. After tossing $ 64^{i} $ coins, the machine uses the leftmost value of the second counter as its answer for $ x_i $. Once this decision is read from the work tape and immediately after the working tape head is placed on the first $ \# $, the current iteration is finished. If (i) an iteration is finished, (ii) there is no more symbol remaining to be read from the input, and (iii) the decision is positive, then the input is accepted, which is the single condition to accept the input. After an iteration is finished, the next one starts and each counter is initialized appropriately and then the same procedure is repeated as long as there are some input symbols to be read.
Since the input is read in realtime mode, the number of computational steps is equal to the length of the input plus two (the end-markers). Now, we provide the details of each iteration step so that we can identify which strings are accepted by $ P_I $.
At the beginning of the $ i $-th iteration, the working tape head is placed in the first $ \# $ and the contents of the counters are as follows: $$\# \underbrace{0 \cdots 0 }_{3(i-1)+3} \# \underbrace{0 \cdots 0 }_{6(i-1)} \#.$$ By reading $ 9i+5 +1$ symbols from the input, the counters are initialized for the current iteration as $$\# \underbrace{0 \cdots 0 }_{3i+3} \# \underbrace{0 \cdots 0 }_{6i} \#$$ by shifting the second and third $ \# $s to 3 and 9 amounts of cells to the right (after initialization the working head is placed on the first zero to the left from the third $\#$).
After the initialization of the counters, the working head goes to the first $ \# $ and then comes back on the third $ \# $ $ 64^{i} -1$ times. In each pass from right to left, the first counter is increased by 1, the coin is flipped, and then the second counter is increased by 1 if the result is head. When all digits of the first counter are 1, which means the number of passes reaches $ 64^{i} - 1 $, the working tape head makes its last pass from the third $ \# $ to the first $ \# $. During the last pass, $ P_I $ flips the coin once more and then determines the leftmost digit of the second counter. Meanwhile, it also sets both counters to zeros.
By also considering the initialization step, $ P_I $ makes $ 64^i $ passes starting from the first $ \# $. So, the total number of steps is $ 64^i \cdot 2 \cdot (9i+5) $ during the $ i $-th iteration. One can easily verify that this is valid also for the case of $ i = 1 $.
Therefore, $ P_I $ can deterministically detect the $ i $-th shortest member of $ {\mathtt{ULOG}}$ after reading $ k_i $ symbols, where $ k_1 = 64 \cdot (28)$ and $ k_i = k_{i-1} + 64^i \cdot (18i+10) $ for $ i > 1 $. Then, by using Fact \[fact:DY16A\], we can follow that $ P_I $ recognizes $ {\mathtt{ULOG(I)}}$ with error bound $ \frac{1}{4} $.
It is known that bounded-error unary one-way PFAs with a single stack cannot recognize any nonregular language [@KGF97]. Therefore, we can check the case of having two stacks.
Bounded-error unary realtime PFA with two stacks using logarithmic amount of space can recognize uncountably many languages.
It is a well-known fact that two stacks can easily simulate a worktape of a TMs without any delay on the running time. Therefore, by using Theorem \[thm:log-uPTM\], we can follow the result in a straightforward way.
It is possible to replace stacks with counters by losing the space efficiency. We start with four counters.
\[thm:4PCA\] Bounded-error realtime unary P4CAs can recognize uncountably many languages.
We start with describing a realtime P4CA, say $ P_I $, that can use a coin landing head with probability $ p_I $ for an $ I \in \mathcal{I} $. Let $ C_i $ ($ 1 \leq i \leq 4 $) represent the values of counters.
The automaton $P_I$ executes an iterative algorithm. We use $ m $ to denote the iteration steps. At the beginning, $ m = 1 $. In each iteration, $ 64^m $ coin tosses are performed. The details are as follows:
- Set $ C_1 = 64^m $ and $ C_2 = 4 \cdot 8^m $.
- Perform $ C_1 $ coin flips and meanwhile increase/decrease the values of $ C_2 $ and $ C_3 $ by 1. If the coin flip result is head, one of the counters is increased by 1 and the other one is decreased by 1. When one of them hits zero, update strategy is changed. Since $ C_3 $ is zero at the beginning, the first strategy is decreasing the value of $ C_2 $ and increasing the value of $ C_3 $. Thus, after each $4 \cdot 8^m $ heads, the update strategy on the counters is changed.
- When $ C_1 $ hits zero, $ C_2 $ and $ C_3 $ are equal to $ X $ and $ 4\cdot 8^m - X $, and, the automaton makes its decision on $ x_m $. If the latest strategy is decreasing the value of $ C_3 $ or $ C_2 = 0 $, then $ x_m $ is determined as 1. Otherwise, it is determined as 0.
The described algorithm is similar to the one that is used in the proof of Theorem \[thm:log-uPTM\]. Here changing the update strategy between $ C_2 $ and $ C_3 $ refers to the change of bit $ x_m $, which is changed after each $4 \cdot 8^m $ heads: it is 0 initially and then changed as $ 1, 0, 1, \ldots $.
At the end of the $ m $-th iteration, we have $C_1 = 0 $, $ C_2 = X $, and $ C_3 = 4 \cdot 8^m -X $. We initialize $ (m+1) $-th iteration as follows:
- By using $ C_2 $ and $C_3$, we can set $ C_1 = 2X + 2 (4 \cdot 8^m - X) = 8^{m+1} $. Now $ C_2 = C_3 = C_4 = 0 $.
- Set $ C_2 = C_3 = 8^{m+1} $ by setting $ C_1 = 0 $. Then, in a loop, until $ C_2 $ hits zero: decrease value of $ C_2 $ by 1, then transfer $ C_3 $ to $ C_4$ (or $ C_4 $ to $ C_3 $ if at the beginning of loop’s iteration $ C_3 = 0$) and meanwhile add $ 8^{m+1} $ to $ C_1 $.
- $ C_1 = 8^{m+1} (8^{m+1}) = 64 ^{m+1} $, $ C_2 = 0 $, $ C_3 = 8^{m+1} $, $ C_4 = 0 $. Then set $ C_2 = 4 \cdot 8^{m+1} $ by setting $ C_3 = 0 $.
After initializing, we execute the coin-flip procedure. Each iteration finalizes after coin-flip procedure.
The input is accepted if there is no more input symbol to be read exactly at the end of an iteration, say $m$-th, and $ x_m $ is guessed as $ 1 $. Otherwise, the input is always rejected.
The coin tosses part is performed in $64^m$ steps. The initialization part for $m$-th iteration is performed in $ 8^m + 8^m + 64^m + 4 \cdot 8^m = 64^m + 6 \cdot 8^m $ steps, where $m>1$. The initialization part for $ m = 1 $ is performed in 64 steps.
Based on this analysis, we can easily formulate the language recognized by $ P_I $, which is subset of the following language $$\mathtt{UP4CA} = \{ 0^{k_i} \mid k_1=128 \mbox{ and } k_i=k_{i-1} + 6 \cdot 8^i + 2 \cdot 64^i \mbox{ for } i > 1 \}.$$ For any $ I \in \mathcal{I} $, the realtime P4CA $ P_I $ can recognize the language $$\mathtt{UP4CA(I)} = \{ a^{k_i} \mid a^{k_i} \in \mathtt{U4PCA} \mbox{ for } i \geq 1 \mbox{ and } i \in I \}$$ with bounded error. The automaton $ P_I $ iteratively determines the values of $ x_1, x_2, \ldots $ with high probability and the number of steps for each iteration corresponds with the members of $\mathtt{U4PCA}$.
Since $ \mathcal{I} $ is an uncountable set and there is a bijection between $ I \in \mathcal{I} $ and $ \mathtt{UP4CA(I)} $, realtime P4CAs can recognize uncountably many unary languages with bounded error.
We can establish a similar result also for realtime P2CAs. For this purpose, we can use the well-known simulating technique of $ k $ counters by 2 counters.
\[thm:2PCA\] Bounded-error unary realtime P2CAs can recognize uncountably many languages.
Let $ P_I $ be the realtime P4CA described above and $ \mathtt{UP4CA(I)} $ be the language recognized by it. Due to the realtime reading mode, the unary inputs to $ P_I $ can also be seen as the time steps. For example, $ P_I $ can be seen as a machine without any input but still making its transition after each time step. Thus, after each step it can be either in an accepting case or a rejecting case.
It is a well-known fact that two counters can simulate any number of counters with big slowdown [@Min67]. The values of $ k $ counters, say $ c_1,c_2,\ldots,c_k $, can be stored on a counter as $$p_1^{c_1} \cdot p_2^{c_2} \cdot \cdots p_k^{c_k},$$ where $ p_1,\ldots,p_k $ are some prime numbers. Then, by the help of the second counter and the internal states, it can be easily detected and stored the status of each simulated counters, and then all updates on the simulated counters are reflected one by one.
Thus, by fixing the above simulation, we can easily simulate $ P_I $ by a P2CA, say $ P_I' $. Then, $ P_I' $ recognizes a language with bounded error, say $ \mathtt{UP2CA(I)} $.
It is easy to see that there is a bijection between $$\{ \mathtt{UP4CA(I)} \mid I \in \mathcal{I} \} \mbox{ and } \{ \mathtt{UP2CA(I)} \mid I \in \mathcal{I} \},$$ and so realtime P2CAs also recognize uncountably many languages with bounded error. Remark that for each member of $ \mathtt{UP4CA(I)} $, the corresponding member of $ \mathtt{UP2CA(I)} $ is much longer.
Generic alphabet languages {#sec:binary}
--------------------------
Here, we focus on non-unary alphabets and establish our result for double logarithmic space. For this purpose, we use a fact given by Freivalds in [@Fre83].
\[fact:Fre83\] Let $P_1 (n)$ be the number of primes not exceeding $2^{ \lceil log_2 n \rceil }$, $P_2 (l,N',N'')$ be the number of primes not exceeding $2^{ \lceil log_2 l \rceil }$ and dividing $|N'-N''|$, and $P_3 (l,n)$ be the maximum of $P_2 (l,N',N'')$ over all $N'<2^n$, $N'' \leq 2^n$, $N' \neq N''$. Then, for any $\epsilon > 0$, there is a natural number $c$ such that $\lim_{n\to\infty} \frac{P_3 (cn,n)}{P_1 (cn)} < \epsilon$.
Let $bin(i)$ denote the unique binary representation of $i>0$ that always starts with digit 1. The language $ {\mathtt{LOGLOG}}$ is composed by the strings $$bin(1) 2 bin(2) 2 bin(3) 2 ...2 bin(s) 4,$$ where $ |bin(s)| = 64^k $ for some positive integer $k$. For any $ I \in \mathcal{I} $, we define language $ {\mathtt{LOGLOG(I)}}= \{ w \mid w \in {\mathtt{LOGLOG}}\mbox{ and } k \in I \} $.
\[fact:dist-of-primes\] Denote by $\pi (x)$ the number of primes not exceeding $x$. The Prime Number Theorem states that $\lim_{x\to\infty} \frac{\pi (x)}{ x / \ln x} = 1 $ [@Chandrasekharan1968].
\[thm:onewayPTM\] Bounded–error one–way PTMs can recognize uncountably many languages in $O(\log \log n)$ space.
By modifying the one-way algorithm given in [@Fre83], we present a PTM, say $P_{c,I}$, shortly $P$, for language $ {\mathtt{LOGLOG(I)}}$ for $I \in \mathcal{I}$ and for a specific $c$ that determines the error bound. $P$ performs different checks by using the separate parts of the work tape.
For each $ i $, $P$ keeps two registers storing $ m = |bin(i)| $ and $ m_0 = |bin(i-1)| $. After reading $bin(i)$, $P$ checks: if $m=m_0$ or ($m=m_0+1$ and $bin(i-1)$ contained only ones), then $P$ continues. Otherwise, $P$ rejects the input.
For each $bin(i)$, $P$ generates a random number of $|m| \cdot c$ bits and tests it for primality. If the generated number is not prime, the same procedure is repeated. Due to Fact \[fact:dist-of-primes\], we can follow that the probability of picking a prime number of $|m| \cdot c$ bits is $ \theta (\frac{1}{|m| \cdot c})$. Therefore, the expected time of finding a prime number is $ O(|m| \cdot c) $. Assume that the generated prime number is $r_i$. For each $bin(i)$, $P$ calculates $bin(i) \mod r_i$ and $bin(i+1) \mod r_i$. If $(bin(i) \mod r_i)+1 \neq bin(i+1) \mod r_i$, $P$ rejects the input. Otherwise, the computation continues.
After reading “4”, $P$ checks whether $m=64^k$ for some integer $k>0$. If so, $m$ is written on the tape as $1(000000)^k$. If $m \neq 64^k$, then the input is rejected.
If all previous checks are successful, $P$ tosses $ 64^k $ coins and meanwhile calculates the number of heads $ \mod (8 \cdot 8^k) $, say $ C $. If after all coin tosses, the leftmost bit of $C$ is 1, then the input is accepted, otherwise it is rejected.
The PTM $P$ reaches symbol “4” without rejecting with probability 1 if the input belongs to ${\mathtt{LOGLOG}}$, and it rejects the input before reaching “4” with probability at least $1-\epsilon$ if the input is not in ${\mathtt{LOGLOG}}$ due to Fact \[fact:Fre83\]. Due to Fact \[fact:DY16A\] the membership of $k \in I$ for ${\mathtt{LOGLOG(I)}}$ will be computed with probability at least $\frac{3}{4}$. Therefore language ${\mathtt{LOGLOG(I)}}$ is recognized correctly with probability at least $(1-\epsilon) \cdot \frac{3}{4}$, which can be arbitrarily close to $\frac{3}{4} $ by picking a suitable $c$.
The space used on the work tape is linear in the length of the counter for $|bin(i)|$. The value of $bin(i)$ is logarithmic to the length of input word, and so the length of the counter is double logarithmic to the input length. Therefore, the space used is in $O(\log \log n)$ throughout the computation.
Let $ L \subseteq \Sigma^* $ be a language recognized by a one-way DTM, say $D$, and $ \sigma $ be a symbol not in $ \Sigma $. We can execute $ D $ in realtime reading mode on the inputs defined on $ \Sigma \cup \{\sigma\} $ as follows [@YS13B]: For each original “wait” move on a symbol from $ \Sigma $, the machine expects to read symbol $\sigma $. If it reads something else or there is no more input symbol, then the input is rejected. If there is more than expected $ \sigma $ symbol, then again the input is rejected. Thus, we can say that this modified machine recognizes a language $ L' $ and there is a bijection between $ L $ and $L'$. Moreover, the space and time bounds for both machines are the same.
The question is whether we can apply a similar idea for one-way PTM given above in order to get a realtime PTM. A DTM follows a single path during its computation and so the aforementioned bijection can be created in a straightforward way. On the other hand, PTMs can follow different paths with different lengths in each run. So, in order to follow a similar bijection, we need some modifications. The main modification is necessary for the task of picking the prime numbers. Except this task, the other ones can be executed with the same number of steps (remember the algorithms in the previous subsection) in every execution of the machine.
Now, we modify PTM $P_{c,I}$ in order to guarantee that each computation path uses the same amount of time steps on the same input. We represent the new PTM as $ P'_{c,I} $ or shortly $P'$.
The PTM $P'$ uses some registers on the work tape separated by “$\#$”: $$\#1st\#2nd\# \cdots \#last\#.$$
- The 1st register keeps both the lengths of the counters $m$ and $m_0$. If $m= x_1 x_2 x_3 \cdots$ and $m_0= y_1 y_2 y_3 \cdots$, then the register keeps the values in the following way: $x_1 y_1 x_2 y_2 x_3 y_3 \cdots$. After reading symbol “2” it is easy to compare $m$ and $m_0$ bit by bit with a single pass.
- The 2nd register keeps the number of heads for the coin-tosses, based on which the bit $x_k$ is determined. It is set to $\lceil |m|/2 \rceil+2$ zeros before any coin-toss procedure and it is updated accordingly when the value of $m$ is changed.
- The 3rd register keeps the track of attempts to generate prime number, it has $|m| \cdot c$ bits.
- The 4th and 5th registers keep the prime numbers with some auxiliary numbers. Each register has $|m| \cdot c \cdot 2$ bits. If the (candidate) prime number is $r = r_1 r_2 r_3 \cdots$ and the auxiliary number is $q = q_1 q_2 q_3 \cdots$, then the register keeps both of them as $r_1 q_1 r_2 q_2 r_3 q_3 \cdots$. The machine uses $r$ to store the prime number that is being checked or computed, and $q$ is used to help to perform tasks with $r$ like storing number modulo $r$ and comparing and copying numbers. For each $j>0$, the machine uses 4th and 5th registers to work with prime numbers and then checks the correctness of the candidates for $bin(2 \cdot j-1)$ and $bin(2 \cdot j)$.
- The 6th and 7th registers are the same as 4th and 5th registers, respectively. Only they are responsible for the correctness of the candidates for $bin(2 \cdot j)$ and $bin(2 \cdot j+1)$.
- The 8th register has a number to keep track of total number of subtractions performed while checking the divisibility of $r$ by $d$. It has $|m| \cdot c$ bits.
- The 9th register has twice of $\lceil |m|/2 \rceil \cdot c$ bits to keep numbers $d$ and $h$ (each is $\lceil |m|/2 \rceil \cdot c$ bits). If $d = d_1 d_2 d_3 \cdots$ and $h = h_1 h_2 h_3 \cdots$, then the register keeps them as $d_1 h_1 d_2 h_2 d_3 h_3 \cdots$. Both numbers are used to check whether the generated number $r$ is prime. The machine uses $d$ to check whether $d$ does not divide $r$, such check is performed for different values $d$. The check is performed by making subtractions. The value of $d$ is subtracted from $r$ multiple times. For this operation, the machine uses $h$ as auxiliary number.
Each member of ${\mathtt{LOGLOG(I)}}$ has parts $bin(i)$ at least up to $bin(2^{63})$. $P'$ deterministically checks input up to $bin(2^{63})$ and prepares the work tape with 9 registers.
Now, we describe the steps of picking prime numbers.
For number $bin(i)$, the prime number is generated in $(6 - 2 \cdot (i \mod 2))$-th register. The number $r$ is generated by using $|m| \cdot c$ random bits (bit by bit). After this, the primality check is performed. For this purpose, the machine checks whether $r$ is divided by any natural number between 2 and $2^{\lceil |m|/2 \rceil \cdot c}-1$, where $2^{\lceil |m|/2 \rceil \cdot c} > sqrt(r)$ because $r < 2^{|m| \cdot c}$. Each candidate natural number is denoted by $d$ below. Remark that the number of $d$s does not depend on $r$ and so for any candidate prime number, the primary test procedure takes the same number of steps.
To begin the check of divisibility of $r$ by $d$, the value of $r$ is copied to $q$ bit after bit, and the value of $d$ is copied to $h$ bit after bit. The 8th register is initialized with zeros before check for pair $r$ and $d$. Then, $2^{|m| \cdot c}$ iterations are performed. In each iteration, the values of $q$ and $h$ are decreased by 1, the value of 8th register is increased by 1. If only $h$ reaches zero, $d$ is again copied into $h$ and the machine continues to perform iterations. When $ q $ reaches zero, if $ h $ reaches zero at the same time, the machine concludes that $ r $ is not a prime number, otherwise, $r$ is not divisible by $d$. After that, $P'$ continues to perform the iterations but without changing $q$ and checks of value of $q$ until the value of the 8th register reaches $2^{|m| \cdot c}$. Then, $P'$ repeats the procedure for the next $d$.
If $r$ is not divisible by any of these $d$s, then the procedure of finding prime is terminated successfully since $r$ is prime, otherwise, the machine continues with the next prime candidate number since $ r $ is not a prime number.
The 3rd register counts the number of attempts to generate a prime number. It is initialized with zeros and is increased by one after each try. If $P'$ finds a prime number before 3rd register reaches $2^{|m| \cdot c}$, $P'$ continues performing the algorithm until the register reaches $2^{|m| \cdot c}$ by fixing the candidate with the already found prime number. If the register reaches value $2^{|m| \cdot c}$ (all bits become zeros) and $P'$ fails to generate a prime number, $P'$ uses the last generated $ r $ for the modular check for pair $bin(i)$ and $bin(i+1)$. $P'$ performs each try to generate (or process already generated) prime number in equal number of steps. For any $bin(i)$ $P'$ performs exactly $2^{|m| \cdot c}$ such operations.
After finding and checking prime $r$, the machine copies $r$ into $(7 - 2 \cdot (i \mod 2))$-th register bit by bit. To perform this operation, the machine sets $q$ to zeros in both registers, copies the bits of $r$ one by one, and marks the copied bit by setting the next bit in $q$ to one.
Now, we describe how the machine calculates the value $bin(i) \mod r$. At the beginning, the register keeps $r$ and zeros for $q$. Assume that $bin(i) = i_1 i_2 \cdots i_m$. When the machine reads $i_j$, the value of $q$ is multiplied by 2 and increased by $i_j$. Therefore, all bits of $q$ are shifted to left by one position, and the machine puts value $i_j$ in leftmost bit. If, after this operation, $q \geq r$, then $r$ is subtracted from $q$. Because both values are interleaved, it is easy to subtract $r$ from $q$ in one pass. In the case when $q < r$ the machine performs one pass through registers without changing the values. This ensures that each iteration for $i_j$ is performed in equal number of steps. The machine performs the calculation while reading $bin(i)$ for the 5th and the 6th registers if $i \mod 2 = 0$, and for the 4th and the 7th registers otherwise.
After these, the machine compares the values of two modules: the 4th and the 5th registers if $i \mod 2 = 0$; the 6th and the 7th registers otherwise. This time machine sets $r$ in both registers to zeros and marks compared bits of $q$’s by setting bits in $r$ to one.
If $ r $ in modular check is not prime, $P'$ cannot guarantee that incorrect pair $bin(i)$ and $bin(i+1)$ will be rejected with probability at least $1-\epsilon$. The probability not to generate a prime number of $|m| \cdot c$ random bits in $2^{|m| \cdot c}$ tries does not exceed ${(1 - \frac{1}{|m| \cdot c})}^{2^{|m| \cdot c}}$ because of Fact \[fact:dist-of-primes\]. Note that $\lim_{n\to\infty} {{(1 - \frac{1}{n})}^n} = \frac{1}{e}$, therefore $\lim_{m\to\infty} {(1 - \frac{1}{|m| \cdot c})}^{2^{|m| \cdot c}} = \lim_{m\to\infty} {\frac{1}{e}}^{\frac{2^{|m| \cdot c}}{|m| \cdot c}} = 0$. The smallest $|m|$ for which a prime number is generated is 7. By picking a suitable $c$, the value ${(1 - \frac{1}{7 \cdot c})}^{2^{7 \cdot c}} = \epsilon_0$ can be arbitrarily close to zero. For each $i>0$, checking the equality of $bin(i)$ and $bin(i+1)$ by using the generated prime number is performed independently. Therefore, any incorrect pair is accepted with probability at most $\epsilon$ due to Fact \[fact:Fre83\]. Since $P'$ can fail to generate a prime number, this probability is increased to at most $\epsilon + \epsilon_0 - \epsilon \cdot \epsilon_0$. If the input belongs to ${\mathtt{LOGLOG(I)}}$, $P'$ is guaranteed to not reject the input before reaching “4” on input tape. If at least one pair $bin(i)$ and $bin(i+1)$ is inacceptable, then $P'$ rejects input right after checking this pair with probability at least $1 - \epsilon - \epsilon_0 \cdot (1 - \epsilon)$. Therefore, the error remains bounded.
The other parts of the algorithm are executed with the same number of steps in every execution of $P'$.
Bounded–error realtime PTMs can recognize uncountably many languages in $O(\log \log n)$ space.
We can obtain a realtime algorithm from $P'$, say $ R_{c,I} $ or shortly $R$, by using aforementioned technique borrowed from [@YS13B]. Let ${\mathtt{LOGLOG(I)}}'$ be the language recognized by $ R $. Then, the language ${\mathtt{LOGLOG(I)}}'$ differs from the language ${\mathtt{LOGLOG(I)}}$ with the presence of symbols “3”: for each “wait” move on “0”, “1”, “2” or “4” by $P'$, $ R $ expects to read one symbol of “3”. If $ R $ fails to read a symbol of “3” when it is expected, the input is rejected.
PTM $P'$ recognizes ${\mathtt{LOGLOG(I)}}$ in $O(\log \log n)$ space, therefore, realtime machine $R$ recognizes ${\mathtt{LOGLOG(I)}}'$ in $O(\log \log n)$ space and there is a bijection between ${\mathtt{LOGLOG(I)}}$ and ${\mathtt{LOGLOG(I)}}'$.
In [@Fre83] Freivalds has proven that only regular languges can be recognized with one-way PTM in $o(\log \log n)$ space and with probability $p > \frac{1}{2}$. Therefore, the presented space bound is tight.
Acknowledgments {#acknowledgments .unnumbered}
===============
Dimitrijevs is partially supported by University of Latvia project AAP2016/B032 “Innovative information technologies”. Yakary[i]{}lmaz is partially supported by ERC Advanced Grant MQC. We thank to the reviewers for their helpful comments.
|
---
author:
- 'A. Coutens'
- 'M. V. Persson'
- 'J. K. Jørgensen'
- 'S. F. Wampfler'
- 'J. M. Lykke'
date: 'Received xxx; accepted xxx'
title: 'Detection of glycolaldehyde towards the solar-type protostar NGC 1333 IRAS2A[^1]'
---
Introduction
============
The inner regions of low-mass protostars are known to harbor a rich complex organic chemistry characterized by the presence of molecules such as methyl formate (CH$_3$OCHO), dimethyl ether (CH$_3$OCH$_3$), and ethyl cyanide (C$_2$H$_5$CN) [e.g., @Cazaux2003; @Bottinelli2004a; @Bisschop2008]. To differentiate them from the hot cores present in high-mass star-forming regions, they were called hot corinos [@Ceccarelli2004; @Bottinelli2004b]. These complex organic molecules are thought to be efficiently formed on grains and then released into the gas phase in the hot corino by thermal desorption [e.g., @Garrod2008; @Herbst2009]. Some of these complex organic molecules are particularly interesting because of their supposed role in the emergence of life. Indeed, the detection of so called prebiotic molecules in low-mass star-forming regions indicates that they can form early during the star formation process and thereby be available for possible later incorporation into solar system bodies, e.g., comets.
Glycolaldehyde (CH$_2$OHCHO) is one of these prebiotic molecules: it is a simple sugar-like molecule and under Earth-like conditions the first product in the formose reaction leading to the formation of ribose, an essential constituent of ribonucleic acid (RNA) [e.g., @Zubay2001; @Jalbout2007]. Glycolaldehyde was first detected towards the Galactic center (Sgr B2(N): @Hollis2000 [@Hollis2001; @Hollis2004; @Halfen2006; @Belloche2013]; molecular clouds: @Requena2008). Later it was shown to be present in the high mass star-forming region G31.41+0.31 [@Beltran2009], in the intermediate mass protostar NGC 7129 FIRS 2 [@Fuente2014], and even in the hot corinos of the Class 0 protostellar binary, IRAS 16293-2422 [hereafter IRAS16293, @Jorgensen2012]. This indicates that this molecule can be synthesized relatively early in the environments of solar-type protostars. Furthermore, glycolaldehyde can easily survive during impact delivery to planetary bodies, and impacts can even facilitate the formation of even more complex molecules [@McCaffrey2014].
Similarly to other complex organic molecules, the formation of glycolaldehyde is thought to occur on grains. In particular, a gas phase formation was excluded by @Woods2012 [@Woods2013], as the produced abundances are too low compared with the observations. Several grain surface formation pathways were proposed in the literature. @Woods2012 modeled their efficiency and showed that the formation by the reaction CH$_3$OH + HCO would be very efficient, but that, from chemical considerations, H$_3$CO + HCO could be more feasible. Another probably efficient way to form glycolaldehyde would be through HCO dimerization (HCO + HCO $\rightarrow$ HOCCOH) followed by two successive hydrogenations [@Woods2013]. A recent experimental study based on surface hydrogenations of CO seems to confirm this pathway [@Fedoseev2014].
A related species to this prebiotic molecule is ethylene glycol ((CH$_2$OH)$_2$). More commonly known as antifreeze, it is the reduced alcohol of glycolaldehyde. This molecule was tentatively detected towards IRAS16293 with one line of the gGg$'$ conformer [@Jorgensen2012]. Interestingly, the aGg$'$ conformer of ethylene glycol (the conformer of lowest energy) was detected in three comets, Hale-Bopp, Lemmon, and Lovejoy, while glycolaldehyde was not, leading to a lower limit of 3–6 for the (CH$_2$OH)$_2$/CH$_2$OHCHO abundance ratio [@Crovisier2004; @Biver2014]. Ethylene glycol was also detected in the Murchison and Murray carbonaceous meteorites, while the presence of aldehyde sugars have not been reported yet [@Cooper2001].
NGC 1333 IRAS2A (hereafter IRAS2A) is another of these famous hot corinos. In particular, methyl formate, the most abundant isomer of glycolaldehyde, was detected towards this source by @Jorgensen2005a and @Bottinelli2007. More recently, ethylene glycol was detected in the framework of the CALYPSO program carried out with the IRAM Plateau de Bure Interferometer (PdBI) by @Maury2014. We here report the detection of glycolaldehyde towards the same low-mass protostar, and present an analysis of the relative abundances of these three species.
Observations {#sect_obs}
============
This work is based on several separate programs carrying out observations of the solar-type protostar IRAS2A with the PdBI. Four spectral ranges (84.9$-$88.5, 223.5$-$227.1, 240.2$-$243.8, and 315.5$-$319.1 GHz) were covered with the WIDEX correlator at a spectral resolution of 1.95 MHz ($d\varv$ = 6.8 kms$^{-1}$ at 86 GHz, $d\varv$ = 2.6 kms$^{-1}$ at 225 GHz, $d\varv$ = 2.4 kms$^{-1}$ at 242 GHz, $d\varv$ = 1.8 kms$^{-1}$ at 317 GHz) and reduced with the GILDAS[^2] software. The synthetized beam sizes obtained with natural weighting are about 3.0$\arcsec$$\times$3.0$\arcsec$ at 86 GHz, 1.2$\arcsec$$\times$1.0$\arcsec$ at 225 GHz, 1.4$\arcsec$$\times$1.0$\arcsec$ at 242 GHz, and 0.9$\arcsec$$\times$0.8$\arcsec$ at 317 GHz. The dust continuum fluxes at 0.9 and 1.3 mm are consistent with previous measurements [e.g., @Jorgensen2007; @Persson2012]. The absolute calibration uncertainty for each dataset is about 20%. Additional information about the observations and their reduction can be found in @Coutens2014 and @Persson2014. The 3mm data are from Wampfler (priv. comm.).
Using the CASSIS[^3] software, we detected 8 lines of glycolaldehyde, 31 lines of the aGg$'$ conformer of ethylene glycol, and 26 lines of methyl formate (see Table \[table\_obs\]). The glycolaldehyde and methyl formate transitions are taken from the JPL spectroscopic database [@Pickett1998], while the ethylene glycol transitions are from the CDMS catalogue [@Muller2001; @Muller2005]. The predictions are based on experimental data from @Butler2001, @Widicus2005 and @Carroll2010 for glycolaldehyde, @Christen1995 and @Christen2003 for ethylene glycol, and @Ilyushin2009 for methyl formate. The frequencies of five of the detected glycolaldehyde lines were directly measured in the laboratory [@Butler2001]. Some of the lines result from a blending of several transitions of the same species. The lines that are strongly blended with other species are not listed in Table\[table\_obs\]. All three species are emitted very compactly at the position of the continuum peak ($\alpha_{2000}$=$03^{\rm h}28^{\rm m}55\fs57$, $\delta_{2000}$=$31\degr14\arcmin37\farcs1$). The angular sizes obtained with a circular Gaussian fit in the ($u$,$\varv$) plane vary from a point source to a maximum of 1$\arcsec$ depending on the transition. The line fluxes listed in Table \[table\_obs\] were measured at the continuum peak position with the CASSIS software using a Gaussian fitting method (Levenberg-Marquardt algorithm). The lines that are contaminated in the wings by other transitions are consequently fitted with a sum of Gaussians. We carefully checked that the derived full widths at half maximum ($FWHM$) are consistent with the other line measurements. The average $FWHM$ is about 4.5 kms$^{-1}$ at 317 GHz, and 5.0 kms$^{-1}$ at 225 and 242 GHz. The widths of the methyl formate lines at 87 GHz are quite broad ($\sim$12 kms$^{-1}$). It is consequently difficult to completely exclude an extra flux contribution from other species. The variation of FWHM with the frequency can be explained by the spectral resolution of the observations that decreases towards the lower frequencies.
----------------------- --------------------------------------------------- ----------------- -------------- ---------------------- -------------- ------------------ -------- -- --
Species Transition Frequency $E_{\rm up}$ $A_{\rm ij}$ $g_{\rm up}$ Flux RD$^a$
(MHz) (K) (s$^{-1}$) (Jy km s$^{-1}$)
CH$_2$OHCHO 13$_{10,3}-$13$_{9,4}$ ($\varv$=0) 240366.34$^*$ 111.3 1.2$\times$10$^{-4}$ 27 0.055 Y
13$_{10,4}-$13$_{9,5}$ ($\varv$=0) 240366.34$^*$ 111.3 1.2$\times$10$^{-4}$ 27
CH$_2$OHCHO 12$_{10,2}-$12$_{9,3}$ ($\varv$=0) 240482.78$^*$ 104.0 1.0$\times$10$^{-4}$ 25 0.038 Y
12$_{10,3}-$12$_{9,4}$ ($\varv$=0) 240482.78$^*$ 104.0 1.0$\times$10$^{-4}$ 25
CH$_2$OHCHO 11$_{5,6}-$10$_{4,7}$ ($\varv$=0) 240890.46$\,$ 51.9 1.8$\times$10$^{-4}$ 23 0.663 N$^b$
CH$_2$OHCHO 22$_{2,20}-$21$_{3,19}$ ($\varv$=0) 241131.84$\,$ 142.8 2.8$\times$10$^{-4}$ 45 0.061 Y
CH$_2$OHCHO 23$_{2,22}-$22$_{1,21}$ ($\varv$=0) 242239.09$\,$ 146.2 3.5$\times$10$^{-4}$ 47 0.128 Y
CH$_2$OHCHO 24$_{0,24}-$23$_{1,23}$ ($\varv$=0) 242957.72$^*$ 148.2 4.2$\times$10$^{-4}$ 49 0.258 Y
24$_{1,24}-$23$_{0,23}$ ($\varv$=0) 242957.98$^*$ 148.2 4.2$\times$10$^{-4}$ 49
CH$_2$OHCHO 19$_{13,7}-$19$_{12,8}$ ($\varv$=0) 315941.48$^*$ 208.1 3.2$\times$10$^{-4}$ 39 0.074 Y
19$_{13,6}-$19$_{12,7}$ ($\varv$=0) 315941.48$^*$ 208.1 3.2$\times$10$^{-4}$ 39
CH$_2$OHCHO 11$_{8,4}-$10$_{7,3}$ ($\varv$=0) 317013.88$^*$ 75.5 6.6$\times$10$^{-4}$ 23 0.363 Y$^c$
11$_{8,3}-$10$_{7,4}$ ($\varv$=0) 317013.90$^*$ 75.5 6.6$\times$10$^{-4}$ 23
CH$_2$OHCHO 27$_{5,23}-$26$_{4,22}$ ($\varv$=0) 317850.44$\,$ 226.2 4.5$\times$10$^{-4}$ 55 0.117 Y$^c$
aGg$'$-(CH$_2$OH)$_2$ 21$_{6,16}$ ($\varv$=1)–20$_{6,15}$ ($\varv$=0) 223741.66$\,$ 132.0 2.5$\times$10$^{-4}$ 387 0.212 Y
aGg$'$-(CH$_2$OH)$_2$ 21$_{6,15}$ ($\varv$=1)–20$_{6,14}$ ($\varv$=0) 224405.85$\,$ 132.1 2.5$\times$10$^{-4}$ 301 0.197 Y
aGg$'$-(CH$_2$OH)$_2$ 24$_{0,24}$ ($\varv$=1)–23$_{1,23}$ ($\varv$=0) 224511.70$^*$ 136.8 5.4$\times$10$^{-5}$ 441 0.145 N
24$_{1,24}$ ($\varv$=1)–23$_{0,23}$ ($\varv$=0) 224512.74$^*$ 136.8 5.4$\times$10$^{-5}$ 343
aGg$'$-(CH$_2$OH)$_2$ 21$_{3,18}$ ($\varv$=1)–20$_{3,17}$ ($\varv$=0) 225688.94$\,$ 121.3 2.4$\times$10$^{-4}$ 301 0.195 Y
aGg$'$-(CH$_2$OH)$_2$ 22$_{3,20}$ ($\varv$=1)–21$_{3,19}$ ($\varv$=0) 225929.69$\,$ 127.8 2.5$\times$10$^{-4}$ 315 0.349 N$^c$
aGg$'$-(CH$_2$OH)$_2$ 22$_{5,17}$ ($\varv$=0)–21$_{5,16}$ ($\varv$=1) 226095.96$\,$ 138.2 2.6$\times$10$^{-4}$ 315 0.303 N
aGg$'$-(CH$_2$OH)$_2$ 22$_{2,20}$ ($\varv$=1)–21$_{2,19}$ ($\varv$=0) 226561.99$\,$ 127.7 3.0$\times$10$^{-4}$ 405 0.329 Y
aGg$'$-(CH$_2$OH)$_2$ 25$_{1,25}$ ($\varv$=0)–24$_{1,24}$ ($\varv$=1) 226643.30$^*$ 147.7 2.9$\times$10$^{-4}$ 357 0.470 Y
25$_{0,25}$ ($\varv$=0)–24$_{0,24}$ ($\varv$=1) 226643.46$^*$ 147.7 2.9$\times$10$^{-4}$ 459
aGg$'$-(CH$_2$OH)$_2$ 25$_{1,25}$ ($\varv$=1)–24$_{1,24}$ ($\varv$=0) 240778.12$^*$ 148.0 3.4$\times$10$^{-4}$ 459 0.360 Y
25$_{0,25}$ ($\varv$=1)–24$_{0,24}$ ($\varv$=0) 240778.30$^*$ 148.0 3.4$\times$10$^{-4}$ 357
aGg$'$-(CH$_2$OH)$_2$ 24$_{8,17}$ ($\varv$=0)–23$_{8,16}$ ($\varv$=1) 240807.88$\,$ 179.2 3.0$\times$10$^{-4}$ 441 0.340 N$^e$
aGg$'$-(CH$_2$OH)$_2$ 24$_{8,16}$ ($\varv$=0)–23$_{8,15}$ ($\varv$=1) 240828.89$\,$ 179.2 3.0$\times$10$^{-4}$ 343 0.149 Y
aGg$'$-(CH$_2$OH)$_2$ 24$_{5,20}$ ($\varv$=0)–23$_{5,19}$ ($\varv$=1) 241291.27$\,$ 160.7 3.1$\times$10$^{-4}$ 441 0.196 Y
aGg$'$-(CH$_2$OH)$_2$ 24$_{7,18}$ ($\varv$=0)–23$_{7,17}$ ($\varv$=1) 241545.26$\,$ 172.1 3.1$\times$10$^{-4}$ 441 0.151 Y
aGg$'$-(CH$_2$OH)$_2$ 24$_{6,19}$ ($\varv$=0)–23$_{6,18}$ ($\varv$=1) 241860.73$\,$ 166.0 2.8$\times$10$^{-4}$ 441 0.137 Y
aGg$'$-(CH$_2$OH)$_2$ 23$_{15,8}$ ($\varv$=1)–22$_{15,7}$ ($\varv$=0) 242244.69$^*$ 246.4 2.0$\times$10$^{-4}$ 329 0.474 N
23$_{15,9}$ ($\varv$=1)–22$_{15,8}$ ($\varv$=0) 242244.69$^*$ 246.4 2.0$\times$10$^{-4}$ 423
23$_{6,17}$ ($\varv$=1)–22$_{6,17}$ ($\varv$=1) 242245.62$^*$ 154.6 1.1$\times$10$^{-5}$ 329
23$_{14,9}$ ($\varv$=1)–22$_{14,8}$ ($\varv$=0) 242246.34$^*$ 232.2 2.2$\times$10$^{-4}$ 329
23$_{14,10}$ ($\varv$=1)–22$_{14,9}$ ($\varv$=0) 242246.34$^*$ 232.2 2.2$\times$10$^{-4}$ 423
aGg$'$-(CH$_2$OH)$_2$ 23$_{13,10}$ ($\varv$=1)–22$_{13,9}$ ($\varv$=0) 242277.72$^*$ 218.9 2.4$\times$10$^{-4}$ 329 0.275 N
23$_{13,11}$ ($\varv$=1)–22$_{13,10}$ ($\varv$=0) 242277.72$^*$ 218.9 2.4$\times$10$^{-4}$ 423
aGg$'$-(CH$_2$OH)$_2$ 23$_{10,14}$ ($\varv$=1)–22$_{10,13}$ ($\varv$=0) 242656.22$^*$ 185.2 2.8$\times$10$^{-4}$ 423 0.305 Y
23$_{10,13}$ ($\varv$=1)–22$_{10,12}$ ($\varv$=0) 242656.24$^*$ 185.2 2.8$\times$10$^{-4}$ 329
aGg$'$-(CH$_2$OH)$_2$ 23$_{9,15}$ ($\varv$=1)–22$_{9,14}$ ($\varv$=0) 242947.99$^*$ 175.9 3.0$\times$10$^{-4}$ 423 0.291 Y
23$_{9,14}$ ($\varv$=1)–22$_{9,13}$ ($\varv$=0) 242948.59$^*$ 175.9 3.0$\times$10$^{-4}$ 329
aGg$'$-(CH$_2$OH)$_2$ 23$_{5,19}$ ($\varv$=1)–22$_{5,18}$ ($\varv$=0) 243636.57$^*$ 149.1 3.4$\times$10$^{-4}$ 423 0.287 Y
aGg$'$-(CH$_2$OH)$_2$ 31$_{8,23}$ ($\varv$=0)–30$_{8,22}$ ($\varv$=1) 315671.33$\,$ 276.6 7.0$\times$10$^{-4}$ 567 0.314 N$^c$
aGg$'$-(CH$_2$OH)$_2$ 31$_{7,25}$ ($\varv$=0)–30$_{7,24}$ ($\varv$=1) 315892.11$\,$ 269.6 6.8$\times$10$^{-4}$ 441 0.144 N$^d$
aGg$'$-(CH$_2$OH)$_2$ 30$_{9,21}$ ($\varv$=1)–29$_{9,20}$ ($\varv$=0) 315961.89$\,$ 269.4 7.0$\times$10$^{-4}$ 549 0.259 Y
aGg$'$-(CH$_2$OH)$_2$ 34$_{1,34}$ ($\varv$=0)–33$_{0,33}$ ($\varv$=0) 316444.07$^*$ 268.5 1.6$\times$10$^{-4}$ 621 0.078 Y
34$_{0,34}$ ($\varv$=0)–33$_{1,33}$ ($\varv$=0) 316444.07$^*$ 268.5 1.6$\times$10$^{-4}$ 483
aGg$'$-(CH$_2$OH)$_2$ 20$_{6,14}$ ($\varv$=0)–19$_{5,15}$ ($\varv$=0) 316698.08$\,$ 121.3 5.1$\times$10$^{-5}$ 287 0.029 Y
aGg$'$-(CH$_2$OH)$_2$ 30$_{7,24}$ ($\varv$=1)–29$_{7,23}$ ($\varv$=0) 316868.23$^*$ 254.4 6.9$\times$10$^{-4}$ 427 0.445 N$^c$
20$_{16,4}$ ($\varv$=0)–20$_{15,5}$ ($\varv$=0) 316870.92$^*$ 228.9 3.8$\times$10$^{-5}$ 287
20$_{16,5}$ ($\varv$=0)–20$_{15,6}$ ($\varv$=0) 316870.92$^*$ 228.9 3.8$\times$10$^{-5}$ 369
aGg$'$-(CH$_2$OH)$_2$ 30$_{8,23}$ ($\varv$=1)–29$_{8,22}$ ($\varv$=0) 316917.19$\,$ 261.4 7.2$\times$10$^{-4}$ 427 0.299 Y
aGg$'$-(CH$_2$OH)$_2$ 16$_{8,9}$ ($\varv$=1)–15$_{7,8}$ ($\varv$=1) 317054.30$^*$ 98.6 8.0$\times$10$^{-5}$ 231 0.116 N
16$_{8,8}$ ($\varv$=1)–15$_{7,9}$ ($\varv$=1) 317055.36$^*$ 98.6 8.0$\times$10$^{-5}$ 297
aGg$'$-(CH$_2$OH)$_2$ 21$_{4,18}$ ($\varv$=1)–20$_{4,18}$($\varv$=0) 317267.77$^*$ 122.1 1.1$\times$10$^{-5}$ 387 0.312 N
14$_{9,5}$ ($\varv$=1)–13$_{8,6}$ ($\varv$=1) 317267.91$^*$ 91.4 1.1$\times$10$^{-4}$ 261
14$_{9,6}$ ($\varv$=1)–13$_{8,5}$ ($\varv$=1) 317267.91$^*$ 91.4 1.1$\times$10$^{-4}$ 203
14$_{9,5}$ ($\varv$=0)–13$_{8,6}$ ($\varv$=0) 317268.88$^*$ 91.4 1.1$\times$10$^{-4}$ 203
14$_{9,6}$ ($\varv$=0)–13$_{8,5}$ ($\varv$=0) 317268.88$^*$ 91.4 1.1$\times$10$^{-4}$ 261
aGg$'$-(CH$_2$OH)$_2$ 32$_{3,30}$ ($\varv$=1)–31$_{3,29}$ ($\varv$=0) 317962.58$\,$ 257.2 7.8$\times$10$^{-4}$ 455 0.310 Y
aGg$'$-(CH$_2$OH)$_2$ 32$_{4,28}$ ($\varv$=0)–31$_{4,27}$ ($\varv$=1) 317982.56$\,$ 272.2 4.8$\times$10$^{-4}$ 455 0.160 Y
aGg$'$-(CH$_2$OH)$_2$ 35$_{0,35}$ ($\varv$=0)–34$_{0,34}$ ($\varv$=1) 318433.40$^*$ 284.1 8.0$\times$10$^{-4}$ 639 0.639 N
35$_{1,35}$ ($\varv$=0)–34$_{1,34}$ ($\varv$=1) 318433.40$^*$ 284.1 8.0$\times$10$^{-4}$ 497
CH$_3$OCHO 7$_{3,4}$–6$_{3,3}$ E ($\varv_T$=0) 87143.28$^*$ 22.6 7.7$\times$10$^{-6}$ 30 0.049 N
21$_{5,16}$–21$_{4,17}$ E ($\varv_T$=1) 87143.65$^*$ 342.0 1.4$\times$10$^{-6}$ 86
CH$_3$OCHO 8$_{0,8}$–7$_{1,7}$ E ($\varv_T$=1) 87160.84$^*$ 207.0 1.3$\times$10$^{-6}$ 34 0.038 N
7$_{3,4}$–6$_{3,3}$ A ($\varv_T$=0) 87161.28$^*$ 22.6 7.8$\times$10$^{-6}$ 30
CH$_3$OCHO 18$_{6,13}$–17$_{6,12}$ E ($\varv_T$=0) 224021.87$^*$ 125.3 1.5$\times$10$^{-4}$ 74 0.752 Y
18$_{6,13}$–17$_{6,12}$ A ($\varv_T$=0) 224024.10$^*$ 125.3 1.5$\times$10$^{-4}$ 74
CH$_3$OCHO 18$_{5,14}$–17$_{5,13}$ E ($\varv_T$=0) 224313.15$\,$ 118.3 1.6$\times$10$^{-4}$ 74 0.351 Y
CH$_3$OCHO 19$_{3,17}$–18$_{3,16}$ E ($\varv_T$=1) 224491.31$\,$ 303.2 1.7$\times$10$^{-4}$ 78 0.212 Y
CH$_3$OCHO 18$_{6,12}$–18$_{6,11}$ E ($\varv_T$=0) 224582.35$\,$ 125.4 1.5$\times$10$^{-4}$ 74 0.463 Y
CH$_3$OCHO 18$_{6,12}$–17$_{6,11}$ A ($\varv_T$=0) 224609.38$\,$ 125.4 1.5$\times$10$^{-4}$ 74 0.466 Y
CH$_3$OCHO 20$_{2,19}$–19$_{2,18}$ A ($\varv_T$=1) 225372.22$\,$ 307.3 1.7$\times$10$^{-4}$ 82 0.186 Y
CH$_3$OCHO 19$_{3,17}$–18$_{3,16}$ E ($\varv_T$=0) 225608.82$\,$ 116.7 1.7$\times$10$^{-4}$ 78 0.426 Y
CH$_3$OCHO 18$_{5,13}$–17$_{5,12}$ A ($\varv_T$=1) 225648.42$^*$ 305.6 1.6$\times$10$^{-4}$ 74 0.163 N
26$_{9,18}$–26$_{8,19}$ A ($\varv_T$=0) 225648.42$^*$ 261.7 1.6$\times$10$^{-5}$ 106
CH$_3$OCHO 21$_{0,21}$–20$_{1,20}$ A ($\varv_T$=1) 226381.36$^*$ 309.6 2.8$\times$10$^{-5}$ 86 0.277 Y
21$_{1,21}$–20$_{1,20}$ A ($\varv_T$=1) 226382.72$^*$ 309.6 1.7$\times$10$^{-4}$ 86
21$_{0,21}$–20$_{0,20}$ A ($\varv_T$=1) 226383.86$^*$ 309.6 1.7$\times$10$^{-4}$ 86
21$_{1,21}$–20$_{0,20}$ A ($\varv_T$=1) 226385.15$^*$ 309.6 2.8$\times$10$^{-5}$ 86
CH$_3$OCHO 21$_{0,21}$–20$_{1,20}$ E ($\varv_T$=1) 226433.26$^*$ 308.9 2.7$\times$10$^{-5}$ 86 0.296 N
21$_{1,21}$–20$_{1,20}$ E ($\varv_T$=1) 226434.47$^*$ 308.9 1.7$\times$10$^{-4}$ 86
21$_{0,21}$–20$_{0,20}$ E ($\varv_T$=1) 226435.52$^*$ 308.9 1.7$\times$10$^{-4}$ 86
25$_{9,16}$–25$_{8,17}$ A ($\varv_T$=0) 226435.52$^*$ 246.2 1.6$\times$10$^{-5}$ 102
21$_{1,21}$–20$_{0,20}$ E ($\varv_T$=1) 226436.66$^*$ 308.9 2.7$\times$10$^{-5}$ 86
CH$_3$OCHO 20$_{2,19}$–19$_{2,18}$ A ($\varv_T$=0) 226718.69$\,$ 120.2 1.7$\times$10$^{-4}$ 82 0.505 Y
CH$_3$OCHO 19$_{2,17}$–18$_{2,16}$ E ($\varv_T$=0) 227019.55$^*$ 116.6 1.7$\times$10$^{-4}$ 78 0.561 N
25$_{9,17}$–25$_{8,18}$ A ($\varv_T$=0) 227021.13$^*$ 246.2 1.6$\times$10$^{-5}$ 102
CH$_3$OCHO 19$_{2,17}$–18$_{2,16}$ A ($\varv_T$=0) 227028.12$\,$ 116.6 1.7$\times$10$^{-4}$ 78 0.436 Y
CH$_3$OCHO 20$_{4,17}$–19$_{4,16}$ A ($\varv_T$=1) 242610.07$\,$ 321.7 2.1$\times$10$^{-4}$ 82 0.092 Y
CH$_3$OCHO 37$_{7,31}$–37$_{6,32}$ A ($\varv_T$=0) 242870.39$^*$ 452.0 1.8$\times$10$^{-5}$ 150 0.714 N
19$_{5,14}$–18$_{5,13}$ E ($\varv_T$=0) 242871.57$^*$ 130.5 2.0$\times$10$^{-4}$ 78
CH$_3$OCHO 19$_{5,14}$–18$_{5,13}$ A ($\varv_T$=0) 242896.02$\,$ 130.4 2.0$\times$10$^{-4}$ 78 0.666 Y
CH$_3$OCHO 21$_{12,9}$–21$_{11,10}$ E ($\varv_T$=0) 316742.00$^*$ 231.8 3.3$\times$10$^{-5}$ 86 0.072 Y
21$_{12,10}$–21$_{11,11}$ E ($\varv_T$=0) 316742.71$^*$ 231.8 3.3$\times$10$^{-5}$ 86
CH$_3$OCHO 21$_{12,9}$–21$_{11,10}$ A ($\varv_T$=0) 316776.74$^*$ 231.8 3.3$\times$10$^{-5}$ 86 0.069 Y
21$_{12,10}$–21$_{11,11}$ A ($\varv_T$=0) 316776.74$^*$ 231.8 3.3$\times$10$^{-5}$ 86
CH$_3$OCHO 26$_{13,13}$–25$_{13,12}$ E ($\varv_T$=1) 317177.16$\,$ 506.5 3.6$\times$10$^{-4}$ 106 0.043 Y
CH$_3$OCHO 9$_{8,2}$–8$_{7,1}$ A ($\varv_T$=1) 318009.06$^{*}$ 256.8 6.8$\times$10$^{-5}$ 38 0.200 N
9$_{8,1}$–8$_{7,2}$ A ($\varv_T$=1) 318009.06$^{*}$ 256.8 6.9$\times$10$^{-5}$ 38
13$_{12,1}$–13$_{11,2}$ E ($\varv_T$=0) 318009.55$^{*}$ 149.2 1.3$\times$10$^{-5}$ 54
26$_{13,13}$–25$_{13,12}$ A ($\varv_T$=1) 318012.17$^{*}$ 506.2 3.7$\times$10$^{-4}$ 106
26$_{13,14}$–25$_{13,13}$ A ($\varv_T$=1) 318012.17$^{*}$ 506.2 3.7$\times$10$^{-4}$ 106
CH$_3$OCHO 13$_{12,2}$–13$_{11,3}$ E ($\varv_T$=0) 318016.90$^{*}$ 149.2 1.3$\times$10$^{-5}$ 54 0.155 N
9$_{8,1}$–8$_{7,1}$ E ($\varv_T$=0) 318017.37$^{*}$ 69.0 6.7$\times$10$^{-5}$ 38
CH$_3$OCHO 12$_{12,0}$–12$_{11,1}$ E ($\varv_T$=0) 318064.54$^{*}$ 141.6 7.0$\times$10$^{-6}$ 50 0.204 N
9$_{8,2}$–8$_{7,1}$ A ($\varv_T$=0) 318065.26$^{*}$ 69.0 6.8$\times$10$^{-5}$ 38
9$_{8,1}$–8$_{7,2}$ A ($\varv_T$=0) 318065.26$^{*}$ 69.0 6.8$\times$10$^{-5}$ 38
CH$_3$OCHO 27$_{4,23}$–26$_{5,22}$ E ($\varv_T$=1) 318139.11$^{*}$ 426.4 3.4$\times$10$^{-5}$ 110 0.192 N
25$_{6,19}$–24$_{6,18}$ A ($\varv_T$=1) 318140.72$^{*}$ 405.1 4.6$\times$10$^{-4}$ 102
26$_{11,15}$–25$_{11,14}$ E ($\varv_T$=1) 318145.25$^{*}$ 474.5 4.0$\times$10$^{-4}$ 106
CH$_3$OCHO 27$_{4,24}$–26$_{4,23}$ E ($\varv_T$=1) 318979.14$\,$ 417.8 4.8$\times$10$^{-4}$ 110 0.208 Y
----------------------- --------------------------------------------------- ----------------- -------------- ---------------------- -------------- ------------------ -------- -- --
: CH$_2$OHCHO, aGg$'$-(CH$_2$OH)$_2$ and CH$_3$OCHO transitions observed towards NGC 1333 IRAS2A. []{data-label="table_obs"}
[ Notes: The symbol $^*$ present after some frequency values indicates that the associated transition is blended with one or more transitions from the same species. $^a$Y indicates that the line was considered in the rotational diagram analysis, while N indicates that it could not be used (for blending reasons). $^b$Blended with an unidentified species. $^c$Blended with CH$_3$OCHO. $^d$Blended with CH$_2$OHCHO. $^e$Potentially blended with the gGg$'$ conformer of ethylene glycol.]{}
Results {#sect_analysis}
=======
![Rotational diagrams for glycolaldehyde, ethylene glycol, and methyl formate.[]{data-label="Fig_RD"}](RDs_glyco_Monte_Carlo.pdf)
We carried out a local thermodynamic equilibrium analysis of the three species through the rotational diagram method [@Goldsmith1999]. We consider that the lines are emitted in a region of 0.5$\arcsec$ size, which is the average size derived for the methyl formate lines when fitting circular gaussians in the ($u$,$\varv$) plane (see also @Maury2014). It is also similar to what we found for deuterated water [@Coutens2014]. It also corresponds to the expected size of the region where the temperature increases above $\sim$100K according to dust radiation transfer models of the envelope [@Jorgensen2002] and where the complex molecules and deuterated water should sublimate from the grains. The line fluxes that result from a combination of several transitions of the same species are used in the rotational diagrams, unless the transitions have different $E_{\rm up}$ values. For glycolaldehyde, we include two lines slightly blended with some methyl formate transitions after subtraction of the predicted flux contribution from methyl formate. As the best-fit model for methyl formate reproduces extremely well the observations, the final fluxes of the glycolaldehyde lines can be trusted, which is also confirmed by their alignment with the other points in the rotational diagram of glycolaldehyde (see upper panel in Figure \[Fig\_RD\]). Assuming a source size of 0.5$\arcsec$, we derive column densities (with 1$\sigma$ uncertainties) of 2.4$^{+0.6}_{-0.3}$$\times$10$^{15}$cm$^{-2}$, 1.3$^{+0.1}_{-0.1}$$\times$10$^{16}$cm$^{-2}$, and 4.8$^{+0.3}_{-0.3}$$\times$10$^{16}$cm$^{-2}$, and excitation temperatures of 103$^{+60}_{-20}$K, 133$^{+23}_{-14}$K, and 126$^{+9}_{-7}$K for glycolaldehyde, ethylene glycol, and methyl formate, respectively. Within the uncertainty range, the excitation temperature seems to be similar between the three species ($\sim$130K), which is consistent if the three species arise from a same region. We checked, for each species, that there is no line flux overpredicted by the model anywhere in the four datasets. In the case of ethylene glycol, the model shows an overproduced flux for some transitions, especially the lines (240.778, 241.545, 241.860, and 316.444 GHz) that correspond to the four lower points in the rotational diagram (see middle panel in Figure \[Fig\_RD\]). A model with a column density of 1.1$\times$10$^{16}$cm$^{-2}$ would be sufficient to produce line fluxes consistent with these observations. Table \[table\_model\] summarizes the parameters used for the line modeling of the three species that can be seen in Figures \[Model\_GA\], \[Model\_aGgglycol\], and \[Model\_CH3OCHO\]. According to these models, all lines are optically thin ($\tau$ $\leq$ 0.1).
\[table\_model\]
----------------------- ------------------ -------------- ---------------------- -------------------
Molecule Source $T_{\rm ex}$ $N$ $\varv_{\rm LSR}$
size ($\arcsec$) (K) (cm$^{-2}$) (kms$^{-1}$)
CH$_2$OHCHO 0.5 130 2.4$\times$10$^{15}$ 7.0
aGg$'$-(CH$_2$OH)$_2$ 0.5 130 1.1$\times$10$^{16}$ 7.0
CH$_3$OCHO 0.5 130 4.8$\times$10$^{16}$ 7.0
----------------------- ------------------ -------------- ---------------------- -------------------
: Parameters used to compute the synthetic spectra of glycolaldehyde, ethylene glycol, and methyl formate.
Although no other species than glycolaldehyde is found at a frequency of 240890.5 MHz, the line ($E_{\rm up}$ = 52K) is probably blended with an unidentified species: the predicted flux is completely underproduced with respect to the observations, and it cannot be due to a different excitation in the cold gas, as a line of glycolaldehyde at 243232.21 MHz ($E_{\rm up}$ = 47K) – blended with a bright CH$_2$DOH line in the red-shifted part of the spectrum and also potentially blended with a DCOOH line ($E_{\rm up}$=106K, $A_{\rm ij}$=1.35$\times$10$^{-4}$s$^{-1}$) – would have a higher flux inconsistent with the observed one.
\[table\_comp\]
Source (CH$_2$OH)$_2$/CH$_2$OHCHO CH$_3$OCHO/CH$_2$OHCHO CH$_3$OCHO/(CH$_2$OH)$_2$ References
----------------------------- ---------------------------- ------------------------ --------------------------- ------------
NGC 1333 IRAS2A $\sim$5 $\sim$20 $\sim$4 1
IRAS 16293-2422 $\sim$1 $\sim$13 $\sim$13 2
C/1995 O1 (Hale-Bopp) $\ge$ 6 $\ge$ 2 $\sim$ 0.3 3
C/2012 F6 (Lemmon) $\ge$ 3 ... $\le$ 0.7 4
C/2013 R1 (Lovejoy) $\ge$ 5 ... $\le$ 0.6 4
Sgr B2(N) 0.7–2.2 $\sim$52 $\sim$30 5, 6, 7
G34.41+0.31 ... $\le$ 34 ... 8
NGC7129 FIRS2 $\sim$2 $\sim$40 $\sim$20 9
G–0.02, G–0.11, and G+0.693 $\sim$1.2–1.6 $\sim$3.3–5.2 $\sim$2.5–4.3 10
is shown in red lines. \[Model\_CH3OCHO\]
Discussion {#sect_discussion}
==========
The relative abundances of the three species are derived from the column densities in Table \[table\_model\] and compared with other star-forming regions and comets in Table \[table\_comp\]. The (CH$_2$OH)$_2$/CH$_2$OHCHO abundance ratio of $\sim$0.3–0.5 previously derived in IRAS16293 by @Jorgensen2012 was revised. Indeed, the assignment in @Jorgensen2012 was based on only one line of the gGg$'$ conformer of ethylene glycol about 200 cm$^{-1}$ ($\sim$290 K, @Muller2004) above the lowest-energy aGg$'$ conformer – and thus tentative. An analysis from observations of 6 transitions of the lower energy conformer from ALMA Cycle 1 observations at 3 mm (4 spectral windows at 89.48–89.73, 92.77–93.03, 102.48–102.73 and 103.18–103.42 GHz; Jørgensen et al. in prep.) results in a higher ethylene glycol-to-glycolaldehyde abundance ratio of 1.0$\pm$0.3. This new estimate is consistent with the ratio expected between the aGg$'$ and gGg$'$ conformers under thermal equilibrium conditions at 300K, the excitation temperature of glycolaldehyde derived in IRAS16293 [@Jorgensen2012]. The (CH$_2$OH)$_2$/CH$_2$OHCHO abundance ratio in IRAS2A is estimated at 5.5$\pm$1.0 if we consider the column densities derived from the rotational diagrams. It is however slightly lower (4.6) if we use the column density of ethylene glycol of 1.1$\times$10$^{16}$ cm$^{-2}$ that does not overproduce the peak intensities of a few lines (see Fig. \[Model\_aGgglycol\]). The (CH$_2$OH)$_2$/CH$_2$OHCHO abundance ratio is consequently a factor $\sim$5 higher than in the Class 0 protostar IRAS16293. It is also higher than in the other star-forming regions (see Table \[table\_comp\]), but comparable to the lower limits derived in comets ($\gtrsim$3–6). This indicates that the glycolaldehyde chemistry may vary among hot corinos in general. It is possible that, like IRAS2A, other very young low-mass protostars show high (CH$_2$OH)$_2$/CH$_2$OHCHO abundance ratios, in agreement with the cometary values. The CH$_3$OCHO/CH$_2$OHCHO column density ratio found in IRAS2A ($\sim$20) ranges between the values derived in the molecular clouds from the Galactic Center ($\sim$3.3–5.2) and the high-mass star-forming regions ($\sim$40–52). A lower limit of 2 was derived for the Hale-Bopp comet.
In contrast to IRAS16293, the (CH$_2$OH)$_2$/CH$_2$OHCHO abundance ratio in IRAS2A is comparable to the lower limits in comets. To explain these different abundance ratios in IRAS2A and IRAS16293, two scenarios are possible: either the (CH$_2$OH)$_2$/CH$_2$OHCHO ratio is similar in the grain mantles of low-mass protostars and it evolves in the gas phase after the sublimation of the molecules in the hot corinos, or this ratio was already different in the grain mantles of the two protostars.
In the first scenario, if we assume that the (CH$_2$OH)$_2$/CH$_2$OHCHO increases until it reaches the cometary value, it would mean that glycolaldehyde can easily be destroyed in the gas phase of the warm inner regions. Another possibility would be that ethylene glycol can form efficiently in the gas phase, but complex organic molecules are generally difficult to form with high abundances in the gas phase. If the evaporation temperature of ethylene glycol is higher than glycolaldehyde, as assumed in the chemical model of @Garrod2013, ethylene glycol would desorb later than glycolaldehyde and the (CH$_2$OH)$_2$/CH$_2$OHCHO abundance ratio would consequently increase with time (until the two molecules have completely desorbed). This chemical model however predicts an abundance of glycolaldehyde significantly higher than those of ethylene glycol and methyl formate, which is inconsistent with the ratios derived in IRAS2A. More theoretical and experimental work would be needed to make the case that these hypotheses are plausible.
In contrast, experimental studies based on irradiation of ices show that the second scenario *is* likely. Such studies show that glycolaldehyde, ethylene glycol, and methyl formate can be synthesized by irradiation of pure or mixed methanol (CH$_3$OH) ices [@Hudson2000; @Oberg2009]. Interestingly, the (CH$_2$OH)$_2$/CH$_2$OHCHO abundance ratio is found to be dependent on the initial ice composition as well as the ice temperature during the UV irradiation. The CH$_3$OH:CO ratio in the ices is a key parameter: for irradiated 20 K ices a composition of pure CH$_3$OH leads to a (CH$_2$OH)$_2$/CH$_2$OHCHO ratio higher than 10, while a CH$_3$OH:CO 1:10 ice mixture produces a (CH$_2$OH)$_2$/CH$_2$OHCHO ratio lower than 0.25 [@Oberg2009]. The difference found between IRAS16293 and IRAS2A could then be related to a different grain mantle composition in the two sources. If the CH$_3$OH:CO ratio in the grain mantles of IRAS 2A was higher than in IRAS 16293, a higher (CH$_2$OH)$_2$/CH$_2$OHCHO abundance ratio would be expected according to the laboratory results. In fact, the CH$_3$OH gas-phase abundance in the inner envelope is found to be higher in IRAS2A ($\sim$4$\times$10$^{-7}$, @Jorgensen2005b) than in IRAS16293 ($\sim$1$\times$10$^{-7}$, @Schoier2002), while the CO abundance is relatively similar ($\sim$(2–3)$\times$10$^{-5}$, @Jorgensen2002 [@Schoier2002]). This could consequently be the result of the desorption of ices with a higher CH$_3$OH:CO ratio in IRAS2A than IRAS16293. The question then arises: how can CH$_3$OH be more efficiently produced on grains in IRAS2A than in IRAS16293? Several scenarios are possible: *i)* The initial conditions may play an important role in the CH$_3$OH:CO ratio. In particular, experiments and simulations show that the efficiency of CH$_3$OH formation through CO hydrogenation on the grains is dependent on temperature, ice composition (CO:H$_2$O), and time [@Watanabe2004; @Fuchs2009]. *ii)* The collapse timescale was longer in IRAS2A than in IRAS16293, enabling to form more CH$_3$OH. *iii)* The H$_2$ density in the prestellar envelope of IRAS2A was lower than that of IRAS16293. Indeed a less dense environment would lead to a higher atomic H density and consequently to a higher efficiency of CO hydrogenation. This was proposed by @Maret2004 and @Bottinelli2007 to explain an anti-correlation found between the inner abundances of H$_2$CO and CH$_3$OH and the submillimeter luminosity to bolometric luminosity ($L_{\rm smm}$/$L_{\rm bol}$) ratios of different low-mass protostars. The $L_{\rm smm}$/$L_{\rm bol}$ parameter is interpreted as an indication of different initial conditions, rather than an evolutionary parameter in this context [@Maret2004]. The $L_{\rm smm}$/$L_{\rm bol}$ ratios of IRAS2A ($\sim$0.005, @Karska2013) and IRAS16293 ($\sim$0.019, @Froebrich2005) are consistent with this hypothesis. The current H$_2$ density profiles of these two sources are also in agreement with this scenario, if they keep the memory of the prestellar conditions. The density derived in the outer envelope of IRAS2A with a power-law model [@Jorgensen2002] is lower than the density derived in IRAS16293 by @Crimier2010 whether it be for a Shu-like model or a power-law model, while the temperature profiles are relatively similar (see Figure \[Fig\_comp\_profile\]). Along the same lines, @Hudson2005 showed with proton irradiation experiments that glycolaldehyde is more sensitive to radiation damage than ethylene glycol. Irradiation would be more important in less dense envelopes, which would also be consistent with a less dense prestellar envelope in IRAS2A. A recent experiment by Fedoseev et al. (submitted) shows that these two species can also be synthesized by surface hydrogenations of CO molecules in dense molecular cloud conditions. They do not directly form from CH$_3$OH, but the results of this experiment show that similarly to CH$_3$OH that results from successive hydrogenations of CO, ethylene glycol forms by two successive hydrogenations of glycolaldehyde. This is consequently in agreement with the proposed scenario.
![Black: H$_2$ density (solid line) and temperature (dotted line) profiles of the protostar IRAS2A from @Jorgensen2002. Red: H$_2$ density (solid line: power-law model, dashed line: Shu-like model) and temperature (dotted line) profiles of the protostar IRAS16293 from @Crimier2010.[]{data-label="Fig_comp_profile"}](comparison_profile_density.pdf)
In conclusion, the (CH$_2$OH)$_2$/CH$_2$OHCHO abundance ratio measured in low-mass protostars can be different from one source to another, and possibly consistent with cometary values. In some cases, the (CH$_2$OH)$_2$/CH$_2$OHCHO ratios determined in comets could consequently be inherited from early stages of star formation. Such a difference between low-mass protostars could be related to a different CH$_3$OH:CO ratio in the grain mantles. A more efficient hydrogenation (due for example to a lower density) on the grains would lead to higher abundances of CH$_3$OH and (CH$_2$OH)$_2$. A determination of (CH$_2$OH)$_2$/CH$_2$OHCHO ratios in larger samples of star-forming regions could help understand how the initial conditions (density, molecular cloud, ...) affect their relative abundances.
The authors are grateful to the IRAM staff, especially Tessel van der Laan, Arancha Castro-Carrizo, Chin-Shin Chang, and Sabine K[ö]{}nig, for their help with the calibration of the data. This research was supported by a Junior Group Leader Fellowship from the Lundbeck Foundation (to JKJ). Centre for Star and Planet Formation is funded by the Danish National Research Foundation. MVP acknowledges EU FP7 grant 291141 CHEMPLAN. The research leading to these results has received funding from the European Commission Seventh Framework Programme (FP/2007-2013) under grant agreement N$^{\rm o}$ 283393 (RadioNet3).
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[^1]: Based on observations carried out with the IRAM Plateau de Bure Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain).
[^2]: <http://www.iram.fr/IRAMFR/GILDAS/>
[^3]: <http://cassis.irap.omp.eu>
|
---
abstract: 'We investigate whether small perturbations can cause relaxation to quantum equilibrium over very long timescales. We consider in particular a two-dimensional harmonic oscillator, which can serve as a model of a field mode on expanding space. We assume an initial wave function with small perturbations to the ground state. We present evidence that the trajectories are highly confined so as to preclude relaxation to equilibrium even over very long timescales. Cosmological implications are briefly discussed.'
author:
- 'Adithya Kandhadai, Antony Valentini'
title: Perturbations and quantum relaxation
---
Introduction
============
The de Broglie-Bohm pilot-wave formulation of quantum theory [@deB28; @BV09; @B52a; @B52b; @Holl93] provides a generalisation of the quantum formalism, in which probabilities may differ from those predicted by the usual Born rule [@AV91a; @AV91b; @AV92; @AV96; @AV01; @PV06]. The Born rule applies only to a statistical state of quantum equilibrium, which may be understood as arising from a process of dynamical relaxation or ‘quantum relaxation’ (analogous to thermal relaxation) [@AV91a; @AV92; @AV01; @VW05; @EC06; @TRV12; @SC12; @ACV14]. Quantum nonequilibrium may have existed in the early universe [@AV91a; @AV91b; @AV92; @AV96; @AV01; @AV09], in which case violations of the Born rule could leave discernible traces in the cosmic microwave background (CMB) [@AV07; @AV10; @CV13; @CV15; @AV15; @CV16] and perhaps even survive until today for certain relic cosmological particles [@AV01; @AV07; @AV08; @UV15; @UV16]. Our current understanding of these cosmological scenarios depends, however, on our understanding of quantum relaxation – which remains incomplete in some important respects. In particular, the effect of small perturbations has not been considered. As we shall see, from a cosmological point of view it is important to establish what the effect of small perturbations might be, in particular over long timescales.
In pilot-wave theory, a system has a definite configuration $q(t)$ which evolves in time according to a law of motion for its velocity, where $\dot
{q}\equiv dq/dt$ is determined by the wave function $\psi(q,t)$. Here $\psi$ satisfies the usual Schrödinger equation $i\partial\psi/\partial t=\hat
{H}\psi$ (taking $\hbar=1$). For standard Hamiltonians, $\dot{q}$ is proportional to the gradient $\partial_{q}S$ of the phase $S$ of $\psi$. More generally, $\dot{q}=j/|\psi|^{2}$ where $j=j\left[ \psi\right] =j(q,t)$ is the Schrödinger current [@SV08]. The ‘pilot wave’ $\psi$ guides the motion of an individual system and in principle has no connection with probability. For an ensemble of systems with the same wave function, we may consider an arbitrary initial distribution $\rho(q,0)$ (at $t=0$) of configurations $q(0)$. By construction, the time evolution $\rho(q,t)$ will obey the continuity equation$$\frac{\partial\rho}{\partial t}+\partial_{q}\cdot\left( \rho\dot{q}\right)
=0\ .$$ Because $\left\vert \psi\right\vert ^{2}$ obeys the same equation, an initial distribution $\rho(q,0)=\left\vert \psi(q,0)\right\vert ^{2}$ will evolve into $\rho(q,t)=\left\vert \psi(q,t)\right\vert ^{2}$. In this equilibrium state, probabilities match the Born rule and pilot-wave theory reproduces the usual predictions of quantum theory [@B52a; @B52b]. But we may just as well consider nonequilibrium distributions $\rho(q,0)\neq\left\vert \psi
(q,0)\right\vert ^{2}$, opening up the possibility of a new and wider physics with violations of the Born rule and new phenomena outside the domain of conventional quantum physics [@AV91a; @AV91b; @AV92; @AV96; @AV01; @AV02; @AV07; @AV08; @AV08a; @AV09; @AV10; @AVPwtMw; @PV06].
Quantum relaxation to the equilibrium state $\rho=\left\vert \psi\right\vert
^{2}$ may be quantified by a coarse-grained $H$-function$$\bar{H}=\int dq\ \bar{\rho}\ln(\bar{\rho}/\overline{\left\vert \psi\right\vert
^{2}})\ ,$$ where $\bar{\rho}$, $\overline{\left\vert \psi\right\vert ^{2}}$ are obtained by coarse-graining $\rho$, $\left\vert \psi\right\vert ^{2}$ respectively. This obeys a coarse-graining $H$-theorem $\bar{H}(t)\leq\bar{H}(0)$ (if the initial state has no fine-grained micro-structure) [@AV91a; @AV92; @AV01]. The minimum $\bar{H}=0$ corresponds to equilibrium $\bar{\rho}=\overline
{\left\vert \psi\right\vert ^{2}}$. While this provides some understanding of how equilibrium is approached, the extent of relaxation depends on the system and on the initial conditions. For two-dimensional systems with wave functions that are evenly-weighted superpositions of energy eigenstates, extensive numerical studies have shown that initial nonequilibrium distributions $\rho$ (with no fine-grained micro-structure) rapidly approach $\left\vert
\psi\right\vert ^{2}$ on a coarse-grained level [@AV92; @AV01; @VW05; @TRV12; @SC12], with $\bar{H}(t)$ decaying approximately exponentially with time [@VW05; @TRV12]. In these examples, the wave function is periodic in time and the simulations were carried out up to one period $T$. More recently, such simulations were extended to longer timescales (up to $50T$) [@ACV14]. It was found that, for some initial wave functions (with certain choices of initial phases), the decay of $\bar{H}(t)$ saturates to a small but non-zero residue – signalling an incomplete relaxation. This was shown to occur when a significant fraction of the trajectories remain confined to sub-regions and do not explore the full support of $\left\vert
\psi\right\vert ^{2}$. The numerical evidence indicated that such confinement (and the associated incomplete relaxation) is less likely to occur for larger numbers of superposed energy states [@ACV14]. These conclusions are consistent with earlier examples studied by Colin [@SC12] and by Contopoulos *et al*. [@CDE12], in which limited relaxation – and an associated confinement of trajectories – was found for some initial wave functions with only three or four energy states.
Previous studies of quantum relaxation have mostly focussed on a coarse-graining approach for isolated systems [@AV91a; @AV92; @AV01], modelled on the analogous classical discussion [@Tol; @Dav].[^1] In this paper we consider instead the effect of small perturbations, in particular over very long timescales (of order $10^{3}T$). This is of interest in its own right, as well as for cosmological reasons.
Consider a system with an unperturbed wave function $\psi$, which generates an unperturbed velocity field $\dot{q}$ and unperturbed trajectories $q(t)$. The system might be subjected to small external perturbations, which in a first approximation we may model as perturbations to the classical potential of the system. The system will then have a perturbed wave function $\psi^{\prime}$ which is close to $\psi$, and a perturbed velocity field $\dot{q}^{\prime}$ which we expect to be close to $\dot{q}$. Will the perturbed trajectories $q^{\prime}(t)$ remain close to $q(t)$? One might expect that even a small difference in the velocity field, acting over sufficiently long periods of time, would yield perturbed trajectories $q^{\prime}(t)$ which deviate greatly from $q(t)$. For example, one might consider a two-dimensional harmonic oscillator with configuration $q=(q_{1},q_{2})$ whose unperturbed wave function is simply the ground state, $\psi(q_{1},q_{2},t)=\phi_{0}(q_{1}%
)\phi_{0}(q_{2})e^{-iE_{0}t}$, where $\phi_{0}(q_{1})\phi_{0}(q_{2})$ is a real Gaussian and $E_{0}$ is the ground-state energy. Because the phase $S=\operatorname{Im}\ln\psi$ is independent of position, the unperturbed velocity field $\dot{q}$ vanishes everywhere and all unperturbed trajectories are static. There can be no relaxation, nor indeed any evolution at all of the unperturbed density $\rho$. Any initial nonequilibrium distribution $\rho(q_{1},q_{2},0)\neq\left\vert \phi_{0}(q_{1})\phi_{0}(q_{2})\right\vert
^{2}$ will remain the same. Now let us consider a perturbed wave function $\psi^{\prime}$ that differs from $\psi$ by the addition of excited states $\phi_{m}(q_{1})\phi_{n}(q_{2})$ with small amplitudes $\epsilon_{mn}$. For small $\epsilon_{mn}$ the perturbed velocity field $\dot{q}^{\prime}$ will be small but generally non-zero. The question is: over arbitrarily long times, will the perturbed trajectories $q^{\prime}(t)$ remain confined to small sub-regions of the support of $\left\vert \phi_{0}(q_{1})\phi_{0}%
(q_{2})\right\vert ^{2}$ or will they wander over larger regions and possibly over the bulk of the support of $\left\vert \phi_{0}(q_{1})\phi_{0}%
(q_{2})\right\vert ^{2}$? In the former case, there could be no relaxation even over arbitrarily long times. In the latter case, relaxation could occur. Indeed, in the latter case it might seem plausible that, no matter how small $\epsilon_{mn}$ may be, over sufficiently long timescales the perturbed distribution $\rho^{\prime}(q_{1},q_{2},t)$ could approach $\left\vert
\phi_{0}(q_{1})\phi_{0}(q_{2})\right\vert ^{2}$ to arbitrary accuracy (where $\left\vert \phi_{0}(q_{1})\phi_{0}(q_{2})\right\vert ^{2}$ coincides with equilibrium as $\epsilon_{mn}\rightarrow0$). The question, then, is whether small perturbations are generally ineffective for relaxation or whether they might conceivably drive systems to equilibrium over sufficiently long times.
Cosmologically, the effect of perturbations over long timescales could be important for several reasons. According to inflationary cosmology, the temperature anisotropies in the CMB were seeded by primordial quantum fluctuations of a scalar field whose quantum state was approximately a vacuum (the Bunch-Davies vacuum) [@LL00; @Muk05; @PU09]. It has been shown that de Broglie-Bohm trajectories for field amplitudes in the Bunch-Davies vacuum are too trivial to allow relaxation [@AV07; @AV10]. On this basis it was concluded that, if quantum nonequilibrium existed at the beginning of inflation, then it would persist throughout the inflationary phase and potentially leave an observable imprint in the CMB today. However, strictly speaking this conclusion depends on the implicit assumption that (unavoidable) small corrections to the Bunch-Davies vacuum can be neglected in the sense that they will not generate relaxation during the inflationary era. Similarly, a cosmological scenario has been developed according to which quantum relaxation occurred during a pre-inflationary (radiation-dominated) phase [@AV07; @AV10; @CV13; @CV15; @AV15; @CV16]. It was shown that during such a phase relaxation proceeds efficiently at short wavelengths but is suppressed at long wavelengths, resulting in a distinctive signature of quantum nonequilibrium at the beginning of inflation – which is then imprinted at later times in the CMB.[^2] The resulting predictions for the CMB depend, however, on the assertion that there will be no significant relaxation during inflation itself, an assertion which again depends on the implicit assumption that small corrections to the Bunch-Davies vacuum may be ignored. Finally, a scenario has also been developed according to which, for certain particles created in the early universe, any nonequilibrium carried by them could conceivably survive (or partially survive) to the present [@UV15]. But such a scenario would fail if small perturbations caused the particles to relax over very long timescales. Indeed, if small perturbations do cause relaxation over long timescales it would be exceedingly difficult to have any hope at all of discovering relic nonequilibrium today.
To discuss these cosmological matters quantitatively, it suffices to consider a free (minimally-coupled) massless scalar field $\phi$ on expanding flat space with scale factor $a(t)$. Here $t$ is standard cosmological time and physical wavelengths are proportional to $a(t)$. In Fourier space we have field components $\phi_{\mathbf{k}}(t)$ which may be written in terms of their real and imaginary parts, $\phi_{\mathbf{k}}=\frac{\sqrt{V}}{(2\pi)^{3/2}%
}\left( q_{\mathbf{k}1}+iq_{\mathbf{k}2}\right) $ (where $V$ is a normalisation volume). The field Hamiltonian then becomes a sum $H=\sum
_{\mathbf{k}r}H_{\mathbf{k}r}$, where $H_{\mathbf{k}r}$ ($r=1,2$) is mathematically the Hamiltonian of a harmonic oscillator with mass $m=a^{3}$ and angular frequency $\omega=k/a$ [@AV07; @AV08; @AV10]. If we consider an unentangled mode $\mathbf{k}$, we have an independent dynamics with a wave function $\psi=\psi(q_{1},q_{2},t)$ (dropping the index $\mathbf{k}$) that satisfies the Schrödinger equation$$i\frac{\partial\psi}{\partial t}=\sum_{r=1,\ 2}\left( -\frac{1}{2m}%
\partial_{r}^{2}+\frac{1}{2}m\omega^{2}q_{r}^{2}\right) \psi$$ (with $\partial_{r}\equiv\partial/\partial q_{r}$). The pilot-wave equation of motion for the actual configuration $(q_{1},q_{2})$ then reads$$\dot{q}_{r}=\frac{1}{m}\operatorname{Im}\frac{\partial_{r}\psi}{\psi
}\label{deB}%$$ and an arbitrary marginal distribution $\rho=\rho(q_{1},q_{2},t)$ will then evolve according to the continuity equation$$\frac{\partial\rho}{\partial t}+\sum_{r=1,\ 2}\partial_{r}\left( \rho\frac
{1}{m}\operatorname{Im}\frac{\partial_{r}\psi}{\psi}\right) =0\ .$$ These equations are just those of pilot-wave dynamics for a two-dimensional harmonic oscillator with (time-dependent) mass $m=a^{3}$ and angular frequency $\omega=k/a$. It may be shown that the resulting time evolution is mathematically equivalent to that of an ordinary harmonic oscillator (with constant mass and angular frequency) but with the time parameter replaced by a ‘retarded time’ that depends on $k$ [@CV13]. It is found, in particular, that relaxation is suppressed at long (super-Hubble) wavelengths while proceeding efficiently at short (sub-Hubble) wavelengths [@AV08; @CV13; @CV15].
Thus cosmological relaxation for a single field mode may be discussed in terms of relaxation for a standard oscillator. By studying the effect of small perturbations on relaxation for a simple two-dimensional harmonic oscillator, then, we may draw conclusions that have application to cosmology.
In Section 2 we present our model, which is obtained simply by setting $m=1$ in the equation of motion (\[deB\]) (for $r=1,2$). This defines our dynamics of trajectories for a two-dimensional harmonic oscillator, with constant mass and constant angular frequency and with a given wave function. We shall take $\psi$ to be the ground state with small perturbations of amplitude $\epsilon_{mn}$ coming from the lowest excited states $\phi_{m}(q_{1})\phi
_{n}(q_{2})$. In Section 3 we discuss our method, where two different techniques are applied to infer the extent of relaxation in the long-time limit, using samples of trajectories evolved over long times. In Section 4 we then study numerically the behaviour of a sample of trajectories over very long timescales, in particular we consider how their behaviour changes as the perturbations become smaller. As we shall see, for sufficiently small perturbations the trajectories become highly confined, and neighboring trajectories are confined to almost the same regions, even over very long timescales – from which we conclude (tentatively, given our numerical evidence) that small perturbations do not cause relaxation. Cosmological implications are briefly discussed in Section 5, where we draw our conclusions.
Oscillator model
================
The system under consideration is the standard two-dimensional harmonic oscillator. We employ units such that $\hbar = m = \omega = 1$. The wave function at $t = 0$ is taken to be the ground state of the oscillator perturbed by a superposition of excited states:
$$\psi(q_1, q_2, 0) = N\left[e^{i\theta_{00}}\phi_0(q_1)\phi_0(q_2) + \sum_{(m,n) \neq (0,0)} \epsilon_{mn}e^{i\theta_{mn}}\phi_m(q_1)\phi_n(q_2) \right] .$$
Here $N$ is a suitable normalization factor.
The $\theta_{mn}$ are randomly chosen initial phases taking values between $0$ and $2\pi$. The function $\phi_m(q_r)$ is the eigenfunction corresponding to the $m^{th}$ energy state of the harmonic oscillator, given by
$$\phi_m(q_r) = \frac{1}{\pi^{\frac{1}{4}}} \frac{1}{\sqrt{2^m m!}} H_m(q_r) e^{-\frac{q_r^2}{2}} ,$$
where $H_m$ is the Hermite polynomial of order $m$. For an energy eigenstate $\phi_m(q_1)\phi_n(q_2)$ of the two-dimensional harmonic oscillator, the corresponding energy eigenvalue is $E_{mn} = (m + n + 1)$ (in our units).
The parameters $\epsilon_{mn}$ take values between $0$ and $1$ and quantify the difference between the initial wave function and the ground state. For $\epsilon_{mn} = 1$, the initial wave function is an equally-weighted superposition of the first four energy states, as studied in [@ACV14]. For small values of $\epsilon_{mn}$, the initial wave function can be thought of as the ground state with small perturbations.
Introducing the quantities $ \alpha_{mn}(t) = \theta_{mn} - E_{mn}t $, the wave function at any time $t$ is given by
$$\psi(q_1, q_2, t) = N\left[e^{i\alpha_{00}(t)}\phi_0(q_1)\phi_0(q_2) + \sum_{(m,n) \neq (0,0)} \epsilon_{mn}e^{i\alpha_{mn}(t)}\phi_m(q_1)\phi_n(q_2) \right] .$$
Note that this wave function is periodic with period $T = 2\pi$.
The velocity field for this wave function is given by (4) (with $m = 1$), which is the equation of motion that determines our trajectories.
In most of our simulations, we shall for simplicity assume that $\epsilon_{mn} = \epsilon$, that is, we assume a ‘homogeneous’ perturbation. Our wave function at time $t$ then reads
$$\psi(q_1, q_2, t) = N\left[e^{i\alpha_{00}}\phi_0(q_1)\phi_0(q_2) + \epsilon \sum_{(m,n) \neq (0,0)} e^{i\alpha_{mn}}\phi_m(q_1)\phi_n(q_2) \right] .$$
We shall however verify with examples that similar results are obtained for unequal values of $\epsilon_{mn}$, so that this simplification is unimportant.
If the wave function is simply one of the energy eigenstates, as opposed to a superposition, then the trajectories are stationary (since the eigenstates of the harmonic oscillator are purely real, apart from an overall phase factor) and no relaxation occurs. However, a superposition of energy eigenstates will usually generate non-trivial trajectories and some degree of relaxation. The question is: if the perturbations are small, will significant relaxation still occur – at least over sufficiently long timescales?
Method
======
As we have noted, our goal is to study the extent of relaxation for harmonic oscillator wave functions of the form (8) consisting of the ground state with small perturbations from excited states. We will be considering long timescales of order 1000 periods. It would be computationally intractable to simulate the evolution of a complete non-equilibrium distribution $\rho$ over such long times. For example, previous simulations of relaxation were carried out for up to 50 periods only [@ACV14], which was already computationally demanding. Instead, for each initial wave function, we shall begin by examining a sample of ten individual trajectories that start at the following points: $(q_1, q_2)$ = (1.5, 1.5), (1.5, -1.5), (-1.5, 1.5), (-1.5, -1.5), (0.5, 0.0), (0.0, -0.5), (-0.5, 0.0), (0.0, 0.5), (0.25, 0.25), and (0.25, -0.25). These points will be referred to as points 1 through 10, respectively. For a given initial wave function, the number of trajectories that travel over the main support of $|\psi|^2$ (as opposed to remaining confined to a small sub-region) and the degree to which they cover it may be used to draw preliminary inferences about the extent of relaxation.
The justification for this method comes from ref. [@ACV14], in which it was shown that limits on relaxation – quantified by a non-zero ‘residue’ of the coarse-grained $H$-function at large times – occur when a significant fraction of the trajectories show a substantial degree of confinement (to a sub-region of the support of $|\psi|^2$). As is to be expected, confinement of trajectories is associated with a lack of complete relaxation (even over long timescales), while if the trajectories tend to wander over the bulk of the support of $|\psi|^2$ then a more complete relaxation can take place. This was shown by means of numerous examples in ref. [@ACV14].
There is of course no sharp dividing line between trajectories that are confined and those that are not, and we do not use a precise quantitative criterion. As in ref. [@ACV14] it will suffice to use our judgment in deciding how well the trajectories cover the space. The effectiveness of this method – at least for present purposes – will become clear in what follows.
We shall also use another method, again inspired by ref. [@ACV14], to infer whether or not relaxation to equilibrium occurs over very long timescales. We shall study trajectories starting from neighbouring initial points to see whether or not they are confined to essentially the same sub-regions. If they are so confined, this will constitute further evidence against relaxation.
To plot a trajectory, we consider a particle starting at the required point and calculate its position every hundredth of a period. We employ a Dormand-Prince adaptive time step algorithm to solve the equation of motion (4) with the wave function (9). The algorithm has an upper bound on the allowable error in each time step (denoted ABSTOL) which is used to choose the step size. Our trajectories are accurate up to an absolute error of 0.01 in the final position. This is confirmed by checking that the final positions with two consecutive values of ABSTOL are not separated by a distance greater than 0.01.
Behaviour of trajectories over very long timescales
===================================================
We now present numerical results for the behaviour of trajectories over very long timescales. We first consider the previously-discussed case $\epsilon$ = 1 with four modes[@ACV14], though now over much longer timescales. We are able to confirm that the behaviour found in [@ACV14] persists over timescales about two orders of magnitude larger than were previously considered. We then examine how the behaviour changes for smaller $\epsilon$. Our results indicate that relaxation is suppressed for small $\epsilon$. We present evidence that small perturbations are unlikely to yield significant relaxation even over arbitrarily long timescales.
Case $\epsilon = 1$
-------------------
We first consider the same four-mode wave function, with $m$, $n$ in equation (9) summing from $0$ to $1$, that was investigated in [@ACV14].[^3] In that work the confinement of the trajectories was studied up to 25 periods. The results are displayed in figure 8 of [@ACV14] and show that certain trajectories remain confined over 25 periods. (It was also found that the coarse-grained $H$-function seemed to saturate, with a small residue indicating incomplete relaxation – caused by the confinement of a significant fraction of the trajectories.) We have now found that, for this same case, the confinement persists all the way up to 3000 periods. Initial points 1, 2, 3, 4 and 6 generate trajectories that wander over the bulk of the support of the wave function, while initial points 5, 7, 8, 9 and 10 generate trajectories that are confined to small sub-regions.
If a trajectory is found numerically to be confined to a sub-region over a given large time interval, it is of course conceivable that over even larger times the trajectory could wander outside of that region and possibly cover the bulk of the support of $|\psi|^2$. We are, however, able to provide evidence that our apparently confined trajectories will remain confined indefinitely, by the following method.
We plot the ‘widths’ and ‘heights’ of the trajectories – that is, differences between the maximal and minimal $q_1$- and $q_2$-coordinates respectively – for various final times. The widths and heights of the unconfined trajectories continue to show an increasing trend even between 2000 and 3000 periods. In contrast, the widths and heights of the confined trajectories quickly saturate and remain essentially constant after the early increase, indicating that those trajectories are bounded indefinitely within a sub-region. Of course this method does not provide a proof of strict and indefinite confinement, but it does provide strong numerical evidence.
To illustrate our results we may consider two specific trajectories, the first unconfined and the second confined.
An example of an unconfined trajectory is shown in Fig. 1. The trajectory is plotted up to 25, 100, 200, 500, 1000 and 3000 periods. Clearly, the trajectory wanders over the bulk of the support of $|\psi|^2$ . (For comparison, the distribution $|\psi|^2$ is shown in Fig. 2.)
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For comparison, the Born distribution is shown below in Fig. 2.
{width="6cm" height="6cm"}
The widths and heights of the trajectory in Fig. 1, calculated for the final times 25, 100, 200, 500, 1000, 2000 and 3000 periods, are shown in Fig. 3. The increasing range of the trajectory is obvious.
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An example of a confined trajectory is shown in Fig. 4. The trajectory is again plotted up to 25, 100, 200, 500, 1000 and 3000 periods. The trajectory is clearly confined to a sub-region of the support of $|\psi|^2$.
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Evidence for strict confinement comes from plotting the width and height of the trajectory at the final times 25, 100, 200, 500, 1000, 2000 and 3000 periods. The results are shown in Fig. 5. There is a quick saturation after the early increase, indicating a strict and indefinite confinement to a sub-region (of the corresponding saturation width and height).
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To understand the overall behavior for $\epsilon = 1$, we have calculated trajectories starting at the same ten points 1-10 but for ten additional sets of randomly-generated initial phases $\theta_{mn}$ (hence a total of 100 additional trajectories). For about half of the initial wave functions, the trajectories are split more or less evenly between unconfined and confined (as was the case for the initial phases given in footnote 3), while for the remaining initial wave functions we find that 7-8 trajectories are unconfined and 2-3 are confined.
Results for smaller $\epsilon$. Effect on confinement
-----------------------------------------------------
We wish to investigate how relaxation over very long timescales will be affected when $\epsilon$ is made smaller. We hope to find behavior that is largely independent of the choice of initial phases. To this end we generate trajectories with the same ten starting points 1-10 and for the same additional ten sets of values of $\theta_{mn}$ as in Section 4, but now with $\epsilon = 0.5$ and then again with $\epsilon = 0.25$, $0.1$ and $0.05$. [^4]
For $\epsilon = 0.5$, it was almost always the case (for almost all choices of initial phases) that points 1-4 had largely unconfined trajectories while points 5-10 were confined to small sub-regions. Examples of both types of trajectory are shown in Fig. 6.
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When $\epsilon$ is decreased to $0.25$, a pattern begins to emerge. The initial points 1-4 orbit the origin while remaining confined to outer annular regions. The trajectories travel over large distances but leave a large empty space in the inner part of the support of $|\psi|^2$, and so may be considered confined. Points 5-10 are again confined to small sub-regions, as we saw for $\epsilon = 0.5$. Examples of both types of behaviour are shown in Fig. 7.
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It is found that the motion of the trajectory in Fig. 7(a) may be decomposed into a quasi-oscillation confined to a small region together with an approximately uniform circular motion around the origin. This is shown in Fig. 8.
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When $\epsilon$ is decreased even further, to $0.1$ and then to $0.05$, the general pattern of behaviour found for $\epsilon = 0.25$ continues to hold.
For $\epsilon = 0.1$ the initial points 1-4 travel in ever narrower annular regions centered on the origin and with support at the edges of the main support of $|\psi|^2$, while points 5-10 are confined to very small regions (Fig. 9).
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For $\epsilon = 0.05$ the inner and outer trajectories are confined to an even greater degree. Examples are shown in Fig. 10.
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For such small values of $\epsilon$, the outer trajectories are confined to annular regions centered on the origin - see for example Fig. 9(a). Close examination of the trajectories at intermediate times shows that they are confined to small regions with a superposed mean angular drift (as observed already in the example of Figure 8, though with a somewhat narrower annular region). This is found to occur for all four outer trajectories for all sets of initial phases. The rate of angular drift is roughly the same for all the outer trajectories for a given set of initial phases – but it varies depending on the choice of initial phases. While these trajectories travel over large distances, as far as relaxation is concerned they are highly confined.
The above results are not specific to a homogeneous perturbation with $\epsilon_{mn} = \epsilon$. To confirm this, we may consider a specific inhomogeneous perturbation, for example with $\epsilon_{01} = 0.2$, $\epsilon_{10} = 0.15$ and $\epsilon_{11} = 0.1$. Illustrative examples of the trajectories for such a wave function are displayed in Fig. 11. The general behaviour is found to be comparable to that seen for homogeneous perturbations.
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If we take the trajectories we have calculated (for many different sets of phases) as representative of the behaviour over very long timescales, we may draw a simple tentative conclusion. As the perturbation in the initial wave function is made smaller, the extent to which the trajectories explore the support of $|\psi|^2$ becomes smaller. We then expect that, for smaller perturbations, the extent of relaxation will be smaller (even in the long time limit). From our results, indeed, we may reasonably conclude that the extent of relaxation at large times will vanish for $\epsilon \rightarrow 0$.
Small perturbations with two additional modes
---------------------------------------------
So far, we have considered wave functions with the ground state perturbed by a superposition of the $|1,0>$, $|0,1>$ and $|1,1>$ energy eigenstates of the two-dimensional harmonic oscillator. However, the $|2,0>$ and $|0,2>$ states have the same energy as the $|1,1>$ state and there is no particular reason to assume that they will be less likely to be present in a perturbation than the $|1,1>$ state. In this section we discuss the results of the simulations for perturbations with all five aforementioned excited states present, with the perturbation parameter $\epsilon = 0.1$.
We find that the trajectories are slightly less confined than they tend to be without the two newly added states (the annular regions are a bit thicker), but still do not come close to traveling over the bulk of the support of $|\psi|^2$. Thus we again find an absence of relaxation for small perturbations. Examples are shown in Fig. 12.[^5]
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Confinement of neighboring trajectories
---------------------------------------
Despite the above results, it might be thought that relaxation could in principle still occur if neighboring initial points generated trajectories covering different sub-regions, in such a way that a small initial region explored the full support of $|\psi|^2$. As further evidence that relaxation does not occur, even over arbitrarily long timescales, we now show that in fact neighboring initial points generate trajectories that cover essentially the same sub-region of the support of $|\psi|^2$. As we shall see, neighboring initial points can generate trajectories that diverge widely, but even so the trajectories remain confined to the same sub-region. In such circumstances, relaxation cannot occur.
We consider 10 small squares of edge 0.04 centered at the points 1-10 listed above, with trajectories calculated for 13 initial points in each square.[^6] This process was repeated for three different sets of phases. For these simulations we use the wave function with five homogeneous perturbative modes added to the ground state (as in section 4.3).
We find that the trajectories for all the points inside a given square explore almost exactly the same sub-region of the support of $|\psi|^2$. This is illustrated in Fig. 13.[^7]
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In some cases, however, a trajectory starting from one of the points in a square covered only a portion of the sub-region explored by its neighbors. An example is shown in Fig. 14.
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To ensure that our results are not specific to a homogeneous perturbation (with a common value of $\epsilon$), we have applied the same analysis for a wave function with the inhomogeneous perturbation parameters $\epsilon_{01} = 0.11$, $\epsilon_{02} = 0.12$, $\epsilon_{10} = 0.13$, $\epsilon_{11} = 0.14$ and $\epsilon_{20} = 0.15$. The results are similar. An example is displayed in Fig. 15.[^8]
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It might be thought that the confinement of neighboring initial points to essentially the same evolved sub-region arises simply from the trajectories remaining close to each other. This is not always the case, however. Remarkably, some initial points within the same small square can become temporarily far apart (within the sub-region) and then very close again – indicating that a confinement mechanism is at play. For example, when two of the trajectories in Fig. 13 were compared, the maximum distance between the positions over 3000 periods was 1.47 - which is roughly 15 percent of the diameter of the evolved sub-region. However, the distance between the final positions was only 0.08, which is of the same order of magnitude as the distance between the initial points.
Conclusion
==========
Our numerical results provide evidence that small perturbations will not, in fact, cause significant relaxation – not even over arbitrarily long timescales. In the examples we have studied, the system trajectories are confined to sub-regions of the support of $|\psi|^2$. Furthermore, neighboring initial points generate trajectories that are confined to essentially the same sub-regions. Such behavior precludes relaxation.
We have restricted ourselves to the harmonic oscillator, which as we explained in Section 1 provides a testing ground applicable to high-energy field theory in the early universe. In future work it would be of interest to consider other systems, as well as to develop an analytical understanding of the results (for which the methods of ref. [@CDE12] may prove useful).
From the point of view of a general understanding of relaxation, our results suggest that in de Broglie-Bohm theory quantum equilibrium cannot be understood as arising from the effects of small perturbations only, not even in the long-time limit. Since all systems we know of have a long and violent astrophysical history (ultimately stretching back to the big bang), their current obedience to the Born rule may nevertheless be understood in terms of the efficient relaxation found in previous simulations (at least at the sub-Hubble wavelengths relevant to laboratory physics) for wave functions with significant contributions from a range of energy states.
As regards cosmology, our results point to the following conclusions. Firstly, the implicit assumption made in refs. [@AV07; @AV10; @CV13; @CV15; @AV15; @CV16] is justified: small corrections to the Bunch-Davies vacuum during inflation are unlikely to cause significant relaxation, and so the derived predictions for the CMB still stand (for the assumed scenario with a pre-inflationary period). Secondly, relic nonequilibrium particles from the early universe surviving to the present day remains a possibility at least in principle (albeit a rather remote one for other reasons, as discussed in ref. [@UV15]).
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[^1]: An exception is an early paper by Bohm [@B53], which considered an ensemble of two-level molecules subject to random external collisions and argued that the molecules would relax to equilibrium.
[^2]: Specifically, the signature amounts to a primordial power deficit at long wavelengths with a specific (inverse-tangent) dependence on wavenumber $k$ [@CV15; @CV16]. A large-scale power deficit has in fact been reported in the *Planck* data [@PlanckXV-2013; @Planck15-XI-PowerSpec], though the extent to which it matches our prediction is still being evaluated [@VVP16].
[^3]: The initial phases were $\theta_{00} = 0.5442$, $\theta_{01} = 2.3099$, $\theta_{10} = 5.6703$, $\theta_{11} = 4.5333$.
[^4]: For the illustrative figures in this subsection, the initial phases were as follows: $\theta_{00} = 4.8157$, $\theta_{01} = 1.486$, $\theta_{10} = 2.6226$, $\theta_{11} = 3.8416$.
[^5]: For the trajectories in Fig. 12, the initial phases were $\theta_{00} = 4.2065$, $\theta_{01} = 0.1803$, $\theta_{02} = 2.0226$, $\theta_{10} = 5.5521$, $\theta_{11} = 3.3361$, $\theta_{20} = 2.6561$.
[^6]: For a square centered at the origin, the coordinates of the 13 points would be as follows: $(-0.02,0.02)$, $(-0.02,0.0)$, $(-0.02,-0.02)$, $(0.0,-0.02)$, $(0.02,-0.02)$, $(0.02,0.0)$, $(0.02,0.02)$, $(0.0,0.02)$, $(-0.01,0.01)$, $(-0.01,-0.01)$, $(0.01,-0.01)$, $(0.01,0.01)$, $(0.0,0.0)$.
[^7]: For the trajectories displayed in Fig. 13, the initial phases used were $\theta_{00} = 1.2434$, $\theta_{01} = 4.411$, $\theta_{02} = 4.3749$, $\theta_{10} = 4.2427$, $\theta_{11} = 1.5574$, $\theta_{20} = 5.7796$.
[^8]: For the trajectories displayed in Fig. 15, the initial phases used were $\theta_{00} = 4.0857$, $\theta_{01} = 0.2194$, $\theta_{02} = 4.6059$, $\theta_{10} = 1.2201$, $\theta_{11} = 0.439$, $\theta_{20} = 4.0563$.
|
---
abstract: 'We present what is to our knowledge the most complete 1-D numerical analysis of the evolution and the propagation dynamics of an ultrashort laser pulse in a Ti:Sapphire laser oscillator. This study confirms the dispersion managed model of mode-locking, and emphasizes the role of the Kerr nonlinearity in generating mode-locked spectra with a smooth and $well-behaved$ spectral phase. A very good agreement with preliminary experimental measurements is found.'
author:
- |
Marco V. Tognetti and Helder M. Crespo\
\
CLOQ/Departamento de Física,\
Faculdade de Ci$\hat{e}$ncias, Universidade do Porto,\
Rua do Campo Alegre 687, 4169-007 Porto, Portugal
title: 'Dispersion managed mode-locking dynamics in a Ti:Sapphire laser'
---
Pulse generation in mode-locked lasers has been a subject of intense research over more than two decades. In more recent descriptions of Kerr-lens mode-locked (KLM) ultrafast lasers, pulse formation is assumed to rely upon a soliton-like mechanism, based on the master equation approximation[@Haus2000-5], where the combined and balanced action of self-phase modulation (SPM) and group-delay dispersion (GDD) is the basis for the generation of a short pulse. In this picture the shortest pulse durations are obtained for cavity configurations with minimum net GDD. This approach gives a good description of actual systems but does not take into account the pulse propagation dynamics in the single optical elements that comprise the laser cavity and usually does not include higher order dispersion terms, which are known to significantly affect sub$-10\,fs$ pulses such as those generated with today’s state-of-the-art laser oscillators[@Morgner1999-5; @Fuji2003-5]. As an alternative to the master equation approach, the evolution of laser pulses has also been modeled numerically [@Christov1994-5; @Christov1996-5], even if the actual sequence of intracavity components was not fully taken into consideration. In fact, the actual ordering of the optical elements is behind a new model for the generation of ultrashort pulses later introduced by Chen $et\,al.$[@Chen1999], on its hand directly related to the nonlinear propagation of pulses in dispersion managed communication fibers. They identify a solid state mode-locked laser as another example of a system where dispersion managed solitons (DMS) can be observed. DMS are stable soliton-like solutions of the nonlinear Schrödinger equation which are known to occur in optical media with a periodical change of sign in the GDD[@Haus1999], such as a femtosecond laser cavity. The main difference from $standard$ soliton propagation is that the spectrum and temporal profile of DMS periodically broaden and recompress as the pulse crosses regions of opposite GDD.
Here we present what is to our knowledge the most detailed 1-D numerical simulation of a prism-dispersion controlled linear laser cavity, which includes build up of the laser pulse from noise through the action of active- and Kerr-lens mode-locking, the measured reflectivity and phase distortion of every optical element in the cavity, the measured gain bandwidth of the Ti:sapphire crystal, and the propagation inside the active medium, by numerically solving the corresponding nonlinear Schrödinger equation in the presence of gain. In particular, great attention is devoted to intracavity pulse formation and propagation, showing how the spectrum and its phase evolve as the pulse crosses the crystal, is reflected off the intracavity mirrors, and goes through the negatively dispersive prism line. This study confirms the dispersion managed model of mode-locking, showing that a spectrum extending from $700$ to $950$nm with a smooth and [*nearly flat*]{} phase can be obtained using commercially available ultrafast optical elements, in agreement with recent experimental work[@Crespo2005].
Figure\[figure\_1\_tognetti\_crespo\](a) shows a schematic diagram of the Ti:Sapphire laser oscillator, which consists of: an active crystal of length $L$ enclosed between two focusing mirrors $M_1$ and $M_2$, a flat folding mirror $M_3$, an output coupler $OC$ and a silver high reflector at the cavity ends, a pair of fused-silica prisms for dispersion compensation, and an active amplitude modulator $M(t)$ to form the initial pulse from noise. The evolution of the pulse inside the cavity is described by the following iterative procedure which connects the spectral amplitude of the field, $\tilde{A}_{k+1}(\omega)$, for the $(k+1)-th$ passage inside the cavity, with the field envelope $A_k(t)$, obtained from the previous passage: $$\begin{aligned}
\tilde{A}_{k_1}(\omega) &=& R_{1}(\omega)^{\frac{1}{2}}e^{i
\phi_{1}(\omega)}\int_{-\infty}^{+\infty} M(t)A_{k}(t)e^{-i \omega t}\nonumber\\
\tilde{A}_{k_2}(\omega) &=& {\cal P}(\tilde{A}_{k_1}(\omega))\nonumber\\
\tilde{A}_{k_3}(\omega) &=& R_{2}(\omega)R_{3}(\omega)e^{2i(\phi_{2}(\omega)+\phi_{3}(\omega)+\phi_{pr}(\omega))}\tilde{A}_{k_2}(\omega)\nonumber\\
\tilde{A}_{k_4}(\omega) &=& {\cal P}(\tilde{A}_{k_3}(\omega))\nonumber\\
\tilde{A}_{k+1}(\omega) &=& (R_{1}(\omega)R_{OC}(\omega))^{\frac{1}{2}} e^{i (\phi_{1}(\omega)+\phi_{OC}(\omega))} \tilde{A}_{k_4}(\omega),\\
\label{eq_it}\nonumber\end{aligned}$$ where $R_{i}$ and $\phi_i$ are the reflectivity and phase of the $i-th$ optical element respectively, $M(t)$ is an initial active modulation used to start mode-locked operation [@Christov1994-5], and $\tilde{A}_{k_{n+1}}(\omega) = {\cal P}(\tilde{A}_{k_n}(\omega))$ is the spectral amplitude at the crystal output, obtained by numerically solving the following propagation equation inside the crystal: $$\begin{aligned}
\frac{\partial A(z,t)}{\partial z} &=& \int_{-\infty}^{+\infty} (i
\beta(\omega)+g(\omega)) \tilde{A}(z,\omega)e^{-i\omega t} d
\omega\nonumber\\
& &+i \gamma |A(z,t)|^2 A(z,t) \label{eq_prop}\end{aligned}$$ with $A(0,t)=A_{k_n}(t)$. Here $\beta(\omega)$ and $g(\omega)$ are the crystal phase distortion and gain profile per unit length, and $\gamma=2.2 \times 10^{-6} W^{-1}\,cm^{-1}$ is the estimated nonlinearity coefficient. Moreover, self-amplitude modulation (SAM) induced by the Kerr nonlinearity is included as a nonlinear intensity discriminator in the time domain[@Christov1994-5], here modeled as a supergaussian: $K(A)=exp[-\frac{1}{2}(\frac{|A|^2}{\sigma P_0})^m]$, with $P_0=max(|A|^2)$, $m=24$ and $\sigma=0.47$. These parameters were determined by trial-and-error, even though the final spectrum will not significantly depend on the exact values, provided that mode-locking operation is established. The crystal gain and the mirror reflectivities and phase distortions were directly obtained from measurements performed on actual optical components, which comprise a relatively thick ($4.5\,mm$) Ti:Sapphire crystal (Crystal Systems Inc.), commercially available standard ultrafast laser mirrors designed for $850\,nm$ (Spectra-Physics and TecOptics) and a $3.5\%$ output-coupler (Spectra-Physics). Figures\[figure\_1\_tognetti\_crespo\](b) and (c) show the total GDD and the total cavity gain $G_{tot}$ for one round trip defined as $G_{tot}(\lambda)=R_1(\lambda)^2R_2(\lambda)^2R_3(\lambda)^2R_{OC}(\lambda)G(\lambda)^2$, with $G(\lambda)=e^{g(\lambda)L}$ the total crystal gain normalized at $1.04$. As initial condition, a $low$ amplitude random noise with flat spectral phase is assumed. The evolution of the intracavity spectral profile and phase of the pulse prior to entering the output coupler are shown in figure\[figure\_2\_tognetti\_crespo\], after (a)$20$,(b)$500$, and (c)$1000$ round-trips of the pulse inside the cavity. After the first $20$ round-trips, the spectral phase of the intracavity radiation shows strong fluctuations due to the accumulated linear phase distortion of the optics, as expected from figure\[figure\_1\_tognetti\_crespo\](b). Then a spectrum with increasingly $smooth$ spectral phase centered around $850\,nm$ builds up from noise mainly through the action of the crystal gain, the active modulation and SAM, resulting in the spectrum shown in figure\[figure\_2\_tognetti\_crespo\](b) at a pulse peak power of $3.5\times10^5\,W$. As the pulse intensity increases, new mode-locked frequencies are generated via SPM and the spectrum is further broadened until a steady-state is reached at a pulse peak power of $5 \times 10^6\,W$ (see figure\[figure\_2\_tognetti\_crespo\](c)), as a result of the interplay between SPM, phase distortion and the finite bandwidth of the total cavity gain (see figure\[figure\_1\_tognetti\_crespo\](c)). To illustrate the validity of the simulation code the measured and the simulated spectra obtained after the OC are given in figure\[figure\_2\_tognetti\_crespo\](d), revealing a very good agreement between experimental measurements and theoretical predictions. The steady-state spectrum of figure\[figure\_2\_tognetti\_crespo\](c) is reproduced every round-trip but it has a different spectral profile in different dispersion regions of the cavity. Figure\[figure\_3\_tognetti\_crespo\] shows how the cavity can be put in analogy with a dispersion managed fiber made of a region of nonlinear propagation and positive GDD (the active medium crossed two times), and a region of linear propagation and negative GDD (the prism-pair crossed two times). The OC and the silver mirror are placed in the middle point of each dispersion region. In correspondence with the points $A,B,C,D,E$ and $F$ of figure\[figure\_3\_tognetti\_crespo\], figure\[figure\_4\_tognetti\_crespo\] shows how the steady-state pulse spectrum, phase, and temporal profile evolve when crossing the crystal, going through the dispersion compensating prism-pair, and coming back to the OC. As the pulse propagates towards the silver mirror, it enters the active medium with a positive chirp (point $A$ in figures\[figure\_3\_tognetti\_crespo\] and\[figure\_4\_tognetti\_crespo\]) and its spectrum broadens while its temporal width increases due to the concomitant action of positive GDD and SPM (points $B$ and $C$). It can be observed that spectral broadening mainly takes place in the first half of the crystal (from $A$ to $B$) since temporal broadening due to GDD decreases the strength of SPM as the pulse penetrates more into the crystal. When the pulse crosses the two prisms and reaches the silver mirror (point $D$), the acquired positive chirp is partially compensated for, while the spectrum remains unaffected, resulting in a shorter and more intense pulse. This is the point of maximum intracavity spectral width, minimum pulse duration and maximum peak intensity. In contrast to the case of point $A$, when the pulse enters the crystal on its way back to the OC (point $E$), it is negatively chirped by a second passage in the prism sequence: nonlinear propagation now results in spectral narrowing and phase flattening until the pulse is transform limited at a depth of 3 mm within the crystal (point $F$). This is the point of minimum spectral width. As the pulse continues its way to the OC and goes back to the silver mirror, its spectrum is broadened, recovering its maximum width at points $C$ and $D$. The pulse behaviour described here confirms the dispersion managed model of mode-locking[@Chen1999], by showing the typical spectral and temporal $breathing$ of DMS, and also explains the experimental fact that spectra are broader when taken in the dispersive end of KLM lasers[@asaki1993]. Furthermore figure\[figure\_4\_tognetti\_crespo\] shows that the pulse propagating in the cavity, while varying its spectrum and temporal profile, mantains a $smooth$ spectral phase due to the joint action of SAM and SPM, which are able to wash out from the spectral phase the modulations inherent in the net intracavity GDD. This is in agreement with recent experimental results, where it was shown that operation of a broadband Ti:Sapphire laser under strong Kerr-lens mode-locking conditions resulted in a smoothing of the spectral phase[@Morgner1999-5; @Fuji2003-5]. Once extracted the pulse from the cavity, such a $well-behaved$ phase appears to be suitable for further extracavity pulse compression, giving for the spectrum of point $D$ a Fourier transform limited pulse duration of $11\,fs$, also in good agreement with recent experimental results[@Crespo2005].
In conclusion we proved numerically the dispersion managed model of mode-locking using what is to our knowledge the most complete 1-D simulation of an actual ultrafast Ti:Sapphire laser. Moreover, we show how the pulse acquires and preserves a $smooth$ spectral phase in its propagation inside the cavity. These results are in very good agreement with experimental measurements of the spectrum outside the OC. A detailed experimental demonstration of pulse evolution as predicted by this model is presently under development. This work was partly supported by FCT Grant No. POCTI/FIS/48709/2002, Portuguese Ministry of Science, co-financed by FEDER. M. V. Tognetti’s e-mail address is marco.tognetti@fc.up.pt.
H. A. Haus, *IEEE J. Select. Topics Quantum Electron.* **6**, 1173 (2000).
U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, *Opt. Lett.* **24**, 411 (1999).
T. Fuji, A. Unterhuber, V. S. Yakovlev, G. Tempea, A. Stingl, F. Krausz, and W. Drexler, *Appl. Phys. B* **77**, 125 (2003).
I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. Zhou, and Ch.-P. Huang, *Opt. Lett.* **19**, 1465 (1994).
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Y. Chen, F. X. Kartner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, *J. Opt. Soc. Am. B*, [**16**]{}, 1999 (1999).
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H. Crespo, M. V. Tognetti, M. A. Cataluna, J. T. Mendonça, and A. dos Santos, to be published in Ultrafast Optics V, Springer.
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List of Figure Captions {#list-of-figure-captions .unnumbered}
=======================
Fig. 1 (a) Schematic diagram of the Ti:sapphire laser oscillator. (b) Total round trip intracavity GDD. (c) Total cavity gain $G_{tot}$ assuming a peak crystal gain of 1.04.
Fig. 2 Normalized intracavity spectrum (solid line) and phase (dotted line) before the OC after $(a)\,20$, $(b)\,500$, $(c)\,1000$ round trips inside the cavity. (d) Simulated (solid line) and measured (dashed line) final spectra outside the OC.
Fig. 3 Laser cavity dispersion map. $A,B,C,D,E$ and $F$ are reference points corresponding to the spectra, phases, and temporal profiles reported in figure\[figure\_4\_tognetti\_crespo\] (see the text for more details).
Fig. 4 Spectra, phases and temporal profiles corresponding to the reference points reported in figure\[figure\_3\_tognetti\_crespo\].
|
---
author:
- 'Elena Boguslavskaya, Yuliya Mishura, and Georgiy Shevchenko'
bibliography:
- 'wtbibl.bib'
title: 'Replication of Wiener-transformable stochastic processes with application to financial markets with memory'
---
Introduction
============
Consider a general continuous time market model with one risky asset. For simplicity, we will work with discounted values. Let the stochastic process $\{X_t,t\in [0,T]\}$ model the discounted price of risky asset. Then the discounted final value of a self-financing portfolio is given by a stochastic integral $$\label{wti:capital}
V^\psi(T) = V^\psi(0) + \int_0^T \psi(t) dX(t),$$ where an adapted process $\psi$ is the quantity of risky asset in the portfolio. Loosely speaking, the self-financing assumption means that no capital is withdrawn or added to the portfolio; for precise definition and general overview of financial market models with continuous time we refer a reader to [@bjork; @karat].
Formula raises several important questions of financial modeling, we will focus here on the following two.
- *Replication*: identifying random variables (i.e. discounted contingent claims), which can be represented as final capitals of some self-financing portfolios. In other words, one looks at integral representations $$\label{wti:represent}
\xi = \int_0^T \psi(t) dX(t)$$ with adapted integrand $\psi$; the initial value may be subtracted from $\xi$, so we can assume that it is zero.
- *Utility maximization*: maximizing the expected utility of final capital over some set of admissible self-financing portfolios.
An important issue is the meaning of stochastic integral in or . When the process $X$ is a semimartingale, it can be understood as Itô integral. In this case is a kind of Itô representation, see e.g. [@KS] for an extensive coverage of this topic. When the Itô integral is understood in some extended sense, then the integral representation may exist under very mild assumptions and may be non-unique. For example, if $X=W$, a Wiener process, and $\psi$ satisfies $\int_0^T \psi_s^2 ds<\infty$ a.s., then, as it was shown by [@dudley], any random variable can be represented as a final value of some self-financing portfolio for any value of initial capital.
However, empirical studies suggest that financial markets often exhibit long-range dependence (in contrast to stochastic volatility that can be both smooth and rough, i.e., can demonstrate both long-and short-range dependence). The standard model for the phenomenon of long-range dependence is the fractional Brownian motion with Hurst index $H>1/2$. It is not a semimartingale, so the usual Itô integration theory is not available. The standard approach now is to define the stochastic integral in such models as a pathwise integral, namely, one usually considers the fractional integral, see [@bender-sottinen-valkeila; @zahle].
The models based on the fractional Brownian motion usually admit arbitrage possibilities, i.e. there self-financing portfolios $\psi$ such that $V_\psi(0)\le 0$, $V_\psi(T)\ge 0$ almost surely, and $V_\psi(T)>0$ with positive probability. In the fractional Black–Scholes model, where $X_t=X_0\exp\{at+bB_t^H\}$, and $B^H$ is a fractional Brownian motion with $H>1/2$, the existence of arbitrage was shown in [@rogers]. Specifically, the strategy constructed there was of a “doubling” type, blowing the portfolio in the case of negative values; thus the potential intermediate losses could be arbitrarily large. It is worth to mention that such arbitrage exists even in the classical Black–Scholes model: the aforementioned result by Dudley allows gaining any positive final value of capital from initial zero by using a similar “doubling” strategy. For this reason, one usually restricts the class of admissible strategies by imposing a lower bound on the running value: $$\label{wti:eq:nds}
V^\psi(t)\ge -a,\quad t\in(0,T),$$ which in particular disallows the “doubling” strategies. However, in the fractional Black–Scholes model, the arbitrage exists even in the class of strategies satisfying , as was shown in [@Cheridito1].
There are several ways to exclude arbitrage in the fractional Brownian model. One possibility is to restrict the class of admissible strategies. For example, in [@Cheridito1] the absence of arbitrage is proved under further restriction that interval between subsequent trades is bounded from below (i.e. high frequency trading is prohibited). Another possibility is to add to the fractional Brownian motion an independent Wiener process, thus getting the so-called mixed fractional Brownian motion $M^H = B^H + W$. The absence in such mixed models was addressed in [@andrmish; @Cheridito]. In [@andrmish], it was shown that there is no arbitrage in the class of self-financing strategies $\gamma_t = f(t,M^H,t)$ of Markov type, depending only on the current value of the stock. In [@Cheridito], it was shown that for $H\in(3/4,1)$ the distribution of mixed fractional Brownian motion on a finite interval is equivalent to that of Wiener process. As a result, in such models there is no arbitrage strategies satisfying the non-doubling assumption . A more detailed exposition concerning arbitrage in models based on fractional Brownian motion is given in [@bender-sottinen-valkeila1].
The replication question, i.e. the question when a random variable can be represented as a pathwise (fractional) integral in the models with long memory was studied in many articles, even in the case where arbitrage opportunities are present. The first results were established in [@msv], where it was shown that a random variable $\xi$ has representation with respect to fractional Brownian motion if it is a final value of some Hölder continuous adapted process. The assumption of Hölder continuity might seem too restrictive at the first glance. However, the article [@msv] gives numerous examples of random variables satisfying this assumption.
The results of [@msv] were extended in [@shev-viita], where similar results were shown for a wide class of Gaussian integrators. The article [@mish-shev] extended them even further and studied when a combination of Hölder continuity of integrator and small ball estimates lead to existence of representation .
For the mixed fractional Brownian motion, the question of replication was considered in [@shev-viita]. The authors defined the integral with respect to fractional Brownian motion in pathwise sense and that with respect to Wiener process in the extended Itô sense and shown, similarly to the result of [@dudley], that any random variable has representation .
It is worth to mention that the representations constructed in [@msv; @mish-shev; @shev-viita] involve integrands of “doubling” type, so in particular they do not satisfy the admissibility assumption .
Our starting point for this article was to see what contingent claims are representable as final values of some Hölder continuous adapted processes. It turned out that the situation is quite transparent whenever the Gaussian integrator generates the same flow of sigma-fields as the Wiener process. As a result, we came up with the concept of Wiener-transformable financial market, which turned out to be a fruitful idea, as a lot of models of financial markets are Wiener-transformable. We consider many examples of such models in our paper. Moreover, the novelty of the present results is that we prove representation theorems that, in financial interpretation, are equivalent to the possibility of hedging of contingent claims, in the class of *bounded* strategies. While even with such strategies the non-doubling assumption may fail, the boundedness seems a feasible admissibility assumption.
More specifically, in the present paper we study a replication and the utility maximization problems for a broad class of asset prices processes, which are obtained by certain adapted transformation of a Wiener process; we call such processes *Wiener-transformable* and provide several examples. We concentrate mainly on non-semimartingale markets because the semimartingale markets have been studied thoroughly in the literature. Moreover, the novelty of the present results is that we prove representation theorems that, in financial interpretation, are equivalent to the possibility of hedging of contingent claims, in the class of bounded strategies. We would like to draw the attention of the reader once again to the fact that the possibility of representation means that we have arbitrage possibility in the considered class of strategies and they may be limited, although in a narrower and more familiar class of strategies the market can be arbitrage-free. Therefore, our results demonstrate rather subtle differences in the properties of markets in different classes of strategies.
The article is organized as follows. In Section \[wti:sec:2\], we recall basics of pathwise integrations in the fractional sense. In Section \[wti:sec:3\], we prove a new representation result, establishing an existence of integral representation with bounded integrand, which is of particular importance in financial applications. We also define the main object of study, Wiener-transformable markets, and provide several examples. Section \[wti:sec:4\] is devoted to application of representation results to the utility maximization problems.
Elements of fractional calculus {#wti:sec:2}
===============================
As announced in the introduction, the integral with respect to Wiener-transformable processes will be defined in pathwise sense, as fractional integral. Here we present the basic facts on fractional integration; for more details see [@samko; @zahle]. Consider functions $f,g:[0,T]\rightarrow \mathbb{R}$, and let $[a,b]\subset [0,T]$. For $\alpha\in (0,1)$ define Riemann-Liouville fractional derivatives on finite interval $[a,b]$ $$\begin{gathered}
\big(\mathcal{D}_{a+}^{\alpha}f\big)(x)=\frac{1}{\Gamma(1-\alpha)}\bigg(\frac{f(x)}{(x-a)^\alpha}+\alpha
\int_{a}^x\frac{f(x)-f(u)}{(x-u)^{1+\alpha}}du\bigg)1_{(a,b)}(x),\end{gathered}$$ $$\begin{gathered}
\label{wti:equ:dif}\big(\mathcal{D}_{b-}^{ \alpha}g\big)(x)=\frac{1} {\Gamma(1-\alpha)}\bigg(\frac{g(x)}{(b-x)^{ \alpha}}+ \alpha
\int_{x}^b\frac{g(x)-g(u)}{(u-x)^{1+\alpha}}du\bigg)1_{(a,b)}(x).\end{gathered}$$ Assuming that $\mathcal{D}_{a+}^{\alpha}f\in L_1[a,b]$, $\mathcal{D}_{b-}^{1-\alpha}g_{b-}\in
L_\infty[a,b]$, where $g_{b-}(x) = g(x) - g(b)$, the generalized Lebesgue–Stieltjes integral is defined as $$\int_a^bf(x)dg(x)= \int_a^b\big(\mathcal{D}_{a+}^{\alpha}f\big)(x)
\big(\mathcal{D}_{b-}^{1-\alpha}g_{b-}\big)(x)dx.$$
Let function $g$ be $\theta$-Hölder continuous, $g\in C^\theta[a,b]$ with $\theta\in(\frac12,1)$, i.e. $$\sup_{t,s\in[0,T],t\neq s}\frac{{\left|g(t)-g(s)\right|}}{{\left|t-s\right|}^\theta}<\infty.$$ In order to integrate w.r.t. function $g$ and to find an upper bound of the integral, fix some $\alpha \in(1-\theta,1/2)$ and introduce the following norm: $$\begin{gathered}
\|f\|_{\alpha,[a,b]} = \int_a^b \left(\frac{|{f(s)}|}{(s-a)^\alpha} + \int_a^s \frac{|{f(s)-f(z)}|}{(s-z)^{1+\alpha}}dz\right)ds.\end{gathered}$$ For simplicity we abbreviate $\|\cdot\|_{\alpha,t} = \|\cdot\|_{\alpha,[0,t]}$. Denote $$\Lambda_\alpha(g):= \sup_{0\le s<t\le T} |{\mathcal{D}_{t-}^{1-\alpha}g_{t-}}(s)|.$$ In view of Hölder continuity, $\Lambda_\alpha(g)<\infty$.
Then for any $t\in(0,T]$ and for any $f$ with $\|f\|_{\alpha,t}<\infty$, the integral $\int_0^t f(s) dg(s)$ is well defined as a generalized Lebesgue–Stieltjes integral, and the following bound is evident: $$\begin{gathered}
\label{wti:equ:ineq}
\Big|{\int_0^t f(s)dg(s)}\Big|\le \Lambda_\alpha(g) \|f\|_{\alpha,t}.\end{gathered}$$ It is well known that in the case if $f$ is $\beta$-Hölder continuous, $f\in C^\beta[a,b]$, with $\beta+\theta>1$, the generalized Lebesgue–Stieltjes integral $\int_a^bf(x)dg(x)$ exists, equals to the limit of Riemann sums and admits bound for any $\alpha \in(1-\theta, \beta\wedge 1/2)$.
Representation results for Gaussian and Wiener-transformable processes {#wti:sec:3}
======================================================================
Let throughout the paper $(\Omega, \mathcal{F}, { {\bf P}})$ be a complete probability space supporting all stochastic processes mentioned below. Let also $\mathbb{F} = \{\mathcal F_t,t\in[0,T]\}$ be a filtration satisfying standard assumptions. In what follows, the adaptedness of a process $X = \{X(t),t\in[0,T]\}$ will be understood with respect to $\mathbb{F}$, i.e. $X$ will be called adapted if for any $t\in[0,T]$, $X(t)$ is $\mathcal{F}_t$-measurable.
We start with representation results, which supplement those of [@mish-shev].
Consider a continuous centered Gaussian process $G$ with incremental variance of $G$ satisfying the following two-sided power bounds for some $H\in (1/2,1)$.
- There exist $C_1, C_2>0$ such that for any $s,t\in [0,T]$ $$\label{wti:eq:helix}
C_1\left|t-s\right|^{2H}\le{ {\bf E}}\left|G(t)-G(s)\right|^2\le C_2 \left|t-s\right|^{2H}.$$
Assume additionally that the increments of $G$ are positively correlated. More exactly, let the following condition hold
- For any $0 \le s_1 \le t_1 \le s_2 \le t_2\le T$ $${ {\bf E}}\left(G({t_1})-G({s_1})\right)\left(G({t_2})-G({s_2})\right)\ge0.$$
A process satisfying is often referred to as a *quasi-helix*.
Note that the right inequality in implies that $$\label{wti:eq:Gmodcont}
\sup_{t,s\in[0,T]}\frac{|G(t)-G(s)|}{|t-s|^{H}|\log(t-s)|^{1/2}} <\infty$$ almost surely (see e.g. p. 220 in [@lifshits]).
We will need the following small deviation estimate for sum of squares of Gaussian random variables, see e.g. [@lishao].
\[wti:lem:small\] Let $\{\xi_i\}_{i=1,\ldots,n}$ be jointly Gaussian centered random variables. For all $x$ such that $0<x<\sum_{i=1}^n { {\bf E}}\xi^2_i$, it holds $$\begin{gathered}
{\mathbf{P}\left(\sum_{i=1}^n\xi_i^2\le x\right)}\leq\exp\left\{-\frac{\left(\sum_{i=1}^n { {\bf E}}{\xi_i^2}-x\right)^2}{\sum_{i,j=1}^n ({ {\bf E}}{\xi_i\xi_j})^2}\right\}.\end{gathered}$$
\[wti:thm:representation\] Let a centered Gaussian process $G$ satisfy $(A)$ and $(B)$ and $\xi$ be a random variable such that there exists an adapted $r$-Hölder continuous process $Z$ with $Z(T) = \xi$. There exists a bounded adapted process $\psi$, such that $\left\|\psi\right\|_{\alpha,T}<\infty$ for some $\alpha\in \left(1-H,1\right)$ and $\xi$ admits the representation $$\label{wti:reprez}
\xi=\int_{0}^{T}\psi(s) dG(s),$$ almost surely.
A similar result was proved in [@mish-shev], Theorem 4.1, which assumed with different exponents in the right-hand side and in the left-hand side of the inequality. Having equal exponents allowed us to establish existence of a *bounded* integrand $\psi$, thus extending previous results.
To construct an integrand, we modify ideas of [@mish-shev] and [@shalaiko]. Throughout the proof, $C$ will denote a generic constant, while $C(\omega)$, a random constant; their values may change between lines.
Choose some $\alpha \in \big(1-H ,(r+1-H)\wedge \frac12\big)$.
We start with the construction of $\psi$. First take some $\theta\in (0,1)$, put $t_n = T-\theta^{n}$, $n\ge 1$, and let $\Delta_n = t_{n+1}-t_n$. It is easy to see that $$\begin{gathered}
T-t_n\le C\Delta_n.
\label{wti:t_n-ineq}\end{gathered}$$ Denote for brevity $\xi_n = Z(t_n)$. Then by Assumption 1, ${\left|\xi_n -\xi_{n+1}\right|}\le C(\omega) \theta^{rn}$. Therefore, there exists some $N_0 = N_0(\omega)$ such that $$\label{wti:eq:deltaxi}
{\left|\xi_n -\xi_{n+1}\right|}\le n \theta^{rn}$$ for all $n\ge N_0(\omega)$.
We construct the integrand $\psi$ inductively between the points $\{t_n,n\ge 1\}$. First let $\psi(t)=0$, $t\in[0,t_1]$. Assuming that we have already constructed $\psi(t)$ on $[0,t_n)$, define $V(t)=\int_0^t \psi(s)dG(s), t\in[0,t_n]$.
Consider some cases.
$V(t_n)\neq \xi_{n-1}$. By Lemma 4.1 in [@mish-shev], there exists an adapted process $\{\phi_n(t),t\in[t_n,t_{n+1}]\}$, bounded on $[t_n,t]$ for any $t\in(t_n,t_{n+1})$ and such that $\int_{t_n}^{t}\phi_n(s) d G(s)\to +\infty$ as $t\to t_{n+1}-$. Define a stopping time $$\begin{gathered}
\tau_n=\inf\left\{t\geq t_n: \int_{t_n}^{t}\phi_n(s) dG(s)\geq |\xi_n-V_{t_n}| \right\},\end{gathered}$$ and set $$\begin{gathered}
\psi(t)=\phi_n(t) {\operatorname{sign}}\big(\xi_n-V(t_n)\big){\mathbf{1}}_{[t_n,\tau_n]}(t), \,t\in[t_n,t_{n+1}).\end{gathered}$$ It is obvious that $\int_{t_n}^{t_{n+1}}\psi(s)dG(s)=\xi_n-V(t_n)$ and $V(t_{n+1})=\xi_n$.
$V(t_n)=\xi_{n-1}$. We consider a uniform partition $s_{n,k} = t_n + k\delta_n$, $k=1,\ldots,n$ of $[t_n,t_{n+1}]$ with a mesh $\delta_n=\Delta_n/n$ and an auxiliary function $$\begin{gathered}
\phi_n(t)=a_n\sum_{k=0}^{n-1}
\big(G(t)-G(s_{n,k})\big){\mathbf{1}}_{[s_{n,k},s_{n,k+1})}(t),\end{gathered}$$ where $a_n = n^{-2}\theta^{(\alpha-H-1)n}$. Since $\phi_n$ is piecewise Hölder continuous of order up to $H$, by the change of variables formula (Theorem 4.3.1 in [@zahle]) $$\int_{t_n}^{t_{n+1}} \phi_n(t) dG(t) = a_n \sum_{k=0}^{n-1} \big(G(s_{n,k+1})-G(s_{n,k})\big)^2.$$
Define a stopping time $$\begin{gathered}
\sigma_n=\inf\Big\{t\geq t_n: \int_{t_n}^t \phi_n(s)dG(s)\geq |\xi_n-\xi_{n-1}|\Big\}\wedge t_{n+1},\end{gathered}$$ and set $$\begin{gathered}
\psi(t)={\operatorname{sign}}(\xi_n-\xi_{n-1})\phi_n(t){\mathbf{1}}_{[t_n,\sigma_n]}(t),\, t\in[t_n,t_{n+1}).\end{gathered}$$ Now we want to ensure that, almost surely, $V(t_{n}) = \xi_{n-1}$ for all $n$ large enough. By construction, Case I is always succeeded by Case II. So we need to ensure that $\sigma_n <t_{n+1}$ for all $n$ large enough, equivalently, that $$a_n \sum_{k=0}^{n-1} \big(G(s_{n,k+1})-G(s_{n,k})\big)^2> {\left|\xi_n - \xi_{n-1}\right|}.$$ Thanks to , it is enough to ensure that $$\sum_{k=0}^{n-1} \big(G(s_{n,k+1})-G(s_{n,k})\big)^2 > a_n^{-1}n\theta^{rn} = n^2 \theta^{(r+H +1-\alpha)n}$$ for all $n$ large enough. Define $\xi_k = G(s_{n,k+1})-G(s_{n,k})$, $k=0,\dots,n-1$. Thanks to our choice of $\alpha$, $r+H+1-\alpha > 2H$, so $n^2 \theta^{(r+H +1-\alpha)n} < C_1 n^{1-2H}\theta^{2Hn}$ for all $n$ large enough. Therefore, in view of , $$\begin{gathered}
\sum_{k=0}^{n-1} { {\bf E}}\xi_k^2 \ge C_1 n \delta_n^{2H} = C_1 n^{1-2H}\theta^{2H n} > n^2 \theta^{(r+H +1-\alpha)n},\end{gathered}$$ so we can use Lemma \[wti:lem:small\]. Using $(A)$ and $(B)$, estimate $$\begin{gathered}
\sum_{i,j=0}^{n-1} \big({ {\bf E}}\xi_i\xi_j \big)^2\le \max_{0\le i,j\le n-1} { {\bf E}}\xi_i\xi_j \sum_{i,j=0}^{n-1} { {\bf E}}\xi_i\xi_j\\
\le C_1 \delta_n^{2H} { {\bf E}}\Big( \sum_{i=0}^{n-1} \xi_i\Big)^2 = C_1 \delta_n^{2H} { {\bf E}}\big(G(t_{n+1}) - G(t_n) \big)^2\\
\le C_1^2 \delta_n^{2H}\Delta_n^{2H}\le C_1^2 n^{-2H} \Delta^{4H} = C_1^2 n^{-2H} \theta^{4H n}.\end{gathered}$$ Hence, by Lemma \[wti:lem:small\], $$\begin{gathered}
{\mathbf{P}\left( \sum_{k=0}^{n-1} \big(G(s_{n,k+1})-G(s_{n,k})\big)^2 \le n^2 \theta^{(r+H +1-\alpha)n}\right)}\\
\le \exp\left\{ - \frac{\big(C_1 n^{1-2H}\theta^{2H n} - n^2 \theta^{(r+H +1-\alpha)n}\big)^2}{C_1^2 n^{-2H} \theta^{4H n}}\right\}
\le \exp \left\{ - C n^{2 -2H}\right\}.\end{gathered}$$ Therefore, by the Borel–Cantelli lemma, almost surely there exists some $N_1(\omega)\ge N_0(\omega)$ such that for all $n\ge N_1(\omega)$ $$\sum_{k=0}^{n-1} \big(G(s_{n,k+1})-G(s_{n,k})\big)^2 > n^2 \theta^{(r+H +1-\alpha)n},$$ so, as it was explained above, we have $V(t_n) = \xi_{n-1}$, $n\ge N_1(\omega)$.
Since all functions $\phi_n$ are bounded, we have that $\psi$ is bounded on $[0,t_N]$ for any $N\ge 1$. Further, thanks to , for $t\in[t_n,t_{n+1}]$ with $n\ge N_1(\omega)$, $$\label{wti:eq:psibound}
\begin{gathered}
{\left|\psi(s)\right|}\le C(\omega) a_n \delta_n^H{\left|\log \delta_n\right|}^{1/2} \le C(\omega) n^{-2}\theta^{(\alpha-H-1)n} n^{-H} \theta^{Hn} n^{1/2}\\ = C(\omega) n^{\alpha - H - 3/2}\theta^{(\alpha-1)n}.
\end{gathered}$$ Therefore, $\psi$ is bounded (moreover, $\psi(t)\to 0$, $t\to T-$).
Further, by construction, ${\left\|\psi\right\|}_{\alpha,t_N}<\infty$ for any $N\ge 1$. Moreover, ${\left|V(t)-\xi_{N-1}\right|}\le {\left|\xi_N - \xi_{N-1}\right|}$, $t\in[t_{N},t_{N+1}]$. Thus, it remains to to verify that ${\left\|\psi\right\|}_{\alpha,[t_N,1]}<\infty$ and $\int_{t_N}^1 \psi(s) dG(s)\to 0$, $N\to\infty$, which would follow from ${\left\|\psi\right\|}_{\alpha,[t_N,1]}\to 0$, $N\to\infty$.
Let $N\ge N_1(\omega)$. Write $${\left\|\psi\right\|}_{\alpha,[t_N,T]} = \sum_{n=N}^\infty \int_{t_n}^{t_{n+1}}\left(\frac{|\psi(s)|}{(s-t_N)^{\alpha}}+\int_{t_N}^s \frac{|\psi(s)-\psi(u)|}{|s-u|^{1+\alpha}}du\right) ds.$$ Thanks to , $$\begin{gathered}
\int_{t_n}^{t_{n+1}}\frac{|\psi(s)|}{(s-t_N)^{\alpha}} ds
\le
C(\omega)\Delta_n^{1-\alpha}n^{\alpha - H - 3/2}\theta^{(\alpha-1)n}
= C(\omega)n^{\alpha - H - 3/2}.\end{gathered}$$ Further, $$\begin{gathered}
\int_{t_N}^{t_{n+1}}\int_{t_n}^s\frac{|\psi(s)-\psi(u)|}{|s-u|^{1+\alpha}}du\, ds\\ =
\sum_{k=1}^n \int_{s_{n,k-1}}^{s_{n,k}}\left(\int_{t_N}^{t_n}+\int_{t_n}^{s_{n,k-1}}+\int_{s_{n,k-1}}^{s} \right)\frac{|\psi(s)-\psi(u)|}{|s-u|^{1+\alpha}}du\, ds=:I_1+I_2+I_3.\end{gathered}$$ Start with $I_1$, observing that $\psi$ vanishes on $(\sigma_n,t_{n+1}]$: $$\begin{gathered}
I_1\leq \int_{t_n}^{t_{n+1}}\sum_{j=N}^n \int_{t_{j-1}}^{t_j}\frac{|\psi(s)|+|\psi(u)|}{|s-u|^{1+\alpha}}du\, ds\\
\leq C(\omega) n^{\alpha - H - 3/2}\theta^{(\alpha-1)n}
\int_{t_n}^{t_{n+1}}(s-t_n)^{-\alpha}ds\\
+ C(\omega)\sum_{j=N}^{n-1} j^{\alpha-H-3/2}\theta^{(\alpha-1)j} \int_{t_n}^{t_{n+1}}(s-t_{j+1})^{-\alpha}ds\\
\leq C(\omega)n^{\alpha - H - 3/2}\theta^{(\alpha-1)n}\Delta_n^{1-\alpha} + C(\omega) \sum_{j=N}^{n-1}j^{\alpha-H-3/2}\theta^{(\alpha-1)j} \Delta_n^{1-\alpha}\\
= C(\omega)n^{\alpha - H - 3/2} + C(\omega) \sum_{j=N}^{n-1}j^{\alpha-H-3/2}\theta^{(\alpha-1)(j-n)}.\end{gathered}$$ Similarly, $$\begin{gathered}
I_2\leq C(\omega)n^{\alpha - H - 3/2}\theta^{(\alpha-1)n} \sum_{k=1}^n \int_{s_{n,k-1}}^{s_{n,k}}\int_{t_n}^{s_{n,k-1}}|s-u|^{-1-\alpha}du\, ds \\
\le C(\omega)n^{\alpha - H - 3/2}\theta^{(\alpha-1)n} \sum_{k=1}^n \int_{s_{n,k-1}}^{s_{n,k}} (s-s_{n,k-1})^{-\alpha}ds\\
\le C(\omega)n^{\alpha - H - 3/2}\theta^{(\alpha-1)n} n \delta_n^{1-\alpha}=C(\omega)n^{2\alpha - H - 3/2}.\end{gathered}$$ Finally, assuming that $\sigma_n\in [s_{n,l-1},s_{n,l})$, $$\begin{gathered}
I_3\leq C(\omega)\sum_{k=1}^{l-1} \int_{s_{n,k-1}}^{s_{n,k}}\int_{s_{n,k-1}}^s a_n\frac{(s-u)^{H}|\log (s-u)|^{1/2}}{(s-u)^{1+\alpha}}du\, ds\\
+ \int_{s_{n,l-1}}^{\sigma_n}\int_{s_{n,l-1}}^s\frac{|\psi(s)-\psi(u)|}{|s-u|^{1+\alpha}}du\, ds+\int_{\sigma_n}^{s_{n,l}}\int_{s_{n,l-1}}^{\sigma_n}\frac{|\psi(s)-\psi(u)|}{|s-u|^{1+\alpha}}du\, ds\\
\le C(\omega)a_n\sum_{k=1}^n \int_{s_{n,k-1}}^{s_{n,k}}(s-s_{n,k-1})^{H-\alpha}|\log(s-s_{n,k-1})|^{1/2}ds\\
+ C(\omega) n^{\alpha - H - 3/2}\theta^{(\alpha-1)n}\int_{\sigma_n}^{s_{n,l}}\int_{s_{n,l-1}}^{\sigma_n}\frac{1}{|s-u|^{1+\alpha}}du\, ds\\ \leq
C(\omega)a_n n\delta_n^{H+1-\alpha}|\log \delta_n|^{1/2} + C(\omega) n^{\alpha - H - 3/2}\theta^{(\alpha-1)n}\delta_n^{-\alpha}\\
= C(\omega)n^{\alpha-H-3/2} + C(\omega)n^{2\alpha - H - 3/2}\le C(\omega)n^{2\alpha - H - 3/2}.\end{gathered}$$
Gathering all estimates we get $$\begin{gathered}
\int_{t_N}^1 |D^\alpha_{t_N+}(\psi)(s)|ds
\leq C(\omega)\sum_{n=N}^\infty \Big(n^{2\alpha - H - 3/2} + \sum_{j=N}^{n-1}j^{\alpha-H-3/2}\theta^{(\alpha-1)(j-n)}\Big)\\
\le C(\omega)\Big( N^{2\alpha - H-1/2} + \sum_{j=N}^
\infty j^{\alpha-H-3/2}\sum_{n=j+1}^\infty \theta^{(1-\alpha)(n-j)} \Big)\\
\le C(\omega) N^{2\alpha - H-1/2},\end{gathered}$$ which implies that ${\left\|\psi\right\|}_{\alpha,[t_N,T]}\to 0$, $N\to\infty$, finishing the proof.
Now we turn to the main object of this article.
\[wti:def1\] A Gaussian process $G=\{G(t), t\in{{\mathbb{R}}}^+\}$ is called $m$-Wiener-transformable if there exists $m$-dimensional Wiener process $W=\{W(t), t\in{{\mathbb{R}}}^+\}$ such that $G$ and $W$ generate the same filtration, i.e. for any $t\in{{\mathbb{R}}}^+$ $$\mathcal{F}_t^G=\mathcal{F}_t^W.$$ We say that $G$ is $m$-Wiener-transformable to $W$ (evidently, process $W$ can be non-unique.)
- In the case when $m=1$ we say that the process $G$ is Wiener-transformable.
- Being Gaussian so having moments of any order, $m$-Wiener-transformable process admits at each time $t\in{{\mathbb{R}}}^+$ the martingale representation $G(t)={ {\bf E}}(G(0))+\sum_{i=1}^m\int_0^tK_i(t,s)dW_i(s),$ where $K_i(t,s)$ is $\mathcal{F}_s^W$-adapted for any $0\leq s\leq t$ and $\int_0^t{ {\bf E}}(K_i(t,s))^2ds<\infty$ for any $t\in{{\mathbb{R}}}^+$.
Now let the random variable $\xi$ be ${{\mathcal{F}}}_T^W$-measurable, ${ {\bf E}}\xi^2<\infty$. Then in view of martingale representation theorem, $\xi$ can be represented as $$\label{wti:mart-repr}
\xi = { {\bf E}}\xi + \int_0^T \vartheta(t) dW(t),$$ where $\vartheta$ is an adapted process with $\int_0^T { {\bf E}}\vartheta(t)^2 dt<\infty$.
As it was explained in introduction, we are interested when $\xi$ can be represented in the form $$\xi = \int_0^T \psi(s) dG(s),$$ where the integrand is adapted, and the integral is understood in the pathwise sense.
\[wti:thm1\] Let the following conditions hold.
- Gaussian process $G$ satisfies condition $(A)$ and $(B)$.
- Stochastic process $\vartheta$ in representation satisfies $$\label{wti:thetaassump}
\int_{0}^{T}|\vartheta(s)|^{2p}ds<\infty$$ a.s. with some $p> 1$.
Then there exists a bounded adapted process $\psi$ such that $\left\|\psi\right\|_{\alpha,T}<\infty$ for some $\alpha\in \left(1-H,\frac{1}{2}\right)$ and $\xi$ admits the representation $$\xi=\int_{0}^{T}\psi(s) dG(s),$$ almost surely.
As it was mentioned in [@mish-shev], it is sufficient to require the properties $(A)$ and $(B)$ to hold on some subinterval $[T-\delta,T]$. Similarly, it is enough to require in $(ii)$ that $\int_{T-\delta}^T |\vartheta(t)|^{2p}dt<\infty$ almost surely.
First we prove a simple result establishing Hölder continuity of Itô integral.
\[wti:lem1\] Let $\vartheta=\{\vartheta(t), t\in [0,T]\}$ be a real-valued progressively measurable process such that for some $p\in(1,+\infty]$ $$\int_{0}^{T}|\vartheta(s)|^{2p}ds<\infty$$ a.s. Then the stochastic integral $\int_{0}^{t}\vartheta(s)dW(s)$ is Hölder continuous of any order up to $\frac{1}{2}-\frac{1}{2p}$.
First note that if there exist non-random positive constants $a,C$ such that for any $s,t\in [0,T]$ with $s<t$ $$\int_{s}^{t}\vartheta^2(u)du \le C(t-s)^a,$$ then $\int_0^t \vartheta(s)dW(s)$ is Hölder continuous of any order up to $a/2$. Indeed, in this case by the Burkholder inequality, for any $r>1$ and $s,t\in [0,T]$ with $s<t$ $${ {\bf E}}\left| \int_s^t \vartheta(u) dW(u)\right|^r\le C_r { {\bf E}}\left( \int_s^t \vartheta^2(u) du\right)^{r/2} \le C (t-s)^{ar/2},$$ so by the Kolmogorov–Chentsov theorem, $\int_0^t \vartheta(s)dW(s)$ is Hölder continuous of order $\frac{1}{r}(\frac{ar}2-1) = \frac{a}2 - \frac{1}{2r}$. Since $r$ can be arbitrarily large, we deduce the claim.
Now let for $n\ge 1$, $\vartheta_n(t) = \vartheta(t)\mathbf{1}_{\int_0^t |\vartheta(s)|^{2p} ds\le n}$, $t\in[0,T]$. By the Hölder inequality, for any $s,t\in [0,T]$ with $s<t$ $$\int_s^t \vartheta_n^2(u)du \le (t-s)^{1-1/p}\left(\int_s^t |\vartheta(u)|^{2p} du\right)^{1/p} \le n^{1/p}(t-s)^{1-1/p}.$$ Therefore, by the above claim, $\int_0^t \vartheta_n(s)dW(s)$ is a.s. Hölder continuous of any order up to $\frac12 - \frac{1}{2p}$. However, $\vartheta_n$ coincides with $\vartheta$ on $\Omega_n = \{\int_0^T |\vartheta(t)|^{2p} dt\le n\}$. Consequently, $\int_0^t \vartheta_n(s)dW(s)$ is a.s. Hölder continuous of any order up to $\frac12 - \frac{1}{2p}$ on $\Omega_n$. Since ${ {\bf P}}(\bigcup_{n\ge 1} \Omega_n) = 1$, we arrive at the statement of the lemma.
**Proof of Theorem \[wti:thm1\].** Define $$Z(t) = { {\bf E}}\xi + \int_0^t \vartheta(s) dW(s).$$ This is an adapted process with $Z(T) = \xi$, moreover, it follows from Lemma \[wti:lem1\] that $Z$ is Hölder continuous of any order up to $\frac{1}{2}- \frac{1}{2p}$. Thus, the statement follows from Theorem \[wti:thm:representation\]. In the case where one looks at improper representation, no assumptions on $\xi$ are needed.
\[wti:thm2\] (Improper representation theorem) Assume that an adapted Gaussian process $G=\{G(t)$, $t\in[0,T]\}$ satisfies conditions $(A),(B)$. Then for any random variable $\xi$ there exists an adapted process $\psi$ that $\left\|\psi\right\|_{\alpha,t}<\infty$ for some $\alpha\in \left(1-H,\frac{1}{2}\right)$ and any $t\in[0,T)$ and $\xi$ admits the representation $$\xi=\lim_{t\to T-}\int_{0}^{t}\psi(s) dG(s),$$ almost surely.
The proof is exactly the same as for Theorem 4.2 in [@shev-viita], so we just sketch the main idea.
Consider an increasing sequence of points $\{t_n,n\ge 1\}$ in $[0,T)$ such that $t_n\to T$, $n\to\infty$, and let $\{\xi_n,n\ge 1\}$ be a sequence of random variables such that $\xi_n$ is $\mathcal{F}_{t_n}$-measurable for each $n\ge 1$, and $\xi_n\to \xi$, $n\to\infty$, a.s. Set for convenience $\xi_0 = 0$. Similarly to Case I in Theorem \[wti:thm:representation\], for each $n\ge 1$, there exists an adapted process $\{\phi_n(t),t\in[t_n,t_{n+1}]\}$, such that $\int_{t_n}^{t}\phi_n(s) d G(s)\to +\infty$ as $t\to t_{n+1}-$. For $n\ge 1$, define a stopping time $$\begin{gathered}
\tau_n=\inf\left\{t\geq t_n: \int_{t_n}^{t}\phi_n(s) dG(s)\geq |\xi_n-\xi_{n-1}| \right\}\end{gathered}$$ and set $$\begin{gathered}
\psi(t)=\phi_n(t) {\operatorname{sign}}\big(\xi_n-\xi_{n-1}\big){\mathbf{1}}_{[t_n,\tau_n]}(t), \,t\in[t_n,t_{n+1}).\end{gathered}$$ Then for any $n\ge 1$, we have $\int_{0}^{t_{n+1}}\psi(s)dG(s)=\xi_n$ and $\int_{0}^{t}\psi(s)dG(s)$ lies between $\xi_{n-1}$ and $\xi_n$ for $t\in[t_{n-1},t_n]$. Consequently, $\int_{0}^{t}\psi(s) dG(s)\to \xi$, $t\to T-$, a.s., as required.
Further we give several examples of Wiener-transformable Gaussian processes satisfying conditions $(A)$ and $(B)$ (for more detail and proofs see, e.g. [@mish-shev]) and formulate the corresponding representation results.
Fractional Brownian motion
--------------------------
Fractional Brownian motion $B^H$ with Hurst parameter $H\in(0,1)$ is a centered Gaussian process with the covariance $${ {\bf E}}B^H(t)B^H(s) = \frac{1}{2}\left(t^{2H}+s^{2H}-|t-s|^{2H}\right);$$ an extensive treatment of fractional Brownian motion is given in [@Mish]. For $H=\frac12$, fractional Brownian motion is a Wiener process; for $H\neq \frac12$ it is Wiener-transformable to the Wiener process $W$ via relations $$\label{wti:fbmviawin}
B^H(t)=\int_0^t K^H(t,s) dW(s)$$ and $$\label{wti:winviafbm}
W(t)=\int_0^t k^H(t,s)dB^H(s),$$ see e.g. [@norros].
Fractional Brownian motion with index $H\in(0,1)$ satisfies condition $(A)$ and satisfies condition $(B)$ if $H\in(\frac{1}{2},1)$.
Therefore, a random variable satisfying with any $p>1$ admits the representation .
Fractional Ornstein–Uhlenbeck process
-------------------------------------
Let $H\in (\frac{1}{2},1)$. Then the fractional Ornstein–Uhlenbeck process $Y=\{Y(t), t\ge 0\}$, involving fractional Brownian component and satisfying the equation $$Y(t)=Y_0+\int_0^t(b-aY(s))ds+\sigma B^H(t),$$ where $a,b\in{{\mathbb{R}}}$ and $\sigma>0$, is Wiener-transformable to the same Wiener process as the underlying fBm $B^H$.
Consider a fractional Ornstein–Uhlenbeck process of the simplified form $$Y(t) = Y_0 + a \int_0^t Y(s) ds + B^H(t), \mbox{ } t \geq 0.$$ It satisfies condition $(A)$; if $a>0$, it satisfies condition $(B)$ as well.
As it was mentioned in [@mish-shev], the representation theorem is valid for a fractional Ornstein-Uhlenbeck process with a negative drift coefficient too. Indeed, we can annihilate the drift of the fractional Ornstein-Uhlenbeck process with the help of Girsanov theorem, transforming a fractional Ornstein-Uhlenbeck process with negative drift to a fractional Brownian motion $\widetilde{B}^H$. Then, assuming , we represent the random variable $\xi$ as $\xi=\int_0^T\psi(s)d\widetilde{B}^H(s)$ on the new probability space. Finally, we return to the original probability space. Due to the pathwise nature of integral, its value is not changed upon changes of measure.
Subfractional Brownian motion
-----------------------------
Subfractional Brownian motion with index $H$, that is a centered Gaussian process $G^H=\left\{G^H(t), t \geq 0 \right\}$ with covariance function $${ {\bf E}}G^H(t) G^H(s) = t^{2H}+s^{2H} -\frac{1}{2}\left(|t+s|^{2H} + |t-s|^{2H} \right),$$ satisfies condition $(A)$ and condition $(B)$ for $H\in(\frac{1}{2},1)$.
Bifractional Brownian motion
----------------------------
Bifractional Brownian motion with indices $A \in (0,1)$ and $K \in (0,1)$, that is a centered Gaussian process with covariance function $${ {\bf E}}G^{A,K}(t) G^{A,K}(s) = \frac{1}{2^K} \left( \left(t^{2A}+s^{2A}\right)^K - |t-s|^{2AK}\right),$$ satisfies condition $(A)$ with $H = AK$ and satisfies condition $(B)$ for $AK>\frac{1}{2}$.
Geometric Brownian motion
-------------------------
Geometric Brownian motion involving the Wiener component and having the form $$S=\left\{S(t)=S(0)\exp\left\{\mu t+\sigma W(t)\right\}, \;\; t\ge 0 \right\},$$ with $S(0)>0$, $\mu\in{{\mathbb{R}}}$, $\sigma>0$, is Wiener-transformable to the underlying Wiener process $W$. However, it does not satisfy the assumptions of Theorem \[wti:thm1\]. One should appeal here to the standard semimartingale tools, like the martingale representation theorem.
Linear combination of fractional Brownian motions
-------------------------------------------------
Consider a collection of Hurst indices $\frac{1}{2}\le H_1< H_2<\ldots<H_m<1$ and independent fractional Brownian motions with corresponding Hurst indices $H_i$, $1\le i \le m$. Then the linear combination $\sum_{i=1}^{m}a_iB^{H_i}$ is $m$-Wiener-transformable to the Wiener process $W=(W_1,\ldots,W_m)$, where $W_i$ is such Wiener process to which fractional Brownian motion $B^{H_i}$ is Wiener-transformable. In particular, the mixed fractional Brownian motion $M^H=W+B^H$, introduced in [@Cheridito], is $2$-Wiener-transformable.
The linear combination $\sum_{i=1}^{m}a_iB^{H_i}$ satisfies condition $(A)$ with $H=H_1$, and condition $(B)$ whenever $H_1>1/2$.
We note that in the case of mixed fractional Brownian motion, the existence of representation cannot be derived from Theorem \[wti:thm1\], as we have $H = \frac12$ in this case. By slightly different methods, it was established in [@shev-viita] that arbitrary $\mathcal{F}_T$-measurable random variable $\xi$ admits the representation $$\xi = \int_0^T \psi(s) d\big(B^H(s) + W(s)\big),$$ where the integral with respect to $B^H$ is understood, as here, in the pathwise sense, the integral with respect to $W$, in the extended Itô sense. In contrast to Theorem \[wti:thm:representation\], we can not for the moment establish this result for the bounded strategies. Therefore, it would be interesting to study which random variables have representations with bounded $\psi$ in the mixed model.
Volterra process
----------------
Consider Volterra integral transform of Wiener process, that is the process of the form $G(t) = \int_0^t K(t,s) dW(s)$ with non-random kernel $K(t, \cdot) \in L_2[0,t]$ for $t\in[0,T]$. Let the constant $r\in[0,1/2)$ be fixed. Let the following conditions hold.
- The kernel $K$ is non-negative on $[0,T]^2$ and for any $s\in [0,T]$ $K(\cdot,s)$ is non-decreasing in the first argument;
- There exist constants $D_i>0, i=2,3$ and $H\in(1/2,1)$ such that $$|K(t_2,s) - K(t_1,s)| \leq D_2 |t_2-t_1|^{H}s^{-r},\quad s, t_1,t_2 \in [0,T]$$ and $$\ K(t,s)\leq D_3(t-s)^{H-1/2}s^{-r};$$
and at least one of the following conditions
- There exist constant $D_1>0$ such that $$D_1|t_2-t_1|^{H}s^{-r}\leq|K(t_2,s) - K(t_1,s)|,\quad s, t_1,t_2 \in [0,T];$$
- There exist constant $D_1>0$ such that $$K(t,s)\geq D_1(t-s)^{H-1/2}s^{-r},\quad s, t \in [0,T].$$
Then the Gaussian process $G(t) = \int_0^t K(t,s) dW(s)$, satisfies condition $(A)$, $(B)$ on any subinterval $[T-\delta, T]$ with $\delta\in (0,1)$.
Expected utility maximization in Wiener-transformable markets {#wti:sec:4}
=============================================================
Expected utility maximization for unrestricted capital profiles
---------------------------------------------------------------
Consider the problem of maximizing the expected utility. Our goal is to characterize the optimal asset profiles in the framework of the markets with risky assets involving Gaussian processes satisfying conditions of Theorem \[wti:thm1\]. We follow the general approach described in [@ekel] and [@karat], but apply its interpretation from [@Foll-Sch]. We fix $T>0$ and from now on consider ${{\mathcal{F}}}_T^W$-measurable random variables. Let the utility function $u:{{\mathbb{R}}}\rightarrow{{\mathbb{R}}}$ be strictly increasing and strictly concave, $L^0(\Omega, {{\mathcal{F}}}_T^W, { {\bf P}})$ be the set of all ${{\mathcal{F}}}_T^W$-measurable random variables, and let the set of admissible capital profiles coincide with $L^0(\Omega, {{\mathcal{F}}}_T^W, { {\bf P}})$. Let ${ {\bf P}}^*$ be a probability measure on $(\Omega, {{\mathcal{F}}}_T^W)$, which is equivalent to ${ {\bf P}}$, and denote $\varphi(T)=\frac{d{ {\bf P}}^*}{d{ {\bf P}}}$. The budget constraint is given by ${ {\bf E}}_{{ {\bf P}}^*}(X)=w$, where $w>0$ is some number that can be in some cases, but not obligatory, interpreted as the initial wealth. Thus the budget set is defined as $$\mathcal{B}=\left\{X\in L^0\left(\Omega, {{\mathcal{F}}}_T^{W}, { {\bf P}}\right)\cap L^1\left(\Omega, {{\mathcal{F}}}_T^W, { {\bf P}}^* \right)|{ {\bf E}}_{{ {\bf P}}^*}(X)=w\right\}.$$ The problem is to find such $X^*\in\mathcal{B}$, for which ${ {\bf E}}( u(X^*))=\max_{X\in\mathcal{B}} { {\bf E}}( u(X))$. Consider the inverse function $I(x)=(u'(x))^{-1}$.
\[wti:Theorem main for max\] Let the following condition hold: \[wti:Follmer-Sch\] Strictly increasing and strictly concave utility function $u:{{\mathbb{R}}}\rightarrow{{\mathbb{R}}}$ is continuously differentiable, bounded from above and $$\lim_{x\downarrow -\infty} u'(x)=+\infty.$$ Then the solution of this maximization problem has a form $$X^*=I(c\varphi(T)),$$ under additional assumption that ${ {\bf E}}_{{ {\bf P}}^*}(X^*)=w$.
To connect the solution of maximization problem with specific $W$-transformable Gaussian process describing the price process, we consider the following items.
1\. Consider random variable $\varphi(T)$, $\varphi(T)>0$ a.s. and let ${ {\bf E}}(\varphi(T))=1.$ Being the terminal value of a positive martingale $\varphi=\{\varphi_t={ {\bf E}}(\varphi(T)|{{\mathcal{F}}}_t^W), t\in[0,T]\}$, $\varphi(T)$ admits the following representation $$\label{wti:fi1}
\varphi(T)=\exp\left\{\int_0^T \vartheta(s)dW_s-\frac{1}{2}\int_0^T \vartheta^2(s)ds\right\},$$ where $\vartheta$ is a real-valued progressively measurable process for which $${ {\bf P}}\left\{\int_{0}^{T}\vartheta^2(s)ds<\infty\right\}=1.$$ Assume that $\vartheta$ satisfies . Then $\varphi(T)$ is a terminal value of a Hölder continuous process of order $\frac{1}{2} - \frac{1}{2p}$.
2\. Consider $W$-transformable Gaussian process $G=\{G(t), t\in[0,T]\}$ satisfying conditions $(A)$ and $(B)$, and introduce the set $$\begin{gathered}
\mathcal{B}_w^G=\bigg\{\psi\colon [0,T]\times \Omega\to {{\mathbb{R}}}\ \Big|\ \text{$\psi$ is bounded ${{\mathcal{F}}}_t^W$-adapted, there exists a generalized}\\ \text{Lebesgue-Stieltjes integral}
\int_0^T \psi(s)dG(s), \;\; \text{and} \;\; { {\bf E}}\bigg(\varphi(T)\int_0^T\psi(s) dG(s)\bigg)=w\bigg\}.\end{gathered}$$
\[wti:TheoremIntRepresentation\] Let the following conditions hold
- Gaussian process $G$ satisfies condition $(A)$ and $(B)$.
- Function $I(x),x\in{{\mathbb{R}}}$ is Hölder continuous.
- Stochastic process $\vartheta$ in representation satisfies with some $p>1$.
- There exists $c\in{{\mathbb{R}}}$ such that ${ {\bf E}}(\varphi(T)I(c\varphi(T)))=w$.
Then the random variable $X^*=I(c\varphi(T))$ admits the representation $$\label{wti:reprmain}
X^*=\int_0^T\overline{\psi}(s)dG(s),$$ with some $\overline{\psi}\in \mathcal{B}_w^G,$ and $$\label{wti:maxim}{ {\bf E}}( u(X^*))=\max_{\psi\in\mathcal{B}_w^G}{ {\bf E}}\left(u\left(\int_0^T\psi(s)dG(s)\right)\right).$$
From Lemma \[wti:lem1\] we have that for any $c \in \mathbb{R}$ the random variable $\xi= I(c \varphi(T))$ is the final value of a Hölder continuous process $$U(t)= I(c \varphi(t)) = I\left(c \exp\left\{\int_0^t \vartheta(s) d W(s) - \frac{1}{2} \int_0^t \vartheta^2(s) ds\right\}\right).$$ and the Hölder exponent exceeds $\rho$. Together with $(i)$–$(iii)$ this allows to apply Theorem \[wti:thm1\] to obtain the existence of representation (\[wti:reprmain\]). Assume now that (\[wti:maxim\]) is not valid, and there exists $\psi_0 \in \mathcal{B}_w^G$ such that ${ {\bf E}}\left(\varphi(T)\int_0^T \psi_0(s) d G(s)\right)=w$, and ${ {\bf E}}u \left( \int_0^T \psi_0(s) d G(s) \right)>{ {\bf E}}u(X^*)$. But in this case $\int_0^T \psi_0(s) d G(s)$ belongs to $\mathcal{B}$, and we get a contradiction with Theorem \[wti:Theorem main for max\].
Assuming only $(i)$ and $(iv)$, one can show in a similar way, but using Theorem \[wti:thm2\] instead of Theorem \[wti:thm1\] that $${ {\bf E}}( u(X^*))=\sup_{\psi\in\mathcal{B}_w^G}{ {\bf E}}\left(u\left(\int_0^T\psi(s)dG(s)\right)\right).$$ However, the existence of a maximizer is not guaranteed in this case.
Let $u(x) = 1 - e^{- \beta x}$ be an exponential utility function with constant absolute risk aversion $\beta>0$. In this case $I(x) = - \frac{1}{\beta} \log ( \frac{x}{\beta})$. Assume that $$\varphi(T) = \exp \left\{ \int_0^T \vartheta(s) dW(s) - \frac{1}{2} \int_0^T\vartheta^2(s) ds \right\}$$ is chosen in such a way that $$\begin{gathered}
\label{wti:eq1ex41}
{ {\bf E}}\left( \varphi(T) |\log \varphi(T)|\right)\\ ={ {\bf E}}\bigg( \exp\left\{ \int_0^T \vartheta(s) dW(s) - \frac{1}{2} \int_0^T\vartheta^2(s) ds \right\}\\ \times
\left|\int_0^T \vartheta(s) dW(s) - \frac{1}{2} \int_0^T\vartheta^2(s) ds\right|
\bigg)<\infty.
\end{gathered}$$ Then, according to Example 3.35 from [@Foll-Sch], the optimal profile can be written as $$\label{wti:ex41optprofile}
X^* = - \frac{1}{\beta} \left( \int_0^T \vartheta(s) dW(s) - \frac{1}{2} \int_0^T\vartheta^2(s) ds \right) + w + \frac{1}{\beta} H({ {\bf P}}^*|{ {\bf P}}),$$ where $H({ {\bf P}}^*|{ {\bf P}}) = { {\bf E}}\left(\varphi(T) \log \varphi(T)\right)$, condition (\[wti:eq1ex41\]) supplies that $H({ {\bf P}}^*|{ {\bf P}})$ exists, and the maximal value of the expected utility is $${ {\bf E}}(u(X^*)) = 1 - \exp\left\{-\beta w - H({ {\bf P}}^*|{ {\bf P}}) \right\}.$$ Let $\varphi(T)$ be chosen in such a way that the corresponding process $\vartheta$ satisfies the assumption of Lemma \[wti:lem1\]. Also, let $W$-transformable process $G$ satisfy conditions $(A)$ and $(B)$ of Theorem \[wti:Follmer-Sch\], and $\vartheta$ satisfy with $p>1$. Then we can conclude directly from representation (\[wti:ex41optprofile\]) that conditions of Theorem \[wti:Follmer-Sch\] hold. Therefore, the optimal profile $X^*$ admits the representation $X^* = \int_0^T \psi (s) d G(s).$
Similarly, under the same conditions as above, we can conclude that for any constant $d\in \mathbb{R}$ there exists $\psi_d$ such that $X^* = d + \int_0^T \psi_d(s) d G(s).$ Therefore, we can start from any initial value of the capital and achieve the desirable wealth. In this sense, $w$ is not necessarily the initial wealth as it is often assumed in the semimartingale framework, but is rather a budget constraint in the generalized sense.
In the case when $W$-transformable Gaussian process $G$ is a semimartingale, we can use Girsanov’s theorem in order to get the representation, similar to . Indeed, let, for example, $G$ be a Gaussian process of the form $G(t)=\int_0^t\mu(s)ds +\int_0^t a(s)dW(s)$, $|\mu(s)|\leq \mu$, $a(s)>a>0$ are non-random measurable functions, and $\xi$ is ${{\mathcal{F}}}_T^W$-measurable random variable, ${ {\bf E}}(\xi^2)<\infty$. Then we transform $G$ into $\widetilde{G}=\int_0^\cdot a(s)d\widetilde{W}(s)$, with the help of equivalent probability measure $\widetilde{{ {\bf P}}}$ having Radon–Nikodym derivative $$\frac{d\widetilde{{ {\bf P}}}}{d{ {\bf P}}}=\exp\left\{-\int_0^T\frac{\mu(s)}{a(s)}dW(s)-\frac{1}{2}
\int_0^T\left(\frac{\mu(s)}{a(s)}\right)^2d s \right\}.$$ With respect to this measure ${ {\bf E}}_{\widetilde{{ {\bf P}}}}|X^*|<\infty$, and we get the following representation $$\begin{gathered}
\label{wti:repraux}
X^*={ {\bf E}}_{\widetilde{{ {\bf P}}}}(X^*)+\int_0^T\psi(s)d\widetilde{W}_s={ {\bf E}}_{\widetilde{{ {\bf P}}}}(X^*)+\int_0^T
\frac{\psi(s)}{a(s)}d\widetilde{G}(s)\\={ {\bf E}}_{\widetilde{{ {\bf P}}}}(X^*)+\int_0^T
\frac{\psi(s)}{a(s)}d{G}(s)
={ {\bf E}}_{\widetilde{{ {\bf P}}}}(X^*)+\int_0^T\psi(s) {\mu(s)} ds+\int_0^T
{\psi(s)} dW(s).\end{gathered}$$ Representations and have the following distinction: “starts” from $0$ (but can start from any other constant) while “starts” exactly from ${ {\bf E}}_{\widetilde{{ {\bf P}}}}(X^*)$.
As we can see, the solution of the utility maximization problem for $W$–transformable process depends on the process in indirect way, through the random variable $\varphi(T)$ such that ${ {\bf E}}\varphi(T)=1$, $\varphi(T)>0$ a.s. Also, this solution depends on whether or not we can choose the appropriate value of $c$, but this is more or less a technical issue. Let us return to the choice of $\varphi(T)$. In the case of the semimartingale market, $\varphi(T)$ can be reasonably chosen as the likelihood ratio of some martingale measure, and the choice is unique in the case of the complete market. The non-semimartingale market can contain some hidden semimartingale structure. To illustrate this, consider two examples.
\[wti:ex4.2\] Let the market consist of bond $B$ and stock $S$, $$B(t)=e^{rt},\;S(t)=\exp\left\{\mu t +\sigma B_t^{H}\right\},$$ $r\geq0$, $\mu\in{{\mathbb{R}}}$, $\sigma>0$, $H>\frac{1}{2}$. The discounted price process has a form $Y(t)=\exp\left\{(\mu-r)t+\sigma B_t^H\right\}$. It is well-known that such market admits an arbitrage, but even in these circumstances the utility maximization problem makes sense. Well, how to choose $\varphi(T)$? There are at least two natural approaches. 1. Note that for $H>\frac{1}{2}$ the kernel $K^H$ from (\[wti:fbmviawin\]) has a form $$K^H(t,s)= C(H) s^{\frac{1}{2}- H}\int_s^t u^{H-\frac{1}{2}}(u-s)^{H-\frac{3}{2}} du,$$ and representation (\[wti:winviafbm\]) has a form $$W(t) = \left(C(H)\right)^{-1}\int_0^t s^{\frac{1}{2}-H} K^*(t,s) d B_s^H,$$ where $$\begin{gathered}
K^*(t,s) = \Big(t^{H-\frac{1}{2}}(t-s)^{\frac{1}{2} - H} \\ -\left(H-\frac{1}{2}\right)\int_s^t u^{H-\frac{3}{2}}(u-s)^{\frac{1}{2}-H}du\Big)\frac{1}{\Gamma\left(\frac{3}{2} - H\right)}.
\end{gathered}$$
Therefore, $$\begin{aligned}
& &\left(C(H)\right)^{-1}\int_0^t s^{\frac{1}{2}-H} K^*(t,s) d \left( (\mu - r) s + \sigma B_s^H\right)\\
&=& \sigma W(t) + \frac{\mu-r}{C(H)} \int_0^t s^{\frac{1}{2}-H} K^*(t,s) ds\\
&=& \sigma W(t) + \frac{\mu-r}{C(H)\Gamma\left(\frac{3}{2} - H\right)} \int_0^t\left( s^{\frac{1}{2}-H} t^{H- \frac{1}{2}} (t-s)^{\frac{1}{2}-H}\right.\\
& &{}- \left.\left(H-\frac{1}{2}\right)s^{\frac{1}{2}-H} \int_s^t u^{H-\frac{3}{2}}(u-s)^{\frac{1}{2}-H}du \right)ds\\
&=&\sigma W_t + \frac{\mu -r}{C(H) \Gamma(\frac{3}{2}-H)}\frac{\Gamma^2(\frac{3}{2}-H)}{(\frac{3}{2}-H)\Gamma (2 - 2 H)} t^{\frac{3}{2}-H}\\
&=& \sigma W_t + (\mu-r) C_1(H) t^{\frac{3}{2}-H},\end{aligned}$$ where $$C_1(H) = \left(\frac{3}{2}-H\right)^{-1} \left(\frac{\Gamma(\frac{3}{2} - H)}{2H\Gamma(2 - 2H)\Gamma(H+\frac{1}{2})} \right)^\frac{1}{2}.$$ In this sense we say that the model involves a hidden semimartingale structure.\
Consider a virtual semimartingale asset $$\begin{aligned}
\hat{Y}(t) &= \exp \left\{ (C(H))^{-1} \int_0^t s^{\frac{1}{2}-H}K^*(t,s) d \log Y(s) \right\} \\
&= \exp\left\{\sigma W_t + (\mu-r) C(H)t^{\frac{3}{2}-H}\right\}.\end{aligned}$$ We see that measure ${ {\bf P}}^*$ such that $$\begin{split}\label{wti:ex42ChangeMeasure}
\frac{d{ {\bf P}}^*}{d{ {\bf P}}} &=\exp \left\{ - \int_0^T \left(\frac{(\mu-r)C_2(H)}{\sigma} s^{\frac{1}{2}-H}+ \frac{\sigma}{2}\right)dW_s\right. \\
& \quad \left.- \frac{1}{2} \int_0^T \left(\frac{(\mu-r)C_2(H)}{\sigma} s^{\frac{1}{2}-H}+ \frac{\sigma}{2}\right)^2 ds\right\},
\end{split}$$ where $C_2(H) = C_1(H) \left(\frac{3}{2}-H\right),$ reduces $\hat{Y}(t)$ to the martingale of the form$\exp \left\{\sigma W_t - \frac{\sigma^2}{2} t\right\}$. Therefore, we can put $\varphi(T) = \frac{d{ {\bf P}}^*}{d{ {\bf P}}}$ from (\[wti:ex42ChangeMeasure\]). Regarding the Hölder property, $\vartheta(s)= s^{\frac{1}{2}-H}$ satisfies with some $p>1$ for any $H\in(\frac12,1)$. Therefore, for the utility function $u(x) = 1 - e^{-\alpha x}$ we have $$X^* = \frac{1}{\alpha} \left(\int_0^T \varsigma(s) dW_s - \frac{1}{2} \int_0^T \varsigma_s^2 ds \right) + W + \frac{1}{2} H({ {\bf P}}^*|{ {\bf P}}),$$ where $\varsigma(s) = \frac{(\mu-r)C_2(H)}{\sigma} s^{\frac{1}{2}-H}+ \frac{\sigma}{2}$, and $|H({ {\bf P}}^*|{ {\bf P}})|<\infty.$
2\. It was proved in [@Dung] that the fractional Brownian motion $B^H$ is the limit in $L_p(\Omega,\mathcal{F},{ {\bf P}})$ for any $p>0$ of the process $$B^{H,\epsilon}(t)=\int_0^tK(s+\epsilon,s)dW(s)+\int_0^t\psi_\epsilon (s)ds,$$ where $W$ is he underlying Wiener process, i.e. $B^{H}(t)=\int_0^tK(t,s)dW(s),$ where $$\begin{gathered}
K(t,s)=C_H s^{\frac{1}{2}-H}\int_s^t u^{H-\frac{1}{2}}(u-s)^{H-\frac{3}{2}}du,\\
\psi_\epsilon(s)=\int_0^s\partial_1K(s+\epsilon,u)dW_u,\\
\partial_1K(t,s)=\frac{\partial K(t,s)}{\partial t}=C_Hs^{\frac{1}{2}-H}t^{H-\frac{1}{2}}(t-s)^{H-\frac{3}{2}}.\end{gathered}$$ Consider prelimit market with discounted risky asset price $Y^{\epsilon}$ of the form $$Y^{\epsilon}(t)=\exp{\left\{(\mu-r)t+\sigma\int_{0}^{t}\psi_\epsilon(s)ds+\sigma\int_0^tK(s+\epsilon,s)dW_s\right\}}.$$ This financial market is arbitrage-free and complete, and the unique martingale measure has the Radon-Nikodym derivative $$\varphi_\epsilon(T)=\exp\left\{-\int_0^T\zeta_\epsilon(t)dW_t-\frac{1}{2}\int_0^T\zeta^2_\epsilon(t)dt\right\},$$ where $$\zeta_\epsilon(t)=\frac{\mu-r+\sigma\psi_\epsilon(t)}{\sigma K(t+\epsilon,t)}+\frac{1}{2}\sigma K(t+\epsilon,t).$$ Note that $K(t+\epsilon,t)\rightarrow 0$ as $\epsilon\rightarrow0$. Furthermore, $\rho_t=\frac{\mu-r+\sigma\psi_\epsilon(t)}{\sigma K(t+\epsilon,t)}$ is a Gaussian process with ${ {\bf E}}\rho_t=0$ and $$\begin{gathered}
{\operatornamewithlimits{\bf var}}\zeta_\varepsilon(t)=\int_{0}^{t}\left(\frac{\partial_1 K(t+\epsilon,u)}{ K(t+\epsilon,t)}\right)^2du\\
=\int_{0}^{t}\left(\frac{u^{1/2-H}(t+\epsilon)^{H-1/2}(t+\epsilon-u)^{H-3/2}}{t^{1/2-H}\int_t^{t+\epsilon}v^{H-1/2}(v-t)^{H-3/2}}\right)^2du\\
\ge\epsilon^{1-2H}\int_0^t(t+\epsilon-u)^{2H-3}du=\frac{\epsilon^{1-2H}t}{2-2H}\left(\epsilon^{2H-2}-(t+\epsilon)^{2H-2}\right)\rightarrow\infty.\end{gathered}$$ Therefore, we can not get a reasonable limit of $\varphi_\epsilon(T)$ as $\epsilon\rightarrow0.$ Thus one should use this approach with great caution.
Expected utility maximization for restricted capital profiles
-------------------------------------------------------------
Consider now the case when the utility function $u$ is defined on some interval $(a,\infty)$. Assume for technical simplicity that $a=0$. Therefore, in this case the set $\mathcal{B}_0$ of admissible capital profiles has a form $$\mathcal{B}_0=\left\{X\in L^0(\Omega,{{\mathcal{F}}},{ {\bf P}}):X\ge0 \;\; \text{a.s. and} \;\; { {\bf E}}(\varphi(T) X)=w\right\}.$$ Assume that the utility function $u$ is continuously differentiable on $(0,\infty)$, introduce $\pi_1=\lim\limits_{x\uparrow\infty}u'(x)\ge0$, $\pi_2=u'(0+)=\lim\limits_{x\downarrow0}u'(x)\le +\infty$, and define $I^+:(\pi_1,\pi_2)\longrightarrow(0,\infty)$ as the continuous, bijective function, inverse to $u'$ on $(\pi_1,\pi_2)$.
Extend $I^+$ to the whole half-axis $\left[0,\infty\right]$ by setting $$I^+(y)=\left\{ \begin{array}{ll}
+\infty,& y\le\pi_1\\
0,& y\ge\pi_2.
\end{array}\right.$$
Let the random variable $X^*\in\mathcal{B}_0$ have a form $X^*=I^+(c\varphi(T))$ for such constant $c>0$ that ${ {\bf E}}( \varphi(T) I^{+}(c\varphi(T)))=w$. If ${ {\bf E}}u(X^*)<\infty$ then $${ {\bf E}}( u(X^*))=\max\limits_{X\in\mathcal{B}_0}{ {\bf E}}(u(X)),$$ and this maximizer is unique.
From here we deduce the corresponding result on the solution of utility maximization problem similarly to Theorem \[wti:TheoremIntRepresentation\]. Define, as before, $$\begin{gathered}
\mathcal{B}_w^G=\bigg\{\psi\colon [0,T]\times \Omega\to {{\mathbb{R}}}\ \Big|\ \text{$\psi$ is bounded ${{\mathcal{F}}}_t^W$-adapted, there exists a generalized}\\ \text{Lebesgue-Stieltjes integral}
\int_0^T \psi(s)dG(s)\ge 0, \;\; \text{and} \;\; { {\bf E}}\bigg(\varphi(T)\int_0^T\psi(s) dG(s)\bigg)=w\bigg\}.\end{gathered}$$
Let the following conditions hold
- Gaussian process $G$ satisfies conditions $(A)$ and $(B)$.
- Function $I^+(x),x\in{{\mathbb{R}}}$ is Hölder continuous.
- Stochastic process $\vartheta$ in representation satisfies with some $p> 1$.
- There exists $c\in{{\mathbb{R}}}$ such that ${ {\bf E}}(\varphi(T)I^+(c\varphi(T)))=w$.
Then the random variable $X^*=I^+(c\varphi(T))$ admits the representation $$X^*=\int_0^T\overline{\psi}(s)dG(s),$$ with some $\overline{\psi}\in \widetilde{\mathcal{B}}_w^G$. If ${ {\bf E}}u(X^*)<\infty$, the $X^*$ is the solution to expected utility maximization problem: and $${ {\bf E}}( u(X^*))=\max_{\psi\in\widetilde{\mathcal{B}}_w^G}{ {\bf E}}\left(u\left(\int_0^T\psi(s)dG(s)\right)\right).$$
Consider the case of CARA utility function $u$. Let first $u(x)=\frac{x^\gamma}{\gamma}$, $x>0$, $\gamma\in(0,1)$. Then, according to [@Foll-Sch Example 3.43], $$I^+(c\varphi(T))=c^{-\frac{1}{1-\gamma}}(\varphi(T))^{-\frac{1}{1-\gamma}}.$$ If $d:={ {\bf E}}(\varphi(T))^{-\frac{\gamma}{1-\gamma}}<\infty$ then unique optimal profile is given by $X^*=\frac{w}{d}(\varphi(T))^{-\frac{1}{1-\gamma}}$, and the maximal value of the expected utility is equal to $${ {\bf E}}( u(X^*))=\frac{1}{\gamma}w^{\gamma}d^{1-\gamma}.$$ As it was mentioned, $$\label{wti:fi}
\varphi=\varphi(T)=\exp\left\{\int\limits_{0}^T\vartheta(s)dW(s)-\frac{1}{2}\int\limits_0^T \vartheta^2(s)ds,\right\}$$ thus $$(\varphi(T))^{-\frac{1}{1-\gamma}}=\exp\left\{-\frac{1}{1-\gamma}\int\limits_0^T \vartheta(s)dW(s)+\frac{1}{2(1-\gamma)}\int\limits_0^T \vartheta^2(s)ds\right\}.$$ Therefore, we get the following result.
Let the process $\vartheta$ in the representation satisfy , and $${ {\bf E}}\exp\left\{-\frac{\gamma}{1-\gamma}\int\limits_0^T\vartheta(s)dW_s+\frac{\gamma}{2(1-\gamma)}\int\limits_0^T\vartheta^2_sds\right\}<\infty.$$ Let the process $G$ satisfy the same conditions as in Theorem \[wti:TheoremIntRepresentation\]. Then $X^*=\int\limits_0^T\psi(s)dG(s)$.
In the case where $u(x)=\log x$, we have $\gamma=0$ and $X^*=\frac{w}{\varphi(T)}$. Assuming that the relative entropy $H\left({{ {\bf P}}}|{{ {\bf P}}^*}\right)={ {\bf E}}(\frac{1}{\varphi(T)}\log \varphi(T))$ is finite, we get that $${ {\bf E}}(\log X^*)=\log w + H\left({{ {\bf P}}}|{{ {\bf P}}^*}\right).$$
Conclusion {#conclusion .unnumbered}
==========
We have studied a broad class of non-semimartingale financial market models, where the random drivers are Wiener-transformable Gaussian random processes, i.e. some adapted transformations of a Wiener process. Under assumptions that the incremental variance of the process satisfies two-sided power bounds, we have given sufficient conditions for random variables to admit integral representations with bounded adapted integrand; these representations are models for bounded replicating strategies. It turned out that these representation results can be applied to solve utility maximization problems in non-semimartingale market models.
**Acknowledgements** Elena Boguslavskaya is supported by Daphne Jackson fellowship funded by EPSRC. The research of Yu. Mishura was funded (partially) by the Australian Government through the Australian Research Council (project number DP150102758). Yu. Mishura acknowledges that the present research is carried through within the frame and support of the ToppForsk project nr. 274410 of the Research Council of Norway with title STORM: Stochastics for Time-Space Risk Models.
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abstract: 'Generalisations of the bent property of a boolean function are presented, by proposing spectral analysis with respect to a well-chosen set of local unitary transforms. Quadratic boolean functions are related to simple graphs and it is shown that the orbit generated by successive Local Complementations on a graph can be found within the transform spectra under investigation. The flat spectra of a quadratic boolean function are related to modified versions of its associated adjacency matrix.'
author:
- 'Constanza Riera[^1], Matthew G. Parker[^2]'
title: 'Generalised Bent Criteria for Boolean Functions (I)'
---
Introduction
============
It is often desirable that a boolean function, $p$, used for cryptographic applications, is be highly [*[nonlinear]{}*]{}, where nonlinearity is determined by examining the spectrum of $p$ with respect to (w.r.t.) the [*[Walsh Hadamard Transform]{}*]{} (WHT), and where the nonlinearity is maximised for those functions that minimise the magnitude of the spectral coefficients. To be precise, define the boolean function of $n$ variables $p : \mbox{GF}(2)^n \rightarrow \mbox{GF}(2)$, and the WHT by the $2^n \times 2^n$ unitary matrix $U = H \otimes H \ldots \otimes H = \bigotimes_{i=0}^{n-1} H$, where the Walsh-Hadamard kernel $H = \frac{1}{\sqrt{2}}\begin{tiny} \left ( \begin{array}{rr}
1 & 1 \\
1 & -1
\end{array} \right ) \end{tiny}$, ’$\bigotimes$’ indicates the tensor product of matrices, and unitary means that $UU^{\dag} = I_n$, where ’$\dag$’ means transpose-conjugate and $I_n$ is the $2^n \times 2^n$ identity matrix. We further define a length $2^n$ vector, $s = (s_{0\ldots00},s_{0\ldots 01},s_{0\ldots 11},\ldots,s_{1\ldots 11})$ such that $s_{{\bf{i}}} = (-1)^{p({\bf{i}})}$, where ${\bf{i}} \in \mbox{GF}(2)^n$. Then the Walsh-Hadamard spectrum of $p$ is given by the matrix-vector product $P = Us$, where $P$ is a vector of $2^n$ real spectral coefficients, $P_{{\bf{k}}}$, where ${\bf{k}} \in \mbox{GF}(2)^n$.
The spectral coefficient, $P_{{\bf{k}}}$, with maximum magnitude tells us the minimum (Hamming) distance, $d$, of $p$ to the set of affine boolean functions, where $d = 2^{n-1} - 2^{\frac{n-2}{2}}|P_{\bf{k}}|$. By Parseval’s Theorem, the extremal case occurs when all $P_{{\bf{k}}}$ have equal magnitude, in which case $p$ is said to have a [*[flat]{}*]{} WHT spectra, and is referred to as [*[bent]{}*]{}. If $p$ is bent, then it is as far away as it can be from the affine functions [@Meier:NL], which is a desirable cryptographic design goal. It is an open problem to classify all bent boolean functions, although many results are known [@Dill:DS; @MacW:Cod; @Car:NewB; @Dob:Bent].
In this paper, we extend the concept of a bent boolean function to some [*[Generalised Bent Criteria]{}*]{} for a boolean function, where we now require that $p$ has flat spectra w.r.t. one or more transforms from a specified set of unitary transforms. The set of transforms we choose is not arbitrary but is motivated by the choice of unitary transforms that are typically used to action a local basis change for a pure $n$-qubit quantum state. We here apply such transforms to a $n$-variable boolean function, and examine the resultant spectra accordingly. In particular we apply all possible transforms formed from $n$-fold tensor products of the identity $I = \begin{tiny} \left ( \begin{array}{rr}
1 & 0 \\
0 & 1
\end{array} \right ) \end{tiny}$, the Walsh-Hadamard kernel, $H$, and the Negahadamard kernel [@Par:Bent], $N = \frac{1}{\sqrt{2}}\begin{tiny} \left ( \begin{array}{rr}
1 & i \\
1 & -i
\end{array} \right ) \end{tiny}$, where $i^2 = -1$. We refer to this set of transforms as the [*[ $\{I,H,N\}^n$ transform set]{}*]{}, i.e. where all transforms are of the form $\{I,H,N\}^n=\bigotimes_{j \in {\bf{R_I}}} I_j \bigotimes_{j \in {\bf{R_H}}} H_j
\bigotimes_{j \in {\bf{R_N}}} N_j$, where the sets ${\bf{R_I}}, {\bf{R_H}}$ and ${\bf{R_N}}$ partition $\{0,\ldots,n-1\}$, and $H_j$, say, is short for
$I \otimes I \otimes \ldots \otimes I \otimes H \otimes I \otimes \ldots \otimes I$
, with $H$ in the $j^{th}$ position. There are $3^n$ such transforms which act on a boolean function of $n$ variables to produce $3^n$ spectra, each spectrum of which comprises $2^n$ spectral elements (complex numbers). By contrast, the WHT can be described as $\{H\}^n$, which is a transform set of size one, where the single resultant output spectrum comprises just $2^n$ spectral elements.
The Quantum Context
-------------------
The choice of $I$, $H$, and $N$, is motivated by their importance for the construction of [*[Quantum Error-Correcting Codes]{}*]{} (QECCs). This is because $I$, $H$, and $N$ are generators of the [*[Local Clifford Group]{}*]{} [@Cald:Qua; @Klapp:Cliff1] which is defined to be the set of matrices that [*[stabilize]{}*]{} the group of Pauli matrices
[ The Pauli matrices are $I$, $\sigma_x = \begin{tiny} \left ( \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right ) \end{tiny}$, $\sigma_z = \begin{tiny} \left ( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right ) \end{tiny}$, and $\sigma_y = i\sigma_x\sigma_z$. ]{}
which, in turn, form a basis for the set of local errors that act on the quantum code. This implies that the set of [*[locally-equivalent]{}*]{} $n$-qubit quantum states, that occur as joint eigenspectra w.r.t. $\{I,H,N\}^n$, are equally robust to quantum errors from the Pauli error set. Stabilizer QECCs can also be interpreted as additive codes over GF$(4)$ [@Cald:Qua].
To evaluate the quantum [*[entanglement]{}*]{} of a pure $n$-qubit state one should really examine the spectra w.r.t. the infinite set of $n$-fold tensor products of all $2 \times 2$ unitary matrices [@Par:QE]. Those states which minimise all spectral magnitudes w.r.t. this infinite transform set are as far away as possible from all generalised affine functions and can be considered to be highly entangled as the probability of observing (measuring) any specific qubit configuration is as small as possible, in any local measurement basis. However it is computationally intractable to evaluate, to any reasonable approximation, this continuous local unitary spectrum beyond about $n = 4$ qubits (although approximate results up to $n = 6$ are given in [@Par:QE]). Therefore we choose, in this paper, a well-spaced subset of spectral points, as computed by the set of $\{I,H,N\}^n$ transforms, from which to ascertain approximate entanglement measures. Complete spectra for such a transform set can be computed up to about $n = 10$ qubits using a standard desk-top computer, although partial results for higher $n$ are possible if the $n$-qubit quantum state is represented by, say, a quadratic boolean function over $n$ variables.
The Graphical Context
---------------------
The graphical description of certain pure quantum states was investigated by Parker and Rijmen [@Par:QE]. They proposed partial entanglement measures for such states and made observations about a [*[Local Unitary (LU) Equivalence]{}*]{} between graphs describing the states w.r.t. the tensor product of $2 \times 2$ local unitary transforms. These graphs were interpreted as quadratic boolean functions and it was noted that bipartite quadratic functions are LU-equivalent to indicators for binary linear error-correcting codes. It was further observed that physical quantum graph arrays are relevant to the work of [@Par:QE] and were already under investigation in the guise of [*[cluster states]{}*]{}, by Raussendorf and Briegel [@Raus:QC; @Brie:Ent]. These clusters form the ’substrate’ for measurement-driven quantum computation.
Measurement-driven quantum computation on a [*[quantum factor graph]{}*]{} has been discussed by Parker [@Par:QFG]. Independent work by Schlingemann and Werner [@Sch:QG], Glynn [@Glynn:Graph; @Glynn:Tome], and by Grassl, Klappenecker, and Rotteler [@Gras:QG] proposed to describe [*[stabilizer]{}*]{} Quantum Error-Correcting Codes (QECCs) using graphs and, for QECCs of dimension zero, the associated graphs can be referred to as [*[graph states]{}*]{}. The graph states are equivalent to the graphs described by [@Par:QE] and therefore have a natural representation using quadratic boolean functions.
In [@Par:QE] it was observed that the complete graph, star graph, and generalised GHZ (Greenberger-Horne-Zeilinger) states are all LU-equivalent. It turns out that LU-equivalence for graph states can be characterised, graphically, via the [*[Vertex-Neighbour-Complement]{}*]{} (VNC) transformation, which was defined by Glynn, in the context of QECCs, in [@Glynn:Graph] (definition 4.2) and also, independently, by Hein, Eisert and Briegel [@Hein:GrEnt], and also by Van Den Nest and De Moor [@VanD:Gr]. VNC is another name for [*[Local Complementation]{}*]{} (LC), as investigated by Bouchet [@Bou:Iso; @Bou:Tree; @Bou:Grph] in the context of [*[isotropic systems]{}*]{}. By applying LC to a graph $G$ we obtain a graph $G'$, in which case we say that $G$ and $G'$ are [*LC-equivalent*]{}. Moreover, the set of all LC-equivalent graphs form an [*[LC-orbit]{}*]{}. LC-equivalence translates into the natural equivalence between $\GF(4)$ additive codes that keeps the weight distribution of the code invariant [@Cald:Qua]. There has been recent renewed interest in Bouchet’s work motivated, in part, by the application of [*[interlace graphs]{}*]{} to the reconstruction of DNA strings [@Arr:DNA; @Arr:Int]. In particular, various [*[interlace polynomials]{}*]{} have been defined [@Arr:Int; @Aig:Int; @Arr:Int1; @Arr:Int2] which mirror some of the quadratic results of part II of this paper [@RP:BCII]. We point out links to this work in part II but defer a thorough exposition of these links to the future.
The Boolean Context
-------------------
Spectral analysis w.r.t. $\{I,H,N\}^n$ also has application to the cryptanalysis of classical cryptographic systems [@DanAPC]. In particular, for a block cipher it models attack scenarios where one has full read/write access to a subset of plaintext bits and access to all ciphertext bits, (see [@DanAPC] for more details). The analysis of spectra w.r.t. $\{I,H,N\}^n$ tells us more about $p$ than is provided by the spectrum w.r.t. the WHT; for instance, identifying relatively high generalised linear biases for $p$ [@Par:SB]. In Part I of this paper our aim is to introduce these new generalised bent criteria. In Part II [@RP:BCII] we enumerate the flat spectra w.r.t. $\{I,H,N\}^n$ and its subsets. We are trying to answer the question: which boolean functions are as far away as possible from the set of generalised affine functions as defined by the rows of $\{I,H,N\}^n$?
[ A row of $U_0 \otimes U_1 \otimes \ldots \otimes U_{n-1}$ for $U_i$ a $2 \times 2$ unitary matrix can always be written as $u = (a_0,b_0) \otimes (a_1,b_1) \otimes \ldots \otimes (a_{n-1},b_{n-1})$, where $a_i,b_i$ are complex numbers. For $\al$ a $r^{th}$ complex root of 1, and $m$ an integer modulus, we can approximate an unnormalised version of $u$ by $u \simeq m({\bf{x}})\al^{p({\bf{x}})}$, for some appropriate choice of integers $s$ and $r$, where $m : \mbox{GF}(2)^n \rightarrow \mbox{GF}(s)$, $p : \mbox{GF}(2)^n \rightarrow \mbox{GF}(r)$, and ${\bf{x}} \in \mbox{GF}(2)^n$, such that the ${\bf{j}}^{th}$ element of $u$, $u_{\bf{j}} = m({\bf{j}})\al^{p({\bf{j}})}$, where ${\bf{j}} \in \mbox{GF}(2)^n$ and $u_{\bf{j}}$ is interpreted as a complex number. When $u$ is fully-factorised using the tensor product then $m$ and $p$ are affine functions and we say that $u$ represents a generalised affine function (see [@Par:QE], Section 5, for more details). ]{}
The classification of bent quadratic (degree-two) boolean functions is well-known [@MacW:Cod], and is facilitated because the bent criteria is an invariant of affine transformation of the input variables. However, the classification of generalised bent criteria for a quadratic boolean function w.r.t. the $\{I,H,N\}^n$ transform set is new, and the generalised bent criteria are not, in general, invariant to affine transformation of the inputs. This paper characterises these generalised bent criteria for both quadratic and more general boolean functions. We associate a quadratic boolean function with an undirected graph, which allows us to interpret spectral flatness with respect to $\{I,H,N\}^n$ as a [*[maximum rank]{}*]{} property of suitably modified adjacency matrices. We interpret LC as an operation on quadratic boolean functions, and as an operation on the associated adjacency matrix, and we also identify the LC-orbit with a subset of the flat spectra w.r.t. $\{I,H,N\}^n$. The spectra w.r.t. $\{I,H,N\}^n$ motivates us to examine the properties of the WHT of all ${\mathbb{Z}}_4$-linear offsets of boolean functions, the WHT of all subspaces of boolean functions that can be obtained by fixing a subset of the variables, the WHT of all ${\mathbb{Z}}_4$-linear offsets of all of the above subspace boolean functions, the WHT of each member of the LC-orbit, and the distance of boolean functions to all ${\mathbb{Z}}_4$-linear functions. This leads us to prove the following: All quadratic boolean functions are [*[bent$_4$]{}*]{}, [*[I-bent]{}*]{} and [*[I-bent$_4$]{}*]{}.\
Not all quadratic boolean functions are [*[LC-bent]{}*]{}.\
All boolean functions are [*[I-bent$_4$]{}*]{}.\
Not all boolean functions are [*[bent$_4$]{}*]{} or [*[I-bent]{}*]{}.\
There are no [*[${\mathbb{Z}}_4$-bent]{}*]{} or [*[Completely I-bent$_4$]{}*]{} boolean functions. where the above terms for generalised bent criteria will be made clear in the sequel. We are able to characterise and analyse the criteria for quadratic boolean functions by considering properties of the adjacency matrix for the associated graph state.
Paper Overview
--------------
For the interested reader, Appendix A reviews the graph state and its intepretations in the literature. In Section \[LCGraph\] we review LC as an operation on an undirected graph [@Glynn:Graph; @Glynn:Tome], and provide an algorithm for LC in terms of the adjacency matrix of the graph. In Section \[LCLUT\], we show that the LC-orbit of a quadratic boolean function lies within the set of transform spectra w.r.t. tensor products of the $2 \times 2$ matrices, $I$, $\sqrt{-i\sigma_x}$, and $\sqrt{i\sigma_z}$, where $\sigma_x$ and $\sigma_z$ are Pauli matrices. We also show, equivalently, that the orbit lies within the spectra w.r.t. $\{I,H,N\}^n$. We show that doing LC to vertex $x_v$ can be realised by the application of the Negahadamard kernel, $N$, to position $v$ (and the identity matrix to all other positions) of the bipolar vector $(-1)^{p({\bf{x}})}$, i.e. $$\omega^{4p'({\bf{x}}) + a({\bf{x}})}(-1)^{p'({\bf{x}})} = U_v(-1)^{p({\bf{x}})} =
I \otimes \cdots \otimes I \otimes N \otimes I \otimes \cdots \otimes
I \ (-1)^{p({\bf{x}})} \enspace,$$ where $p({\bf{x}})$ and $p'({\bf{x}})$ are quadratic, $p'({\bf{x}})$ is obtained by applying LC to variable $x_v$, $\omega = \sqrt{i}$, and $a({\bf{x}})$ is any offset over ${\mathbb{Z}}_8$. In Appendix B we identify spectral symmetries that hold for $p({\bf{x}})$ of any degree w.r.t. $\{I,H,N\}^n$. In Section \[BentSection\], we introduce the concepts of [*bent$_4$*]{}, [*${\mathbb{Z}}_4$-bent*]{}, [*(Completely) I-bent*]{}, [*LC-bent*]{}, and [*(Completely) I-bent$_4$*]{} boolean functions, and show how, for quadratic boolean functions, these properties can be evaluated by examining the ranks of suitably modified versions of the adjacency matrix.
Local Complementation (LC) {#LCGraph}
==========================
Given a graph $G$ with adjacency matrix $\Gamma$, define its [*complement*]{} to be the graph with adjacency matrix $\Gamma+I + {\bf 1}\ (\mo 2)$, where $I$ is the identity matrix and ${\bf 1}$ is the all-ones matrix. Let ${\cal{N}}(v)$ be the set of neighbours of vertex, $v$, in the graph, $G$, i.e. the set of vertices connected to $v$ in $G$.
Define the action of [*LC*]{} (or [*vertex-neighbour-complement (VNC)*]{}) on a graph $G$ at vertex $v$ as the graph transformation obtained by replacing the subgraph $G[{\cal{N}}(v)]$ by its complement.
By Glynn (see [@Glynn:Graph]), a self-dual quantum code $[[n,0,d]]$ corresponds to a graph on $n$ vertices, which may be assumed to be connected if the code is indecomposable. It is shown there that two graphs $G$ and $H$ give equivalent self-dual quantum codes if and only if $H$ and $G$ are LC-equivalent.
For a study of the group of compositions of local complementations, see [@Bou:Iso; @Bou:Grph; @Bou:Tree; @Cou:Ver], which describe the relation between local complementation and [*isotropic systems*]{}. Essentially, a suitably-specified isotropic system has graph presentations $G$ and $G'$ iff $G$ and $G'$ are locally equivalent w.r.t. local complementation.
LC in terms of the adjacency matrix
-----------------------------------
Let $p({\bf{x}}) : F_2^n \rightarrow F_2$ be a (homogeneous) quadratic boolean function, defined by, $$p({\bf{x}}) = \sum_{0\leq i < j\leq n-1} a_{ij}x^ix^j \enspace.$$ We can express $p({\bf{x}})$ by the adjacency matrix of its associated graph, $\Gamma$, such that $\Gamma(i,j) = \Gamma(j,i) = a_{ij}$, $i < j$, $\Gamma(i,i) = 0$. The LC operation on the graph associated to $p({\bf{x}})$ can be expressed in terms of the adjacency matrix. Without loss of generality, we show how the matrix changes from $\Gamma$ to $\Gamma_0$ after doing LC on vertex $x_{0}$:
$$\Gamma_{0}=\left(\begin{array}{cccccc}
0 & a_{01} & a_{02} & a_{03} & \ldots & a_{0n}\\
a_{01} & 0 & a_{12}+a_{01}a_{02} & a_{13}+a_{01}a_{03} & \ldots & a_{1n}+a_{01}a_{0,n-1}\\
a_{02} & a_{12}+a_{01}a_{02} & 0 & a_{23}+a_{02}a_{03} & \ldots & a_{2n}+a_{02}a_{0,n-1}\\
a_{03} & a_{13}+a_{01}a_{03} & a_{23}+a_{02}a_{03} & 0 & \ldots & a_{3n}+a_{03}a_{0,n-1}\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots\\
a_{0,n-1} & a_{1,n-1}+a_{01}a_{0,n-1} & a_{2,n-1}+a_{02}a_{0,n-1} & a_{3,n-1}+a_{03}a_{0,n-1} & \ldots & 0\end{array}\right) \enspace.$$
The general algorithm, mod 2, is $$\left\{\begin{array}{l}
\Gamma_{v}(i,j)=\Gamma(i,j)+\Gamma(v,i)*\Gamma(v,j),\ \ i<j,\ \ \ i,j=1,\ldots,n\\
\Gamma_{v}(i,i)=0\ \ \forall i\\
\Gamma_{v}(j,i)=\Gamma_{v}(i,j),\ \ \ i>j
\end{array}\right.$$ where $\Gamma_{v}$ is the adjacency matrix of the function after doing LC to the vertex $x_{v}$.
Local Complementation (LC) and Local Unitary (LU) Equivalence {#LCLUT}
=============================================================
Hein et al [@Hein:GrEnt] state that LC-Equivalence (and therefore Local Unitary (LU) Equivalence) of graph states is obtained via successive transformations of the form, U\_v(G) = (-i\_x\^[(v)]{})\^[1/2]{} \_[b \_v]{} (i\_z\^[(b)]{})\^[1/2]{} \[LUEquiv\] ,where $\sigma_x = \begin{tiny} \left ( \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right ) \end{tiny}$ and $\sigma_z = \begin{tiny} \left ( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right ) \end{tiny}$ are Pauli matrices, the superscript $(v)$ indicates that the Pauli matrix acts on qubit $v$ (with $I$ acting on all other qubits)
[For instance, $\sigma_x^{(2)} =
I \otimes I \otimes \sigma_x \otimes I \otimes \ldots \otimes I$. ]{}
, and ${\cal{N}}_v$ comprises the neighbours of qubit $v$ in the graphical representation. Define matrices $x$ and $z$ as follows: $$x = (-i\sigma_x)^{1/2} = \frac{1}{\sqrt{2}}
\begin{small} \left ( \begin{array}{rr} -1 & i \\ i & -1 \end{array} \right ) \end{small}$$ and $$z = (i\sigma_z)^{1/2} =
\begin{small} \left ( \begin{array}{rr} w & 0 \\ 0 & w^3 \end{array} \right ) \end{small} \enspace,$$ where $w = e^{2\pi i/8}$. Furthermore, let $I = \begin{tiny} \left ( \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right ) \end{tiny}$.
Define ${\bf{D}}$ to be the set of $2 \times 2$ diagonal or anti-diagonal local unitary matrices, i.e. of the form $\begin{tiny} \left ( \begin{array}{rr} a & 0 \\ 0 & b \end{array} \right ) \end{tiny}$ or $\begin{tiny} \left ( \begin{array}{rr} 0 & a \\ b & 0 \end{array} \right ) \end{tiny}$, for some $a$ and $b$ in $\mathbb{C}$. We make extensive use of the fact that a final multiplication of a spectral vector by tensor products of members of ${\bf{D}}$ does not change spectral coefficient magnitudes. In this sense a final multiplication by tensor products of members of ${\bf{D}}$ has no effect on the final spectrum and does not alter the underlying graphical interpretation. For instance, applying $x$ twice to the same qubit is the same as applying $x^2 = \begin{tiny} \left ( \begin{array}{rr} 0 & -i \\ -i & 0 \end{array} \right ) \end{tiny}$, which is in ${\bf{D}}$. Therefore we can equate $x^2$ with the identity matrix, i.e. $x^2 \simeq I = \begin{tiny} \left ( \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right ) \end{tiny}$. Similarly, the action of any $2\times 2$ matrix from ${\bf{D}}$ on a specific vertex is ’equivalent’ the action of the identity on the same vertex. Note that $z \in {\bf{D}}$. The same equivalence holds over $n$ vertices, so we define an equivalence relation with respect to a tensor product of members of ${\bf{D}}$ by the symbol ’$\simeq$’.
Let $u$ and $v$ be two $2 \times 2$ unitary matrices. Then, $$u \simeq v \mv \Leftrightarrow \mv u = dv, \mf d \in {\bf{D}} \enspace.$$\[equiv\]
This equivalence relation allows us to simplify the concatenation of actions of $x$ and $z$ on a specific qubit.
[**[Remark:]{}**]{} Note that $u \simeq v$ cannot be deduced from (and does not imply) $u=vd$ for some $d \in {\bf{D}}$.
We now show that the LC-orbit of an $n$-node graph is found as a subset of the transform spectra w.r.t. $\{I,x,xz\}^n$. Subsequently, it will be shown that we can alternatively find the LC-orbit as a subset of the transform set w.r.t. $\{I,H,N\}^n$. We then re-derive the single LC operation on a graph from the application of $x$ (or $N$) on a single vertex.
The LC-orbit Occurs Within the $\{I,x,xz\}^n$ Set of Transform Spectra {#LCMultispectra}
----------------------------------------------------------------------
We summarise the result of (\[LUEquiv\]) as follows.
Given graphs $G$ and $G'$ as represented by the quadratic boolean functions, $p({\bf{x}})$ and $p'({\bf{x}})$, then $G$ and $G'$ are in the same LC-orbit iff $(-1)^{p'({\bf{x}})} \simeq
U_{v_{t-1}}U_{v_{t-2}}\ldots U_{v_{0}}(-1)^{p({\bf{x}})}$ for some series of $t$ local unitary transformations, $U_{v_i}$. \[GEq\]
From Lemma \[GEq\] we see that, by applying $U_v(G)$ successively for various $v$ to an initial state, one can generate all LC-equivalent graphs within a finite number of steps. (It is evident from the action of LC on a graph that any LC-orbit must be of finite size). Instead of applying $U$ successively, it would be nice to identify a (smaller) transform set in which all LC-equivalent graphs exist as the spectra, to within a post-multiplication by the tensor product of matrices from ${\bf{D}}$. One can deduce from definition \[equiv\] that $zx \simeq x$, and it is easy to verify that
$zxx \simeq I$, and $xzx \simeq zxz$ \[lem1\]
With these definitions and observations we can derive the following theorem.
To within subsequent transformation by tensor products of matrices from ${\bf{D}}$, the LC-orbit of the graph, $G$, over $n$ qubits occurs within the spectra of all possible tensor product combinations of the $2 \times 2$ matrices, $I$, $x$, and $xz$. There are $3^n$ such transform spectra. \[thm1\]
For each vertex in $G$, consider every possible product of the two matrices, $x$, and $z$. Using the equivalence relationship and lemma \[lem1\],
---------------------------------------------------- -- --------------------------
$xxx \simeq x$ $zxx \simeq I$
$xxz \simeq I$ $zxz \simeq xz$
$xzx \simeq zxz \simeq xz$ $zzx \simeq x$
$xzz \simeq zxzz \simeq xzxz \simeq xxzx \simeq x$ $zzz \simeq I \enspace.$
---------------------------------------------------- -- --------------------------
Thus, any product of three or more instances of $x$ and/or $z$ can always be reduced to $I$, $x$, or $xz$. Theorem \[thm1\] follows by recursive application of (\[LUEquiv\]) with these rules, and by noting that the rules are unaffected by the tensor product expansion over $n$ vertices.
For instance, for $n = 2$, the LC-orbit of the graph represented by the quadratic function $p({\bf{x}})$ is found as a subset of the $3^2 = 9$ transform spectra of $(-1)^{p({\bf{x}})}$ w.r.t. the transforms $I \otimes I$, $I \otimes x$, $I \otimes xz$, $x \otimes I$, $x \otimes x$, $x \otimes xz$, $xz \otimes I$, $xz \otimes x$, and $xz \otimes xz$. Theorem \[thm1\] gives a trivial and very loose upper bound on the maximum size of any LC-orbit over $n$ qubits, this bound being $3^n$. It has been computed in [@DanQECC] that the number of LC-orbits for connected graphs for $n = 1$ to $n = 12$ are $1,1,1,2,4,11,26,101,440,3132,40457$, and $1274068$, respectively (see also [@Hein:GrEnt; @Glynn:Tome; @Hohn:Klein; @Dan:Dat; @Slo:Seq]).
The LC-orbit Occurs Within the $\{I,H,N\}^n$ Set of Transform Spectra {#IHNsection}
---------------------------------------------------------------------
One can verify that $N \simeq x$ and $H \simeq xz$. Therefore one can replace $x$ and $xz$ with $N$ and $H$, respectively, so the transform set, $\{I,xz,x\}$ becomes $\{I,H,N\}$. This is of theoretical interest because $H$ defines a 2-point (periodic) Discrete Fourier Transform matrix, and $N$ defines a 2-point negaperiodic Discrete Fourier Transform matrix. In other words a basis change from the rows of $x$ and $xz$ to the rows of $N$ and $H$ provides a more natural set of multidimensional axes in some contexts. For $t$ a non-negative integer, N\^[3t]{} I, N\^[3t+1]{} N, N\^[3t+2]{} H, N\^[24]{} = I \[NPowers\] ,so $N$ could be considered a ’generator’ of $\{I,H,N\}$. The $\{I,H,N\}^n$ transform set over $n$ binary variables has been used to analyse the resistance of certain S-boxes to a form of generalised linear approximation in [@Par:SB]. It also defines the basis axes under which aperiodic autocorrelation of boolean functions is investigated in [@DanAPC]. The [*[Negahadamard Transform]{}*]{}, $\{N\}^n$, was introduced in [@Par:Bent]. Constructions for boolean functions with favourable spectral properties w.r.t. $\{H,N\}^n$ (amongst others) have been proposed in [@Par:LowPAR], and [@Par:QE] showed that boolean functions that are LU-equivalent to indicators for distance-optimal binary error-correcting codes yield favourable spectral properties w.r.t. $\{I,H\}^n$.
A Spectral Derivation of LC {#SpecDeriv}
---------------------------
We now re-derive LC by examining the repetitive action of $N$ on the vector form of the graph states, interspersed with the actions of certain matrices from $D$. We will show that, as with Lemma \[GEq\], these repeated actions not only generate the LC-orbit of the graph, but also generate the $\{I,H,N\}^n$ transform spectra. The LC-orbit can be identified with a subset of the flat transform spectra w.r.t. $\{I,H,N\}^n$. Let $s = (-1)^{p({\bf{x}})}$, where $p({\bf{x}})$ is quadratic and represents a graph $G$. Then the action of $N_v$ on $G$ is equivalent to $U_vs$, where: $$U_v \simeq U'_v = I \otimes \cdots \otimes I \otimes N \otimes I \otimes \cdots \otimes I \enspace,$$ where $N$ occurs at position $v$ in the tensor product decomposition. Let us write $p({\bf{x}})$, uniquely, as, $$p({\bf{x}}) = x_v{\cal{N}}_v({\bf{x}}) + q({\bf{x}}) \enspace,$$ where $q({\bf{x}})$ and ${\cal{N}}_v({\bf{x}})$ are independent of $x_v$ (${\cal{N}}_v({\bf{x}})$ has nothing to do with the Negahadamard kernel, $N_v$). We shall state a theorem that holds for $p({\bf{x}})$ of any degree, not just quadratic, and then show that its specialisation to quadratic $p({\bf{x}})$ gives the required single LC operation. Express ${\cal{N}}_v({\bf{x}})$ as the sum of $r$ monomials, $m_i({\bf{x}})$, as follows, $${\cal{N}}_v({\bf{x}}) = \sum_{i=0}^{r-1} m_i({\bf{x}}) \enspace.$$ For $p({\bf{x}})$ of any degree, the $m_i({\bf{x}})$ are of degree $\le n - 1$. In the sequel we mix arithmetic, mod 2, and mod 4 so, to clarify the formulas for equations that mix moduli, anything in square brackets is computed $(\mo 2)$. The $\{0,1\}$ result is then embedded in $(\mo 4)$ arithmetic for subsequent operations outside the square brackets. We must also define, $${\cal{N}}'_v({\bf{x}}) = \sum_{i=0}^{r-1} [m_i({\bf{x}})] \mf (\mo 4) \enspace.$$
Let $s' = U_vs$, where $s = (-1)^{p({\bf{x}})}$ and $s' = i^{p'({\bf{x}})}$. Then, p’([**[x]{}**]{}) = 2+3’\_v([**[x]{}**]{}) + 3\[x\_v\] (4) \[LC1\] .\[thm2\]
Assign to $A$ and $B$ the evaluation of $p({\bf{x}})$ at $x_v = 0$ and $x_v = 1$, respectively. Thus, $$A = p({\bf{x}})_{x_v = 0} = q({\bf{x}}) \enspace.$$ Similarly, $$B = p({\bf{x}})_{x_v = 1} = {\cal{N}}_v({\bf{x}}) + q({\bf{x}}) \enspace.$$ We need the following equality between mod 2 and mod 4 arithmetic.
$$\sum_{i=1}^n [A_i] \mz (\mo 4)\mf = \mf [\sum_{i=1}^n A_i] + 2[\sum_{i\neq j} A_i A_j] \mz (\mo 4) \wh A_i \in {\mathbb{Z}}_2 \enspace.$$ \[lem2\]
Observe the following action of $N$:
$$\begin{array}{lcr}
\frac{1}{\sqrt{2}}\left ( \begin{array}{rr} 1 & i \\ 1 & -i \end{array} \right )
\left ( \begin{array}{r} 1 \\ 1 \end{array} \right ) & = &
w\left ( \begin{array}{r} 1 \\ -i \end{array} \right ) \\
\frac{1}{\sqrt{2}}\left ( \begin{array}{rr} 1 & i \\ 1 & -i \end{array} \right )
\left ( \begin{array}{r} -1 \\ 1 \end{array} \right ) & = &
w\left ( \begin{array}{r} i \\ -1 \end{array} \right ) \\
\frac{1}{\sqrt{2}}\left ( \begin{array}{rr} 1 & i \\ 1 & -i \end{array} \right )
\left ( \begin{array}{r} 1 \\ -1 \end{array} \right ) & = &
w\left ( \begin{array}{r} -i \\ 1 \end{array} \right ) \\
\frac{1}{\sqrt{2}}\left ( \begin{array}{rr} 1 & i \\ 1 & -i \end{array} \right )
\left ( \begin{array}{r} -1 \\ -1 \end{array} \right ) & = &
w\left ( \begin{array}{r} -1 \\ i \end{array} \right )
\end{array}$$
where $w = e^{2\pi i/8}$. We ignore the global constant, $w$, so that $N$ maps $(-1)^{00}$ to $i^{03}$, $(-1)^{10}$ to $i^{12}$, $(-1)^{01}$ to $i^{30}$ and $(-1)^{11}$ to $i^{21}$. In general, for $A,B \in {\mathbb{Z}}_2$, $\al,\beta \in {\mathbb{Z}}_4$, $(-1)^{AB}$ is mapped by $N_v$ to $i^{\al \beta}$, where, $$\begin{array}{lr}
\al = 2[AB] + [A] + 3[B] & (\mo 4)\\
\beta = 2[AB] + 3[A] + [B] + 3 & (\mo 4)
\end{array}$$ Substituting the previous expressions for $A$ and $B$ into the above and making use of Lemma \[lem2\] gives, $$\begin{array}{lr}
\al({\bf{x}}) = 2[q({\bf{x}})] + 3[{\cal{N}}_v({\bf{x}})] & (\mo 4)\\
\beta({\bf{x}}) = 2[q({\bf{x}})] + [{\cal{N}}_v({\bf{x}})] + 3 & (\mo 4)
\end{array}$$ $p'({\bf{x}})$ can now be written as, $$p'({\bf{x}}) = (3[x_v] + 1)\al({\bf{x}}) + [x_v]\beta({\bf{x}}) \mf (\mo 4) \enspace.$$ Substituting for $\al$ and $\beta$ gives, $$p'({\bf{x}}) = 2[q({\bf{x}})]+2[x_v{\cal{N}}_v({\bf{x}})]
+3[{\cal{N}}_v({\bf{x}})]
+ 3[x_v] \mf (\mo 4)$$ Applying Lemma \[lem2\] to the term $3[{\cal{N}}_v({\bf{x}})]$, $$3[{\cal{N}}_v({\bf{x}})] = 2 \left[ \sum_{j\neq k}
m_j({\bf{x}})m_k({\bf{x}}) \right] + 3{\cal{N}}'_v({\bf{x}}) \mf (\mo 4) \enspace.$$ Furthermore, Lemma \[lem2\] implies that, $$2\left[\sum_{i=1}^n A_i \right] (\mo 4) = 2 \sum_{i=1}^n [A_i] (\mo 4) \wh A_i \in {\mathbb{Z}}_2 \enspace.$$
For $p({\bf{x}})$ a quadratic function, ${\cal{N}}_v({\bf{x}})$ has degree one, so ${\cal{N}}'_v({\bf{x}})$ is a sum of degree-one terms over ${\mathbb{Z}}_4$. Therefore the ${\mathbb{Z}}_4$ degree-one terms, ${\cal{N}}'_v({\bf{x}})$ and $3[x_v]$, can be eliminated from (\[LC1\]) by appropriate subsequent action by members of $\{{\bf{D}}\}^n$ to $s'$. As all monomials, $m_i({\bf{x}})$, are then of degree one, (\[LC1\]) reduces to, p’([**[x]{}**]{}) p([**[x]{}**]{}) + \_[j,k \_v, j k]{} x\_jx\_k (2) \[LC2\] .(\[LC2\]) precisely defines the action of a single LC operation at vertex $v$ of $G$, where we have used $\simeq$ to mean that $(-1)^{p'({\bf{x}})} = BU(-1)^{p({\bf{x}})}$, for some fully tensor-factorisable matrix, $U$, and some $B \in \{{\bf{D}}\}^n$. As $p'({\bf{x}})$ is also quadratic boolean, we can realise successive LC operations on chosen vertices in $G$ via successive actions of $N$ at these vertices, where each action of $N$ be interspersed with the action of a matrix from $\{{\bf{D}}\}^n$ to eliminate ${\mathbb{Z}}_4$-linear terms from (\[LC1\]). In particular, one needs to intersperse with tensor products of $\begin{tiny} \left ( \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array} \right )
\end{tiny}$ and $\begin{tiny} \left ( \begin{array}{ll} 1 & 0 \\ 0 & i \end{array} \right )
\end{tiny}$.
Given a graph, $G$, as represented by $s = (-1)^{p({\bf{x}})}$, with $p({\bf{x}})$ quadratic, the LC-orbit of $G$ comprises graphs which occur as a subset of the spectra w.r.t. $\{I,H,N\}^n$ acting on $s$. \[LCinIHN\]
Define $D_1 \subset D$ such that $$D_1 =
\{\begin{tiny} \left ( \begin{array}{ll} a & 0 \\ 0 & b \end{array} \right )
\end{tiny},\begin{tiny} \left ( \begin{array}{ll} 0 & a \\ b & 0 \end{array} \right )
\end{tiny} \mz | \mz a = 1, b = \pm 1 \} \enspace.$$ Similarly, define $D_2 \subset D$ such that $$D_2 =
\{\begin{tiny} \left ( \begin{array}{ll} a & 0 \\ 0 & b \end{array} \right )
\end{tiny},\begin{tiny} \left ( \begin{array}{ll} 0 & a \\ b & 0 \end{array} \right )
\end{tiny} \mz | \mz a = 1, b = \pm i \}, \wh i^2 = -1 \enspace.$$ Then it is straightforward to establish that, for any $\Delta_1,\Delta_1' \in D_1$, any $\Delta_2,\Delta_2' \in D_2$, and any $c \in \{1,i,-1,-i\}$,
[l]{} N\_1 = c \_1’N H\_1 = c\_1’H\
N\_2 = c \_1H H\_2 = c \_1N .
\[DMove\] Let $\Delta_* \in D_1 \bigcup D_2$. Then, for a vertex, succesive applications of $\Delta_*N$ can, using (\[DMove\]), be re-expressed as, $$\prod (\Delta_*N) = c \Delta_* \prod N \simeq \prod N \enspace.$$ But, from (\[NPowers\]), successive powers of $N$ generate $I$, $H$, or $N$, to within a final multiplication by a member of $D$. It follows that successive LC actions on arbitrary vertices can be described by the action on $s$ of a member of the transform set, $\{I,H,N\}^n$, and therefore that the LC-orbit occurs within the $\{I,H,N\}^n$ transform spectra of $s$.
LC on Hypergraphs
-----------------
For $p({\bf{x}})$ of degree $> 2$, ${\cal{N}}_v({\bf{x}})$ will typically have degree higher than 1, and therefore the expansion of the sum will contribute higher degree terms. For such a scenario we can no longer eliminate the nonlinear and non-boolean term, ${\cal{N}}'_v({\bf{x}})$, from the right-hand side of (\[LC1\]) by subsequent actions from ${\bf{D}}$. Therefore, it is typically not possible to iterate LC graphically beyond one step. We would like to identify hypergraph equivalence w.r.t. unitary transforms, in particular w.r.t. $\{I,H,N\}^n$. Computations have shown that orbits of boolean functions of degree $> 2$ and size greater than one do sometimes exist with respect to $\{I,H,N\}^n$, although they appear to be significantly smaller in size compared to orbits for the quadratic case [@DanAPC].
[*[An interesting open problem is to characterise a ’LC-like’ equivalence for hypergraphs.]{}*]{}
Further spectral symmetries of boolean functions w.r.t. $\{I,H,N\}^n$ are discussed in Appendix B.
Generalised Bent Properties of Boolean Functions {#BentSection}
================================================
Bent Boolean Functions
----------------------
A bent boolean function can be defined by using the WHT. Let $p({\bf{x}})$ be our function over $n$ binary variables. Define the WHT of $p({\bf{x}})$ by, P\_[**[k]{}**]{} = 2\^[-n/2]{}\_[[**[x]{}**]{} GF(2)\^n]{} (-1)\^[p([**[x]{}**]{}) + [**[k x]{}**]{}]{} , \[WHT\] where ${\bf{x,k}} \in {\mbox{GF}}(2)^n$, and $\cdot$ implies the scalar product of vectors.
The WHT of $p({\bf{x}})$ can alternatively be defined as a multiplication of the vector $(-1)^{p({\bf{x}})}$ by $H \otimes H \otimes \ldots \otimes H$. Thus, P = 2\^[-n/2]{}(H H …H)(-1)\^[p([**[x]{}**]{})]{} = 2\^[-n/2]{}(\_[i=0]{}\^[n-1]{} H)(-1)\^[p([**[x]{}**]{})]{} , \[WHTMatrix\] where $P=(P_{(0,\ldots,0)},\ldots,P_{(1,\ldots,1)}) \in {\mathbb{C}}^{2^n}$. $p({\bf{x}})$ is defined to be [*[bent]{}*]{} if $|P_{\bf{k}}| = 1$ $\forall {\bf{k}}$, in which case we say that $p({\bf{x}})$ has a [*[flat spectra]{}*]{} w.r.t. the WHT. In other words, $p({\bf{x}})$ is bent if $P$ is [*[flat]{}*]{}.
Let $\Gamma$ be the binary adjacency matrix associated to $p({\bf{x}})$ when $p({\bf{x}})$ is a quadratic.
[@MacW:Cod] $$p({\bf{x}}) \m{ is {\em{bent}} } \mv \Leftrightarrow \mv \Gamma \m{ has maximum rank, mod 2} \enspace.$$ \[MaxRank\]
It is well-known [@MacW:Cod] that all bent quadratics are equivalent under affine transformation to the boolean function $\left ( \sum_{i=0}^{\frac{n}{2}-1} x_{2i}x_{2i+1} \right ) + {\bf{c \cdot x}} + d$ for $n$ even, where ${\bf{c}} \in {\mbox{GF}}(2)^n$, and $d \in {\mbox{GF}}(2)$. More generally, bent boolean functions only exist for $n$ even. It is interesting to investigate other bent symmetries where affine symmetry has been omitted. In particular, in the context of LC, we are interested in the existence and number of flat spectra of boolean functions with respect to the $\{H,N\}^n$-transform set ([*[bent$_4$]{}*]{}), the $\{I,H\}^n$-transform set ([*[I-bent]{}*]{}), and the $\{I,H,N\}^n$-transform set ([*[I-bent$_4$]{}*]{}). In the following subsections we investigate the bent$_4$, ${\mathbb{Z}}_4$-bent, (Completely) I-bent, LC-bent, and (Completely) I-bent$_4$ properties of connected quadratic boolean functions, where affine symmetry is omitted, and make some general statements about these properties for more general boolean functions.
Bent Properties with respect to $\{H,N\}^n$
-------------------------------------------
We now investigate certain spectral properties of boolean functions w.r.t. $\{H,N\}^n$, where $\{H,N\}^n$ is the set of $2^n$ transforms of the form $\bigotimes_{j \in {\bf{R_H}}} H_j
\bigotimes_{j \in {\bf{R_N}}} N_j,$ where the sets ${\bf{R_H}}$ and ${\bf{R_N}}$ partition $\{0,\ldots,n-1\}$.
The following is trivial to verify: $$p({\bf{x}}) \m{ is bent } \Leftrightarrow p({\bf{x}}) + {\bf{k \cdot x}} + d
\mz \m{ is bent} \enspace,$$ where ${\bf{k}} \in {\mbox{GF}}(2)^n$ and $d \in {\mbox{GF}}(2)$. In other words, if $p({\bf{x}})$ is bent then so are all its affine offsets, mod 2. However the above does not follow if one considers every possible ${\mathbb{Z}}_4$-linear offset of the boolean function. The WHT of $p({\bf{x}})$ with a ${\mathbb{Z}}_4$-linear offset can be defined as follows. P\_[**[k]{},**[c]{}****]{} = 2\^[-n/2]{}\_[[**[x]{}**]{} GF(2)\^n]{} (i)\^[2\[p([**[x]{}**]{}) + [**[k x]{}**]{}\] + \[[**[c x]{}**]{}\]]{} ,[**[c]{}**]{} (2)\^n . \[WHT4\]
$$p({\bf{x}}) \m{ is } \m{{\em{bent$_4$}} } \mv \Leftrightarrow \mv
\exists {\bf{c}} \m{ such that } |P_{{\bf{k},\bf{c}}}| = 1 \mf
\forall {\bf{k}} \in {\m{GF}}(2)^n \enspace.$$ \[Bent4Def\]
Let ${\bf{R_N}}$ and ${\bf{R_H}}$ partition $\{0,1,\ldots,n-1\}$. Let, $$U = \bigotimes_{j \in {\bf{R_H}}} H_j \bigotimes_{j \in {\bf{R_N}}} N_j \enspace.$$ s’ = U(-1)\^[p([**[x]{}**]{})]{} \[HNeq\] .
$p({\bf{x}})$ is bent$_4$ if there exists one or more partitions, ${\bf{R_N}},{\bf{R_H}}$ such that $s'$ is [*[flat]{}*]{}. \[HN\]
The rows of $U$ can be described by $(i)^{f({\bf{x}})}$, where ${\bf{x}} = (x_0,x_1,\ldots,x_{n-1})$, where $f$ is linear, $f : {\mbox{GF}}(2)^n \rightarrow {\mbox{GF}}(4)$, and the coefficient of $x_j$ in $f \in \{0,2\}$ for $j \in {\bf{R_H}}$ and $f \in \{1,3\}$ for $j \in {\bf{R_N}}$. Therefore $s'$ can always, equivalently, be expressed as $s' = (\bigotimes H)(i)^{2p[{\bf{x}}] + [f'({\bf{x}})]}$, where $f'$ is linear, $f':{\mbox{GF}}(2)^n \rightarrow {\mbox{GF}}(2)$, and the coefficient of $x_j$ in $f'$ is $0$ for $j \in {\bf{R_H}}$, and $1$ for $j \in {\bf{R_N}}$.
An alternative way to define the bent$_4$ property for $p({\bf{x}})$ quadratic is via a modified form of the adjacency matrix.
For quadratic $p({\bf{x}})$, $$p({\bf{x}}) \m{ is bent$_4$ } \mv \Leftrightarrow \mv \Gamma_{\bf{v}}
\m{ has maximum rank, mod 2, for some } {\bf{v}} \in {\mbox{GF}}(2)^n \enspace.$$ \[MaxRank4\]
where $\Gamma_{\bf{v}}$ is a modified form of $\Gamma$ with $v_i$ in position $[i,i]$, where ${\bf{v}} = (v_0,v_1,\ldots,v_{n-1})$.
We first show that the transform of $(-1)^{p({\bf{x}})}$ by tensor products of $H$ and $N$ produces a flat spectra if and only if the associated periodic and negaperiodic autocorrelation spectra have zero out-of-phase values. We then show how these autocorrelation constraints lead directly to constraints on the associated adjacency matrix.
Consider a function, $p$, of just one variable, $x_0$, and let $s = (-1)^{p({x_0})}$. Define the periodic autocorrelation function as follows, $$a_k = \sum_{x_0 \in {GF}(2)} (-1)^{p(x_0) + p(x_0 + k)}, \mf k \in {\mbox{GF}}(2) \enspace.$$ Then it is well-known that $s' = Hs$ is a flat spectrum if and only if $a_k = 0$ for $k \ne 0$.
Define the negaperiodic autocorrelation function as follows, $$b_k = \sum_{x_0 \in {GF}(2)} (-1)^{p(x_0) + p(x_0 + k) + k(x_0 + 1)}, \mf k \in {\mbox{GF}}(2) \enspace.$$ Then $s' = Ns$ is a flat spectrum if and only if $b_k = 0$ for $k \ne 0$. (For $p$ a boolean function of just one variable, $Hs$ is never flat and $Ns$ is always flat, but this only holds for one variable).
We now elaborate on the above two claims. Define $s(z) = s_0 + s_1z$, $a(z) = a_0 + a_1z$, and $b(z) = b_0 + b_1z$. Then the periodic and negaperiodic relationships between autocorrelation and fourier spectra, as claimed above, follow because periodic autocorrelation can be realised by the polynomial multiplication, $a(z) = s(z)s(z^{-1})$ mod $(z^2 - 1)$, with associated residue reduction, mod $(z-1)$ and mod $(z+1)$, realised by $s' = Hs =
\begin{tiny} \frac{1}{\sqrt{2}}\left ( \begin{array}{rr}
1 & 1 \\
1 & -1
\end{array} \right ) \end{tiny}s$ (with the Chinese Remainder Theorem realised by $H^{\dag}s'$, where ’$\dag$’ means transpose conjugate). By Parseval, $s'$ can only be flat if $a_1 = 0$. Similarly, negaperiodic autocorrelation can be realised by the polynomial multiplication, $b(z) = s(z)s(z^{-1})$ mod $(z^2 + 1)$, with associated residue reduction, mod $(z-i)$ and mod $(z+i)$, realised by $s' = Ns =
\begin{tiny} \frac{1}{\sqrt{2}}\left ( \begin{array}{rr}
1 & i \\
1 & -i
\end{array} \right ) \end{tiny}s$ (with the Chinese Remainder Theorem realised by $N^{\dag}s'$). By Parseval, $s'$ can only be flat if $b_1 = 0$.
We extend this autocorrelation $\leftrightarrow$ Fourier spectrum duality to $n$ binary variables by defining multivariate forms of the above polynomial relationships. If we choose periodic autocorrelation for indices in ${\bf{R_H}}$ and negaperiodic autocorrelation for indices in ${\bf{R_N}}$, we obtain the autocorrelation spectra, A\_[[**[k]{}**]{},[**[R\_H]{}**]{},[**[R\_N]{}**]{}]{} = \_[[**[x]{}**]{} GF(2)\^n]{} (-1)\^[p([**[x]{}**]{}) + p([**[x]{}**]{}+ [**[k]{}**]{}) + \_[i=0]{}\^[n-1]{} \_[\_[**[R\_N]{}**]{}]{}(i)k\_i (x\_i + 1)]{} , \[spectra\] where ${\bf{k}}=(k_0,k_1,\ldots,k_{n-1}) \in {\mbox{GF}}(2)^n$, and $\chi_{_{\bf{R_N}}}(i)$ is the characteristic function of ${\bf{R_N}}$, i.e, $$\chi_{_{\bf{R_N}}}(i)=\left\{\begin{array}{l}
1,\ i\in{\bf{R_N}}\\
0,\ i\notin{\bf{R_N}}\end{array}\right.$$ In polynomial terms, with ${\bf{z}} \in {\mbox{GF}}(2)^n$ and $s({\bf{z}}) =
\sum_{{\bf{j}} \in GF(2)^n} s_{\bf{j}} \prod_{i=0}^{n-1} z_i^{j_i}$, we have,
[ll]{} A\_[[**[R\_H]{}**]{},[**[R\_N]{}**]{}]{}([**[z]{}**]{}) & = \_[[**[k]{}**]{} GF(2)\^n]{} A\_[[**[k]{}**]{},[**[R\_H]{}**]{},[**[R\_N]{}**]{}]{} \_[i=0]{}\^[n-1]{} z\_i\^[k\_i]{}\
& = s(z\_0,z\_1,…,z\_[n-1]{})s(z\_0\^[-1]{},z\_1\^[-1]{},…,z\_[n-1]{}\^[-1]{}) \_[i=0]{}\^[n-1]{} (z\_i\^2 - (-1)\^[\_[**[R\_N]{}**]{}(i)]{}) .
\[ACFPol\]
Then, by appealing to a multivariate version of Parseval’s Theorem, $s'$ as defined in (\[HNeq\]) is flat if and only if $A_{{\bf{k}},{\bf{R_H}},{\bf{R_N}}} = 0$, $\forall\ {\bf{k}} \ne {\bf{0}}$.
These constraints on the autocorrelation coefficients of $s$ translate to requiring a maximum rank property for a modified adjacency matrix, as follows. The condition $A_{\bf{k},{\bf{R_H}},{\bf{R_N}}} = 0$ for ${\bf{k}} \ne {\bf{0}}$ is equivalent to requiring that, if we compare the function with its multidimensional periodic and negaperiodic rotations (but for the identity rotation), the remainder should be a balanced function. When dealing with quadratic boolean functions, the remainder is always linear or constant. This gives us a system of linear equations represented by the binary adjacency matrix, $\Gamma$, of $p({\bf{x}})$, with a modified diagonal, that is with $\Gamma_{i,i} = 1$ for all $i \in {\bf{R_N}}$, and $\Gamma_{i,i} = 0$ otherwise. Let $$p(x_{0},x_{1},\ldots,x_{n-1})=a_{01}x_{0}x_{1}+a_{02}x_{0}x_{2}+\cdots+a_{ij}x_{i}x_{j}+\cdots+a_{n-2,n-1}x_{n-2}x_{n-1} \enspace.$$
Therefore, $$\begin{array}{lcl}
p({\bf{x}}) + p({\bf{x}} + {\bf{k}})+\sum_{i=0}^{n-1} \chi_{_{\bf{R_N}}}(i)k_i x_i & = & k_0(\chi_{_{\bf{R_N}}}(0)x_0+a_{01}x_{1}+a_{02}x_{2}+\cdots+a_{0,n-1}x_{n-1})\\
& + & k_1(a_{01}x_{0}+\chi_{_{\bf{R_N}}}(1)x_1+a_{02}x_{2}+\cdots+a_{0,n-1}x_{n-1})+\cdots\\
& + & k_{n-1}(a_{0,n-1}x_{0}+\cdots+a_{n-2,n-1}x_{n-2}+\chi_{_{\bf{R_N}}}(n-1)x_{n-1}) \enspace.
\end{array}$$ This is equal to: $$\begin{array}{l}
x_{0}(\chi_{_{\bf{R_N}}}(0)k_0+a_{01}k_1+\cdots+a_{0n}k_n)+x_{1}(a_{01}k_0+\chi_{_{\bf{R_N}}}(1)k_1+
\cdots a_{1,n-1}k_{n-1})\\
+ \cdots+x_{n-1}(a_{0,n-1}k_0+a_{1,n-1}k_1+\cdots+a_{n-2,n-1}k_{n-2}+\chi_{_{\bf{R_N}}}(n-1)k_{n-1}) \enspace,
\end{array}$$ which is balanced unless constant. The constant $\sum_{i=0}^{n-1} \chi_{_{\bf{R_N}}}(i)k_i$ will not play any role in the equation $A_{\bf{k}}=0$, and can be ignored. We have the the following system of equations: $$\begin{array}{cl}
&\chi_{_{\bf{R_N}}}(0)k_0+a_{01}k_1+a_{02}k_2+\cdots+a_{0,n-1}k_{n-1}=0\\
&a_{01}k_0+\chi_{_{\bf{R_N}}}(1)k_1+a_{12}k_2+\cdots+a_{1,n-1}k_{n-1}=0\\
&.................................................................................\\
&a_{0,n-1}k_0+a_{1,n-1}k_1+\cdots+a_{n-2,n-1}k_{n-2}+\chi_{_{\bf{R_N}}}(n-1)k_{n-1}=0 \enspace.
\end{array}$$ Writing this system as a matrix, we have:
$$\left(\begin{array}{ccccc}
\chi_{_{\bf{R_N}}}(0) & a_{01} & a_{02} & \ldots & a_{0,n-1}\\
a_{01} & \chi_{_{\bf{R_N}}}(1) & a_{12} & \ldots & a_{1,n-1}\\
a_{02} & a_{12} & \chi_{_{\bf{R_N}}}(2) & \ldots & a_{2,n-1}\\
\vdots &\vdots &\vdots & \ddots & \vdots\\
a_{0,n-1} & a_{1,n-1} & a_{2,n-1} & \ldots & \chi_{_{\bf{R_N}}}(n-1)\end{array}\right) \enspace.$$
This is a modification of $\Gamma$, with 1 or 0 in position $i$ of the diagonal depending on whether $i\in{\bf{R_N}}$ or $i\in{\bf{R_H}}$.
In general, $$p({\bf{x}}) \m{ is bent }
\begin{array}{l} \Rightarrow \\
{\not \Leftarrow} \end{array} \mf p({\bf{x}}) \m{ is bent$_4$} \enspace.$$
All boolean functions of degree $\le 2$ are bent$_4$. \[Z4Bent\]
Degree zero and degree one functions are trivial. Consider the adjacency matrix, $\Gamma$, associated with the quadratic boolean function, $p({\bf{x}})$. We now prove that $\Gamma_{\bf{v}}$ has maximum rank (mod 2) for at least one choice of ${\bf{v}}$, where $\Gamma_{\bf{v}} = \Gamma + \m{diag}({\bf{v}})$ as before. Let $M$ be the minor associated with the first entry of $\Gamma$; in other words, let
${\Gamma}= \begin{tiny} \left ( \begin{array}{ll}
0 & \ \\
\ & M
\end{array} \right ) \end{tiny} \enspace.$
We prove by induction that there exists at least one choice of $v$ such that $\Gamma_{\bf{v}}$ has maximum rank (mod 2). The theorem is true for $n=2$: in this case,
${\Gamma}= \begin{tiny} \left ( \begin{array}{ll}
0 & a \\
a & 0
\end{array} \right ) \end{tiny} \enspace.$
Then, either det$({\Gamma})=1$, in which case we choose ${\bf{v}}=(0,0)$, or we have $a=0$ (empty graph). In the last case we choose ${\bf{v}}=(1,1)$, so $det(\Gamma_{\bf{v}})=1+a=1$. Suppose the theorem is true for $n-1$ variables. We will see that it is true for $n$ variables. If the determinant of ${\Gamma}$ is 1 we take ${\bf{v}}=(0,\ldots,0)$ and we are done. If det$({\Gamma})=0$, then we have two cases:
- det$(M)=1$: Take ${\bf{v}}=(1,0,\ldots,0)$.
- $det(M)=0$: By the induction hypothesis there is at least one choice of ${\bf{v}}(M) \in {\mbox{GF}}(2)^{n-1}$, where ${\bf{v}}(M)=(v_1,\ldots,v_{n-1})$ such that $M_{{\bf{v}}(M)}$ has full rank. Let ${\bf{v}}'=(0,v_1,\ldots,v_{n-1}) \in {\mbox{GF}}(2)^n$. If det$({\Gamma_{\bf{v'}}}) = 1$ we have finished. If det$({\Gamma_{\bf{v'}}}) = 0$ we are in the first case again, so we take ${\bf{v}}=(1,v_1,\ldots,v_{n-1})$, and we are done.
The theorem follows from lemma \[MaxRank4\].
[**[Remark:]{}**]{} Theorem \[Z4Bent\] is true even for boolean functions associated with non-connected or empty graphs.
Not all boolean functions of degree $ > 2$ are bent$_4$. \[NotAllBent4\]
Counter-example - by computation there are no bent$_4$ cubics of three variables.
Further computations show that there are no bent$_4$ boolean functions of four variables of degree $> 2$. Similarly, there are only $252336$ bent$_4$ cubic boolean functions in five variables (out of a possible $2^{20} - 2^{10}$, not including affine offsets), and no bent$_4$ boolean functions of degree $\ge 4$ in five variables. bent$_4$ cubics of six variables do exist. Lemma \[NotAllBent4\] identifies an open problem:
$$p({\bf{x}}) \m{ is } {\em{{\mathbb{Z}}_4\m{-bent }}} \Leftrightarrow
|P_{{\bf{k}},\bf{c}}| = 1 \mf \forall {\bf{c,k}} \in {\mbox{GF}}(2)^n \enspace.$$ \[Z4BentDef\]
The definition requires that [**[all]{}**]{} ${\mathbb{Z}}_4$-linear offsets of the boolean function, $p({\bf{x}})$, are flat w.r.t. the WHT. WE prove that no such boolean functions exist, first for all boolean functions of degree $\le 2$, and then for all boolean functions.
There are no ${\mathbb{Z}}_4$-bent boolean functions of degree $\le 2$. \[NoZ4BentQuad\]
This is trivial for degree zero and degree one functions. Consider the adjacency matrix, $\Gamma$, associated with the quadratic boolean function, $p({\bf{x}})$. The theorem is equivalent to proving that there is a ${\bf{v}}$ such that $\Gamma_{\bf{v}}$ has rank less than maximal. Then:
1. if $p({\bf{x}})$ is not bent, then we take ${\bf{v}}=(0,\ldots,0)$ and we are done.
2. if $p({\bf{x}})$ is bent, we take $M$ as in the proof for Theorem \[Z4Bent\]. If $det(M)=1$, we take ${\bf{v}}=(1,0,\ldots,0)$ and we are done; if $det(M)=0$, modify the diagonal as in the proof for Theorem \[Z4Bent\]. If the determinant of the new matrix is equal to $0$, we are done; if not, we are in case 1.
There are no ${\mathbb{Z}}_4$-bent boolean functions. \[NoZ4Bent\]
Consider the proof of Lemma \[MaxRank4\]. We have established that, for a fixed choice of ${\bf{R_H}}$ and ${\bf{R_N}}$, $s'$, as defined in (\[HNeq\]), is flat if and only if $A_{{\bf{k}},{\bf{R_H}},{\bf{R_N}}} = 0$, $\forall {\bf{k}}$, ${\bf{k \ne 0}}$. Therefore $p({\bf{x}})$ is ${\mathbb{Z}}_4$-bent iff $A_{{\bf{k}},{\bf{R_H}},{\bf{R_N}}} = 0$, $\forall {\bf{k}}$, ${\bf{k \ne 0}}$, for [**[all]{}**]{} partitions $\{{\bf{R_H,R_N}}\}$. In particular, if $p({\bf{x}})$ is ${\mathbb{Z}}_4$-bent, then the polynomials, $A_{{\bf{R_H}},{\bf{R_N}}}({\bf{z}})$, as defined in (\[ACFPol\]), satisfy $A_{{\bf{R_H}},{\bf{R_N}}}({\bf{z}}) = 2^n$ for all choices of ${\bf{R_H}}$ and ${\bf{R_N}}$ (i.e. their out-of-phase coefficients are all zero). By the Chinese Remainder Theorem (CRT) we can combine these polynomials for each choice of ${\bf{R_H}}$ and ${\bf{R_N}}$ to construct the polynomial, r([**[z]{}**]{}) \_[j=0]{}\^n (z\_j\^4 - 1) = {A\_[[**[R\_H]{}**]{},[**[R\_N]{}**]{}]{}([**[z]{}**]{}) | ,[**[R\_N]{}**]{}} \[FullCRT\] ,where $r({\bf{z}}) =
s(z_0,z_1,\ldots,z_{n-1})s(z_0^{-1},z_1^{-1},\ldots,z_{n-1}^{-1})$.
But as $r({\bf{z}})$ comprises monomials containing only $z_i^{-1},z_i^0,z_i^1$, the modular restriction in (\[FullCRT\]) has no effect on coefficient magnitudes, and $$r({\bf{z}}) \equiv r({\bf{z}}) \mo \prod_{j=0}^n (z_j^4 - 1) \enspace.$$ to within a multiplication of the coefficients by $\pm 1$. It follows, by application of the CRT to (\[FullCRT\]) that, if $A_{{\bf{R_H}},{\bf{R_N}}}({\bf{z}}) = 2^n$, $\forall {\bf{R_H}},{\bf{R_N}}$, then $r({\bf{z}}) = 2^n$ also, i.e. $r({\bf{z}})$ is integer. But this is impossible as the coefficients of the maximum degree terms, $\prod_j z_j^{-1^{u_j}}$, $u_j \in {\mathbb{Z}}_2$, in $r({\bf{z}})$ can never be zero, but are always $\pm 1$. Therefore $p({\bf{x}})$ can never be ${\mathbb{Z}}_4$-bent.
[**[Remark: ]{}**]{} Although we proved for boolean functions, it is possible to generalise the proof so as to state that no function from ${\mbox{GF}}(2)^n \rightarrow {\mbox{GF}}(q)$ can be ${\mathbb{Z}}_4$-bent, for any even integer $q$.
Bent Properties with respect to $\{I,H\}^n$
-------------------------------------------
We now investigate certain spectral properties of boolean functions w.r.t. $\{I,H\}^n$, where $\{I,H\}^n$ is the set of $2^n$ transforms of the form $\bigotimes_{j \in {\bf{R_I}}} I_j \bigotimes_{j \in {\bf{R_H}}} H_j,$ where the sets ${\bf{R_I}}$ and ${\bf{R_H}}$ partition $\{0,\ldots,n-1\}$. [@Par:QE] has investigated other spectral properties w.r.t. $\{I,H\}^n$, such as [*[weight hierarchy]{}*]{} if the graph is bipartite.
The WHT of the subspace of a function from ${\mbox{GF}}(2)^n$ to ${\mbox{GF}}(2)$, obtained by fixing a subset, ${\bf{R_I}}$, of the input variables, can be defined as follows. Let ${\bf{\theta }} \in {\mbox{GF}}(2)^n$ be such that $\theta_j = 1$ iff $j \in {\bf{R_I}}$. Let ${\bf{r \preceq \theta }}$, where ’$\preceq$’ means that $\theta$ ’[*[covers]{}*]{}’ ${\bf{r}}$, i.e. $r_i \le \theta_i$, $\forall i$. Then, P\_[**[k]{},**[r]{},********]{} = 2\^[-(n - ())/2]{}\_[[**[x]{}**]{} = [**[r + y]{}**]{} | [**[y]{}**]{} ]{} (-1)\^[p([**[x]{}**]{}) + [**[k x]{}**]{}]{} , [**[r]{}**]{} **\[IHEQU\] .**
$$p({\bf{x}}) \m{ is } \m{{\em{I-bent}} } \mv \Leftrightarrow \mv
\exists \theta \m{ such that } |P_{{\bf{k},\bf{r},\bf{\theta}}}| = 1 \mf
\forall {\bf{k}} \preceq {\bf{\bar{\theta}}}, \forall \bf{r} \preceq \bf{\theta} \enspace,$$ where $\m{wt}(\bf{\theta}) < n$. \[IBentDef\]
Let U = \_[j ]{} I\_j \_[j ]{} H\_j . \[UIH\] s’ = U(-1)\^[p([**[x]{}**]{})]{} \[IHeq\] .
$p({\bf{x}})$ is I-bent if there exist one or more partitions, ${\bf{R_I}},{\bf{R_H}}$ such that $s'$ is flat, where $|{\bf{R_I}}| < n$. \[DefIH\]
An alternative way to define the I-bent property of $p({\bf{x}})$ is via its associated adjacency matrix, $\Gamma$. Let $\Gamma_I$ be the adjacency matrix obtained from $\Gamma$ by deleting all rows and columns of $\Gamma$ with indices in ${\bf{R_I}}$.
For quadratic $p({\bf{x}})$, $$p({\bf{x}}) \m{ is I-bent } \mv \Leftrightarrow \mv \Gamma_I
\m{ has maximum rank, mod 2}$$ for one or more choices of ${\bf{R_I}}$ where $|{\bf{R_I}}| < n$. \[MaxRankIH\]
In general, $$p({\bf{x}}) \m{ is bent }
\begin{array}{l} \Rightarrow \\
{\not \Leftarrow} \end{array} \mf p({\bf{x}}) \m{ is I-bent} \enspace.$$
All boolean functions in two or more variables of degree $\le 2$ are I-bent. \[I-Bent\]
Degree zero and degree one functions are trivial. It is easy to show that all quadratic boolean functions of 2 variables are I-bent. The theorem follows by observing that all adjacency matrices, $\Gamma$, representing quadratic functions of $n > 2$ variables contain $2 \times 2$ submatrices, obtained from $\Gamma$ by deleting all rows and columns of $\Gamma$ with indices ${\bf{R_I}}$, for $|{\bf{R_I}}| = n - 2$.
Not all boolean functions of degree $ > 2$ are I-bent. \[NotAllIBent\]
Counter-example - by computation there are no I-bent cubics of three variables.
Further computations show that there are only 416 I-bent cubics in four variables, and no I-bent quartics in four variables. There are only 442640 I-bent cubics, only 1756160 I-bent quartics in five variables, and no I-bent quintics in five variables. I-bent cubics in six variables do exist. Lemma \[NotAllIBent\] indicates an open problem:
$$p({\bf{x}}) \m{ is } \m{{\em{Completely I-bent}} } \Leftrightarrow
|P_{{\bf{k}},\bf{r},\bf{\theta}}| = 1 \mf \forall \bf{\theta},{\bf{k}},{\bf{r}}, \mv
{\bf{k}} \preceq {\bar{\theta}}, {\bf{r}} \preceq \theta \enspace.$$ \[CIBentDef\]
There are no Completely I-bent boolean functions. \[NoCIBent\]
Let $s = (-1)^{p({\bf{x}})}$. Let $|{\bf{R_I}}| = n - 1$. Then for $U$ as defined in (\[UIH\]), $s'$ cannot be flat.
Bent Properties with respect to $\{I,H,N\}^n$
---------------------------------------------
The $\{H,N\}^{n - |{\bf{R_I}}|}$ set of transforms of the subspace of a function from ${\mbox{GF}}(2)^n$ to ${\mbox{GF}}(2)$, obtained by fixing a subset, ${\bf{R_I}}$, of the input variables, is defined as follows. Let ${\bf{\theta }} \in {\mbox{GF}}(2)^n$ be such that $\theta_j = 1$ iff $j \in {\bf{R_I}}$. Let ${\bf{r \preceq \theta }}$. Then, P\_[**[k]{},**[c]{},**[r]{},[****]{}******]{} = 2\^[-(n - ())/2]{}\_[[**[x]{}**]{} = [**[r + y]{}**]{} | [**[y]{}**]{} ]{} (i)\^[2\[p([**[x]{}**]{}) + [**[k x]{}**]{}\] + \[[**[c x]{}**]{}\]]{} , [**[r]{}**]{} . \[IHNEQU\]
$$p({\bf{x}}) \m{ is } \m{{\em{I-bent$_4$}} } \mv \Leftrightarrow \mv
\exists {\bf{c}}, {\bf{\theta }} \m{ such that }
|P_{{\bf{k},\bf{c},\bf{r},{\bf{\theta }}}}| = 1 \mf
\forall {\bf{k}} \preceq {\bar{\bf{\theta }}}, \forall \bf{r} \preceq {\bf{\theta }} \enspace,$$ where $\m{wt}({\bf{\theta }}) < n$. \[I-Bent4Def\]
Let ${\bf{R_I}}$, ${\bf{R_H}}$ and ${\bf{R_N}}$ partition $\{0,1,\ldots,n-1\}$. Let, U = \_[j ]{} I\_j \_[j ]{} H\_j \_[j ]{} N\_j . \[UIHN\] s’ = U(-1)\^[p([**[x]{}**]{})]{} . \[IHNeq\]
$p({\bf{x}})$ is I-bent$_4$ if there exists one or more partitions, ${\bf{R_I}},{\bf{R_H}},{\bf{R_N}}$ such that $s'$ is [*[flat]{}*]{}, where $|{\bf{R_I}}| < n$. \[IHN\]
As a generalization of (\[spectra\]), we get flat spectra for one or more partitions ${\bf{R_I}},{\bf{R_H}},{\bf{R_N}}$ iff $$A_{k,{\bf{R_I}},{\bf{R_H}},{\bf{R_N}}}=
\sum_{{\bf x} = {\bf{r + y}} | {\bf{y}} \preceq {\bar{\bf{\theta }}}}(-1)^{p({\bf{x}}) +
p({\bf{x}}+ {\bf{k}})
+ \sum_{i=0}^{n-1} \chi_{_{\bf{R_N}}}(i)k_i (x_i + 1)}=0, \mf \forall {\bf{k}}\neq{\bf{0}} \enspace,$$ where $\theta_j = 1$ iff $j \in {\bf{R_I}}$, ${\bf{r}} \preceq \theta$, and $r_j = k_j$ if $j \in {\bf{R_I}}$.
An alternative way to define the I-bent$_4$ property when $p({\bf{x}})$ is quadratic is via its associated adjacency matrix, $\Gamma$. Let $\Gamma_{I,{\bf{v}}}$ be the matrix obtained from $\Gamma_{\bf{v}}$ when we erase the $i^{th}$ row and column if $i\in{\bf{R_I}}$.
For quadratic $p({\bf{x}})$, $$p({\bf{x}}) \m{ is I-bent$_4$ } \mv \Leftrightarrow \mv \Gamma_{I,{\bf{v}}}
\m{ has maximum rank, mod 2}, \wh {\bf{v}} \preceq \bf{\bar{\theta}}$$ for one or more choices of ${\bf{v}}$ and $\theta$ where $\m{wt}(\theta) < n$. \[MaxRankIHN\]
In general, $$p({\bf{x}}) \m{ is bent }
\begin{array}{l} \Rightarrow \\
{\not \Leftarrow} \end{array}
\begin{array}{l}
\mf p({\bf{x}}) \m{ is bent$_4$} \\
\mf p({\bf{x}}) \m{ is I-bent}
\end{array}
\begin{array}{l} \Rightarrow \\
{\not \Leftarrow} \end{array}
\mf p({\bf{x}}) \m{ is I-bent$_4$} \enspace.$$
All boolean functions of degree $\le 2$ are I-bent$_4$. \[I-Bent4\]
Follows from Theorems \[Z4Bent\] and \[I-Bent\].
All boolean functions are I-bent$_4$. \[NotAllIBent4\]
From Theorem \[thm2\], the action of a single $U_v$ on a boolean function, $p({\bf{x}})$, of any degree, always gives a flat output spectra, for any value of $v$. This gives (at least) $n$ flat spectra for any boolean function.
$$p({\bf{x}}) \m{ is } \m{{\em{Completely I-bent$_4$}} } \Leftrightarrow
|P_{{\bf{k}},{\bf{c}},\bf{r},\bf{\theta}}| = 1 \mf \forall
\bf{\theta},{\bf{c}},{\bf{k}},{\bf{r}}, \mv
{\bf{k}},{\bf{c}} \preceq {\bar{\theta}}, {\bf{r}} \preceq \theta \enspace.$$ \[CIBent4Def\]
There are no completely I-bent$_4$ boolean functions. \[NoCIBent4\]
Follows from theorems \[NoZ4Bent\] or \[NoCIBent\].
It is natural to ask whether, for a given quadratic, $p({\bf{x}})$, there exists at least one member of its LC-orbit which is bent. If so, then we state that the graph state, $p({\bf{x}})$, and its associated LC-orbit, is [*[LC-bent]{}*]{}. More formally,
The graph state, $p({\bf{x}})$ (a quadratic boolean function), and its associated LC-orbit is [*[LC-bent]{}*]{} if $\exists\ p'({\bf{x}})$ such that $p'({\bf{x}}) \in \m{LC-orbit}(p({\bf{x}}))$, and such that $p'({\bf{x}})$ is bent. \[LC-Bent\]
For example, the bent function $x_0x_1 + x_0x_2 + x_0x_3 + x_1x_2 + x_1x_3 + x_2x_3$ is in the same LC-orbit as $x_0x_1 + x_0x_2 + x_0x_3$ so, although $x_0x_1 + x_0x_2 + x_0x_3$ is not bent, it is LC-bent.
In general, for $p({\bf{x}})$ quadratic, $$p({\bf{x}}) \m{ is bent}
\begin{array}{l} \Rightarrow \\
{\not \Leftarrow} \end{array} \mf p({\bf{x}}) \m{ is LC-bent} \enspace.$$
Not all quadratic boolean functions are LC-bent. \[notallLCBent\]
By computation, the LC-orbit associated with the $n = 6$-variable boolean function, $x_0x_4 + x_1x_5 + x_2x_5 + x_3x_4 + x_4x_5$ is not LC-bent.
By computation it was found that all quadratic boolean functions of $n \le 5$ variables are LC-bent. Table I lists orbit representatives for those orbits which are not LC-bent, for $n = 2$ to $9$, and provides a summary for $n = 10$, where the boolean functions are abbreviated so that, say, $ab,de,fg$ is short for $x_ax_b + x_dx_e + x_fx_g$. For those orbits which are not LC-bent we provide the maximum rank satisfied by a graph within the orbit.
$n$ ANF for the orbit representative Max. Rank within Orbit
----- ---------------------------------- ------------------------
2-5 - -
6 04,15,25,34,45 4
7 - -
8 07,17,27,37,46,56,67 6
06,17,27,37,46,56,67 6
07,17,25,36,46,57,67 6
06,17,27,36,45,46,47,56,57,67 6
07,16,26,35,45,47,67 6
9 08,18,28,38,47,57,67,78 6
08,18,26,37,47,56,68,78 6
10 08,19,29,39,49,58,68,78,89 6
51 other orbits 8
: Representatives for all LC-Orbits which are not LC-bent for $n = 2$ to $10$
\[notLCBent\]
Conclusion
==========
This paper has examined the spectral properties of boolean functions with respect to the transform set formed by tensor products of the identity, $I$, the Walsh-Hadamard kernel, $H$, and the Negahadamard kernel, $N$ (the $\{I,H,N\}^n$ transform set). In particular, the idea of a bent boolean function was generalised in a number of ways to $\{I,H,N\}^n$. Various theorems about the generalised bent properties of boolean functions were established. It was shown how a quadratic boolean function maps to a graph and it was shown how the local unitary equivalence of these graphs can be realised by successive application of the LC operation - Local Complementation - or, alternatively, by identifying a subset of the flat spectra with respect to $\{I,H,N\}^n$. For quadratic boolean functions it was further shown how the $\{I,H,N\}^n$ set of transform spectra could be characterised by looking at the ranks of suitably modified versions of the adjacency matrix. In the second part of the paper, we will apply this method to enumerate the flat spectra w.r.t. $\{I,H\}^n$, $\{H,N\}^n$ and $\{I,H,N\}^n$ for certain concrete functions [@RP:BCII].
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Appendix A - Various Interpretations of the Graph States {#graphstates}
========================================================
In this section we briefly characterise graph states.
Interpretation as a Quantum Error Correcting Code
-------------------------------------------------
Let $E$ be a $2n$-dimensional binary vector space, whose elements are written as $(a|b)$, where $a,\ b\ \in \mbox{GF}(2)^n$, and $E$ is equiped with the (symplectic) inner product $((a|b),(a'|b'))=a\cdot b'+a'\cdot b$. Define the [*weight*]{} of $(a|b)=(a_1,\ldots,a_n|b_1,\ldots,b_n)$ as the number of coordinates $i$ such that at least one of the $a_i$ or $b_i$ is 1. The distance between two elements $(a|b)$ and $(a'|b')$ is defined to be the weight of their difference.
\[bin\] [@Cald:Qua] Let $S$ be a $(n-k)$ - dimensional linear subspace of $E$, contained in its dual $S^\perp$ (with respect to the inner product), such that there are no vectors of weight $< d$ in $S\setminus S^\perp$. By taking an eigenspace of $S$ (for any chosen linear character) we obtain a quantum error-correcting code mapping $k$ qubits to $n$ qubits that corrects $[(d-1)/2]$ errors. Such a code is called an [*additive quantum error-correcting code (QECC)*]{}, and is described by its parameters, $[[n,k,d]]$, where $d$ is the [*minimal distance*]{} of the code.
We show, later, that a $[[n,0,d]]$ QECC can be represented by a graph. First we re-express the QECC as a $\GF(4)$ additive code.
Interpretation as a $\GF(4)$ Additive Code
------------------------------------------
From [@Cald:Qua] we see how to interpret the binary space $E$ as the space GF(4)$^n$ and thereby how to derive a QECC from an additive (classical) code over GF(4)$^n$. Let $GF(4)=\{0,1,\omega,\bar{\omega}\}$, with $\omega^2=\omega+1$, $\omega^3=1$; and conjugation defined by $\bar{\omega}=\omega^2=\omega+1$. The [*Hamming weight*]{} of a vector in GF(4)$^n$, written $wt(u)$, is the number of non-zero components, and the [*Hamming distance*]{} between $u,u'\in$ GF(4)$^n$ is dist$(u,u')=wt(u + u')$. Define the [*trace function*]{} as: $tr(x): GF(4) \rightarrow GF(2),\ tr(x)=x+\bar{x}$. To each vector $v=(a|b)\in E$ we associate the vector $\phi(v)=a\omega+b\bar{\omega}$. The weight of $v$ is the Hamming weight of $\phi(v)$, and the distance between two vectors in $E$ is the Hamming distance of their images. If $S$ is a subspace of $E$ then $C=\phi(S)$ is a subset of GF$(4)^n$ that is closed under addition (defining thus an additive code). The [*trace inner product*]{} of $u,v\in \mbox{ GF}(4)^n$ is $$u\star v=Tr(u\cdot \bar{v})=\sum_{i=1}^n(u_i\bar{v_i}+\bar{u_i}v_i) \enspace,$$ Define the [*dual code*]{} $C^\perp$ as $$C^\perp=\{u\in \mbox{ GF}(4)^n:u\star v=0\ \forall v\in C\} \enspace.$$ Now one can reformulate Theorem \[bin\].
Let $C$ be an additive self-orthogonal subcode of GF$(4)^n$, containing $2^{n-k}$ vectors, such that there are no vectors of weight $<d$ in $C\setminus C^\perp$. Then any eigenspace of $\phi^{-1}(C)$ is a QECC with parameters $[[n,k,d]]$.
By Glynn (see [@Glynn:Graph; @Glynn:Tome]), we have: Let $S$ be a stabilizer matrix, that is $(n-k)\times n$ over GF(4) and such that its rows are GF(2)-linearly independent. Then we define a QECC with parameters $[[n,k,d]]$ as the set of all GF(2)-linear combinations of the rows of $S$. The code is [*self-dual*]{} when $k=0$.
The QECC as a Graph
-------------------
Assume that each column of $S$ contains at least two non-zero values, for the columns that do not have this property may be deleted to obtain a better code. Following [@Glynn:Graph], a self-dual quantum code $[[n,0,d]]$ corresponds to a graph on $n$ vertices, which may be assumed to be connected if the code is indecomposable. Let PG$(m,q)$ be the finite projective space defined from the vector space of rank $m+1$ over the field GF$(q)$. Then, the [*Grassmannian*]{} of lines of PG$(n-1,2)$, $G_1(\mbox{PG}(n-1,2))$, regarded as a variety immersed in PG$(\tiny{\left(\begin{array}{c}n\\
2
\end{array}\right)},2)$ is as follows: each line $l_i$ is defined by two points, $a_i$ and $b_i$. We associate to the set of lines all products $a_ib_j+a_jb_i,\ i\neq j\ (\mbox{mod} 2)$. Define a mapping from a column of an $n\times n$ stabilizer matrix $S$ over GF(4) to a vector of length $\begin{small} {\left(\begin{array}{c}n\\
2
\end{array}\right)} \end{small}$ with coefficients in GF(2): We write each column over GF(4) as $a+b\omega$, where $a,b\in$ GF$(2)^n$.
$$\left(\begin{array}{c}x_1\\
x_2\\
\vdots\\
x_n
\end{array}\right)=\left(\begin{array}{c}a_1\\
a_2\\
\vdots\\
a_n
\end{array}\right)+\omega\left(\begin{array}{c}b_1\\
b_2\\
\vdots\\
b_n
\end{array}\right) \enspace.$$
Taking all the $2\times 2$ subdeterminants found when we put the two vectors into a matrix, we get the points of the Grassmannian. A point in $G_1(\mbox{PG}(n-1,2))\equiv$ a line in PG$(n-1,2)\equiv$ a column of length $n$ over GF(4) (with at least two different non-zero components). A quantum self-dual code $[[n,0,d]]$ corresponds to some set of $n$ lines that generate PG$(n-1,2)$. As each line of PG$(n-1,2)$ corresponds to a (star) kind of graph, the set corresponds to a graph in $n$ vertices.
Interpretation as a Modified Adjacency Generator Matrix over GF(2) and GF(4)
----------------------------------------------------------------------------
From any connected graph we obtain an indecomposable code. Let $\Gamma$ be the adjacency matrix of a graph $G$ in $n$ variables. Then, $G_T=(I\ |\ \Gamma)$ (where $I$ is the $n\times n$ identity matrix) is the generator matrix of a binary linear code [@Tonc:Err]. In other words,
$$G_T=\left(\begin{array}{cccccccc}
1 & 0 & \ldots & 0 & 0 & a_{01} & \ldots & a_{0n}\\
0 & 1 & \ldots & 0 & a_{01} & 0 & \ldots & a_{1n}\\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & \ldots & 1 & a_{0n} & a_{1n} & \ldots & 0\end{array}\right)$$
generates a code over GF(2)$^n$. We can further interpret $G_T$ as a generating matrix of a code over GF(4)$^n$, as follows [@Cald:Qua]:
$$G=\Gamma+\omega I=\left(\begin{array}{cccc}
\omega & a_{01} & \ldots & a_{0n}\\
a_{01} & \omega & \ldots & a_{1n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{0n} & a_{1n} & \ldots & \omega\end{array}\right)$$
is the generating matrix of an additive code over GF(4)$^n$. Different graphs may define the same code, but this relation is 1-1 with respect to LC-equivalence between graphs, as defined in section \[LCGraph\].
Interpretation as a Modified Adjacency Matrix over ${\mathbb{Z}}_4$
-------------------------------------------------------------------
Define from a graph with adjacency matrix, $\Gamma$, the generating matrix of an additive code over ${\mathbb{Z}}_4^n$ as $2\Gamma+I$. This code has the same weight distribution over ${\mathbb{Z}}_4^n$ as $\Gamma+\omega I$ over GF$(4)^n$. Once again, LC-equivalent graphs define equivalent ${\mathbb{Z}}_4$ codes.
Interpretation as an Isotropic System
-------------------------------------
The graph state can also be viewed as an isotropic system (see [@Bou:Iso; @Bou:Grph; @Bou:Tree; @Cou:Ver; @MonSar]).
Let $A$ be a 2-dimensional vector space over GF(2). For $x,y \in A$, define a bilinear form, $<,>$, by $$<x,y>=\left\{\begin{array}{l}
1\ \m{ if } x\neq y,x\neq0\m{ and }y\neq0\\
0,\m{ otherwise}
\end{array}\right.$$
Let $V$ be a finite set. Define the space of GF(2)-homomorphisms $A^V:V\rightarrow A$. Define in this GF(2)-vector space a bilinear form as: $$\m{for }\phi,\psi\in A^V,\ <\phi,\psi>=\sum_{v\in V}<\phi(v),\psi(v)>\ (\m{mod }2) \enspace.$$
Let $L$ be a subspace of $A^V$. Then, $I=(V,L)$ is an [*isotropic system*]{} if dim $(L)=|V|$ and $<\phi,\psi>=0\ \forall\ \phi,\psi\in L$.
For a graph $G$, $V(G)$ denotes the set of vertices of $G$. If $v\in V(G)$, ${\cal{N}}(v)$ denotes the [*neighbourhood*]{} of vertex $v$, that is, the set of all its neighbours. For $P\subseteq V$, we set ${\cal{N}}(P)=\sum_{v\in P}{\cal{N}}(v)$. Let $K=\{0,x,y,z\}$ be the Klein group, which is a 2-dimensional vector space, and set $K'=K\setminus \{0\}$. Note that $x+y+z=0$.
([@Bou:Grph]) Let $G$ be a simple graph with vertex set $V$. Let $\phi,\psi\in K'^V$ such that $\phi(v)\neq\psi(v)\ \forall v\in V$, and set $L=\{\phi(P)+\psi({\cal{N}}(P))\ :\ P\subseteq V\}$. Then $S=(V,L)$ is an isotropic system.
The triple $\Pi=(G,\phi,\psi)$ is called a [*graphic presentation*]{} of $S$.
For $\phi\in K^V$, we set $\widehat{\phi}=\{\phi(P)\ :\ P\subseteq V\}$. $\widehat{\phi}$ is a vector subspace of $K^V$.
For $\psi\in K'^V$, the restricted Tutte-Martin polynomial $m(S,\psi;x)$ is defined by $$m(I,\psi;x)=\sum(x-1)^{dim(L\cup \widehat{\phi})} \enspace,$$ where the sum is over $\phi\in K'^V$ such that $\phi(v)\neq\psi(v),\ v\in V$.
([@Bou:Grph]) If $G$ is a simple graph and $I$ is the isotropic system defined by a graphic presentation $(G,\phi,\psi)$, then $$q(G;x)=m(I,\phi+\psi;x) \enspace,$$ where $q(G;x)$ is the interlace polynomial of $G$.
We mention the interlace polynomial and its relation to our work in Part II of this paper [@RP:BCII].
Interpretation as a Quadratic Boolean Function
----------------------------------------------
Let $p({\bf{x}}) : GF(2)^n \rightarrow GF(2)$ be a quadratic boolean function, defined by its Algebraic Normal Form (ANF), $ p({\bf{x}}) = \sum_{0\leq i < j\leq n-1} a_{ij}x^ix^j +
\sum_{i =0}^{n-1}b_ix_i+\sum_{i =0}^{n-1}c_i \enspace.$ We associate to $p({\bf{x}})$ the non-directed graph that has variables as vertices (and vice-versa) [@Par:QE] The adjacency matrix, $\Gamma$, associated to $p({\bf{x}})$, satisfies $\Gamma(i,j) = \Gamma(j,i) = a_{ij}$, $i < j$, $\Gamma(i,i) = 0$.
Interpretation of a Bipartite Quadratic Boolean Function as a Binary Linear Code
--------------------------------------------------------------------------------
Quadratic ANFs, as represented by bipartite graphs, have an interpretation as binary linear codes [@Par:QE]: Let ${\bf{T_C}},\ {\bf{T_{C^\perp}}}$ be a bipartite splitting of $\{0,\ldots,n-1\}$, and let us partition the variable set ${\bf{x}}$ as ${\bf{x}}={\bf{x_C}}\cup{\bf{x_{C^\perp}}}$, where ${\bf{x_C}}=\{x_i: i\in {\bf{T_C}}\}$, and ${\bf{x_{C^\perp}}}=\{x_i: i\in {\bf{T_{C^\perp}}}\}$. Let $p({\bf{x}})=\sum_k q_k({\bf{x_C}})r_k({\bf{x_{C^\perp}}})$, where $\mbox{deg}(q_k({\bf{x_C}}))=\mbox{deg}(r_k({\bf{x_{C^\perp}}}))=1\ \forall k$ (clearly, such a function corresponds to a bipartite graph), and let $s({\bf{x}})=(-1)^{p({\bf{x}})}$. Then the action of the transform $\bigotimes_{i\in {\bf{T}}}H_i$, with ${\bf{T}}={\bf{T_C}}$ or ${\bf{T_{C^\perp}}}$, on $s({\bf{x}})$ gives $s'({\bf{x}})=m({\bf{x}})$, with $m$ the ANF of a Boolean function. $s'$ is the binary indicator for a binary linear $[n, n-|{\bf{T}}|,d]$ error correcting code.
[There is also an equivalent interpretation of bipartite graphs as [*[binary matroids]{}*]{} (e.g. [@Cam:Cyc]). ]{}
Appendix B - Further Spectral Symmetries of Boolean Functions with respect to $\{I,H,N\}^n$
===========================================================================================
The [*[power spectrum]{}*]{} of the WHT of a boolean function is invariant to within a re-ordering of the spectral elements after an invertible affine transformation of the variables of the boolean function
[ The [*[power]{}*]{} of the $k^{th}$ spectral element, $P_k$, is given by $|P_k|^2$, where $P_k$ is defined in (\[WHT\]). ]{}
. This implies that bent boolean functions remain bent after affine transform (see Section \[BentSection\] for a discussion of bent properties). However, the set of $\{I,H,N\}^n$ power spectra are not an invariant of affine transformation. In this section we ascertain for which binary transformations (other than LC) the power spectra of the $\{I,H,N\}^n$ transform remains invariant to within a re-ordering of the spectral elements within each spectrum. We refer to the complete set of $3^n \times 2^n$ power spectral values w.r.t. $\{I,H,N\}^n$ as ${\bf{S_{IHN}}}$. Moreover, ’invariance’ is to within any re-ordering of the $3^n \times 2^n$ spectral elements. From the discussion of sections \[LCMultispectra\] and \[IHNsection\] it is evident that ${\bf{S_{IHN}}}$ of a quadratic boolean function is LC-invariant. However the LC-orbit is not the only spectral symmetry exhibited with respect to ${\bf{S_{IHN}}}$. We identify the following symmetries.
Let $p({\bf{x}})$ be a boolean function of [*[any]{}*]{} degree. Then ${\bf{S_{IHN}}}$ of $p({\bf{x}})$ and ${\bf{S_{IHN}}}$ of $p({\bf{x}}) + l({\bf{x}})$ are equivalent, where $l$ is any affine function of its arguments. \[AffOff\]
Let $p({\bf{x}})$ be a boolean function of [*[any]{}*]{} degree over $n$ variables. Then ${\bf{S_{IHN}}}$ of $p({\bf{x}})$ and ${\bf{S_{IHN}}}$ of $p({\bf{x + a}})$ are equivalent, where ${\bf{a}} \in {\mbox{GF}}(2)^n$. \[CycSym\]
Replacing $x_j$ with $x_j + 1$ within any $p({\bf{x}})$ is equivalent to the action of the ’bit-flip’ operator, $\sigma_x = \begin{tiny} \left ( \begin{array}{ll}
0 & 1 \\
1 & 0
\end{array} \right ) \end{tiny}$, at position $j$ of the transform on $(-1)^{p({\bf{x}})}$, applying $I$ in the rest of the positions.
We can rewrite $H\sigma_x$ as follows, $$H\sigma_x =
\begin{tiny} \frac{1}{\sqrt{2}}\left ( \begin{array}{ll}
1 & 1 \\
1 & -1
\end{array} \right ) \end{tiny}
\begin{tiny} \left ( \begin{array}{ll}
0 & 1 \\
1 & 0
\end{array} \right ) \end{tiny}
=
\begin{tiny} \frac{1}{\sqrt{2}} \left ( \begin{array}{ll}
1 & 0 \\
0 & -1
\end{array} \right ) \end{tiny}
\begin{tiny} \left ( \begin{array}{ll}
1 & 1 \\
1 & -1
\end{array} \right ) \end{tiny}
=
\begin{tiny} \left ( \begin{array}{ll}
1 & 0 \\
0 & -1
\end{array} \right ) \end{tiny}H = \sigma_zH \enspace.$$ In other words, a bit-flip (or periodic shift) followed by the action of $H$ is identical to the action of $H$ followed by a ’phase-flip’. (This is well-known to quantum code theorists). The final phase-flip is a member of the set ${\bf{D}}$ (see Section \[LCLUT\] for a definition of ${\bf{D}}$) so does not change the magnitude of the spectral values produced by $H$. Therefore the power spectra produced by $H$ is invariant to prior periodic shift.
We can rewrite $N\sigma_x$ as follows, $$N\sigma_x =
\begin{tiny} \frac{1}{\sqrt{2}}\left ( \begin{array}{ll}
1 & i \\
1 & -i
\end{array} \right ) \end{tiny}
\begin{tiny} \left ( \begin{array}{ll}
0 & 1 \\
1 & 0
\end{array} \right ) \end{tiny}
=
\begin{tiny} \frac{1}{\sqrt{2}} \left ( \begin{array}{ll}
0 & i \\
-i & 0
\end{array} \right ) \end{tiny}
\begin{tiny} \left ( \begin{array}{ll}
1 & i \\
1 & -i
\end{array} \right ) \end{tiny}
=
\begin{tiny} \left ( \begin{array}{ll}
0 & i \\
-i & 0
\end{array} \right ) \end{tiny}N = -\sigma_yN \enspace,$$ where $\sigma_y$ is one of the four Pauli matrices. In other words, a bit-flip (or periodic shift) followed by the action of $N$ is identical to the action of $N$ followed by a member of the set ${\bf{D}}$. Therefore the power spectra produced by $N$ is invariant to a prior periodic shift.
The above argument is trivial with respect to $I$. The argument extends naturally to any $n$-dimensional tensor product of $I$, $H$, and $N$.
Let $p({\bf{x}})$ be a boolean function of [*[any]{}*]{} degree over $n$ variables. We perform a combination of affine offset and periodic shift on $p({\bf{x}})$ by the following operation: $$p({\bf{x}}) \Rightarrow p({\bf{x}} + {\bf{a}}) +
{\bf{c \cdot x}}
+ d \enspace,$$ where ${\bf{a}},{\bf{c}} \in {\mbox{GF}}(2)^n$, $d \in {\mbox{GF}}(2)$, and ’$\cdot$’ is the scalar product.
The symmetries generated by affine offset and periodic shift include all symmetries generated by any combination of periodic and negaperiodic shift, because we perform periodic and negaperiodic shifts on $p({\bf{x}})$ by the following operation: $$p({\bf{x}}) \Rightarrow p({\bf{x}} + {\bf{a}}) +
{\bf{c \cdot x}}
+ \m{ wt}({\bf{c}}), \mf {\bf{c \preceq a}} \enspace,$$ where ${\bf{a}},{\bf{c}} \in {\mbox{GF}}(2)^n$, ’${\bf{c \preceq a}}$’ means that $c_i \le a_i$, $\forall i$ (i.e. $a$ [*[covers]{}*]{} $c$), and $\m{wt}({\bf{c}})$ is the binary weight of ${\bf{c}}$. The one positions in ${\bf{a}}$ identify variables $x_i$ which are to undergo periodic or negaperiodic shift, and the one positions in ${\bf{c}}$ identify the variables $x_i$ which are to undergo negaperiodic shift. The combined periodic and negaperiodic symmetry induced by $\{I,H,N\}^n$ implies an aperiodic symmetry, as discussed further in [@DanAPC].
[^1]: C. Riera is with the Depto. de Álgebra, Facultad de Matemáticas, Universidad Complutense de Madrid, Avda. Complutense s/n, 28040 Madrid, Spain. E-mail: `criera@mat.ucm.es`. Supported by the Spanish Government Grant AP2000-1365, and the Marie Curie Scholarship.
[^2]: M.G.Parker is with the Selmer Centre, Inst. for Informatikk, H[ø]{}yteknologisenteret i Bergen, University of Bergen, Bergen 5020, Norway. E-mail: `matthew@ii.uib.no`. Web: `http://www.ii.uib.no/~matthew/`
|
---
abstract: 'In the last decade, the LEP and SLD experiments played a central role in the study of B hadrons (hadrons containing a $b$ quark). New B hadrons have been observed ($B^0_s$, $\Lambda_b$, $\Xi_b$ and $B^{**}$) and their production and decay properties have been measured. In this paper we will focus on measurements of the CKM matrix elements : $|V_{cb}|$, $|V_{ub}|$, $|V_{td}|$ and $|V_{ts}|$. We will show how all these measurements, together with theoretical developments, have significantly improved our knowledge on the flavour sector of the Standard Model.'
author:
- |
Achille Stocchi\
Laboratoire de l’Accélérateur Linéaire,\
IN2P3-CNRS et Université de Paris-Sud, BP34, F-91898 Orsay Cedex, France.\
title: Final Results on Heavy Quarks at LEP and SLD
---
Introduction {#sec:intro}
============
B physics studies are exploiting a unique laboratory for testing the Standard Model in the fermion sector, for studying the QCD in the non-perturbative regime and for searching for New Physics through virtual processes.\
In the last decade, the LEP and SLD experiments played an important role in the study of B hadrons. At the start of the LEP and SLC accelerator in 1989, only the $B_d$ and the $B^+$ hadrons were known and their properties were under study. New weakly decaying B hadrons have been observed ($B^0_s$, $\Lambda_b$, $\Xi_b$) for the first time and their production and decay properties have been measured. New strongly decaying hadrons, the orbitally (L=1) excited B ($B^{**}$) mesons have been also observed and their mass and production rates measured.\
In this paper we will focus on the measurements of the CKM matrix elements : $V_{cb}$ and $V_{ub}$ through B decays and $V_{td}$ and $V_{ts}$ using $B^0-\bar{B}^0$ oscillations. On the other hand many additional measurements on B meson properties (mass, branching fractions, lifetimes...) are necessary to constrain the Heavy Quark theories (Operator Product Expansion (OPE) /Heavy Quark Effective Theory (HQET) /Lattice QCD (LQCD)) to allow for precise extraction of the CKM parameters. We finally show how these measurements constrain the Standard Model in the fermion sector, through the determination of the unitarity triangle parameters.\
In this paper we try to compare the LEP/SLD results with those obtained from other collaborations (CLEO at Cornell, CDF at TeVatron and the asymmetric B-factories: BaBar and Belle) and to present, when available, the world average result. A detailed description of the results and of the averaging techniques can be found in [@ref:bphys; @ref:stocchi].
B physics at the $Z^0$
======================
At the $Z^0$ resonance, B hadrons are produced from the coupling of the $Z^0$ to a $b \bar{b}$ quark pair. The production cross section is of $\sim$ 6 nb, which is five times larger than at the $\Upsilon(4S)$. Because of the specific (V-A) behaviour of the electroweak coupling at the $Z^0$ pole, hadronic events account for about 70$\%$ of the total production rate; among these, the fraction of $b \overline{b}$ events is $\sim 22\%$. Because of the energy available only $B^+$ and $B_d^0$ mesons can be produced at the $\Upsilon(4S)$. The B particles are produced almost at rest (the average momentum is of about 350 MeV/c), with no accompanying additional hadrons, and the decay products of the two B particles are spread isotropically over the space. At the Z pole, the primary $b \bar{b}$ pair, picks up from the vacuum other quark-antiquarks pairs and hadronizes into B hadrons plus few other particles. Therefore not only $B^{\pm}$ and $B_d$ mesons are produced, but also $B_s^0$ mesons or $B$ baryons can be present in the final state. The $b$ and the $\bar{b}$ hadronize almost independently. $b$ quarks fragment differently from light quarks, because of their high mass as compared with $\Lambda_{QCD}$. B hadrons carry, on average, about 70$\%$ of the available beam energy, whereas the rest of the energy is distributed among the other fragmenting particles. As a consequence, the two B hadrons fly in opposite directions and their decay products form jets situated in two opposite hemispheres.\
The hard fragmentation and the long lifetime of the b quark make that the flight distance of a B hadron at the Z pole, defined as $L = \gamma \beta c \tau$, on average of the order of 3 mm. As decay products have a mean charged multiplicity of 5 [^1], it was possible to tag B hadrons using a lifetime tag.
Most of the precision measurements in B physics performed at LEP/SLC, Tevatron and B factories, would not have been possible without the development of Silicon micro vertex detectors. In practice the averaged flight distance of the B hadrons becomes measurable thanks to the precision of silicon detectors, located as close as possible to the beam interaction point. These detectors determine with a precision better than 10 $\mu m$, the position of a charged particle trajectory. In particular the separation between $b$ quarks and other quarks is mainly based on the use of vertex detectors. Charged particles produced at the B vertex (secondary vertex) can be separated from those produced at the interaction point (primary vertex) using the precise tracking information. In spite of the relatively small statistics collected by the SLD experiment, it gave very important and competitive contributions to B physics, because of its silicon vertex detectors, which is located very close to the interaction point. A typical LEP $b\bar{b}$ event is shown in Figure \[fig:aleph\_event\].
![*[The three plots from top-left to bottom-left show the invariant mass spectra of ${\Lambda}$, ${((D^0 \pi)-D^0)}$ and ${D_s}$ which are obtained in correlation with an opposite sign lepton. These events are attributed mainly to the semileptonic decays of ${\Lambda_B}$, ${B_d^0}$ and ${B_s^0}$ hadrons, respectively. The bottom-right figure shows the possibility of distinguishing the charged and neutral B mesons based on inclusive techniques.]{}*[]{data-label="fig:4figures"}](4figures.eps){width="11cm"}
Because of the large B mass, B hadrons are expected to decay into several decay modes with branching ratio of the order of a per mil.
According to the registered statistics, at LEP, inclusive or semi-exclusive $b$-hadron decays had to be studied in place of exclusive channels for which very few events are expected[^2].\
Semileptonic decays benefit of a large branching ratio ( of the order of 10$\%$ ) and of clean and easily distinguishable final states. Semileptonic decays allow also to distinguish between different types of $b$ hadrons, by reconstructing charmed hadrons. As an example, a $\Lambda_c^+$ accompanied by a lepton with negative electric charge, in a jet, signs a $b$-baryon. For baryons, it is not even necessary to completely reconstruct the $\Lambda_c^+$ charmed baryon, correlations as $p \ell^-$ or $\Lambda \ell^-$ are sufficient. Similarly, $D_s^+ \ell^-$ or $D^* \ell^-$ events in a jet, provide event samples enriched in $\bar{B}^0_s$ and $\bar{B}^0_d$ mesons respectively.\
An overview of the signals used to study these new states is given in Figure \[fig:4figures\].
Example of historical evolution
===============================
[![*[ The left plot shows the first signal of the $\bar{B}_s^0$ meson in 1992, seen in the semileptonic decay : $\bar{B}_s^0 \rightarrow D_s^+ \ell^- \overline{\nu_{\ell}}$, whereas the right plot shows the same signal few years later. ]{}*](delphi-bulletin.eps "fig:"){width="6cm"}]{} [![*[ The left plot shows the first signal of the $\bar{B}_s^0$ meson in 1992, seen in the semileptonic decay : $\bar{B}_s^0 \rightarrow D_s^+ \ell^- \overline{\nu_{\ell}}$, whereas the right plot shows the same signal few years later. ]{}*](bs_delphi_bis.eps "fig:"){width="8cm"}]{}
We take the example of the ${B}_s^0$ meson to illustrate how our knowledge on the properties of B hadrons has evolved during the last ten years. In 1992, 7 events $\bar{B}_s^0 \rightarrow D_s^+ \ell^- \overline{\nu_{\ell}}$, constituted the first evidence for the $B_s^0$ meson. A few years later the same signal consists of more than 200 events.\
In the mean time our knowledge has much improved: the fraction of $B^0_s$ mesons in b jets is precisely measured as well as the $B^0_s$ mass and lifetime.
- the $\bar{B}_s^0$ rate in $b$-jets amounts to: $f_s ~=~(9.7\pm 1.2)\%$,
- the $B_s^0$ meson mass is $m_{B_s^0}=(5369.6 \pm 2.4)$ MeV (CDF mainly)
- the lifetime is $\tau(\bar{B}_s^0) = 1.464 \pm 0.057~ps$.
- the studies on $B_s^0-\bar{B}_s^0$ oscillations give $\Delta m_s > 15 ps^{-1} ~~~(95\%~\rm C.L.)$
- the ratio $\Delta \Gamma_{B_s^0}/\Gamma_{B_s^0}<0.31~~~(95\%~{\rm C.L.})$
Heavy hadron lifetimes
======================
Measurements of B lifetimes test the decay dynamics, giving important information on non-perturbative QCD corrections induced by the spectator quark (or diquark). Decay rates are expressed using the OPE formalism, as a sum of operators developed in series of order $O(\Lambda_{QCD}/m_Q)^n$. In this formalism, no term of order $1/m_Q$ is present and spectator effects contribute at order $1/m_Q^3$ [^3]. Non-perturbative operators are evaluated, most reliably, using lattice QCD calculations.
Beauty hadron lifetimes
-----------------------
Since the beginning of the LEP/SLD data taking an intense activity has been concentrated on the studies of B hadron lifetimes.\
Results are given in Table \[table:life\] [@lifeWG].
\[table:life\]
[@ll]{} B Hadrons & Lifetime \[ps\]\
$\tau({B^0_d})$ & 1.540 $\pm$ 0.014 (0.9 $\%$)\
$\tau({B^+})$ & 1.656 $\pm$ 0.014 (0.8 $\%$)\
$\tau({B^0_s})$ & 1.461 $\pm$ 0.057 (3.9 $\%$)\
$\tau({\Lambda^0_b})$ & 1.208 $\pm$ 0.051 (4.2 $\%$)\
\
\
\
\
Figure \[fig:liferatio\] gives the ratios of different B hadron lifetimes, as compared with theory predictions (dark(yellow) bands).
![*[B hadrons lifetime ratios [@lifeWG], compared with the theoretical predictions as given by the dark(yellow) bands.]{}*[]{data-label="fig:liferatio"}](liferatio.ps){width="120mm"}
The achieved experimental precision is remarkable and LEP results are still dominating the scene. The fact that charged B mesons live longer than neutral B mesons is now established at 5$\sigma$ level and is in agreement with theory. The $B^0_d$ and $B^0_s$ lifetimes are expected (at $\simeq$1$\%$) and found (at $\simeq$4$\%$) to be equal. A significant measurement in which this ratio differs from unity will have major consequences for the theory. The lifetime of the b-baryons is measured to be shorter than the $B^0_d$ lifetime, but the size of this effect seems to be more important than predicted (2-3$\sigma$). Recent calculations of high order terms give an evaluation of the b-baryon lifetime in better agreement with the experimental result[@vittorio].\
New results are expected from B-Factories (which could decrease the relative error on the lifetimes of the $B^0_d$ and $B^+$ to 0.4-0.5$\%$) and mainly from Tevatron (Run II) which could precisely measure all B hadron lifetimes, including those for the $\Xi_b$, $\Omega_b$ and the $B_c$.\
In figure \[fig:life\_story\] the improvement on the precision of the measured B hadron lifetimes over the years is shown.
![*[Evolution of the combined measurement of the different B hadron lifetimes over the years. The vertical band, in each plot, indicates the end of the data taking at LEP.]{}*[]{data-label="fig:life_story"}](life_story.eps){width="11cm"}
Determination of the CKM element: $|V_{cb}|$
============================================
The $|V_{cb}|$ element of the CKM matrix can be accessed by studying the rates of inclusive and exclusive semileptonic $b$-decays.
$|V_{cb}|$ inclusive analyses.
-------------------------------
The first method to extract $|V_{cb}|$ makes use of the inclusive semileptonic decays of B-hadrons and of the theoretical calculations done in the framework of the OPE. The inclusive semileptonic width $\Gamma_{s.l.}$ is expressed as: $$\begin{aligned}
\Gamma_{s.l.} = \frac{BR(b \rightarrow c l \nu)}{\tau_b} = \gamma_{theory} |V_{cb}|^2 ; & \nonumber \\
\gamma_{theory} = f(\alpha_s,m_b,\mu_{\pi}^2,1/m_b^3...).
\label{eq:vcbtheo}\end{aligned}$$ From the experimental point of view the semileptonic width has been measured by the LEP/SLD and $\Upsilon(4S)$ experiments with a relative precision of about 2$\%$ [@vcbWG]: $$\begin{aligned}
\Gamma_{sl} = & (0.431 \pm 0.008 \pm 0.007) 10^{-10} MeV & \small{\Upsilon(4S)} \nonumber \\
\Gamma_{sl} = & (0.439 \pm 0.010 \pm 0.007) 10^{-10} MeV & \small{\rm{LEP/SLD}} \nonumber \\
\Gamma_{sl} = & (0.434 \times (1 \pm 0.018)) 10^{-10} MeV & \small{\rm{ave.} }
\label{eq:gammsl}\end{aligned}$$ The precision on the determination of $|V_{cb}|$ is mainly limited by theoretical uncertainties on the parameters entering in the expression of $\gamma_{theory}$ in equation \[eq:vcbtheo\].
Moments analyses
----------------
Moments of the hadronic mass spectrum, of the lepton energy spectrum and of the photon energy in the $b \rightarrow s \gamma$ decay are sensitive to the non perturbative QCD parameters contained in the factor $\gamma_{theory}$ of equation \[eq:vcbtheo\] and in particular to the mass of the $b$ and $c$ quarks and to the Fermi motion of the heavy quark inside the hadron, $\mu_{\pi}^2$ [^4].\
Results from DELPHI collaboration are shown in Figure \[fig:moments\].\
Similar results (and with comparable precision) have been obtained by CLEO (which did a pioneering work in this field) and by the BaBar Coll. [@ref:stocchi].
![*[Constraints in the $\bar{\Lambda}-\lambda_1$ plane obtained: by the DELPHI Coll. using the measured values of the first two moments of the hadronic mass and lepton energy spectra. The bands represent the 1$\sigma$ regions selected by each moment and the ellipses show the 39$\%$, 68$\%$ and 90$\%$ probability regions of the global fit.]{}*[]{data-label="fig:moments"}](moments_delphi.eps){width="95mm"}
Using the experimental results on $\bar{\Lambda}$ and $\lambda_1$: $$|V_{cb}| = (40.7 \pm 0.6 \pm 0.8(\rm{theo.})) 10^{-3} \rm{(inclusive)}
\label{eq:vcbinclres}$$ This result corresponds to an important improvement on the determination of the $|V_{cb}|$ element. Part of the theoretical errors (from $m_b$ and $\mu_{\pi}^2$) is now absorbed in the experimental error and the theoretical error is reduced by a factor two. The remaining theoretical error could be further reduced if the parameters controlling the $1/m_b^3$ corrections are extracted directly from experimental data.
$|V_{cb}|$: $B \rightarrow D^* \ell \nu$ analyses.
--------------------------------------------------
An alternative method to determine $|V_{cb}|$ is based on exclusive $\overline{ B^0_d} \rightarrow D^{*+} \ell^-
\overline{\nu_l}$ decays. Using HQET, an expression for the differential decay rate can be derived $$\frac{d\Gamma}{dw} = \frac{G_F^2}{48 \pi^2} |V_{cb}|^2 |F(w)|^2 G(w) ~;~ w = v_B.v_D
\label{eq:exclvcb}$$
![*[Summary of the measurements of $F(1) \times |V_{cb}|$ [@vcbWG].]{}*[]{data-label="fig:vcb_ave"}](vcb_ave.eps){width="100mm"}
$w$ is the relative velocity between the B ($v_B$) and the D systems ($v_D$). G($w$) is a kinematical factor and F($w$) is the form factor describing the transition. At zero recoil ($w$=1) F(1) goes to unity. The strategy is then to measure $d\Gamma/dw$, to extrapolate at zero recoil and to determine $F(1) \times |V_{cb}|$.\
The experimental results are summarised in Figure \[fig:vcb\_ave\]. Using F(1) = 0.91 $\pm$ 0.04 [@latticeF1], it gives [@vcbWG]: $$|V_{cb}| = (41.9 \pm 1.1 \pm 1.9(F(1)) 10^{-3} \rm{(exclusive)}
\label{eq:vcbexclres}$$ Combining the two determinations of $|V_{cb}|$ (a possible correlation between the two determinations has been neglected) it gives [@ref:stocchi]: $$\begin{aligned}
|V_{cb}| = (40.9 \pm 0.8) 10^{-3} \tiny{\rm{(exclusive+inclusive)}}
\label{eq:vcbave}\end{aligned}$$
Measurement of $|V_{ub}|$. {#sec:vub}
==========================
![*[Summary of $|V_{ub}|$ measurements [@marco].]{}*[]{data-label="fig:vub_ave"}](vub_ave.eps){width="100mm"}
The CKM matrix element $|V_{ub}|$ has been measured at LEP using semileptonic $b$ to $u$ decays. This measurement is rather difficult because one has to suppress the large background from the more abundant semileptonic $b$ to $c$ quark transitions. By using kinematical and topological variables, the LEP experiments have succeeded in measuring the semileptonic $b$ to $u$ branching ratio [@ref:ckmworkshop], and obtain :
$ BR(b\rightarrow l^-\bar{\nu} X_u) = (1.71 \pm 0.53 ) \:10^{-3} $
Using models based on the Operator Product Expansion, a value for $|V_{ub}|$ is obtained : $$\begin{aligned}
|V_{ub}| &=& ( 40.9 \pm 6.1 \pm 3.1 ) \times 10^{-4} \quad \mbox{LEP} \ .\end{aligned}$$ Prior to this analysis, the $V_{ub}$ matrix element was firstly obtained, by CLEO and ARGUS collaborations, by looking at the spectrum of the lepton in B semileptonic decays. The difference between D meson and $\pi$ masses is reflected in the momentum of the lepton from the B decays. This analysis has been recently revised by the CLEO Coll.. An alternative method to determine $|V_{ub}|$ consists in the reconstruction of the charmless semileptonic B decays: $B \rightarrow \pi (\rho) \ell \nu$. This analysis has been performed by the CLEO Coll. and now by the b-factories.\
Figure \[fig:vub\_ave\] shows the full set of results on $V_{ub}$ [@marco].
Study of $B^0-\overline{B^0}$ oscillations {#sec:oscillations}
==========================================
The probability that a $B^0$ meson oscillates into a $\overline{B}^0$ or remains a ${B}^0$ is given by: $$P_{{B}^0_q \rightarrow {B}^0_q(\overline{{B}}^0_q)} =
\frac{1}{2}e^{-t/\tau_q} (1 \pm \cos \Delta {m}_q t)
\label{eq:oscillation}$$ Where $t$ is the proper time, $\tau_q$ the lifetime of the ${B}^0_q$ meson, and $\Delta {m}_q = {m}_{B^0_1}- m_{{B}^0_2}$ the mass difference between the two physical mass eigenstates [^5]. To derive this formula the effects of CP violation and lifetime differences for the two states have been neglected.\
Integrating expression \[eq:oscillation\], over the decay time, the probability to observe a $\bar{B}^0_{d(s)}$ meson starting from a $B^0_{d(s)}$ meson is given by $\chi_d(s) = x^2_{d(s)}/(2+x^2_{d(s)})$, where $x_{d(s)} =
\Delta m_{d(s)} \tau(B_{d(s)}^0)$. At Z energies, both $B_d^0$ and $B^0_s$ mesons are produced with fractions $f_{B_d}$ and $f_{B_s}$. The average mixing parameter $\chi$ is defined as : $\chi = f_{B_d} \chi_d + f_{B_s} \chi_s$. It has to be noted that for fast $B^0_s$ oscillations $\chi_s$ takes values close to 0.5 and $\chi_s$ becomes very insensitive to $x_s$. Even a very precise measurement of $\chi_s$ does not allow a determination of $\Delta m_s$.\
It is then clear that only the time evolution of the $B^0-\overline{B}^0$ oscillations allow to measure $\Delta m_d$ and $\Delta m_s$.\
A time dependent study of $B ^0 - \overline{B}^0$ oscillations requires:
- the measurement of the proper time t,
- to know if a $B ^0$ or a $\overline{B}^0$ decays at time t (decay tag)
- to know if a $b$ or a $\overline{b}$ quark has been produced at t = 0 (production tag).
In the Standard Model, $B^0-\bar{B}^0$ oscillations occur through a second-order process - a box diagram - with a loop of W and up-type quarks. The box diagram with the exchange of a $top$ quark gives the dominant contribution : $$\begin{aligned}
\Delta {m}_d & ~\propto ~V_{td}^2 f^2_{{B}_d} {B}_{B_d}
~\propto ~ V_{cb}^2 \lambda^2 [(1 - \bar{\rho})^2 + \bar{\eta}^2]f^2_{{B}_d} {B}_{B_d} & \nonumber \\
\Delta {m}_s &~\propto ~V_{ts}^2 f^2_{{B}_s} {B}_{B_s}
~\propto ~ V_{cb}^2 f^2_{{B}_s} {B}_{B_s} &\nonumber \\
\frac{\Delta m_d}{\Delta m_s} &~\propto 1/\xi^2|\frac{V_{td}}{V_{ts}}|^2 ~\propto ~
1/\xi^2 \lambda^2 [(1 - \bar{\rho})^2 + \bar{\eta}^2] &
\label{eq:dmddmsxi}\end{aligned}$$
where $\xi=\frac{ f_{B_s}\sqrt{B_{B_s}}}{ f_{B_d}\sqrt{B_{B_d}}}$ .
Thus, the measurement of $\Delta m_d$ and $\Delta m_s$ gives access to the CKM matrix elements $|V_{td}|$ and $|V_{ts}|$ respectively. The difference in the $\lambda$ dependence of these expressions ($\lambda \sim 0.22$) implies that $\Delta {m}_s \sim 20 ~\Delta {m}_d$. It is then clear that a very good proper time resolution is needed to measure the $\Delta {m}_s$ parameter. On the other hand the measurement of the ratio $\Delta m_d/\Delta m_s$ gives the same constraint as $\Delta m_d$ but this ratio is expected to have smaller theoretical uncertainties since the ratio $\xi$ is better known than the absolute value of $f_B \sqrt B_B$.
$\Delta {m}_d$ measurements {#delta-m_d-measurements .unnumbered}
---------------------------
Analyses using different events sample have been performed at LEP. A typical time distribution is shown in Figure \[fig:dmd\]. $B^0_d - \overline{B}^0_d$ oscillations with a frequency $\Delta {m}_d$ are clearly visible. This can be a textbook plot ! The present summary of these results on $\Delta {m}_d$, is shown in Figure \[fig:dmdsummary\]. Combining LEP, CDF and SLD measurements it follows that [@osciWG]: $$\Delta m_d = (0.498 \pm 0.013) \:ps^{-1}$$
$\Delta {m}_d$ has been first measured with high precision by the LEP/SLD/CDF experiments. The new and precise measurements performed at the B-Factories confirmed these measurements and improved the precision by a factor two. the combined result is now : $\Delta m_d = (0.503 \pm 0.006) \: ps^{-1}$. The evolution, over the years, of the combined $\Delta m_d$ frequency measurement is shown in Figure \[fig:dmd\_story\].
![*[The evolution of the combined $\Delta m_d$ frequency measurement over the years.]{}*[]{data-label="fig:dmd_story"}](dmd_story.eps){width="8cm"}
Analyses on $\Delta {m}_s$ {#analyses-on-delta-m_s .unnumbered}
--------------------------
The search for $B^0_s-\overline{B^0_s}$ oscillations is more difficult because the oscillation frequency is much higher. In the Standard Model one expects $\Delta {m}_s \sim 20 ~\Delta {m}_d$. The proper time resolution will therefore play an essential role. Five different types of analyses have been performed at LEP/SLD. An overview is given in Table \[tab1\].
Analysis N(events) $P({B}_S)$ $\varepsilon_1$ $\varepsilon_2$ $\sigma_t (t~<~1 \rm{ps})$
---------------------- --------------- -------------- ----------------- ----------------- ----------------------------
Dipole $\sim 700000$ $\sim 10\%$ $\sim 70\%$ $\sim 60\%$ $\sim 0.25$ ps
Inclusive lepton $\sim 50000$ $\sim 10\%$ $\sim 70\%$ $\sim 90\%$ $\sim 0.25$ ps
${D}^\pm_s h^\mp$ $\sim3000$ $\sim 15\%$ $\sim 72\%$ $\sim 90\%$ $\sim 0.22$ ps
${D}^\pm_s \ell^\mp$ $\sim 400$ $\sim 60 \%$ $\sim 78 \%$ $\sim 90 \%$ $\sim 0.18$ ps
Exclusive ${B}^0_S$ $\sim 25$ $\sim 70 \%$ $\sim 78 \%$ $\sim 100 \%$ $\sim 0.08$ ps
: *[Characteristics of the different analyses are given in terms of statistics (N), ${B}^0_s$ purity \[$(P({B}_s)$\] , tagging purities - i.e. the fraction of correctly tagged events - at the production and decay time $(\varepsilon_1, \varepsilon_2)$ and average time resolution within the first picosecond.]{}*[]{data-label="tab1"}
The so-called amplitude method [@ref:amp] has been developed to combine data from different experiments. It corresponds to the following change in equation \[eq:oscillation\]: $$1 \pm \cos \Delta {m}_s t \rightarrow 1 \pm A \cos \Delta {m}_s t$$ A and $\sigma_A$ are measured at fixed values of $\Delta {m}_s$. In case of a clear oscillation signal, the measured amplitude is compatible with A = 1 at the corresponding value of $\Delta m_s$. With this method it is also easy to set an exclusion limit. The values of $\Delta {m}_s$ excluded at 95% C.L. are those satisfying the condition A($\Delta{m}_s$) + 1.645 $\sigma_A (\Delta {m}_s) < 1$. Furthermore, the sensitivity of the experiment can be defined as the value of $\Delta {m}_s$ corresponding to 1.645 $\sigma_A (\Delta {m}_s) = 1$ (for A($\Delta {m}_s) = 0$, namely supposing that the “true” value of $\Delta {m}_s$ is well above the measurable value of $\Delta {m}_s$).
![*[The evolution, over the years, of the combined $\Delta m_s$ sensitivity.]{}*[]{data-label="fig:dms_story"}](dms_story.eps){width="11cm"}
During the last seven years impressive improvements in the analysis techniques allowed to improve the sensitivity of this search, as it can be seen in Figure \[fig:dms\_story\].\
The combined result of the LEP/SLD/CDF [@osciWG] analyses, displayed as an amplitude vs $\Delta m_s$ plot, is shown in Figure \[fig:dms\] and is: $$\Delta {m}_s > 14.4~\rm{ps}^{-1}~~\rm{at}~~95\%~~\rm{C.L.}$$ The sensitivity is at $19.2~\rm{ps}^{-1}$.\
The summary of the present results on $\Delta m_s$ is shown in Figure \[fig:dmssummary\].
The present combined limit implies that $B_s^0$ oscillates at least 30 times faster than $B_d^0$ mesons.\
The significance of the “signal”, appearing around 17 ps$^{-1}$, is about 2.5 $\sigma$ and no claim can be made on the observation of $B^0_s-\bar{B^0_s}$ oscillations.\
Tevatron experiments, are thus expected to measure soon $B^0_s-\bar{B^0_s}$ oscillations...
The CKM Matrix {#sec:ickm}
==============
In the Standard Model, the weak interactions among quarks are encoded in a 3 $\times$ 3 unitary matrix: the CKM matrix.\
The existence of this matrix conveys the fact that quarks weak interaction eigenstates are a linear combination of their mass eigenstates [@Cabibbo; @km].\
$$V_{CKM} =
\left ( \begin{array}{ccc}
V_{ud} ~~ V_{us} ~~ V_{ub} \\
V_{cd} ~~ V_{cs} ~~ V_{cb} \\
V_{td} ~~ V_{ts} ~~ V_{tb}
\end{array} \right )$$
The CKM matrix can be parametrized in terms of four free parameters. Here, the improved Wolfenstein [@ref:Wolf] parametrization, expressed in terms of the four parameters $\lambda$, $A$, $\rho$ and $\eta$ (which accounts for the CP violating phase) , will be used:
$$\begin{array}{cccc}
\hspace{-15mm}
V_{CKM} =
&
\left ( \begin{array}{cccc}
~~~~1 - \frac{\lambda^{2}}{2} - \frac{\lambda^4}{8} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \lambda ~~~~~~~~~~~~~~~~~~~~~~~~
A \lambda^{3} (\rho - i \eta) \\
- \lambda +\frac{A^2 \lambda^5}{2}(1-2 \rho) -i A^2 \lambda^5 \eta~~~~~~~~~~ 1 - \frac{\lambda^{2}}{2}
-\lambda^4(\frac{1}{8}+\frac{A^2}{2})
~~~~~~~~ A \lambda^{2} \\
A \lambda^{3} [1 - (1-\frac{\lambda^2}{2})(\rho +i \eta)] ~~~ -A \lambda^{2}(1-\frac{\lambda^2}{2})(1 + \lambda^{2}(\rho +i \eta))
~~~ 1-\frac{A^2 \lambda^4}{2}
\end{array} \right )
&
+ O(\lambda^{6}).
\end{array}
\label{eq:eq8}$$
The CKM matrix elements can be expressed as: $$V_{us}~=~ \lambda ~
V_{cb}~=~~A \lambda^2,~
V_{ub}~=~A \lambda^3(\overline{\rho}-i \overline{\eta})/(1-\lambda^2/2),~
V_{td}~=~A \lambda^3(1-\overline{\rho}+i \overline{\eta})$$ where the parameters $\overline{\rho}$ and $\overline{\eta}$ have been introduced [@ref:blo] [^6].
The parameter $\lambda$ is precisely determined to be $0.2210 \pm 0.0020$ [^7] using semileptonic kaon decays. The other parameters: $A$, $\overline{\rho}$ and $\overline{\eta}$ were rather unprecisely known.\
The Standard Model predicts relations between the different processes which depend upon these parameters; CP violation is accommodated in the CKM matrix and its existence is related to $\bar{\eta} \neq 0$. The unitarity of the CKM matrix can be visualized as a triangle in the $\bar{\rho}-\bar{\eta}$ plane. Several quantities, depending upon $\bar{\rho}$ and $\bar{\eta}$ can be measured and they must define compatible values for the two parameters, if the Standard Model is the correct description of these phenomena. Extensions of the Standard Model can provide different predictions for the position of the upper vertex of the triangle, given by the $\bar{\rho}$ and $\bar{\eta}$ coordinates.
![*[The allowed regions for $\overline{\rho}$ and $\overline{\eta}$ (contours at 68%, 95%) are compared with the uncertainty bands for $\left | V_{ub} \right |/\left | V_{cb} \right |$, $\epsilon_K$, $\Delta {m_d}$,the limit on $\Delta {m_s}/\Delta {m_d} $ and sin2$\beta$.]{}*[]{data-label="fig:bande"}](chiral_2d.eps){width="120mm"}
----------------------------------------------------------------------- -------------------------------- ------------------------ ----------------------- -----------------------
Parameter Value Gaussian Uniform Ref.
$\sigma$ half-width
$\lambda$ $0.2210$ 0.0020 - [@ref:ckmworkshop]
$\left | V_{cb} \right |$(excl.) $ 42.1 \times 10^{-3}$ $ 2.1 \times 10^{-3}$ - [@ref:ArtusoBarberio]
$\left | V_{cb} \right |$(incl.) $ 40.4 \times 10^{-3}$ $ 0.7 \times 10^{-3}$ $ 0.8 \times 10^{-3}$ [@ref:ArtusoBarberio]
$\left | V_{ub} \right |$(excl.) $ 32.5 \times 10^{-4}$ $ 2.9 \times 10^{-4}$ $ 5.5 \times 10^{-4}$ [@ref:ckmworkshop]
$\left | V_{ub} \right |$(incl.) $ 40.9 \times 10^{-4}$ $ 4.6 \times 10^{-4}$ $ 3.6 \times 10^{-4}$ [@ref:ckmworkshop]
$\Delta m_d$ $0.503~\mbox{ps}^{-1}$ $0.006~\mbox{ps}^{-1}$ – [@osciWG]
$\Delta m_s$ $>$ 14.4 ps$^{-1}$ at 95% C.L. [@osciWG]
$m_t$ $167~GeV$ $ 5~GeV$ – [@ref:top]
$f_{B_d} \sqrt{\hat B_{B_d}}$ $235~MeV$ $33~MeV$ $^{+0}_{-24}~MeV$ [@ref:lellouch]
$\xi=\frac{ f_{B_s}\sqrt{\hat B_{B_s}}}{ f_{B_d}\sqrt{\hat B_{B_d}}}$ 1.18 0.04 $^{+0.12}_{-0.00}$ [@ref:lellouch]
$\hat B_K$ 0.86 0.06 0.14 [@ref:lellouch]
sin 2$\beta$ 0.734 0.054 - [@ref:sin2b]
----------------------------------------------------------------------- -------------------------------- ------------------------ ----------------------- -----------------------
Different constraints can be used to select the allowed region for the apex of the triangle in the $\bar{\rho}$-$\bar{\eta}$ plane. Five have been used so far: $\epsilon_k$, $|V_{ub}|/|V_{cb}|$, $\Delta m_d$, the limit on $\Delta m_s$ and sin 2$\beta$ from the measurement of the CP asymmetry in $J/\psi K^0$ decays. These constraints are shown in Figure \[fig:bande\] [@ref:bello]. These measurements provide a set of constraints which are obtained by comparing measured and expected values of the corresponding quantities, in the framework of the Standard Model (or of any other given model). In practice, theoretical expressions for these constraints involve several additional parameters such as quark masses, decay constants of B mesons and bag-factors. The values of these parameters are constrained by other measurements (e.g. the top quark mass) or using theoretical expectations.\
Different statistical methods have been defined to treat experimental and theoretical errors. The methods essentially differ in the treatment of the latter and can be classified into two main groups: frequentist and Bayesian. The net result is that, if the same inputs are used, the different statistical methods select quite similar values for the different CKM parameters [@ref:ckmfits]. The results in the following are shown using the Bayesian approach.\
Central values and uncertainties taken for the relevant parameters used in these analyses are given in Table \[tab:inputs\] [@parodi].\
The most crucial test is the comparison between the region selected by the measurements which are sensitive only to the sides of the Unitarity Triangle and the regions selected by the direct measurements of the CP violation in the kaon ($\epsilon_K$) or in the B (sin2$\beta$) sector. This test is shown in Figure \[fig:testcp\].
![*[The allowed regions for $\overline{\rho}$ and $\overline{\eta}$ (contours at 68%, 95%) as selected by the measurement of $\left | V_{ub} \right |/\left | V_{cb} \right |$, $\Delta {m_d}$, the limit on $\Delta {m_s}/\Delta {m_d} $ are compared with the bands (at 1 and 2$\sigma$) selected from CP violation in the kaon ($\epsilon_K$) and in the B (sin2$\beta$) sectors.]{}*[]{data-label="fig:testcp"}](chiral_2d_noepsk.eps){width="120mm"}
It can be translated quantitatively in the comparison between the value of sin2$\beta$ obtained from the measurement of the CP asymmetry in $J/\psi K^0$ decays and the one determined from triangle “sides“ measurements [@ref:stocchi],[@parodi]: $$\begin{aligned}
\sin 2 \beta = & 0.725^{+0.055}_{-0.065} & \rm {triangle~sides~ only} \nonumber \\
\sin 2 \beta = & 0.734 \pm 0.054 & \rm \quad B^0 \rightarrow J/\psi K^0.
\label{eq:sin2beta}\end{aligned}$$ The spectacular agreement between these values shows the consistency of the Standard Model in describing the CP violation phenomena in terms of one single parameter $\eta$. It is also an important test of the OPE,HQET and LQCD theories which have been used to extract the CKM parameters.\
Including all five constraints the results are [@ref:stocchi],[@parodi]: $$\begin{aligned}
\bar {\eta} = & 0.357 \pm 0.027 & ~(0.305-0.411) \nonumber \\
\bar {\rho} = & 0.173 \pm 0.046 & ~(0.076-0.260) \nonumber \\
\sin 2\beta = & 0.725 ^{+0.035}_{-0.031} & ~(0.660-0.789) \nonumber \\
\sin 2\alpha = & -0.09 \pm 0.25 & ~(-0.54-0.40) \nonumber \\
\gamma = & (63.5 \pm 7.0)^{\circ} & ~(51.0-79.0)^{\circ} \nonumber \\
\Delta m_s = & (18.0^{+1.7}_{-1.5}) ps^{-1} & ~(15.4-21.7) ps^{-1}.
\label{eq:allres}\end{aligned}$$
The ranges within parentheses correspond to 95$\%$ probability.\
The results on $\Delta m_s$ and $\gamma$ are predictions for those quantities which will be measured in near future.
Conclusions
===========
During the last ten years, our understanding of the flavour sector of the Standard Model improved. LEP and SLD played a central role.\
At the start of LEP and SLD, only the $B_d$ and the $B^+$ hadrons were known and their properties were under study. Today B hadrons have been carefully studied and many quantities have already been measured with good precision. The hadron lifetimes are now measured at the one/few percent level. LEP experiments are the main contributors for the measurement of $|V_{cb}|$, which is known with a relative precision better than 2$\%$. In this case, not only, the decay width has been measured, but also some of the non-perturbative QCD parameters entering in its expression. It is a great experimental achievement and a success for the theory description of the non-perturbative QCD phenomena in the framework of the OPE.\
LEP experiments have been pioneering in determining $|V_{ub}|$ using inclusive methods and reaching a precision of about 10$\%$, defining a road for future measurements at B-factories.\
The time behaviour of $B^0-\bar{B^0}$ oscillations has been studied and precisely measured in the $B_d^0$ sector. The new and precise measurements performed at the B-Factories confirmed these measurements and improved the precision by a factor two. The oscillation frequency $\Delta m_d$ is known with a precision of about 1$\%$. $B_s^0-\bar{B_s^0}$ oscillations have not been measured so far, but this search has pushed the experimental limit on the oscillation frequency $\Delta m_s$ well beyond any initial prediction for experimental capabilities. SLD experiment has played a central role in this search. Today we know that $B_s^0$ oscillates at least 30 times faster than $B_d^0$ mesons. The frequency of the $B_s^0-\bar{B_s^0}$ oscillations will be soon measured at the Tevatron. Nevertheless the impact of the actual limit on $\Delta m_s$ for the determination of the unitarity triangle parameters is crucial.\
The unitarity triangle parameters are today known within good precision. The evolution of our knowledge concerning the allowed region in the $\overline{\rho}$-$\overline{\eta}$ plane is shown in Figure \[fig:storia\]. The reduction in size of the error bands, from the year 1995 to 2000, is essentially due to the analyses described in this paper and to the progress in the OPE, HQET and LQCD theories. The reduction between 2000 and 2002 is also driven by the precise measurements of sin 2 $\beta$ at the $b$-factories.\
A crucial test has been already done: the comparison between the unitarity triangle parameters, as determined with quantities sensitive to the sides of the unitarity triangle (semileptonic B decays and oscillations), with the measurements of CP violation in the kaon ($\epsilon_K$) and in the B (sin2$\beta$) sectors. This agreement tells us that the Standard Model is also working in the flavour sector and it is also an important test of the OPE,HQET and LQCD theories which have been used to extract the CKM parameters. On the other hand, these tests are at about 10$\%$ level accuracy, the current and the next facilities can surely push these tests to a 1$\%$ level.
Acknowledgements
================
I would like to thank the organisers for the invitation and for having set up a very interesting topical conference in a stimulating and nice atmosphere during and after the talks.\
Thanks to all the LEP and SLD members which have made all of it possible ! I would also like to remember the important work made from the members of the Heavy Flavour Working Groups who prepared a large fraction of the averages quoted in this note. They are all warmly thanked.\
Thanks to P. Roudeau for the careful reading of the manuscript.
[ref99]{}
ALEPH, CDF, DELPHI, L3, OPAL, SLD Collaborations, CERN-EP/2001-050 A. Stocchi, plenary talk given at the XXXI$^{st}$ ICHEP, Amsterdam 24-31 July 2002, hep-ph/0211245. LEP B Lifetime Working Group:\
http:$\backslash\backslash$lepbosc.web.cern.ch/LEPBOSC/lifetimes/lepblife.html E. Franco, V. Lubicz, F. Mescia, C. Tarantino, Nucl. Phys. B633 (2002) 212. (hep-ph/02030890). $V_{cb}$ Working Group: http:$\backslash\backslash$lepvcb.web.cern.ch/LEPVCB/ L. Lellouch, plenary talk given at the XXXI$^{st}$ ICHEP, Amsterdam 24-31 July 2002 ; A. S. Kronfeld, P. B. Mackenzie, J. N. Simone, S. Hashimoto, S. M. Ryan, proceedings of FPCP, May 16–18, Philadelphia, Pennsylvania (hep-ph/0110253); Phys. Rev. D66 (2002) 014503. Results presented at the CKM Unitarity Triangle Workshop, CERN Feb. 2002:\
http:$\backslash\backslash$ckm-workshop.web.cern.ch/ckm-workshop/ in particular see also:\
LEP Working group on $|V_{cb}|$: http:$\backslash\backslash$lepvcb.web.cern.ch/LEPVCB/ Winter 2002 averages.\
LEP Working group on $|V_{ub}|$:\
http:$\backslash\backslash$battagl.home.cern.ch/battagl/vub/vub.html. M. Battaglia, talk given at the XXXI$^{st}$ ICHEP, Amsterdam 24-31 July 2002. LEP Working group on oscillations:\
http:$\backslash\backslash$lepbosc.web.cern.ch/LEPBOSC/combined$\_$results/amsterdam$\_$2002/. H.G. Moser and .A. Roussarie, Nucl. Instrum. Meth. A384 (1997) 491. N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531. M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1973) 652. L. Wolfenstein, Phys. Rev. Lett. [**51**]{} (1983) 1945. A.J. Buras, M.E. Lautenbacher and G. Ostermaier, [Phys. Rev.]{} [**D50**]{} (1994) 3433. For details see : M. Ciuchini et al., JHEP 0107:013, 2001 hep-ph/0012308. M. Artuso and E. Barberio, hep-ph/0205163. F. Abe [*et al.,*]{} CDF Collaboration, [ Phys. Rev. Lett.]{} [**74**]{} (1995) 2626.\
S. Abachi [*et al.,*]{} D0 Collaboration, [ Phys. Rev. Lett.]{} [**74**]{} (1995) 2632. L. Lellouch, plenary talk given at the XXXI$^{st}$ ICHEP, Amsterdam 24-31 July 2002. Average from Y. Nir see these proceedings based on: R. Barate et al., (ALEPH Collaboration) [*Phys. Lett.*]{} [**B492**]{} (2000), 259-274; K. Ackerstaff et al., (OPAL Collaboration) [*Eur. Phys.*]{} [**C5**]{} (1998) 379; T. Affolder at al., Phys. Rev. D61 (2000) 072005; B. Aubert et al., (Babar Collaboration) hep-ex/0207042; K. Abe at al., (Belle Collaboration) hep-ex/0207098. F. Parodi, talk given at the XXXI$^{st}$ ICHEP, Amsterdam 24-31 July 2002. Results presented at the CKM Unitarity Triangle Workshop, CERN Feb 2002. http:$\backslash\backslash$ckm-workshop.web.cern.ch/ckm-workshop/\
For the description of the methods see:\
M. Ciuchini [*et al.*]{}, JHEP [**0107**]{} (2001) 013 (hep-ph/0012308).\
A. Höcker, H. Lacker, S. Laplace, F. Le Diberder, Eur. Phys. J. [**C21**]{}, 225 (2001).\
G. P. Dubois-Felsmann D. G. Hitlin, F. C. Porter and G. Eigen, CALT 68-2396 June 2002.
[^1]: On average there are as many particles originating from $b$-quark fragmentation and from B decay.
[^2]: with the final LEP statistics, B rare decays with branching fraction of the order of a few 10$^{-5}$ could be accessed.
[^3]: Terms at order 1/$m_Q$ would appear if in this expansion the mass of the heavy hadron was used instead of the mass of the quark. The presence of such a term would violate the quark-hadron duality.
[^4]: In another formalism, based on pole quark masses, the $\bar{\Lambda}$ and $\lambda_1$ parameters are used, which can be related to the difference between hadron and quark masses and to $\mu_{\pi}^2$, respectively.
[^5]: $\Delta {m}_q$ is usually given in ps$^{-1}$: 1 ps$^{-1}$ corresponds to 6.58 10$^{-4}$eV.
[^6]: $ \overline{\rho} = \rho ( 1-\frac{\lambda^2}{2} ) ~~~;~~~ \overline{\eta} =
\eta ( 1-\frac{\lambda^2}{2} ).$
[^7]: due to the disagreement between the different determinations $\lambda$ has been recently evaluated to be [@ref:ckmworkshop]: $0.2237 \pm 0.0033$
|
---
abstract: 'We prove the contractibility of the dual complexes of weak log Fano pairs. As applications, we obtain a vanishing theorem of Witt vector cohomology of Ambro-Fujino type and a rational point formula in dimension three.'
address: 'Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan.'
author:
- Yusuke Nakamura
title: Dual complex of log Fano pairs and its application to Witt vector cohomology
---
Introduction
============
In the first half of this paper, we discuss the dual complex of Fano pairs. A dual complex is a combinatorial object which expresses how the components of $\Delta ^{=1}$ intersect for a dlt pair $(X, \Delta)$. In [@dFKX17], they study the dual complex of a dlt modification of a pair $(X, \Delta)$, and show that the dual complex is independent of the choice of a dlt modification (minimal dlt blow-up) up to PL homeomorphism. In [@KX16] and [@Mau], the dual complexes of log canonical dlt pairs $(X, \Delta)$ with $K_X + \Delta \sim _{\mathbb{Q}} 0$ are studied.
Our main theorem is the contractibility of the dual complexes of weak Fano pairs (see [@KX16 22] for a similar result).
\[thm:main1\] Let $(X, \Delta)$ be a projective pair over an algebraic closed field of characteristic zero. Assume that $-(K_X + \Delta)$ is nef and big. Then for any dlt blow-up $g: (Y, \Delta _Y) \to (X, \Delta)$, the dual complex $\mathcal{D}(\Delta _Y ^{\ge 1})$ is contractible, where we define $\Delta _Y$ by $K_Y + \Delta _{Y} = g^* (K_X + \Delta)$.
In this theorem, the coefficients of $\Delta$ might be larger than one contrary to the setting in [@dFKX17]. We prove that the dual complex is independent of the choice of a dlt blow-up (which is possibly not minimal) up to homotopy equivalence in our setting (Proposition \[prop:indep\], see also Remark \[rmk:independence\]). The proof of Proposition \[prop:indep\] depends on the weak factorization theorem [@AKMW02] and does not work in positive characteristic even in dimension three. Hence, we get the following weaker theorem in positive characteristic.
\[thm:main1’\] Let $(X, \Delta)$ be a three dimensional projective pair over an algebraic closed field of characteristic larger than five. Assume that $-(K_X + \Delta)$ is nef and big. Then, there exists a dlt blow-up $g: (Y, \Delta _Y) \to (X, \Delta)$ such that the dual complex $\mathcal{D}(\Delta _Y ^{\ge 1})$ is contractible, where we define $\Delta _Y$ by $K_Y + \Delta _{Y} = g^* (K_X + \Delta)$.
In the latter half of this paper, we discuss an application of the above result on the dual complex to the study on Witt vector cohomology in positive characteristic. In [@Esn03], it is shown that the vanishing $H^i(X, W \mathcal{O}_{X, \mathbb{Q}}) = 0$ holds for $i > 0$ and a geometrically connected smooth Fano variety $X$ defined over an algebraic closed field $k$. This vanishing theorem is very impressive because it is not known whether $H^i(X, \mathcal{O}_{X}) = 0$ holds or not by the lack of the Kodaira vanishing theorem in positive characteristic. In [@GNT], Esnault’s result is generalized to klt pairs of dimension $3$ and in $\mathrm{char} (k) > 5$. In [@NT], the result in [@GNT] is generalized as a vanishing theorem of Nadel type (see Theorem \[thm:WNV\]). The following main theorem of this paper is another generalization of the result in [@GNT] (we note that the result in [@GNT] is Theorem \[thm:main2\] with the additional restriction that the pair $(X, \Delta)$ is klt).
\[thm:main2\] Let $k$ be a perfect field of characteristic $p > 5$. Let $(X, \Delta)$ be a three-dimensional projective $\mathbb{Q}$-factorial log canonical pair over $k$ with $-(K_X + \Delta)$ ample. Then $H^i(X,W \mathcal{O}_{X, \mathbb{Q}}) = 0$ holds for $i > 0$.
By the vanishing theorem of Nadel type (Theorem \[thm:WNV\]), the proof of Theorem \[thm:main2\] is reduced to the topological study of the non-klt locus of the pair $(X, \Delta)$ (Proposition \[prop:normality\] and Proposition \[prop:tree\]). For the proof of Proposition \[prop:normality\] and Proposition \[prop:tree\], we use the result on the dual complex (Theorem \[thm:main1’\]) to obtain the topological information.
An important application of Witt vector cohomology is the rational point formula on varieties defined over a finite field (cf. [@Esn03; @GNT; @NT]). One of the motivation of the papers [@Esn03; @GNT; @NT] is to generalize the Ax-Katz theorem [@Ax64; @Kat71], which states that any hypersurface $H \subset \mathbb{P}^n$ of degree $d \le n$ defined over $\mathbb{F}_q$ has a rational point. Theorem \[thm:main2\] suggests that the Ax-Katz theorem might be generalized to singular ambient spaces, and we actually obtain the following theorem in dimension three.
\[thm:main3\] Let $k$ be a finite field of characteristic $p > 5$. Let $(X, \Delta)$ be a geometrically connected three-dimensional projective $\mathbb{Q}$-factorial log canonical pair over $k$ with $-(K_X + \Delta)$ ample. Then the number of the $k$-rational points on the non-klt locus of $(X, \Delta)$ satisfies $$\# \mathrm{Nklt}(X, \Delta) (k) \equiv 1 \mod {|k|}.$$ In particular, there exists a $k$-rational point on $\mathrm{Nklt}(X, \Delta)$.
If $X$ is klt and $\Delta$ is a reduced divisor, then $\mathrm{Nklt}(X, \Delta) = \mathrm{Supp} (\Delta)$ holds and Theorem \[thm:main3\] claims that there exists a $k$-rational point on $\mathrm{Supp} (\Delta)$. This formulation can be seen as a generalization of the Ax-Katz theorem from the view point of birational geometry.
We would like to thank Professors Yoshinori Gongyo and Hiromu Tanaka for the discussions and Mirko Mauri for useful comments and suggestions. The author is partially supported by the Grant-in-Aid for Young Scientists (KAKENHI No. 18K13384).
Preliminaries {#section:prelimi}
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Notation {#subsection:notation}
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- We basically follow the notations and the terminologies in [@Har77] and [@Kol13].
- For a field $k$, we say that $X$ is a *variety over* $k$ if $X$ is an integral separated scheme of finite type over $k$.
- A *sub log pair* $(X, \Delta)$ over a filed $k$ consists of a normal variety $X$ over $k$ and an $\mathbb{R}$-divisor $\Delta$ such that $K_X + \Delta$ is $\mathbb{R}$-Cartier. A sub log pair is called *log pair* if $\Delta$ is effective. Note that the coefficient of $\Delta$ may be larger than one in this definition.
- Let $\Delta = \sum r_i D_i$ be an $\mathbb{R}$-divisor where $D_i$ are distinct prime divisors. We define $\Delta ^{\ge 1} := \sum_{r_i \ge 1} r_i D_i$ and $\Delta ^{\wedge 1} := \sum r_i ' D_i$ where $r_i ' := \min \{ r_i, 1 \}$. We also define $\Delta ^{> 1}, \Delta ^{< 1}$, and $\Delta ^{= 1}$ similarly.
- Let $X$ be a variety over $k$. We denote by $(\star)$ the condition that $k$ and $X$ satisfy one of the following conditions:
- $\mathrm{ch} (k) = 0$, or
- $\mathrm{ch} (k) > 5$ and $\dim X = 3$.
This condition is necessary for running a certain MMP appeared in this paper [@BCHM10; @HX15; @Bir16; @BW17; @Wal18; @HNT].
Results on minimal model program
--------------------------------
In this subsection we review results on minimal model program. In this subsection, $k$ is an algebraic closed field.
First, we review the definition of singularities of log pairs. In this paper, we treat the following definitions only under the condition $(\star)$, because we do not know whether the definitions perform well also in $\dim X > 3$ and $\mathrm{ch} (k) >0$.
1. Let $(X, \Delta)$ be a log pair over $k$. For a proper birational $k$-morphism $f: X' \to X$ from a normal variety $X'$ and a prime divisor $E$ on $X'$, the *log discrepancy* of $(X, \Delta)$ at $E$ is defined as $$a_E (X, \Delta) := 1 + \mathrm{coeff}_E (K_{X'} - f^* (K_X + \Delta)).$$
2. A log pair $(X, \Delta)$ is called *klt* (resp. *log canonical*) if $a_E (X, \Delta) > 0$ (resp. $\ge 0$) for any prime divisor $E$ over $X$.
3. A log pair $(X, \Delta)$ is called *dlt* if the coefficients of $\Delta$ are at most one and there exists a log resolution $g: Y \to X$ of the pair $(X, \Delta)$ such that $a_E (X, \Delta) > 0$ holds for any $g$-exceptional prime divisor $E$ on $Y$.
4. Let $(X, \Delta)$ be a dlt pair and let $\Delta ^{=1} = \sum _{i \in I} E_i$ be the irreducible decomposition. For any non-empty subset $J \subset I$, an connected component of $\bigcap _{i \in J} E_i$ is called a *stratum* of $\Delta ^{=1}$.
The above definition (3) is equivalent to the definition in [@KM98 Definition 2.37]:
- The coefficients of $\Delta$ are at most one. Moreover, there exists an open subset $U \subset X$ such that $(U, \Delta |_U)$ is log smooth and no non-klt center of $(X, \Delta)$ is contained in $X \setminus U$.
This equivalence is shown in [@Sza94] in characteristic zero. The equivalence is also true in positive characteristic (in dimension three) since the Szabó’s resolution lemma ([@Fuj17 Lemma 2.3.19]) also holds by [@CP08 Proposition 4.1] (see also [@Fuj17 Proposition 2.3.20] and [@Bir16 2.4, 2.5]). Hence, even in our definition, being dlt is preserved under the MMP ([@KM98 Corollary 3.44]).
The following proposition is necessary for defining the dual complexes of dlt pairs.
\[prop:dltstrata\] Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial dlt pair over $k$ and $\Delta ^{=1} = \sum _{i \in I} E_i$ be the irreducible decomposition. We assume the condition $(\star)$ (defined in Subsection \[subsection:notation\]). Then the following hold.
1. Let $J \subset I$ be a subset. If $\bigcap _{i \in J} E_i \not= \emptyset$, then each connected component of $\bigcap _{i \in J} E_i$ is normal (hence irreducible) and has codimension $\# J$.
2. Let $J \subset I$ be a subset, and let $j \in J$. Then each connected component of $\bigcap _{i \in J} E_i$ is contained in the unique connected component of $\bigcap _{i \in J \setminus \{ j \}} E_i$.
See [@Kol13 Theorem 4.16]. The assertion that each connected component of $\bigcap _{i \in J} E_i$ is irreducible is not explicitly written in [@Kol13 Theorem 4.16]. However it follows from the fact that the intersection of any two log canonical centers is also a union of log canonical centers (cf. [@Fuj11 Theorem 9.1], [@DH16 Lemma 1]).
\(2) is trivial.
In this paper, we will use the terminology “dlt blow-up" in the following sense.
Let $(X, \Delta)$ be a log pair over $k$ and let $g:Y \to X$ be a projective birational $k$-morphism. We call $g$ a *dlt blow-up* of $(X, \Delta)$ if the following conditions hold:
- $a_E(X, \Delta) \le 0$ holds for any $g$-exceptional prime divisor $E$.
- $(Y, \Delta _Y ^{\wedge 1})$ is a $\mathbb{Q}$-factorial dlt pair, where $\Delta _Y$ is the $\mathbb{R}$-divisor defined by $K_Y + \Delta _Y = g^*(K_X + \Delta)$.
\[thm:dltmodif\] Let $(X, \Delta)$ be a log pair over $k$ with the condition $(\star)$. Then a dlt blow-up of $(X, \Delta)$ exists. Further, we can take a dlt blow-up $g: V \to X$ with the following additional condition:
- $g^{-1}(\mathrm{Nklt} (X, \Delta)) = \mathrm{Nklt} (V, \Delta _V)$ holds.
By Step 2 in the proof of [@HNT Proposition 3.5], in order to show the existence of a dlt blow-up with the condition (3), it is sufficient to show the existence of a usual dlt blow-up (that is with the only conditions (1) and (2)). For the existence of a dlt blow-up, the same proof as in [@Fuj11 Theorem 10.4] works as follows.
Let $f: Y \to X$ be a log resolution of $(X, \Delta)$. Let $F = \sum F_i$ be the sum of the $f$-exceptional divisors $F_i$ with $a_{F_i} (X, \Delta) \le 0$, and let $G = \sum G_i$ be the sum of the $f$-exceptional divisors $G_i$ with $a_{G_i} (X, \Delta) > 0$. Let $\widetilde{\Delta}$ be the strict transform of $\Delta$ on $Y$. We may assume that there exists an effective $\mathbb{R}$-divisor $H$ on $Y$ such that $\mathrm{Supp}\, H = \mathrm{Supp}\, (F+G)$ and that $-H$ is $f$-ample. We set an $\mathbb{R}$-divisor $\Omega$ as $$\Omega = \widetilde{\Delta}^{\wedge 1} + F + (1 - \epsilon) G - \delta H$$ for sufficiently small $\epsilon, \delta > 0$. Since $-H$ is $f$-ample, there exists an effective ample $\mathbb{R}$-divisor $A$ on $Y$ such that $- \delta H \sim _{/X,\, \mathbb{R}} A$. We set $$\overline{\Omega} = \widetilde{\Delta}^{\wedge 1} + F + (1 - \epsilon) G + A.$$ We may assume that $(Y, \overline{\Omega})$ is dlt. Note that $K_Y + \Omega \sim _{/X,\, \mathbb{R}} K_Y + \overline{\Omega}$.
Since $A$ is ample, there exists an $\mathbb{R}$-divisor $\overline{\Omega}'$ such that $\overline{\Omega}' \sim _{\mathbb{R}} \overline{\Omega}$ and $(Y, \overline{\Omega}')$ is klt. Hence, we may run a $(K_Y + \Omega)$-MMP over $X$ and it terminates. Let $Y'$ be the end result and let $h: Y' \to X$ be the induced morphism. We shall show that $h: Y' \to X$ is a dlt blow-up of $(X, \Delta)$. We have $$\begin{aligned}
K_Y + \Omega
& \sim _{/X,\, \mathbb{R}} K_Y + \Omega - f^*(K_X + \Delta) \\
&= - (\widetilde{\Delta} - \widetilde{\Delta}^{\wedge 1}) + \sum a_i F_i + \sum b_i G_i - \epsilon G - \delta H, \end{aligned}$$ where $a_i = a_{F_i} (X, \Delta) \le 0$ and $b_i = a_{G_i} (X, \Delta) >0$. Since $b_i > 0$ and $\epsilon$ and $\delta$ are sufficiently small, it follows that $$\mathrm{coeff}_{G_j} \left( \sum a_i F_i + \sum b_i G_i - \epsilon G - \delta H \right) > 0$$ for each $G_j$. Hence by the negativity lemma, all the divisors $G_i$’s are contracted in this MMP. Therefore $a_E (X, \Delta) \le 0$ holds for any $h$-exceptional prime divisor $E$. Since the $(K_Y + \Omega)$-MMP is also a $(K_Y + \overline{\Omega})$-MMP, the pair $(Y', \overline{\Omega} _{Y'})$ is still dlt where $\overline{\Omega} _{Y'}$ is the push forward of $\overline{\Omega}$. Define $\Delta _{Y'}$ by $K_{Y'} + \Delta _{Y'} = h^* (K_X + \Delta)$. Then $\Delta _{Y'} ^{\wedge 1}$ is the push forward of $\widetilde{\Delta}^{\wedge 1} + F$ on $Y'$, and hence $0 \le \Delta _{Y'} ^{\wedge 1} \le \overline{\Omega} _{Y'}$ holds. Therefore $(Y', \Delta _{Y'} ^{\wedge 1})$ is also dlt. We have proved that $g$ is a dlt blow-up of $(X, \Delta)$.
\[rmk:dltbup\]
1. When $X$ is $\mathbb{Q}$-factorial, any dlt blow-up of $(X, \Delta)$ satisfies condition (3) in Theorem \[thm:dltmodif\].
2. If $V \to X$ is a log resolution of $(X, \Delta)$, then we can construct (by the proof above) a dlt blow-up $Y \to X$ of $(X, \Delta)$ such that the induced birational map $Y \dasharrow V$ does not contract any divisor on $Y$.
Dual complexes {#subsection:dual_cpx}
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In this subsection, we explain how to define a CW complex from a dlt pair, and we also prove an invariant property (Proposition \[prop:indep\]).
First, we briefly review the notion of $\Delta$-complexes following [@Hat02].
1. Let $X = \cup _{\varphi _{\alpha}} F_{\alpha}$ be a CW complex with the attaching maps $\varphi _{\alpha} : F_{\alpha} \to X$. We call $X$ a *$\Delta$-complex* when each cell $F_{\alpha}$ is a simplex and the restriction of $\varphi _{\alpha}$ to each face of $F_{\alpha}$ is equal to the attaching map $\varphi _{\beta} : F_{\beta} \to X$ for some $\beta$.
2. A $\Delta$-complex $X$ is called *regular* if the attaching maps are injective, or equivalently, if every $d$-cell in $X$ has $d+1$ distinct vertices.
3. A regular $\Delta$-complex $X$ is called a *simplicial complex* if the intersection of any two cells in $X$ is a face of both cells, or equivalently, every $k+1$ vertices in $X$ is incident to at most one $k$-cell.
For a dlt pair $(X, \Delta)$, we define the dual complex $\mathcal{D}(\Delta ^{=1})$.
1. Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial dlt pair over $k$ with the condition $(\star)$ and let $\Delta ^{=1} = \sum _{i \in I} E_i$ be the irreducible decomposition. Then the *dual complex* $\mathcal{D}(\Delta ^{=1})$ is a CW complex obtained as follows. The vertices of $\mathcal{D}(\Delta ^{=1})$ are the set of $\{ E_i \} _{i \in I}$. To each $k$-codimensional stratum $S$ of $\Delta ^{=1}$ we associate a $k$-dimensional cell. The attaching map is uniquely defined by Proposition \[prop:dltstrata\] (2).
2. Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial pair over $k$ with the condition $(\star)$. Suppose that $(X, \Delta^{\wedge 1})$ is dlt. Then we define $\mathcal{D} (\Delta ^{\ge 1}) = \mathcal{D} ((\Delta^{\wedge 1}) ^{= 1})$
\[prop:regular\] Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial dlt pair over $k$ with the condition $(\star)$. Then the dual complex $\mathcal{D}(\Delta ^{=1})$ is a regular $\Delta$-complex.
The assertion follows from Proposition \[prop:dltstrata\] (1). See [@dFKX17] for more detail.
The following theorem from [@dFKX17] says that the dual complex is preserved under a certain MMP up to simple-homotopy equivalence.
\[thm:inv\_mmp\] Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial dlt pair over $k$ with condition $(\star)$ and let $f:X \dasharrow Y$ be a divisorial contraction or flip corresponding to a $(K_X + \Delta)$-negative extremal ray $R$. Assume that there is a prime divisor $D_0 \subset \Delta ^{=1}$ such that $D_0 \cdot R >0$. Then $\mathcal{D}(\Delta ^{=1})$ collapses to $\mathcal{D}(\Delta _Y ^{=1})$ where we set $\Delta _Y := f_* \Delta$. In particular $\mathcal{D}(\Delta ^{= 1})$ and $\mathcal{D}(\Delta _Y ^{=1})$ are simple-homotopy equivalent.
\[lem:szabo\] Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial pair over $k$ with the condition $(\star)$. Suppose that $(X, \Delta ^{\wedge 1})$ is dlt. Let $f: Y \to X$ be a log resolution of $(X, \Delta ^{\wedge 1})$ such that $a_E (X, \Delta ^{\wedge 1}) > 0$ holds for any $f$-exceptional divisor $E$. Let $\Delta _Y$ be the (not necessarily effective) $\mathbb{R}$-divisor defined by $K_Y + \Delta _Y = f^*(K_X + \Delta)$. Then the dual complexes $\mathcal{D}(\Delta ^{\ge 1})$ and $\mathcal{D}(\Delta _Y ^{\ge 1})$ are simple-homotopy equivalent.
Let $F = \sum F_i$ be the sum of the $f$-exceptional divisors $F_i$ with $a_{F_i} (X, \Delta) \le 0$, and let $G = \sum G_i$ be the sum of the $f$-exceptional divisors $G_i$ with $a_{G_i} (X, \Delta) > 0$. Let $\widetilde{\Delta}$ be the strict transform of $\Delta$ on $Y$. Let $H$ be an effective $\mathbb{R}$-divisor on $Y$ such that $\mathrm{Supp}\, H = \mathrm{Supp}\, (F+G)$ and that $-H$ is $f$-ample. We set an $\mathbb{R}$-divisor $\Omega$ as $$\Omega = \widetilde{\Delta}^{\wedge 1} + F + (1 - \epsilon) G - \delta H$$ for sufficiently small $\epsilon, \delta > 0$. Since $-H$ is $f$-ample, there exists an effective ample $\mathbb{R}$-divisor $A$ on $Y$ such that $- \delta H \sim _{/X,\, \mathbb{R}} A$. We set $$\overline{\Omega} = \widetilde{\Delta}^{\wedge 1} + F + (1 - \epsilon) G + A.$$ We may assume that $(Y, \overline{\Omega})$ is dlt. Note that $K_Y + \Omega \sim _{/X,\, \mathbb{R}} K_Y + \overline{\Omega}$ and $\mathcal{D}(\overline{\Omega} ^{=1}) = \mathcal{D}(\Delta _Y ^{\ge 1})$.
Since $A$ is ample, there exists an $\mathbb{R}$-divisor $\overline{\Omega}'$ such that $\overline{\Omega}' \sim _{\mathbb{R}} \overline{\Omega}$ and $(Y, \overline{\Omega}')$ is klt. Hence, we may run a $(K_Y + \Omega)$-MMP over $X$ and it terminates. First, we prove that this MMP ends with $X$. $$\begin{aligned}
K_Y + \Omega
& \sim _{/X,\, \mathbb{R}} K_Y + \Omega - f^*(K_X + \Delta^{\wedge 1}) \\
&= \sum a_i F_i + \sum b_i G_i - \epsilon G - \delta H, \end{aligned}$$ where $a_i = a_{F_i} (X, \Delta ^{\wedge 1})$ and $b_i = a_{G_i} (X, \Delta ^{\wedge 1})$. Since $a_i, b_i > 0$ and $\epsilon$ and $\delta$ are sufficiently small, it follows that $$\mathrm{coeff}_E \left( \sum a_i F_i + \sum b_i G_i - \epsilon G - \delta H \right) > 0$$ for any $f$-exceptional divisor $E$. Hence by the negativity lemma, all the divisors $F_i$’s and $G_i$’s are contracted in this MMP. Since $X$ is $\mathbb{Q}$-factorial, this MMP ends with $X$.
Let $Y_j \dasharrow Y_{j+1}$ be the step of the $(K_Y + \Omega)$-MMP over $X$, and let $R$ be the corresponding extremal ray, and $Y_j \to Z_j$ be its contraction. For a divisor $D$ on $Y$, we denote $D_{Y_j}$ the strict transform of $D$ on $Y_j$. We also denote $g: Y_j \to X$ the induced morphism. Then, $$\begin{aligned}
& K_{Y_j} + \Omega _{Y_j} \\
\sim & _{/X,\, \mathbb{R}} K_{Y_j} + \Omega _{Y_j} - g^*(K_X + \Delta) \\
= & - \left( \widetilde{\Delta}_{Y_j} - (\widetilde{\Delta} _{Y_j}) ^{\wedge 1} \right) +
\sum a' _i F_{i, Y_j} + \sum b' _i G_{i, Y_j} - \epsilon G_{Y_j} - \delta H_{Y_j}, \end{aligned}$$ where $a' _i = a_{F_i} (X, \Delta)$ and $b' _i = a_{G_i} (X, \Delta)$. Since $a' _i \le 0$, it follows that $$\mathrm{coeff}_{F_{i, Y_j}} \left( \sum a' _i F_{i, Y_j} + \sum b' _i G_{i, Y_j} - \epsilon G_{Y_j} - \delta H_{Y_j} \right) < 0.$$ Since $b' _i > 0$ and $\epsilon$ and $\delta$ are sufficiently small, it follows that $$\mathrm{coeff}_{G_{i, Y_j}} \left( \sum a' _i F_{i, Y_j} + \sum b' _i G_{i, Y_j} - \epsilon G_{Y_j} - \delta H_{Y_j} \right) > 0.$$ Since $(K_{Y_j} + \Omega_{Y_j}) \cdot R < 0$, at least one of the following conditions hold:
1. $D \cdot R > 0$ holds for some component $D \subset \mathrm{Supp} (\widetilde{\Delta} _{Y_j} ^{>1})$.
2. $F_{i, Y_j} \cdot R > 0$ for some $i$.
3. $G_{i, Y_j} \cdot R < 0$ for some $i$.
Here, we have $K_{Y_j} + \Omega _{Y_j} \sim _{/X,\, \mathbb{R}} K_{Y_j} + \overline{\Omega} _{Y_j}$ and that a component of $\overline{\Omega} _{Y_j} ^{=1}$ is one of $F_{i, Y_j}$’s or a component of $(\widetilde{\Delta} _{Y_j}) ^{\ge 1}$. Hence in the case (1) or (2), by Theorem \[thm:inv\_mmp\], the dual complexes $\mathcal{D}((\overline{\Omega} _{Y_j}) ^{=1})$ and $\mathcal{D}((\overline{\Omega} _{Y_{j+1}}) ^{=1})$ are simple-homotopy equivalent. In the case (3), the exceptional locus $L$ of $Y_j \to Z_j$ is contained in $\mathrm{Supp} (G_{i, Y_j})$. Since $(Y_j, \overline{\Omega} _{Y_j})$ is dlt, any stratum of $(\overline{\Omega} _{Y_j}) ^{=1}$ is not contained in $\mathrm{Supp} (G_{i, Y_j})$ and neither in $L$. Hence in the case (3), by [@dFKX17 Lemma 16], it follows that $\mathcal{D}((\overline{\Omega} _{Y_j}) ^{=1}) = \mathcal{D}((\overline{\Omega} _{Y_{j+1}}) ^{=1})$.
Hence by induction on $j$, the dual complexes $\mathcal{D}(\overline{\Omega} ^{=1})$ and $\mathcal{D}((\overline{\Omega} _X) ^{=1})$ are simple-homotopy equivalent. Since $\mathcal{D}(\overline{\Omega} ^{=1}) = \mathcal{D}(\Delta _Y ^{\ge 1})$ and $\mathcal{D}((\overline{\Omega} _X) ^{=1}) = \mathcal{D}(\Delta ^{\ge 1})$, it follows that $\mathcal{D}(\Delta _Y ^{\ge 1})$ and $\mathcal{D}(\Delta ^{\ge 1})$ are homopoty equivalent.
\[lem:bup\] Let $(X, \Delta)$ be a sub log pair over $k$ with the condition $(\star)$ such that $(X, \mathrm{Supp}\ \Delta)$ is log smooth. Let $Z$ be a smooth irreducible subvariety of $X$ which has only simple normal crossing with $\mathrm{Supp}\, \Delta$. Let $f : Y \to X$ be the blow up along $Z$, and let $\Delta _Y$ be the $\mathbb{R}$-divisor defined by $K_Y + \Delta _Y = f^*(K_X + \Delta)$. Then the dual complexes $\mathcal{D}(\Delta ^{\ge 1})$ and $\mathcal{D}(\Delta _Y ^{\ge 1})$ are simple-homotopy equivalent.
Set $F = (\Delta ^{\ge 1})^{\wedge 1}$ and $F_Y = (\Delta_Y ^{\ge 1})^{\wedge 1}$. Let $E$ be the $f$-exceptional divisor and $\widetilde{F}$ be the strict transform of $F$ on $Y$. We divide the case as follows:
1. $Z$ is a stratum of $F$.
2. $Z \subset \mathrm{Supp}\, F$ but $Z$ is not a stratum of $F$.
3. $Z \not \subset \mathrm{Supp}\, F$ (in particular, $Z$ is not a stratum of $F$).
Suppose (1). Then $F_Y = \widetilde{F} + E$ holds. On the other hand, $\mathcal{D}(\widetilde{F} + E)$ and $\mathcal{D}(F)$ are PL homeomorphic each other by [@dFKX17 9].
Suppose (2). Then $F_Y = \widetilde{F}$ or $F_Y = \widetilde{F} + E$ holds. On the other hand, $\mathcal{D}(\widetilde{F}) = \mathcal{D}(F)$ holds and $\mathcal{D}(\widetilde{F} + E)$ and $\mathcal{D}(F)$ are simple-homotopy equivalent by [@dFKX17 9].
Suppose (3). Then $F_Y = \widetilde{F}$ holds and $\mathcal{D}(\widetilde{F}) = \mathcal{D}(F)$.
\[prop:indep\] Let $(X, \Delta)$ be a pair over $k$ with condition (i) in $(\star)$ and let $f_1: Y_1 \to X$ and $f_2: Y_2 \to X$ be two dlt blow-ups of $(X, \Delta)$. Define $\mathbb{R}$-divisors $\Delta _{Y_i}$ by $K_{Y_i} + \Delta _{Y_i} = f_i ^* (K_X + \Delta)$. Then the dual complexes $\mathcal{D}(\Delta _{Y_1} ^{\ge 1})$ and $\mathcal{D}(\Delta _{Y_2} ^{\ge 1})$ are simple-homotopy equivalent.
The same proof of [@dFKX17 Proposition 11] works. For the reader’s convenience, we give a sketch of proof.
By definition of dlt pairs, we can take a log resolution $g_i: W_i \to Y_i$ of $(Y_i, \Delta _{Y_i})$ such that $a_E (Y_i, \Delta _{Y_i}^{\wedge 1}) > 0$ holds for any $g_i$-exceptional divisor $E$. Define $\mathbb{R}$-divisors $\Delta _{W_i}$ on $W_i$ by $K_{W_i} + \Delta _{W_i} = g_i ^* (K_{Y_i} + \Delta _{Y_i})$. By Lemma \[lem:szabo\], the dual complexes $\mathcal{D} (\Delta _{Y_i} ^{\ge 1})$ and $\mathcal{D} (\Delta _{W_i} ^{\ge 1})$ are simple-homotopy equivalent. By the weak factorization theorem [@AKMW02], the pairs $(W_1, \Delta _{W_1})$ and $(W_2, \Delta _{W_2})$ can be connected by a sequence of blow up as in Lemma \[lem:bup\]. Hence $\mathcal{D}(\Delta _{W_1} ^{\ge 1})$ and $\mathcal{D}(\Delta _{W_2} ^{\ge 1})$ are simple-homotopy equivalent.
\[rmk:independence\]
1. In the proof of this proposition, the weak factorization theorem [@AKMW02] is used. So, the proof does not work in positive characteristic even in dimension three.
2. If $(X, \Delta)$ is log canonical in this proposition, then $\mathcal{D}(\Delta _{Y_1} ^{\ge 1})$ and $\mathcal{D}(\Delta _{Y_2} ^{\ge 1})$ are PL homeomorphic each other ([@dFKX17 Proposition 11]).
Results on the Witt vector cohomologies
---------------------------------------
For the definition of the Witt vector cohomology and its basic properties, we refer to [@GNT] and [@CR12]. The following vanishing theorem of Nadel type will be used in this paper.
\[thm:WNV\] Let $(X, \Delta)$ be a projective log pair over a perfect field $k$ with condition (ii) in $(\star)$. Then $$H^i(X, WI_{\mathrm{Nklt}(X, \Delta), \mathbb{Q}}) = 0$$ holds for $i >0$, where $\mathrm{Nklt}(X, \Delta)$ denotes the reduced closed subscheme of $X$ consisting of the non-klt points of $(X, \Delta)$ and $I_{\mathrm{Nklt}(X, \Delta)}$ is the coherent ideal sheaf on $X$ corresponding to $\mathrm{Nklt}(X, \Delta)$.
Dual complex of weak Fano varieties
===================================
Dual complex of dlt pairs with a Mori fiber space structure
-----------------------------------------------------------
\[lem:MFS2\] Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial dlt pair over $k$ with condition $(\star)$ and let $f: X \to Z$ be a projective surjective $k$-morphism to a quasi-projective $k$-scheme $Z$ such that
1. $\dim X >\dim Z$,
2. $f$ has connected fibers,
3. $-(K_X+\Delta)$ is $f$-ample, and
4. there exists an irreducible component $D_0$ of $\Delta ^{=1}$ such that $D_0$ is $f$-ample.
Then $\mathcal{D}(\Delta ^{=1})$ is contractible.
Let $g:D_0 \to Z$ be the induced morphism. Let $\Delta ^{=1} = \sum_{i=0}^m D_i$ be the irreducible decomposition.
What we want to show is the following:
- Any stratum $S$ of $\sum_{i=1}^m D_i$ intersects with $D_0$, and
- $S \cap D_0$ is connected.
These conditions imply that the dual complex of $\sum_{i=0}^m D_i$ is the cone of the dual complex of $\sum_{i=1}^m D_i$ with the vertex $D_0$. Therefore $\mathcal{D}(\Delta ^{=1})$ is contractible. We show (1) and (2) by induction on $\dim X$.
First we prove the following claim.
\[claim:majiwaru\] For any $i \in \{1, \ldots, m\}$, it holds that
1. $\dim f(D_i) < \dim D_i$, and
2. $f(D_i) =f(D_i \cap D_0)$ holds, in particular $D_i \cap D_0 \not= \emptyset$.
3. $D_i \cap D_0$ is connected and irreducible.
We prove (a). Since the assertion is clear if $\dim Z \leq \dim X - 2$, we may assume that $\dim Z = \dim X - 1$.
Suppose that $f(D_i)=Z$ holds for some $i \in \{1, \dots, m\}$. For a general fiber $F$ of $f$, $\dim F = 1$ holds and $(D_i \cup D_0) \cap F$ is not connected. This contradicts the Koll[á]{}r-Shokurov connectedness lemma (cf. [@NT Theorem 1.2]). Therefore, $D_i$ does not dominate $Z$ for any $i \in \{1, \dots, m\}$.
We prove (b). Let $x \in f(D_i)$ be a closed point. Since $\dim f(D_i) < \dim D_i$, there exists a curve $C$ on $X$ contained in $D_i \cap f^{-1}(x)$. Since $D_0$ is ample over $Z$, the contracted curve $C$ intersects with $D_0$. This implies $x \in f(D_i \cap D_0)$. Thus we get $f(D_i) = f(D_i \cap D_0)$.
We prove (c). Suppose that $D_i \cap D_0$ is not connected. Let $S$ be a connected component of $D_i \cap D_0$ which satisfies $f(S) = f(D_i \cap D_0)$. Let $G$ be another connected component of $D_i \cap D_0$. Then for any closed point $x \in f(G)$, $D_i \cap D_0$ is not connected over $x$. However, this contradicts the Koll[á]{}r-Shokurov connectedness lemma. Therefore $D_i \cap D_0$ is connected. Then the irreducibility follows from Proposition \[prop:dltstrata\] (1).
Let $S$ be a stratum of $\sum_{i=1}^m D_i$. Then $S$ is a connected component of $\bigcap _{i \in I} D_i$ for some $I \subset \{ 1, \ldots ,m \}$. We may assume that $1 \in I$ possibly changing the indices.
Let $C:= f(D_1)$, and let $D_1 \xrightarrow{f'} C' \xrightarrow{s} C$ be the Stain factorisation of $D_1 \to C$. Let $\Delta _{D_1}$ be the effective $\mathbb{R}$-divisor on $D_1$ defined by adjunction $(K_X+\Delta)|_{D_1}=K_{D_1}+\Delta _{D_1}$. Then the following properties hold.
1. $(D_1, \Delta _{D_1})$ is dlt.
2. $-(K_{D_1} + \Delta _{D_1})$ is $f'$-ample.
3. $- D_0|_{D_1}$ is $f'$-ample.
We prove (1) and (2) by induction on $\# I$. If $\# I = 1$, then (1) and (2) follow from Claim \[claim:majiwaru\]. Suppose $\# I \ge 2$. Then $S$ is also a stratum of $\Delta _{D_1} ^{=1}$. Hence (1) and (2) holds by induction on the dimension.
\[lem:MFS\] Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial dlt pair over $k$ with the condition $(\star)$ and let $f: X \to Z$ be a $(K_X+\Delta)$-Mori fiber space to a quasi-projective $k$-variety $Z$. Suppose that $f(\mathrm{Supp}\, \Delta ^{=1})=Z$. Then $\mathcal{D}(\Delta ^{=1})$ is contractible.
Since $f(\mathrm{Supp}\, \Delta ^{=1})=Z$, some irreducible component $D_0$ of $\Delta ^{=1}$ satisfies $f(D_0)=Z$. Since $\rho(X/Z)=1$, it follows that $D_0$ is $f$-ample. Hence the assertion follows from Lemma \[lem:MFS2\].
Dual complex of weak Fano varieties
-----------------------------------
\[thm:scc2\] Let $(X, \Omega)$ be a $\mathbb{Q}$-factorial projective log pair. Assume that
- $(X, \Omega ^{\wedge 1})$ is dlt but not klt.
- $K_X + \Omega \sim _{\mathbb{R}} 0$.
- $\mathrm{Supp}\, \Omega ^{>1} = \mathrm{Supp}\, \Omega ^{\ge 1}$.
Then, the dual complex $\mathcal{D}(\Omega ^{\ge 1})$ is contractible.
Since $(X, \Omega ^{\wedge 1})$ is not klt, it follows that $$\mathrm{Supp}\, \Omega ^{>1} = \mathrm{Supp}\, \Omega ^{\ge 1} \not = \emptyset.$$ Therefore, $K_X+\Omega^{\wedge 1} \sim _{\mathbb{R}} - (\Omega-\Omega^{\wedge 1})$ is not pseudo-effective. Further, by (3), $\left( X, \Omega^{\wedge 1} - \epsilon (\Omega-\Omega^{\wedge 1}) \right)$ is klt for some small $\epsilon >0$. Hence we may run a $(K_X+\Omega^{\wedge 1})$-MMP and ends with a Mori fiber space $f: X_{\ell} \to Z$: $$X=:X_0 \dashrightarrow X_1 \dashrightarrow \cdots \dashrightarrow X_{\ell}.$$
Let $\Omega_{i}$ be the push-forward of $\Omega$ on $X_{i}$. Then $\Omega _{i}$ also satisfies the conditions (1)–(3). Further, since $\Omega _{\ell} -\Omega _{\ell} ^{\wedge 1} \sim _{\mathbb{R}}
-(K_{X_{\ell}}+\Omega _{\ell} ^{\wedge 1})$ is ample over $Z$, it follows that $f(\mathrm{Supp}\, \Omega _{\ell} ^{\ge 1}) = Z$. Hence, by Lemma \[lem:MFS\], the dual complex $\mathcal{D} ( (\Omega _{\ell} ^{\wedge 1}) ^{=1})$ is contractible.
Let $R_i$ be the extremal ray of $\overline{\mathrm{NE}} (X_i)$ corresponding to the step of the MMP $X_i \dashrightarrow X_{i+1}$. Since $- (\Omega _{i} -\Omega _{i} ^{\wedge 1}) \cdot R_i < 0$, it follows that some component $D_i$ of $\mathrm{Supp} (\Omega _{i} ^{> 1})$ ($= \mathrm{Supp} \left( (\Omega ^{\wedge 1}) ^{=1} \right)$) satisfies $D_i \cdot R_i > 0$. By Theorem \[thm:inv\_mmp\], $\mathcal{D} ( (\Omega _{i} ^{\wedge 1}) ^{=1})$ and $\mathcal{D} ( (\Omega _{i+1} ^{\wedge 1}) ^{=1})$ are homotopy equivalent. Hence, $\mathcal{D}(\Omega ^{\ge 1}) = \mathcal{D} ( (\Omega ^{\wedge 1}) ^{=1})$ is contractible.
\[prop:ample\] Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial projective pair over $k$ with the condition $(\star)$. Suppose that $-(K_X + \Delta)$ is ample. Then for any dlt blow-up $g: (Y, \Delta _Y) \to (X, \Delta)$, the dual complex $\mathcal{D}(\Delta _Y ^{\ge 1})$ is contractible, where we define $\Delta _Y$ by $K_Y + \Delta _{Y} = g^* (K_X + \Delta)$.
By Proposition \[thm:scc2\], it suffices to find an effective $\mathbb{R}$-divisor $\Omega _Y$ on $Y$ such that
1. $(Y, \Omega _Y ^{\wedge 1})$ is dlt,
2. $K_Y + \Omega _Y \sim _{\mathbb{R}} 0$,
3. $\mathrm{Supp} (\Omega _Y^{>1}) = \mathrm{Supp} (\Omega _Y ^{\ge 1})$, and
4. $\mathrm{Supp} (\Omega _Y ^{\ge 1}) = \mathrm{Supp} (\Delta _Y ^{\ge 1})$.
Since $X$ is $\mathbb{Q}$-factorial, there exists an effective $\mathbb{R}$-divisor $F$ on $Y$ such that $-F$ is $g$-ample and $\mathrm{Supp}\, F = \mathrm{Excep} (g)$. Since $-(K_Y + \Delta _Y)$ is the pullback of an ample $\mathbb{R}$-divisor $-(K_X + \Delta)$ on $X$, it follows that $-(K_Y + \Delta _Y)- \epsilon F$ is ample for any sufficiently small $\epsilon > 0$.
Note that $\mathrm{Supp}\, F \subset \mathrm{Supp} (\Delta _Y ^{\ge 1})$. Thus, we can find an effective $\mathbb{R}$-divisor $B$ on $Y$ such that
- $B \ge \epsilon F$,
- $-(K_Y + \Delta _Y) - B$ is still ample, and
- $\mathrm{Supp}\, B = \mathrm{Supp} (\Delta _Y ^{\ge 1})$.
Then there exists an effective $\mathbb{R}$-divisor $A$ on $Y$ such that
- $A \sim_{\mathbb{R}} -(K_Y + \Delta _Y)- B$, and
- $(Y, \Delta _Y ^{\wedge 1} + 2A)$ is dlt (cf. [@NT Lemma 2.8] in positive characteristic case).
In particular, it follows that
- $(Y, \Delta _Y ^{\wedge 1} + A)$ is dlt, and
- $(\Delta _Y + A) ^{\ge 1} = \Delta _Y ^{\ge 1}$.
Set $\Omega _Y := \Delta _Y + A + B$. Then (2) holds. Since
- $\Omega _Y ^{\wedge 1} = (\Delta _Y + A) ^{\wedge 1} = \Delta _Y ^{\wedge 1} + A$,
\(1) also holds. (3) and (4) hold by the way of taking $A$ and $B$.
\[lem:Qnefbig\] Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial projective pair over $k$ with the condition $(\star)$. Assume that $-(K_X + \Delta)$ is nef and big. Then there exists a dlt blow-up $g: (Y, \Delta _Y) \to (X, \Delta)$ such that the dual complex $\mathcal{D}(\Delta _Y ^{\ge 1})$ is contractible, where we define $\Delta _Y$ by $K_Y + \Delta _{Y} = g^* (K_X + \Delta)$.
Since $-(K_X + \Delta)$ is nef and big, there exists an effective $\mathbb{R}$-divisor $E$ such that $-(K_X + \Delta) - \epsilon E$ is ample for any real number $\epsilon$ satisfying $0<\epsilon \le 1$. Let $h: W \to X$ be a log resolution of $(X, \Delta + E)$. For sufficiently small $\epsilon >0$, we can assume that
- For any $h$-exceptional prime divisor $F$ with $a_F (X, \Delta) > 0$, it still holds that $a_F(X, \Delta + \epsilon E) > 0$.
Let $g: Y \to X$ be a dlt blow-up of $(X, \Delta + \epsilon E)$ such that the birational map $Y \dasharrow W$ does not contract any divisor on $Y$ (Remark \[rmk:dltbup\] (2)). By Proposition \[prop:ample\], the dual complex $\mathcal{D}((\Delta _Y + \epsilon g^* E) ^{\ge 1})$ is contractible. Hence it is sufficient to check the following three conditions (the conditions (2) and (3) below mean that $g$ is also a dlt blow-up of $(X, \Delta)$):
- $a_F(X, \Delta) \le 0$ for any $g$-exceptional divisor $F$.
- $(Y, \Delta _Y ^{\wedge 1})$ is dlt.
- $\mathrm{Supp} \left( (\Delta _Y + \epsilon g^* E) ^{\ge 1} \right) = \mathrm{Supp} (\Delta _Y ^{\ge 1})$.
Since any $g$-exceptional divisor is $h$-exceptional, the conditions (2) and (4) follow from (1). By (2), it follows that $\Delta _Y \ge 0$ and hence (3) is obvious.
\[thm:nefbig\] Let $(X, \Delta)$ be a projective pair over $k$ with the condition $(\star)$. Assume that $-(K_X + \Delta)$ is nef and big. Then there exists a dlt blow-up $g: (Y, \Delta _Y) \to (X, \Delta)$ such that the dual complex $\mathcal{D}(\Delta _Y ^{\ge 1})$ is contractible, where we define $\Delta _Y$ by $K_Y + \Delta _{Y} = g^* (K_X + \Delta)$. Further, we can take such $g$ with the additional condition (3) in Theorem \[thm:dltmodif\].
Let $h: (X', \Delta _{X'}) \to (X, \Delta)$ be a dlt blow-up of $(X, \Delta)$ with the condition (3) in Theorem \[thm:dltmodif\]. Then, $X'$ is $\mathbb{Q}$-factorial and $-(K_{X'} + \Delta _{X'})$ is still nef and big.
By Lemma \[lem:Qnefbig\], there exists a dlt blow-up $(X'', \Delta _{X''}) \to (X', \Delta _{X'})$ such that the dual complex $\mathcal{D}(\Delta _{X''} ^{\ge 1})$ is contractible. Since the composition $X'' \to X$ is again a dlt blow-up of $(X, \Delta)$ with condition (3) in Theorem \[thm:dltmodif\], we complete the proof.
In characteristic zero, we can conclude the following stronger statement.
\[thm:nefbig0\] Let $(X, \Delta)$ be a projective pair over $k$ with the condition (i) in $(\star)$. Assume that $-(K_X + \Delta)$ is nef and big. Then for any dlt blow-up $g: (Y, \Delta _Y) \to (X, \Delta)$, the dual complex $\mathcal{D}(\Delta _Y ^{\ge 1})$ is contractible, where we define $\Delta _Y$ by $K_Y + \Delta _{Y} = g^* (K_X + \Delta)$.
The assertion follows from Theorem \[thm:nefbig\] and Proposition \[prop:indep\].
Vanishing theorem on log canonical Fano varieties
=================================================
In this section, we prove a vanishing theorem of Witt vector cohomology of Fano varieties of Ambro-Fujino type (Theorem \[thm:WAFV\]).
Non-klt locus of log Fano three-folds
-------------------------------------
In this subsection, we prove Lemma \[lem:rationality\], Proposition \[prop:normality\], and Proposition \[prop:tree\] which will be used in the proof of Theorem \[thm:WAFV\].
\[lem:rationality\] Let $k$ be an algebraic closed field of characteristic $p > 5$. Let $(X, \Delta)$ be a three-dimensional projective log canonical pair over $k$ with $-(K_X + \Delta)$ ample. Suppose that $\mathrm{Nklt}(X, \Delta)$ is pure dimension one. Then each irreducible component of $\mathrm{Nklt}(X, \Delta)$ is a rational curve.
Let $C_0$ be an irreducible component of $\mathrm{Nklt}(X, \Delta)$. Let $f: Y \to X$ be a dlt blow-up of $(X, \Delta)$ (Theorem \[thm:dltmodif\]). We define $\Delta _Y$ by $K_Y + \Delta _Y = f^* (K_X + \Delta)$. Let $E \subset \mathrm{Supp}\, \Delta _Y ^{=1}$ be a component which dominates $C_0$. We define $\Delta _E$ by $K_{E} + \Delta _E = (K_Y + \Delta _Y)|_{E}$. Then $K_E + \Delta _E$ is not nef since $-(K_X + \Delta)$ is ample. By the cone theorem for surfaces (cf. [@Tan14 Proposition 3.15]), there exists a $(K_E + \Delta _E)$-negative rational curve $B$. Since $(K_E + \Delta _E) \cdot B < 0$ and $-(K_E + \Delta _E)$ is the pulled back of a divisor on $C_0$, it follows that $B$ dominates $C_0$, which proves the rationality of $C_0$.
We prove two properties (Proposition \[prop:G1\], Proposition \[prop:G2\]) on the dual complex $\mathcal{D}(\Delta _{Y} ^{\ge 1})$ of a dlt blow-up. They will be used in the proof of Proposition \[prop:normality\].
First, we introduce some notations. When we write that $G$ is a $\Delta$-complex, we regard $G$ as the set of its simplices. Hence, when we write $S \in G$, then $S$ is a simplices of $G$.
1. For a $\Delta$-complex $G$, we denote by $|G|$ the topological space of $G$. Hence, $|G| = \bigcup _{S \in G} S$ as set.
2. For a $\Delta$-complex $G$ and its subset $G' \subset G$, we define the *star* $\mathrm{st}(G', G)$ by $$\mathrm{st}(G', G) = \{ S \in G \mid \text{$S \cap S' \not = \emptyset$ for some $S' \in G'$} \}.$$
3. For a $\Delta$-complex $G$, and $S, S' \in G$, we write $S < S'$ when $S$ is a face of $S'$.
\[prop:G1\] Let $k$ be an algebraic closed field of characteristic $p > 5$. Let $(X, \Delta)$ be a three-dimensional projective log canonical pair over $k$ with $-(K_X + \Delta)$ nef and big. Suppose that $C := \mathrm{Nklt}(X, \Delta)$ is of pure dimension one. Let $f: (Y, \Delta_Y) \to (X, \Delta)$ be a dlt blow-up such that $\mathrm{Supp}\, \Delta _Y ^{\ge 1} = f^{-1}(C)$ and that $G := \mathcal{D}(\Delta _Y ^{\ge 1})$ is contractible. Let $C_0$ be an irreducible component of $C = \mathrm{Nklt}(X, \Delta)$.
Let $G'$ be the subcomplex of $G$ which consists of the strata of $\Delta _Y ^{\ge 1}$ which dominate $C_0$. Let $U := |G| \setminus |G'|$ be the open subset of $|G|$, and let $V \subset |G|$ be a sufficiently small open neighborhood of $|G'|$. Then following hold.
- $G'$ is a connected subcomplex of $G$ of dimension at most one.
- For each connected component $U'$ of $U$, it follows that $V \cap U'$ is also connected.
We prove (1). Only the connectedness of $G'$ is non-trivial. Let $q \in C_0$ be a general closed point. Since $q$ is general, we may assume that
- for a strata $S$ of $\Delta _Y ^{\ge 1}$, $q \in f(S)$ holds if and only if $f(S) = C_0$ holds.
Then the connectedness of the fiber $f^{-1}(q)$ shows that the connectedness of $G'$.
For (2), consider the Mayer–Vietoris sequence $$H_1(|G| , \mathbb{Q}) \to H_0(U \cap V, \mathbb{Q}) \to
H_0(U, \mathbb{Q}) \oplus H_0(V, \mathbb{Q}) \to
H_0(|G| , \mathbb{Q}) \to 0.$$ Here, $H_1 (|G|, \mathbb{Q}) = 0$ follows because $|G|$ is contractible, $H_0(|G| , \mathbb{Q}) = H_0(V, \mathbb{Q}) = 0$ follows because $|G|$ and $V$ are connected. Therefore, it follows that $H_0(U \cap V, \mathbb{Q}) \cong H_0(U, \mathbb{Q})$, which proves the claim.
\[prop:G2\] Let $(X, \Delta), (Y, \Delta _Y), C, C_0, G, G', U, V$ be as in the Proposition \[prop:G1\]. Let $U'$ be a connected component of $U$. Consider a pair $(B, S)$ with the following conditions:
- $B \in \mathrm{st}(G', G) \setminus G'$ is an edge, and $S \in G'$ is a vertex.
- $S < B$ and $B \cap U' \not = \emptyset$.
For any two pairs $(B, S)$ and $(B', S')$ with the conditions (a) and (b) above, the assertion is that there exists a sequence of pairs $$(B, S) = (B_0, S_0), (B_1, S_1), \ldots , (B_k, S_k)=(B', S')$$ with $k \ge 0$ and the following conditions:
- Each $(B_j, S_j)$ satisfies the conditions (a) and (b).
- For each $0 \le i \le k-1$, the pairs $(B_i, S_i), (B_{i+1}, S_{i+1})$ satisfy one of the following conditions:
- $S_i = S_{i+1}$ holds, and $B_i, B_{i+1} < F$ for some $2$-simplex $F \in G$.
- $S_i \not = S_{i+1}$ holds, and $S_i, S_{i+1} < E$ for some edge $E \in G'$. Further, $B_i, B_{i+1}, E < F$ for some $2$-simplex $F \in G$.
Note that $U' \cap V$ is a connected component of $U \cap V$ by Proposition \[prop:G1\] and that $U \cap V \subset \bigcup _{A \in \, \mathrm{st}\, (G', G) \setminus G'} \mathrm{int}\, A$.
We also note that giving a pair $(B, S)$ with conditions (a) and (b) is equivalent to giving $B ^{\circ}$ in the following set. $$\begin{aligned}
\Gamma :=
\left\{ B^{\circ} \ \middle |
\begin{array}{l}
\text{$B^{\circ}$ is a connected component of $B \cap (U' \cap V)$} \\
\text{for some edge $B \in \mathrm{st}(G, G') \setminus G'$}
\end{array}\right \}\end{aligned}$$ Indeed, for a pair $(B, S)$ with conditions (a) and (b), there exists the unique connected component of $B \cap (U' \cap V)$ which is around $S$. Inversely, for $B^ {\circ}$ in the set above, the corresponding $B$ and $S$ are uniquely determined.
Let $B^{\circ}$ (resp. ${B'} ^{\circ}$) be the connected component of $B \cap (U' \cap V)$ (resp. $B' \cap (U' \cap V)$) which is around $S$ (resp. $S'$).
Since $B^{\circ}, {B'} ^{\circ} \subset U' \cap V$ and $U' \cap V$ is connected, we can take the following sequence $$B^{(0)} := B^{\circ}, F^{(0)}, B^{(1)}, F^{(1)}, \ldots , B^{(k-1)}, F^{(k-1)}, B^{(k)}:= {B'} ^{\circ}$$ with the following conditions:
- Each $B^{(i)}$ is a connected component of $B_i \cap (U' \cap V)$ for some edge $B_i \in \mathrm{st}(G', G) \setminus G'$ (equivalently $B^{(i)} \in \Gamma$).
- Each $F^{(i)}$ is a connected component of $F_i \cap (U' \cap V)$ for some $2$-simplex $F_i \in \mathrm{st}(G', G) \setminus G'$.
- $B^{(i)}, B^{(i+1)} \subset F^{(i)}$
Possibly passing to a subsequence, we may also assume that
- $B^{(i)} \not = B^{(i+1)}$ for each $0 \le i \le k-1$.
We denote by $S_i$ the unique vertex of $B_i$ which is around $B^{(i)}$. Obviously, $S_i$ is a vertex of $G'$.
For each $i$, there are two possibilities.
- The connected component of $F_i \cap |G'|$ which is around $F^{(i)}$ is zero dimensional.
- The connected component of $F_i \cap |G'|$ which is around $F^{(i)}$ is one dimensional.
Suppose $i$ satisfies (e-1). Then $S_i = S_{i+1}$ and $S_i$ is the common vertex of $B_i$ and $B_{i+1}$. Therefore $(B_i, S_i)$ and $(B_{i+1}, S_{i+1})$ satisfy (d-1).
Suppose $i$ satisfies (e-2). Then there exists an edge $E_i \in G'$ such that $E_i, B_i, B_{i+1} < F_i$. Then $S_i$ (resp. $S_{i+1}$) is the common vertex of $E_i$ and $B_i$ (resp. $E_i$ and $B_{i+1}$). Then $(B_i, S_i)$ and $(B_{i+1}, S_{i+1})$ satisfy (d-2).
\[prop:normality\] Let $k$ be an algebraic closed field of characteristic $p > 5$. Let $(X, \Delta)$ be a three-dimensional projective log pair over $k$ with $-(K_X + \Delta)$ nef and big. Suppose that $\mathrm{Nklt}(X, \Delta)$ is of pure dimension one. Then for each irreducible component $C_0$ of $\mathrm{Nklt}(X, \Delta)$, its normalization $\overline{C_0} \to C_0$ is a universal homeomorphism.
Set $C := \mathrm{Nklt}(X, \Delta)$. By contradiction, suppose that $p \in C_0$ is a singular point such that the normalization $\overline{C_0} \to C_0$ is not a universal homeomorphism around $p$. Let $p^{(1)}, \ldots , p^{(m)}$ be the inverse image of $p$. By the assumption, $m \ge 2$.
Let $f: (Y, \Delta_Y) \to (X, \Delta)$ be a dlt blow-up such that $\mathrm{Supp}\, \Delta _Y ^{\ge 1} = f^{-1}(C)$ (Theorem \[thm:dltmodif\]). Let $G = \mathcal{D}(\Delta _Y ^{\ge 1})$. We may assume that $G$ is contractible by Theorem \[thm:nefbig\].
We define $\{ S_i \}_{i \in I}$ and $\{ T_j \}_{j \in J}$ as follows:
- Let $\{ S_i \}_{i \in I}$ be the set of the irreducible components of $\Delta _Y ^{\ge 1}$ which dominate $C_0$.
- Let $\{ T_j \}_{j \in J}$ be the set of the irreducible components $T_j$ of $\Delta _Y ^{\ge 1}$ which do not dominate $C_0$ but $p \in f(T_j)$.
For each $S \in \{S_i \}_{i \in I}$, since $S$ is normal and dominates $C_0$, $S \to C_0$ factors through $S \to \overline{C_0} \to C_0$. We denote by $S_{p^{(k)}}$ the fiber $S \to \overline{C_0}$ over $p^{(k)}$. Obviously, we have
- $S_{p^{(k)}} \cap S_{p^{(j)}} = \emptyset$ holds for each $k \not = j$.
On the other hand, $f^{-1}(p)$ is connected. Since $f^{-1}(p) \cap T \not = \emptyset$ for each $T \in \{ T_j \}_{j \in J}$, $$\left( \bigcup_{i \in I,\ k} S_{i, p^{(k)}} \right) \cup \left( \bigcup _{j \in J} T_j \right)
=
f^{-1}(p) \cup \bigcup _{j \in J} T_j$$ is also connected. Hence, by (1), possibly changing the indices of $p^{(1)}, \ldots , p^{(m)}$, we can conclude the following:
- There exist $S, S' \in \{ S_i \} _{i \in I}$ and a sequence $T_1, \ldots , T_{\ell} \in \{ T_j \}_{j \in J}$ with $\ell \ge 0$ such that $$S_{p^{(1)}} \cap T_1 \not = \emptyset,\ T_1 \cap T_2 \not = \emptyset, \ \ldots ,\ T_{\ell - 1} \cap T_{\ell} \not = \emptyset,\
T_{\ell} \cap S'_{p^{(2)}} \not = \emptyset.$$
We shall rephrase (2) into a combinatorial condition (STEP 1) and lead a contradiction (STEP 2, 3).
For STEP 1, we introduce some notations which are same as in Proposition \[prop:G1\]. Let $G'$ be the subcomplex of $G$ which consists of the strata of $\Delta _Y ^{\ge 1}$ which dominate $C_0$. Let $U := |G| \setminus |G'|$ be the open subset of $|G|$, and let $V$ be a sufficiently small open neighborhood of $G'$.
**STEP 1.** In this step, we prove the following statement from the condition (2).
- There exist a connected component $U'$ of $U$, and pairs $(B, S)$ and $(B', S')$ with the condition (a) and (b) in Proposition \[prop:G2\] such that $S_{p^{(1)}} \cap B \not= \emptyset$ and $S' _{p^{(2)}} \cap B' \not = \emptyset$ hold. Here we allow $B = B'$ and $S=S'$.
Suppose $\ell = 0$ in (2), that is, $S_{p^{(1)}} \cap S' _{p^{(2)}} \not = \emptyset$. Then, we can take a one dimensional stratum $B \subset S \cap S'$ such that $S_{p^{(1)}} \cap S' _{p^{(2)}} \cap B \not= \emptyset$ holds. If $B$ dominates $C_0$, then $B \to C_0$ also factors through $\overline{C_0}$ since $B$ is normal. We define $B_{p^{(1)}}$ and $B_{p^{(2)}}$ as the fibers of $B \to \overline{C_0}$ over $p^{(1)}$ and $p^{(2)}$. Then $S_{p^{(1)}} \cap B = B_{p^{(1)}}$ and $S' _{p^{(2)}} \cap B = B_{p^{(2)}}$ hold, and they contradict the fact that $B_{p^{(1)}} \cap B_{p^{(2)}} = \emptyset$. Therefore, $B$ does not dominate $C_0$, that is, $B \not \in G'$. Hence the condition (3) holds when we set $B' = B$.
Suppose $\ell \ge 1$ in (2). Then we can take a one dimensional stratum $B \subset S \cap T_1$ (resp. $B' \subset T_{\ell} \cap S'$) such that $S_{p^{(1)}} \cap B \not = \emptyset$ (resp. $S' _{p^{(2)}} \cap B' \not = \emptyset$). Since $T_1$ and $T_{\ell}$ do not dominate $C_0$, neither do $B$ and $B'$, hence $B, B' \not \in G'$. Moreover, $T_1 \cup \cdots \cup T_{\ell}$ is connected, $B$ and $B'$ have intersection with a same connected component $U'$ of $U$.
We have proved (3). Here, we note that the condition $S_{p^{(1)}} \cap B \not= \emptyset$ (resp. $S' _{p^{(2)}} \cap B' \not = \emptyset$) in (3) actually implies that $B \subset S_{p^{(1)}}$ (resp. $B' \subset S' _{p^{(2)}}$). Indeed, it follows from the facts that $B \subset S$ (resp. $B' \subset S'$) and that $S$ (resp. $S'$) dominates $C_0$, but $B$ (resp. $B'$) does not dominate $C_0$. Therefore we conclude the following.
- There exist a connected component $U'$ of $U$, and pairs $(B, S)$ and $(B, S')$ with the condition (a) and (b) in Proposition \[prop:G2\] such that $B \subset S_{p^{(1)}}$ and $B' \subset S' _{p^{(2)}}$ hold. Here we allow $B = B'$ and $S=S'$.
**STEP 2.** Let $(B, S), (B', S')$ be pairs which satisfy (a) and (b) in Proposition \[prop:G2\]. Suppose that one of the following conditions holds:
- $S = S'$ holds, and $B, B' < F$ for some $2$-simplex $F \in G$.
- $S \not = S'$ holds, and $S, S' < E$ for some edge $E \in G'$. Further, $B, B', E < F$ for some $2$-simplex $F \in G$.
In this step, we prove the condition $B \subset S_{p^{(1)}}$ implies $B' \subset S' _{p^{(1)}}$.
Suppose (d-1). $S=S'$ in this case. Since $B, B' < F$ for some $2$-simplex $F \in G$, it follows that $B \cap B' \not= \emptyset$. Since $B \subset S_{p^{(1)}}$, it holds that $B' \cap S_{p^{(1)}} \not = \emptyset$. Then $B' \subset S_{p^{(1)}}$ holds because of the facts that $B' \subset S$ and that $S$ dominates $C_0$, but $B'$ does not dominate $C_0$.
Suppose (d-2). Since $B, B', E < F$ for some $2$-simplex $F \in G$, $B \cap B' \cap E \not = \emptyset$ holds. Since $B \subset S_{p^{(1)}}$, it follows that $$S_{p^{(1)}} \cap B' \cap E \not = \emptyset.$$ Here we note that $E$ dominates $C_0$ and hence $E \to C_0$ factors through $\overline{C_0}$ because $E$ is normal. We denote by $E_{p^{(1)}}$ the fiber of $E \to \overline{C_0}$ over $p^{(1)}$. Since $E \subset S, S'$, it holds that $$S_{p^{(1)}} \cap E = E_{p^{(1)}} \subset S' _{p^{(1)}}.$$ Hence we obtain that $B' \cap S' _{p^{(1)}} \not = \emptyset$. It implies that $B' \subset S' _{p^{(1)}}$ by the same reason as before.
**STEP 3.** In this step, we assume the condition (4) in STEP 1, and lead a contradiction.
Let $(B, S), (B', S')$ be the pairs in (4) in STEP 1. Then $(B, S), (B', S')$ satisfy the condition (a), (b) in Proposition \[prop:G2\]. Hence by Proposition \[prop:G2\], we can take a sequence $(B, S) = (B_0, S_0), (B_1, S_1), \ldots , (B_k, S_k)=(B', S')$ as in Proposition \[prop:G2\].
By STEP 2 and the assumption that $B \subset S_{p^{(1)}}$, it follows that $B_k \subset S_{k, p^{(1)}}$ for each $k$ by induction. Therefore $B' \subset S'_{p^{(1)}}$ holds and it contradicts the assumption that $B' \subset S' _{p^{(2)}}$ and the fact that $S'_{p^{(1)}} \cap S'_{p^{(2)}} = \emptyset$.
In order to state Proposition \[prop:tree\], we introduce a notation.
\[def:tree\] Let $C$ be a scheme of finite type of pure dimension one over an algebraic closed field $k$. Let $C = C_1 \cup C_2 \cup \cdots \cup C_{\ell}$ be the irreducible decomposition. We define whether $C$ *forms a tree* or not by induction on the number $\ell$ of the irreducible components.
Any irreducible curve *forms a tree*. We call that the union of irreducible curves $C = C_1 \cup C_2 \cup \cdots \cup C_{\ell}$ *forms a tree*, if there exists $i$ such that $C' = \bigcup _{j \not = i} C_j$ forms a tree and $\# (C' \cap C_i) = 1$.
First, we prepare some notation in combinatorics.
For a sequence $(a_1, \ldots , a_n)$ of length $n$, we define the operations (a), (b) as follows. Here we define $a_{n+1} := a_1$ and $a_{n+2} := a_2$ by convention.
- If $a_i = a_{i+1}$ for some $1 \le i \le n$, we remove $a_i$ and get a new sequence $(a_1, \ldots , a_{i-1}, a_{i+1}, \ldots , a_n)$ of length $n-1$.
- If $a_i = a_{i+2}$ for some $1 \le i \le n$, then we remove $a_i$ and $a_{i+1}$, and get a new sequence $(a_1, \ldots , a_{i-1}, a_i = a_{i+2} ,a_{i+3}, \ldots , a_n)$ of length $n-2$.
For a sequence $(a_1, \ldots , a_n)$ of length $n$, applying the operation (a) (resp. the operations (a) and (b)) repeatedly, we get a sequence $(b_1, \ldots , b_m)$ with the condition that $b_i \not = b_{i+1}$ for each $i$ (resp. the condition that $b_i \not = b_{i+1}$ and $b_i \not = b_{i+2}$). We call such $(b_1, \ldots , b_m)$ *the (a)-reduction* (resp. *(a,b)-reduction*) of $(a_1, \ldots , a_n)$.
We say that a sequence $(a_1, \ldots, a_n)$ has the *trivial (a)-reduction* (resp. *trivial (a,b)-reduction*) if the (a)-reduction (resp. the (a,b)-reduction) has length $0$. We note here that the (a)-reduction of a sequence of length $1$ has length $0$ by definition.
\[defi:curve\_seq\] Let $C = C_1 \cup C_2 \cup \cdots \cup C_{\ell}$ be in Definition \[def:tree\]. We call that a sequence $(a_1, \ldots, a_n)$ is a *cycle sequence* if the following conditions (1), (2) hold.
- Each $a_i$ is an irreducible component of $C$ or a closed point on $C$.
- The (a)-reduction $(b_1, \ldots , b_m)$ of $(a_1, \ldots , a_n)$ satisfies the following conditions:
- $\dim b_i \not = \dim b_{i+1}$ for each $1 \le i \le m$. We set $b_{m+1} = b_1$ by convention.
- If $\dim b_i = 0$, then $b_i \in b_{i-1} \cap b_{i+1}$. We set $b_0 = b_m$ by convention.
Moreover, we say that a cycle sequence $(a_1, \ldots , a_n)$ is *trivial* if it has the trivial (a,b)-reduction.
Consider curves $C=C_1 \cup \cdots \cup C_5$ in the following figure.
Set sequences $Q_1, Q_2$ as follows: $$\begin{aligned}
Q_1&=(P_1, C_1, P_2, C_2, P_3, C_3, P_3, C_4, P_4, C_5, P_5, C_3, P_3, C_2, P_2, C_1), \\
Q_2&=(P_1, C_1, P_2, C_2, P_3, C_3, P_3, C_4, P_4, C_5, P_5, C_5, P_4, C_4, P_3, C_2, P_2, C_1). \end{aligned}$$ Their (a)-reductions are themselves. These two sequences satisfy the condition (2-1), (2-2), hence they are cycle sequences. $Q_1$ is not trivial, but $Q_2$ is trivial. Indeed the (a,b)-reduction of $Q_1$ is $(C_3, P_3, C_4, P_4, C_5, P_5)$.
A typical example of cycle sequences we will see in the proof of Proposition \[prop:tree\] is as follows.
\[ex:edge\_loop\] Let $k$ be an algebraic closed field of characteristic $p > 5$. Let $(X, \Delta)$ be a three-dimensional projective log pair over $k$ with $-(K_X + \Delta)$ nef and big. Suppose that $\mathrm{Nklt}(X, \Delta)$ is pure dimension one. Let $f: (Y, \Delta_Y) \to (X, \Delta)$ be a dlt blow-up, and let $G := \mathcal{D} (\Delta _Y ^{\ge 1})$ be the dual complex of $\Delta _Y ^{\ge 1}$. Then $G$ is a regular $\Delta$-complex by Proposition \[prop:regular\].
For an edge $C$ in $G$, its two vertices $S$ and $S'$ are distinct because $G$ is regular. We denote by $C(S,S')$ the oriented $1$-cell corresponding to $C$ with initial point $S$ and final point $S'$.
When we write $$P: S_1 \overset{C_1}{\longrightarrow} S_2 \overset{C_2}{\longrightarrow} \cdots
\overset{C_{n-1}}{\longrightarrow} S_n \overset{C_n}{\longrightarrow} S_{n+1},$$ we assume the following condition.
- For each $1 \le i \le n$, $S_i \not = S_{i+1}$ holds and $S_i$ and $S_{i+1}$ are the two vertices of $C_i$.
We denote by $P$ the edge path obtained by joining the oriented $1$-cell $C_1(S_1, S_2), \ldots, C_n(S_n, S_{n+1})$. The edge path $P$ is called an *edge loop* when $S_1 = S_{n+1}$.
Let $P$ as above be an edge loop in $G$. Then $(f(S_1), f(C_1), \ldots , f(S_n), f(C_n))$ is a cycle sequence because $f(S_{n-1}) \supset f(C_n) \subset f(S_n)$ holds. We say that the sequence $(f(S_1), f(C_1), \ldots , f(S_n), f(C_n))$ is the *image* of the edge loop $P$ for simplicity.
\[lem:cycle\_seq\] Let $C = C_1 \cup C_2 \cup \cdots \cup C_{\ell}$ be in Definition \[def:tree\]. Suppose that $C$ is connected but does not form a tree. Then there exists a non-trivial cycle sequence $(a_1, \ldots , a_n)$ such that $a_i$’s are distinct.
Since $C$ does not form a tree, there exist irreducible components $B_1, \ldots, B_k$ of $C$ with the condition
- $\# \left( B_i \cap (\bigcup _{j \not = i} B_j) \right) \ge 2$ for each $1 \le i \le k$.
We set $b_1 = B_1$ and take a point $b_2$ in $B_1 \cap (\bigcup _{j \not = 1} B_j)$. Inductively, we set $b_{2i+1}$ and $b_{2i+2}$ for $i \ge 1$ as follows:
- We take an arbitrary $B_m$ among $\{ B_1, \ldots , B_k \} \setminus \{ b_{2i-1} \}$ such that $b_{2i} \in b_{2i-1} \cap B_m$.
- We take an arbitrary point $p$ in $\left( B_{m} \cap (\bigcup _{j \not = m} B_j) \right) \setminus \{ b_{2i} \}$.
- We set $b_{2i+1} = B_m$ and $b_{2i+2} = p$.
We can repeat this process by the condition (3). Since $\{ B_1, \ldots , B_k \}$ is a finite set, there exist $m_1, m_2$ with $m_2 \ge m_1 + 4$ such that
- $b_{m_1} = b_{m_2}$ but $b_{m_1}, \ldots , b_{m_2 -1}$ are distinct.
Then the sequence $(b_{m_1}, \ldots , b_{m_2 -1})$ satisfies the condition (1), (2) in Definition \[defi:curve\_seq\]. Since $b_{m_1}, \ldots , b_{m_2 -1}$ are distinct, the sequence has the non-trivial (a,b)-reduction.
\[prop:tree\] Let $k$ be an algebraic closed field of characteristic $p > 5$. Let $(X, \Delta)$ be a three-dimensional projective log pair over $k$ with $-(K_X + \Delta)$ nef and big. Suppose that $\mathrm{Nklt}(X, \Delta)$ is pure dimension one. Then $\mathrm{Nklt}(X, \Delta)$ forms a tree.
Note that $C := \mathrm{Nklt}(X, \Delta)$ is connected by [@NT Theorem 1.2]. Suppose that $C = \mathrm{Nklt}(X, \Delta)$ does not form a tree. Let $C = C_1 \cup \cdots \cup C_{\ell}$ be the irreducible decomposition.
**STEP 1.** Let $f: (Y, \Delta_Y) \to (X, \Delta)$ be a dlt blow-up such that $\mathrm{Supp}\, \Delta _Y ^{\ge 1} = f^{-1}(C)$. Let $G := \mathcal{D} (\Delta _Y ^{\ge 1})$ be the dual complex. We may assume that $G$ is contractible by Theorem \[thm:nefbig\].
The assertion in this step is that there exists an edge loop (see Example \[ex:edge\_loop\] for the notation) $$S_1 \overset{C_1}{\longrightarrow} S_2 \overset{C_2}{\longrightarrow} \cdots
\overset{C_{n-1}}{\longrightarrow} S_n \overset{C_n}{\longrightarrow} S_1$$ in $G$ such that
- its image $(f(S_1), f(C_1), \ldots , f(S_n), f(C_n))$ is a non-trivial cycle sequence.
By Lemma \[lem:cycle\_seq\], there exists a non-trivial cycle sequence $(a_1, \ldots , a_n)$ such that $a_i$’s are distinct. Since $a_i$’s are distinct, the (a)-reduction of this sequence is itself. Therefore, this sequence itself satisfies the conditions (2-1) and (2-2) in Definition \[defi:curve\_seq\]. We may assume that $\dim a_1 = 0$.
Suppose that $i$ is odd.
Then, we can take an edge path $P_i$: $$P_i: S^{(i)}_1 \overset{C^{(i)}_1}{\longrightarrow} S^{(i)}_2 \overset{C^{(i)}_2}{\longrightarrow} \cdots
\overset{C^{(i)}_{m_i -1}}{\longrightarrow} S^{(i)}_{m_i}$$ in $G$ with the following conditions:
- $f(S^{(i)}_1) = a_{i-1}$ and $f(S^{(i)} _{m_i}) = a_{i+1}$.
- $a_i \in f(S^{(i)}_j)$ but $f(S^{(i)}_j) \not = a_{i-1}, a_{i+1}$ for $2 \le j \le m_i -1$.
Such $P_i$ can be taken by the connectedness of the subcomplex of $G$ which consists of the simplices corresponding to the stratum $S$ of $\Delta _Y ^{\ge 1}$ which satisfies $a_i \in f(S)$.
Suppose that $i$ is even.
Then, we can take an edge path $P_i$: $$P_i: S^{(i)}_1 \overset{C^{(i)}_1}{\longrightarrow} S^{(i)}_2 \overset{C^{(i)}_2}{\longrightarrow} \cdots
\overset{C^{(i)}_{m_i -1}}{\longrightarrow} S^{(i)}_{m_i}$$ in $G$ with the following conditions:
- $S_1 ^{(i)} = S_{m_{i-1}} ^{(i-1)}$ and $S_{m_i} ^{(i)} = S_1 ^{(i+1)}$ (here, $S_{m_{i-1}} ^{(i-1)}$ and $S_1 ^{(i+1)}$ were already taken).
- $f(S^{(i)}_j) = a_i$ for $1 \le j \le m_i$ and $f(C^{(i)}_j) = a_i$ for $1 \le j \le m_i - 1$.
Such $P_i$ can be taken by the connectedness of the subcomplex of $G$ which consists of the simplices corresponding to the stratum $S$ of $\Delta _Y ^{\ge 1}$ which satisfies $a_i = f(S)$ (Proposition \[prop:G1\] (1)).
Connecting the edge paths $P_1, P_2, \ldots , P_n$, we get an edge loop $P$, which is possibly not simple.
We prove that the image of $P$ is a non-trivial cycle sequence. Note that the image of any edge loop in $G$ is a cycle sequence (Example \[ex:edge\_loop\]). Therefore, it is sufficient to show that the (a,b)-reduction of the image of $P$ is non-trivial.
For an even number $i$, it follows that $$f(S^{(i)} _1) = f(C^{(i)} _1) = \cdots =
f(C^{(i)} _{m_i -1}) = f(S^{(i)} _{m_i}) = a_{i}$$ by the condition (7). Hence, applying the operation (a) to the image of $P$ $$\left( \ldots, f(C_{m_{i-1} - 1}^{(i-1)}) , f(S_{m_{i-1}}^{(i-1)}) = f(S_1 ^{(i)}),
\ldots , f(S_{m_i} ^{(i)}) = f(S_{1} ^{(i+1)}), f(C_{1} ^{(i+1)} ) , \ldots \right),$$ it can be reduced to $$\left( \ldots, f(C_{m_{i-1} - 1}^{(i-1)}) , f(S_{m_{i-1}}^{(i-1)}) = a_i = f(S_{1} ^{(i+1)}),
f(C_{1} ^{(i+1)} ) , \ldots \right).$$
For an odd number $i$, we set $$b_1 ^{(i)} = f(S^{(i)} _1), \quad b_2^{(i)} = f(C^{(i)} _1), \quad \ldots , \quad
b_{2(m_i-1)}^{(i)} = f(C^{(i)} _ {m_i -1}), \quad b^{(i)} _{2m_i -1} = f(S^{(i)} _{m_i}).$$ Then, they satisfy the following conditions:
- $b_1 ^{(i)} = a_{i-1}$ and $b^{(i)} _{2m_i -1} = a_{i+1}$ by the condition (4).
- $a_i \in b_j ^{(i)}$ but $b_j ^{(i)} \not = a_{i-1}, a_{i+1}$ for $2 \le j \le 2m_i -2$ by the condition (5).
- If $b_j ^{(i)} \not = b_{j+1} ^{(i)}$, then either $b_j ^{(i)} = a_i$ or $b_{j+1} ^{(i)} = a_i$ (This is because $a_i$ is a point and we have an inclusion either $a_i \in b_j ^{(i)} \subset b_{j+1} ^{(i)}$ or $a_i \in b_{j+1} ^{(i)} \subset b_{j} ^{(i)}$).
Hence, applying the operation (a) to the image of $P$ $$\left( \ldots , f(C^{(i-1)} _{m_{i-1} -1}),
b_1^{(i)}, \ldots, b_{2m_i -1} ^{(i)}, f(C^{(i+1)} _1) \ldots \right),$$ it can be reduced to $$\left( \ldots , f(C^{(i-1)} _{m_{i-1} -1} ),
a_{i-1}, a_i, c_1, a_i, \cdots, a_i, c_{n_i} ,a_i, a_{i+1}, f(C^{(i+1)} _1), \ldots \right),$$ for some $c_1, \ldots, c_{n_i}$. Applying the operation (b), it can be reduced to $$\left( \ldots , f(C^{(i-1)} _{m_{i-1} -1}),
f(S_{m_i}^{(i-1)})= a_{i-1}, a_i, a_{i+1}= f(S_{1}^{(i+1)}), f(C^{(i+1)} _1), \ldots \right).$$ Therefore, the (a,b)-reduction of the image of $P$ is $(a_1, a_2, \ldots , a_n)$, which is non-trivial since $a_i$’s are distinct. We have proved that the image of $P$ is a non-trivial cycle sequence.
**STEP 2.** Let $Q$ be an edge loop $$Q: S'_1 \overset{C'_1}{\longrightarrow} S'_2 \overset{C'_2}{\longrightarrow} \cdots
\overset{C'_{n-1}}{\longrightarrow} S'_n \overset{C'_n}{\longrightarrow} S'_1$$ in $G$. Suppose that $C'_i \not = C' _{i+1}$ there exist $i$ and a $2$-simplex $F$ in $G$ such that $C' _i, C' _{i+1} < F$. Let $C' < F$ be the edge which is different from $C' _i$ and $C' _{i+1}$.
Then we have a new edge loop $Q'$: $$Q': S'_1 \overset{C'_1}{\longrightarrow} S'_2 \overset{C'_2}{\longrightarrow} \cdots
\overset{C'_{i-1}}{\longrightarrow} S'_i \overset{C'}{\longrightarrow} S'_{i+2} \overset{C'_{i+2}}{\longrightarrow}
\cdots
\overset{C'_{n-1}}{\longrightarrow} S'_n \overset{C'_n}{\longrightarrow} S'_1.$$
We claim in this step that
- the image of $Q$ and the image of $Q'$ have the same (a,b)-reduction.
The image of $Q$ is $$R: \left( f(S' _1), f(C' _1), \ldots , f(S' _i), f(C' _i), f(S' _{i+1}), f(C' _{i+1}), f(S' _{i+2}), \ldots , f(C'_n) \right),$$ and the image of $Q'$ is $$R': \left( f(S' _1), f(C' _1), \ldots , f(S' _i), f(C'), f(S' _{i+2}), \ldots , f(C'_n) \right).$$ We have four cases.
- $f(S' _i) = f(S' _{i+1}) = f(S' _{i+2})$.
- $f(S'_i) = f(S' _{i+2}) \not = f(S'_{i+1})$.
- $f(S'_i) = f(S' _{i+1}) \not = f(S' _{i+2})$ or $f(S'_i) \not = f(S' _{i+1}) = f(S' _{i+2})$.
- $f(S' _i), f(S' _{i+1}), f(S' _{i+2})$ are distinct.
Suppose (i). Applying the operation (b) twice to $R$, and once to $R'$, we get the same sequence $$\left( f(S' _1), f(C' _1), \ldots , f(C'_{i-1}),f(S'_i) = f(S'_{i+2}) ,f(C'_{i+2}), \ldots , f(C'_n) \right).$$
Suppose (ii). Since $f(C'_i) \subset f(S' _i) \cap f(S' _{i+1})$ and $f(S' _i) \not = f(S' _{i+1})$, it follows that $\dim f(C'_i) = 0$. By the same reason, it follows that $\dim f(C'_{i+1}) = 0$. Since $f(F) \subset f(C' _i) \cap f(C' _{i+1})$, it follows that $f(C' _i) = f(F) = f(C' _{i+1})$. Applying the operation (b) twice to $R$, and once to $R'$, we get the same sequence $$\left( f(S' _1), f(C' _1), \ldots , f(C'_{i-1}),f(S'_i)=f(S'_{i+2}),f(C'_{i+2}), \ldots , f(C'_n) \right).$$
Suppose (iii). We may assume that $f(S'_i) = f(S' _{i+1}) \not = f(S' _{i+2})$. By the same reason as in the case (ii), $f(C' _{i+1}) = f(F) = f(C')$. Applying the operation (b) once to $R$, we get the sequence $R'$ $$\left( f(S' _1), f(C' _1), \ldots , f(C'_{i-1}),f(S'_i)=f(S'_{i+1}),f(C' _{i+1})=f(C'),
f(S' _{i+2}), f(C'_{i+2}), \ldots , f(C'_n) \right).$$
Suppose (iv). In this case, $f(C' _i) = f(C' _{i+1}) = f(C')$. Applying the operation (b) once to $R$, we get the sequence $R'$ $$\left( f(S' _1), f(C' _1), \ldots , f(C'_{i-1}),f(S'_i), f(C' _i) = f(C' _{i+1}) = f(C'), f(S' _{i+2}),
f(C'_{i+2}), \ldots , f(C'_n) \right).$$
In any case, $R$ and $R'$ have the same (a,b)-reduction.
**STEP 3.** Let $Q$ be an edge loop $$Q: S'_1 \overset{C'_1}{\longrightarrow} S'_2 \overset{C'_2}{\longrightarrow} \cdots
\overset{C'_{i-1}}{\longrightarrow} S' _i \overset{C'_i}{\longrightarrow} S'_{i+1} \overset{C'_{i+1}}{\longrightarrow} S'_{i+2}
\overset{C'_{i+2}}{\longrightarrow} \cdots
\overset{C'_{n-1}}{\longrightarrow} S'_n \overset{C'_n}{\longrightarrow} S'_1$$ in $G$. Suppose that there exist $i$ such that $C' _i = C' _{i+1}$. Then $S'_i = S'_{i+2}$ holds.
Then we have a new edge loop $Q'$: $$Q': S'_1 \overset{C'_1}{\longrightarrow} S'_2 \overset{C'_2}{\longrightarrow} \cdots
\overset{C'_{i-1}}{\longrightarrow} S'_i \overset{C'_{i+2}}{\longrightarrow} S'_{i+3} \overset{C'_{i+3}}{\longrightarrow}
\cdots
\overset{C'_{n-1}}{\longrightarrow} S'_n \overset{C'_n}{\longrightarrow} S'_1.$$
We claim in this step that
- the image of $Q$ and the image of $Q'$ have the same (a,b)-reduction.
The image of $Q$ is $$R: \left( f(S' _1), f(C' _1), \ldots , f(S' _i), f(C' _i), f(S' _{i+1}), f(C' _{i+1}), f(S' _{i+2}),f(C' _{i+2}), \ldots , f(C'_n) \right),$$ and the image of $Q'$ is $$R': \left( f(S' _1), f(C' _1), \ldots , f(S' _i), f(C'_{i+2}), \ldots , f(C'_n) \right).$$
Since $C' _i = C' _{i+1}$ and $S'_i = S'_{i+2}$, applying the operation (b) twice to $R$, we get $R'$. Hence $R$ and $R'$ have the same (a,b)-reduction.
**STEP 4.** By STEP 1, there exists an edge loop $Q$ $$Q: S_1 \overset{C_1}{\longrightarrow} S_2 \overset{C_2}{\longrightarrow} \cdots
\overset{C_{n-1}}{\longrightarrow} S_n \overset{C_n}{\longrightarrow} S_1$$ in $G$ such that
- its image $(f(E_1), f(C_1), \ldots , f(E_n), f(C_n))$ is a non-trivial cycle sequence.
Since $G$ is simply connected, applying the operation in STEP 2 and STEP 3 (and the reversing operation) repeatedly, we get a trivial path $$Q': S'$$ for some vertex $S'$ [@Geo08 Theorem 3.4.1]. The (a,b)-reduction of the image of $Q$ is not trivial but that of $Q'$ is trivial. This contradicts STEP 2 and STEP 3.
Vanishing theorem of Witt vector cohomology of Ambro-Fujino type
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\[thm:WAFV\] Let $k$ be a perfect field of characteristic $p > 5$. Let $(X, \Delta)$ be a three-dimensional projective $\mathbb{Q}$-factorial log canonical pair over $k$ with $-(K_X + \Delta)$ ample. Then $H^i(X,W \mathcal{O}_{X, \mathbb{Q}}) = 0$ holds for $i > 0$.
By replacing $\Delta$ smaller, we may assume that $\dim \mathrm{Nklt}(X, \Delta) \le 1$.
By the exact sequence $$0 \to WI_{\mathrm{Nklt}(X, \Delta), \mathbb{Q}} \to W \mathcal{O}_{X, \mathbb{Q}} \to
W \mathcal{O}_{\mathrm{Nklt}(X, \Delta), \mathbb{Q}} \to 0,$$ and the Nadel type vanishing $H^i(X, WI_{\mathrm{Nklt}(X, \Delta), \mathbb{Q}}) = 0$ for $i >0$ (Theorem \[thm:WNV\]), it is sufficient to show that $$H^1 (\mathrm{Nklt}(X, \Delta), W \mathcal{O}_{\mathrm{Nklt}(X, \Delta), \mathbb{Q}}) = 0.$$ Here, we may assume that $k$ is algebraically closed by [@NT Lemma 2.15]. Since $\dim \mathrm{Nklt}(X, \Delta) \le 1$ and $\mathrm{Nklt}(X, \Delta)$ is connected ([@NT Theorem 1.2]), we may assume that $\mathrm{Nklt}(X, \Delta)$ is a union of curves.
Let $C := \mathrm{Nklt}(X, \Delta) = C_1 \cup C_2 \cup \cdots \cup C_l$ be the irreducible decomposition. By Lemma \[lem:rationality\], Proposition \[prop:normality\], and Proposition \[prop:tree\], the curve $C$ satisfies the following conditions.
1. Each $C_i$ is a rational curve.
2. Each normalization of $C_i$ is a universal homeomorphism.
3. $C = C_1 \cup \ldots \cup C_l$ forms a tree (see Definition \[def:tree\]).
Then $H^1 (C_i, W \mathcal{O}_{C_i, \mathbb{Q}}) = 0$ follows from (1) and (2) (cf. [@GNT Lemma 2.21, 2.22]). Hence the desired vanishing $H^1 (C, W \mathcal{O}_{C, \mathbb{Q}}) = 0$ follows from (3).
Rational point formula
----------------------
As an application of Theorem \[thm:WAFV\], we obtain the following rational point formula.
\[thm:RPF\] Let $k$ be a finite field of characteristic $p > 5$. Let $(X, \Delta)$ be a geometrically connected three-dimensional projective $\mathbb{Q}$-factorial log canonical pair over $k$ with $-(K_X + \Delta)$ ample. Then the number of the $k$-rational points on the non-klt locus on $(X, \Delta)$ satisfies $$\# \mathrm{Nklt}(X, \Delta) (k) \equiv 1 \mod {|k|}.$$ In particular, there exists a $k$-rational point on $\mathrm{Nklt}(X, \Delta)$.
Let $Z = \mathrm{Nklt}(X, \Delta)$ and let $I_Z$ be the corresponding coherent ideal sheaf. By Theorem \[thm:WNV\] and Theorem \[thm:WAFV\], $$H^i(X, WI_{Z, \mathbb{Q}}) = 0,\ \text{and}\ H^i(X,W \mathcal{O}_{X, \mathbb{Q}}) = 0$$ hold for $i > 0$. By the exact sequence $$0 \to WI_{Z} \to W \mathcal{O}_{X, \mathbb{Q}} \to
W \mathcal{O}_{Z, \mathbb{Q}} \to 0,$$ $H^i(Z, W \mathcal{O}_{Z, \mathbb{Q}}) = 0$ holds for $i >0$. By [@BBE07 Proposition 6.9 (i)], it follows that $\# Z (k) \equiv 1 \pmod {|k|}$.
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author:
- 'Takashi <span style="font-variant:small-caps;">Tonegawa</span>$^{1}$, Kiyomi <span style="font-variant:small-caps;">Okamoto</span>$^{2}$ and Makoto <span style="font-variant:small-caps;">Kaburagi</span>$^{3}$'
title: 'Finite-Field Ground State of the ${{\mib S}{\mib =}{\mib 1}}$ Antiferromagnetic-Ferromagnetic Bond-Alternating Chain'
---
Introduction
============
There has been a considerable current interest in the study of the ground-state properties of various one-dimensional quantum spin systems in a finite magnetic field. In this paper we discuss the case of the $S\!=\!1$ bond-alternating chain, for which we express the Hamiltonian in the following form: $$\begin{aligned}
{\calH}&&\!\!\!\!\!\!\!\!\!\!={\calH}_{\rm ex} + {\calH}_{\rm Z}\,, \\
&& {\calH}_{\rm ex}
= \sum\nolimits_{\ell} \Bigl\{\vecS_{2\ell-1}\cdot\vecS_{2\ell}
{} + J \vecS_{2\ell}\cdot\vecS_{2\ell+1}\Bigr\} \nonumber \\
&&\qquad\qquad {} + D \sum\nolimits_{\ell}\,\bigl(S_{\ell}^z)^2\,, \\
&& {\calH}_{\rm Z} = - H \sum\nolimits_\ell\,S_\ell^z\,,\end{aligned}$$ where $\vecS_\ell$ is the operator at the $\ell$th site; $J$ () is the parameter representing the bond alternation of the nearest-neighbor interactions; $D$ () is the uniaxial single-ion-type anisotropy constant; $H$ () is the magnitude of the external magnetic field applied along the $z$-direction. Hereafter, we denote by $M$ the $z$ component of the total spin $\vecS_{\rm tot}\!\equiv\!\sum_{\ell=1}^N \vecS_\ell$, where $N$, being assumed to be a multiple of four, is the total number of spins in the system.
The finite-field as well as the zero-field ground-state properties of the present system in the case of the antiferromagnetic-antiferromagnetic bond-alternating chain with have already been investigated by using mainly numerical methods. [@TNK; @CHS] As for the zero-field properties, the ground-state phase diagram on the $J$ versus $D$ plane, in which the N[é]{}el, Haldane, large-$D$, and dimer phases appear, has been determined. [@TNK; @CHS] On the other hand, the results for the ground-state magnetization curve show that the magnetization plateau at a half of the saturation magnetization, which is called the ${1\over 2}$-plateau, appears for arbitrary values of $J$ except for $J\!=\!1$ at least when $D\!\geq\!0$. [@TNK]
In this study we explore the finite-field ground-state properties of the system in the case of the antiferromagnetic-ferromagnetic bond-alternating chain with , focusing our attention mainly upon the $\frac{1}{2}$-plateau appearing in the magnetization curve. When $D\!=\!0$, the present system is mapped onto the isotropic uniform antiferromagnetic chain in the limit of . Thus, it is considered that, no plateau appears in the ground-state magnetization curve in this limit when . As we have already discussed, [@TNK] on the other hand, the ${1\over 2}$-plateau appears when and . These imply that the \`${1\over 2}$-plateau’-\`no-plateau’ transition occurs at a finite value of $J$ for a certain region of $D$ including . Furthermore, as has already been shown, [@KO] the magnetization plateau at one third of the saturation magnetization, the ${1\over 3}$-plateau, appearing in the ground-state magnetization curve of the uniform antiferromagnetic chain with single-ion-type anisotropy changes its character from the Haldane-type to the large-$D$-type at a finite value of the anisotropy constant. We expect that this type of \`Haldane-type-${1\over 2}$-plateau’-\`large-$D$-type-${1\over 2}$-plateau’ transition may take place also in the present $S\!=\!1$ antiferromagnetic-ferromagnetic bond-alternating chain. To discuss quantitatively these transitions and to complete the ${1\over 2}$-plateau phase diagram on the $D$ versus $J$ plane are our main purposes. We also aim at calculating the magnetization phase diagram as well as the ground-state magnetization curve.
In the next section ([§]{}2) we discuss the ${1\over 2}$-plateau phase diagram. In determining the phase diagram, we apply the twisted-boundary-condition level spectroscopy analyses of the numerical diagonalization data, developed by Kitazawa [@K] and also by Nomura and Kitazawa. [@NK] Then, we discuss in [§]{}3 the ground-state magnetization curve, which we calculate by using the density-matrix renormalization-group (DMRG) method proposed originally by White. [@White] We also discuss the magnetization phase diagram which is obtained from the result for the magnetization curve. Finally, concluding remarks are given in [§]{}4.
${{\mib 1}\over{\mib 2}}$-Plateau Phase Diagram
===============================================
From the considerations presented in the previous section, it is expected that three phases, i.e., the \`no-plateau’, \`Haldane-type-${1\over 2}$-plateau’, and \`large-$D$-type-${1\over 2}$-plateau’ phases appear in the ${1\over 2}$-plateau phase diagram. Schematical pictures in terms of a kind of the valence-bond-solid picture, which represents the latter two phases, are shown in Fig. \[fig:1\].
![Schematical pictures representing (a) the \`Haldane-type-${1\over 2}$-plateau’ phase and (b) the \`large-$D$-type-${1\over 2}$-plateau’ phase. The thick solid and dotted lines stand, respectively, for the antiferromagnetic and ferromagnetic bonds, and the small open circles for the spins. Each large open circle surrounding two spins stands for an operation of constructing an spin by symmetrization. The spin with an upward arrow is in the up state $\uparrow$. Furthermore, two spins in a flat ellipse form a singlet dimer , and two spins in a rectangular form a triplet dimer []{data-label="fig:1"}](fig1a.eps){width="7.0cm"}
![Schematical pictures representing (a) the \`Haldane-type-${1\over 2}$-plateau’ phase and (b) the \`large-$D$-type-${1\over 2}$-plateau’ phase. The thick solid and dotted lines stand, respectively, for the antiferromagnetic and ferromagnetic bonds, and the small open circles for the spins. Each large open circle surrounding two spins stands for an operation of constructing an spin by symmetrization. The spin with an upward arrow is in the up state $\uparrow$. Furthermore, two spins in a flat ellipse form a singlet dimer , and two spins in a rectangular form a triplet dimer []{data-label="fig:1"}](fig1b.eps){width="7.0cm"}
We can determine the transition lines between two of the above thee phases rather precisely by applying the TBCLS methods in the following way. Both the \`Haldane-type-${1\over 2}$-plateau’-\`no-plateau’ transition and the \`large-$D$-type-${1\over 2}$-plateau’-\`no-plateau’ transition are the Berezinskii-Kosterlitz-Thouless transition [@BKT] accompanying no spontaneous breaking of the translational symmetry of the Hamiltonian $\calH$. Therefore, the critical points $J_{\rm c}^{\Hno}$ and $J_{\rm c}^{\LDno}$ of $J$, which are, respectively, for the former and latter transitions, for a given value of $D$ (or, $D_{\rm c}^{\Hno}$ and $D_{\rm c}^{\LDno}$ of $D$ for a given value of $J$) can be estimated by means of Nomura and Kitazawa’s TBCLS method. [@NK] We denote by $E_0^{({\rm P})}(N,M)$ the lowest energy eigenvalue of $\calH_{\rm ex}$ with the periodic boundary condition within the subspace determined by the values of $N$ and $M$, and also by $E_0^{({\rm T})}(N,M,P)$ the lowest energy eigenvalue of $\calH_{\rm ex}$ with the twisted boundary condition within the subspace determined by the values of $N$, $M$, and $P$, where $P$ is the eigenvalue of the space inversion operator. Then, Nomura and Kitazawa’s TBCLS method [@NK] predicts that $J_{\rm c}^{\Hno}$ ($D_{\rm c}^{\Hno}$) is obtained by extrapolating $J_{\rm c}^{\Hno}(N)$ $\bigl(D_{\rm c}^{\Hno}(N)\bigr)$ determined from the equation $$\begin{aligned}
E_0^{({\rm T})}&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\Bigl(N,\frac{N}{2},-1\Bigr)
\nonumber \\
&&\!\!\!\!\!\!\!\!\!\!=\frac{1}{2}
\Bigl[E_0^{({\rm P})}\Bigl(N,\frac{N}{2}+2\Bigr)
+ E_0^{({\rm P})}\Bigl(N,\frac{N}{2}-2\Bigr)\Bigr]\end{aligned}$$ to . Similarly, $J_{\rm c}^{\LDno}$ ($D_{\rm c}^{\LDno}$) is obtained by extrapolating $J_{\rm c}^{\LDno}(N)$ $\bigl(D_{\rm c}^{\LDno}(N)\bigr)$ determined from $$\begin{aligned}
E_0^{({\rm T})}&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\Bigl(N,\frac{N}{2},+1\Bigr)
\nonumber \\
&&\!\!\!\!\!\!\!\!\!\!=\frac{1}{2}
\Bigl[E_0^{({\rm P})}\Bigl(N,\frac{N}{2}+2\Bigr)
+ E_0^{({\rm P})}\Bigl(N,\frac{N}{2}-2\Bigr)\Bigr]\end{aligned}$$ to .
On the other hand, the \`Haldane-type-${1\over 2}$-plateau’-\`large-$D$-type-${1\over 2}$-plateau’ transition is expected to be of the Gaussian type as in the case of the \`Haldane-type-${1\over 3}$-plateau’-\`large-$D$-type-${1\over 3}$-plateau’ transition in the the uniform antiferromagnetic chain with single-ion-type anisotropy, [@KO] discussed in the previous section. Thus, we can apply Kitazawa’s TBCLS method [@K] to estimate the critical point $J_{\rm c}^{\HLD}$ of $J$ for a given $D$ (or, $D_{\rm c}^{\HLD}$ for a given $J$) in this transition. According to this method, $J_{\rm c}^{\HLD}$ ($D_{\rm c}^{\HLD}$) can be estimated by extrapolating $J_{\rm c}^{\HLD}(N)$ $\bigl(D_{\rm c}^{\HLD}(N)\bigr)$ determined from $$E_0^{({\rm T})}\Bigl(N,\frac{N}{2},+1\Bigr)
= E_0^{({\rm T})}\Bigl(N,\frac{N}{2},-1\Bigr)$$ to . It is noted that in the limit of , the value of $E_0^{({\rm T})}\Bigl(N,\frac{N}{2},-1\Bigr)$ is smaller or larger than that of $E_0^{({\rm T})}\Bigl(N,\frac{N}{2},+1\Bigr)$ depending upon whether in the \`Haldane-type-${1\over 2}$-plateau’ region or in the \`large-$D$-type-${1\over 2}$-plateau’ region.
Determining the ${1\over 2}$-plateau phase diagram on the $D$ versus $J$ plane, we have first calculated numerically the energy eigenvalues , , $E_0^{({\rm T})}\Bigl(N,\frac{N}{2},+1\Bigr)$, and $E_0^{({\rm T})}\Bigl(N,\frac{N}{2},-1\Bigr)$ for a variety values of $J$ and $D$ for finite-size systems with , $12$, and $16$ spins, employing the computer program package KOBEPACK [@KP] coded by means of the Lancz[ö]{}s technique. Then, we have solved numerically eqs.$\,$(4), (5), and (6), respectively, to evaluate $J_{\rm c}^{\Hno}(N)$ $\bigl(D_{\rm c}^{\Hno}(N)\bigr)$, $J_{\rm c}^{\LDno}(N)$ $\bigl(D_{\rm c}^{\LDno}(N)\bigr)$, and $J_{\rm c}^{\HLD}(N)$ $\bigl(D_{\rm c}^{\HLD}(N)\bigr)$. Finally, we have extrapolated these finite-size values to to estimate $J_{\rm c}^{\Hno}$ ($D_{\rm c}^{\Hno}$), $J_{\rm c}^{\LDno}$ ($D_{\rm c}^{\LDno}$), and $J_{\rm c}^{\HLD}$ ($D_{\rm c}^{\HLD}$) by assuming that the $N$-dependences of the finite-size values are quadratic functions of $N^{-2}$. Plotting the extrapolated results on the $D$ versus $J$ plane, we have obtained the $\frac{1}{2}$-plateau phase diagram as depicted in Fig. \[fig:2\]. It is noted that, for example, for $D\!=\!0$. Furthermore, the three transition lines in the phase diagram meet at the point ($J\!=\!-8.13\pm0.01$,$\,$$D\!=\!1.19\pm0.01$).
![${1\over 2}$-plateau phase diagram on the $D$ versus $J$ plane. Here, NO, H, and LD stand, respectively, for the \`no-plateau’, \`Haldane-type-${1\over 2}$-plateau’, and \`large-$D$-type-${1\over 2}$-plateau’ regions. The three transition lines meet at the point (, ), which is marked by the double circle.[]{data-label="fig:2"}](fig2.eps){width="7.0cm"}
As can be seen from Fig. \[fig:2\], the \`large-$D$-type-${1\over 2}$-plateau’-\`no-plateau’ transition line is almost parallel to the $J$-axis in the whole region of $J\!\lsim\!-8.13$. This result may be attributed to the fact that the present system in this region is almost equivalent to the uniform antiferromagnetic chain with single-ion-type anisotropy, since $\vert J\vert$ is fairly large with . Thus, we may expect a direct transition from the \`no-plateau’ phase to the \`large-$D$-type-${1\over 2}$-plateau’ one in the above chain, in contrast to the case of the ${1\over 3}$-plateau in the corresponding $S\!=\!{3\over 2}$ chain, [@KO] mentioned in [§]{}1. Figure 2 also shows that the \`large-$D$-type-${1\over 2}$-plateau’ phase appears when both $\vert J\vert$ ($J\!<\!0$) and $D$ ($D\!>\!0$) are sufficiently large. The fitting of several values of $J_{\rm c}^{\HLD}$ for sufficiently large $D$’s to a quadratic function of $D^{-1}$ suggests that the \`Haldane-type-${1\over 2}$-plateau’-\`large-$D$-type-${1\over 2}$-plateau’ transition line approaches $J\!=\!-2.05\pm0.05$ in the limit of $D\!\to\!\infty$.
Magnetization Curve and Magnetization Phase Diagram
===================================================
Let us begin with discussing the saturation field $H_{\rm s}$ for the present system in the case of $J\!<\!0$. It is given by $$H_{\rm s} = E_0^{({\rm P})}(N,N) - E_0^{({\rm P})}(N,N-1)
= D + 2$$ or $$H_{\rm s}
= \frac{1}{2} \lim_{N\to\infty}\Big\{E_0^{({\rm P})}(N,N)
- E_0^{({\rm P})}(N,N-2)\Bigr\}
= \frac{1}{2} E_{\rm B}$$ depending on whether $2(D\!+\!2)\!>\!E_{\rm B}$ or $2(D\!+\!2)\!<\!E_{\rm B}$, where $E_{\rm B}$ is the maximum energy of the ferromagnetic-two-magnon bound state. In principle, the energy $E_{\rm B}$ can be calculated analytically. However, this calculation in the case of $J\!\ne\!1$ has not yet been done as far as we know, [@T] and therefore we estimate $H_{\rm s}$ numerically when $2(D\!+\!2)\!<\!E_{\rm B}$.
We have applied the DMRG method [@White] to calculate the ground-state magnetization curve, which we define here as the average magnetization per spin versus the reduced field $H/H_{\rm s}$ curve. In this application, we usually have to impose open boundary conditions for the Hamiltonian treated. In the present system, we have two kinds of open boundary conditions, Oa and Of, which are, respectively, the cases where the edge bonds are the antiferromagnetic and ferromagnetic bonds. We adopt, respectively, the Oa and Of conditions when we calculate the magnetization curves in the \`Haldane-type-${1\over 2}$-plateau’ and \`large-$D$-type-${1\over 2}$-plateau’ regions. This is because, as can be seen from the schematical pictures for these phases shown in Fig. \[fig:1\], no degrees of freedom appear at both edges of the chain in the above adoption, which leads to a less boundary effect.
We have calculated the lowest energy eigenvalue $E_0^{({\rm Ox})}(N,M)$ (x$=$f or a) under the open boundary condition Ox of $\calH_{\rm ex}$ within the subspace determined by the values of $N$ and $M$ for , $64$, and $84$ and for all values of $M$ (, $1$, $\cdots$, $N$). From these results the ground-state magnetization curves for , $64$, and $84$ can be readily obtained. [@BF] The resulting magnetization curve for each $N$ is a stepwisely increasing function of $H$, starting from at and reaching to the saturation value at . We note that $H_{\rm s}\!=\!\lim_{N\to\infty}H_{\rm s}(N)$.
![Ground-state magnetization curves in the limit obtained for the cases of (a) and and of (b) and []{data-label="fig:3"}](fig3a.eps){width="7.0cm"}
![Ground-state magnetization curves in the limit obtained for the cases of (a) and and of (b) and []{data-label="fig:3"}](fig3b.eps){width="7.0cm"}
Except for plateau regions, a satisfactorily good approximation to the magnetization curve in the limit, i.e., the versus $H/H_{\rm s}$ curve, may be obtained by drawing a smooth curve through the midpoints of the steps in the finite-size magnetization curves. [@BF] As for the estimations of , which is nothing but the energy gap between the ground state and a first excited state in the case of , as well as of $H_1$ and $H_2$, which are the lowest and highest values of $H$ giving the $\frac{1}{2}$-plateau, respectively, we extrapolate the corresponding results for , $64$, and $84$ to , assuming again the $N$-dependences of the finite-size values are quadratic functions of $N^{-2}$. The magnetization curve thus obtained for the cases of and and of and , which are, respectively, in the \`Haldane-type-${1\over 2}$-plateau’ and \`large-$D$-type-${1\over 2}$-plateau’ regions (see Fig. \[fig:2\]), are depicted in Fig. \[fig:3\].
The magnetization phase diagram on the $H/H_{\rm s}$ versus $J$ plane for a given value of $D$ is obtained by plotting $H_0$, $H_1$, $H_2$, and $H_{{\rm s}}$ as a function of $J$. The results for the cases of and are shown in Fig. \[fig:4\]. Figure \[fig:4\](a) demonstrates that in the case of , the $\frac{1}{2}$-plateau, which is of the \`Haldane-type’, appears when , and Fig. \[fig:4\](b) shows that in the case of , the $\frac{1}{2}$-plateau disappears at the critical point for the \`Haldane-type-${1\over 2}$-plateau’-\`large-$D$-type-${1\over 2}$-plateau’ phase transition (see Fig. \[fig:2\]).
![Magnetization phase diagrams on the $H/H_{\rm s}$ versus $J$ plane obtained for the cases of (a) and (b) , where $H_0/H$ (open squares), $H_1/H$ (solid circles), and $H_2/H$ (open circles) ate plotted as functions of $J$. In the regions A, B$_{\rm X}$ (X$=$H or LD), and C, the average magnetization $m$ per spin is equal to $0$ (the $0$-plateau region), $\frac{1}{2}$ (the $\frac{1}{2}$-plateau region), and $1$ (the saturated magnetization region); the B$_{\rm H}$ and B$_{\rm LD}$ regions are, respectively, the \`Haldane-type-${1\over 2}$-plateau’ and \`large-$D$-type-${1\over 2}$-plateau’ regions. In other regions, $m$ increases continuously with the increase of $H/H_{\rm s}$.[]{data-label="fig:4"}](fig4a.eps){width="7.0cm"}
![Magnetization phase diagrams on the $H/H_{\rm s}$ versus $J$ plane obtained for the cases of (a) and (b) , where $H_0/H$ (open squares), $H_1/H$ (solid circles), and $H_2/H$ (open circles) ate plotted as functions of $J$. In the regions A, B$_{\rm X}$ (X$=$H or LD), and C, the average magnetization $m$ per spin is equal to $0$ (the $0$-plateau region), $\frac{1}{2}$ (the $\frac{1}{2}$-plateau region), and $1$ (the saturated magnetization region); the B$_{\rm H}$ and B$_{\rm LD}$ regions are, respectively, the \`Haldane-type-${1\over 2}$-plateau’ and \`large-$D$-type-${1\over 2}$-plateau’ regions. In other regions, $m$ increases continuously with the increase of $H/H_{\rm s}$.[]{data-label="fig:4"}](fig4b.eps){width="7.0cm"}
Concluding Remarks
==================
We have investigated the finite-field ground state of the $S\!=\!1$ antiferromagnetic-ferromagnetic bond-alternating chain described by the Hamiltonian ${\calH}$ \[see eqs.$\,$(1-3)\] with and , focusing our attention mainly upon the $\frac{1}{2}$-plateau appearing in the magnetization curve. We have found that two kinds of magnetization plateaux, i.e., the \`Haldane-type-${1\over 2}$-plateau’ and the \`large-$D$-type-${1\over 2}$-plateau’ \[Fig. \[fig:1\]\] appear. We have determined the $\frac{1}{2}$-plateau phase diagram on the $D$ versus $J$ plane \[Fig. \[fig:2\]\], applying the TBCLS methods. [@K; @NK] We have also calculated, by means of the DMRG method [@White], the ground-state magnetization curves for a variety values of $J$ with $D$ fixed at $D\!=\!0.0$ and $3.0$, two examples of which being presented \[Fig. \[fig:3\]\]. Using the results for the magnetization curves, we finally obtained the magnetization phase diagrams on the $H/H_{\rm s}$ versus $J$ plane for and $3.0$ \[Fig. \[fig:4\]\].
For the purpose of exploring the effect of frustration, we are now studying the case where the antiferromagnetic next-nearest-neighbor interaction term $J_2\sum_\ell\vecS_{\ell}\cdot\vecS_{\ell+2}$ () is added to the Hamiltonian of eqs.$\,$(1-3), assuming either $J\!>\!0$ or $J\!<\!0$. It is noted that when $J\!=\!0$, the system is reduced to the $S\!=\!1$ antiferromagnetic ladder, where the ratio of the leg interaction constant to the rung one is $J_2$. The results will be published in the near future.
Acknowledgments {#acknowledgments .unnumbered}
===============
We wish to thank Professor T. Hikihara by whom the DMRG program used in this study is coded. We also thank the Supercomputer Center, Institute for Solid State Physics, University of Tokyo, the Information Synergy Center, Tohoku University, and the Computer Room, Yukawa Institute for Theoretical Physics, Kyoto University for computational facilities. The present work has been supported in part by Grants-in-Aid for Scientific Research (C) (No. 16540332, No. 14540329, and No. 14540358) and a Grant-in-Aid for Scientific Research on Priority Areas (B) (\`Field-Induced New Quantum Phenomena in Magnetic Systems’) from the Ministry of Education, Culture, Sports, Science and Technology.
[99]{}
T. Tonegawa, T. Nakao and M. Kaburagi: J. Phys. Soc. Jpn. [**65**]{} (1996) 3317.
W. Chen, H. Hida and B. C. Sanctuary: J. Phys. Soc. Jpn. [**69**]{} (2000) 237.
A. Kitazawa and K. Okamoto: Phys. Rev. B [**62**]{} (2000) 940; K. Okamoto and A. Kitazawa: J. Phys. Chem. Solids [**62**]{} (2001) 365.
A. Kitazawa: J. Phys. A: Math. Gen. [**30**]{} (1997) L285.
K. Nomura and A. Kitazawa: J. Phys. A: Math. Gen. [**31**]{} (1998) 7341.
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The case of $J\!=\!1$ has already been studied by T. Tonegawa: Prog. Theor. Phys., Suppl., No.$\,$46 (1970) 61.
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abstract: 'The theory of angular momentum connects physical rotations and quantum spins together at a fundamental level. Physical rotation of a quantum system will therefore affect fundamental quantum operations, such as spin rotations in projective Hilbert space, but these effects are subtle and experimentally challenging to observe due to the fragility of quantum coherence. Here we report a measurement of a single-electron-spin phase shift arising directly from physical rotation, without transduction through magnetic fields or ancillary spins. This phase shift is observed by measuring the phase difference between a microwave driving field and a rotating two-level electron spin system, and can accumulate nonlinearly in time. We detect the nonlinear phase using spin-echo interferometry of a single nitrogen-vacancy qubit in a diamond rotating at 200,000rpm. Our measurements demonstrate the fundamental connections between spin, physical rotation and quantum phase, and will be applicable in schemes where the rotational degree of freedom of a quantum system is not fixed, such as spin-based rotation sensors and trapped nanoparticles containing spins.'
author:
- 'A. A. Wood'
- 'L. C. L. Hollenberg'
- 'R. E. Scholten'
- 'A. M. Martin'
title: Observation of a quantum phase from classical rotation of a single spin
---
Physical rotation is one of the most ubiquitous elements of classical physics. Everything from galaxies to individual molecules rotate on a myriad of timescales, with fundamental effects on the physical processes in each system. Quantum systems are also affected by physical rotation, but in most cases meaningful observation or exploitation of the effects is challenging due to the difficulties associated with controllably rotating an addressable quantum system at rates comparable to the coherence time of the system. Solid-state spin qubits such as the nitrogen-vacancy (NV) center in diamond [@doherty_nitrogen-vacancy_2013; @schirhagl_nitrogen-vacancy_2014] provide promising testbeds of how classical rotation affects quantum systems [@wood_quantum_2018], due to their long coherence times of up to several milliseconds [@balasubramanian_ultralong_2009] and the robust nature of the host substrate. As the natural quantization axis of the NV is set by the host diamond crystal orientation, rotating the crystal rotates the qubit, and the effects of other phenomena such as magnetic fields can be examined independently . In this work we observe a phase shift between a single NV qubit in a classically rotating diamond and an external microwave field used to drive quantum rotations. Depending on the angle between the instrinsic axis of the NV, the microwave field and the axis of rotation, this rotationally-induced phase can accumulate nonlinearly, and can therefore be detected in a spin-echo measurement of the NV electron spin.
Detecting phase shifts arising from physical rotation is of significant interest to quantum sensing [@zhang_inertial_2016; @degen_quantum_2017]. Theoretical work [@maclaurin_measurable_2012; @ledbetter_gyroscopes_2012; @song_nanoscale_2018] has proposed using the Berry phase [@berry_quantal_1984] arising from physical rotation of a tilted NV qubit as a gyroscopic sensor, while other proposals envisage gyroscopes based on detecting rotational shifts of the Larmor precession frequency of a proximal nuclear spin [@ajoy_stable_2012]. Experimentally, realizations of Berry phase in NV systems has been constrained to stationary systems. Yale *et. al.* used optical transitions to drive the NV spin along closed paths on the Bloch sphere [@yale_optical_2016], while other work used an additional off-resonant microwave driving field varied along a circuit in the rotating frame [@zhang_experimental_2016; @arai_geometric_2018], similar to the first observations of Berry’s phase in solid-state qubits [@leek_observation_2007] and NMR systems [@suter_berrys_1987]. Recent experimental work has simulated rotation with phase-shifted microwave pulses, emulating rotation of the microwave field [@jaskula_cross-sensor_2019].
The principle challenge associated with detecting the effect of rotation on an NV electron spin, from Berry’s phase or otherwise, even in the case of rapidly rotating proof-of-principle experiments rests on discerning the small rotational phase shifts from the effects of magnetic fields and temperature variations. In our work, the phase shift from rotation can be made to accumulate non-linearly due to the effective rotation of the microwave field in the physically rotating frame of the NV qubit. We are thus able to use the decoupling properties of the spin-echo sequence to extend the interferometric interrogation time so that direct measurement of a rotationally-induced phase shift of the electron spin becomes possible in a noisy environment.
A schematic of our system is depicted in Fig. \[fig:fig1\] and the specific experimental configuration we use is similar to that described previously in Ref.[@wood_quantum_2018]. A $99.99\,\%$ $^{12}$C diamond is mounted on its $(100)$ face to an electric motor that spins at $\omega_\text{rot} = 3.33\,$kHz about an axis $z$. The diamond hosts individually resolvable NV centers with typical coherence times of $T_2^\ast \approx 50\,\upmu$s and $T_2 \approx 1\,$ms. Single NV centers are imaged by a scanning confocal microscope with illumination pulsed synchronously with the rotation of the motor. We consider a particular NV located about $3\,\upmu$m from the rotation center. The N-V axis is tilted by $\theta_\text{NV} = 54.7^\circ$ from $z$. Application of a magnetic field parallel to the rotation axis (not shown) breaks the degeneracy of the NV $m_S = \pm1$ states and allows us to consider the $|m_S=0\rangle$ and $|m_S = -1\rangle$ states as a pseudospin-1/2 system (see the Supplementary Information for full details). We introduce rotation operators $R_i(\theta) = \exp\left(-i S_i \theta\right)$ with $S_i = \frac{1}{2}\sigma_i$ the spin-1/2 Pauli matrices and $i\in\{x,y,z\}$, $\hbar = 1$. In the stationary laboratory frame, the rotating NV Hamiltonian is given by $R_\text{NV} H_0 R^{-1}_\text{NV}$ with $R_\text{NV} = R_z(\phi)R_y(\theta_\text{NV})$, $H_0 = D_\text{zfs} S_z^2$, $D_\text{zfs} = 2.870\,\text{GHz}$ the zero-field splitting and $\phi = \omega_\text{rot} t$ the azimuthal coordinate. The linearly-polarized microwave driving field can be represented as an oscillating magnetic field tilted from the $z$-axis by an angle $\theta_\text{mw}$, as depicted in Fig. \[fig:fig1\](c), and is represented by the operator $H_\text{mw}(\theta_\text{mw}) = R_y(\theta_\text{mw}) \Omega_0 S_z\,\cos\left(\omega t - \phi_0\right)R^{-1}_y(\theta_\text{mw})$, with $\Omega_0$ the Rabi frequency, $\omega$ the microwave oscillation frequency and $\phi_0$ an arbitrary initial phase. In the frame of the rotating NV, the interaction Hamiltonian is $$H_I = R_\text{NV}^{-1}(\theta_\text{NV}, \phi) H_\text{mw}(\theta_\text{mw}) R_\text{NV}(\theta_\text{NV}, \phi).
\label{eq:hint}$$ Under the rotating wave approximation (RWA), the diagonal terms of $H_I$ are neglected and only slowly-varying terms ($\ll\omega$) in the off-diagonal elements remain. A near-resonant microwave field is therefore approximated as an azimuthal vector in the rotating frame with components $\boldsymbol{\Omega} = (\Omega\,\cos\phi_\text{eff},\Omega\,\sin\phi_\text{eff},0)$, with $\Omega = |H^{(i,j)}_I|$ and $\phi_\text{eff} = \text{Arg}(H^{(i,j)}_I)$, *i.e.* the modulus and argument of the complex numbers in the off-diagonal components of $H^{(i,j)}_I = \left(H^{(j,i)}_I\right)^\dagger$: $$\begin{aligned}
H^{(i,j)}_I = & \Omega_0 e^{-i\phi_0}\left(\cos\theta_{\text{NV}}\cos\phi\sin\theta_{\text{mw}}\right. \nonumber \\
- & \left. \cos\theta_\text{mw}\sin\theta_\text{NV}+ i\sin\theta_\text{mw}\sin\phi\right)/2.
\label{eq:absarg}\end{aligned}$$ In the frame rotating at the microwave frequency, the spin vector of the two-level system $S_i = \frac{1}{2}\langle \sigma_i\rangle$ precesses about the driving field $\boldsymbol{\Omega}$, and in a pulsed interferometric sequence it can be seen from Eq.(\[eq:absarg\]) that the azimuthal rotation angle $\phi$ sets the effective phase $\phi_\text{eff}$ and determines the axis around which each pulse rotates the spin vector. For example, in a simple $\frac{\pi}{2}$-$\tau$-$\frac{\pi}{2}$ Ramsey sequence the relative phase of the second pulse of the sequence is therefore set by $\phi_\text{eff}$.
The effective phase becomes more complicated when considering $\theta_\text{mw}\neq0$. The tilt angle of the microwave field $\theta_\text{mw}$ is varied by translating the position of a wire producing a microwave magnetic field, as shown in Fig.\[fig:fig1\](b). Figure \[fig:fig1\](c) depicts the tilted microwave field in the laboratory, rotating-NV frame and projective Hilbert space (the Bloch sphere, under the RWA), and the resulting effective phase $\phi_\text{eff}$ for different microwave tilt angles $\theta_\text{mw}$ when rotating. The geometric path followed by the microwave field in the NV frame (Fig \[fig:fig1\](c,ii)) is projected onto the azimuthal plane in the RWA Hilbert space (Fig \[fig:fig1\](c,iii)), with $\phi_\text{eff}$ identified as the azimuthal angle of $\boldsymbol{\Omega}$. For $\theta_\text{mw}\neq0$, $\phi_\text{eff}$ exhibits nonlinear behaviour, with $\boldsymbol{\Omega}$ oscillating in one hemisphere for $\theta_\text{mw} < \theta_\text{NV}$ before rotating about the entire azimuthal plane for $\theta_\text{mw} > \theta_\text{NV}$. Thus, for $\theta_\text{mw}\neq0$, $\phi_\text{eff}$ accumulates nonlinearly in rotation angle, allowing measurement with spin-echo interferometry. Spin-echo allows for longer interrogation times and therefore spans larger azimuthal angles, as well as insensitivity to temperature drifts compared to the simple Ramsey sequence.
A spin-echo sequence measures the difference in phase accumulation on either side of the refocusing $\pi$-pulse occuring midway between the initial and final $\pi/2$-pulses. Spin-echo is therefore insensitive to linearly accumulating phase shifts, whether originating in the quantum system itself (due to a static magnetic field, for instance) or the azimuthal precession of $\boldsymbol{\Omega}$ due to a linear $\phi_\text{eff}$. In the latter case, since the microwave field is switched off during the free evolution periods, it is the instantaneous $\phi_\text{eff}$ sampled by each microwave pulse that imparts the phase shift we seek to measure, and if $\phi_\text{eff}$ varies linearly across the pulse sequence, then it is cancelled in analogy with dc magnetic field shifts in conventional spin-echo measurements. Any deviation from linear effective phase accumulation (due to $\theta_\text{mw}\neq0$) when the $\pi$-refocusing pulse is applied (Figure \[fig:fig1\]d) is reflected in the final populations of the two-level system, allowing us to define the nonlinear component of $\phi_\text{eff}$ as $$\delta\phi = \frac{\phi_\text{eff}(\tau)}{2} - \phi_\text{eff}\left(\frac{\tau}{2}\right),
\label{eq:pphase}$$ assuming $\phi_\text{eff}(\tau = 0) = 0$. The population of the $|m_S = 0\rangle$ state then varies as $\cos^2\left(\delta\phi\right)$. Practically, we measure the rotationally-induced $\delta\phi$ by observing the phase of spin-echo interference fringes traced out as a function of some applied magnetic field. In a rotating-NV spin-echo measurement, any dc magnetic field not parallel to the rotation axis is effectively up-converted to an oscillating field in the rotating frame . Since the effective microwave phase $\phi_\text{eff}$ can be controlled by varying the microwave tilt angle $\theta_\text{mw}$, the rotationally-induced phase shift manifests as a tilt-angle dependent fringe phase shift in a rotating-NV spin-echo measurement.
As depicted in Fig.\[fig:fig1\](d), and according to Eq.(\[eq:pphase\]), the specific behaviour of $\phi_\text{eff}$ sampled in a spin-echo measurement of duration $\tau$ may be approximately linear for a particular selection of microwave tilt angles, in which case $\delta\phi$ will be small, or the effective phase may be very nonlinear, resulting in a large $\delta\phi$. To estimate the expected $\phi_\text{eff}$ behaviour for our system we measured the angular dependence of the microwave coupling strength $\Omega(\theta_\text{mw}, \phi)$, which informs the vector behaviour of the microwave field and then reconstructed the expected phase behaviour. We measured the microwave Rabi frequency by performing a discrete Fourier transform of time-domain Rabi oscillations collected at different stationary ‘park’ angles $\phi$, for various microwave wire positions ($\theta_\text{mw}$). Using a simple approximation of the microwave field emanating from the wire as ${\hat{\boldsymbol{B}}\xspace}_\text{mw} = {\hat{\boldsymbol{\varphi}}\xspace}$, where $\varphi$ is the azimuthal coordinate in the cylindrical coordinate system $({\hat{\boldsymbol{r'}}\xspace}, {\hat{\boldsymbol{\varphi}}\xspace}, {\hat{\boldsymbol{x}}\xspace})$ of the wire, is sufficient to accurately reproduce the angular dependence of the measured Rabi frequency $\Omega$. The only free parameter in the plotted model depicted in Fig \[fig:rabisphases\] is the calibration between absolute azimuthal orientation of the NV axis and the arbitrary motor park angle. These measurements also determine the microwave pulse durations required to execute the $\pi/2$- and $\pi$-pulses used in the rotating spin-echo sequence, which can differ significantly in a single sequence for certain $\theta_\text{mw}$. The maximum variation of Rabi frequency across a single spin-echo sequence was between $1$ and $6\,$MHz. Figure \[fig:rabisphases\] shows the measured microwave Rabi frequency and reconstructed effective phase as a function of motor park angle.
![Reconstruction of effective phase accumulation from stationary Rabi frequency. Top: Measured normalized Rabi frequency vs. stationary rotation angle of motor for different microwave tilt angles, lines denote model derived from Eq.(\[eq:absarg\]) and error bars denote uncertainty in calculated Rabi frequency derived from a Fourier transform of time-domain Rabi oscillations. Bottom: reconstructed effective phase, using the same model. Vertical lines denote the Rabi frequency and effective phase sampled where microwave pulses are applied - for minimum (blue) and maximum (orange) nonlinearity of $\phi_\text{eff}$.[]{data-label="fig:rabisphases"}](fig2.pdf)
For our measurements of $\phi_\text{eff}$ while rotating, we chose $\tau =100\,\upmu$s ($120^\circ$ at $3.33\,$kHz) and two initial configurations: a ‘null’ measurement where $\phi_\text{eff}$ is approximately linearly varying ($\delta_\phi \approx 0$), and a second configuration sampling the highly nonlinear behaviour in the vicinity of $\phi = 250^\circ$ to observe maximum $\delta\phi$. The experimental sequence was synchronized to the rotation of the diamond, and a delay time between the motor trigger and the start of the experimental sequence was used to change the starting angle in the spin-echo experiment. The experimental sequence consisted of measuring spin-echo fringes as a function of an applied dc magnetic field ${\bf B} = (B_x,0,0)$ for different microwave tilt angles $\theta_\text{mw}$. One preparation-interrogation-readout cycle was performed every rotation period, and was repeated $>10^5$ times over an approximate duration of two to three minutes.
Figure \[fig:fig3\](a,b) shows spin-echo fringes for several different microwave tilt angles in the regions of linear and nonlinear $\phi_\text{eff}$ accumulation. We extract the phase shift, plotted in Fig \[fig:fig3\](c), by fitting functions of the form $\cos^2(2\pi f_0-\delta\phi)$ to the fringe data, where $f_0$ is the average fringe frequency across the whole dataset. Drifts of the apparatus temperature, which in turn lead to drifts in the background magnetic field environment, are particularly problematic, since a drift in the transverse magnetic fields will result in a spurious shift of the spin-echo signal phase. For this reason, we sample the spin echo signal over a narrow range of $B_x$ (one period) and as rapidly as possible. A more detailed analysis of drifts and other sources of error is provided in the Supplementary Material. The results agree very well with the model developed from the stationary data in Fig \[fig:rabisphases\], and constitute the first demonstration of a single, room-temperature spin system with a quantum phase determined by classical rotation.
In measuring a rotationally-induced phase, we have performed a detection of actual physical rotation using the electron spin of the NV center. The nonlinear aspect of $\phi_\text{eff}$, key to our measurement working in a noisy environment, is a somewhat overlooked effect that may offer new possibilities for precision rotation sensing or an additional control handle for quantum systems. While practical deployment of a diamond-based rotation sensor is a considerable technological challenge, our results will be applicable to other systems where physical rotation on quantum timescales plays an important role. On a much slower timescale than the work reported here, studies of Brownian motion and rotational diffusion of nanodiamonds in fluidic environments [@mcguinness_quantum_2011; @maclaurin_nanoscale_2013; @yoshinari_observing_2013] offers an intriguing example of multi-axis physical rotations in quantum systems, and our work offers the prospect of taking such measurements a step further and probing rapid rotation and motion on quantum-relevant timescales. In a more general sense, our results serve to further highlight the remarkable physical effects that simple physical rotation can impart to sophisticated systems. Further work could consider quantum measurements of rotation on faster timescales in different systems, such as optically and electrically-trapped nanodiamonds [@horowitz_electron_2012; @hoang_torsional_2016; @delord_electron_2017; @delord_ramsey_2018] which have interesting prospects for realising sensitive torque detectors and probing fundamental quantum mechanics [@delord_strong_2017; @stickler_probing_2018; @delord_spin-cooling_2019].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank L. P. McGuinness for provision of the diamond sample and L. D. Turner and R. P. Anderson for fruitful discussion. This work was supported by the Australian Research Council Discovery Scheme (DP150101704, DP190100949).
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|
---
abstract: 'We use the evolution operator method to find the Schwinger pair-production rate at finite temperature in scalar and spinor QED by counting the vacuum production, the induced production and the stimulated annihilation from the initial ensemble. It is shown that the pair-production rate for each state is factorized into the mean number at zero temperature and the initial thermal distribution for bosons and fermions.'
author:
- Sang Pyo Kim
- Hyun Kyu Lee
- Yongsung Yoon
title: Schwinger Pair Production at Finite Temperature in QED
---
Introduction
============
Vacuum polarization and pair production have been issues of continuous concern since the early works by Sauter, Heisenberg and Euler, and Weisskopf [@Sauter], and then by Schwinger [@Schwinger] (for a review and references, see Ref. [@Dunne]). The task of directly computing, without relying on the electromagnetic duality, the effective action in electric field backgrounds, however, has been a challenging problem due to the vacuum instability. Dunne and Hall used the resolvent method to directly find the effective action in time-dependent electric fields [@Dunne-Hall]. In the previous paper [@KLY], employing the evolution operator method, we found the exact one-loop effective actions of scalar and spinor QED at zero temperature in a constant or a pulsed electric field of Sauter-type, which satisfy the exact relation $2
{\rm Im} {\cal L}_{\rm eff} =\pm \sum_{n} \ln (1\pm \overline{\cal N}_{n})$ (with $+$ for scalar and $-$ for spinor) between the imaginary part of the effective Lagrangian density ${\cal L}_{\rm eff}$ and the mean number of created pairs $\overline{\cal N}_{n}$ at state $n$. Even finding the pair production rate by time-dependent or spatially localized electric fields is methodologically nontrivial, which has recently been intensively studied [@Kim-Page; @DSWG; @KRX].
To calculate the effective action and thereby Schwinger pair production at finite temperature is another challenging problem in QED. In a constant pure magnetic field the QED effective action was studied at finite temperature [@Dittrich] and at finite temperature and density [@EPS]. However, the presence of an additional electric field raised Schwinger pair production at debate depending on the formalism employed. Some calculations in thermal field theory reported that the effective action in both a constant magnetic and electric field had an imaginary part having dependence on temperature [@Loewe-Rojas; @GKP; @Hallin-Liljenberg]. However, in the real-time formalism, no imaginary part was found in the QED effective action in the presence of both magnetic and electric field [@Elmfors-Skagerstam]. The recent calculation of effective action in the imaginary-time formalism shows an imaginary part only at two-loop but not at one-loop in both a constant electric and magnetic field [@Gies99]. In nonequilibrium quantum field theory of scalar QED, a calculation in the real-time formalism shows thermal enhancement of pair production [@Kim-Lee07].
In this paper, using the evolution operator method, we find the pair-production rate at finite temperature in time-dependent electric fields both in scalar and spinor QED. The evolution operator, unitarily transforming the particle and antiparticle operators from the ingoing vacuum to the outgoing vacuum, carries all the information of quantum evolution. In fact, the evolution operator is completely determined by the Bogoliubov coefficients. The advantage of the evolution operator is the readiness to calculate the probability for transitions among multiparticle states. This allows us to compute the mean number of created pairs at finite temperature in scalar and spinor QED by counting the pairs from the vacuum and the induced production and the stimulated annihilation in a thermal ensemble of bosons and fermions. We find that the mean number of created pairs is factorized into the mean number of created pairs at zero temperature and the initial thermal distribution for bosons and fermions.
The organization of this paper is as follows. In Sec. II, rewriting the Bogoliubov transformation as a unitary transformation by the evolution operator, we find the mean number of created pairs at zero temperature in scalar and spinor QED. In Sec. III, we calculate the mean number of created pairs at finite temperature and apply it to the Sauter-type electric field.
Evolution Operator and Pair Production at $T = 0$
=================================================
We first consider scalar QED for spinless charged bosons under an external electric field with the gauge field $A_{\mu}$. The electric field is assumed to be acting on for a finite period of time so that the ingoing and the outgoing vacua are well-defined. Thus, at $t_{\rm in} = - \infty$, before the external electric field being turned on, the scalar field is free and the Hamiltonian takes the usual form $$\begin{aligned}
H_{\rm in}^{\rm sc} =\int \frac{d^3 {\bf k}}{(2 \pi)^3} \omega_{{\bf
k}, {\rm in}} N_{{\bf k},{\rm in}} =\int \frac{d^3 {\bf k}}{(2
\pi)^3} \omega_{{\bf k}, {\rm in}} (a_{{\bf k}, {\rm in}}^{\dagger}
a_{{\bf k}, {\rm in}} + b_{{\bf k}, {\rm in}}^{\dagger} b_{{\bf k},
{\rm in}}), \label{sc-in-ham}\end{aligned}$$ where $\omega_{{\bf k}, {\rm in}}$ is the initial frequency at momentum ${\bf k}$. Here, the gauge is chosen $A_{\mu} = 0$, so that the ingoing vacuum $\vert 0; t_{\rm in} \rangle$ is the Minkowski vacuum $\vert 0 \rangle_{\rm M}$, annihilated by $a_{\bf
k} (t_{\rm in})$ and $b_{\bf k} (t_{\rm in})$ for each momentum ${\bf k}$. Similarly, the outgoing vacuum at $t_{\rm out} =
\infty$ is defined by $a_{\bf k} (t_{\rm out})$ and $b_{\bf k}
(t_{\rm out})$. These operators are related through the Bogoliubov transformations [@Kim-Lee07] $$\begin{aligned}
a_{{\bf k}, {\rm out}} = \mu_{\bf k} a_{{\bf k}, {\rm in}} +
\nu^*_{\bf k} b^{\dagger}_{{\bf k}, {\rm in}}, ~~~ b_{{\bf k},
{\rm out}} = \mu_{\bf k} b_{{\bf k}, {\rm in}} + \nu^*_{\bf k}
a^{\dagger}_{{\bf k}, {\rm in}}, \label{sc-out-in}\end{aligned}$$ where $ |\mu_{\bf k}|^2 - |\nu_{\bf k}|^2 = 1$.
To express the outgoing vacuum as multiparticle states of the ingoing vacuum, we rewrite the Bogoliubov transformations (\[sc-out-in\]) as a unitary transformation [@KLY] $$\begin{aligned}
a_{{\bf k}, {\rm out}} (A) = U_{\bf k} (A) a_{{\bf k}, {\rm in}}
(0) U^{\dagger}_{\bf k} (A), ~~~ b_{{\bf k}, {\rm out}} (A) =
U_{\bf k} (A) b_{{\bf k}, {\rm in}} (0) U^{\dagger}_{\bf k} (A).
\label{sc-U-tr}\end{aligned}$$ Here, the evolution operator $U_{\bf k}$ is factorized into the overall phase factor and the two-mode squeeze operator as [@Caves-Schumaker1; @Caves-Schumaker2] $$\begin{aligned}
U_{\bf k} (A) = e^{ i \theta_{\bf k} (a^{\dagger}_{{\bf k},
{\rm in}}a_{{\bf k}, {\rm in}} + b^{\dagger}_{{\bf k}, {\rm in}}
b_{{\bf k}, {\rm in}} + 1 ) } e^{ \xi_{\bf k} a^{\dagger}_{{\bf k}, {\rm in}}
b^{\dagger}_{{\bf k}, {\rm in}}} e^{
\frac{\gamma_{\bf k}}{2} ( a^{\dagger}_{{\bf k}, {\rm in}}
a_{{\bf k}, {\rm in}} + b^{\dagger}_{{\bf k}, {\rm in}} b_{{\bf k},
{\rm in}}+1 ) } e^{ - \xi_{\bf k}^* a_{{\bf k},
{\rm in}} b_{{\bf k}, {\rm in}} },\end{aligned}$$ where $$\begin{aligned}
e^{2 i \theta_{\bf k}} = \frac{\mu^*_{\bf k}}{\mu_{\bf k}}, \quad \xi_{\bf k} = \frac{\nu^*_{\bf k}}{\mu_{\bf k}}, \quad \gamma_{\bf k}
= - 2 \ln (\mu_{\bf k} ).\end{aligned}$$
From the charge neutrality of the vacuum, equal numbers of particles and antiparticles are produced at zero temperature and they carry the opposite momenta due to the momentum conservation. The multiparticle state of $n$-pairs can be concisely denoted as $\vert n_{\bf k}, t \rangle = \vert n_{\bf k};\bar{n}_{\bf k}; t
\rangle$. The probability for $n$-pairs with momentum ${\bf k}$ to be created from the vacuum is $$\begin{aligned}
P_n ({\bf k}) = | \langle n_{\bf k}, {\rm out} \vert 0, {\rm in}
\rangle |^2 = | \langle n_{\bf k}, {\rm in} \vert
U^{\dagger}_{\bf k} \vert 0, {\rm in} \rangle |^2 = e^{\gamma_{\bf
k}} |\xi_{\bf k}|^{2n}.\end{aligned}$$ Note that $P_0 = e^{\gamma_{\bf k}}$ and $P_1 = e^{\gamma_{\bf k}}
|\xi_{\bf k}|^{2}$ so that $P_n = P_0 (P_1/P_0)^n$ and $\sum_{n =
0}^{\infty} P_n = 1$ for each ${\bf k}$. Thus, at zero temperature, the mean number of pairs created from the vacuum for each momentum per unit volume is $$\begin{aligned}
\overline{\cal N}_{\bf k}^{\rm sc}(T=0) = \sum_{n = 0}^{\infty} n
P_n ({\bf k}) = |\nu_{\bf k}|^{2}. \label{sc-mean}\end{aligned}$$
Next, in spinor QED, before the interaction of an external electric field, the spinor field is free without the gauge potential $(A_{\mu}=0)$, and has the Hamiltonian given by $$\begin{aligned}
H_{\rm in}^{\rm sp} =\sum_{\sigma} \int \frac{d^3 {\bf k}}{(2
\pi)^3} \omega_{n,{\rm in}}N_{n,{\rm in}} = \sum_{\sigma} \int
\frac{d^3 {\bf k}}{(2 \pi)^3} \omega_{n,{\rm in}} (b_{n,{\rm
in}}^{\dagger} b_{n,{\rm in}} + d_{n,{\rm in}}^{\dagger} d_{n,{\rm
in}}), \label{sp-in-ham}\end{aligned}$$ where $b_{n,{\rm in}}$ and $d_{n,{\rm in}}$ are particle and antiparticle operators in the ingoing vacuum. After the interaction of the electric field, the ingoing vacuum evolves to the outgoing vacuum, whose particle and antiparticle operator are $b_{n,{\rm out}}$ and $d_{n,{\rm out}}$. The Bogoliubov transformations between the ingoing and the outgoing particle and antiparticle operators, $b_{n}, d_{n}$, are similarly given by $$\begin{aligned}
b_{n,{\rm out}} = \mu_{n} b_{n,{\rm in}} + i\nu_{n}^{*}d_{n,{\rm
in}}^{\dagger}, ~~~ d_{n,{\rm out}} = \mu_{n} d_{n,{\rm in}} -
i\nu_{n}^{*}b_{n,{\rm in}}^{\dagger}, \label{sp-out-in}\end{aligned}$$ where $|\mu_{n}|^{2}+|\nu_{n}|^{2}=1$ and $n=({\bf k},\sigma)$ with $\sigma = \pm 1/2$. The Bogoliubov transformation can be also written as a unitary transformation [@Fan] $$\begin{aligned}
b_{n,{\rm out}} = U_{n}b_{n,{\rm in}}U_{n}^{\dagger}, ~~~
d_{n,{\rm out}} = U_{n}d_{n,{\rm in}}U_{n}^{\dagger},
\label{sp-U-tr}\end{aligned}$$ where $$U_{n}=e^{\xi_{n}b_{n,{\rm in}}^{\dagger}d_{n,{\rm in}}^{\dagger}}
e^{(\frac{\gamma_{n}}{2}+i\theta_{n})(b_{n,{\rm
in}}^{\dagger}b_{n,{\rm in}}+d_{n,{\rm in}}^{\dagger}d_{n,{\rm
in}}-1)} e^{e^{2i\theta_{n}}\xi_{n}^{*}b_{n,{\rm in}}d_{n,{\rm
in}}}. \label{sp-U}$$ Here, the three parameters $\xi_{n}, \gamma_{n}$ and $ \theta_{n}$ are determined by the Bogoliubov coefficients as $$\begin{aligned}
\xi_{n} = - i\frac{\nu^*_{n}}{\mu_{n}}, \quad \gamma_{n} = -2\ln(|\mu_n|), \quad e^{2 i\theta_{n}} = \frac{\mu_n^*}{\mu_n}. \label{spinor-parameters}\end{aligned}$$ Note that the pair production on spinor QED is restricted to only one pair of particle and antiparticle for a given quantum number $n$ due to the Pauli exclusion principle. Thus, the mean number of pairs created from the vacuum for each state $n$ at zero temperature is calculated as $$\begin{aligned}
\overline{\cal N}_{n}^{\rm sp}(T=0) = | \langle 1_{n},{\rm out} \vert 0,{\rm in} \rangle |^{2} = |\nu_{n}|^{2}.
\label{sp-mean}\end{aligned}$$
Pair Production at $T \neq 0$
=============================
We now calculate the mean number of pairs at finite temperature from the probability for each transition, as in the case of zero temperature. As there is no mode-mixing, we separately calculate the mean number of created pairs for each mode. For an initial thermal ensemble at $\beta =
1/kT$, which might not be charge neutral, the mean number of produced pairs consists of the vacuum pair production, the induced pair production, and the stimulated pair annihilation as shown in Fig. 1: $$\begin{aligned}
\overline{\cal N}_{n} (T) &=& \frac{1}{Z_{n}} \Biggl[\sum_{n_{n} >
0}^{\infty} e^{-\beta E_{0_{n},0_{n}}} n_{n} P_{0_{n},0_{n}
\rightarrow n_{n},n_{n}} - \sum_{p_{n}
> n_{n} \ge 0, q_{n}
> m_{n} \ge 0}^{\infty} e^{-\beta E_{p_{n},q_{n}}} (p_{n} - n_{n})
P_{p_{n},q_{n} \rightarrow n_{n},m_{n}} \nonumber \\ &~&~~~~~~~~ +
\sum_{p_{n} > n_{n} \ge 0, q_{n}
> m_{n} \ge 0, n_{n} + m_{n} \neq 0}^{\infty} e^{-\beta E_{n_{n},m_{n}}} (p_{n}
- n_{n}) P_{n_{n},m_{n} \rightarrow p_{n},q_{n}} \Biggr],
\label{mean-T1}\end{aligned}$$ where $Z_n$ is the partition function for the initial ensemble and $P_{n_{n},m_{n} \rightarrow p_{n},q_{n}} = |\langle
p_{n},q_{n}, {\rm in} \vert U_{n} \vert n_{n},m_{n}, {\rm in}
\rangle|^2$ is the transition probability from $\vert n_{n},m_{n},
{\rm in} \rangle$ to $\vert p_{n}, q_{n},{\rm in} \rangle$. Using $|\langle p_{n},q_{n}, {\rm in} \vert U_{n} \vert n_{n},m_{n},
{\rm in} \rangle|^2 = |\langle p_{n},q_{n}, {\rm in} \vert
U_{n}^{\dagger} \vert n_{n},m_{n}, {\rm in} \rangle|^2$, which implies $P_{p_{n},q_{n} \rightarrow n_{n},m_{n}} = P_{n_{n},m_{n}
\rightarrow p_{n},q_{n}}$, the mean number of pairs created at finite temperature, Eq. (\[mean-T1\]), can be written as $$\begin{aligned}
\overline{\cal N}_{n} (T) &=& \frac{1}{2Z_{n}} \sum_{p_{n} >
n_{n},q_{n} > m_{n}}^{\infty} \Biggl[ \langle n_{n},m_{n}, {\rm
in} \vert e^{ - \beta H_{n,{\rm in}}} U_{n} \vert p_{n},q_{n},
{\rm in} \rangle \langle p_{n},q_{n}, {\rm in}\vert (N_{n,{\rm
in}}U_{n}^{\dagger}-U_{n}^{\dagger} N_{n,{\rm in}}) \vert
n_{n},m_{n}, {\rm in} \rangle \nonumber \\ &~&~~~~~~~~ - \langle
n_{n},m_{n}, {\rm in} \vert U_{n}^{\dagger} e^{-\beta H_{n,{\rm
in}}} \vert p_{n},q_{n}, {\rm in} \rangle \langle p_{n},q_{n},
{\rm in}\vert (N_{n,{\rm in}}U_{n}-U_{n} N_{n,{\rm in}}) \vert
n_{n},m_{n}, {\rm in} \rangle \Biggr] \nonumber \\ ~ &=&
\frac{1}{2Z_{n}} {\rm Tr}(U_{n,{\rm in}} N_{n,{\rm in}}U_{n,{\rm
in}}^{\dagger} -N_{n,{\rm in}}) e^{-\beta H_{n,{\rm in}}},
\label{mean-T2}\end{aligned}$$ where $N_{n, {\rm in}} = a_{{\bf k}, {\rm in}}^{\dagger} a_{{\bf k},
{\rm in}} + b_{{\bf k}, {\rm in}}^{\dagger} b_{{\bf k}, {\rm in}}$ for scalar particles and $N_{n, {\rm in}} = b_{n, {\rm
in}}^{\dagger} b_{n, {\rm in}} + d_{n, {\rm in}}^{\dagger} d_{n,
{\rm in}}$ for spinor particles and $H_{n,{\rm in}}=\omega_{n,{\rm in}} N_{n, {\rm in}}$ for both.
{width="5.2in"}
From the Bogoliubov transformations (\[sc-U-tr\]) and (\[sp-U-tr\]), we have $N_{n, {\rm out}}
= U_{n} N_{n, {\rm in}} U_{n}^{\dagger}$. Thus, the mean number of pairs created at finite temperature can be written concisely as $$\begin{aligned}
\overline{\cal N}_{n}(T) &=& \frac{1}{2Z_{n}} {\rm Tr}(N_{n,{\rm
out}} - N_{n,{\rm in}}) e^{-\beta H_{n,{\rm in}}}. \label{mean-T3}\end{aligned}$$ First, for scalar particles, from Eq. (\[sc-out-in\]), $$\begin{aligned}
N_{{\bf k},{\rm out}}-N_{{\bf k},{\rm in}}= 2|\nu_{\bf
k}|^{2}(a_{{\bf k},{\rm in}}^{\dagger}a_{{\bf k},{\rm in}}+b_{{\bf
k},{\rm in}}^{\dagger}b_{{\bf k},{\rm in}}+1)+2\mu_{\bf k}\nu_{\bf
k} a_{{\bf k},{\rm in}} b_{{\bf k},{\rm in}}+2\mu_{\bf
k}^{*}\nu_{\bf k}^{*} a_{{\bf k},{\rm in}}^{\dagger} b_{{\bf k},{\rm
in}}^{\dagger},\end{aligned}$$ and $Z_{\bf k}= e^{\beta\omega_{{\bf k},{\rm in}}}(2\sinh
\beta\omega_{{\bf k},{\rm in}}/2)^{-2}$, the mean number of scalar pairs created in momentum ${\bf k}$ at finite temperature is given by $$\begin{aligned}
\overline{\cal N}_{\bf k}^{\rm sc}(T)=\frac{\vert\nu_{\bf
k}\vert^{2}}{Z_{\bf k}}\sum_{n,m=0}^{\infty}\langle
n,m\vert(n+m+1)e^{-(n+m)\beta\omega_{{\bf k},{\rm in}}}|n,m
\rangle= \vert \nu_{\bf k}\vert^{2}\coth\frac{\beta\omega_{{\bf
k},{\rm in}}}{2}. \label{sc-result}\end{aligned}$$
Second, for spinor particles, from Eq. (\[sp-out-in\]), $$\begin{aligned}
N_{n,{\rm out}}-N_{n,{\rm in}} = -2|\nu_{n}|^{2}(b_{n,{\rm
in}}^{\dagger}b_{n,{\rm in}}+d_{n,{\rm in}}^{\dagger}d_{n,{\rm
in}}-1)+2i\mu_{n}\nu_{n} b_{n,{\rm in}} d_{n,{\rm
in}}+2i\mu_{n}^{*}\nu_{n}^{*} b_{n,{\rm in}}^{\dagger} d_{n,{\rm
in}}^{\dagger},\end{aligned}$$ and $Z_{n}= (1+e^{-\beta\omega_{n,{\rm in}}})^{2}$, we find the mean number of spinor pairs created in state $n$ at finite temperature $$\begin{aligned}
\overline{\cal N}_{n}^{\rm
sp}(T)=-\frac{|\nu_{n}|^{2}}{Z_{n}}\sum_{n,m=0}^{1} \langle n,m
\vert (n+m-1)e^{-(n+m)
\beta\omega_{n,{\rm in}}} \vert n,m \rangle = \vert\nu_{n}\vert^{2} \tanh\frac{\beta\omega_{n,{\rm in}}}{2}.
\label{sp-result}\end{aligned}$$
{width="2.8in"} {width="2.8in"}\
As an interesting model for discussions, we consider the Sauter-type electric field ${\cal E}(t) = E {\rm sech}^2
(t/\tau)$ with the gauge choice, $A_z (t) = - E \tau (1+ \tanh (t/\tau))$, which allows the exact solution leading to the exact mean number of pairs created, $|\nu_{\bf k}|^{2}$, at zero temperature both in scalar and spinor QED [@NN; @GG; @KLY; @Kim-Lee07]. Also the approximation scheme of the WKB or worldline instanton method has been developed [@Kim-Page; @DSWG; @KRX]. Because the Sauter-type electric field acts effectively for a finite period of time $\tau$, the ingoing thermal states are stable and well defined as required in our formalism. Fig. 2 shows the mean number of created pairs Eqs.(\[sc-result\],\[sp-result\]) at finite temperature with the electric field $E$, the duration $\tau$ and temperature $k_B T = 1/\beta$ scaled in terms of the critical strength, the Compton time and the electron mass, respectively. The production of scalar pairs is thermally enhanced, while the production of fermion pairs is thermally suppressed as expected by the Pauli blocking, which is consistent with the calculation by density matrix method [@GGT].
The pure thermal effect on the mean number of created pairs, which is $\Delta \overline{\cal N}_{\bf k}(T) =
\overline{\cal N}_{\bf k}(T) - \overline{\cal N}_{\bf k}(0)$, is given by $\Delta \overline{\cal N}_{\bf k}(T) = \pm 2
|\nu_{\bf k}|^2 f^\frac{B}{F}_{\bf k}(T)$ with $f^\frac{B}{F}_{\bf k}(T)$ being the Bose-Einstein or Fermi-Dirac distribution. For $k_B T \ll \omega_{{\bf k}, {\rm in}}$, the distribution $f^\frac{B}{F}_{\bf k}(T)$ approximately equals to the Boltzmann factor $f_{\bf k} \approx e^{-\sqrt{m^{2}+{\bf k}^{2}} /k_B T}$. Thus, the mean number of created pairs at finite temperature, Eqs.(\[sc-result\],\[sp-result\]), is reduced to the zero temperature result, Eqs.(\[sc-mean\],\[sp-mean\]), for a temperature much lower than the rest mass.
Our result Eq.(\[sp-result\]) could be compared with the calculation in imaginary-time formalism [@Gies99], where the thermal effect appears only at two-loop because thermal one-loop fluctuations are on-shell. On the other hand, Eq. (\[sp-result\]), an off-shell calculation, is equivalent to the thermal loop times the vacuum one-loop, in fact, part of two-loops. One interesting comment to be pointed out is that the momentum integral over the distribution function in $\Delta \overline{\cal N}_{\bf k}(T) = \pm 2 |\nu_{\bf k}|^2 f^\frac{B}{F}_{\bf k} (T)$ leads to the factor $T^4$ in Ref. [@Gies99], though the momentum integral of the distribution is intertwined with the vacuum pair-production rate. To show rigorously the connection between our result and Ref. [@Gies99] requires calculating the effective action at finite temperature along the line of Ref.[@KLY], which will be addressed in the future.
Conclusion
==========
In this paper, using the evolution operator method, we found that the mean number of created pairs in state $n$ at finite temperature is given by $$\begin{aligned}
\overline{\cal N}_{n} (T) &=& \frac{1}{2Z_{n}} {\rm Tr}(N_{n,{\rm
out}} - N_{n,{\rm in}}) e^{-\beta H_{n,{\rm in}}} .\end{aligned}$$ For scalar and spinor QED in external electric fields, the total mean number density of created pairs at finite temperature is given by $$\begin{aligned}
\overline{\cal N}^{\rm sc} (T) &=& \int \frac{d^{3} {\bf
k}}{(2\pi)^{3}} \overline{\cal N}^{\rm sc}_{\bf k} (0)
\coth\frac{\beta\omega_{{\bf k},{\rm in}}}{2} \quad ({\rm for~
scalar}), \\ \overline{\cal N}^{\rm sp} (T) &=& \sum_{\sigma}\int
\frac{d^{3} {\bf k}}{(2\pi)^{3}}\overline{\cal N}^{\rm sp}_{n} (0)
\tanh\frac{\beta\omega_{n,{\rm in}}}{2} \quad ({\rm for~ spinor}),\end{aligned}$$ where $\overline{\cal N}^{\rm sc}_{\bf k} (0)$ and $\overline{\cal
N}^{\rm sp}_{n} (0)$ are the mean number of created pairs at zero temperature for scalar and spinor QED, respectively.
S. P. K. would like to appreciate the hospitality of Hanyang University, and H. K. L. and Y. Y. would like to appreciate the hospitality of Kunsan National University. The work of S. P. K. was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2007-C00167) and the work of H. K. L. was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R01-2006-000-10651-0).
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|
---
abstract: 'We develop a phenomenological mean field theory for the strain in through its hidden order transition. Several experimental features are reproduced when the order parameter has $\Bog$ symmetry: the topology of the temperature–pressure phase diagram, the response of the strain stiffness tensor above the hidden-order transition at zero pressure, and orthorhombic symmetry breaking in the high-pressure antiferromagnetic phase. In this scenario, the hidden order is characterized by the order parameter in the high-pressure antiferromagnetic phase modulated along the symmetry axis, and the triple point joining those two phases with the paramagnetic phase is a Lifshitz point.'
author:
- 'Jaron Kent-Dobias'
- Michael Matty
- Brad Ramshaw
bibliography:
- 'hidden\_order.bib'
- 'library.bib'
title: 'Elastic properties of hidden order in are reproduced by modulated $\Bog$ order'
---
The study of phase transitions is central to condensed matter physics. Phase transitions are often accompanied by a change in symmetry whose emergence can be described by the condensation of an order parameter () that breaks the same symmetries. Near a continuous phase transition, the physics of the can often be qualitatively and sometimes quantitatively described by Landau–Ginzburg mean field theories. These depend on little more than the symmetries of the , and coincidence of their predictions with experimental signatures of the is evidence of the symmetry of the corresponding ordered state.
A paradigmatic example of a material with an ordered state whose broken symmetry remains unknown is in . is a heavy fermion superconductor in which superconductivity condenses out of a symmetry broken state referred to as *hidden order* () [@hassinger_temperature-pressure_2008], and at sufficiently large hydrostatic pressures, both give way to local moment antiferromagnetism (). Despite over thirty years of effort, the symmetry of the state remains unknown, and modern theories [@kambe_odd-parity_2018; @haule_arrested_2009; @kusunose_hidden_2011; @kung_chirality_2015; @cricchio_itinerant_2009; @ohkawa_quadrupole_1999; @santini_crystal_1994; @kiss_group_2005; @harima_why_2010; @thalmeier_signatures_2011; @tonegawa_cyclotron_2012; @rau_hidden_2012; @riggs_evidence_2015; @hoshino_resolution_2013; @ikeda_theory_1998; @chandra_hastatic_2013; @harrison_hidden_nodate; @ikeda_emergent_2012] propose a variety of possibilities. Many of these theories rely on the formulation of a microscopic model for the state, but since there has not been direct experimental observation of the broken symmetry, none can been confirmed.
Recent work that studied the transition using *resonant ultrasound spectroscopy* () was able to shed light on the symmetry of the ordered state without the formulation of any microscopic model [@ghosh_single-component_nodate]. is an experimental technique that measures mechanical resonances of a sample. These resonances contain information about the sample’s full strain stiffness tensor. Moreover, the frequency locations of the resonances are sensitive to symmetry breaking at an electronic phase transition due to electron-phonon coupling [@shekhter_bounding_2013]. Ref. [@ghosh_single-component_nodate] uses this information to place strict thermodynamic bounds on the dimension of the independent of any microscopic model.
Motivated by these results, we construct a phenomenological mean field theory for an arbitrary coupled to strain and the determine the effect of its phase transitions on the elastic response in different symmetry channels. We find that only one symmetry reproduces the anomalous features of the experimental strain stiffness. That theory associates the state with a $\Bog$ *modulated along the rotation axis*, the state with uniform $\Bog$ order, and the triple point between them with a Lifshitz point. Besides the agreement with data in the state, the theory predicts uniform $\Bog$ strain in the state, which was recently seen in x-ray scattering experiments [@choi_pressure-induced_2018]. The theory’s implications for the dependence of the strain stiffness on pressure and doping strongly motivates future experiments that could either further support or falsify it.
The point group of is , and any coarse-grained theory must locally respect this symmetry. Our phenomenological free energy density contains three parts: the free energy for the strain, the , and their interaction. The most general quadratic free energy of the strain $\epsilon$ is $f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the bare strain stiffness tensor $C$ tensor is constrained by both the index symmetry of the strain tensor and by the point group symmetry [@landau_theory_1995]. The six independent components of strain can written as linear combinations that each behave like irreducible representations under the action of the point group, or $$\begin{aligned}
\epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth}
\epsilon_\Aog^{(2)}=\epsilon_{33} \\
\epsilon_\Bog^{(1)}=\epsilon_{11}-\epsilon_{22} &&
\epsilon_\Btg^{(1)}=2\epsilon_{12} \\
\epsilon_\Eg^{(1)}=2\{\epsilon_{11},\epsilon_{22}\}.
\end{aligned}
\label{eq:strain-components}$$ All quadratic combinations of these irreducible strains that transform like $\Aog$ are included in the free energy, $$f_\e=\frac12\sum_\X C_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)},$$ where the sum is over irreducible representations of the point group and the bare stiffnesses $C_\X^{(ij)}$ are $$\begin{aligned}
&C_{\Aog}^{(11)}=\tfrac12(C_{1111}+C_{1122}) &&
C_{\Aog}^{(22)}=C_{3333} \\
&C_{\Aog}^{(12)}=C_{1133} &&
C_{\Bog}^{(11)}=\tfrac12(C_{1111}-C_{1122}) \\
&C_{\Btg}^{(11)}=C_{1212} &&
C_{\Eg}^{(11)}=C_{1313}.
\end{aligned}$$ The interaction between strain and an $\eta$ depends on the representation of the point group that $\eta$ transforms as. If this representation is $\X$, the most general coupling to linear order is $$f_\i=-b^{(i)}\epsilon_\X^{(i)}\eta.$$ If the representation $\X$ is not present in the strain there can be no linear coupling, and the effect of the condensing at a continuous phase transition is to produce a jump in the $\Aog$ strain stiffness if $\eta$ is single-component [@luthi_sound_1970; @ramshaw_avoided_2015; @shekhter_bounding_2013], and jumps in other strain stiffnesses if multicompenent [@ghosh_single-component_nodate]. Because we are interested in physics that anticipates the phase transition, we will focus our attention on symmetries that can produce linear couplings to strain. Looking at the components present in , this rules out all of the u-reps (which are odd under inversion) and the $\Atg$ irrep.
If the transforms like $\Aog$, odd terms are allowed in its free energy and any transition will be abrupt and not continuous without fine-tuning. Since this is not a feature of physics, we will henceforth rule it out as well. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quadratic free energy density is $$\begin{aligned}
f_\op=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2
+c_\perp(\nabla_\perp\eta)^2 \\
&\qquad\qquad\qquad\quad+D_\perp(\nabla_\perp^2\eta)^2\big]+u\eta^4
\end{aligned}
\label{eq:fo}$$ where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$ and $\nabla_\perp=\partial_3$ transforms like $\Atu$. Other quartic terms are allowed—especially many for an $\Eg$ —but we have included only those terms necessary for stability when either $r$ or $c_\perp$ become negative. The full free energy functional of $\eta$ and $\epsilon$ is $$\begin{aligned}
F[\eta,\epsilon]
&=F_\op[\eta]+F_\e[\epsilon]+F_\i[\eta,\epsilon] \\
&=\int dx\,(f_\op+f_\e+f_\i)
\end{aligned}$$ The only strain relevant to the is $\epsilon_\X$, which can be traced out of the problem exactly in mean field theory. Extremizing with respect to $\epsilon_\X$, $$0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_\X(x)}\bigg|_{\epsilon=\epsilon_\star}=C_\X\epsilon^\star_\X(x)
-b\eta(x)$$ gives the optimized strain conditional on the as $\epsilon_\X^\star[\eta](x)=(b/C_\X)\eta(x)$ and $\epsilon_\Y^\star[\eta]=0$ for all other $\Y$. Upon substitution into the free energy, the resulting effective free energy $F[\eta,\epsilon_\star[\eta]]$ has a density identical to $f_\op$ with $r\to\tilde r=r-b^2/2C_\X$.
![ Phase diagrams for (a) from experiments (neglecting the superconducting phase) [@hassinger_temperature-pressure_2008] (b) mean field theory of a one-component ($\Bog$ or $\Btg$) Lifshitz point (c) mean field theory of a two-component ($\Eg$) Lifshitz point. Solid lines denote continuous transitions, while dashed lines denote abrupt transitions. Later, when we fit the elastic stiffness predictions for a $\Bog$ to data along the zero (atmospheric) pressure line, we will take $\Delta\tilde r=\tilde
r-\tilde r_c=a(T-T_c)$. []{data-label="fig:phases"}](phase_diagram_experiments){width="\columnwidth"}
![ Phase diagrams for (a) from experiments (neglecting the superconducting phase) [@hassinger_temperature-pressure_2008] (b) mean field theory of a one-component ($\Bog$ or $\Btg$) Lifshitz point (c) mean field theory of a two-component ($\Eg$) Lifshitz point. Solid lines denote continuous transitions, while dashed lines denote abrupt transitions. Later, when we fit the elastic stiffness predictions for a $\Bog$ to data along the zero (atmospheric) pressure line, we will take $\Delta\tilde r=\tilde
r-\tilde r_c=a(T-T_c)$. []{data-label="fig:phases"}](phases_scalar "fig:"){width="0.51\columnwidth"} ![ Phase diagrams for (a) from experiments (neglecting the superconducting phase) [@hassinger_temperature-pressure_2008] (b) mean field theory of a one-component ($\Bog$ or $\Btg$) Lifshitz point (c) mean field theory of a two-component ($\Eg$) Lifshitz point. Solid lines denote continuous transitions, while dashed lines denote abrupt transitions. Later, when we fit the elastic stiffness predictions for a $\Bog$ to data along the zero (atmospheric) pressure line, we will take $\Delta\tilde r=\tilde
r-\tilde r_c=a(T-T_c)$. []{data-label="fig:phases"}](phases_vector "fig:"){width="0.51\columnwidth"}
With the strain traced out, describes the theory of a Lifshitz point at $\tilde r=c_\perp=0$ [@lifshitz_theory_1942; @lifshitz_theory_1942-1]. For a one-component ($\Bog$ or $\Btg$) it is traditional to make the field ansatz $\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$. For $\tilde r>0$ and $c_\perp>0$, or $\tilde r>c_\perp^2/4D_\perp$ and $c_\perp<0$, the only stable solution is $\eta_*=q_*=0$ and the system is unordered. For $\tilde r<0$ there are free energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and this system has uniform order. For $c_\perp<0$ and $\tilde r<c_\perp^2/4D_\perp$ there are free energy minima for $q_*^2=-c_\perp/2D_\perp$ and $$\eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{12D_\perp u}
=\frac{\tilde r_c-\tilde r}{3u}
=\frac{|\Delta\tilde r|}{3u}$$ with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The transition between the uniform and modulated orderings is abrupt for a one-component field and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde
r/5}$. For a two-component ($\Eg$) we must also allow a relative phase between the two components of the field. In this case the uniform ordered phase is only stable for $c_\perp>0$, and the modulated phase is now characterized by helical order with $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$. The uniform–modulated transition is now continuous. This does not reproduce the physics of , which has an abrupt transition between and , and so we will henceforth neglect the possibility of a multicomponent order parameter. The schematic phase diagrams for this model are shown in Figure \[fig:phases\].
We will now proceed to derive the *effective strain stiffness tensor* $\lambda$ that results from the coupling of strain to the . The ultimate result, found in , is that $\lambda_\X$ differs from its bare value $C_\X$ only for the symmetry $\X$ of the . Moreover, the effective strain stiffness does not vanish at the unordered–modulated transition, but exhibits a *cusp*. To show this, we will first compute the susceptibility of the , which will both be demonstrative of how the stiffness is calculated and prove useful in expressing the functional form of the stiffness. Then we will compute the strain stiffness using some tricks from functional calculus.
The susceptibility of a single component ($\Bog$ or $\Btg$) to a thermodynamically conjugate field is given by $$\begin{aligned}
&\chi^\recip(x,x')
=\frac{\delta^2F[\eta,\epsilon_\star[\eta]]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\langle\eta\rangle}
=\big[\tilde r-c_\parallel\nabla_\parallel^2 \\
&\qquad\qquad-c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+12u\langle\eta(x)\rangle^2\big]
\delta(x-x'),
\end{aligned}
\label{eq:sus_def}$$ where $\recip$ indicates a *functional reciprocal* in the sense that $$\int dx''\,\chi^\recip(x,x'')\chi(x'',x')=\delta(x-x').$$ Taking the Fourier transform and integrating over $q'$ we have $$\chi(q)
=\big(\tilde r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+D_\perp q_\perp^4
+12u\sum_{q'}\langle\tilde\eta_{q'}\rangle\langle\tilde\eta_{-q'}\rangle\big)^{-1}.$$ Near the unordered–modulated transition this yields $$\begin{aligned}
\chi(q)
&=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
+|\Delta\tilde r|} \\
&=\frac1{D_\perp}\frac{\xi_\perp^4}
{1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2},
\end{aligned}
\label{eq:susceptibility}$$ with $\xi_\perp=(|\Delta\tilde r|/D_\perp)^{-1/4}=\xi_{\perp0}|t|^{-1/4}$ and $\xi_\parallel=(|\Delta\tilde
r|/c_\parallel)^{-1/2}=\xi_{\parallel0}|t|^{-1/2}$, where $t=(T-T_c)/T_c$ is the reduced temperature and $\xi_{\perp0}=(D_\perp/aT_c)^{1/4}$ and $\xi_{\parallel0}=(c_\parallel/aT_c)^{1/2}$ are the bare correlation lengths. Notice that the static susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde
r|)^{-1}$ does not diverge at the unordered–modulated transition. Though it anticipates a transition with Curie–Weiss-like divergence at $\Delta\tilde
r=-D_\perp q_*^4$, this is cut off with a cusp at $\Delta\tilde r=0$. We must emphasize that this is *not* the magnetic susceptibility because a $\Bog$ or $\Btg$ cannot couple linearly to a uniform magnetic field. The object defined in is most readily interpreted as proportional to the two-point connected correlation function $\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$.
The strain stiffness is given in a similar way to the inverse susceptibility: we must trace over $\eta$ and take the second variation of the resulting effective free energy functional of $\epsilon$. Extremizing over $\eta$ yields $$0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}=
\frac{\delta F_\op[\eta]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}-b\epsilon_\X(x),
\label{eq:implicit.eta}$$ which implicitly gives $\eta_\star[\epsilon]$, the optimized conditioned on the strain. Since $\eta_\star$ is a functional of $\epsilon_\X$ alone, only the stiffness $\lambda_\X$ can be modified from its bare value $C_\X$. Though this differential equation for $\eta_*$ cannot be solved explicitly, we can make use of the inverse function theorem. First, denote by $\eta_\star^{-1}[\eta]$ the inverse functional of $\eta_\star$ implied by , which gives the function $\epsilon_\X$ corresponding to each solution of it receives. This we can immediately identify from as $\eta^{-1}_\star[\eta](x)=b^{-1}(\delta F_\op[\eta]/\delta\eta(x))$. Now, we use the inverse function theorem to relate the functional reciprocal of the derivative of $\eta_\star[\epsilon]$ with respect to $\epsilon_\X$ to the derivative of $\eta^{-1}_\star[\eta]$ with respect to $\eta$, yielding $$\begin{aligned}
\bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}\bigg)^\recip
&=\frac{\delta\eta_\star^{-1}[\eta](x)}{\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]}
=b^{-1}\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]}.
\end{aligned}
\label{eq:inv.func}$$ Next, and can be used in concert with the ordinary rules of functional calculus to yield the second variation
$$\begin{aligned}
\frac{\delta^2F[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}
&=C_\X\delta(x-x')-
2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
-b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') +\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}\frac{\delta F_\op[\eta]}{\delta\eta(x'')}\bigg|_{\eta=\eta_\star[\epsilon]}\\
&\qquad\qquad+\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')}\frac{\delta^2F_\op[\eta]}{\delta\eta(x'')\delta\eta(x''')}\bigg|_{\eta=\eta_\star[\epsilon]} \\
&=C_\X\delta(x-x')-
2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
-b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') +\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}(b\epsilon_\X(x''))\\
&\qquad\qquad+b\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')} \bigg(\frac{\partial\eta_\star[\epsilon](x'')}{\partial\epsilon_\X(x''')}\bigg)^\recip\\
&=C_\X\delta(x-x')-
2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
+b\int dx''\,\delta(x-x'')\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x')}
=C_\X\delta(x-x')-b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}.
\end{aligned}
\label{eq:big.boy}$$
The strain stiffness is given by the second variation evaluated at the extremized strain $\langle\epsilon\rangle$. To calculate it, note that evaluating the second variation of $F_\op$ in at $\langle\epsilon\rangle$ (or $\eta_\star(\langle\epsilon\rangle)=\langle\eta\rangle$) yields $$\bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}\bigg)^\recip\bigg|_{\epsilon=\langle\epsilon\rangle}
=b^{-1}\chi^\recip(x,x')+\frac{b}{C_\X}\delta(x-x'),
\label{eq:recip.deriv.op}$$ where $\chi^\recip$ is the susceptibility given by . Upon substitution into and taking the Fourier transform of the result, we finally arrive at $$\lambda_\X(q)
=C_\X-b\bigg(\frac1{b\chi(q)}+\frac b{C_\X}\bigg)^{-1}
=C_\X\bigg(1+\frac{b^2}{C_\X}\chi(q)\bigg)^{-1}.
\label{eq:elastic.susceptibility}$$ Though not relevant here, this result generalizes to multicomponent s. At $q=0$, which is where the stiffness measurements used here were taken, this predicts a cusp in the static strain stiffness $\lambda_\X(0)$ of the form $|\Delta\tilde r|^\gamma$ for $\gamma=1$.
![ Measurements of the effective strain stiffness as a function of temperature for the six independent components of strain from . The vertical lines show the location of the transition. []{data-label="fig:data"}](fig-stiffnesses){width="\columnwidth"}
experiments [@ghosh_single-component_nodate] yield the strain stiffness tensor; the data broken into the irrep components defined in is shown in Figure \[fig:data\]. The $\Btg$ stiffness doesn’t appear to have any response to the presence of the transition, exhibiting the expected linear stiffening with a low-temperature cutoff [@varshni_temperature_1970]. The $\Bog$ stiffness has a dramatic response, softening over the course of roughly $100\,\K$. There is a kink in the curve right at the transition. While the low-temperature response is not as dramatic as the theory predicts, mean field theory—which is based on a small-$\eta$ expansion—will not work quantitatively far below the transition where $\eta$ has a large nonzero value and higher powers in the free energy become important. The data in the high-temperature phase can be fit to the theory , with a linear background stiffness $C_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is shown in Figure \[fig:fit\]. The data and theory appear quantitatively consistent in the high temperature phase.
![ Strain stiffness data for the $\Bog$ component of strain (solid) along with a fit of to the data above $T_c$ (dashed). The fit gives $C_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $b^2/D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp
q_*^4\simeq0.0038\,\K^{-1}$. The failure of the Ginzburg–Landau prediction below the transition is expected on the grounds that the is too large for the free energy expansion to be valid by the time the Ginzburg temperature is reached. []{data-label="fig:fit"}](fig-fit){width="\columnwidth"}
We have seen that the mean-field theory of a $\Bog$ recreates the topology of the phase diagram and the temperature dependence of the $\Bog$ strain stiffness at zero pressure. This theory has several other physical implications. First, the association of a modulated $\Bog$ order with the phase implies a *uniform* $\Bog$ order associated with the phase, and moreover a uniform $\Bog$ strain of magnitude $\langle\epsilon_\Bog\rangle^2=b^2\tilde
r/4uC_\Bog^2$, which corresponds to an orthorhombic phase. Orthorhombic symmetry breaking was recently detected in the phase of using x-ray diffraction, a further consistency of this theory with the phenomenology of [@choi_pressure-induced_2018]. Second, as the Lifshitz point is approached from low pressure, this theory predicts that the modulation wavevector $q_*$ should vanish continuously. Far from the Lifshitz point we expect the wavevector to lock into values commensurate with the space group of the lattice, and moreover that at zero pressure, where the data here was collected, the half-wavelength of the modulation should be commensurate with the lattice spacing $a_3\simeq9.68\,\A$, or $q_*=\pi/a_3\simeq0.328\,\A^{-1}$ [@meng_imaging_2013; @broholm_magnetic_1991; @wiebe_gapped_2007; @bourdarot_precise_2010]. In between these two regimes, the ordering wavevector should shrink by jumping between ever-closer commensurate values in the style of the devil’s staircase [@bak_commensurate_1982]. This motivates future experiments done at pressure, where the depth of the cusp in the $\Bog$ stiffness should deepen (perhaps with these commensurability jumps) at low pressure and approach zero like $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. The presence of spatial commensurability is not expected to modify the critical behavior otherwise [@garel_commensurability_1976].
There are two apparent discrepancies between the orthorhombic strain in the phase diagram presented by [@choi_pressure-induced_2018] and that predicted by our mean field theory. The first is the apparent onset of the orthorhombic phase in the state prior to the onset of . As [@choi_pressure-induced_2018] notes, this could be due to the lack of an ambient pressure calibration for the lattice constant. The second discrepancy is the onset of orthorhombicity at higher temperatures than the onset of . Susceptibility data sees no trace of another phase transition at these higher temperatures [@inoue_high-field_2001], and therefore we don’t expect there to be one. We do expect that this could be due to the high energy nature of x-rays as an experimental probe: orthorhombic fluctuations could appear at higher temperatures than the true onset of an orthorhombic phase.
Three dimensions is below the upper critical dimension $4\frac12$, and so mean field theory should break down sufficiently close to the critical point due to fluctuations, at the Ginzburg temperature [@hornreich_lifshitz_1980; @ginzburg_remarks_1961]. Magnetic phase transitions tend to have Ginzburg temperature of order one. Our fit above gives $\xi_{\perp0}q_*=(D_\perp
q_*^4/aT_c)^{1/4}\simeq2$, which combined with the speculation of $q_*\simeq\pi/a_3$ puts the bare correlation length $\xi_{\perp0}$ at about what one would expect for a generic magnetic transition. The agreement of this data in the $t\sim0.1$–10 range with the mean field exponent suggests that this region is outside the Ginzburg region, but an experiment may begin to see deviations from mean field behavior within around several degrees Kelvin of the critical point. A experiment with more precise temperature resolution near the critical point may be able to resolve a modified cusp exponent $\gamma\simeq1.31$ [@guida_critical_1998], since the universality class of a uniaxial modulated scalar is $\mathrm O(2)$ [@garel_commensurability_1976]. We should not expect any quantitative agreement between mean field theory and experiment in the low temperature phase since, by the point the Ginzburg criterion is satisfied, $\eta$ is order one and the Landau–Ginzburg free energy expansion is no longer valid.
We have preformed a general treatment of phenomenological s with the potential for linear coupling to strain. The possibilities with consistent mean field phase diagrams are $\Bog$ and $\Btg$, and the only of these consistent with zero-pressure data is $\Bog$, with a cusp appearing in the associated stiffness. In this picture, the phase is characterized by uniaxial modulated $\Bog$ order, while the phase is characterized by uniform $\Bog$ order. The corresponding prediction of uniform $\Bog$ symmetry breaking in the phase is consistent with recent diffraction experiments [@choi_pressure-induced_2018]. This work motivates both further theoretical work regarding a microscopic theory with modulated $\Bog$ order, and preforming experiments at pressure that could further support or falsify this idea.
This research was supported by NSF DMR-1719490 and DMR-1719875.
|
---
abstract: 'We describe the late-time acceleration of the Universe within the paradigm of the brane-world scenario. More precisely, we show how a phantom-like behaviour or a crossing of the cosmological constant line can be achieved safely in a dilatonic brane-world model with an induced gravity term on the brane. The brane tension plays the role of dark energy which is coupled to the dilaton bulk scalar field. The phantom mimicry as well as the crossing of the cosmological constant line are achieved without invoking any phantom matter either on the brane or in the bulk.'
author:
- 'Mariam Bouhmadi-López'
title: 'Exploring the dark side of the Universe in a dilatonic brane-world scenario[^1]'
---
Introduction {#sec1}
============
Understanding the recent acceleration of the universe is a challenge and a landmark problem in physics. Its resolution may affect in the short term our understanding of a fundamental interaction like gravity as well as enlarge the framework of particle physics. The smoking gun of the acceleration of the universe (if we assume it is homogeneous and isotropic on large scales) was provided by the analysis of the Hubble diagram of SNe Ia a decade ago [@Perlmutter:1998np]. This discovery, together with (i) the measurement of the fluctuations in the CMB which implied that the universe is (quasi) spatially flat and (ii) that the amount of matter which clusters gravitationally is much less than the critical energy density, implied the existence of a “dark energy component” that drives the late-time acceleration of the universe. Subsequent precision measurements of the CMB anisotropy by WMAP [@Spergel:2003cb] and the power spectrum of galaxy clustering by the 2dFGRS and SDSS surveys [@Cole:2005sx; @Tegmark:2003uf] have confirmed this discovery.
A plethora of different theoretical models have been so far proposed to explain this phenomenon [@Copeland], although unfortunately none of the models advanced so far is both completely convincing and well motivated. A cosmological constant corresponding to roughly two thirds of the total energy density of the universe is perhaps the simplest [*phenomenological*]{} way to explain the late-time speed up of the universe –and in addition match rather well the observational data. However, the expected theoretical value of the cosmological constant is about 120 orders of magnitude larger than the value needed to fit the data [@Durrer:2007re].
Alternative approaches to explain the late-time acceleration invoke (i) a dark energy component in the universe which would provide a negative pressure or (ii) an infrared modification of general relativity on large scales (like in some brane-world scenarios [@Dvali:2000hr] or f(R) models [@frmodels]) which, by weakening the gravitational interaction on those scales, allows the recent speed up of the universal expansion. The second approach is also motivated by the fact that we only have precise measurements of gravity from sub-millimiter scales up to solar system scales while the Hubble radius, which is the scale relevant for the cosmic acceleration, is many orders of magnitude larger.
A pioneering scheme in the second approach is the Dvali, Gabadadze and Porrati (DGP) model [@Dvali:2000hr] which corresponds to a 5-dimensional (5D) induced gravity brane-world model [@Deffayet; @IG; @Sahni:2002dx; @LDGP2; @Bouhmadi-Lopez:2004ys], where a low-energy modification occurs with respect to general relativity; i.e. an infrared effect takes place, leading to two branches of solutions: (i) the self-accelerating branch and (ii) the normal branch.
The self-accelerating branch solution gives rise to an asymptotically de Sitter brane; i.e. a late-time accelerating brane universe. The acceleration of the brane expansion arises naturally, i.e. without invoking the presence of any dark energy on the brane to produce the speed-up. Most importantly, it has recently been shown that by embedding the DGP model in a higher dimensional space-time the ghost problem in the original model [@Koyama:2007za] may be cured [@deRham:2007xp] while preserving the existence of a self-accelerating solution [@Minamitsuji:2008fz].
The normal branch also constitutes in itself an extremely interesting physical setup of the DGP model however, as it can mimic a phantom behaviour on the brane by means of the $\Lambda$DGP scenario [@Sahni:2002dx]. We would like to highlight that observational data do not seem incompatible with a phantom-like behaviour [@Percival:2007yw] and therefore we should keep an open mind about what is producing the recent inflationary era of our universe. Furthermore, this phantom-like behaviour may well be a property acquired only recently by dark energy. This leads to an interest in modelling the so called crossing of the phantom divide line $w=-1$; for example in the context of the brane-world scenario [@LDGP2; @crossing; @BouhmadiLopez:2008bk]. The most important aspect of the $\Lambda$DGP model is that the phantom-like mimicry is obtained without invoking any real phantom-matter [@phantom] which is known to violate the null energy condition and induce quantum instabilities[^2] [@Cline:2003gs]. In the DGP scenario it is as well possible to get a mimicry of the crossing of the phantom divide, however, at the cost of invoking a dynamical dark energy on the brane [@LDGP2], for example modelled by a quiessence fluid or a (generalised) Chaplygin gas.
One aim of this paper is to show that a dilatonic brane-world model with an induced gravity term in the brane can mimic a phantom-like behaviour without including matter on the brane that violates the null energy condition. A second aim of this paper is to show that there is an alternative form (to the one introduced in [@Sahni:2002dx; @LDGP2]) of mimicking the crossing the cosmological constant line $w=-1$ in the brane-world scenario. More precisely, we consider a 5D dilatonic bulk with a brane endowed with an induced gravity term, a brane matter content corresponding to cold dark matter, and a brane tension $\lambda$ that depends on the minimally coupled bulk scalar field. We will show that in this set-up the vacuum generalised self-accelerating branch expands in a super-accelerating way and mimics a phantom-behaviour. On the other hand, the generalised normal branch expands in an accelerated way due to $\lambda$ playing the role of *dark energy* –through its dependence on the bulk scalar field. Furthermore, in this case, it turns out that the brane tension grows with the brane scale factor until it reaches a maximum positive value and then starts decreasing. Therefore, in our model the brane tension mimics a crossing of the phantom divide. Most importantly no matter violating the null energy density is invoked in our model.
The paper is organised as follows. In section 2, we present our dilatonic brane-world model with induced gravity. The bulk scalar field potential is an exponential potential. The matter content of the brane is coupled to the dilaton field. We deduce the modified Friedmann equation for both branches, the junction condition of the dilaton across the brane, which constrains the brane tension, and the energy balance on the brane. In section 3, we then analyse the vacuum (i.e., $\rho_{m}=0$) solutions in both branches. We show that the brane tension has a phantom-like behaviour on the generalised self-accelerating branch in the sense that the brane tension grows as the brane expands. In this branch, the brane hits a singularity in its future evolution which may be interpreted as a “big rip” singularity pushed towards an infinite cosmic time. Then, in section 4, we show that, under some assumptions on the nature of the coupling parameters between $\lambda$ and $\phi$, $1+w_{\rm{eff}}$ changes sign as the normal brane evolves, with $w_{\rm{eff}}$ the effective equation of state for the brane tension. Our conclusions are presented in section 5.
The framework {#sec2}
=============
We consider a brane, described by a 4D hyper-surface ($h$, metric g), embedded in a 5D bulk space-time ($\mathcal{B}$, metric $g^{(5)}$), whose action is given by $$\begin{aligned}
\mathcal{S} = \,\,\, \frac{1}{\kappa_5^2}\int_{\mathcal{B}} d^5X\, \sqrt{-g^{(5)}}\;
\left\{\frac{1}{2}R[g^{(5)}]\;+\;\mathcal{L}_5\right\}%\nonumber \\
+ \int_{h} d^4X\, \sqrt{-g}\; \left\{\frac{1}{\kappa_5^2} K\;+\;\mathcal{L}_4 \right\}\,, \label{action}\end{aligned}$$ where $\kappa_5^2$ is the 5D gravitational constant, $R[g^{(5)}]$ is the scalar curvature in the bulk and $K$ the extrinsic curvature of the brane in the higher dimensional bulk, corresponding to the York-Gibbons-Hawking boundary term.
We consider a dilaton field $\phi$ living on the bulk [@Chamblin:1999ya; @Maeda:2000wr] and we choose $\phi$ to be dimensionless. Then, the 5D Lagrangian $\mathcal{L}_5$ can be written as $$\begin{aligned}
\mathcal{L}_5=-\frac12 (\nabla\phi)^2 -V(\phi).\end{aligned}$$
The 4D Lagrangian $\mathcal{L}_4$ corresponds to $$\mathcal{L}_4= \alpha {R}[g] -\lambda(\phi)+\Omega^{4}\mathcal{L}_m(\Omega^2 g_{\mu\nu}).%e^{4b\phi}\mathcal{L}_m(e^{2b\phi}g_{\mu\nu}),
\label{L4}$$ The first term on the right hand side (rhs) of the previous equation corresponds to an induced gravity term [@Dvali:2000hr; @Deffayet; @IG; @Sahni:2002dx], where $R[g]$ is the scalar curvature of the induced metric on the brane and $\alpha$ is a positive parameter which measures the strength of the induced gravity term and has dimensions of mass squared. The term $\mathcal{L}_m$ in Eq. (\[L4\]) describes the matter content of the brane and $\lambda(\phi)$ is the brane tension, and we will restrict ourselves to the case where they are homogeneous and isotropic on the brane. We allow the brane matter content to be non-minimally coupled on the (5D) Einstein frame but to be minimally coupled respect to a conformal metric ${\tilde g}^{(5)}_{AB}=\Omega^2\;g^{(5)}_{AB}$, where $\Omega=\Omega(\phi)$ [@Maeda:2000wr].
We are interested in the cosmology of this model. It is known that for an expanding FLRW brane the unique bulk space-time in Einstein gravity (in vacuum) is a 5D Schwarzschild-anti de Sitter space-time. This property as far as we know cannot be extended to a 5D dilatonic bulk. On the other hand, the presence of an induced gravity term in the brane-world scenario affects only the dynamics of the brane, through the junction conditions at the brane, and does not affect the bulk field equations. Therefore, in order to study the effect of an induced gravity term in a brane-world dilatonic model, it is possible to consider a bulk corresponding to a dilatonic 5D space-time and later on impose the junction conditions at the brane location. The junction conditions will then determine the cosmological evolution of the brane and constrain the brane tension. This is the approach we will follow.
From now on, we consider a 5D dilatonic solution obtained by Feinstein et al [@Feinstein:2001xs; @Kunze:2001ji] *without an induced gravity term on the brane*. The 5D dilatonic solution reads [@Kunze:2001ji] $$ds^2_5=\frac{1}{\xi^2}r^{2/3(k^2-3)}dr^2 +r^2(-d{t}^2+\gamma_{ij}dx^idx^j),
\label{bulkmetric}$$ where $\gamma_{ij}$ is a 3D spatially flat metric. The bulk potential is $$V(\phi)=\Lambda\exp[-(2/3) k\phi].
\label{liouville}$$ The parameters $k$ and $\xi$ in Eq. (\[bulkmetric\]) define the 5D cosmological constant $\Lambda$ $$\Lambda= \frac12 (k^{2}-12)\xi^2.$$ The 5D scalar field grows logarithmically with the radial coordinate $r$ [@Kunze:2001ji] $$\phi=k\log (r). \label{phi}$$ Now, we consider a FLRW brane filled only with cold dark matter (CDM); i.e pressureless matter, and the brane tension $\lambda(\phi)$. On the other hand, the brane is considered to be embedded in the previous 5D dilatonic solution and its trajectory in the bulk is described by the following parametrisation $${t}={t}(\tau),\,\,\,\, r=a(\tau),\,\,\,\, x_i= constant,\,\, i=1 \ldots 3.$$ Here $\tau$ corresponds to the brane proper time. Then the brane metric reads $$ds^2_4\,=\,g_{\mu\nu}\,dx^{\mu}dx^{\nu}\,=\,-d\tau^2+a^2(\tau)\gamma_{ij}dx^idx^j.$$ For an induced gravity brane-world model [@Deffayet; @Bouhmadi-Lopez:2004ys], there are two physical ways of embedding the brane in the bulk when a $\mathbb{Z}_2$-symmetry across the brane is assumed: the generalised normal branch[^3] and the generalised self-accelerating branch. For example, in the first case the brane is moving in the bulk away from the bulk naked singularity located at $r=0$ [@Bouhmadi-Lopez:2004ys].
For simplicity, we will consider that the matter content of the brane is minimally coupled respect to the conformal metric ${\tilde g}^{(5)}_{AB}=\exp(2b\phi)\;g^{(5)}_{AB}$; i.e. $\Omega=\exp(b\phi)$, where $b$ is a constant. We will also consider only the case[^4] $k > 0$; i.e. the scalar field is a growing function of the coordinate $r$. Then, the Israel junction condition at the brane [@Chamblin:1999ya] describes the cosmological evolution of the brane through the modified Friedmann equation, which in our case reads $$\sqrt{{\xi^2}{a^{-\frac23 k^2}}+H^2} =-\epsilon\,\frac{\kappa_5^2}{6}
\left[\lambda(\phi)+\rho_m-6\alpha H^2\right],
\label{Friedmann1}$$ where $\epsilon=1$ for the self-accelerating brane and $\epsilon=-1$ for the normal branch. The modified Friedmann equation can be more conveniently expressed as $$\begin{aligned}
H^2=%&&
\frac{1}{6\alpha}\left\{\lambda+\rho_m%\right.\label{Friedmann}\\
%&&\hspace*{-0.9cm}+\left.
+\frac{3}{\kappa_5^4\alpha}
\left[1+\epsilon \sqrt{1+4\kappa_5^4\alpha^2\xi^2a^{-2k^2/3}+\frac{2}{3}\kappa_5^4\alpha(\lambda+\rho_m)}\right]\right\},%\nonumber
\label{Friedmann}\end{aligned}$$ where $\lambda$ is the brane tension and $\rho_m$ is the energy density of CDM.
On the other hand, as it is usual in a dilatonic brane-world scenario, matter on the brane –in this case CDM– is not conserved due to the coupling $\Omega$ (see Eq. (\[L4\])). In fact, we have $$\dot\rho_m=-3H\left(1-\frac13 kb\right)\rho_m,
\label{conservationrho}$$ where a dot stands for derivatives respect to the brane cosmic time $\tau$. Therefore, CDM on the brane scales as $$\rho_m=\rho_0 a^{-3+kb}.
\label{dust}$$
Finally, the junction condition of the scalar field at the brane [@Chamblin:1999ya] constrains the brane tension $\lambda(\phi)$. In our model this is given by $$a\frac{d\lambda}{da}=-kb\rho_m\, {+\epsilon}\frac{2k^2}{\kappa_5^2}\sqrt{{\xi^2}{a^{-\frac23 k^2}}+H^2},
\label{constraint}$$ where for convenience we have rewritten the scalar field (valued at the brane) in terms of the scale factor of the brane. At this respect we remind the reader that at the brane $\phi=k\log(a)$.
Vacuum solutions {#sec3}
================
The vacuum solutions, i.e. in absence of matter on the brane, depends crucially on the embedding of the brane in the bulk, therefore, which branch we are considering.
The self-accelerating branch
----------------------------
For the vacuum self-accelerating branch; i.e. $\epsilon=1$ and $\rho_m=0$, the brane tension is an increasing function of the scale factor of the brane[^5] (see Eq. (\[constraint\])). For small values of the scale factor, the brane tension reaches infinite negative values. On the other hand, for very large value of the scale factor the brane tension approaches infinite positive values. Therefore, when the brane tension acquires positive values, it mimics a phantom energy component in a standard FLRW universe. We remind at this respect that we have not included any matter that violates the null energy condition; i.e. any explicit phantom energy in the model.
The Hubble parameter is an increasing function of the scale factor, i.e the brane super-accelerates. In fact, $$\dot H=-\frac{ k^2 H^2}{\kappa_5^4\alpha(\lambda-6\alpha H^2)+3}, \label{Rayeq}$$ while the modified Friedmann equation (\[Friedmann\]) implies that the denominator of the previous equation has to be negative (see also footnote \[footnote3new\]), consequently $\dot H >0$. At small scale factors, $H$ reaches a constant positive value. Therefore, in the vacuum self-accelerating brane there is no big bang singularity on the brane; indeed, the brane is asymptotically de Sitter. On the other hand, at very large values of the scale factor, the Hubble parameter diverges.
The divergence of the Hubble parameter for very large values of the scale factor might point out the existence of a big rip singularity in the future evolution of the brane; i.e. the scale factor and Hubble parameter blow up in a finite cosmic time in the future evolution of the brane. However, it can be shown that the divergence of $H$ and $a$ (and also of $\lambda$) occur in an infinite cosmic time in the future evolution of this branch. This can be easily proven by noticing that the asymptotic behaviour of the Hubble parameter at large value of the scale factor is $$H\sim\frac{k^2}{\kappa_5^2 \alpha}\ln (a).$$ Consequently, the Hubble rate does not grow as fast as in phantom energy models with a constant equation of state where a big rip singularity takes place on the future evolution of a homogeneous and isotropic universe [@phantom].
In summary, we have proven that in the vacuum self-accelerating branch the brane tension mimics a phantom behaviour. On the other hand, there is a singularity in the future evolution of the brane. The singularity is such that for large value of the cosmic time, the scale factor and the Hubble parameter diverge. This kind of singularity can be interpreted as a “big rip” singularity pushed towards an infinite cosmic time of the brane.
The normal branch
-----------------
For the vacuum normal branch; i.e. $\epsilon=-1$ and $\rho_m=0$, the brane tension is a decreasing function of the scale factor of the brane[^6] (see Eq. (\[constraint\])). For small values of the scale factor, the brane tension reaches infinite positive values. On the other hand, for very large value of the scale factor the brane tension vanishes.
The Hubble parameter is a decreasing function of the scale factor, i.e the brane is never super-accelerating. In fact, Eq. (\[Rayeq\]) and the Israel junction condition (\[Friedmann1\]) implies that the denominator of the previous equation has to be positive (see also footnote \[footnote3\]), therefore $\dot H<0$. At high energy, $H$ reaches a constant positive value. Consequently, in the vacuum brane there is no big bang singularity on the brane; indeed, the brane is asymptotically de Sitter. On the other hand, at very large values of the scale factor, the Hubble parameter vanishes (the brane is asymptotically Minkowski in the future). Although the brane never super-accelerates, the brane always undergoes an inflationary period.
The brane behaves in two different ways depending on the value taken by $k^2$ (cf. Fig. \[inflation2\]). Thus, for $k^2\leq 3$ the brane is eternally inflating. A similar behaviour was found in [@Kunze:2001ji]. On the other hand, for $k^2>3$ the brane undergoes an initial stage of inflation and later on it starts decelerating. This second behaviour contrasts with the results in [@Kunze:2001ji] for a vacuum brane without an induced gravity term on the brane. Then, the inclusion of an induced gravity term on a dilatonic brane-world model with an exponential potential in the bulk allows for the normal branch to inflate in a region of parameter space where the vacuum dilatonic brane alone would not inflate. This behaviour has some similarity with steep inflation [@Copeland:2000hn], where high energy corrections to the Friedmann equation in RS scenario [@Randall:1999vf] permit an inflationary evolution of the brane with potentials too steep to sustain it in the standard 4D case, although the inflationary scenario introduced by Copeland et al in [@Copeland:2000hn] is supported by an inflaton confined in the brane while in our model inflation on the brane is induced by a dilaton field on the bulk.
![This figure shows the behaviour of the dimensionless acceleration parameter given by $\alpha^2\kappa_5^4\ddot{a} /a$ as a function of the time (see the left hand side arrow). The solid (darker grey), dotted and dashed-dotted (lighter grey) lines correspond to the acceleration parameter for $k^2=2,3,15$ respectively. For $k^2= 2,3$ the negative branch is eternally inflating. On the other hand, for $k^2= 15$ the brane undergoes an initial transient inflationary epoch.[]{data-label="inflation2"}](figura2-final.eps){width="50.00000%"}
Crossing the cosmological constant line {#sec4}
=======================================
We now address the following question: is it possible to mimic a crossing of the phantom divide in particular in the model introduced in section \[sec2\]? Unlike the vacuum case –which can be solved analytically [@Bouhmadi-Lopez:2004ys]– in this case we cannot exactly solve the constraint (\[constraint\]). Nevertheless, we can answer the previous question based in some physical and reasonable assumptions and as well as on numerical methods. For simplicity, we will restrict to the normal branch.
In order to answer the previous question, it is useful to introduce the following dimensionless quantities $$\begin{aligned}
&&\bar\lambda\equiv\frac23\kappa_5^4\alpha\lambda,\,\, x\equiv \frac23 k\phi-\ln d, \,\, d\equiv 4\alpha^2\kappa_5^4 \xi^2,\,\, m\equiv 3-kb,\nonumber\\
&&\beta_0\equiv\frac{9\beta_2}{2k^2},\,\,\beta_1\equiv\frac{2\kappa_5^4\alpha}{m}\rho_0d^{-\beta_0},\,\, \beta_2\equiv\frac{m}{3}. \label{def}\end{aligned}$$ In terms of these new variables, the constraint given in Eq. (\[constraint\]) reads $$\begin{aligned}
\frac{d\bar\lambda}{dx}=1-\beta_0\beta_1(1-\beta_2)e^{-\beta_0x}-
\sqrt{1+\bar\lambda+e^{-x}+\beta_1\beta_2 e^{-\beta_0x}}.
\label{dvdx}\end{aligned}$$
![The figure shows the effective equation of state of the brane tension defined in Eq. (\[eqstate\]) against the variable $x$ defined in Eq. (\[def\]). Notice that $x$ grows as the brane expands and therefore $dx/d\tau>0$ where $\tau$ corresponds to the cosmic time of the brane. This illustrative numerical solution has been obtained for $b=-1$, $k=1$ and $\beta_1=1$. The last parameter is defined in Eq. (\[def\]). In order to impose the right initial condition, we started the integration well in the past where CDM dominated over the scalar field on the brane and we took as a good approximated solution the dark matter solution given in Eq. (\[asymlambda11\]).[]{data-label="efectomenweff"}](weffplot.eps){width="45.00000%"}
The assumptions we make are the following:
1. We assume that CDM dominates over the vacuum term ($a^{-2/3
k^2}$) at early times on the brane. This implies that the parameter $\beta_0$ introduced in Eq. (\[def\]) must satisfy $\beta_0 > 1$. On the other hand, the brane tension will play the role of dark energy (through its dependence on the scalar field) in our model. This first assumption assumes that dark matter dominates over dark energy at high redshift which is a natural assumption to make. Indeed, at high redshift the brane tension would scale as
$$\bar\lambda\sim\beta_1(1-\beta_2)e^{-\beta_0 x} + \ldots.
\label{asymlambda11}$$
2. We also assume that CDM redshifts away a bit faster than usual; i.e. $bk<0$ or $\beta_2$ introduced in Eq. (\[def\]) is such that $\beta_2 > 1$. This lost energy will be used to increase the value of the scalar field $\phi(a)$ on the brane. That is, to push the brane to higher values of $a$.
3. Finally, we also assume that $\beta_2< 2\beta_0(\beta_2-1)$. This condition, together with $\beta_0, \beta_2 > 1$, is sufficient to prove the non existence of a local minimum of the brane tension during the cosmological evolution of the brane. In fact, we can show the existence of a unique maximum for an even larger set of parameters $\beta_0>1/2$, $\beta_2 > 1$ and $\beta_2< 2\beta_0(\beta_2-1)$. Therefore, the set of allowed parameter $k$ and $b$ that fulfil the last three inequalities are such that $$k< {\rm min}\left\{-3b,\frac32\left[-b+\sqrt{b^2+4}\right]\right\}=-3b.$$ where $b$ is positive.
Under these three assumptions, it can be shown that the brane tension has a local maximum which must be positive (we refer the reader to [@BouhmadiLopez:2008bk] for a detailed proof). In fact, what happens under these conditions is that the brane tension increases until it reaches its maximum positive value and then it starts decreasing. It is precisely at this maximum that the brane tension mimics crossing the phantom divide. Around the local maximum of the brane tension we can always define an effective equation of state in analogy with the standard 4D relativistic case:
$$1+w_{\textrm{eff}}=-\frac{1}{3H}\frac{1}{\lambda}\frac{d\lambda}{d\tau}.
\label{eqstate}$$
As we mentioned earlier, the constraint equation (\[constraint\]) cannot be solved analytically and therefore we have to resort to numerical methods. We show in Fig. \[efectomenweff\] an example of our numerical results where it can be seen clearly that $1+w_{\textrm{eff}}$ changes sign. It is precisely at that moment that the crossing takes place.
![The figure shows the deceleration parameter $q=-\ddot a a/\dot a^2$ against the variable $x$ defined in Eq. (\[def\]). The brane is accelerating in the future when $q$ is negative. Notice that $x$ grows as the brane expands and therefore $dx/d\tau>0$ where $\tau$ corresponds to the cosmic time of the brane. This numerical example has been obtained for $b=-1$, $k=1$ and $\beta_1=1$. The last parameter is defined in Eq. (\[def\]). Again in order to impose the right initial condition, we started the integration well in the past where CDM dominated over the scalar field on the brane and we can take as a good approximated solution the dark matter solution given in Eq. (\[asymlambda11\]).[]{data-label="efectomenq"}](qplot.eps){width="45.00000%"}
Another important question to address is whether the brane is accelerating at the time that the crossing takes place. We know that the vacuum term dominates at late times (see the first assumption). Thus, at that point the brane tension will be adequately described by the vacuum solution; i.e. $$\bar\lambda\sim C\exp(-x/2) + \ldots\,,\quad C=constant\,>0.
\label{asymlambda21}$$ The constant $C$ is positive because for the vacuum solution the brane tension is always positive [@Bouhmadi-Lopez:2004ys]. Now, from the results in the previous section, we can conclude that the brane will be speeding up at late times as long as $k^2\leq3$. On the other, hand it can be checked numerically that the brane can be accelerating at the crossing as Fig. \[efectomenq\] shows.
Conclusions
===========
In this paper we analyse the behaviour of dilatonic brane-world models with an induced gravity term on the brane with a constant induced gravity parameter. We assume a $\mathbb{Z}_2$-symmetry across the brane. The dilatonic potential is an exponential function of the bulk scalar field and the matter content of the brane is coupled to the dilaton field. We deduce the modified Friedmann equation for the generalised self-accelerating and generalised normal branch (which specifies the way the brane is embedded in the bulk), the junction condition for the scalar field across the brane and the energy balance on the brane.
We describe the vacuum solutions; i.e. the matter content of the brane is specified by the brane tension, for a FLRW brane:
1. In the vacuum self-accelerating branch, the brane tension is a growing function of the scale factor and, consequently, mimics the behaviour of a phantom energy component on the brane. This phantom-like behaviour is obtained without including a phantom fluid on the brane. In fact, the brane tension does not violate the null energy condition. The expansion of the brane is super-inflationary; i.e. the Hubble parameter is a growing function of the cosmic time. At high energy (small scale factors), the brane is asymptotically de Sitter. The brane faces a curvature singularity in its infinite future evolution, where the Hubble parameter, brane tension and scale factor diverge. The singularity happens in an infinite cosmic time. Therefore, the singularity can be interpreted as a “big rip” singularity pushed towards an infinite future cosmic time.
2. On the other hand, in the vacuum normal branch, the brane tension is a decreasing function of the scale factor. Unlike the positive branch, the branch is not super-inflating. However, it always undergoes an inflationary expansion (see Fig. \[inflation2\]). The inflationary expansion can be eternal ($k^2\leq 3$) or transient ($k^2> 3$), where $k$ is related to the slope of bulk scalar field. For large values of the scale factor, the negative branch is asymptotically Minkowski.
Furthermore, we have shown the existence of a mechanism that mimics the crossing of the cosmological constant line $w=-1$ in the brane-world scenario introduced in section 2, and which is different from the one introduced in Refs. [@Sahni:2002dx; @LDGP2]. More precisely, we have shown that if we consider the 5D dilatonic bulk with an induced gravity term on the normal branch, a brane tension $\lambda$ which depends on the minimally coupled bulk scalar field, and a brane matter content corresponding only to cold dark matter, then under certain conditions the brane tension grows with the brane scale factor until it reaches a maximum positive value at which it mimics crossing the phantom divide, and then starts decreasing. Most importantly no matter violating the null energy condition is invoked in our model. Despite the transitory phantom-like behaviour of the brane tension no big rip singularity is hit along the brane evolution (unlike the vacuum self-accelerating branch).
In this model for the normal branch or non-self-accelerating branch, the constraint equation fulfilled by the brane tension is too complicated to be solved analytically (see Eqs. (\[Friedmann\]) and (\[constraint\])). However, we have shown that under certain physical and mathematical conditions -cold dark matter dominates at higher redshifts and it dilutes a bit faster than dust during the brane expansion as well as a mathematical condition that guarantees the non-existence of a local minimum of the brane tension- it is possible for the brane tension to cross the cosmological constant line. The analytical proof has been confirmed by numerical solutions. Furthermore, we have shown that for some values of the parameters the normal branch inflates eternally to the future due to the brane tension $\lambda$ playing the role of dark energy through its dependence on the bulk scalar field.
In summary, in the models presented here the mimicry of a phantom-like behaviour and the phantom divide crossing is based on the interaction between the brane and the bulk through the brane tension that depends explicitly on the scalar field that lives in the bulk. We have also shown that in both cases the brane undergoes a late-time acceleration epoch.
Acknowledgements {#acknowledgements .unnumbered}
================
I am grateful to Prof P. M. Lavrov and Prof. V. Ya. Epp for a kind invitation to submit this article to the anniversary volume *The Problems of Modern Cosmology*, on the occasion of the 50th birthday of . I am also grateful to my collaborator A. Ferrera for collaborations upon which some of the work presented here is based. MBL is supported by the Portuguese Agency Fundação para a Ciência e Tecnologia through the fellowship SFRH/BPD/26542/2006.
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[^1]: Contribution to the anniversary volume “*The Problems of Modern Cosmology*”, on the occasion of the 50th birthday of . Editor: Prof. P. M. Lavrov, Tomsk State Pedagogical University.
[^2]: We are referring here to a phantom energy component described through a minimally coupled scalar field with the wrong kinetic term.
[^3]: We will refer to the normal DGP branch also as the non-self-accelerating DGP branch.
[^4]: The main conclusions do not depend on the sign of $k$ but on the sign of the parameter $kb$. Therefore, we can always describe the same physical situation on the brane for $k<0$ by changing the sign of $b$.
[^5]: The constraint equation (\[constraint\]) (after substituting the Hubble rate given in Eq. (\[Friedmann\])) can be solved analytically in this case [@Bouhmadi-Lopez:2004ys] and it can be explicitly shown that the brane tension increases as the brane expands. In the same way a parametric expression can be found for the Hubble rate and its cosmic derivative. \[footnote3new\]
[^6]: The constraint equation (\[constraint\]) (after substituting the Hubble rate given in Eq. (\[Friedmann\])) can be solved analytically in this case [@Bouhmadi-Lopez:2004ys] and it can be explicitly shown that the brane tension decreases as the brane expands. In the same way a parametric expression can be found for the Hubble rate and its cosmic derivative. \[footnote3\]
|
---
abstract: 'Existing neural machine translation (NMT) systems utilize sequence-to-sequence neural networks to generate target translation word by word, and then make the generated word at each time-step and the counterpart in the references as consistent as possible. However, the trained translation model tends to focus on ensuring the accuracy of the generated target word at the current time-step and does not consider its future cost which means the expected cost of generating the subsequent target translation (i.e., the next target word). To respond to this issue, we propose a simple and effective method to model the future cost of each target word for NMT systems. In detail, a time-dependent future cost is estimated based on the current generated target word and its contextual information to boost the training of the NMT model. Furthermore, the learned future context representation at the current time-step is used to help the generation of the next target word in the decoding. Experimental results on three widely-used translation datasets, including the WMT14 German-to-English, WMT14 English-to-French, and WMT17 Chinese-to-English, show that the proposed approach achieves significant improvements over strong Transformer-based NMT baseline.'
author:
- 'Chaoqun Duan$^1$[^1]'
- Kehai Chen$^2$
- Rui Wang$^2$
- Masao Utiyama$^2$
- |
\
Eiichiro Sumita$^2$
- |
Conghui Zhu$^1$Tiejun Zhao$^1$ $^1$Harbin Institute of Technology, Harbin, China\
$^2$National Institute of Information and Communications Technology (NICT), Kyoto, Japan\
cqduan@stu.hit.edu.cn, {conghui, tjzhao}@hit.edu.cn\
{khchen, wangrui, mutiyama, eiichiro.sumita}@nict.go.jp
bibliography:
- 'ijcai20.bib'
title: Modeling Future Cost for Neural Machine Translation
---
[UTF8]{}[gbsn]{}
Introduction {#Intro}
============
The future cost estimation plays an important role in traditional phrase-based statistical machine translation (PBSMT) [@koehn2009statistical]. Typically, it utilizes the pre-learned translation knowledge (i.e., translation model and language model) to compute a future cost of any span of input words in advance for one source sentence. The computed future cost estimates how hard it is to translate the untranslated part of the source sentence. For example, for all translation options that have the same number of input words, the higher future cost means that the untranslated part of the source sentence is more difficult to be translated. During the decoding, PBSMT adds up the partial translation probability score of the current span and its future cost to measure the quality of each translation option. As a result, a (or several) translation hypothesis, which is extended by translation options with cheaper future cost, is remained in the beam-search stack as the best paths to generate subsequent translation.
Neural machine translation (NMT) systems [@bahdanau2014neural; @vaswani2017attention] often utilize sequence-to-sequence neural networks to model translation between the source language and the target language, and achieve state-of-the-art performance on most of the translation tasks [@barrault-etal-2019-findings]. Compared with the traditional PBSMT, NMT systems model translation knowledge through neural networks. This means that there is no need to learn large-scale translation rules as traditional PBSMT. However, lack of translation rules prevents the future cost from being estimated in advance for NMT systems. Therefore, it is difficult to directly use this effective future cost mechanism in PBSMT to enhance the beam-search stack decoding in NMT systems.
In addition, the NMT systems generally model translation between a source language and a target language in an auto-regressive way, that is, based on the previously translated target word (or context) and the source representation to generate target translation word by word. However, this makes the trained translation model only focus on ensuring the accuracy of the generated target word at the current time-step and do not consider its future cost as PBSMT. In other words, there is no mechanism to estimate the future cost of the current generated target word for generating subsequent target translation (i.e., next target word) in NMT systems.
In this paper, we propose a future cost mechanism to learn the expected cost of generating the next target word for NMT systems, for example, state-of-the-art Transformer-based NMT system [@vaswani2017attention]. Specifically, the future cost is dynamically estimated based on the current target word and its contextual representation instead of pre-estimated in PBSMT. We then use the estimated future cost to compute an additional loss item to boost the training of the Transformer-based NMT model. This allows the Transformer-based NMT model to preview the future cost of the current generated target word for the generation of the target word at the next time-step. In addition, the learned future context representation at the current time-step is further used to help the generation of the next target word in the decoding. This allows the future cost information to be applied to the beam-search stack decoding in the auto-regressive way instead of in the isolation way in PBSMT, and thereby enhances translation performance of Transformer-based NMT model.
This paper primarily makes the following contributions:
- It introduces a novel future cost mechanism to estimate the impact of the current generated target word for generating subsequent target translation (i.e., next target word) in NMT.
- The proposed two models can integrate the proposed future cost mechanism into the state-of-the-art Transformer-based NMT system to improve translation performance.
- Experiment results on the WMT14 English-to-German, WMT14 English-to-French, and WMT17 Chinese-to-English translation tasks verify the effectiveness and universality of the proposed future cost mechanism.
Background
==========
An advanced Transformer-based NMT model [@vaswani2017attention], which solely relies on self-attention networks (SANs), generally consists of a SAN-based encoder and a SAN-based decoder. Formally, given an source input sequence $\textbf{x}$={$x_1$, $\cdots$, $x_J$} with length of $J$, this encoder is adopted to encode the source input sequence $\textbf{x}$. In particular, each layer includes an SAN sub-layer $\textup{SelfATT}(\cdot)$ and a position-wise fully connected feed-forward network sub-layer $\textup{FFN}(\cdot)$. A residual connection [@he2016deep] is applied between the SAN sub-layer and the FFN syb-layer, followed by layer normalization $\textup{LN}(\cdot)$ [@ba2016layer]. Thus, the output of the first sub-layer $\textbf{C}_{e}^{n}$ and the second sub-layer $\textbf{H}_{e}^{n}$ are sequentially calculated as Eq. and Eq.: $$\begin{aligned}
\label{eq1:encoder_att} \textbf{C}_{e}^{n}&=\textup{LN}(\textup{SelfATT}(\textbf{H}_{e}^{n-1})+\textbf{H}_{e}^{n-1}), \\
\label{eq2:encoder_fnn}
\textbf{H}_{e}^{n}&=\textup{LN}(\textup{FFN}(\textbf{C}_{e}^{n})+\textbf{C}_{e}^{n}).\end{aligned}$$ Typically, this encoder is composed of a stack of $N$ identical layers. As a result, $\textbf{H}_{e}^{N}$ is the final source sentence representation to model translation.
Furthermore, this decoder, which is also composed of a stack of $N$ identical layers, models the context information for predicting translations. In addition to two sub-layers in each decoder layer, the decoder inserts a third sub-layer $\textup{ATT}(\textbf{C}_{i}^{n}, \textbf{H}_{e}^{N})$ perform attention over the output of the encoder $\textbf{H}_{e}^{N}$: $$\begin{aligned}
\label{eq3:decoder_self_att} \textbf{C}_{i}^{n}&=\textup{LN}(\textup{SelfATT}(\textbf{H}_{i}^{n-1})+\textbf{H}_{i}^{n-1}), \\
\label{eq4:decoder_att} \textbf{D}_{i}^{n}&=\textup{LN}(\textup{ATT}(\textbf{C}_{i}^{n}, \textbf{H}_{e}^{N})+\textbf{C}_{i}^{n}), \\
\label{eq5:decoder_fnn} \textbf{H}_{i}^{n}&=\textup{LN}(\textup{FFN}(\textbf{D}_{i}^{n})+\textbf{D}_{i}^{n}).\end{aligned}$$ At the $i$-th time-step, the top layer of the decoder $\textbf{H}_{i}^{N}$ is then used to generate the target word $y_i$ by a linear, potentially multi-layered function (or a softmax function): $$P(y_i|\textbf{y}_{<i}, \textit{x}) \propto
\textup{exp}(\textbf{\textit{W}}_\textit{o}\textup{tanh}(\textbf{\textit{W}}_\textit{w}\textbf{H}_{i}^{N}),
\label{eq6:Probabilities_SANs}$$ where $\textbf{\textit{W}}_{o}$ and $\textbf{\textit{W}}_{w}$ are projection matrices. Thus, the cross entropy loss is computed over a bilingual parallel sentence pair $\{[\textbf{x}, \textbf{y}]\}$: $$\mathcal{L}(\theta)=\operatorname*{arg\,max}_{\theta}\{\sum_{i=1}^{I}\textup{log}P(y_i|\textbf{y}_{<i}, \textbf{x})\}.
\label{eq7:NMT_LossItem}$$
Proposed Future Cost Mechanism
==============================
In the traditional PBSMT, the future cost aims to estimate the difficulty of each translation option for one source sentence. Generally, PBSMT takes the beam-search stack decoding algorithm to select translation options to expand the current hypotheses. In detail, given a source sentence, all available translation options for any span of input words are collected in advance from the pre-learned translation model and language model. The future cost of each translation option is then computed based on the statistical scores of the translation model and the language model. By adding up the partial translation score and the future cost, PBSMT selects a translation option with the cheapest future cost to expand the current translation hypothesis. Finally, this makes a much better basis for pruning decisions in the beam-search stack decoding.
However, it is difficult to directly apply this future cost of PBSMT to the existing NMT system due to lack of pre-learned translation rules and its auto-regressive characteristic. Compared with the traditional PBSMT system, the NMT system models the translation knowledge as a time-dependent context vector for translation prediction through large-scale neural networks instead of translation rules. In particular, the time-dependent context representation is input to Eq. to compute the translation probability of the target word. Actually, the translation probability is an important part of computing the future cost in PBSMT. Meanwhile, NMT is seen as a neural network language model with attention mechanism [@kalchbrenner-blunsom-2013-recurrent; @bahdanau2014neural]. For example, the time-dependent context representation of NMT is regarded as that of the neural network language model [@10.5555/944919.944966].
Based on the above analysis, we propose a new method to model the future cost for the existing NMT systems. Specifically, the proposed approach utilizes the current target word and its context representation to learn a future context representation. Thus, this future context representation is input to a softmax layer to compute its future cost for the current target word. Formally, we utilize the top layer $\textbf{H}_{i}^{N}$ learned by the stacked Eq.$\sim$Eq. to model the current context information. The $\textbf{H}_{i}^{N}$ is together with the current generated target word $y_{i}$ to learn a future context representation $\textbf{F}_{i}$ as follows: $$\begin{aligned}
\label{eq8:gru_gate_r} &\textbf{R}_{i}=\sigma(\textbf{W}_{r}\cdot \textbf{E}[y_i]+\textbf{U}_{r}\cdot \textbf{H}_{i}^{N}), \\
\label{eq9:gru_gate_z} &\textbf{Z}_{i}=\sigma(\textbf{W}_{z}\cdot \textbf{E}[y_i]+\textbf{U}_{z}\cdot \textbf{H}_{i}^{N}), \\
\label{eq10:gru_cell} &\textbf{S}_{i}=\textup{ReLU}(\textbf{W}\cdot \textbf{E}[y_i]+\textbf{U}\cdot (\textbf{R}_{i}\odot \textbf{H}_{i}^{N})), \\
\label{eq11:gru_hidden_state} &\textbf{F}_{i}=\textbf{Z}_{i}\odot \textbf{S}_{i}+(1-\textbf{Z}_{i})\odot \textbf{H}_{i}^{N},\end{aligned}$$ where $\textbf{E}$ is the embedding matrix of target vocabulary, $\textbf{W}_{r}$, $\textbf{U}_{r}$, $\textbf{W}_{z}$, $\textbf{U}_{z}$, $\textbf{W}$, and $\textbf{U}$ are model parameters. $\sigma(\cdot)$ is sigmoid function in which $\odot$ means the element-wise dot. Note that for the initial future context representation, we use the special source end token “$<$/s$>$" and the mean of vectors in the source representation $\textbf{H}_{e}^{N}$ as the input to Eq.$\sim$Eq. to learn $\textbf{F}_{0}$.
Finally, the learned future context representation $\textbf{F}_{i}$ is as the input to a softmax layer to compute approximate probabilities of temporary target word $\hat{y}_{i+1}$ at the current time-step, called as the future cost of the current generated target word: $$\hat{P}(\hat{y}_{i+1}|\textbf{y}_{<i}, y_{i}, \textbf{x}) \propto
\textup{exp}(\bm{\mathcal{W}}_\textit{o}\textup{tanh}(\bm{\mathcal{W}}_\textit{w}\textbf{F}_{i})),
\label{eq12:future_cost}$$ where $\bm{\mathcal{W}}_{o}$ and $\bm{\mathcal{W}}_{w}$ are projection matrices. Later, $\hat{P}(\hat{y}_{i+1}|\textbf{y}_{<i}, y_{i}, \textbf{x})$ will be used to guide the training of NMT.
Neural Machine Translation with Future Cost Mechanism
=====================================================
In this section, we design two NMT models as Figure \[fig:model\] to make use of the proposed future cost in the previous section. For the first model, we compute an additional loss item of future cost at each time-step, and thereby gain a future cost-ware translation model. In addition to the additional loss item of future cost, the second model utilizes the learned future context representation to help the generation of target word at the next time-step, thus improving the translation performance of the Transformer-based NMT model.
Model I
-------
The training objective of NMT is to minimize the loss between the words in the translated sentences and those in the references. Specifically, the word-level cross-entropy between the generated target word by NMT and the reference serves as the loss item at each time-step. However, as we analyzed in Section \[Intro\], this existing training objective does not consider the future cost of the current generated target word for generating the next target word.
Therefore, we introduce an addition loss term $\mathcal{F}(\theta)$ to preview the future cost of the generated target word at the current time-step according to Eq.(\[eq12:future\_cost\]): $$\mathcal{F}(\theta)=\operatorname*{arg\,max}_{\theta}\sum_{i=1}^{I}\textup{log}\hat{P}(\hat{y}_{i+1}|\textbf{y}_{<i}, y_{i}, \textbf{x}; \theta).
\label{eq13:FutureCostAwareLoss}$$ The $\mathcal{F}(\theta)$ encourages the translation model to select a target word that is beneficial to the generation of target word at next time-step. Thus, the loss of the proposed Model I is computed over a bilingual parallel sentence pair $\{[\textbf{x}, \textbf{y}]\}$: $$\mathcal{J}(\theta)=\mathcal{L}(\theta) + \lambda\cdot\mathcal{F}(\theta),
\label{eq14:FutureCostAwareTraining}$$ where $\lambda$ is a hyper-parameter to weight the expected importance of the future cost loss in relation to the trained translation model. Finally, the trained Model I performs the translation decoding according to Eq..
In addition, the other candidate decoding adopts the probability computed in Eq.(\[eq6:Probabilities\_SANs\]) and the future cost computed in Eq.(\[eq12:future\_cost\]) to predict the target word $y_i$ jointly: $$\mathcal{P}(y_{i}|\textbf{y}_{<i}, \textbf{x})=(P(y_{i}|\textbf{y}_{<i}, \textbf{x})+ \hat{\mathcal{P}}(\hat{y}_{i}|\textbf{y}_{<i-1}, y_{i-1}, \textbf{x}))/2,$$ where $\beta$ is a coefficient to balance the weight of probability and future cost in the word prediction.
![The proposed Transformer-based NMT architecture.[]{data-label="fig:model"}](model_architecture.pdf "fig:"){width="3.3in" height="3.0in"}\
Model II
--------
In PBSMT, the future cost mechanism can help the generation of next target word (or phrase) in addition to minimize search errors in the beam-search stack decoding. However, the proposed Model I only focuses on learning future cost-aware NMT model to remain optimal translation hypotheses into the search stack. In other words, this future cost information may be not adequately utilized to predict target translation in NMT. Therefore, we further make use of the learned future context representation to help the generation of target word at the next time-step.
Formally, at the (i+1)-*th* time-step, the future context representation $\textbf{F}_{i}$ learned at the i-*th* time-step is first concatenated with the top layer of the decoder $\textbf{H}_{i+1}^{N}$ as the input to the sigmoid function to learn a gate scalar $g_{i+1}$: $$g_{i+1} = \sigma([\textbf{H}_{i+1}^{N}:\textbf{F}_{i}]\textbf{W}_{g}),
\label{eq15:gate}$$ where $\sigma$ is a function and $g_{i+1}$$\in$$[0, 1]$ is used to weight the expected importance of the learned future context representation $\textbf{F}_{i}$ to gain a fused context representation $\overline{\textbf{H}}_{i+1}^{N}$ as follows: $$\overline{\textbf{H}}_{i+1}^{N} = \textbf{H}_{i+1}^{N}+g_{i+1}\odot \textbf{F}_{i},
\label{eq16:FusedContext}$$ where $\textbf{W}_{g}\in \mathbb{R}^{2d_{model}\times 1}$ is a trainable parameter, and $\odot$ is the element-wise dot product.
Finally, the fused context representation $\overline{\textbf{H}}_{i+1}^{N}$ is as the input to a softmax layer to compute translation probabilities of the target word $y_{i+1}$ at the (i+1)-*th* time-step: $$P(y_{i+1}|\textbf{y}_{<i+1}, \textbf{x}) \propto
\textup{exp}(\textbf{\textit{W}}_\textit{o}\textup{tanh}(\textbf{\textit{W}}_\textit{w}\overline{\textbf{H}}_{i+1}^{N}).
\label{eq17:Probabilities_Model2}$$ Meanwhile, the training objective of Model II is the same to that of Model I as Eq.(\[eq14:FutureCostAwareTraining\]). Note that the future cost is estimated over the ground-truth target word during the training, and is estimated over the generated target word during the decoding.
[l|l||l|r|r|l|r|l|r]{} & & & &\
& & & & & & & &\
\
@vaswani2017attention & Trans.base & 27.30 & N/A & 65.0M & 38.10 & N/A & N/A & N/A\
@zheng2019dynamic & +Future and Past & 28.10 & N/A & N/A & N/A & N/A & N/A & N/A\
@hao-etal-2019-modeling & +BIARN & 28.21 & N/A & 97.4M & N/A & N/A & 24.70 & 117.3M\
@li-etal-2019-information & +Aggregation & 28.26 & N/A & 92.0M & N/A & N/A & 24.68 & 112.0M\
@li2020datadependent & +D2GPo & 27.93 & N/A & N/A & 39.23 & N/A & N/A & N/A\
@vaswani2017attention & Trans.big & 28.40 & N/A & 213.0M & 41.00 & N/A & N/A & N/A\
@hao-etal-2019-modeling & +BIARN & 28.98 & N/A & 333.5M & N/A & N/A & 25.10 & 373.3M\
@li-etal-2019-information & +Aggregation & 28.96 & N/A & 297.0M & N/A & N/A & 25.00 & 337.0M\
@li2020datadependent & +D2GPo & 29.10 & N/A & N/A & 41.77 & N/A & N/A & N/A\
\
& Trans.base & 27.42 & 12.7k & 66.5M & 39.13 & 66.9M & 23.93 & 70.7M\
& +Model I & 27.97+ & 12.5k & 68.4M & 39.68+ & 68.8M & 24.48+ & 72.5M\
& +Model II & 28.17++ & 12.0k & 68.4M & 39.96++ & 68.8M & 24.84++ & 72.5M\
& Trans.big & 28.45 & 10.1k & 221.1M & 41.07 & 221.9M & 24.55 & 229.4M\
& +Model I & 28.98+ & 9.8k & 228.4M & 41.79+ & 229.2M & 24.96+ & 236.8M\
& +Model II & 29.12++ & 9.4k & 228.4M & 42.02++ & 229.2M & 25.13++ & 236.8M\
Experiments
===========
Datasets
--------
The proposed methods were evaluated on the WMT14 English-to-German (EN-DE), WMT14 English-to-French (EN-FR), and WMT17 Chinese-to-English (ZH-EN) translation tasks. The EN-DE training set contains 4.5M bilingual sentence pairs, and the *newstest2013* and *newstest2014* data sets were used as the validation and test sets, respectively. The EN-FR training set contains 36M bilingual sentence pairs, and the *newstest2012* and *newstest2013* datasets were combined for validation and *newstest2014* was used as the test set. The ZH-EN training set contains 22M bilingual sentence pairs, where the *newsdev2017* and the *newstest2017* data sets were used as the validation and test sets, respectively. The baselines are involved:
**Trans.base/big**: a vanilla Transformer-based NMT system without future cost [@vaswani2017attention], for example Transformer (base) and Transformer (big) models.
**+Future and Past** [@zheng2019dynamic]: introduce a capsule network into the Transformer NMT system which is adopted to recognize the translated and untranslated contents, and pay more attention to untranslated parts.
Besides, we reported results of the state-of-the-art works [@hao-etal-2019-modeling; @li-etal-2019-information; @li2020datadependent] for the three translation tasks.
Settings
--------
We implemented the proposed method in the *fairseq* [@ott2019fairseq] toolkit, following most settings in @vaswani2017attention. In training the NMT model (base), the byte pair encoding (BPE) [@sennrich-haddow-birch:2016:P16-12] was adopted. The vocabulary size of EN-DE and EN-FR was set to 40K and ZH-EN was set to 32k. The dimensions of all input and output layers were set to *512*, and that of the inner feedforward neural network layer was set to *2,048*. The total heads of all multi-head modules were set to *8* in both the encoder and decoder layers. In each training batch, there was a set of sentence pairs containing approximately *4,096$\times$8* source tokens and *4,096$\times$8* target tokens. The value of label smoothing was set to 0.1, and the attention dropout and residual dropout were $p$ = $0.1$. We adopt the Adam optimizer [@kingma2014adam] to learn the parameters of the model. The learning rate was varied under a warm-up strategy with warmup steps of 8,000. For evaluation, we validated the model with an interval of 2,000 batches on the dev set. Following the training of 300,000 batches, the model with the highest BLEU score for the validation set was selected to evaluate the test sets. We used multi-bleu.perl[^2] script to obtain the case-sensitive 4-gram BLEU score. All models were trained on eight V100 GPUs.
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Overall Results
---------------
The main results of the translation are shown in Tables \[tb1:main\_result\]. We made the following observations:
1\) The performance of our implemented Trans.base/big is slightly superior to that of the original Trans.base/big in the EN-DE and EN-FR dataset. This indicates that it is a strong baseline NMT system and it makes the evaluation convincing.
2\) The proposed Model I and Model II significantly outperformed the baseline Trans.base. This indicated that the future cost information was beneficial for the Transformer-based NMT. Meanwhile, the Model II outperformed the comparison system +Future and Past [@zheng2019dynamic], which means that the future cost estimated at the previous time-step is further used to help the generation of the current target word in the decoding.
3\) Compared with Model I, Model II achieved a slight advantage on all tasks. This means that it is more effective to enhance the translation of the next target word by integrating the learned future hidden representation into the contextual representation of the next word.
4\) We also compared our methods with the baseline Trans.big model. In particular, the proposed models yielded similar improvements on the three translation tasks, indicating that the proposed future cost mechanism is a universal method for improving the performance of the Transformer-based NMT model.
5\) The proposed models contain approximately $3\%$ additional parameters. Training and decoding speeds are nearly the same as Trans.base. This indicates that the proposed method is efficient by only adding a few training and decoding costs.
Translating Sentences of Different Lengths
------------------------------------------
The proposed future cost mechanism focuses on capturing its future cost for the generated target word at each time-step, thus measuring how good it is to generate the next target word. Thus, we show the translation performance of source sentences with different sentence lengths, further verifying the effectiveness of our method. Specifically, we divided each test set into 6 groups according to the length of the source sentence. Figure \[fig:TranslationDifferentLength\] shows the results of the proposed models and Trans.base model on the three translation tasks. We observed as follows:
1\) Model I and Model II were superior to the Trans.base model in almost every length group on all three tasks. This means that the future cost information capturing by the proposed approach is beneficial to Transformer-based NMT.
2\) Compared with Model I, Model II achieved a slight advantage in most groups on each task. This indicates that this future cost information also helps the generation of the next target word in addition to that of the current target word.
3\) BLEU scores of all models decreased when the length was greater than 30 over ZH-EN task. In contrast, the trend of BLEU scores increased with the sentence length for EN-DE and EN-FR tasks. We think that NMT may be good at modeling translation between distant language pairs (i.e., ZH-EN) than similar language pairs (i.e., EN-DE and EN-FR).
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\
Learning Curve of Hyper-parameter $\lambda$
-------------------------------------------
In Eq.(\[eq14:FutureCostAwareTraining\]), we introduce a hyper-parameter $\lambda$ to adjust the weight of the future cost loss in relation to the trained translation model. To tune the value, we conducted experiments with different $\lambda$ on validate set for the three tasks. As shown in Figure \[fig:TranslationDifferentLambda\], the proposed models achieved an advantage over the Trans.base model with different $\lambda$ on the two tasks. As a result, for the EN-DE task, Model I and Model II achieved the best BLEU score with $\lambda$=$0.7$ respectively. The trend on EN-FR (not shown) is similar with that on EN-DE and we set $\lambda$=$0.7$ for EN-FR. For the ZH-EN task, Model I achieved the best BLEU score with $\lambda$=$0.3$ while Model II with $\lambda$=$0.5$. Finally, the results of Table \[tb1:main\_result\] are obtained according to these optimized hyper-parameter $\lambda$.
Translation Cases
-----------------
Figure \[fig:baseline\_example\](a) shows a translation case to observe the effect of the proposed method. Our method translated “具有\[had\] 额外的\[additional\] 政治\[political\] 价值\[value\]” into “had additional political value” which are same as the reference, while Trans.base translated it into “has added political value” which is different from the reference in the meaning. We think the reason is that +Model II not only predicts “had" accurately at the current step, but also captures future cost information that is beneficial for generating “additional" at the next time-step. Concretely, Figure \[fig:baseline\_example\](b) and Figure \[fig:baseline\_example\](c) illustrate the beam-search processing for the Trans.base model and +Model II, respectively. In the proposed +Model II, no matter after “had” or after “has", the candidate “additional” is produced. However, in the Trans.base model, only “added” follows “has”. This indicates that the learned future contextual representation is beneficial for NMT.
Related Work
============
Modeling translated and untranslated information in a source sentence is beneficial to generate target translation in NMT. @tu-etal-2016-modeling employed a coverage vector to track the translation part in the source sequence. Similarly, @mi-etal-2016-coverage proposed to use a coverage embedding to model the degree of translation for each word in the source sentence. Later, @li-etal-2018-simple presented a coverage score to describe to what extent the source words are translated. Recently, @zheng-etal-2018-modeling introduced two extra recurrent layers in the decoder to maintain the representations of the past and future translation contents. Furthermore, @zheng2019dynamic adopted a capsule network to model the past and future translation contents explicitly.
Compared with the source information, @lin-etal-2018-deconvolution proposed to adopt a deconvolution network to model the global information of the target sentence. @NIPS2017_6622 applied a value network to dynamically compute a BLEU score for the rest part of the target sequence based on the difference between the generated sub-sequence and the source sequence. @NIPS2017_6775 designed a deliberation network to preview future words through multi-pass decoding. @li-etal-2018-target enhanced the attention model by the implicit information of target foresight word oriented to both alignment and translation. @zhou-etal-2019-synchronous employed two bidirectional decoders to generate a target sentence in an interactive translation setting. Closely related to our work, @weng-etal-2017-neural proposed to adopt a word prediction mechanism to enhance the contextual representation during the training. The main difference is that the proposed future cost mechanism can not only minimize search errors but also help the generation of the next word explicitly while @weng-etal-2017-neural just used it as an extra training objective.
In short, the above mentioned works focused on adopting the future context to enhance the contextual information of the target word at the current time-step. Inspired by the future cost in PBSMT, we propose a simple and effective method to estimate the future cost of the current generated target word. This approach enables the NMT model to preview the translation cost of the subsequent target word at the current time-step and thereby helps the generation of the target word at the next time-step. This proposed future cost mechanism is integrated into the existing Transformer-based NMT model to improve translation performance.
Conclusion and Future Work
==========================
In this paper, we propose a simple and effective future cost mechanism to enable the translation model to preview the translation cost of next target word at the current time-step. We empirically demonstrate that such explicit future cost mechanism benefits NMT with considerable and consistent improvements on three language pairs. In the future, we will further extend this work to other NLP tasks.
[^1]: Contribution during internship at NICT.
[^2]: https://github.com/moses-smt/mosesdecoder/blob/RELEASE-4.0/scripts/generic/multi-bleu.perl
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---
abstract: 'The wide bandwidth and large number of antennas used in millimeter wave systems put a heavy burden on the power consumption at the receiver. In this paper, using an additive quantization noise model, the effect of analog-digital conversion (ADC) resolution and bandwidth on the achievable rate is investigated for a multi-antenna system under a receiver power constraint. Two receiver architectures, analog and digital combining, are compared in terms of performance. Results demonstrate that: (i) For both analog and digital combining, there is a maximum bandwidth beyond which the achievable rate decreases; (ii) Depending on the operating regime of the system, analog combiner may have higher rate but digital combining uses less bandwidth when only ADC power consumption is considered, (iii) digital combining may have higher rate when power consumption of all the components in the receiver front-end are taken into account.'
author:
-
title: 'Low Power Analog-to-Digital Conversion in Millimeter Wave Systems: Impact of Resolution and Bandwidth on Performance'
---
Introduction {#intro}
============
The millimeter wave (mmWave) bands, roughly between 30 and 300 GHz, are an attractive candidate for next-generation cellular systems due to the vast quantities of spectrum and the potential for exploiting very high-dimensional antenna arrays [@KhanPi:11-CommMag]-[@Rappaport2014-mmwbook]. However, a significant issue in realizing these systems is power consumption, particularly in handheld mobile devices. In addition to the high power consumed baseband processing, one critical concern is the power consumption in the analog-digital conversion (ADC) due to the need to process large number of antenna outputs and very wide bandwidths.
Classical information theoretic formulation [@cover] characterizes the maximum achievable communication rate on a channel as a function on the signal-to-noise ratio (SNR) and bandwidth. The broad goal of this paper is to understand what the information theoretic limits on communication in the presence of constraints on the ADC power consumption are and how we can design communication systems to meet these limits. Although a closed form expression for the optimal input and capacity of a 1-bit quantizer is given in [@madhow], it is difficult to obtain the capacity and input distributions of multi-bit quantizer. Therefore, we restrict our analysis to an additive quantization noise model (AQNM) [@gray]. Under this assumption, ADC power constraints can be easily abstracted as constraints on the sampling rate and quantizer noise level. We can then obtain lower bounds on the capacity and investigate the optimal bandwidth and resolution of the ADC such that the achievable rate is maximized.
We first study the effect of bandwidth and sampling rate of ADC on the performance of single-input and single output (SISO) system under the total receiver power budget including the ADC power cost. We next consider multiple-input multiple-output (MIMO) point-to-point systems operating over a large bandwidth. Our analysis considers both spatial multiplexing and beamforming – the two basic methods for MIMO systems [@TseV:07]. Spatial multiplexing is favorable when the channel is bandwidth-limited and the SNR is high. However, beamforming provides power gain and is beneficial if the channel is power-limited as in most of the mmWave systems due to available wide bandwidth and high path loss [@sun].
For mmWave MIMO systems, there are three receiver combining methods that need to be considered: analog combining, digital combining and hybrid analog/digital combining [@sparse]. Digital combining is the method in use in conventional cellular transceivers today where each antenna element has a separate pair of ADCs. This method offers the greatest flexibility, but highest power consumption for a given ADC resolution and sampling rate. An alternate design is to combine the signals in analog (either in RF or IF) so that only one pair of ADCs is required per stream [@KohReb:07]-[@Heath:partialBF]. Hybrid beamforming [@Heath:partialBF] uses a two stage combination of these designs. See [@sun] for a general discussion.
Since power dissipation of ADC scales linearly in sampling rate and exponentially in the number of bits per sample [@lee], it may not be desirable to operate the system over the full bandwidth and high resolution. Therefore, digital combining can save energy by using narrow band (low sampling rate) and low resolution ADCs while increasing its effective received SNR and allowing the spatial multiplexing at the transmitter for better performance. As a result, analysis of performance trade-off of analog and digital combining is essential for power limited receivers.
To compare these architectures, we derive achievable rates of the MIMO system with both analog combining and digital combining and study the effect of bandwidth, ADC resolution and the number of antennas on the achievable rate. Constraints are placed on the total front-end receiver power or only the ADC power. We expect most of the simulation results for the hybrid analog/digital combining to be somewhere between analog and digital combining, and hence we leave detailed analysis of this as a future work.
Under these power constraints, our results show that:
- For a point-to-point SISO system when bandwidth is large as in mmWave systems, there exists an optimal bandwidth and resolution of the ADC. As a consequence, it may not be desirable to operate the system over the full bandwidth.
- For a point-to-point MIMO system when only ADC power consumption is considered, we see that at least in some operating regimes, analog combining may have higher maximum achievable rate than digital combining. However, in these cases, digital combining can attain its maximum rate at much less bandwidth. Indeed, as the number of antennas at the transmitter and receiver $N$ increases, the utilized bandwidth decreases by a factor of $\frac{1}{N}$.
- For a point-to-point MIMO system when the power consumption of all receiver components including ADCs are considered, digital combining may be able to achieve a higher rate when the channel state information is available at the transmitter.
Previous work {#previous-work .unnumbered}
-------------
In recent years, energy efficient transceiver architectures such as use of low resolution ADCs and hybrid analog/digital precoding (combining) has attracted significant interest. The limits of communication over additive white Gaussian channel with low resolution (1-3) bits ADCs at the receiver is studied in [@madhow]. This is extended in [@fettweis] to the capacity of the 1-bit ADC with oversampling at the receiver. The bounds on the capacity of the MIMO channel with 1-bit ADC at high and low SNR regimes are derived in [@heath] and [@mezgani], respectively. Using AQNM of ADC, the joint optimization of ADC resolution with the number of antennas in a MIMO channel is studied in [@nossek]. There is also rich literature on the hybrid analog/digital transceivers [@sparse]-[@robert]. While [@sparse] provides efficient hybrid precoding and combining algorithms for sparse mmWave channels which performs close to full digital solution, [@robert] combines efficient channel estimation with the hybrid precoding and combining algorithm in [@sparse]. However, to the best of our knowledge, there is no work which considers the effects of both sampling rate (bandwidth) and resolution of ADC on the performance of the system under a total receiver power constraint. In addition, we investigate and compare the performance of analog and digital combining and optimize the resolution, bandwidth and the number of antennas.
It should be noted that our analysis would likely apply most closely to data plane traffic where the overhead for channel tracking is relatively small. For initial synchronization, where the channel or even the presence of the transmitting base station may not be known, digital combining can have distinct advantages not accounted for in this analysis – see [@BarHosCellSearch:14-spawc].
System Model {#sys model}
============
We consider a point-to-point mmWave communication system operating over a bandwidth $W_{tot}$ Hz. We assume that there are $N_t$ antennas at the transmitter and $N_r$ antennas at the receiver. We assume additive white Gaussian noise (AWGN) with power spectral density $\frac{N_0}{2}$ Watts/Hz. The transmitted signal has average power constraint $P$ Watts. The channel exhibits frequency selective fading, which is independent across frequency bands and across antennas. The instantaneous fading realizations are assumed to be known at the receiver.
We consider two receiver architectures as shown in Figure \[sys\]: digital combining and analog combining. As shown in the figure, for digital combining, ADCs are employed to quantize the signal before the baseband combiner. Each block labeled “I/Q ADC" represents two ADCs – one for the inphase and another for the quadrature components, both with sampling rate equal to the Nyquist rate. In the analog combining architecture, the signals are combined in analog with phase shifters, and then digitized with a single I/Q ADC. Thus, for RF analog combining, we have $N_r$ low noise amplifiers (LNAs), $N_r$ phase shifters, one combiner, one mixer and one I/Q ADC, while the digital combining architecture consists of only $N_r$ LNAs and $N_r$ mixers [@Taghivand], but $N_r$ ADCs.
We assume that for either analog or digital combing, each ADC consists of a $b$-bin scalar quantizer. Note that in [@madhow], it is proved that for $b$-bin output quantization, at most $b+1$ mass points at the input are enough to achieve the capacity. However, obtaining the optimal input distribution for arbitrary number of quantization bins is difficult. Therefore, in this paper, in order to get insights, we use an AQNM for the quantizer and find a lower bound to the capacity of the corresponding channel by assuming Gaussian quantization noise and Gaussian inputs. Further benefits of AQNM include ease of implementation since the decoder can use standard linear processing and Gaussian decoding.
Additive Quantization Noise Model (AQNM)
----------------------------------------
We denote the output of the ADC corresponding to input $z$ by $Q(z)$. We consider that the quantizer output $z_q=Q(z)$ is chosen such that $z_q=E[z|z_q]$. The quantizer $Q(\cdot)$ can be represented by the following AQNM [@rangan]: $$\begin{aligned}
\label{eq0}
z_q &=&\alpha z+n_q,\end{aligned}$$ where $n_q$ is the additive quantization noise such that $z$ and $n_q$ are uncorrelated. Note that $$\begin{aligned}
\label{eq1}
E[n_q]&=&(1- \alpha) E[z]\\\label{eq2}
\sigma_{n_q}^2&=&(1-\alpha)\alpha \sigma_z^2,\end{aligned}$$ Here, $\sigma_{n_q}^2$ is variance of additive quantization noise. Accordingly, $\alpha$ can be computed as: $$\begin{aligned}
\label{eq3}
\alpha =1-\beta,\end{aligned}$$ where $\beta=\frac{\sigma_{e_q}^2}{\sigma_z^2}$ where $\sigma_{e_q}^2$ is the variance of quantization error, $e_q=z-z_q$, and $\sigma_z^2$ is variance of the quantization input. In [@gray], $\frac{1}{\beta}$ is referred as the coding gain.
Motivated by [@cover] we assume that $\mathbf{n_q}$ has the Gaussian distribution. Furthermore, since the Gaussian input maximizes the mutual information between the input and the output when the noise is Gaussian, we assume that the input to the MIMO channel is jointly Gaussian. Accordingly, for non-uniform scalar MMSE quantizer of a Gaussian random variable, $\beta$ can be approximated to $\beta=\frac{\pi \sqrt{3}}{2}b^{-2}$, where $b$ is the number of quantization bins [@gray]. In this paper, without loss of generality we assume that $\beta=ab^{-2}\leq 1$ for some constant $a>0$. Note that as $b \rightarrow \infty$, $\beta \rightarrow 0$.
Power Consumption of the Receiver
---------------------------------
We consider two scenarios to account for the power consumption at the receiver. In the first one, we assume a total processing power budget. Denoting the power consumption of the LNA, phase shifter, combiner, mixer and ADC by $P_{LNA}$, $P_{PS}$, $P_C$, $P_M$ and $P_{ADC}$, respectively, the total power consumption $P_{tot}$ of analog and digital combining in terms of $N_r$ are respectively given by $$\begin{aligned}
\label{pow1}
P_{tot}=N_r (P_{LNA}+P_{PS})+P_C+ P_M +2 P_{ADC},\end{aligned}$$ and $$\begin{aligned}
\label{pow2}
P_{tot}=N_r (P_{LNA}+P_{M}+2 P_{ADC}).\end{aligned}$$
We assume that the power consumption of the LNA, phase shifter, combiner, and mixer are independent of the bandwidth. The power dissipation of ADC scales linearly in sampling rate and exponentially in the number of bits per sample [@lee]. Assuming sampling at the Nyquist rate, the figure of merit of ADC power consumption is modeled as $$\begin{aligned}
\label{adc_power}
P_{ADC}=c W b,\end{aligned}$$ where $W$ is sampling rate (bandwidth), $b$ is the number of quantization bins of ADC, and $c$ is the energy consumption per conversion step, e.g. $c=494$ fJ [@Murmann].
In the second scenario, we only put a constraint on the total power used by all the ADCs in the receiver. While this helps avoid comparing power consumption of different components (which are designed using possibly different technologies), the analysis may only give a partial indication of what could happen in practice.
Accuracy of AQNM {#accuray}
================
In order to illustrate the accuracy of the AQNM, we consider a simple SISO system, i.e., $N_t=N_r=1$, with fixed complex channel gain $h$. We compare the rate achieved by Gaussian inputs in the AQNM with the capacity computed for the 2, 4, and 8-bin quantizers [@madhow]. Using the AQNM in (\[eq0\]), the equivalent received signal after the quantizer can be written as $$\begin{aligned}
y=(1-ab^{-2}) h x +(1-ab^{-2}) n + n_q,\end{aligned}$$ where $\sigma_{n_q}^2=ab^{-2}(1-ab^{-2})(|h|^2 P + N_0)$. Then, the following rate is achievable when operating over a bandwidth $W$: $$\begin{aligned}
\label{ch1}
R=W\log_2\left(1+\frac{(1-ab^{-2}) |h|^2 P}{ab^{-2} |h|^2 P +WN_0}\right) \quad \text{bits/sec}.\end{aligned}$$ Note that the rate given in (\[ch1\]) is monotonically increasing and concave function of $P$ for a given bandwidth and resolution.
As shown in Figure 2, the rate in (\[ch1\]) lower bounds the capacity. At low SNRs the gap between the capacity and the rate under the AQNM is small, and as the quantization resolution increases the gap decreases. For example, at $-10$dB SNR, the rate assuming AQNM is $96\%$ and $99\%$ of the capacity for $2$-bin and $8$-bin quantization, respectively. However, at high SNR, the gap between the capacity and the rate of the AQNM increases. For example, at $20$dB SNR, the rate under the AQNM is $72\%$ and $77\%$ of the capacity for $2$-bin and $8$-bin quantization, respectively. In this regime, the rate achieved by an equiprobable $b$ point input distribution is close to the capacity which can be approximated by $W\log_2(b)$ bits/s [@madhow]. Similarly, in the high SNR regime the rate in (\[ch1\]) can be approximated by $W\log_2(\frac{b}{a})/2$, or $-W\log_2(\beta)/2$, bits/s. Therefore, the gap between the capacity and the rate under the AQNM becomes $W\log_2(\sqrt{a})$, or $W\log_2(b\sqrt{\beta})$ as SNR increases. However, based on the fact that the capacity and the achievable rate under the AQNM show similar trends, we believe that the conclusions obtained in this paper would be valid when optimal input distribution is used, even though the actual numerical values of optimal bandwidth, resolution etc. may be slightly different.
\[sim1\]
The Optimal Bandwidth and Resolution of ADC for SISO Systems {#SISO}
============================================================
In order to get insights on the bandwidth usage and optimal resolution of ADCs, in this section we limit our investigation to the SISO channel. As in Section \[accuray\], we assume the channel gain is constant and we investigate how the achievable rate in (\[ch1\]) can be maximized by optimally choosing the bandwidth $W$ and the number of quantization bins $b$ under a total receiver power $P_{tot}$ constraint. For $P_{tot}$, we follow the model in (\[pow2\]). The optimal bandwidth and resolution can be computed by the following optimization problem.
\[ppp:4\] $$\begin{aligned}
\label{ppp:4a}
\underset{b, W}{\operatorname{max}} && W\log_2\left(1+\frac{\left(1-a b^{-2}\right) |h|^2 P}{a b^{-2} |h|^2 P +W N_0}\right) \\\label{ppp:4b}
\text{s.t.}~ && P_{LNA}+ P_{M} + 2cWb \leq P_{tot}, \\ \label{ppp:4c}
&& 0\leq W \leq W_{tot}, \quad 0 \leq b.\end{aligned}$$
\[lemma1\] For the optimization problem in (\[ppp:4\]) when $W_{tot}$ is large, there exists an optimal bandwidth beyond which the achievable rate in (\[ppp:4a\]) starts decreasing.
Since the rate in (\[ch1\]) is monotonically increasing function of $b$, the constraint in (\[ppp:4b\]) must be satisfied with equality. Therefore, from (\[ppp:4b\]) we can argue that $b$ is equal to $\frac{P_{tot}-P_{LNA}-P_{M}}{2Wc}$. Accordingly, the optimization problem in (\[ppp:4\]) can be reformulated as follows: $$\begin{aligned}
\label{eq5a}
\underset{W}{\operatorname{max}} && W \log_2\left(\frac{|h|^2P + W N_0}{\frac{a 2^2 c^2 W^2 |h|^2P}{(P_{tot}-P_{LNA}-P_{M})^2}+WN_0}\right),\end{aligned}$$ where $0 \leq W \leq W_{tot}$. Here we assume $W_{tot} \rightarrow \infty$. The objective function in (\[eq5a\]) is a concave function of $W$ when $P_{tot}-P_{LNA}-P_{M}>0$, $W>0$ and $a>0$. We can argue that as $W\rightarrow 0$, the objective function in (\[eq5a\]) goes to 0, as $W\rightarrow \infty$ the objective function in (\[eq5a\]) goes to $-\infty$. In addition, the objective function in (\[eq5a\]) is positive when $\frac{a 2^2 c^2 W^2}{(P_{tot}-P_{LNA}-P_{M})^2}=ab^{-2}<1$. Therefore we can conclude that when $ab^{-2}< 1$, there exists an optimal finite bandwidth $W^*$ for which rate in (\[ppp:4\]) is maximized.
Lemma \[lemma1\] shows that while for AWGN channels (even for those whose outputs are quantized) the capacity increases as the bandwidth gets larger, under receiver power constraints, this is no longer the case. This suggests that even though the bandwidth is abundant in mmWave systems, since ADC is power hungry, it may not be desirable to operate the system over its full bandwidth. Similarly, there exits an optimal number of quantization levels that maximizes capacity under receiver power constraints.
In order to show the effect of SNR on the optimal bandwidth and the number of quantization bins, we set $P_{tot}-P_{LNA}-P_{M}=2P_{ADC}= 20$ mW and the energy consumption per conversion step $c=494$ fJ [@Murmann]. The total bandwidth is $W_{tot}= 7$ GHz [@daniel]. We consider scalar non-uniform MMSE quantizer at the ADC and compute $\beta$ using Lloyd-Max algorithm [@max]. As shown in Figure 3, in the low SNR regime it is optimal to use a lower bandwidth and larger number of quantization bins. This allows the transmitter to spread the available power over a smaller band and increase effective SNR. Here, SNR refers to the received SNR at the full bandwidth $W_{tot}$, i.e., $|h|^2{P}/{W_{tot}N_0}$. Note that beyond 5dB SNR, the optimal resolution and bandwidth are $b=3$ and $W=6.75$ GHz, which is slightly less than the total available bandwidth 7 Ghz.
\[sim4\]
MIMO System {#MIMO}
===========
In this section, we investigate the achievable rate for the MIMO system with digital and analog combining. For both receiver architectures we consider two scenarios: i) Perfect channel state information at the transmitter (CSIT) where instantaneous channel realizations are known at the transmitter; ii) no CSIT, where the transmitter only knows the channel state.
Digital Combining {#MIMO_dig}
-----------------
Using AQNM, we can obtain the equivalent channel per frequency band as follows: $$\begin{aligned}
\label{eq20}
\mathbf{y_q} =(1-a b^{-2})\mathbf{H}\mathbf{x}+(1-a b^{-2})\mathbf{n} +\mathbf{n_q},\end{aligned}$$ where $R_{n_qn_q}=a b^{-2}(1-a b^{-2})\text{diag}(\mathbf{H} R_{xx}\mathbf{H^H}+W R_{nn})$. Here $R_{nn}$ is the noise covariance matrix and $R_{xx}$ is the input covariance matrix. Then, the following rate is achievable with digital combining operating over a bandwidth $W \leq W_{tot}$: $$\label{eq17}
E_H\left[W\log_2\left|\mathbf{I}+\frac{(1-a b^{-2})\mathbf{H}R_{xx}\mathbf{H^H}}{a b^{-2}\text{diag}(\mathbf{H} R_{xx}\mathbf{H^H})+W R_{nn}}\right|\right].$$ Since we consider independent identically distributed fading across antennas and frequency bands, the optimal input covariance matrix under no CSIT is $R_{xx}=\frac{P}{N_t}\mathbf{I}$ [@telatar].
Under CSIT, since the input covariance matrix appears as noise in (\[eq17\]), using singular value decomposition (SVD) of the channel matrix for $R_{xx}$ does not immediately convert the MIMO channel into multiple parallel channels. However, in this section for ease of implementation we use the SVD of the channel matrix to determine $R_{xx}$ which provides a further lower bound to the achievable rate. We also assume constant power allocation across frequency bands. Then, denoting SVD of $\mathbf{H}$ by $\mathbf{U}\Lambda \mathbf{V^H}$, we can rewrite (\[eq20\]) as $$\begin{aligned}
\mathbf{y_q}&=&(1-a b^{-2})\mathbf{U}\Lambda \mathbf{V^H}\mathbf{x}+(1-a b^{-2})\mathbf{n} +\mathbf{n_q},\end{aligned}$$ Using the transmit beamforming matrix $\mathbf{V}$ and the digital combining matrix $\mathbf{U^H}$, we obtain the received signal as $$\begin{aligned}
\hspace{-0.05in}\mathbf{y}& = & \mathbf{U^H}\mathbf{y_q} \\
&=&(1-a b^{-2})(\mathbf{U^H}\mathbf{U}\Lambda \mathbf{V^H}\mathbf{V}\mathbf{\tilde{x}}+\mathbf{U^H}\mathbf{n}) +\mathbf{U^H}\mathbf{n_q}\\\label{eq21}
&=&(1-a b^{-2})\Lambda \mathbf{\tilde{{x}}}+(1-a b^{-2})\mathbf{\tilde{n}} +\mathbf{\tilde{n}_q},\end{aligned}$$ where $\mathbf{\tilde{x}}=\mathbf{V^H}\mathbf{x}$, $\mathbf{\tilde{n}}=\mathbf{U^H}\mathbf{n}$ and $\mathbf{\tilde{n}_q}=\mathbf{U^H}\mathbf{n_q}$. Here $\mathbf{\tilde{n}}$ has the same distribution as $\mathbf{n}$. Allocating power across the eigenvalues of $\Lambda$ according to the waterfilling solution, we obtain the following achievable rate: $$\label{eq11}
E_H\left[W\log_2\frac{|R_{n'n'}+(1-a b^{-2})\Lambda \mathbf{Q}\Lambda |}{|R_{n'n'}|}\right],$$ where $R_{n'n'}=a b^{-2}\mathbf{U^H}\text{diag}(\mathbf{U}\Lambda \mathbf{Q} \Lambda \mathbf{U^H})\mathbf{U}+W R_{nn}$. Here $\mathbf{Q}$ denotes the diagonal matrix that contains the power levels. Note that the above waterfilling strategy would be optimal in the case of no quantization, where the power dependent quantization noise term is zero.
Analog Combining:
-----------------
Using AQNM, we can obtain the equivalent channel of analog combining per frequency band as follows: $$\begin{aligned}
\label{eq13}
\mathbf{y_q} = (1-a b^{-2})\mathbf{w_r^H}\mathbf{H}\mathbf{w_t}x+(1-a b^{-2})\mathbf{w_r^H}\mathbf{n} +\mathbf{n_q},\end{aligned}$$ where $\sigma_{n_q}^2=a b^{-2}(1-a b^{-2})(|\mathbf{w_r^H}\mathbf{H}\mathbf{w_t}|^2P+N_r N_0)$. Here, $\mathbf{w_r}$ is the analog combining vector such that $|w_{r,i}|=1$ $i=1, \ldots, N_r$, and $\mathbf{w_t}\in \mathbb{C}^{N_t \times 1}$ is the digital beamforming vector at the transmitter. Then, the achievable rate of analog combining operating over a bandwidth $W \leq W_{tot}$ is given by: $$\label{eq19}
E_H\left[\underset{\mathbf{w_r},\mathbf{w_t}}{\operatorname{max}} W\log_2\left(1+\frac{(1-a b^{-2})P|\mathbf{w_r^H}\mathbf{H}\mathbf{w_t}|^2}{a b^{-2}P|\mathbf{w_r^H}\mathbf{H}\mathbf{w_t}|^2+W N_r N_0}\right)\right].$$ Note that the above expression is monotonically increasing function of the received power $P|\mathbf{w_r^H}\mathbf{H}\mathbf{w_t}|^2$. Therefore, the achievable rate is maximized when the term $|\mathbf{w_r^H}\mathbf{H}\mathbf{w_t}|^2$ is maximized. The maximum value that can be obtained depends of the availability of CSIT.
### Analog Combining without CSIT
Since there is no CSIT and the channel is symmetric, the optimal transmit beamforming vector is $w_{t,i}=\frac{1}{\sqrt{N_t}}$, $i=1,..., N_t$. We the have [@love] $$\begin{aligned}
|\mathbf{w_r^H}\mathbf{H}\mathbf{w_t}|^2 &= & \left|\sum_{i=1}^{N_r} e^{j\phi_i}\frac{1}{\sqrt{N_t}}\sum_{j=1}^{N_t} h_{ij}\right|^2\\
&\leq & \frac{1}{N_t}\left(\sum_{i=1}^{N_r}\left|\sum_{j=1}^{N_t} h_{ij}\right|\right)^2,\end{aligned}$$ where $\phi_i$ is phase of the $i$th element of $\mathbf{w_r}$. The inequality above holds with equality when $\phi_i$ is chosen as the phase of $\sum_{j=1}^{N_t} h_{ij}$ plus $\pi$.
### Analog Combining with CSIT
At the transmitter digital beamforming in the form of maximum ratio transmission maximizes the received power. For a given analog combining vector $\mathbf{w_r}$ we have $$\begin{aligned}
|\mathbf{w_r^H}\mathbf{H}\mathbf{w_t}|^2 \leq \|\mathbf{w_r^H}\mathbf{H}\|^2 \|\mathbf{w_t}\|^2.\end{aligned}$$ The above inequality satisfied with equality when $\mathbf{w_t}=\frac{\mathbf{H^H}\mathbf{w_r}}{\|\mathbf{H^H}\mathbf{w_r}\|}$. Note that the normalization is due to the power constraint at the transmitter, where, as in Section \[MIMO\_dig\] we assume no power allocation across frequency bands. Then, we have $|\mathbf{w_r^H}\mathbf{H}\mathbf{w_t}|^2=\|\mathbf{w_r^H}\mathbf{H}\|^2$.
Illustration of the Results
===========================
In this section, we provide numerical results to show the effects of ADC bandwidth and resolution, the number of antennas, receiver architecture and power consumption on the achievable rate. We consider scalar non-uniform MMSE quantizer at each ADC and compute $\beta$ using Llyod-Max algorithm [@max]. We set the energy consumption per conversion step $c$ to $494$ fJ [@Murmann]. We consider that the maximum available bandwidth is $W_{tot}=7$ Ghz for operation in the 57-64 Ghz band [@daniel]. Throughout SNR refers to SNR at the full bandwidth, that is, ${P}/{W_{tot}N_0}$. All results are obtained under independent Rayleigh fading across space and frequency.
We first consider only the total ADC power consumption which is set to 20 mW. Hence for analog combining $2P_{ADC}=20$ mW while for digital $2N_r P_{ADC}=20$ mW. We examine the effect of SNR on the rates of $1 \times 3$ SIMO, and $ 3 \times 3$ MIMO systems with and without CSIT for analog and digital combining as shown in Figure 4. The figure includes achievable rates for only the best resolution levels. Note that performances of the SIMO and the MIMO systems with no CSIT are the same for analog combining. We observe that the optimal resolution is 3-bin quantization for all scenarios investigated except for SIMO with digital combining for which 2-bin quantization optimal. As shown in the figure, analog combining achieves higher rates than digital combining for all the cases studied even though the MIMO system with digital combining has the advantage of spatial multiplexing. The benefit of spatial multiplexing can be seen from the gap between the rates of the SIMO and MIMO systems with digital combining. However, MIMO system with digital combining utilizes one-third ($1/N_r$) of the bandwidth used by the analog combining architecture. This is due to fact that the total ADC power budget in the case of digital combining is shared among antennas lowering the bandwidth usage. For the scenario investigated, optimal bandwidth usage for analog combining is $W=6.75$ GHz while the usage for the digital combining architecture is 2.25 Ghz and 3.37 Ghz for MIMO and SIMO respectively.
\[sim5\]
\[table\]
[|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|]{} & & & &\
-----------
$P_{tot}$
(mW)
-----------
&
-----------
Rate
(Gbits/s)
-----------
& $b$ & $N$ &
-------
W
(Ghz)
-------
&
-----------
Rate
(Gbits/s)
-----------
& $b$ & $N$ &
-------
W
(Ghz)
-------
&
-----------
Rate
(Gbits/s)
-----------
& $b$ & $N$ &
-------
W
(Ghz)
-------
&
-----------
Rate
(Gbits/s)
-----------
& $b$ & $N$ &
-------
W
(Ghz)
-------
\
100 & 5.52 & 5 & 1 & 7 & 2.16 & 2 & 1 & 1.73 & 5.40 & 5 & 1 & 7 & 2.21 & 2 & 1 & 1.73\
150 & 10.03 & 2 & 2 & 6.4 & 5.70 & 6 & 1 & 7 & 7.20 & 2 & 2 & 6.4 & 5.64 & 7 & 1 & 6.9\
200 & 13.09 & 5 & 2 & 7 & 11.97 & 6 & 2 & 6.67 & 9.21 & 5 & 2 & 7 & 8.40 & 6 & 2 & 6.67\
250 & 17.85 & 3 & 3 & 6.88 & 14.91 & 4 & 3 & 7 & 11.00 & 3 & 3 & 6.88 & 10.11 & 4 & 3 & 7\
300 & 20.51 & 2 & 4 & 6.4 & 17.68 & 11 & 3 & 7 & 11.98 & 5 & 3 & 7 & 11.58 & 11 & 3 & 7\
350 & 24.43 & 4 & 4 & 6.34 & 20.58 & 10 & 4 & 7 & 13.23 & 3 & 4 & 7 & 13.36 & 10 & 4 & 7\
400 & 28.07 & 3 & 5 & 6.05 & 22.47 & 9 & 5 & 7 & 14.24 & 5 & 4 & 7 & 14.85 & 9 & 5 & 7\
450 & 31.85 & 4 & 5 & 6.84 & 23.73 & 16 & 5 & 7 & 15.40 & 4 & 5 & 6.84 & 15.89 & 8 & 6 & 6.96\
500 & 36.40 & 3 & 6 & 6.88 & 25.62 & 15 & 6 & 7 & 16.31 & 3 & 6 & 6.88 & 16.73 & 15 & 7 & 3.38\
Next, we investigate the effect of the number of antennas on the achievable rates of the SIMO and the MIMO systems at 0dB SNR. The MIMO system has $N$ transmit and $N$ receive antennas, while for SIMO, $N_r=N$. We assume total ADC power budget of $20$ mW as in Figure 4. Our results are shown in Figure 5, where we only include the best quantization resolution (3-bins for most of the scenarios investigated, 2- and 3-bins for SIMO system with digital combining). We observe that the rate of the MIMO system with analog and digital combining increases with the number of antennas. In addition, although analog combining has higher capacity for all the cases, the optimal bandwidth of digital combining scales as ${1}/{N}$ as opposed to the constant bandwidth usage of analog combining.
\[sim6\]
\[sim6\]
Finally, we consider all components of the front-end receiver, namely the LNA, combiner, phase shifter, mixer and ADC, in computing the total receiver power consumption. We set $P_{LNA}=39$mW, $P_C=19.5$mW, $P_{PS}=19.5$mW [@Kramer], and $P_M=16.8$mW [@yu]. We optimize the rate of each MIMO architecture with analog and digital combining, with and without CSIT, over the number of antennas $N=N_t=N_r$, bandwidth and ADC resolution for a given total receiver power budget $P_{tot}$. We fix the SNR to be 0 dB. In Figure 6, we provide the achievable rates versus the total receiver power budget $P_{tot}$. In addition, Table I provides the maximum achievable rate in Gbits/s, the optimal resolution $b$, the optimal number of antennas $N=N_t=N_r$, and bandwidth $W$ in Ghz for different values of the total receiver power budget $P_{tot}$ under different combining and CSIT scenarios.
From Figure 6 we observe that digital combining with CSIT has higher rate than analog combining with CSIT, and the gap in between increases as the power budget increases. When there is no CSIT, analog combining performs better than the digital one only when the total power budget is greater than 350mW, and even in that case the performance improvement is very small. These observations are in contrast with Figure 4, where only the ADC power budget was considered. Comparing equations (\[pow1\]) and (\[pow2\]), we see that under a total receiver budget constraint, the power consumption of analog combining also starts increasing with the number of antennas, which helps emphasize the spatial multiplexing advantage of digital architectures. Table I shows that optimal resolution level of digital combining is always less than that of the analog one except for $P_{tot}=100$ mW. As expected, the optimal number of antennas increases with $P_{tot}$. However, we observe that the optimal bandwidth utilization may fluctuate as $P_{tot}$ increases. This is due to fact that in some cases it is better to spend the additional power to increase the number of antennas or the increase the resolution while reducing the bandwidth.
Conclusions {#s:conc}
===========
In this paper, we have studied a point-to-point MIMO mmWave communication system under two receiver architectures: Digital combining where each antenna element has a separate pair of ADCs, or analog combining where the signals are combined in analog domain so that only one ADC pair is required. For these receiver architectures we have investigated the effects of the bandwidth, resolution of ADCs and the number of antennas at the receiver on the achievable rate under the constraint of the total front-end receiver power or only the ADC power. Using AQNM, first, we have shown that there is an optimal bandwidth and resolution of ADC for the SISO channel. Then, we have derived achievable rates of the MIMO system with both analog combining and digital combining, with and without CSIT. When only ADC power budget is constrained, we have illustrated that depending on the operating regime analog combining may have higher rate but digital combining utilizes less bandwidth. However, when power consumption of all the receiver components are taken into account, we have shown that digital combining may have higher rate, and in some cases it can be optimal to increase the number of antennas or the increase the resolution while reducing the bandwidth.
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|
---
abstract: 'Several intriguing phenomena, unlikely within the standard inflationary cosmology, were reported in the cosmic microwave background (CMB) data from WMAP and appear to be uncorrelated. Two of these phenomena, termed CMB anomalies, are representative of their disparate nature: the North-South asymmetry in the CMB angular-correlation strength, inconsistent with an isotropic universe, and the cold spot, producing a significant deviation from Gaussianity. We find a correlation between them, at medium angular scales ($\ell = 11 - 20$): we show that a successive diminution of the cold spot (absolute-value) temperature implies a monotonic decrease of the North-South asymmetry power, and moreover we find that the cold spot supplies 60% of such power.'
author:
- Armando Bernui
title: 'Is the cold spot responsible for the CMB North-South asymmetry?'
---
Introduction {#Introduction}
============
Successive data releases from the Wilkinson Microwave Anisotropy Probe (WMAP) [@wmap] have confirmed the validity of the standard inflationary cosmological model $\Lambda$CDM, which asserts that the observed cosmic microwave background (CMB) temperature fluctuations are a stochastic realization of an isotropic Gaussian random field on the celestial sphere. This means that the CMB sky should exhibit statistical isotropy and statistical Gaussianity attributes.
Close scrutiny of the CMB WMAP maps have revealed highly significant departures both from statistical isotropy and from Gaussianity at large and medium angular scales. Evidences of an anomalous power asymmetry of the CMB angular correlations between the northern and southern ecliptic hemispheres (termed the NS-asymmetry) indicate that the CMB temperature field is inconsistent with the statistical isotropy expected in the model [@NS-asymmetry]. Another intriguing detection concerns an anomalously cold and large spot (termed the cold spot), centered at $(l,b) \simeq (209^{\circ},-57^{\circ})$ in galactic coordinates, which causes a significant deviation from Gaussianity [@cold-spot].
Many attempts have been made to explain these phenomena, termed CMB anomalies, in particular, within the standard inflationary $\Lambda$CDM scenario where they are unlikely at $\lesssim 1 \%$ of probability. Hypotheses like instrumental noise, systematic errors (e.g., in the mapmaking process), the inhomogeneous exposure function of the probe, incomplete sky-data (due to the cut-sky mask), unmodeled foreground emissions [@foregrounds], and physical mechanisms that break statistical isotropy (e.g., during the epoch of inflation or during the decoupling era, as for instance, primordial magnetic fields) [@mechanisms], have been extensively investigated. Unfortunately, none of them seems to explain satisfactorily the reported CMB anomalies.
Another way to comprehend these phenomena is to look for possible correlations between them, or their absence. Clearly, finding a cause-effect relationship between CMB anomalies would simplify the search for their origin. For instance, [@correl-anom1] proved the absence of correlation between the alignment of low order multipoles and the observed lack of CMB angular correlations on scales $> 60^{\circ}$. On the other hand, [@correl-anom2] found that the alignment of the CMB quadrupole and octopole is not responsible for the anomalous NS-asymmetry at large angles. Likewise, using needlets [@correl-anom3] detected anomalous spots in the needlets’ coefficients map. However, it seems possible that one of such spots could be caused by the CMB cold spot. In such a case the hemispherical asymmetry found in the needlets’ power spectrum [@correl-anom3] could be related to the CMB cold spot.
In this work we show that the cold spot is responsible for 60% of the NS-asymmetry strength in WMAP maps at medium angular scales (i.e., maps with multipoles $\ell = 11 - 20$). Firstly, we prove that the NS-asymmetry is present at 94%$-$98% C.L. in these maps. Then we show that gradually reducing the cold-spot temperature (turning it less cold) turns the NS-asymmetry effect less and less statistically significant. When the cold spot is suppressed the statistical significance of the NS-asymmetry phenomenon goes to the level found in the average from statistically-isotropic Gaussian CMB maps.
WMAP data {#WMAP data}
=========
Substantial efforts done by the WMAP science team to minimize foregrounds and to limit systematic errors resulted in the five-year foreground-reduced single-frequency Q, V, and W band-maps [@wmap]. Our aim is to investigate the angular distribution and statistical features of the CMB temperature field in these Q, V, and W maps at medium angular scales, that is, maps with multipole components $\ell = 11 - 20$. To obtain such data we first perform the multipolar decomposition of the original maps, applying the recommended KQ75 mask [@wmap], by using the [anafast]{} code [@Gorski]. After that, we select the multipole components $11 \leq \ell \leq 20$ and generate the corresponding CMB map with the [synfast]{} code [@Gorski].
In the top panel of Fig. \[fig1\] we exhibit the V map containing the multipoles $\ell = 11 - 20$. The Q and W maps are fully similar to this V map, as corroborated by Pearson’s coefficient $p_{_{\mbox{\sc\footnotesize ab}}}$ that measures the pixel-to-pixel correlation between A and B maps: $p_{_{\mbox{\sc\footnotesize vq}}}=0.9985$, $p_{_{\mbox{\sc\footnotesize vw}}}=0.9982$, and $p_{_{\mbox{\sc\footnotesize qw}}}=0.9976$. In this V map one clearly observes a large blue spot in the lower right corner: this is the cold spot with radius about $8^{\circ}$ and centered at $(l,b) \!\simeq\! (209^{\circ},-57^{\circ})$.
A statistical analysis (outside the KQ75 region) reveals that the skewness of the Q, V, and W maps is -0.124, -0.125, and -0.127, respectively, indicating that the temperature distribution is skewed to the left with a longer tail for negative CMB values. For comparison, notice that Monte Carlos Gaussian maps (described in detail below) have skewness mean: $0.0012 \pm 0.0853$. Notice also that in less than 5% of the Monte Carlos Gaussian maps one finds skewness values larger than those values found in WMAP data and this fact just represent the ergodicity of the data. For completeness, we shall analyze the temperature distribution in original and modified WMAP maps, termed X-cases, in which the cold-spot temperature is reduced. Consider the set of $n$ cold-spot pixels, that is, those pixels of the WMAP map belonging to the cap centered at $(209^{\circ},-57^{\circ})$ and within $8^{\circ}$ of radius. Let $\{ T_i^{\mbox{\bf\tiny CS}}\,;\, i=1,\ldots,n \}$ be the set of cold spot pixels’ temperatures of a WMAP map: the X-case corresponds to reducing these temperatures to X% of the original ones, while leaving all the other map pixels intact. For instance, X $\!$= $\!$90 means that the cold-spot pixels’ temperatures were reduced to the values $0.9 \times T_i ^{\mbox{\bf\tiny CS}}$. Clearly, X=100 or 100%-case refers to the original WMAP map. For X = 0 we replace the cold-spot temperature by the mean temperature of the WMAP map outside the KQ75 region, that is, $\sim -10^{-5\,}$mK.
We exhibit the temperature distribution for the V map in the bottom panel of Fig. \[fig1\], although in an atypical format in order to enhance possible deviations from a Gaussian distribution. In the horizontal axis we plot the square of the temperatures $T^2$ and in the vertical axis we have the number of pixels in logarithmic scale. In this picture, a Gaussian curve $= (1/\sigma_{\mbox{\sc\footnotesize g}} \sqrt{2\pi}) \,
e^{-\frac{1}{2} (T^2 / \sigma_{\mbox{\sc\tiny G}}^2)}$ is transformed into a straight line. We plotted four data sets: (i) The straight (blue) line is the expected Gaussian distribution, for positive and negative CMB temperature fluctuations because a Gaussian is symmetric with respect to zero, with variance $\sigma_{\mbox{\sc\footnotesize g}}^2 = 4.65 \times 10^{-4}$ mK$^{2}$. (ii) The histogram (red) curve corresponds to the negative temperatures of the V map. (iii) The dashed (violet) line corresponds to the positive temperatures of the V map. (iv) The triangles represent the negative temperatures of the case X=70 for the V map. As predicted by the skewness value, the (red) histogram curve reveals a non-linear tail due to pixels with anomalously large negative temperatures $T \lesssim -0.07$ mK (or $T^2 \gtrsim 0.0045$ mK$^{2}$), and we want to know if the cold spot [@cold-spot] is related to this effect. Instead of masking the cold spot in a given WMAP map (which, at the angular scales we are interested, shall introduce a spurious signal in the next NS-asymmetry analysis) we investigate several X-cases. In fact, we discover that when the cold-spot temperature is 70% of its original value, or lower, the non-linear tail disappears. Consistently, in the 70%-case the skewness is -0.0323, -0.0294, and -0.0334, for the Q, V, and W maps, respectively. In conclusion, the diminution of the cold-spot temperature implies the suppression of the non-linear tail in the temperature distribution plot.
![\[fig1\] Top: This is the V map with multipole components $\,\ell = 11 - 20$, obtained using the KQ75 mask. Bottom: This plot shows the temperature distribution of this V map, considering data outside the KQ75 region, where: (i) the straight (blue) line is the expected Gaussian distribution with variance $\sigma_{\mbox{\sc\footnotesize g}}^2 = 4.65 \times 10^{-4}$ mK$^{2}$, (ii) the histogram (red) curve corresponds to the negative pixel’s temperatures, (iii) the dashed (violet) line corresponds to the positive pixel’s temperatures, and (iv) the triangles represent the distribution of the negative pixels’ temperatures corresponding to a modified V map where the cold-spot temperature is 70% of its original value. ](fig1a.ps){width="7.5cm" height="13cm"}
sigma-map method {#method}
================
Now we are interested in studying the effect of these cold-spot temperature changes on the angular-correlation strengths in WMAP maps. For this we use a geometrical-statistical method that leads us to quantify, in intensity and direction, the CMB angular correlations, in particular, to search for an hemispherical asymmetry (for details of this method see Ref. [@BFW]).
Let $\Omega_{\gamma_0}^J \equiv \Omega(\theta_J,\phi_J;\gamma_0)
\subset {\cal S}^2$ be a spherical cap region on the celestial sphere ${\cal S}^2$, of $\gamma_0$ degrees of aperture, with vertex at the $J$th pixel, $J = 1, \ldots, N_{\mbox{\footnotesize caps}}$, where $(\theta_J,\phi_J)$ are the angular coordinates of the $J$th pixel’s center. Both the number of spherical caps $N_{\mbox{\footnotesize caps}}$ and the coordinates of their centers $(\theta_J,\phi_J)$ are defined using the [healpix]{} pixelization scheme [@Gorski]. The union of the $N_{\mbox{\footnotesize caps}}$ spherical caps covers completely ${\cal S}^2$.
Given a pixelized CMB map, the 2-point angular-correlation function (2PACF) of the temperature fluctuations, $T=T(\theta,\phi)$, corresponding to the pixels located in the spherical cap $\Omega_{\gamma_0}^J$ is defined by $\mbox{\rm C}(\gamma)^J \equiv \langle\, T(\theta_i,\phi_i) T(\theta_{i'},\phi_{i'}) \,\rangle$, where $\cos\gamma = \cos\theta_i \cos\theta_{i'}
+ \sin\theta_i \sin\theta_{i'} \cos(\phi_i\!-\!\phi_{i'})$, and $\gamma \in (0,2\gamma_0]$ is the angular distance between the $i$th and the $i'$th pixels’ centers. The average $\langle \,\, \rangle$ in the above definition is done over all the products $T(\theta_i,\phi_i) T(\theta_{i'},\phi_{i'})$ such that $\gamma_k \equiv \gamma \in ((k-1)\delta,\, k\delta]$, for $k = 1,..., N_{\mbox{\footnotesize bins}}$, where $\delta \equiv 2\gamma_0 / N_{\mbox{\footnotesize bins}}$ is the bin width. We denote by $\mbox{\rm C}_k^J \equiv \mbox{\rm C}(\gamma_k)^J$ the value of the 2PACF for the angular distances $\gamma_k \in ((k-1)\delta,\, k\delta]$. Now, consider a scalar function $\sigma: \Omega_{\gamma_0}^J \subset {\cal S}^2 \mapsto {\Re}^{+}$, for $J = 1, \ldots, N_{\mbox{\footnotesize caps}}$, which assigns to the $J$ cap, centered at $(\theta_J,\phi_J)$, a real positive number $\sigma_{J} \equiv \sigma(\theta_J,\phi_J) \in \Re^+$. We define $\sigma_{J}$ as $$\label{sigma}
\sigma^2_J \equiv \frac{1}{N_{\mbox{\footnotesize bins}}}
\sum_{k=1}^{N_{\mbox{\footnotesize bins}}} (\mbox{\rm C}_k^J)^2 \,\, .$$ To obtain a quantitative measure of the angular correlations in a CMB map, we cover the celestial sphere with $N_{\mbox{\footnotesize caps}}$ spherical caps, and calculate the set of sigma values $\{ \sigma_{J}, \, J=1,...,N_{\mbox{\footnotesize caps}} \}$ using Eq. (\[sigma\]). Associating the sigma value $\sigma_J$ to the $J$th pixel, for $J=1, \ldots, N_{\mbox{\footnotesize caps}}$, one fills the celestial sphere with positive real numbers. Then, according to a linear scale (where $\sigma^{\mbox{\footnotesize min}} \rightarrow blue$, $\sigma^{\mbox{\footnotesize max}} \rightarrow red$), one converts this sigma-values map into a colored map: this is the sigma map. Finally, we find the multipole components of a sigma map $\sigma(\theta,\phi) = \sum_{\ell,\, m} A_{\ell\, m} Y_{\ell\, m}(\theta,\phi)$, and calculate its angular power spectrum $\{ \mbox{\sc S}_{\ell},\, \ell=1,2,... \}$, that is $$\begin{aligned}
\label{aps}
\mbox{\sc S}_{\ell} \equiv \frac{1}{2\ell+1} \sum_{m={\mbox{\small -}}\ell}^{\ell} \, |A_{\ell\, m}|^2 \, . \end{aligned}$$
Accordingly, the power spectrum $\mbox{\sc S}_{\ell}$ of a sigma map computed from a WMAP map provides quantitative information of its statistical anisotropy features as compared with the mean of the sigma-map power spectra obtained from simulated isotropic Gaussian CMB maps. With this aim we produced a set of $1\,000$ Monte Carlo (MC) CMB maps, which correspond to random realizations seeded by the $\Lambda$CDM angular power spectra [@wmap], with $\ell = 2 - 512$ ($N_{\mbox{\footnotesize\rm side}}=256$). From these full-spectrum MC maps we generate, similarly as we have done with the WMAP data, a set of MC maps containing the multipoles $\ell = 11 - 20$ ($N_{\mbox{\footnotesize\rm side}}=32$). We then calculate the sigma maps of these MC maps (hereafter sigma-maps MC) and their corresponding angular power spectra using eq. (\[aps\]). Last, we compute statistical significance by comparing the power spectra of the sigma-maps WMAP with the spectra from the sigma-maps MC. For these spectra, $\mbox{\sc S}_{\ell} \simeq 0$, for $\ell \ge 5$, thus we only consider the multipole range $\{ \mbox{\sc S}_{\ell}, \,\ell=1,\!...,5 \}$. $\mbox{\sc S}_{1}$ corresponds to the dipolar anisotropy strength.
![\[fig2\] Top: This is the sigma-map WMAP-V obtained using $\gamma_0 = 45^{\circ}$, $N_{\mbox{\footnotesize bins}}=45$, and $N_{\mbox{\footnotesize caps}}=3\,072$. The NS-asymmetry phenomenon is clearly seen in the uneven hemispherical distribution of the angular correlation’s strengths, depicted as red and blue pixels (large and small $\sigma_J$, respectively). Bottom: Angular power spectra of the sigma map shown above (100%-case) and sigma maps from several cases: X = 90, 70, 50, 30, and 0, where the cold-spot temperature in the V map is a fraction, X%, of its original value. The solid (dashed) line corresponds to the mean (95.4% CL) spectra of $1\,000$ sigma-maps MC. The statistical significance of the NS-asymmetry is revealed at 97.6% CL in the 100%-case, while for the 70%-case it is 91.5% CL. The dipole component of the above sigma map, displayed in the inset, points toward $(l,b) \simeq (225^{\circ},-45^{\circ})$. ](fig2a.ps){width="7.5cm" height="13cm"}
Results and Conclusions {#results and conclusions}
=======================
As observed in Fig. \[fig2\] top’s panel, this sigma-map WMAP-V clearly exhibits a dipolar red-blue region indicating a hemispherical asymmetry in the distribution of the angular correlations power, this is the NS-asymmetry anomaly at medium angular scales ($\ell=11 - 20$). In fact, in the bottom panel of Fig. \[fig2\] we show its angular power spectrum and confirm, for the 100%-case, that its dipole term is indeed anomalous, unexpected in 97.6% of the sigma-maps MC for the sigma-map V (97.6% and 97.5% for sigma-map Q and sigma-map W, respectively). Moreover, we studied the effects caused by a systematic reduction of the cold-spot temperature. Our results, for the cases X = 90, 70, 50, 30, and 0, shown also in the bottom panel of Fig. \[fig2\], illustrate that successively lowering the cold-spot temperature implies the monotonic decrease of the dipole intensity, $\mbox{\sc S}_{1}$, which in turn means the loss of statistical significance of the NS-asymmetry phenomenon. Consistently, the triangles data corresponding to the 70%-case shown in Figs. \[fig1\] and \[fig2\], illustrate the correlation found here: the weakening of non-Gaussian features in the cold spot implies the fading of the anomalous NS-asymmetry in WMAP maps. A correlation between a non-Gaussian feature (i.e., the cold spot) and a large-scale anisotropy (i.e., the NS-asymmetry) has been established. Another crucial issue concerns the case X=0, which corresponds to the absence of the cold spot in CMB maps. For X=0 the dipole intensity $\mbox{\sc S}_{1}$ decreases 59.6% with respect to the case X=100 for the sigma-map V (while it decreases 58.7% and 60.4% for the sigma-map Q and sigma-map W, respectively), achieving a value very close to the mean obtained averaging the sigma-maps-MC spectra. Additionally, we have investigated nine other spots, the coldest and hottest ones in the southern hemisphere of WMAP maps, to know their influence on NS-asymmetry phenomenon. Considering separately the case X=0 for these nine spots, our results show that none of them reduces the confidence level of the hemispherical asymmetry below 90% C.L. In conclusion, the (non-Gaussian) cold spot is responsible for 60% of the anomalous NS-asymmetry power observed in WMAP data, at medium angular scales.
maps [$\!\!\backslash$]{} $\!\!(N_{\mbox{\footnotesize caps}}, \gamma_{_0})$ (768,$30^{\circ}$) (768,$45^{\circ}$) (3072,$45^{\circ}$) (768,$60^{\circ}$)$\!$
------------------------------------------------------------------------------ -------------------- -------------------- --------------------- ------------------------
Q 94%$-$96% 97%$-$98% 97%$-$98% 94%$-$96%
V 94%$-$96% 97%$-$98% 97%$-$98% 94%$-$96%
$\:$W 94%$-$96% 97%$-$98% 97%$-$98% 94%$-$96%
: \[table1\] Robustness tests for sigma-map analyses using several parameters. Here are the confidence levels (CL) for $\mbox{\sc S}_{1}$ in the NS-asymmetry analyses in Q, V, and W maps ($N_{\mbox{\footnotesize side}}=32$, pixel size $1.8^{\circ} \!\Rightarrow$ $\delta \equiv 2\gamma_{_0} / N_{\mbox{\footnotesize bins}}$ [$\gtrsim$]{} $2^{\circ}$). The C.L. intervals correspond to different binning tests: $N_{\mbox{\footnotesize bins}} = 15, 30, 45, 60$, provided $\gamma_{_0} \!\ge \!N_{\mbox{\footnotesize bins}}$ to minimize the statistical noise in $\sigma_{J}$ calculations.
It is well known [@wmap] that CMB foregrounds are frequency dependent. For this, in order to find out possible foreground signatures in our findings we used the Q, V, and W band maps. In fact, robustness tests (see Table \[table1\]) and sigma-map outcomes studying these three maps are not significantly different, meaning that our results are unlikely due to residual foregrounds. Pixel noise in WMAP maps is another possible source of incorrect results. However, at the angular scales we are working, pixel noise artifact is dominated by the cosmic variance [@wmap], and this effect was already included in Monte Carlo maps. In fact, pixel noise in WMAP data for $\ell=11 - 20$ is of the order $1\mu$K, while temperature uncertainty due to cosmic variance is $\sim 5\mu$K. Consequently, the robustness of our analyses supports the validity of our NS-asymmetry results.
There is strong evidence for NS-asymmetry at several angular scales in WMAP data [@NS-asymmetry]. In addition, we know that different angular scales in the CMB field encode information corresponding to distinct physical phenomena. Therefore, it is highly plausible that NS-asymmetry at different scales has not a common origin. One manifestation of this is the fact that the dipolar direction of hemispherical asymmetry is distinct for large ($\ell = 2 - 10$) and medium ($\ell = 11 - 20$) angular scales, i.e., $\sim(220^{\circ},-20^{\circ})$ [@NS-asymmetry] and $\sim(225^{\circ},-45^{\circ})$, respectively. Here we focused on those angular scales compatible with cold-spot dimensions, that is $\theta \simeq 9^{\circ} \!-\! 16^{\circ}$ [@cold-spot] (i.e., $\ell=11 - 20$, where $\ell \sim \pi / \theta$). Clearly, the effect of the cold spot does not starts at $\ell=11$ neither does it end at $\ell=20$. For $\ell \le 10$ the cold-spot influence contends with colder and hotter spots, this is because the angular power spectra $C_{\ell} \propto 1/(\ell (\ell+1))$ (Sachs-Wolfe effect) is larger for smaller $\ell$, and seems difficult to cause the NS-asymmetry at large-angles. On the other hand, the impact of the cold spot on NS-asymmetry for smaller scales, $\ell \ge 21$, is unknown and deserves further investigation. These facts lead us to conclude that it might be possible that distinct phenomena are causing the reported hemispherical asymmetry at several angular scales [@NS-asymmetry], and what is really happening is that different phenomena are predominant at distinct angular scales. Of course, the origin of the cold spot, including the possibility that it is just an anomalous statistical fluke, is still an open question.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank A. F. F. Teixeira, M. J. Rebouças, and G. D. Starkman for critical reading and suggestions. This work was supported by CNPq (309388/2008-2). We are grateful for the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA) [@wmap]. Some of the results in this paper have been derived using the [healpix]{} package [@Gorski].
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abstract: 'Precise calibration is a must for high reliance 3D computer vision algorithms. A challenging case is when the camera is behind a protective glass or transparent object: due to refraction, the image is heavily distorted; the pinhole camera model alone can not be used and a distortion correction step is required. By directly modeling the geometry of the refractive media, we build the image generation process by tracing individual light rays from the camera to a target. Comparing the generated images to their distorted – observed – counterparts, we estimate the geometry parameters of the refractive surface via model inversion by employing an RBF neural network. We present an image collection methodology that produces data suited for finding the distortion parameters and test our algorithm on synthetic and real-world data. We analyze the results of the algorithm.'
author:
- '[Sz]{}abolcs Pável'
- '[Cs]{}anád Sándor'
- 'Lehel [Cs]{}ató'
title: Distortion Estimation Through Explicit Modeling of the Refractive Surface
---
Introduction
============
Video-cameras are widely used in different robotic and automated driving applications. These applications frequently employ the pinhole camera model to make the association between the outside world and image pixels. The parameters of the model are found using a camera calibration procedure: done either statically, using calibration patterns (e.g. checkerboards, see Fig. \[fig:boards\]), or with self-calibration, where the geometric constraints of the scene are leveraged. The camera model is a first step towards image analysis [@szeliski2010computer]: it considers the way a 3D scene – including geometry, lights, materials, etc. – is mapped to a 2D image. This complex mapping is further split into three stages: a geometric, a photo-metric and a sampling stage. Our work considers the first stage, the geometric part: the association of 3D points with 2D pixels by explicitly modeling the refraction caused by a refractive material present between the camera and the object (e.g. protective covers made out of glass or transparent plastic materials). To find this association, we have to follow light rays hitting a given pixel of the sensor. Due to the refractive material, tracing is more complex: as light enters or leaves a denser media, it changes direction, resulting in deviations from the pinhole model; called *image distortions*, as shown in Fig. \[fig:schematic\](a). We construct the *forward model* $f_{\theta}({\boldsymbol{p}}): \Omega \rightarrow \mathbb{R}^3$, where – knowing the camera parameters, the refractive media and scene characteristics, jointly denoted as $\theta$ – we map a pixel ${\boldsymbol{p}}$ from the image $\Omega \subset \mathbb{R}^2$ to a point in the scene. We implement this function as a raycasting algorithm – see Sect. \[sec:raycast\] –, allowing us to generate images given a set of parameters. After constructing the forward model, by using model inversion, we fit the parameters of the refractive media to a set of observations, given as displaced points. We build an RBF-network [@haykin2009neural] based parametric model of the thickness of the refractive media and use ML estimation [@bishop2006] to infer the optimal parameters that generated the distortions.
Our contribution in this work is three-fold: (1) we introduce a parametric model of the refractive media and derive the geometric image formation process in the presence of the refractive media (2) we provide a methodology to estimate the model parameters using a static calibration setup with checkerboard patterns (3) we present our experiments where we estimate the image distortions induced by a conic glass surface in a synthetic, as well as in a real-world scenario.
Related Work {#sec:related}
============
Most 3D computer vision algorithms assume that the pinhole camera model precisely describes image optics but this is not the case in the presence of *geometric* image distortions, where pixels are displaced compared to their expected positions. Without estimating and correcting the images subject to these distortions, 3D computer vision algorithms often loose performance or simply fail to produce meaningful results. To put our work into context, we review some methods for correcting image distortions.
Estimating image distortions is usually a building block of the **camera calibration algorithm**. These algorithms can be static or can use self-calibration. Static calibration [@tsai1987versatile; @zhang2000flexible] techniques use objects with previously known patterns and sizes to extract camera – distortion – parameters. These algorithms provide the highest accuracy, but require the presence of a calibration object. Self-calibration methods [@devernay2001straight; @cefalu2016structureless; @fitzgibbon2001simultaneous] use geometric constraints of the imaging system to estimate camera parameters. These methods are less accurate then static methods, but are more flexible and can be used for online calibration. In our method we perform static calibration using a checkerboard pattern.
Specific algorithms use explicit **distortion models** of differing complexity; the most popular being Brown’s polynomial distortion model [@brown1966decentering] for radial and tangential distortions. The division model [@fitzgibbon2001simultaneous] uses an even simpler model with a single parameter to estimate radial distortions only. Using the radial distortion model as above, one can estimate the center of the distortions [@hartley2007parameter]. Fish-eye lenses create specific image distortions and the *field of view* model explicitly considers those distortions [@devernay2001straight], describing the FOV of an ideal fish-eye lens. Lastly, we mention the *rational function* distortion model [@claus2005rational], the most similar to our model, as they lift the 2D pixels into the 3D space and associate rays to individual pixels. Our method also traces 3D light rays, but we instead directly parameterize the refractive surfaces which generate the distortions.
The work of Agrawal et. al. [@agrawal2012theory] sets up distortions for images taken through flat refractive surfaces (they use a water tank for their experiments), therefore modeling explicitly the **distortions from light refraction**. They use the theory of non-central cameras [@sturm2004generic] and multi-view geometry in the presence of refractive media [@chari2009multiple]. Morinaka et. al. [@morinaka20183d] presents static camera calibration and 3D reconstruction in challenging setups like images taken through a wine glass. They use the “raxel” imaging model [@grossberg2005raxel] with two calibration planes, with polynomial mappings between image pixels and corresponding points on the planes. This model is closest to ours, the main difference being that in their method the refractive media is treated as a black box, while we directly model the refractive surface, giving us a global model.
Refractive Surface Model {#sec:ref_surf}
========================
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We model our refractive media as a “thick” cone slice, as in Fig. \[fig:schematic\](a). The inner and outer cones have the same aperture, and the centers are such that the thickness $\Delta r$ of the media is constant. We constrain the position of the cone such that the main axis is parallel with the $y$-axis of the camera coordinate system and that the cones shrink in the positive (downwards pointing) $y$ direction, shown in Fig. \[fig:schematic\](b). We parameterize the cone surface with its height relative to the apex $s_1 \in [0, h]$ and a polar angle $s_2 \in [-\pi, \pi]$, where a polar angle of $0$ describes the points on the $YOZ$ plane, with positive $z$ values. We use bold notation for the two-dimensional vectors: ${\boldsymbol{s}}=(s_1,s_2)$.
![Sample offsets generated as a linear combination of $4\times4$ RBF kernels distributed on a regular grid. Dot sizes denote the – positive – kernel amplitudes.[]{data-label="fig:rbfsample"}](img/rbfsample.pdf)
To model the uneven refractive media – implicit the distortions – we add a parametric surface in the radial direction to the cone; this radial offset is defined as a Radial Basis Function network (RBF) [@park1991universal] where the inputs are the cone coordinates. We choose an RBF network because they are universal function approximators given a sufficient number of centers, and can be used to estimate arbitrary surfaces. In practice we show that a relatively small number of parameters is enough to model complex surfaces – see Sect. \[sec:exp:synth\]. The RBF centers $\left\{{\boldsymbol{s}}_{ij}\right\}$, ${i,j=\overline{1, N}}$ are on a regular grid over the input region, as shown in Fig. \[fig:schematic\](b). We keep the RBF centers fixed and tune the amplitudes $a_{ij} \in \mathbb{R}$. The radial offset $\Phi({\boldsymbol{s}}^\prime)$ at a given normalized cone point ${\boldsymbol{s}}^\prime$ is defined as the output of the RBF network with Gaussian kernels, an example of which is shown in Fig. \[fig:rbfsample\]: $$\Phi({\boldsymbol{s}}^\prime) = \sum_{i,j=1}^{N} a_{ij} \mathrm{RBF}({\boldsymbol{s}}_{ij},{\boldsymbol{s}}^\prime),
\quad \text{where}\
\mathrm{RBF}( {\boldsymbol{s}} , {\boldsymbol{s}}^\prime) =
\exp\left( -\frac{\|{\boldsymbol{s}} - {\boldsymbol{s}}^\prime\|^2}{2 \beta} \right)$$ When computing the Cartesian coordinates for a point on the cone, parameterized by a height $s_1$ and an angle $s_2$, first we compute the RBF offset $\Phi({\boldsymbol{s^\prime}})$, and then add this offset to the radius. The surface normals of the outer cone can be computed as the cross product of the two partial derivatives of the Cartesian coordinates w.r.t. the parameters. This cross product also has a dependence on the amplitudes associated with the RBF centers, which will be used as the model parameters during minimization. Changing the RBF amplitudes causes a change in the surface normals, which in turn changes the direction of the refracted light rays, and by a consequence the direction and length of the distortion vectors.
Raycasting Model {#sec:raycast}
================
The ray casting model describes how we associate a pixel from the image with a 3D point on an object, in our case on a checkerboard pattern. In a distortion-free setup this can be achieved using the pinhole camera model, which uses a perspective projection and a set of linear operations to describe this relationship. In the presence of a refractive surface this simple perspective geometric description does not hold, and we need additional steps to associate pixels with world points. The complete computational graph can be seen on Fig. \[fig:depgraph\]. Each ray starts at the camera center and we assume that the intrinsic camera parameters – the focal lengths and the principal point, ie. the intrinsic camera matrix – are known. All 3D points are expressed in the camera coordinate system. Using the camera intrinsics, we can convert any pixel coordinate ${\boldsymbol{p}}$ to metric coordinates, which after normalization correspond to the direction vector ${\boldsymbol{r_{cam}}}$ of the light ray that passes through the selected pixel.
The light ray coming from the camera first hits the inner side of the refractive surface. Using the fact that the inner surface is a regular cone, we first compute the intersection with the cone ${\boldsymbol{x}}_i$ and its normal ${\boldsymbol{n}}_i$. The direction ${\boldsymbol{r}}_m$ of the refracted light-ray inside the media is computed using Snell’s law [@pharr2016physically]. Knowing the geometry of the refractive body and the new direction of the refracted ray, we first identify the location ${\boldsymbol{x}}_o$ where the ray hits the outer surface, and is refracted for the second time. Using the surface normal ${\boldsymbol{n}}_o$ at this point we compute the direction of the outgoing light-ray, which we denote with ${\boldsymbol{r}}_o$. Note that this second refraction is modulated by the *direction* of the normal, that is parameterized by the RBF network. At the same time, we ignore the changes caused by the RBF network in the *thickness* of the material when computing ${\boldsymbol{x}}_o$ and we argue that this approximation holds, as the offsets are significantly smaller than the distance between the two cones (the thickness of the media), and the difference is negligible.
Finally, this outgoing light ray hits the calibration target – in our case the checkerboard pattern – whose position is defined through a 3D rotation and the 3D translation of the board center relative to the camera coordinate system. The intersection point ${\boldsymbol{x}}_t$ is computed as an intersection of a line and a plane. For an easier handling of the checkerboard, we define a local 2D coordinate system on the object plane, with its origin at the board center, and the two axes being the horizontal and vertical directions of the square grid. We denote the local coordinates of a 3D point ${\boldsymbol{x}}_t$ as ${\boldsymbol{x}}_{cb}$.
Optimization of the Surface Parameters {#sec:optim}
======================================
The estimation of image distortions is equivalent to finding the surface parameters that generated a set of calibration images. We use a square checkerboard pattern as calibration target, and take multiple images of the same target in different positions. We use gradient descent minimization to find the amplitudes ${\boldsymbol{a}} = \left\{a_{ij}\right\}$, $i,j=\overline{1, N}$, of the RBF centers, the parameters of the refractive surface. During the minimization all other parameters, including the camera intrinsics, the sizes of the inner and outer cones, as well as the calibration pattern pose and size are assumed to be known.
For each calibration image ${\boldsymbol{I}}^k$, $k = \overline{1, N_i}$ we find the pixel coordinates and ordering of the checkerboard pattern corners $\left\{{\boldsymbol{p}}_{ij}^k\right\}$, $i,j=\overline{1, N_c}$. The corresponding local coordinates of the detected corners on the object plane – denoted with $\left\{{\boldsymbol{x}}_{ij}^{cb}\right\}$, $i,j=\overline{1, N_c}$ – are given by their distance from the board center. Let $f_{{\boldsymbol{a}},k}(\cdot)$ be the raycasting function described in Sect. \[sec:raycast\], parameterized by the RBF amplitudes ${\boldsymbol{a}}$, which takes an input pixel, and computes the local coordinates of the corresponding world point on the $k-th$ target checkerboard pattern. Then the loss function used for the minimization is the ${\cal{L}}_2$ loss between the local coordinates of a corner estimated by the raycast function, and the ground-truth local coordinates of the corners: $$\mathcal{L}({\boldsymbol{a}}) = \sum_{k=1}^{N_i} \sum_{i,j = 1}^{N_c} \left\lVert f_{{\boldsymbol{a}}, k} \left({\boldsymbol{p}}_{ij}^k \right) - {\boldsymbol{x}}_{ij}^{cb} \right\rVert^2$$ The optimal parameters ${\boldsymbol{a}}^\star$ are found through a Gradient Descent minimization of the loss function: $${\boldsymbol{a}}^\star = \arg\min_{{\boldsymbol{a}}} \mathcal{L}({\boldsymbol{a}})$$ The optimization algorithm including the ray-casting model is implemented using the PyTorch [@paszke2017automatic] deep learning framework. This framework makes the implementation easier as it provides backward automatic differentiation and implements gradient descent minimization.
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Experiments {#sec:exp}
===========
We evaluate our algorithm on two data sets: a noise-free synthetic one and a real experimental setup. In the synthetic case we show that our algorithm is capable of finding the optimal parameters that generated a given image even for large irregularities on the outer surface. In the second case we present an experimental setup, and show that the algorithm is able to reduce reconstruction errors in real-world scenarios.
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Synthetic Dataset {#sec:exp:synth}
-----------------
In a first set of experiments we applied the forward image generation model to render synthetic images. We set the parameters of the camera, refractive surface, and the checkerboard pattern to similar values as in the real-world experiment, as shown in Fig. \[fig:boards\](a).
In our first synthetic experiment, using an RBF grid of size $4\times4$, we sample the amplitudes ${\boldsymbol{a}}$ from a Gaussian with mean $\mu_a \in [\num{1e-6}, \num{1e-4}]$ and standard deviation $\sigma_a = \mu_a/4$. With the sampled surface we generate $10$ synthetic images using random positions for the calibration target – as shown in Fig. \[fig:boards\]. Using the batch of $10$ images we run the optimization from Sect. \[sec:optim\] for 500 steps and store the final error as the *root mean squared error* between the predicted and ground-truth checkerboard corners.
For each amplitude distribution we repeat the whole process $10$ times and show the results in Fig. \[fig:err\_synth\_ncenter\](a): *Pinhole model* is the error without any distortion model, while *Regular cone* and *Cone + RBF* show the results using a *perfect* cone and one where parameters are inferred. We see that with small RBF amplitudes the error is a result of the cone geometry, while with higher amplitudes the errors due to the uneven surface dominate the distortion. An important conclusion is that for this case, where is no observation noise, the optimization algorithm works well, reducing the errors almost to zero.
In the second experiment we generate random refraction patterns using $10\times10$ RBF centers with amplitudes drawn randomly from Gaussian distributions, and show the errors after performing the optimization – corresponding to the *Cone + RBF* case from Fig. \[fig:err\_synth\_ncenter\](a). Instead of changing the amplitudes, we vary the numbers of RBF centers – both horizontally and vertically, results are shown in Fig. \[fig:err\_synth\_ncenter\](b). We observe that – in this artificial setup – the refractive surface is well approximated with a smaller complexity model: $7\times5$ parameters are enough to approximate the generated surface (the asymmetry is due to our setup of the refractive surface, which is a cone with larger horizontal curvature).
Real Dataset
------------
Set $RMSE$ initial () $RMSE$ final () Rel. imp.
----- ------------------- ----------------- -----------
1 0.1364 0.0772 43.35%
2 0.1649 0.0941 42.90%
3 0.1570 0.0973 38.04%
: Root Mean Squared Errors for the 3 real sets of images. Initial error considers no distortion model, final error is obtained using Cone + RBF surface model with optimized parameters. Last column shows the relative improvement.
\[tab:rmse\_real\]
For a real-world experiment we use a Raspberry Pi Camera Module v2 to capture the checkerboard images. The camera has a $3.68 \times 2.76$ sensor and registers images on a $3280 \times 2464$ pixel resolution. Prior to the experiment we calibrated the camera using Zhang’s method [@zhang2000flexible] and we registered a $2558.36$ pixel focal length and a principal point at the $(1666.03, 1273.65)$ location. After calibration we placed a cone shaped glass in the front of the camera, with approximate parameters shown on Fig. \[fig:schematic\](b). In our experiments we use $45$ images of a checkerboard pattern that were randomly split into $3$ non-overlapping sets of $15$ items each, which we will refer to as Sets 1, 2 and 3.
Since the positions of the checkerboard patterns are unknown, we have to estimate them. We do this in two steps: (1) we estimate the object pose by running Zhang’s method on the distorted images, while we fix the camera intrinsic parameters to the calibrated values; and (2) we apply the *perfect* cone refraction model and with the same objective function we minimize for the calibration pattern positions. Keeping the above values fixed, we then run the optimizer for the RBF center amplitudes.
With our real dataset we use $8 \times 8$ RBF centers – found using experimentation as in Fig. \[fig:err\_synth\_ncenter\] – and run the minimization for $1000$ steps and we report results for the different sets separately. Table \[tab:rmse\_real\] shows the RMSE for the 3 sets of images. The errors can be interpreted as the 3D distance in centimeters between the ground-truth and the predicted position of the same corner on the checkerboard pattern. The first column shows the mean error in the case where we consider no distortion model at all, while the second column shows the results after the minimization using the *Cone + RBF* surface model. Last column shows the relative improvements between the two error values. Our method is able to improve the 3D distances in each case, resulting in a more precise generative model of the image formation process. Fig. \[fig:err\_scatter\] shows all corner errors in the 2D local coordinate system of the checkerboard, with different colors for different images. We see that in the uncorrected scatter plot (a) for each image – denoted by the color – there is a dominant direction, which is caused by the horizontal curvature of the cone. We highlight that the effect of minimization – right subplot in Fig. \[fig:err\_scatter\] – is that the errors are around zero, without dominant directions for individual images.
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Analyzing Distortions
---------------------
Most of the distortion estimation methods directly model the pixel displacements on the image plane, and define a single, fixed distortion map for a given camera. In contrast, our model estimates the distortion map by explicitly modeling the refractive material and using a ray-tracing. We consider this advantageous since the – usually separate directional distortions – are given a unified and *consistent* generative model. A consequence is that the physical model introduces a depth-dependent component in the distortion map, where the distance of a 3D point has to be known in order to find the image distortion, that is usually expressed in pixels.
In order to compute the image distortion vector, we start from a distorted pixel on the image. Using the distorted pixel ${\boldsymbol{p}}_d$, we use the ray-casting algorithm to obtain a 3D coordinate ${\boldsymbol{x}}_t$ of the object point at a given distance. The undistorted pixel coordinates ${\boldsymbol{p}}_u$ for an object point can be computed using the pinhole camera model. The distortion vector $\Delta{\boldsymbol{p}}$ for a given pixel ${\boldsymbol{p}}_d$ is given by the difference between the distorted and undistorted coordinates, i.e. $\Delta{\boldsymbol{p}} = {\boldsymbol{p}}_u - {\boldsymbol{p}}_d$.
Fig. \[fig:dist\_field\] shows the image distortion vector field using the optimal parameters for the images from Set 1. To better visualize our model and to be able to compute a distortion field, we fixed the object distance to for the whole scene; this is comparable to the range of the calibration objects on the images. Rays corresponding to the middle pixels are almost orthogonal to the refractive surface, resulting in little or no distortion, while the rays located on the two lateral sides have a large angle compared to the surface normal, resulting in large refraction and large image distortions.
A consequence of the physical *generative* model is the possibility to inspect the dependence of the distortions on pixel depth, as shown in Fig. \[fig:dist\_depth\]. The image shows the change of distortion norm for two individual pixels with coordinates $(820, 1232)$ and $(410, 1232)$, as well as the change of the mean distortion norm. We can observe, that the distortion norm shows a linear dependence on the inverse depth of the pixel. The figure shows the distortions for the inverse depth interval of $0.5$ (corresponding to a depth of ) to $10$ (depth of ). We can also observe that the slope of the line showing the change of distortion increases as we get closer to the edges of the image, where the change of distortion is $2.4$ pixels over the analyzed interval, compared to the change of $1.64$ pixels for the point closer to the image center.
Conclusions
===========
In our work we presented a model for geometric image distortions caused by refractive surfaces being placed between the camera and the scene. Based on an explicit model of the refractive surface, we presented a forward generative model of the distortions and the image generation; the generating process used the ray-casting mechanism. We assumed a conic refractive surface and used an additive model for the imperfections of the surface; the used model was a restricted Radial Basis Function network. Using model inversion and automated differentiation, we estimated the refractive surface with a set of checkerboard calibration target images. We validated our algorithm on synthetic and real-world data, and analyzed the observed image distortions.
The benefit of the method is that the model we proposed *is parametric*: despite being strongly non-linear and complex, using a small set of calibration images, the algorithm is able to find the global distortion map. A second important aspect is that the data collection methodology allows for heterogeneous data: by estimating the view angles, we can use the *whole* dataset for estimation.
A weak limitation of our method is the fixed base shape, which in the current formulation was a cone. We chose this shape as it was the closest to the actual object used in the experiments. A work-around to the strict constraint of the shape is to parameterize it and – added to the parameters of the irregularities – to optimize for an extended set of parameters. Evidently, as the number of free parameters grows, this introduces further modeling difficulties, as the degree of freedom increases and the optimization can be much harder.
Future work will focus on (1) developing a model that is general enough to be suitable for a wide range of applications without significant changes, and (2) will explore whether there is a possibility for our method to infer distortions using other types of inputs to the system. An extension possibility is to use – under specific constraints – other data, like the estimated optical flows with certain constraints on the collection procedure.
|
---
abstract: |
The large-distance behaviors of the random field and random anisotropy $O(N)$ models are studied with the functional renormalization group in $4-\epsilon$ dimensions. The random anisotropy Heisenberg $(N=3)$ model is found to have a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law $\langle
{\bf m}({\bf r}_1) {\bf m}({\bf r}_2)\rangle\sim| {\bf r}_1-{\bf
r}_2|^{-0.62\epsilon}$. The magnetic susceptibility diverges at low fields as $\chi\sim H^{-1+0.15\epsilon}$. In the random field $O(N)$ model the correlation length is found to be finite at the arbitrarily weak disorder for any $N>3$. The random field case is studied with a new simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization group equations.
address: |
Department of Condensed Matter Physics, Weizmann Institute of Science, 76100 Rehovot, Israel\
Landau Institute for Theoretical Physics, 142432 Chernogolovka, Moscow region, Russia
author:
- 'D.E. Feldman'
title: 'Quasi-long-range order in the random anisotropy Heisenberg model: functional renormalization group in $4-\epsilon$ dimensions'
---
Introduction
============
The effect of impurities on the order in condensed matter is interesting, since the disorder is almost inevitably present in any system. If the disorder is weak the short-range order is the same as in the pure system. However, the large-distance behavior can be strongly modified by the arbitrarily weak disorder. This happens in the systems of continuous symmetry in the presence of the random symmetry breaking field [@IM]. The first experimental example of this kind is the amorphous magnet [@HPZ; @OS]. During the last decade a lot of other related objects were found. These are liquid crystals in the porous media [@LQ], nematic elastomers [@NE], He-3 in aerogel [@He3] and vortex phases of impure superconductors [@HTSC]. The nature of the low-temperature phases of these systems is still unclear. The only reliable statement is that a long-range order is absent [@IM; @L; @P; @AW]. However, other details of the large-distance behavior are poorly understood.
The neutron scattering [@Xray] reveals sharp Bragg peaks in impure superconductors at low temperatures and weak external magnetic fields. Since the vortices can not form a regular lattice [@L] it is tempting to assume that there is a quasi-long-range order (QLRO), that is the correlation length is infinite and correlation functions depend on the distance slow. Recent theoretical [@RFXY; @12a] and numerical [@numXY] studies of the random field XY model, which is the simplest model of the vortex system in the impure superconductor [@HTSC], support this picture. The theoretical advances [@RFXY; @12a] are afforded by two new technical approaches: the functional renormalization group [@FRG] and the replica variational method [@MP]. These methods are free from drawbacks of the standard renormalization group and give reasonable results. The variational method regards the possibility of spontaneous replica symmetry breaking and treats the fluctuations approximately. On the other hand the functional renormalization group provides a subtle analysis of the fluctuations about the replica symmetrical ground state. Surprisingly, the methods give close and sometimes even the same results.
Both techniques were originally suggested for the random manifolds [@FRG; @MP] and then allowed to obtain information about some other disordered systems with the Abelian symmetry [@RFXY; @12a; @F; @EN; @RT]. Less is known about the non-Abelian systems. The simplest of them are the random field (RF) [@IM] and random anisotropy (RA) [@HPZ] Heisenberg models. The latter was introduced as a model of the amorphous magnet [@HPZ; @OS]. In spite of a long discussion, initiated by Ref. [@AP], the question of QLRO in these models is still open. There is an experimental evidence in favor of no QLRO [@B]. On the other hand recent numerical simulations [@num] support the possibility of QLRO in these systems. The only theoretical approach, developed up to now, is based on the spherical approximation [@P; @spher; @G] and predicts the absence of QLRO at $N\gg 1$ magnetization components. However, there is no reason for this approximation to be valid at $N\sim 1$.
In this paper we study the RF and RA $O(N)$ models in $4-\epsilon$ dimensions with the functional renormalization group. The large-distance behaviors of the systems are found to be quite different. Whereas in the RF $O(N)$ model with $N>3$ the correlation length is always finite, the RA Heisenberg $(N=3)$ model has a phase with QLRO. In that phase the correlation function of the magnetization obeys a power law and the magnetic susceptibility diverges at low fields.
The paper has the following structure. In the second section the models are formulated and a qualitative discussion is given. The third section contains a derivation of the one-loop renormalization group (RG) equations. The 4th section is devoted to the RF model. The absence of QLRO in that model at $N>3$ is shown with a new method, based on a rigorous inequality. This approach simplifies tedious RG calculations and can be useful in other problems. The RA case is considered in the 5th section. The stable RG fixed point corresponds to QLRO in $4-\epsilon$ dimensions at $N\sim 1$. In particular, at the weak disorder the correlation length is infinite in the low temperature phases of the RA XY ($N=2$) and Heisenberg ($N=3$) models. However, QLRO is absent at $N\ge 10$. In 4 dimensions the correlation functions of the RA Heisenberg model depend on the distance logarithmically. The exact result for the two-spin correlator is given by the $\ln^{-0.62}R$ law. The Conclusion contains a discussion of the results. Appendix A is devoted to a generalization of the Schwartz-Soffer inequality [@SwSo]. The generalized inequality is applied to the stability analyses of the RG fixed points. Appendix B describes a simple Migdal-Kadanoff renormalization group approach that reproduces qualitatively the results of the rigorous method. This approximation provides good estimations of the critical exponents of the RA XY and Heisenberg models. Appendix C includes some technical details of the functional RG in the spherical model.
Model
=====
To describe the large-distance behavior at low temperatures we use the classical nonlinear $\sigma$-model with the Hamiltonian
\[1\] H=d\^D x\[J\_\_([**x**]{}) \_([**x**]{}) + V\_[imp]{}([**x**]{})\], where ${\bf n}({\bf x})$ is the unit vector of the magnetization, $V_{\rm imp}({\bf x})$ the random potential. In the RF case it has the form
\[2\] V\_[imp]{}=-\_h\_([**x**]{})n\_([**x**]{}); =1,...,N, where the random field ${\bf h}({\bf x})$ has a Gaussian distribution and $\langle h_\alpha({\bf x})h_\beta({\bf
x}')\rangle=A^2\delta({\bf x}-{\bf x}')\delta_{\alpha\beta}$. In the RA case the random potential is given by the equation
\[3\] V\_[imp]{}=-\_[,]{}\_([**x**]{})n\_([**x**]{})n\_([**x**]{}); ,=1,...,N, where $\tau_{\alpha\beta}({\bf x})$ is a Gaussian random variable, $\langle\tau_{\alpha\beta}({\bf x})\tau_{\gamma\delta}({\bf
x}')\rangle=A^2\delta_{\alpha\gamma}\delta_{\beta\delta}\delta({\bf
x}-{\bf x}')$. The random potential (\[3\]) corresponds to the same symmetry as the more conventional choice $V_{\rm imp}=-({\bf hn})^2$ but is more convenient for the further discussion.
We assume that the temperature is low and the thermal fluctuations are negligible. The Imry-Ma argument [@IM; @P] suggests that in our problem the long-range order is absent at any dimension $D<4$. One can estimate the Larkin length, up to which there are strong ferromagnetic correlations, with the following qualitative RG approach. Let one remove the fast modes and rewrite the Hamiltonian in terms of the block spins, corresponding to the scale $L=ba$, where $a$ is the ultraviolet cut-off, $b>1$. Then let one make rescaling such that the Hamiltonian would restore its initial form with new constants $A(L), J(L)$. Dimensional analysis provides estimations
\[4\] J(L)\~b\^[D-2]{} J(a); A(L)\~b\^[D/2]{}A(a) To estimate the typical angle $\phi$ between neighbor block spins, one notes that the effective field, acting on each spin, has two contributions: the exchange contribution and the random one. The exchange contribution of order $J(L)$ is oriented along the local average direction of the magnetization. The random contribution of order $A(L)$ may have any direction. This allows one to write at low temperatures that $\phi(L)\sim A(L)/J(L)$. The Larkin length corresponds to the condition $\phi(L)\sim 1$ and equals $L\sim
(J/A)^{2/(4-D)}$ in agreement with the Imry-Ma argument [@IM]. If Eq. (\[4\]) were exact the Larkin length could be interpreted as the correlation length. However, there are two sources of corrections to Eq. (\[4\]). Both of them are relevant already at the derivation of the RG equation for the pure system in $2+\epsilon$ dimensions [@Pol]. The first source is the renormalization due to the interaction and the second one results from the magnetization rescaling which is necessary to ensure the fixed length condition ${\bf n}^2=1$. The leading corrections to Eq. (\[4\]) are proportional to $\phi^2 J, \phi^2 A$. Thus, the RG equation for the combination $(A(L)/J(L))^2$ reads
\[6\] ()\^2= ()\^2+ c()\^4, =4-D If the constant $c$ in Eq. (\[6\]) is positive the Larkin length is the correlation length indeed. But if $c<0$ the RG equation has a fixed point, corresponding to the phase with an infinite correlation length. As seen below, both situations are possible, depending on the system.
The large-distance behaviors of the RF and RA $O(N)$ models are known in two limit cases: $N=2$ and $N=\infty$. In the spherical limit ($N=\infty$) QLRO is absent (Appendix C, [@spher]) while the XY model possesses QLRO [@RFXY; @12a; @prim]. Hence, the ordering disappears at some critical number $N_c$ of the magnetization components. This critical number is larger in the RA model, since the fluctuations of the magnetization are stronger in the RF case. Indeed, in the RF model the magnetization tends to be oriented along the random field, whereas in the RA case there are two preferable local magnetization directions so that the spins tend to lie in the same semispace.
RG equations
============
In the previous section the RG equations are discussed from the qualitative point of view. Eq. (\[6\]) corresponds to the Migdal-Kadanoff approach of Appendix B. In the present section we derive the RG equations in a systematic way.
The one-loop RG equations for the $N$-component RF and RA models in $4+\epsilon$ dimensions were already derived in Ref. [@DF]. We can directly use that result, since the RG equations in dimensions $D<4$ can be obtained by just changing the sign of $\epsilon$. However, the approach [@DF] is cumbersome and we provide below a simpler derivation.
We use the method, suggested by Polyakov [@Pol] for the pure system in $2+\epsilon$ dimensions. This method is technically simpler and closer to the intuition than the other approaches. A disadvantage of the method is the difficulty of the generalization for the higher orders in $\epsilon$. This generalization requires the field-theoretical approach [@ZJ].
The same consideration as in the XY [@12a] and random manifold [@FRG] models suggests that near a zero-temperature fixed point in $4-\epsilon$ dimensions there is an infinite set of relevant operators. Let us show that after the replica averaging the relevant part of the effective replica Hamiltonian can be represented in the form
\[7\] H\_R=d\^D x\[\_a\_\_\_a\_\_a - \_[ab]{}\], where $a,b$ are replica indices, $R(z)$ is some function, $T$ the temperature. We ascribe to the field ${\bf n}$ the scaling dimension $0$. We also assume that the coefficient before the gradient term in (\[7\]) is $1/(2T)$ at any scale. Then in the $(4-\epsilon)$-dimensional space the scaling dimension of the temperature $\Delta_T=-2+O(\epsilon)$. Any operator $A$ containing $m$ different replica indices is proportional [@FRG] to $1/T^m$. Hence, the scaling dimension $\Delta_A$ of the operator $A$ satisfies the relation $\Delta_A=4-n+m\Delta_T+O(\epsilon)$, where $n$ is the number of the spatial derivatives in the operator. The relevant operators have $\Delta_A\ge 0$. Hence, the relevant operators can not contain more than two different replica indices. A symmetry consideration shows that all the possible relevant operators are included into Eq. (\[7\]). The function $R(z)$ is arbitrary in the RF case. In the RA case $R(z)$ is even due to the symmetry with respect to changing the sign of the magnetization.
The one-loop RG equations for the $N$-component model in $4-\epsilon$ dimensions are obtained by a straightforward combination of the methods of Refs. [@FRG] and [@Pol]. We express each replica ${\bf n}^a({\bf x})$ of the magnetization as a combination of fast fields $\phi_i^a({\bf x}), i=1,...,N-1$ and a slow field ${\bf n}'^a({\bf x})$ of the unit length. We use the representation
\[dec\] [**n**]{}\^a([**x**]{})=[**n**]{}’\^a([**x**]{})+ \_i\_i\^a([**x**]{})[**e**]{}\_i\^a([**x**]{}), where the unit vectors ${\bf e}_i^a({\bf x})$ are perpendicular to each other and the vector ${\bf n}'^a({\bf x})$. The slow field ${\bf n}'^a$ can be chosen in different ways. For example, one can cut the system into blocks of size $L\gg a$, where $a$ is the ultra-violet cut-off. In the center ${\bf x}$ of a block the vector ${\bf n}'^a({\bf x})$ should be parallel to the total magnetization of the block. In the other points the field ${\bf n}'^a$ should be interpolated. We assume that the fluctuations of the magnetization are weak, that is $\langle\phi_i^2\rangle\ll 1$. Then the fluctuations of the field ${\bf n}^a$ are orthogonal to the vector ${\bf n}'^a$ because of the fixed length constraint $({\bf n}^a)^2=1$.
To substitute the expression (\[dec\]) into the Hamiltonian we have to differentiate the vectors ${\bf e}_i^a$. We use the following definition
\[dife\] = -c\^a\_[i]{}[**n**]{}’\^a+\_k f\^a\_[, ik]{}[**e**]{}\_k\^a. It is easy to show [@Pol] that $\sum_{\mu i}(c^a_{\mu i})^2=
\sum_{\mu}(\partial_{\mu}{\bf n}'^a)^2$. With the accuracy up to the second order in $\phi$ the replica Hamiltonian (\[7\]) can be represented as follows
\[Hphi\] H\_[R]{}=d\^D x \[ \_[a]{} { (\_[**n**]{}’\^a)\^2 (1-(\_i\^a)\^2)+ c\^a\_[i]{}c\^a\_[k]{}\^a\_i\^a\_k + ( \_ \_i\^a - f\^a\_[, ik]{} \^a\_k)\^2 } & &\
- \_[ab]{} { R([**n**]{}’\^a[**n**]{}’\^b)+A\^[ab]{}(\^a\_i)\^2+ B\^[ab]{}\_[ij]{}\^a\_i\^a\_j + C\^[ab]{}\_[ij]{}\_i\^a\^b\_j } \], & & where the summation over the repeated indices $i,j,k,\mu$ is assumed and
\[coef\] A\^[ab]{}=-([**n**]{}’\^a[**n**]{}’\^b)R’([**n**]{}’\^a[**n**]{}’\^b); B\^[ab]{}\_[ij]{}= ([**n**]{}’\^b[**e**]{}\^a\_i)([**n**]{}’\^b[**e**]{}\^a\_j) R”([**n**]{}’\^a[**n**]{}’\^b); & &\
C\^[ab]{}\_[ij]{}= ([**e**]{}\^a\_i[**e**]{}\_j\^b)R’([**n**]{}’\^a[**n**]{}’\^b) + ([**n**]{}’\^a[**e**]{}\^b\_j)([**n**]{}’\^b[**e**]{}\^a\_i) R”([**n**]{}’\^a[**n**]{}’\^b). & & In the last formula $R'$ and $R''$ denote the first and second derivatives of the function $R(z)$. We have omitted the terms of the first order in $\phi$ in Eq. (\[Hphi\]). These terms are proportional to the products of the fast field $\phi$ and some slow fields. Hence, they are non-zero only in narrow shells of the momentum space. One can show that their contributions to the RG equations are negligible.
To obtain the RG equations we have to integrate out the fast variables $\phi^a_i$. Near a zero-temperature fixed point the Jacobian of the transformation ${\bf n}\rightarrow ({\bf n}', \phi_i)$ can be ignored, since the Jacobian does not depend on the temperature. The integration measure is determined from the condition that the field ${\bf n}'^a$ is a slow part of the magnetization. This condition imposes restrictions on the fields $\phi$. The expression (\[Hphi\]) depends on the choice of the vectors ${\bf e}^a_i$ (\[dec\]). However, after integrating out the fields $\phi$ the Hamiltonian can depend only on the slow part ${\bf n}'^a$ of the magnetization. One can make the calculations simpler, considering special realizations of the field ${\bf n}'^a$. To find the renormalization of the disorder-induced term $R(z)$ (\[7\]) we can assume that the field ${\bf n}'^a$ does not depend on the coordinates. The renormalization of the gradient energy can be determined, assuming that the vectors ${\bf n}'^a({\bf x})$ depend on one spatial coordinate only and lie in the same plane. In both cases the vectors ${\bf e}_i^a$ can be chosen such that in Eq. (\[dife\]) $f^a_{\mu, ik}=0$. At such a choice the integration measure can be omitted and the fields $\phi_i^a$ can be considered as weakly interacting fields with the wave vectors from the interval $1/a>q>1/L$.
To derive the one-loop RG equations we express the free energy via the Hamiltonian (\[Hphi\]). Then we expand the exponent in the partition function up to the second order in $(H_R-\int d^D x \sum_{\mu i}(\partial_{\mu}\phi_i)^2/(2T))$ and integrate over $\phi^a_i$. Finally, we make a rescaling. The vectors ${\bf e}^a_i$ can be excluded from the final expressions with the relation $\sum_i({\bf ae}^a_i)({\bf be}^a_i)=
({\bf ab})-({\bf an}'^a)({\bf bn}'^a)$. In a zero-temperature fixed point the RG equations are
\[Tz\] = -(D-2) + 2(N-2)R’(1)+O(R\^2,T); $$\begin{aligned}
\label{Rz}
0=\frac{dR(z)}{d \ln L}=\epsilon R(z) + 4(N-2)R(z)R'(1)-
2(N-1)zR'(1)R'(z)
+2(1-z^2)R'(1)R''(z) & & \nonumber \\
+
(R'(z))^2(N-2+z^2)-2R'(z)R''(z)z(1-z^2)+
(R''(z))^2(1-z^2)^2, & &\end{aligned}$$ where the factor $1/(8\pi^2)$ is absorbed into $R(z)$ to simplify notations. The RG equations become simpler after the substitution of the argument of the function $R(z)$: $z=\cos\phi$. In terms of this new variable one has to find even periodic solutions $R(\phi)$. The period is $2\pi$ in the RF case and $\pi$ in the RA case due to the additional symmetry of the RA model. The one-loop equations get the form
\[8\] = -(D-2) - 2(N-2)R”(0)+O(R\^2,T); $$\begin{aligned}
0=\frac{dR(\phi)}{d \ln L}=\epsilon R(\phi) + (R''(\phi))^2 - 2R''(\phi)
R''(0) - & & \nonumber\\
\label{9}
(N-2)[4R(\phi)R''(0)+2{\rm ctg}\phi R'(\phi) R''(0) -
\left(\frac{R'(\phi)}{\sin\phi}\right)^2] + O(R^3,T) & &\end{aligned}$$ Eq. (\[8\]) provides the following result for the scaling dimension $\Delta_T$ of the temperature
\[DT\] \_T=-2+-2(N-2)R”(0).
The two-spin correlation function is given in the one-loop order [@Pol] by the expression
\[cf\] \^a([**x**]{})[**n**]{}\^a([**x**]{}’)= ’\^a([**x**]{})[**n**]{}’\^a([**x**]{}’)(1-\_i(\_i\^a)\^2). Hence, in the fixed point $\langle{\bf n}({\bf x}){\bf n}({\bf x}')\rangle\sim|{\bf x}-{\bf
x}'|^{-\eta}$, where
\[11\] =-2(N-1)R”(=0)
Let us find the magnetic susceptibility in the weak uniform external field $H$. We add to the Hamiltonian (\[7\]) the term $-\sum_a\int d^Dx Hn^a_z/T$ (the field is directed along the z-axis). The renormalization of the field $H$ is determined by the renormalization of the temperature (\[8\]) and the field ${\bf n}$. In the zero-loop order the renormalized magnetic field $h(L)$ depends on the scale as $h(L)=H\times(L/a)^2$. Hence, the correlation length $R_c\sim H^{-1/2}$. The magnetization, averaged over a block of size $R_c$, is oriented along the field. The absolute value of this average magnetization is proportional to $R_c^{-\eta/2}$. This allows us to calculate the critical exponent $\gamma$ of the magnetic susceptibility $\chi(H)\sim H^{-\gamma}$ in a zero-temperature fixed point:
\[15\] =1+(N-1)R”(=0)/2 .
In Ref. [@DF] Eqs. (\[Tz\],\[Rz\]) were derived with a different method. In that paper the critical behavior in $4+\epsilon$ dimensions was studied by considering analytical fixed point solutions $R(z)$. In the Heisenberg model, analytical solutions are absent and they are unphysical for $N\ne 3$ [@DF]. In $4-\epsilon$ dimensions appropriate analytical solutions are absent for any $N$. To demonstrate this let us differentiate Eq. (\[Rz\]) over $z$ at $z=1$. For any analytical $R(z)$ we obtain the following flow equation
\[flow\] =R’(z=1) + 2(N-2)(R’(z=1))\^2. At $N> 2$ the fixed point of this equation $R'(z=1)=-\epsilon/[2(N-2)]<0$. It corresponds to the negative critical exponent $\eta$ (\[11\]) and hence is unphysical. However, we shall see that in the RA model some appropriate non-analytical fixed points $R(z)$ appear. In these fixed points $R''(z=1)=\infty$. In Ref. [@DF] the RG charges are the derivatives of the function $R(z)$ at $z=1$. Thus, in a non-analytical fixed point these charges diverge. In the systems with a finite number of the charges their divergence implies the absence of a fixed point. However, if the number of the RG charges is infinite such a criterion does not work and is even ambiguous. Indeed, the set of charges can be chosen in different ways and e.g. the coefficients of the Taylor expansion about $z=0$ remain finite in our problem.
Random field {#sec:IV}
============
For the RF XY model the one-loop RG equations (\[8\],\[9\]) can be solved exactly [@12a]. The solution corresponds to QLRO with the critical exponents $\eta=\pi^2/9\epsilon, \gamma=1-\pi^2/18\epsilon$. In the first order in $\epsilon$ the exponent $\eta$ equals the prefactor $C$ before the logarithm in the correlation function [@12a] of the angles $\phi({\bf x})$ between the spins ${\bf n}({\bf x})$ and some fixed direction: $\langle(\phi({\bf x}_1)-\phi({\bf x}_2))^2\rangle=C\ln|{\bf x}_1-{\bf
x}_2|$. We expect that this coincidence does not extend to the higher orders.
If $N\ne 2$ the RG equation (\[9\]) is more complex. Fortunately, at $N>3$ there is still a simple method to study the large-distance behavior. The method is based on the Schwartz-Soffer inequality [@SwSo] and shows that QLRO is absent.
In Ref. [@SwSo] the inequality is proven for the Gaussian distribution of the random field. It can also be proved for the arbitrary RF distribution (Appendix A).
Let us demonstrate the absence of physically acceptable fixed points in the RF case at $N>3$. We derive some inequality for critical exponents. Then we show that the inequality has no solutions. We use a rigorous inequality for the connected and disconnected correlation functions [@SwSo]
\[17\] ([**q**]{})[**n**]{}(-[**q**]{}) = \_a([**q**]{})[**n**]{}\_a(-[**q**]{})- \_a([**q**]{})[**n**]{}\_b(-[**q**]{}), where ${\bf n}({\bf q})$ is a Fourier-component of the magnetization, $a,b$ are replica indices. The disconnected correlation function is described by the critical exponent (\[11\]). The large-distance behavior of the connected correlation function in a zero-temperature fixed point can be derived from the expression
\[eqchi\] \~([**0**]{})[**n**]{}([**x**]{})d\^D x and the critical exponent of the susceptibility (\[15\]). The integral in the right hand side of Eq. (\[eqchi\]) is proportional to $R_c^{D-\eta_1}$, where $R_c$ is the correlation length in the external field $H$, $\eta_1$ the critical exponent of the connected correlation function. For the calculation of the exponent $\gamma$ (\[15\]) we used the zero-loop expression of $R_c$ via $H$. Now we need the one-loop accuracy. In this order $R_c\sim H^{-1/[2-(N-3)R''(0)]}$. This allows us to get the following equation for the exponent $\eta_1$
\[e1\] \_1=D-2-2R”(0). In a fixed point Eq. (\[17\]) provides an inequality for the critical exponents of the connected and disconnected correlation functions [@SwSo]. The inequality has the form
\[ein\] 2(2-D+\_1)4-D+. This allow us to obtain the following relation
\[18\] 4-D +o(R), where $\eta$ is given by Eq. (\[11\]). The two-spin correlation function can not increase up to the infinity as the distance increases. Hence, the critical exponent $\eta$ is positive. At $N>3$ this is incompatible with Eq. (\[18\]) at small $\epsilon$. Thus, there are no accessible fixed points for $N>3$. This suggests the strong coupling regime with a presumably finite correlation length.
Certainly, in the RF XY model [@RFXY; @12a] Eq. (\[18\]) is satisfied. However, the unstable fixed points of the RG equations [@12a] do not satisfy the inequality.
The marginal Heisenberg case $N=3$ is the most difficult, since in the one-loop order the right hand side of Eq. (\[18\]) equals zero at $N=3$. Hence, the two-loop corrections may be relevant. The RF Heisenberg model is beyond the scope of the present paper.
Random anisotropy
=================
In this section we investigate the possibility of QLRO in the RA $O(N)$ model. The first subsection is devoted to the simplest case of the XY model. The second subsection contains an inequality for the critical exponent $\eta$. The derivation of the inequality is analogous to Eq. (\[18\]). This inequality is applied in the next subsections. The third subsection contains the results for the Heisenberg model. In the last subsection we consider the case $N>3$.
$N=2$
-----
This case is studied analogously to the RF XY model [@12a]. At $N=2$ the RG equation (\[9\]) can be solved analytically. Its solution is a periodical function with period $\pi$. In interval $0<\phi<\pi$ the fixed point solution $R(\phi)$ is given by the formula
\[RAXYsol\] R()=. It is a stable fixed point. This can be verified with the linearization of the flow equation (\[9\]) for the small deviations from the fixed point. Another proof of the stability is based on the inequality of the next subsection.
The stable fixed point corresponds to the QLRO phase at low temperatures and weak disorder. The critical exponents $\eta=\pi^2\epsilon/36,
\gamma=1-\pi^2\epsilon/72$.
The solution (\[RAXYsol\]) is non-analytical at $\phi=0$, since $R^{IV}(\phi=0)=\infty$. Hence, the Taylor expansion over $\phi$ is absent. However, a power expansion over $|\phi|$ exists. We shall see below that the same behavior at small $\phi$ conserves also at other $N$.
An inequality for a critical exponent {#sec:V.B}
-------------------------------------
We use the same approach as in the RF model. Since in the RA case the random field is conjugated with a second order polynomial of the magnetization, the Schwartz-Soffer inequality [@SwSo] should be applied to correlation functions of the field $m({\bf x})=(n_z({\bf x}))^2-1/N$, where $n_z$ denotes one of the magnetization components, $1/N$ is subtracted to ensure the relation $\langle m\rangle=0$.
To calculate the critical exponent $\mu$ of the disconnected correlation function we use the representation (\[dec\]) and obtain the relation
\[RGm\] m\^a([**x**]{})m\^a([**x**]{}’)= m’\^a([**x**]{})m’\^a([**x**]{}’)(1-), where $a$ is a replica index, $m'=(n'_z)^2-1/N$ the slow part of the field $m$. One finds $\mu=-4NR''(0)$.
The critical exponent $\mu_1$ of the connected correlation function is determined analogously to the RF case. We apply a weak uniform field $\tilde H$, conjugated with the field $m$, and calculate the susceptibility $dm/d\tilde H$ in two ways. The result for the critical exponent is $\mu_1=D-2-2(N+2)R''(0)$.
The Schwartz-Soffer inequality provides a relation between the exponents $\mu$ and $\mu_1$. It has the same structure as Eq. (\[ein\]). Finally, we obtain the following equation
\[RAineq\] (N-1)+o(R). In terms of the RG charge $R(\phi)$ this inequality can be rewritten in the form \[Rin\] R”(0)-/8+o(R).
$N=3$ {#sec:V.C}
-----
In this case we solve Eq. (\[9\]) numerically. Since coefficients of Eq. (\[9\]) are large as $\phi\rightarrow 0$, it is convenient to use a series expansion of the fixed-point solution $R(\phi)$ at small $\phi$. At the larger $\phi$ the equation can be integrated with the Runge-Kutta method. The following expansion over $t=\sqrt{(1-z)/2}=|\sin(\phi/2)|$ holds
\[smphi\] R()/=+2a\^2 |\^3| & &\
+(-)\^4+ O(|\^5|), & & where $a=R''(\phi=0)/\epsilon$. We see that the RG charge $R(\phi)$ is non-analytical at small $\phi$. Similar to the random manifold [@FRG] and random field XY [@12a] models $R^{IV}(0)=
\infty$.
Numerical calculations show that at any $N$ the solutions, compatible with the inequality (\[Rin\]), have sign “$+$” before the third term of Eq. (\[smphi\]). The solutions to be found are even periodical functions with period $\pi$. Hence, their derivative is zero at $\phi=\pi/2$. At $N=3$ there is only one solution that satisfies Eq. (\[Rin\]). It corresponds to $R''(\phi=0)=-0.1543\epsilon$. If this solution is stable Eqs. (\[11\],\[15\]) provide the following results for the critical exponents
\[crexp\] =0.62; =1-0.15. All the other solutions of Eq. (\[9\]) do not satisfy Eq. (\[Rin\]) and hence are unstable.
We have still to test the stability of the solution found. For this aim we use an approximate method. First, we find an approximate analytical solution of Eq. (\[9\]). We rewrite Eq. (\[9\]), substituting $\omega(R''(\phi))^2$ for $(R''(\phi))^2$. The case of interest is $\omega=1$ but at $\omega=0$ the equation can be solved exactly. The solution at $\omega=1$ can then be found with the perturbation theory over $\omega$. The exact solution at $\omega=0$ is $R_{\omega=0}(\phi)=\epsilon(\cos
2\phi/24+1/120)$. The corrections of order $\omega^k$ are trigonometric polynomials of order $2(k+1)$. The first correction is
\[fc\] R\_1()=-2+4+[const]{} After the calculation of the corrections we can write an asymptotic series for the critical exponent $\eta$ (\[11\]): $\eta=\epsilon(0.67-0.08\omega+0.14\omega^2-\dots)$. The resulting estimation $\eta=\epsilon(0.67\pm0.08)$ agrees with the numerical result (\[crexp\]) well. This allows us to expect that the stability analysis of the solution $R_{\omega=0}$ of the equation with $\omega=0$ provides information about the stability of the solution of Eq. (\[9\]).
To study the stability of the exact solution of the equation with $\omega=0$ is a simple problem. We introduce a small deviation $r(\phi)$: $R(\phi)=R_{\omega=0}(\phi)+r(\phi)$ and write the flow equation for this deviation:
\[fl\] =(5r()+r”()+ r”(0)2)/3+[const]{}r”(0). It is convenient to use the Fourier expansion $r(\phi)=
\sum_m a_m\cos2m\phi$. The flow equations for the Fourier harmonics can be easily integrated. We see that $a_m\rightarrow 0$ as $L\rightarrow\infty$ for any $m>0$. The solution is unstable with respect to the constant shift $a_0$. However, this instability has no interest for us, since the correlation functions do not change at such shifts [@FRG]. Indeed, the constant shift corresponds to the addition of just a random term, independent of the magnetization, to the Hamiltonian (\[1\]). Thus, the RG equation possesses a stable fixed point. This fixed point describes the QLRO phase with the critical exponents (\[crexp\]).
In the Abelian systems the results of the functional RG are supported by the variational method [@MP]. In our problem this method can not be applied. However, it is interesting that in the Abelian systems the functional RG equations without $(R''(\phi))^2$ reproduce the variational results.
As usual in critical phenomena, in $4$ dimensions the one-loop RG equations allow one to obtain the exact large-distance asymptotics of the correlation function. In the 4-dimensional case $R(\phi)=\tilde R(\phi)/\ln L$, where $\tilde R(\phi)$ satisfies Eq. (\[9\]) at $\epsilon=1$. We obtain the following result for the two-spin correlation function with Eq. (\[cf\])
\[elg\] ([**x**]{})[**n**]{}([**x**]{}’)\~\^[-0.62]{}|[**x**]{}-[**x**]{}’|.
$N>3$ {#n3}
-----
Numerical analysis of Eq. (\[9\]) shows that solutions, compatible with Eq. (\[Rin\]), are absent at $N\ge 10$. Hence, QLRO is absent for any $N\ge 10$. In the spherical model ($N=\infty$) the absence of fixed points can be demonstrated analytically (Appendix C). This agrees with the previous results [@P; @spher]. For each integer $N<10$ the RG equation (\[9\]) has exactly one solution, satisfying the inequality of section \[sec:V.B\]. These solutions are described in Table I. In the table, $\eta$ is the critical exponent of the two-spin correlation function, $\Delta_T$ the scaling dimension of the temperature (\[DT\]).
Unfortunately, it is not clear if the fixed points, found at $N>3$, survive in 3 dimensions. A zero-temperature fixed point can exist only if the scaling dimension of the temperature is negative. Table \[table1\] shows that scaling dimension is positive in the one-loop approximation at $\epsilon=1$ and $N\ge 5$. In the 3-dimensional $O(4)$ model the one-loop correction to the scaling dimension $-2(N-2)R''(0)\approx 0.7\epsilon$ is close to the zero-loop approximation $-2+\epsilon$. Thus, the next orders of the perturbation theory are crucial to understand what happens in 3 dimensions.
In the $O(2)$ model the scaling dimension $\Delta_T=-2+\epsilon$ is exact [@12a; @FRG]. Hence, QLRO disappears in 2 dimensions. In systems with a larger numbers of magnetization components fluctuations become stronger. Thus, one expects the absence of QLRO in all the two-dimensional $O(N)$ models.
At the zero temperature Eq. (\[9\]) is valid independently of the scaling dimension $\Delta_T$. It is tempting to assume that at the zero temperature QLRO still exists in the RA $O(N>3)$ models below the critical dimension, in which $\Delta_T=0$. However, the experience of the two-dimensional RF XY model does not support such an expectation. Recent numerical simulations show that QLRO is absent even in the ground state of that model [@2DGSsim].
------------ -------------------- ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ -----------------
$N$ 2 3 4 5 6 7 8 9
$\eta$ $\pi^2\epsilon/36$ $0.62\epsilon$ $1.1\epsilon$ $1.7\epsilon$ $2.7\epsilon$ $4.6\epsilon$ $9.0\epsilon$ $33\epsilon$
$\Delta_T$ -2+$\epsilon$ -2+$1.3\epsilon$ -2+$1.7\epsilon$ -2+$2.3\epsilon$ -2+$3.2\epsilon$ -2+$4.8\epsilon$ -2+$8.7\epsilon$ -2+$30\epsilon$
------------ -------------------- ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ -----------------
: Critical exponents of the RA $O(N)$ model.[]{data-label="table1"}
Conclusion
==========
We have obtained QLRO in the RA Heisenberg model. This is the first example of QLRO in a non-Abelian system. The RF disorder tends to destroy the ordering which exists in the RA case. This difference between the RF and RA models is not surprising, since the same difference was already obtained in Ref. [@F] for the two-dimensional RF and RA XY models with the dipole forces.
We have not yet discussed the role of the topological defects. The contribution of the topological excitations to the RG equations (\[8\],\[9\]) is determined by the rare regions where the random field is sufficiently strong to compensate the core energy. Hence, similar to the pure system in $2+\epsilon$ dimensions they are responsible for the non-perturbative corrections of order $\exp(-1/\epsilon)$. Thus, their effect is negligible at small $\epsilon$. Several studies were devoted to the role of the vortices in the RF XY model [@vortex]. The theoretical prediction of QLRO in this system is based on the vortexless version of the model [@RFXY; @12a]. A qualitative estimation [@12a] and variational calculations [@vortex] suggest that the topological defects do not change the behavior of the RF XY model at the weak disorder. Our approach allows us to consider the XY model, including vortices. We see that QLRO does exist in the model with the defects.
However, in our problem there may be a more important source of the non-perturbative corrections. The effect of the multiple energy minima can lead to corrections of order $\epsilon^{5/2}$ to the RG equations [@FRG]. Unfortunately, the non-perturbative effects in the RF systems are not well understood.
The present paper uses a systematic RG approach. However, some results can be reproduced more simply with an approximate Migdal-Kadanoff renormalization group (Appendix B).
The question of the large-distance behavior of the RF and RA Heisenberg models was discussed in Ref. [@AP] on the basis of an approximate equation of state. In that paper QLRO was also obtained in the RA case. However, we believe that this is an accidental coincidence, since the equation of state [@AP] is valid only in the first order in the strength of the disorder, while higher orders are crucial for critical properties [@G]. In particular, the approach [@AP] incorrectly predicts the absence of QLRO in the RF XY model and its presence in the exactly solvable RA spherical model. It also provides incorrect critical exponents in the Heisenberg case. The reason of the mistakes is the fact that in the weak external uniform field the perturbation parameter of Ref. [@AP] is large.
The RA Heisenberg model is relevant for the amorphous magnets [@HPZ]. At the same time, for their large-distance behavior the dipole interaction may be important [@B]. Besides, a weak nonrandom anisotropy is inevitably present due to mechanical stresses.
In conclusion, we have found that the random anisotropy Heisenberg model has an infinite correlation length and a power dependence of the correlation function of the magnetization on the distance at low temperatures and weak disorder in $4-\epsilon$ dimensions. On the other hand, the correlation length of the random field $O(N>3)$ model is always finite.
The author thanks E. Domany, G. Falkovich, M.V. Feigelman, Y. Gefen, S.E. Korshunov, Y.B. Levinson, A.I. Larkin, V.L. Pokrovsky and A.V. Shytov for useful discussions. This work was supported by RFBR grant 96-02-18985 and by grant 96-15-96756 of the Russian Program of Leading Scientific Schools.
INEQUALITY FOR CORRELATION FUNCTIONS
====================================
In this appendix we derive an inequality for the correlation functions of the disordered systems. We consider the system with the Hamiltonian
\[A1\] H=dx\^D \[ H\_1(([**x**]{})) - h([**x**]{})m(([**x**]{}))\], where $\phi$ is the order parameter, $h$ the random field with short range correlations, $H_1$ may depend on some other random fields. We prove an inequality for the Fourier components of the field $m$:
\[A2\] G\_[con]{}([**q**]{}), where $G_{dis}({\bf q})=\overline{\langle{\bf m}({\bf q}){\bf m}(-{\bf q})
\rangle} ,
G_{con}({\bf q})= \overline{\langle{\bf m}({\bf q}){\bf m}(-{\bf
q})\rangle} - \overline{\langle{\bf m}({\bf q})\rangle\langle{\bf
m}(-{\bf q})\rangle}$, the angular brackets denote the thermal averaging, the bar denotes the disorder averaging.
This inequality can be easily obtained in the case of the Gaussian distribution $P(h)$ of the field $h$ [@SwSo]. Indeed, in the Gaussian case
G\_[dis]{}([**q**]{})= ( P(h) m\_[**q**]{}(h)) D{h}= & &\
\[A3\] -(P(h) m\_[**q**]{}(h)) D{h} =[const]{}(P(h)h(-[**q**]{})m\_[**q**]{}(h))D{h}, where $\int D\{h\}$ denotes the integration over the realizations of the random field, $m_{\bf q}(h)$ $=$ ${\int D\{\phi\}\exp(-{H}/{T})m({\bf q})}$ $/$ ${\int D\{\phi\}\exp(-{H}/{T})}$. Applying the Cauchy-Bunyakovsky inequality to Eq. (\[A3\]) one gets Eq. (\[A2\]).
However, the assumption about the Gaussian distribution of the random field is not necessary. The inequality (\[A2\]) can also be extended to a more general situation, corresponding to the effective replica Hamiltonian (\[7\]). Indeed, if one adds to any Hamiltonian a weak Gaussian random field $\tilde h$ , conjugated with the field $m$, it suffices for Eq. (\[A2\]) to become valid. The addition of the Gaussian random field corresponds to the transformation $R({\bf n}_a{\bf
n}_b)\rightarrow R({\bf n}_a{\bf n}_b)+\Delta{\bf n}_a{\bf n}_b$ in Eq. (\[7\]) where $\Delta\sim\tilde h^2$ is a positive constant. Thus, Eq. (\[A2\]) is invalid only, if for the arbitrarily small $\Delta$ the replica Hamiltonians can not contain the two-replica contribution $\tilde R({\bf n}_a{\bf n}_b) = R({\bf n}_a{\bf n}_b) - \Delta{\bf
n}_a{\bf n}_b$. This corresponds to the border of the region of the possible Hamiltonians and has zero probability.
For systems in the critical domain there is a simple way to understand why the inequality is valid not only in the Gaussian case but also in the general situation. This is just a consequence of the universality.
MIGDAL-KADANOFF RENORMALIZATION GROUP
=====================================
This appendix contains a simple approximate version of the renormalization group. The results for the critical exponents of the XY and Heisenberg models have a very good accuracy. The value of the magnetization component number $N_c$, at which QLRO disappears in the RF model, is probably exact. However, the critical number of the components in the RA model is underestimated.
Random field {#random-field}
------------
We use the following ansatz for the disorder-induced term in the Hamiltonian (\[1\]): $R({\bf n}_a{\bf n}_b)=\alpha{\bf n}_a
{\bf n}_b+\beta$, where $\alpha$ and $\beta$ are constants. This expression corresponds to the Gaussian RF disorder (\[2\]). Below we ignore the generation of the other contributions to the function $R(z)$. The missed contributions are related with random anisotropies of different orders. In terms of the angle variable $\phi$ (\[8\],\[9\])
\[B1\] R()=+. To ensure consistency we have to truncate the RG equation (\[9\]). We substitute the ansatz (\[B1\]) into Eq. (\[9\]) and retain only the terms, proportional to $\cos\phi$ or independent of $\phi$. This leads to the following RG equation for the constant $\alpha$ (\[B1\])
\[B2\] =+2\^2(N-3). For $N<3$ Eq. (\[B2\]) has a stable solution $\alpha=\epsilon/[2(3-N)]$. The critical exponent (\[11\]) equals
\[B3\] =. At $N=2$ this result has less than ten percent difference with the systematic theory [@12a]. QLRO disappears at $N=3$. This is the same critical number which is found in section \[sec:IV\].
For $N>3$ a fixed point exists in $4+\epsilon$ dimensions. It describes the transition between the ferromagnetic and paramagnetic phases. In this fixed point the critical exponent (\[B3\]) satisfies the modified dimensional reduction hypothesis [@mdr]. However, we believe that this is an artifact of the Migdal-Kadanoff approximation, since the correct value of the critical exponent differs form Eq. (\[B3\]).
Random anisotropy
-----------------
In this case we use the ansatz $R({\bf n}_a{\bf n}_b)=
A({\bf n}_a{\bf n}_b)^2+B$. In terms of the variable $\phi$ (\[8\],\[9\]) $R(\phi)=\alpha\cos2\phi+\beta$. We again substitute our ansatz into Eq. (\[9\]) and retain the terms, proportional to $\cos 2\phi$, and the terms, independent of $\phi$. The RG equation for the constant $\alpha$ has the form
\[B4\] =+8(N-6)\^2. The fixed point solution of this equation is $\alpha=\epsilon/[8(6-N)]$. It describes the QLRO phase at $N<6$. At $N=3$ the function $R(\phi)=\alpha\cos2\phi+\beta$ is just $R_{\omega=0}$ of section \[sec:V.C\]. The critical exponent of the two-spin correlation function is given by the following equation
\[B5\] =. At $N=2,3$ this value is close to the results of the systematic approach (Table I).
SPHERICAL MODEL
===============
In this appendix we consider the spherical RA model with the functional RG. We show that QLRO is absent in this model. In the spherical limit $N=\infty$ only the terms, proportional to $N$, and the term $\epsilon R(z)$ should be retained in the right hand side of Eq. (\[Rz\]). After the change of the variable $R(z)=\epsilon r(z)/N$ one obtains
\[RGr\] 0=r(z) \[1+4r’(1)\] - 2zr’(1)r’(z) + (r’(z))\^2.
It is convenient to differentiate Eq. (\[RGr\]) over $z$. One gets
\[difRG\] 0=r’(z)\[1+2r’(1)\]+2r”(z)\[r’(z)-zr’(1)\]. Analytical functions $r(z)$ can satisfy Eq. (\[difRG\]) at $z=1$ only if $r'(1)=0$ or $r'(1)=-1/2$. In both cases Eq. (\[difRG\]) can be easily solved. There are three analytical non-zero solutions: $r(z)=-z/2+1/4; r(z)=-(1-z)^2/4; r(z)=-z^2/4$. The last solution only has the necessary symmetry.
The non-analytical solutions are absent. Indeed, Eq. (\[difRG\]) can be integrated with the substitution $r'(z)=zt(z)$. The general integral has the form
\[genint\] =Cz. Besides, there are special solutions. They all satisfy the relation $t(z)=t(1)$. Hence, the special solutions are analytical. Thus, the function $t(z)$ can be non-analytical at $z=1$ only under the condition that $z=1$ is a peculiar point of Eq. (\[genint\]). This means that $t(1)=0$ or $t(1)=-1/2$. However, it is easy to verify that in both cases the solution is one of the found above.
We see that the only fixed point of the spherical RA model is $R(z)=-\epsilon z^2/(4N)$. With Eq. (\[11\]) one finds the critical exponent $\eta=-\epsilon/2$. Since $\eta>0$ the solution found is applicable at $D>4$. At $D<4$ the fixed points are absent. Thus, QLRO is absent too.
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|
---
abstract: 'This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be considered via coarsenings of the resulting polyhedral decompositions of ${{\mathbb R}}^d$. This leads to natural cell decompositions of the tropical projective space ${\mathbb{TP}_{\min}}^{d-1}$. Our method is to employ a known class of ordinary convex polyhedra naturally associated with weighted digraphs. This way we can relate to and use results from combinatorics and optimization. One outcome is the solution of a conjecture of Develin and Yu (2007).'
address: 'Institut f[ü]{}r Mathematik, TU Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany'
author:
- Michael Joswig
- Georg Loho
bibliography:
- 'main.bib'
title: Weighted digraphs and tropical cones
---
tropical convexity ,directed graphs ,regular subdivisions
14T05 ,52B12 ,05C20
Introduction
============
The tradition of max-plus linear algebra in optimization and related areas goes back several decades; for an overview e.g., see Litvinov, Maslov and Shpiz [@LitvinovMaslovShpiz:2001], Cohen, Gaubert and Quadrat [@CohenGaubertQuadrat:2004] or Butkovič [@Butkovic:10] and their references. Develin and Sturmfels connected max-plus linear algebra under the name of *tropical convexity* to geometric combinatorics in their landmark paper [@DevelinSturmfels:2004]; see also [@TropicalBook Chapter 5]. This line of research has been continued in [@MJ:2005], [@DevelinYu:2007], [@ArdilaDevelin:2009], [@FinkRincon:1305.6329] and other references. The interest in a more geometric perspective comes from several directions. One source is tropical geometry, which e.g., relates tropical convexity to the combinatorics of the Grassmannians [@SpeyerSturmfels04], [@HerrmannJoswigSpeyer14], [@FinkRincon:1305.6329]. A second independent source is the study of tropical analogues of linear programming [@ABGJ-Simplex:A] which, e.g., is motivated by its connections to deep open problems in computational complexity [@AkianGaubertGutermann12].
Since the paper [@DevelinSturmfels:2004] by Develin and Sturmfels more than ten years ago some of the strands of research still seem to diverge. The main purpose of this paper is to help bridging this gap. Our point of departure is [@DevelinSturmfels:2004 Theorem 1], which establishes a fundamental correspondence between the configurations of $n$ points in the *tropical projective torus* ${{\mathbb R}}^d/{{\mathbb R}}{\mathbf 1}$ and the regular subdivisions of the product of simplices $\Delta_{d-1}\times\Delta_{n-1}$. We suggest to call this result the *Structure Theorem of Tropical Convexity*. It was recently extended by Fink and Rincón [@FinkRincon:1305.6329 Corollary 4.2] to include regular subdivisions of subpolytopes of products of simplices. For the tropical point configurations this amounts to taking $\infty$ as a coordinate into account. Our first contribution is a new proof of that result (Corollary \[coro:dual\_sub\_tcone\]). Moreover, in [@DevelinSturmfels:2004] and [@FinkRincon:1305.6329] only tropical convex hulls of points (or dually, arrangements of tropical hyperplanes) are considered, whereas here we also bring exterior descriptions in terms of tropical half-spaces [@MJ:2005], [@GaubertKatz:11] into the picture. Arrangements of max-tropical halfspaces correspond to the ‘two-sided max-linear systems’ in the max-plus literature [@Butkovic:10 §7]. As an additional benefit our methods allow us to resolve a previously open question raised by Develin and Yu, who conjectured that a finitely generated tropical convex hull is pure and full-dimensional if and only if it has a half-space description in which the apices of these tropical half-spaces are in general position [@DevelinYu:2007 Conjecture 2.11]. We show that, indeed, general position implies pureness and full-dimensionality (Theorem \[thm:pure\]), and we give a counter-example to the converse (Example \[exmp:pure\]). The approach through tropical convex hulls on the one hand and the approach through systems of tropical inequalities on the other hand gives rise to two interesting cell decompositions of the *tropical projective spaces* (Theorem \[thm:covector\_decomposition\] and Corollary \[cor:signed\_cell\_decomposition\]). This ties in with compactifications of tropical varieties; see Mikhalkin [@Mikhalkin:2006 §3.4].
As in [@DevelinSturmfels:2004] it turns out to be convenient to examine the regular subdivisions of products of simplices and their subpolytopes in terms of a dual ordinary convex polyhedron, which we call the *envelope* of the tropical point configuration. In fact, it is even fruitful to see this envelope as a special case of a more general class of ordinary polyhedra which are associated with directed graphs with weighted arcs. These *weighted digraph polyhedra* are defined by linear inequalities of the form $$x_i - x_j \ \leq \ w_{ij} \enspace ,$$ where $w_{ij}$ is the weight on the arc from the node $i$ to the node $j$. Their feasible points are well known as *potentials* in the optimization literature, and the weighted digraph polyhedra are sometimes called ‘shortest path polyhedra’; e.g., see [@Schrijver:CO:A §8.2] for an overview. Recently potentials and weighted digraph polyhedra stared prominently in the work of Khachiyan and al. [@Khachiyan:2008] on hardness results in the context of vertex enumeration. Specializing all arc weights to zero yields the *digraph cones* which are closely related to *order polytopes* of partially ordered sets. By applying a celebrated result of Stanley [@Stanley:1986 Theorem 1.2] we obtain a combinatorial characterization of the entire face lattice of any digraph cone (Theorem \[thm:partition\]).
Our paper is organized as follows. Section \[sec:weighted\] starts out with investigating a general weighted digraph polyhedron $Q(W)$ associated with a $k{\times}k$-matrix $W$, which we read as a directed graph $\Gamma=\Gamma(W)$ equipped with a weight function. The digraph cones, with all finite entries equal to zero, naturally come in as their recession cones. We show that the face lattice of a digraph cone is isomorphic to a face figure of the order polytope associated with the acyclic reduction of $\Gamma$ and, via Stanley’s result [@Stanley:1986 Theorem 1.2], to a partially ordered set of partitions of the node set of $\Gamma$ ordered by refinement. It is a key observation that the faces of a weighted digraph polyhedron are again weighted digraph polyhedra. The envelope of an arbitrary $d{\times}n$-matrix $V$ is the weighted digraph polyhedron for a specific $(d{+}n){\times}(d{+}n)$-matrix constructed from $V$.
In Section \[sec:tropical\] we direct our attention to tropical convexity, which is essentially the same as linear algebra over the tropical semi-ring ${{\mathbb T}}_{\min}=({{\mathbb R}}\cup\{\infty\},\min,+)$. Clearly, it is just a matter of taste if one prefers $\min$ or $\max$ as the tropical addition. More importantly though, it turns out to be occasionally convenient to use both these operations together to be able to phrase some of our results in a natural way. So we usually consider tropical linear spans of vectors in the $\min$-tropical setting and intersections of tropical half-spaces in the $\max$-setting. With any matrix $V\in{{\mathbb R}}^{d\times n}$ Develin and Sturmfels associate a polyhedral decomposition of the tropical projective torus ${{\mathbb R}}^d/{{\mathbb R}}{\mathbf 1}$ [@DevelinSturmfels:2004 §3]; here ${\mathbf 1}$ denotes the all ones vector. We follow Fink and Rincón [@FinkRincon:1305.6329] in calling this polyhedral complex the *covector decomposition*. The cells of the covector decomposition are naturally indexed by subgraphs of the digraph $\Gamma(W)$, where $W$ is the $(d{+}n){\times}(d{+}n)$-matrix mentioned above. Moreover, these cells arise as orthogonal projections of the faces of the envelope of $V$. If $V$ is finite then (in the tropical projective torus) the union of the bounded cells of the type decomposition is exactly the tropical convex hull of the columns of $V$. Further, the covector decomposition is dual to a regular subdivision of the product of simplices $\Delta_{d-1}\times\Delta_{n-1}$. If $V$ has infinite coordinates, it still makes sense to talk about the *tropical cone* generated by the columns, but $\Delta_{d-1}\times\Delta_{n-1}$ gets replaced by the subpolytope corresponding to the finite entries of $V$; see [@FinkRincon:1305.6329]. This leads to studying point configurations in the *tropical projective space*; see Mikhalkin [@Mikhalkin:2006 §3.4] and Section \[sec:projective\] below. Another way of interpreting the matrix $V$, with coefficients in ${{\mathbb T}}_{\min}$, is as an arrangement of max-tropical hyperplanes. The covector decomposition arises as the common refinement of the affine fans corresponding to these tropical hyperplanes. Equipping such a tropical hyperplane arrangement with a certain graph encoding the feasibility of a cell gives rise to a max-tropical cone described as the intersection of finitely many *tropical half-spaces*; see [@MJ:2005] and [@GaubertKatz:11]. This is how tropical cones naturally arise in the context of tropical linear programming. In [@ABGJ-Simplex:A] a tropical version of the simplex method is described. The pivoting operation proposed there can be explained in terms of operations on the graph $\Gamma(W)$, the crucial object being the *tangent digraph* from [@ABGJ-Simplex:A §3.1], which carries the same information as the ‘tangent hypergraphs’ of Allamigeon, Gaubert and Goubault [@AGG:2013]. We show how the tangent digraph encodes the local combinatorics of the covector decomposition induced by $V$ in the neighborhood of a given point. Finally, we recall the *signed cell decompositions* from [@ABGJ-Simplex:A §3.2] which form the tropical analogues of the polyhedral complexes generated from a system of ordinary affine hyperplanes.
The upshot is that all the remarkable combinatorial properties of tropical convexity can be inferred from the weighted digraph polyhedra. It is worth noting that the facet normals of their defining inequalities are precisely the roots of a type A root system. Lam and Postnikov [@LamPostnikov:2007] introduced ‘alcoved polytopes’ which are exactly the weighted digraph polyhedra which are bounded (modulo projecting out the subspace ${{\mathbb R}}{\mathbf 1}$). These are also the *polytropes* in [@JoswigKulas:2010]. Section \[sec:polytropes\] gives more details. The paper closes with a few open problems.
Weighted digraph polyhedra {#sec:weighted}
==========================
The construction
----------------
Let $W=(w_{ij})$ be an arbitrary $k{\times}k$-matrix with coefficients in ${{\mathbb T}}_{\min}={{\mathbb R}}\cup\{\infty\}$. This yields a digraph $\Gamma(W)$ with node set $[k]$ and an arc from $i$ to $j$ whenever the coefficient $w_{ij}$ is finite. Notice that $\Gamma(W)$ may have loops, corresponding to finite entries on the diagonal. Also $(i,j)$ and $(j,i)$ both may be arcs, but there are no other multiple edges. The matrix $W$ induces a map, $\gamma$, which assigns to each arc $(i,j)$ of $\Gamma(W)$ its *weight* $w_{ij}$. We call the pair $(\Gamma(W),\gamma(W))$ the *weighted digraph* associated with $W$. Conversely, each finite directed graph $\Gamma$ endowed with a weight function $\gamma$ on its arcs has a *weighted adjacency matrix* $W(\Gamma,\gamma)$. Often we will not distinguish between the matrix $W$ and the digraph $\Gamma$ equipped with the weight function $\gamma$.
Our key player is the *weighted digraph polyhedron* $Q(W)$ in ${{\mathbb R}}^k$ which is defined by the linear inequalities $$\label{eq:defining}
x_i - x_j \ \leq \ w_{ij} \qquad \text{ for each arc $(i,j)$ in $\Gamma(W)$} \enspace .$$ For a directed graph $\Gamma$ with a weight function $\gamma$ we also write $Q(\Gamma,\gamma)$ instead of $Q(W(\Gamma,\gamma))$. Observe that $-Q(W) = Q({{W}^\top})$. A feasible point in $Q(W)$ is sometimes called a *potential* on the digraph $\Gamma$; e.g., see [@Schrijver:CO:A §8.2]. The following result of Gallai [@Gallai:58] clarifies the feasibility of the constraints; see also [@Schrijver:CO:A Theorem 8.2] and [@Butkovic:10 §2.1].
\[lemma:feasible\] The weighted digraph polyhedron $Q(W)$ is empty if and only if the weighted digraph $(\Gamma,\gamma)$ has a negative cycle.
If the weighted digraph $(\Gamma,\gamma)$ does not have any negative cycle there is a directed shortest path between any two nodes. Let $W^*=(w_{ij}^*)$ be the $k{\times}k$-matrix which records the weights of these shortest paths. Following Butkovič [@Butkovic:10 §1.6.2] we call the shortest path matrix $W^*$ the *Kleene star* of $W$. The tropical addition $\oplus=\min$ extends to vectors and matrices coefficientwise. Moreover, the tropical addition and the tropical multiplication give rise to a tropical matrix multiplication, which we also write as $\odot$. Matrix powers of $W$ with respect to $\odot$ are written as $W^{\odot \ell}$ where $W^{\odot 0}=I$ is the min-tropical unit matrix, which has zero coefficients on the diagonal and $\infty$ otherwise, and $W^{\odot (\ell+1)}=W^{\odot \ell}\odot W$. With this notation we have the formula $$W^* \ = \ I \oplus W \oplus W^{\odot 2} \oplus \cdots \oplus W^{\odot k} \enspace ,$$ whose direct evaluation amounts to applying the Bellman-Ford method for computing all shortest paths [@Schrijver:CO:A §8.3]. The next lemma points out a special property of the inequality description given by $W^*$; see [@Schrijver:CO:A Theorem 8.3].
\[lem:Kleene\_tight\] Each of the defining inequalities from for the weighted digraph polyhedron of the matrix $W^*$ is tight.
Let $x_i - x_j \leq w_{ij}^*$ be an inequality defining $Q(W^*)$. The vector of weights $w_{pj}^*$ for $p \in [k]$, i.e., the $j$th column of $W^*$, satisfies each inequality by the shortest path property $w_{pj}^* \leq w_{pq}^* +
w_{qj}^*$. Equivalently we have $w_{pj}^* - w_{qj}^* \leq w_{pq}^*$. Due to $w_{jj}^* = 0$, this vector satisfies the equality $x_i - x_j = w_{ij}^*$.
Throughout the following we assume that $(\Gamma,\gamma)$ does not have a negative cycle. In view of Lemma \[lemma:feasible\] this is equivalent to the feasibility of $Q(W)$, and the Kleene star $W^*$ is defined. Further, let $E(W)$ be the *equality graph* of $W$, which is the undirected graph on the node set $[k]$ and which has an edge between $i$ and $j$ if $Q(W)$ satisfies $x_i-x_j=w_{ij}^*<\infty$ or $x_j-x_i=w_{ji}^*<\infty$.
\[lemma:zero\_cycles\]
1. We have $Q(W^*) = Q(W)$ and $E(W^*)=E(W)$.
2. \[lemma:item:zero\_cycles\] Two distinct nodes $i$ and $j$ are contained in a directed cycle of weight zero in $\Gamma(W)$ if and only if $\{i,j\}$ is contained in the equality graph $E(W)$ if and only if $w_{ij}^*=-w_{ji}^*<\infty$.
The proof for both statements is essentially the same. Let $\pi=(i_0,i_1,\dots,i_m)$ be a directed path in $\Gamma$. This corresponds to the inequalities $x_{i_{\ell-1}} \leq x_{i_{\ell}} + w_{i_{\ell-1}i_{\ell}}$ for $\ell \in \{1, \ldots, m\}$. By transitivity we obtain $$x_{i_0} \ \leq \ x_{i_m} + \sum_{\ell = 1}^{m}w_{i_{\ell-1}i_{\ell}}$$ as a valid inequality for $Q(W)$. Restricting to shortest paths shows $Q(W^*)\supseteq Q(W)$. The other inclusion is obvious. Notice that this readily implies that the equality graphs $E(W)$ and $E(W^*)$ are the same.
Now suppose that $\pi$ is a directed cycle of weight zero. In particular, $i_0=i_m$ is the same node and because of the presumed feasibility, the cycle contains the shortest path for any pair of its nodes. The above yields for each $\mu \in \{0, \ldots, m\}$ the inequalities $$x_{i_0} \ \leq \ x_{i_{\mu}} + \sum_{\ell = 1}^{\mu}w_{i_{\ell-1}i_{\ell}} \ = \ x_{i_{\mu}} + w_{i_0,i_{\mu}}^* \text{ and } x_{i_{\mu}} \ \leq \ x_{i_m} + \sum_{\ell = \mu+1}^{m}w_{i_{\ell-1}i_{\ell}} \ = \ x_{i_0} + w_{i_{\mu},i_{0}}^*\enspace .$$ With $w_{i_0,i_{\mu}}^* + w_{i_{\mu},i_0}^* = 0$ we obtain $$x_{i_0} - x_{i_{\mu}} \ \leq \ w_{i_0,i_{\mu}}^* \ = \ -w_{i_{\mu},i_0}^* \ \leq \ x_{i_0} - x_{i_{\mu}}$$ and hence the equality $x_{i_0}-x_{i_{\mu}} = w_{i_0,i_{\mu}}^*$. This shows that the edge $\{i_0,i_{\mu}\}$ is contained in the equality graph $E(W^*)=E(W)$.
Finally, let $\{i,j\}$ be an edge in $E(W)=E(W^*)$. Then $x_{i}-x_{j} = w_{ij}^*<\infty$, and it follows that also $x_{j}-x_{i}=-w_{ij}^*$ is finite. Since the inequality $x_j - x_i \leq w_{ji}^*$ is tight by Lemma \[lem:Kleene\_tight\] we obtain $w_{ji}^* = -w_{ij}^*$. Therefore, there is a directed path from $j$ to $i$ in $\Gamma(W)$, and hence $(i,j,i)$ is a directed cycle of weight zero in $\Gamma(W^*)$. From this we infer our claim.
\[example:weighted\_digraph\_W\] The $3{\times}3$ matrix $$\label{eq:W33}
W \ = \
\left(
\begin{array}[c]{ccc}
1 & 4 & 1 \\
-1 & 0 & -2 \\
3 & \infty & 2
\end{array}
\right)$$ defines a directed graph without any cycles of weight zero. Its Kleene star is the matrix $$W^* \ = \
\left(
\begin{array}[c]{ccc}
0 & 4 & 1 \\
-1 & 0 & -2 \\
3 & 7 & 0
\end{array}
\right) \enspace .$$ The graphs of $W$ and $W^*$ are displayed in Figure \[figure:weighted\_digraph\_W\], while Figure \[figure:polyhedron\_W\] shows the corresponding weighted digraph polyhedron. Our convention for drawing digraphs is to omit loops of weight zero and arbitrary arcs of infinite weight. Since each weighted digraph polyhedron contains the one-dimensional linear subspace ${{\mathbb R}}{\mathbf 1}$ in its lineality space, throughout we draw pictures in the quotient ${{\mathbb R}}^d/{{\mathbb R}}{\mathbf 1}$, which is called the *tropical projective $(d{-}1)$-torus* in [@TropicalBook §5.2]. More precisely, for a feasible point $x+{{\mathbb R}}{\mathbf 1}$ in the quotient we draw the unique representative with $x_1=0$. This is the same as drawing the intersection of $Q(W)$ with the hyperplane $x_1=0$. As the polyhedron $Q(W)$ corresponding to the matrix is not contained in any hyperplane its equality graph $E(W)$ is the undirected graph with three isolated nodes.
We return to studying general matrices $W$.
\[lemma:dim equality graph\] The connected components of the equality graph of $E(W)$ are complete graphs, and their number is the dimension of the polyhedron $Q(W)$.
The equalities $x_i - x_j = w_{ij}^*$ and $x_j - x_{\ell} = w_{j \ell}^*$ imply $x_i - x_{\ell} = w_{ij}^* + w_{j \ell}^* \geq
w_{i\ell}^*$ and therefore $x_i - x_{\ell} = w_{i\ell}^*$ for any three nodes $i,j,\ell$ in the equality graph. So there is an edge between any two nodes in a connected component of $E(W)$. The statement about the dimension follows as the equality graph summarizes exactly those inequalities which are attained with equality and the connected components form a partition of the node set.
The lemma above says that the equality graph encodes an equivalence relation on the node set $[k]$. The partition into the connected components is the *equality partition*. Abusing our notation, again we denote this partition as $E(W)$.
Intersections and faces
-----------------------
Throughout the following we will frequently consider several graphs which share the same set of nodes. In this case it makes sense to identify such a graph with its set of edges (or arcs, in the directed case). This allows to talk about intersections and unions of such graphs.
\[lemma: intersection wgp\] Let $U$ and $W$ be $k{\times}k$-matrices. The intersection of the weighted digraph polyhedra $Q(U)$ and $Q(W)$ is the weighted digraph polyhedron $Q(U \oplus W)$. The arc set of the graph $\Gamma(U \oplus W)$ is the union of $\Gamma(U)$ and $\Gamma(W)$.
The intersection of two polyhedra is given by the union of their defining inequalities. The two inequalities of the form $x_i - x_j \leq u_{ij}$ and $x_i - x_j \leq w_{ij}$ are both satisfied if and only if the inequality $x_i - x_j
\leq \min(u_{ij},w_{ij})$ holds.
Again we assume that the graph $\Gamma(W)$ does not contain any negative cycle, and thus $Q(W)$ is feasible. Each face of the polyhedron $Q(W)$ is obtained by turning some of the defining inequalities into equalities. More precisely, for any subgraph $G$ of $\Gamma$ let $$F_G \ = \ F_G(W)\ = \ F_G(\Gamma,\gamma)\ = \ {\left\{\left.x\in Q(W)\vphantom{x_i-x_j=w_{ij} \text{ for all } (i,j)\in G}\ \right|\ x_i-x_j=w_{ij} \text{ for all } (i,j)\in G\vphantom{x\in Q(W)}\right\}} \enspace .$$ By construction $F_G$ is a face of $Q(W)$, and conversely each face of $Q(W)$ arises in this way. We define a new $k{\times}k$-matrix, denoted $W\#G$; it is constructed from $W$ by replacing the entries $w_{ji}$ with $-w_{ij}$ for each $(i,j)\in G$. If $G$ contains both $(i,j)$ and $(j,i)$ as arcs, this operation is only defined provided that $w_{ij} +
w_{ji} = 0$. The reason is that this equality is implied by $x_i-x_j=w_{ij}$ combined with $x_j-x_i=w_{ji}$. The following is immediate.
\[lemma: face of wdp\] Faces of weighted digraph polyhedra are weighted digraph polyhedra. More precisely, $$\begin{aligned}
F_G(W) \ &= \ Q(W) \cap {\left\{\left.x \in {{\mathbb R}}^k\vphantom{x_i - x_j=w_{ij} \text{ for $(i,j)\in G$}}\ \right|\ x_i - x_j=w_{ij} \text{ for $(i,j)\in G$}\vphantom{x \in {{\mathbb R}}^k}\right\}} \\
&= \ Q(W) \cap {\left\{\left.x \in {{\mathbb R}}^k\vphantom{x_j - x_i \leq -w_{ij} \text{ for $(i,j)\in G$}}\ \right|\ x_j - x_i \leq -w_{ij} \text{ for $(i,j)\in G$}\vphantom{x \in {{\mathbb R}}^k}\right\}} \ = \ Q(W\#G) \enspace .
\end{aligned}$$ Furthermore, the equality partition $E(W\#G)$ of a face $F_G(W)$ is obtained from the equality partition $E(W)$ by uniting the two parts which contain $i$ and $j$ if $(i,j)$ is an arc in $G$.
By Lemma \[lemma:dim equality graph\] the dimension of the face $F_G(W)$ equals the size of the partition $E(W\#G)$.
\[example:face\] If $W$ is the matrix from Example \[example:weighted\_digraph\_W\] and $G$ consists of the single arc $(2,3)$ then we have $$W\#G \ = \
\begin{pmatrix}
1 & 4 & 1 \\
-1 & 0 & -2 \\
3 & 2 & 2
\end{pmatrix} \enspace .$$ The equality graph $E(W\#G)$ consists of the isolated node $1$, and the nodes $2$ and $3$ are joined by an edge. This reflects that $Q(W\#G)$ is contained in the supporting hyperplane induced by the equality from $G$. Finally, the equality partition is $\{\{1\},\{2,3\}\}$.
Digraph Cones
-------------
Let us now consider the situation where the weight function is constantly zero on the arcs. Then for an arbitrary digraph $\Gamma$ the weighted digraph polyhedron $$Q(\Gamma,{\mathbf 0}) \ = \ {\left\{\left. x \in {{\mathbb R}}^k \vphantom{ x_i\ \leq\ x_j \text{ for all } (i,j) \in \Gamma }\ \right|\ x_i\ \leq\ x_j \text{ for all } (i,j) \in \Gamma \vphantom{ x \in {{\mathbb R}}^k }\right\}}$$ is a polyhedral cone, the *digraph cone* of $\Gamma$. All points in the subspace ${{\mathbb R}}{\mathbf 1}$ are feasible. Since every cycle has weight zero, applying Lemma \[lemma:zero\_cycles\]\[lemma:item:zero\_cycles\] to the cone $Q(\Gamma,{\mathbf 0})$ yields the following.
\[prop:cones\_components\] The parts of the equality partition $E(W(\Gamma,{\mathbf 0}))$ are exactly the strong components of $\Gamma$. In particular, the dimension of the digraph cone $Q(\Gamma,{\mathbf 0})$ equals the number of strong components of $\Gamma$.
Any hyperplane of the form $x_i=x_j$ defines a *split* of the unit cube $[0,1]^k$, i.e., it defines a (regular) subdivision of the unit cube into two subpolytopes; see [@HerrmannJoswig:2008]. Notice that such a split hyperplane does not separate any edge of the unit cube. Let us look at the map $\kappa$ which sends each face $F$ of the digraph cone $Q(\Gamma,{\mathbf 0})$ to the intersection $F \cap [0,1]^k$. Clearly, this intersection is never empty (unless $F$ is).
Now suppose that $\Gamma$ is acyclic. Then those inequalities which define facets of $Q(\Gamma,{\mathbf 0})$ correspond to the covering relations of the partially ordered set $P(\Gamma)$ on the node set $[k]$ of $\Gamma$ induced by the arcs. It follows that $\kappa(Q(\Gamma,{\mathbf 0}))=Q(\Gamma,{\mathbf 0})\cap[0,1]^k$ is the *order polytope* $\operatorname{Ord}(\Gamma)$ of the poset $P(\Gamma)$. The poset $P(\Gamma)$ describes the transitive closure of the relation defined on the set $[k]$ by the arcs of $\Gamma$. Conversely, each finite poset gives rise to a directed graph whose nodes are the elements and the arcs are given by the covering relations directed, say, upwards.
The order polytope $\operatorname{Ord}(\Gamma)$ contains the points ${\mathbf 0}$ and ${\mathbf 1}$ as vertices. Therefore there exists a unique minimal face which contains both of them; denote this face by $F_{01}$. Note that the dimension of $F_{01}$ can be any number between $1$ (if $F_{01}$ is the edge $[{\mathbf 0},{\mathbf 1}]$) and $k$ (if the graph $\Gamma$ does not contain any edges). The *face figure* of $F_{01}$, written as ${{\mathcal F}}_{01}$, is the principal filter of the element $F_{01}$ in the face poset of the order polytope $\operatorname{Ord}(\Gamma)$. The subposet ${{\mathcal F}}_{01}$ is the face poset of a polytope of dimension $k-\dim
F_{01}-1$. The face figure ${{\mathcal F}}_{01}$ consists of exactly those faces of $\operatorname{Ord}(\Gamma)$ which are not contained in any facet of the cube $[0,1]^k$. It is immediate that $\kappa$ maps faces of the digraph cone $Q(\Gamma,{\mathbf 0})$ to the faces of the order polytope $\operatorname{Ord}(\Gamma)$ which lie in the face figure ${{\mathcal F}}_{01}$.
\[lem:face\_figure\] If $\Gamma$ is acyclic then the map $\kappa$ is a poset isomorphism from ${{\mathcal F}}(Q(\Gamma,{\mathbf 0}))$ to the face figure ${{\mathcal F}}_{01}$ of the face $F_{01}$ of the order polytope $\operatorname{Ord}(\Gamma)$.
For any face $G\in{{\mathcal F}}_{01}$ let $\lambda(G)$ be the cone $\operatorname{pos}(G)+{{\mathbb R}}{\mathbf 1}$. Since $G$ is a face which is not contained in any facet of $[0,1]^k$ it is the intersection of facets of type $x_i\leq x_j$. These inequalities are homogeneous, and so they also hold for $\lambda(G)$. Those inequalities are tight for $Q(\Gamma,{\mathbf 0})$, and so $\lambda$ defines a map from ${{\mathcal F}}_{01}$ to ${{\mathcal F}}(Q(\Gamma,{\mathbf 0}))$. This also shows that, for any face $F$ of $Q(\Gamma,{\mathbf 0})$ we have $\lambda(\kappa(F))=F$ which means that $\kappa$ is one-to-one. Conversely, let $G$ be a face of $\operatorname{Ord}(\Gamma)$ which is contained in ${{\mathcal F}}_{01}$. Then $G$ is defined in terms of split equations of the form $x_i=x_j$. These equations are valid for $\lambda(G)=\operatorname{pos}(G)+{{\mathbb R}}{\mathbf 1}$, which yields $\kappa(\lambda(G))=G$. Hence $\kappa$ is surjective, and $\lambda$ is the inverse map.
Stanley gave a concise description of the face lattices of order polytopes in terms of partitions [@Stanley:1986 Theorem 1.2], and this can be used to derive the following characterization.
\[thm:partition\] Let $\Gamma$ be an arbitrary directed graph on the node set $[k]$. Then a partition $E$ of $[k]$ is the equality partition of a face of the digraph cone $Q(\Gamma,{\mathbf 0})$ if and only if
1. for each part $K$ of $E$ the induced subgraph of $\Gamma$ on $K$ is weakly connected, and
2. the minor of $\Gamma$ which results from simultaneously contracting each part of $E$ does not contain any directed cycle.
Let us first assume that $\Gamma$ is acyclic. By Lemma \[lemma: face of wdp\], together with the fact that every cycle has weight zero, the faces of $Q(\Gamma,{\mathbf 0})$ are given in terms of the equality partitions of $[k]$. In the acyclic case Lemma \[lem:face\_figure\] translates faces of $Q(\Gamma,{\mathbf 0})$ into faces of the order polytope $\operatorname{Ord}(\Gamma)$ which contain the special face $F_{01}$. The property (i) is the connectedness, and property (ii) is the ‘compatibility’ condition in Stanley’s result [@Stanley:1986 Theorem 1.2].
We now turn to the general case. If $\Gamma$ has directed cycles we consider its *acyclic reduction*. The latter graph, occasionally also called ‘condensation’ in the literature, is obtained by identifying the nodes in each strong component. Since strong components are weakly connected and gather all the directed cycles the same reasoning applies as before. It is easy to see that this digraph is indeed acyclic [@Sharir:81 Corollary 5]. Each partition of $[k]$ which describes a face of $Q(\Gamma,{\mathbf 0})$ refines the partition by strong components.
Notice that there are always two partitions which trivially satisfy the conditions above: The partition of $[k]$ by weak components corresponds to the unique minimal face (which is the lineality space); the partition by strong components corresponds to the entire cone.
\[example:order\_polytope\] The smallest non-trivial case is $k=2$, and $\Gamma$ is the directed graph with two nodes, labeled $1$ and $2$, with one arc from $1$ to $2$. The order polytope is the triangle $\operatorname{conv}\{00,01,11\}$, and the face $F_{01}$ is the edge from $00$ to $11$. The digraph cone $Q(\Gamma,{\mathbf 0})$ is the linear half-space $x_1 \leq x_2$, and its lineality space is ${{\mathbb R}}{\mathbf 1}$. The digraph cone and the order polytope are shown in Figure \[figure:cone\_order\_polytope\]. The node set of $\Gamma$ only admits the two trivial partitions. The Hasse diagram of the face lattice of $\operatorname{Ord}(\Gamma)$ and the face figure ${{\mathcal F}}_{01}$ are displayed in Figure \[figure:digraph\_order\_polytope\].
Figure \[figure: unweighted graph representing cone\] shows a digraph on eight nodes and its acyclic reduction, which has six nodes. Figure \[figure: Hasse-Diagram cone\] shows the Hasse diagram of the weighted digraph cone. That cone is $6$-dimensional with a $1$-dimensional lineality space. Modulo its lineality space every cone is projectively equivalent to a pyramid over its face at infinity. In this case the digraph cone inherits the combinatorics of a $4$-simplex.
[![Digraph (left) and its acyclic reduction (right)[]{data-label="figure: unweighted graph representing cone"}](ex41_cone1 "fig:")]{} [![Digraph (left) and its acyclic reduction (right)[]{data-label="figure: unweighted graph representing cone"}](ex41_cone2 "fig:")]{}
Two distinct digraphs on the node set $[k]$ may induce the same digraph cone. This is the case if and only if they induce the same poset. For instance, in Figure \[figure: unweighted graph representing cone\] the arc $(1,3)$ in the graph on the left and the arc $(1,378)$ in the graph on the right are redundant. In the acyclic reduction (on the right) we obtain a tree with directed edges. Every tree on $\ell$ nodes has $\ell-1$ edges, and the digraph cone is a simplex cone of dimension $\ell-1$.
Weyl-Minkowski decomposition
----------------------------
Now we want to use the Theorem \[thm:partition\] on digraph cones to describe digraph polyhedra for arbitrary weights. Again we pick a $k{\times}k$-matrix $W$, and we assume that $Q(W)$ is feasible. The classical theorem of Weyl and Minkowski states that any ordinary polyhedron $Q$ decomposes as the Minkowski sum $$\label{eq:Weyl-Minkowski}
Q \ = \ P + L + C \enspace ,$$ where $P$ is a polytope, $L$ is a linear subspace and $C$ is a pointed polyhedral cone. An ordinary polyhedral cone is *pointed* if it does not contain any affine line (and thus no affine subspace of positive dimension). In the decomposition the maximal linear subspace $L$ is unique, while, in general, there may be many choices for $C$ and $P$. The *recession cone* (which is again unique) is the Minkowski sum of the two unbounded parts, $L$ and $C$. The *pointed part* is the Minkowski sum $P+C$ (which is unique up to an affine transformation). Next we will decompose a weighted digraph polyhedron in this fashion. We decompose $W$ into the graph $\Gamma$ and the weight function $\gamma$ such that $W=W(\Gamma,\gamma)$.
\[lemma:recession\_cone\] The recession cone of the weighted digraph polyhedron $Q(\Gamma,\gamma)$ is the digraph cone $Q(\Gamma,{\mathbf 0})$, and $Q(W(\Gamma,{\mathbf 0})\#\Gamma)$ forms the maximal linear subspace.
Let $x$ be some point in the recession cone of $Q$. Then there exists a vector $t$ such that $x+\lambda t\in Q$ for all $\lambda\geq 0$. This means that $$x_i-x_j+\lambda(t_i-t_j)\ \leq \ w_{ij} \qquad \text{ for all } (i,j)\in \Gamma \text{ and } \lambda\geq 0 \enspace .$$ This forces $t_i-t_j \leq 0$ for all $(i,j)\in \Gamma$, and so $t$ lies in $Q(\Gamma,{\mathbf 0})$. The reverse inclusion is similar, and we conclude that the digraph cone $Q(\Gamma,{\mathbf 0})$ is the recession cone of $Q$.
Again let $t \in Q(\Gamma,{\mathbf 0})$. Then its negative $-t$ is also contained in $Q(\Gamma,{\mathbf 0})$ if and only if $$t_i - t_j \ = \ 0 \qquad \text{for all } (i,j)\in \Gamma \enspace$$ if and only if $t \in Q(W(\Gamma,{\mathbf 0})\#\Gamma)$. We infer that the digraph cone $Q(W(\Gamma,{\mathbf 0})\#\Gamma)$ forms the maximal linear subspace of $Q$.
As a corollary we obtain a slight generalization of [@DevelinSturmfels:2004 Corollary 12].
\[coro:boundedness\_wdp\] The weighted digraph polyhedron $Q(\Gamma, \gamma)$ is bounded in ${{\mathbb R}}^d / {{\mathbb R}}{\mathbf 1}$ if and only if $\Gamma$ consists of one strong component.
If $\Gamma$ has only one strong component, then the recession cone $(\Gamma, {\mathbf 0})$ is exactly the one-dimensional lineality space ${{\mathbb R}}{\mathbf 1}$ by Proposition \[prop:cones\_components\]. Hence, $Q(\Gamma, \gamma)$ is bounded in ${{\mathbb R}}^d / {{\mathbb R}}{\mathbf 1}$. Otherwise, the recession cone is higher-dimensional and the weighted digraph polyhedron is unbounded.
Our next goal is to describe a minimal system of generators for a digraph cone. Recall that a pointed cone is projectively equivalent to a pyramid over its far face. The minimal generators of a pointed cone correspond to the vertices of the far face. For any subset $K \subseteq [k]$, let $\chi(K)\in {{\mathbb R}}^k$ be the characteristic vector. That is, the $i$th coordinate of $\chi(K)$ is one if $i\in K$, and it is zero otherwise. With this notation, e.g., we have $\chi([k])={\mathbf 1}$ and $\chi(\emptyset)={\mathbf 0}$.
\[prop:rays\_of\_cone\] A minimal system of generators of the pointed part of the digraph cone $Q(\Gamma,{\mathbf 0})$ is given by the vectors $\chi(K)$ with $K\subseteq [k]$ so that the induced subgraph on $K$ is connected, its complement in its weak component in $\Gamma$ is also connected and every arc in the cut-set of this partition is directed from $[k]\setminus K$ to $K$.
Let $K_1, \ldots, K_{\ell}$ be the weak components of $\Gamma$. In particular, by applying Proposition \[prop:cones\_components\] to $Q(W(\Gamma,{\mathbf 0})\#\Gamma)$, the dimension of the lineality space of $Q(\Gamma,{\mathbf 0})$ equals $\ell$. Let $F$ be a minimal non-trivial face of the cone $Q(\Gamma,{\mathbf 0})$. This is a Minkowski sum of the lineality space with a single ray. By Theorem \[thm:partition\] the latter corresponds to a partition with $\ell + 1$ parts. Among these exactly $\ell-1$ parts are weak components of $\Gamma$, while the remaining weak component is split into two. Let us assume that the remaining component decomposes as $K_u = K \cup (K_u\setminus K)$, where every arc in the cut-set is directed from $K_u \setminus K$ to $K$. The characteristic vectors $\chi(K_i)$ for $i \in [\ell]$ linearly span the lineality space of $Q(\Gamma,{\mathbf 0})$, while $\chi(K)$ generates the pointed part of $F$.
Envelopes and duality
---------------------
We now turn to the construction of a special class of digraph polyhedra which were introduced by Develin and Sturmfels for studying tropical convexity from the viewpoint of geometric combinatorics [@DevelinSturmfels:2004]. For a $d{\times}n$-matrix $V$ with coefficients in ${{\mathbb T}}_{\min}={{\mathbb R}}\cup\{\infty\}$ we look at the ordinary polyhedron $$\begin{aligned}
{{\mathcal E}}(V) \ &= \ {\left\{\left.(y,z)\in{{\mathbb R}}^d\times{{\mathbb R}}^n\vphantom{y_i - z_j \leq v_{ij} \text{ for all } i\in[d] \text{ and } j\in[n]}\ \right|\ y_i - z_j \leq v_{ij} \text{ for all } i\in[d] \text{ and } j\in[n]\vphantom{(y,z)\in{{\mathbb R}}^d\times{{\mathbb R}}^n}\right\}}\\
&= \ {\left\{\left.(y,z)\in{{\mathbb R}}^d\times{{\mathbb R}}^n\vphantom{y_i - z_j \leq v_{ij} \text{ for all } (i,j)\in {{\rm B}}}\ \right|\ y_i - z_j \leq v_{ij} \text{ for all } (i,j)\in {{\rm B}}\vphantom{(y,z)\in{{\mathbb R}}^d\times{{\mathbb R}}^n}\right\}} \enspace ,\end{aligned}$$ where $$\label{eq:finite}
{{\rm B}}(V) \ = \ {\left\{\left.(i,j)\in [d]\times[n]\vphantom{v_{ij} \neq \infty}\ \right|\ v_{ij} \neq \infty\vphantom{(i,j)\in [d]\times[n]}\right\}}$$ is a (bipartite) directed graph recording the finite entries of $V$. We call ${{\mathcal E}}(V)$ the *envelope* of the matrix $V$. We may see the envelope as a weighted digraph polyhedron via the matrix $(d+n)\times(d+n)$-matrix $W$ which is defined as $$\label{eq:W}
W \ =\
\left(
\begin{array}{cc}
\infty_{d\times d} & V \\
\infty_{n\times d} & \infty_{n\times n}
\end{array}
\right) \enspace .$$ Up to an obvious relabeling of the nodes ${{\rm B}}(V)$ is the same as $\Gamma(W)$ for the matrix $W$ defined above, and thus we can identify ${{\mathcal E}}(V)$ with $Q(W)$. Applying Lemma \[lemma:recession\_cone\] and Proposition \[prop:rays\_of\_cone\] to the envelope we obtain the following.
\[coro:generators\_cone\_envelope\] The minimal generators of the pointed part of the recession cone of the envelope are given by the partitions $D' {\sqcup}D''=[d]$ and $N' {\sqcup}N''=[n]$ so that
1. the induced subgraph on $D'{\times}N'$ has the same number of weak components as ${{\rm B}}$,
2. the induced subgraph on $D''{\times}N''$ is connected, and
3. there are no arcs from $D''$ to $N'$.
The characteristic vector of $D''\times N''$ now yields one such generator.
Similarly we obtain from Proposition \[prop:rays\_of\_cone\] the following corollary which will be helpful in section \[sec:projective\]. A ray can be scaled modulo ${\mathbf 1}$ so that it has only non-negative entries and at least one zero entry. Then the *support* of the ray is the set of indices of the non-zero entries. We keep the notation of the former corollary and consider a face of the envelope ${{\mathcal E}}(V)$ defined by the graph $G$ that contains a minimal generator with support $D'' \times N''$. Let $M$ be the index set of the columns of $V$ with $v_{ij} = \infty$ for all $i \in D''$ and $j \in M$.
\[coro:inf\_env\] None of the shortest paths between any two nodes in $D'$ contains a node in $D'' {\sqcup}([n] \setminus M)$.
Observe that $M$ is exactly the subset of the nodes in $[n]$ without an arc between $D''$ and $M$ in $\Gamma(W\#G)$. Hence, we obtain $N' \subseteq M$ and further $[n] \setminus M \subseteq N''$. Since there is no arc from $N''$ to $D'$ in $\Gamma(W\#G)$ by Proposition \[prop:rays\_of\_cone\] we conclude that there is no arc from $[n] \setminus M$ to $D'$ in $\Gamma(W\#G)$. This implies that every shortest path between two nodes in $(D' {\sqcup}M)$ avoids $D'' {\sqcup}([n] \setminus M)$.
The graph ${{\rm B}}(V)$ has two kinds of nodes, those which correspond to the rows and those which represent columns of $V$. In our drawings, like Figure \[figure:bipartite\_graph\_recession\_cone\], we show row nodes as rectangles and column nodes as circles. Moreover, we always draw the row nodes above the column nodes. Therefore, if we want to distinguish them we sometimes talk about the *top* and the *bottom shore* of the bipartite graph.
\[example:envelope\_with\_infty\] For $d=n=3$ consider the $3{\times}3$-matrix $$V \ = \ \begin{pmatrix} 0 & 0 & 0 \\ 1 & 1 & \infty \\ 0 & 2 & \infty \end{pmatrix} \enspace .$$ The lineality space of the envelope ${{\mathcal E}}(V)$ is spanned by ${\mathbf 1}$. The quotient ${{\mathcal E}}(V)/{\mathbf 1}$ is $5$-dimensional, and it has exactly two vertices: ${(0,1,0;0,0,0)}$ and ${(0,1,2;2,0,0)}$. Its recession cone has six minimal generators, which arise from partitioning the bipartite graph ${{\rm B}}(V)$, which is a subgraph of $K_{3,3}$, into two induced subgraphs which meet the criteria of Corollary \[coro:generators\_cone\_envelope\]. The sets of the form $D'' \times N''$ read $$\emptyset{\times}1 \,, \quad \emptyset{\times}2 \,, \quad \emptyset{\times}3 \,,
\quad 12{\times}123 \,, \quad 13{\times}123 \,, \quad 23{\times}12 \enspace .$$ The complementary parts are given by $D'=\{1,2,3\}\setminus D''$ and $N'=\{1,2,3\}\setminus N''$. Notice that, e.g., $23{\times}123$ does not occur in the list above since $v_{23}=\infty=v_{33}$; this implies that the induced subgraph is not connected. For instance, $23{\times}12$ yields the generator ${(0,1,1;1,1,0)}$.
A *subpolytope* of a polytope $P$ is the convex hull of some subset of the vertices of $P$. Each face is a subpolytope, but the converse does not hold. We write $e_i$ for the $i$th standard basis vector of ${{\mathbb R}}^k$, for any $k$, and we write vectors in the product space ${{\mathbb R}}^d\times{{\mathbb R}}^n$ as ${({x},{y})}$ where $x\in{{\mathbb R}}^d$ and $y\in{{\mathbb R}}^n$. With this notation $$\Delta_{d-1}\times\Delta_{n-1} \ = \ \operatorname{conv}{\left\{\left.{({e_i},{e_j})}\vphantom{(i,j)\in[d]\times[n]}\ \right|\ (i,j)\in[d]\times[n]\vphantom{{({e_i},{e_j})}}\right\}}$$ is a product of simplices. Develin and Sturmfels established that a tropical configuration of $n$ points induces a polyhedral subdivision of ${{\mathbb R}}^d$ which is dual to a regular subdivision of $\Delta_{d-1}\times\Delta_{n-1}$ [@DevelinSturmfels:2004 Theorem 1]. A polytopal subdivision is *regular* if it is induced by a height function; for details see [@Triangulations]. The following statement will be instrumental in Section \[subsec:type\] below for obtaining a natural generalization to subpolytopes of products of simplices. Notice that those subpolytopes naturally correspond to subgraphs of the complete bipartite graph $[d]\times[n]$.
\[thm:regular\_subdivison\] The boundary complex of the envelope ${{\mathcal E}}(V)$ is dual to the regular subdivision of the polytope $$\operatorname{conv}{\left\{\left.{({e_i},{e_j})} \in {{\mathbb R}}^d \times {{\mathbb R}}^n\vphantom{(i,j) \in {{\rm B}}(V)}\ \right|\ (i,j) \in {{\rm B}}(V)\vphantom{{({e_i},{e_j})} \in {{\mathbb R}}^d \times {{\mathbb R}}^n}\right\}}$$ with height function $V$.
We abbreviate ${{\rm B}}={{\rm B}}(V)$. Homogenizing the envelope ${{\mathcal E}}(V)$ (with leading homogenizing coordinate) yields the cone $${\left\{\left.{({\alpha},{y},{z})} \in {{\mathbb R}}_{\geq 0}\times{{\mathbb R}}^d\times{{\mathbb R}}^n\vphantom{ \langle {({v_{ij}},{-e_i},{e_j})}, {({\alpha},{y},{z})} \rangle
\geq 0 \text{ for all } (i,j) \in {{\rm B}}}\ \right|\ \langle {({v_{ij}},{-e_i},{e_j})}, {({\alpha},{y},{z})} \rangle
\geq 0 \text{ for all } (i,j) \in {{\rm B}}\vphantom{{({\alpha},{y},{z})} \in {{\mathbb R}}_{\geq 0}\times{{\mathbb R}}^d\times{{\mathbb R}}^n}\right\}} \enspace .$$ Hence the polar cone with the dual face lattice can be written as $$\operatorname{pos}\left\{{({1},{{\mathbf 0}},{{\mathbf 0}})}\right\} + \operatorname{pos}{\left\{\left.{({v_{ij}},{-e_i},{e_j})}\vphantom{(i,j) \in {{\rm B}}}\ \right|\ (i,j) \in {{\rm B}}\vphantom{{({v_{ij}},{-e_i},{e_j})}}\right\}} \enspace .$$ Intersecting with the affine hyperplane $H={\{{{({\alpha},{y},{z})}}\,|\,{\langle {({0},{-{\mathbf 1}},{{\mathbf 1}})}, {({\alpha},{y},{z})}
\rangle = 2}\}}$ gives the polytope $$P \ = \ \operatorname{conv}{\left\{\left.{({v_{ij}},{-e_i},{e_j})}\vphantom{(i,j) \in {{\rm B}}}\ \right|\ (i,j) \in {{\rm B}}\vphantom{{({v_{ij}},{-e_i},{e_j})}}\right\}} \enspace ,$$ because all these vectors lie in $H$ and the origin does not.
The orthogonal projection of the lower convex hull of $P$ with respect to ${({1},{{\mathbf 0}},{{\mathbf 0}})}$ defines a regular subdivision of the subpolytope of $\Delta_{d-1}\times\Delta_{n-1}$ corresponding to ${{\rm B}}$. If ${{\rm B}}$ is the complete bipartite graph or equivalently no entry of $V$ is $\infty$, that subpolytope is the entire product of simplices.
Any regular subdivision of a subpolytope extends to a regular subdivision of the superpolytope, e.g., by successive placing of the remaining vertices [@Triangulations §4.3.1]. In our situation a regular subdivision of the superpolytope $\Delta_{d-1}\times\Delta_{n-1}$ is obtained by replacing the infinite coefficients in the matrix $V$ with sufficiently large real numbers. Note that this extension is not unique.
Projections
-----------
In this section we investigate orthogonal projections of weighted digraph polyhedra and envelopes into the coordinate directions. To this end we let $\pi_I$ be the projection onto the coordinates in $[k]\setminus I$ for $I\subseteq[k]$. For a $k{\times} k$-matrix $W$ we define $W/I$ by removing the rows and columns whose indices lie in $I$. We write $\pi_i$ and $W/i$ if $I=\{i\}$ is a singleton.
\[lemma:projection\_wdp\] The image of $Q(W)=Q(W^*)$ under the linear projection $\pi_I$ is the weighted digraph polyhedron $Q(W^*/I)$.
By induction it suffices to consider the case where $I=\{k\}$. That $\pi_{k}(Q(W^*))$ is contained in $Q(W^*/k)$ is clear. We want to show the reverse inclusion. For $(x_1, \ldots, x_{k-1}) \in Q(W^*/k)$ we need to find a real number $y$ so that $(x_1,\dots,x_{k-1} ,y ) \in Q(W)=Q(W^*)$. The latter condition is equivalent to $$x_i - w_{ik}^* \ \leq \ y \quad \text{ and } \quad y \ \leq \ x_i + w_{ki}^* \quad \text{ for all } \quad i \in [k-1] \enspace .$$ So, the claim follows if we can show that $$\label{eq:maxmin}
\max_{i \in [k-1]} (x_i - w_{ik}^*) \ \leq \ \min_{i \in [k-1]}(x_i + w_{ki}^*) \enspace .$$ Let $p$ and $q$ be indices for which the maximum and the minimum in , respectively, are attained. Now $w^*_{pq}$ is the length of the shortest path from $p$ to $q$ in the weighted digraph $\Gamma(W)$. This yields $$x_p - x_q \ \leq \ w^*_{pq} \ \leq \ w_{pk}^* + w_{kq}^* \quad\text{and hence}\quad x_p - w_{pk}^* \ \leq \ x_q + w_{kq}^* \enspace .$$
Now we turn to studying projections of faces of the envelope ${{\mathcal E}}(V)$ of a not necessarily square $d{\times}n$-matrix. With $W$ defined as in we have ${{\mathcal E}}(V)=Q(W)$. By Lemma \[lemma: face of wdp\] for any face $F$ of the envelope there is a subgraph $G$ of $\Gamma=\Gamma(W)$ such that $F=Q(W\#G)$. Since, up to a relabeling of the nodes, we can identify the directed graph $\Gamma$ with the bipartite graph ${{\rm B}}={{\rm B}}(V)$ and we may read $G$ as a subgraph of ${{\rm B}}$. We define the $n{\times}d$-matrix $V[G]$ with coefficients $$v_{ji}' \ = \ \begin{cases} -v_{ij} & \text{if } (i,j)\in G\\ \infty & \text{otherwise \enspace .} \end{cases}$$ The following lemma is similar to [@DevelinSturmfels:2004 Lemma 10]. Notice that the tropical matrix product $V
\odot V[G]$ yields a $d{\times d}$-matrix.
\[lemma: projection envelope\] The image of the face $F$ of ${{\mathcal E}}(V)\subset{{\mathbb R}}^d\times{{\mathbb R}}^n$ under the orthogonal projection $\pi_{[n]}$ onto the first component is the weighted digraph polyhedron $Q(V \odot V[G])$.
For $i,\ell\in[d]$ let $u_{i\ell}$ be a coefficient of $V\odot V[G]$. We have $$u_{i\ell} \ = \ \min_{j\in[n]}(v_{ij}+v_{j\ell}') \ = \ \min_{j\in[n],\, v_{ij}\neq\infty,\, v_{\ell
j}\neq\infty}(v_{ij}-v_{\ell j}) \enspace ,$$ which is exactly the length of a shortest path from $i$ to $\ell$ with two arcs in the digraph $\Gamma(W\#G)$. Since the directed graph $\Gamma(W\#G)$ is bipartite the shortest path from $i$ to $\ell$ (over arbitrarily many arcs) is a concatenation of the two-arc-paths above. Now the claim follows from the previous lemma.
[![Weighted digraphs corresponding to a face of ${{\mathcal E}}(V)$ from Example \[example:projection\_face\_envelope\]. The first graph corresponds to a face in ${{\mathbb R}}^d{\times}{{\mathbb R}}^n$ whereas the second corresponds to its projection onto ${{\mathbb R}}^d$. The nodes on the bottom shore are not in their natural ordering to reduce the number of arcs crossing[]{data-label="figure:face_envelope"}](ex_face_envelope1 "fig:")]{} [![Weighted digraphs corresponding to a face of ${{\mathcal E}}(V)$ from Example \[example:projection\_face\_envelope\]. The first graph corresponds to a face in ${{\mathbb R}}^d{\times}{{\mathbb R}}^n$ whereas the second corresponds to its projection onto ${{\mathbb R}}^d$. The nodes on the bottom shore are not in their natural ordering to reduce the number of arcs crossing[]{data-label="figure:face_envelope"}](ex_face_envelope2 "fig:")]{}
\[example:projection\_face\_envelope\] We consider the same matrix $V$ as in Example \[example:envelope\_with\_infty\]. For the bipartite graph $G$ on the six nodes $\{1,2,3\} {\sqcup}\{1,2,3\}$ with arcs $(1,3),(2,2),(3,1)$ we obtain $$V[G] \ = \ \begin{pmatrix} \infty & \infty & 0 \\ \infty & -1 & \infty \\ 0 & \infty & \infty \end{pmatrix} \enspace .$$ This yields the product $$V\odot V[G] \ = \ \begin{pmatrix} 0 & 0 & 0 \\ 1 & 1 & \infty \\ 0 & 2 & \infty \end{pmatrix} \odot
\begin{pmatrix} \infty & \infty & 0 \\ \infty & -1 & \infty \\ 0 & \infty & \infty \end{pmatrix} \ = \ \begin{pmatrix} 0 & -1 & 0 \\ \infty & 0 & 1 \\ \infty & 1 & 0 \end{pmatrix} \enspace .$$ The corresponding graph is depicted in Figure \[figure:face\_envelope\] on the right whereas the left one shows the graph $\Gamma(W\#G)$.
Tropical cones and polyhedral cells {#sec:tropical}
===================================
Polyhedral sectors
------------------
As before let $V$ be a $d{\times}n$-matrix with coefficients in ${{\mathbb T}}_{\min}$. We write $v^{(j)}$ for the $j$th column of $V$, and therefore we can identify $V$ with $(v^{(1)},v^{(2)},\dots,v^{(n)})$, the sequence of column vectors. The $(\min,+)$-linear span of the columns of $V$ is the *$\min$-tropical cone* $$\operatorname{tcone}(V) \ = \ {\left\{\left.(\lambda_1 \odot v^{(1)}) \oplus \dots \oplus (\lambda_n \odot v^{(n)})\vphantom{\lambda_j\in{{\mathbb T}}_{\min}}\ \right|\ \lambda_j\in{{\mathbb T}}_{\min}\vphantom{(\lambda_1 \odot v^{(1)}) \oplus \dots \oplus (\lambda_n \odot v^{(n)})}\right\}} \enspace .$$ Put in a more algebraic language, a tropical cone is the same as a finitely generated subsemimodule of the semimodule $({{\mathbb T}}_{\min}^d,\oplus,\odot)$. A subset $M$ of ${{\mathbb R}}^d$ is *$\min$-tropically convex* if for any two points $u,v\in
M$ we have $\operatorname{tcone}(u,v)\subseteq M$. Any tropically convex set contains ${{\mathbb R}}{\mathbf 1}$, and so we can study its image under the canonical projection to the tropical projective torus. Up to this projection tropical cones generated by vectors with finite entries are precisely the ‘tropical polytopes’ of Develin and Sturmfels [@DevelinSturmfels:2004]. In this section we will generalize key results from that paper to the case where $\infty$ may occur as a coordinate. By homogenization our results also apply to the formally more general ‘tropical polyhedra’ studied, e.g., in [@AkianGaubertGutermann12] and [@ABGJ-Simplex:A].
\[rem:wdp\_min\_cone\] For an arbitrary $k{\times}k$-matrix with coefficients in ${{\mathbb T}}_{\min}$ the weighted digraph polyhedron $Q(W)=Q(W^*)$ coincides with the $\min$-tropical span $\operatorname{tcone}(W^*)$. See also [@Butkovic:10 Theorem 2.1.1] and the Section \[sec:polytropes\] on polytropes below.
For $u \in {{\mathbb T}}_{\min}^d$ and $i \in [d]$ with $u_i \not= \infty$ we define the $i$th *sector* $S_i(u)$ with respect to *max* as $${\left\{\left.z \in {{\mathbb R}}^d\vphantom{\max_{\ell \in [d]}(z_{\ell} - u_{\ell}) = z_i - u_i}\ \right|\ \max_{\ell \in [d]}(z_{\ell} - u_{\ell}) = z_i - u_i\vphantom{z \in {{\mathbb R}}^d}\right\}} \ = \ {\left\{\left.z \in {{\mathbb R}}^d\vphantom{\min_{\ell \in [d]}(u_{\ell} - z_{\ell}) = u_i - z_i}\ \right|\ \min_{\ell \in [d]}(u_{\ell} - z_{\ell}) = u_i - z_i\vphantom{z \in {{\mathbb R}}^d}\right\}} \enspace .$$ Notice that the above equality of sets is a consequence of the elementary fact $$-\max(u,v) \ = \ \min(-u,-v) \enspace .$$ Moreover, the equation $\min_{\ell \in [d]}(u_{\ell} - z_{\ell}) = u_i - z_i$ is equivalent to $z_{\ell} - z_i \leq
u_{\ell} - u_i$ for each $\ell \in [d]$. As $u_i<\infty$ that minimum cannot be attained for any $\ell \in [d]$ with $u_{\ell} = \infty$. We have $$\label{eq:sector}
S_{i}(u) \ = \ \bigcap_{\ell\in[d],\ u_\ell\neq\infty}{\left\{\left.z\in{{\mathbb R}}^d\vphantom{z_{\ell} - z_i \leq u_{\ell} - u_i}\ \right|\ z_{\ell} - z_i \leq u_{\ell} - u_i\vphantom{z\in{{\mathbb R}}^d}\right\}}
\enspace ,$$ which means that this sector is the weighted digraph polyhedron for the graph with node set $[d]$ and arc set ${\{{(\ell,i)}\,|\,{\ell \in [d], u_{\ell} \not= \infty}\}}$, where the arc $(\ell,i)$ has weight $u_{\ell} - u_i$.
\[lem:sectors\] The sectors ${\{{S_i(u)}\,|\,{u_i\neq\infty}\}}$ are the maximal cells of a polyhedral decomposition of ${{\mathbb R}}^d$.
Considering the column vector $u$ as a $d{\times}1$-matrix, we obtain the envelope ${{\mathcal E}}(u)$ as a subset of ${{\mathbb R}}^{d+1}$. The sector $S_i(u)$ is the orthogonal projection of the face defined by the single arc $(i,1)$ in the bipartite graph ${{\rm B}}(u)$.
We denote the polyhedral complex arising from the previous lemma by ${\Delta}(u)$; see also [@DevelinSturmfels:2004 Proposition 16]. The negative $-u$ of the vector $u\in{{\mathbb T}}_{\min}^d$ defines a $\max$-tropical linear form and thus a $\max$-tropical hyperplane. The sectors $S_i(u)$ for $u_i\neq\infty$ are precisely the topological closures of the connected components of the complement of that tropical hyperplane.
The following result characterizes the solvability of a system of tropical linear equations in ${{\mathbb R}}^d$. For matrices with finite coordinates this is the Tropical Farkas Lemma [@DevelinSturmfels:2004 Proposition 9], a version of which already occurs in [@Vorobyev67]. We indicate a short proof for the sake of completeness.
\[lemma:sectors\_tropical\_hull\] A point $z \in {{\mathbb R}}^d$ is contained in $\operatorname{tcone}(V)$ if and only if for every $i \in [d]$ there is an index $s \in [n]$ with $z \in S_i(v^{(s)})$.
Let $z \in {{\mathbb R}}^d$ be a point in $\operatorname{tcone}(V)$. Then there is a vector $\lambda \in {{\mathbb T}}_{\min}^n$ so that $
\bigoplus_{j = 1}^{n} \lambda_j \odot v^{(j)} \ = \ z
$ or, equivalently, $$\label{eq:sectors}
\min{\left\{\left.\lambda_j + v_{ij}\vphantom{j \in [n]}\ \right|\ j \in [n]\vphantom{\lambda_j + v_{ij}}\right\}} \ = \ z_i \qquad \text{for each } i \in [d] \enspace.$$ Now fix $i\in[d]$ and let $s$ be an index $j$ for which the minimum in is attained; that is, $z_i=\lambda_s+v_{is}$. If $\ell\in[d]$ with $v_{\ell s}\neq\infty$ this gives $$z_\ell-z_i \ = \ z_\ell - \lambda_s-v_{is} \ \leq \ \lambda_j + v_{\ell j} - \lambda_s - v_{is} \qquad \text{for each }
j\in[n] \enspace .$$ Specializing to $j=s$ entails $z_\ell-z_i \leq v_{\ell s}-v_{is}$ and thus $z\in S_i(v^{(s)})$. The entire argument can be reversed to prove the converse.
The covector decomposition {#subsec:type}
--------------------------
Again let $V\in{{\mathbb T}}_{\min}^{d\times n}$, and let $W\in{{\mathbb T}}_{\min}^{(d+n)\times(d+n)}$ be the matrix which is associated via . We assume in the following that $V$ has no column equal to the all $\infty$ vector ${{(\infty,\ldots,\infty)}^\top}$; hence, none of the complexes ${\Delta}(v^{(j)})$ is empty. We do admit rows which solely contain $\infty$ entries. They add to the lineality of the occurring polyhedra. However, there may also be other contributions to the lineality space; see Lemma \[lemma:recession\_cone\]. The weighted bipartite graph ${{\rm B}}={{\rm B}}(V)$ and the weighted digraph $\Gamma=\Gamma(W)$ are defined as before. For an arbitrary subgraph $G$ of ${{\rm B}}$ we define the polyhedron $$\label{eq:def_type_cell}
X_G(V) \ = \ \bigcap_{(i,j)\in G} S_i(v^{(j)})$$ in ${{\mathbb R}}^d$.
\[rem:type\_relations\] Right from the definition, we obtain $X_{G \cup H}(V) = X_G(V) \cap X_H(V)$ for any two graphs $G,H \subseteq
{{\rm B}}(V)$ . If, furthermore, $G \subseteq H$ then $X_H(V) \subseteq X_G(V)$. This occurs also in [@DevelinSturmfels:2004 Corollary 11 and 13]. It should be stressed that the cells $X_G(V)$ and $X_H(V)$ may coincide even if the graphs $G$ and $H$ are distinct.
\[prop:projection\_face\_sector\] Let $G$ be an arbitrary subgraph of ${{\rm B}}$ (which we may also read as a subgraph of $\Gamma$). Then the orthogonal projection of the face $F_G(W)$ onto ${{\mathbb R}}^d$ equals $X_G(V)$. If no node in $[n]$ is isolated in $G$ that projection is an affine isomorphism.
Our goal is to exploit what we know about weighted digraph polyhedra. To this end we define several digraphs with the same node set $[d]{\sqcup}G$. Recall that we identify the subgraph $G$ of $\Gamma$ with its set of edges. However, in the class of digraphs to be defined now, those edges (along with the nodes in $[d]$) play the role of nodes.
Pick $(i,j)\in G$. We let $\Phi_{ij}$ be the weighted digraph which results from ${{\rm B}}(v^{(j)})$, which has $[d]{\sqcup}\{1\}$ as its node set, by renaming the node $1$ on the bottom shore by $(i,j)$ and adding an isolated node for each other arc in $G$. The graph $\Phi_{ij}$ has one extra arc in the reverse direction, namely from $(i,j)$ to $i$. The weights on the arcs from top to bottom are the same as in ${{\rm B}}(v^{(j)})$, while the weight on the single reverse arc is $-v_{ij}$. By construction the weighted digraph $\Phi_{ij}$ is bipartite and thus can be identified with a square matrix of size $d+|G|$. By Lemma \[lemma: projection envelope\] the weighted digraph polyhedron $Q(\Phi_{ij})\subset{{\mathbb R}}^d\times{{\mathbb R}}^G$ projects orthogonally onto the sector $S_i(v^{(j)})\subset{{\mathbb R}}^d$.
Let $\Phi$ be the digraph with node set $[d]{\sqcup}G$ which is obtained as the union of the digraphs $\Phi_{ij}$ for $(i,j)\in G$. Notice that by our construction the choice of the weights for the individual graphs $\Phi_{ij}$ is consistent. This way we obtain a natural weight function on $\Phi$. Due to Lemma \[lemma: intersection wgp\] we have $$\pi_G\bigl(Q(\Phi)\bigr) \ = \ \pi_G\bigl(\bigcap_{(i,j)\in G} Q(\Phi_{ij})\bigr) \ = \ \bigcap_{(i,j)\in G} S_i(v^{(j)}) \enspace .$$ If $\Gamma(W\#G)$ has a negative cycle, so has $\Phi$ and by Lemma \[lemma:feasible\] then $F_G(W)$ as well as $X_G(V)$ are empty. If there are no negative cycles, there exists a shortest path between two nodes $i$ and $\ell$ in $[d]$, and it does not matter if we consider $\Gamma(W\#G)$ or $\Phi$. So, the claim follows with Lemma \[lemma:projection\_wdp\].
For the rest, assume that $\Gamma(W\#G)$ has no negative cycle. Since $\Gamma(W\#G)$ is bipartite, any two nodes $i, \ell \in [d]$ are contained in a directed cycle of weight zero of $G$ if this also holds for the graph $\Gamma(\pi_{[n]}(Q(W\#G))$ of the projection of $F_G(W)$ by Lemma \[lemma:projection\_wdp\]. If no node in $[n]$ is isolated in $G$, every node in $[n]$ is contained in a directed cycle of weight zero, as every arc from $[n]$ to $[d]$ in $\Gamma(W\#G)$ induces a cycle of length zero. Hence, the equality partition of $F_G(W)$ and of its projection $\Gamma(\pi_{[n]}(Q(W\#G))$ have the same number of parts by Lemma \[lemma:zero\_cycles\]\[lemma:item:zero\_cycles\]. Therefore, if no node in $[n]$ is isolated in $G$, we get that $F_G(W)$ has the same dimension as $X_G(V)$.
The *covector decomposition* ${{\mathcal T}(V)}$ of ${{\mathbb R}}^d$ is the common refinement of the polyhedral complexes ${\Delta}(v^{(j)})$ for $j \in [n]$. For every cell $C$ in the covector decomposition there is a unique maximal subgraph ${{\rm T}(C)}$ of the complete bipartite graph $[d]\times[n]$, called the *covector graph* of $C$, such that $C =
X_{{{\rm T}(C)}}(V)$. This graph is equivalent to the *covector* $(t_1,t_2,\ldots,t_d) \in [n]^d$ where $t_i
\subseteq [n]$ consists of the nodes adjacent to $i$. While the covector notation is concise in most proofs it is convenient to keep the interpretation as a directed bipartite graph. Notice that our cells are closed by definition. By the former lemma, each covector (graph) also uniquely determines a face of ${{\mathcal E}}(V)$ and every face, for which no node in $[n]$ is isolated, occurs in this way. By Lemma \[lemma:sectors\_tropical\_hull\] the *covector decomposition* ${{\mathcal T}(V)}$ of ${{\mathbb R}}^d$ induces a covector decomposition of the tropical cone $\operatorname{tcone}(V)$. The covector graphs correspond to the ‘types’ of [@DevelinSturmfels:2004].
\[ex:type\_decomposition\] Figure \[figure:tropical\_types\] shows an example for the matrix $$V \ = \
\begin{pmatrix}
0 & 0 & 0 \\
1 & 0 & \infty \\
2 & -1 & \infty
\end{pmatrix} \enspace .$$ The points corresponding to the columns of $V$ are marked $1$, $2$ and $3$. Notice that the third column has $\infty$ as a coordinate, which is why this point lies outside the tropical projective torus. In fact, it is a boundary point of the *tropical projective plane*; see Section \[sec:projective\] and Figure \[fig:signed\_cells\] below.
Only the covectors of the full-dimensional cells are indicated since the covectors of the other cells can directly be deduced from them by Remark \[rem:type\_relations\].
The covector decomposition of $\operatorname{tcone}(V)$ has precisely two cells which are maximal with respect to inclusion: the $2$-dimensional cell with covector $(3,2,1)$ and the $1$-dimensional cell with covector $(13,2,2) = (13,-,2) \cup (13,2,-)$.
From the viewpoint of tropical geometry the decomposition ${{\mathcal T}(V)}$ can be deduced from the $\max$-tropical linear forms corresponding to the columns of $V$. For this, we pick variables $x_{1j},x_{2j},\ldots,x_{dj}$ for each column $v^{(j)}$ of $V$. The product of the tropical linear forms $\max(x_{1j}-v_{1j},x_{2j}-v_{2j},\ldots,x_{dj}-v_{dj})$ yields a homogeneous tropical polynomial $p$ in $d \cdot n$ variables $x_{ij}$. This defines a tropical hypersurface in ${{\mathbb R}}^{d \cdot n}/{{\mathbb R}}{\mathbf 1}$ where the covectors come into play as the exponent vectors of (tropical) monomials in $p$. Substituting $x_{ij}$ by $y_i$ gives rise to the tropical hypersurface in ${{\mathbb R}}^d/{{\mathbb R}}{\mathbf 1}$ which induces the cell decomposition of this space.
\[thm:envelope\] The orthogonal projection from the boundary complex of ${{\mathcal E}}(V)$ onto ${{\mathbb R}}^d$ induces a bijection between the envelope faces whose covector graph have no isolated node in $[n]$ and the cells in the covector decomposition ${{\mathcal T}(V)}$ of ${{\mathbb R}}^d$. This map is a piecewise linear isomorphism of polyhedral complexes.
Each face whose covector graph neither has an isolated node in $[d]$ (nor an isolated node in $[n]$) maps to a cell in the covector decomposition of $\operatorname{tcone}(V)$.
Ranging over all the faces whose covector graph has no isolated node in $[n]$ we obtain the bijection with Proposition \[prop:projection\_face\_sector\]. The definition of the covector of a cell combined with Lemma \[lemma:sectors\_tropical\_hull\] characterizes when a cell in ${{\mathcal T}(V)}$ is contained in the tropical cone generated by the columns of $V$.
With Theorem \[thm:regular\_subdivison\] the former implies the following.
\[coro:dual\_sub\_tcone\] The covector decomposition ${{\mathcal T}(V)}$ of ${{\mathbb R}}^d$ is dual to the regular subdivision of the polytope $$\operatorname{conv}{\left\{\left.{({e_i},{e_j})} \in {{\mathbb R}}^d \times {{\mathbb R}}^n\vphantom{(i,j) \in {{\rm B}}(V)}\ \right|\ (i,j) \in {{\rm B}}(V)\vphantom{{({e_i},{e_j})} \in {{\mathbb R}}^d \times {{\mathbb R}}^n}\right\}}$$ with weights given by $V$. Moreover, the covector decomposition of $\operatorname{tcone}(V)$ is dual to the poset of interior cells.
The result above is the same as [@FinkRincon:1305.6329 Corollary 4.2]; their proof is based on mixed subdivisions and the Cayley Trick [@Triangulations §9.2].
Note that the envelope of a matrix whose coefficients are $0$ or $\infty$ is a digraph cone, and so Theorem \[thm:partition\] applies to describe the combinatorics. The min-tropical cones corresponding to these matrices are tropical analogues of ordinary $0/1$-polytopes.
Let $V$ be a $d\times n$-matrix whose coefficients are $\infty$ or $0$. A partition $E$ of $[d]{\sqcup}[n]$ defines a face of the polyhedral fan ${{\mathcal T}(V)} \subseteq {{\mathbb R}}^d$ with apex ${\mathbf 0}$ if and only if
1. for each part $K$ of $E$ the induced subgraph of ${{\rm B}}(V)$ on $K$ is weakly connected,
2. the minor of ${{\rm B}}(V)$ which results from simultaneously contracting each part of $E$ does not contain any directed cycle, and
3. no part of $E$ is a single element of $[n]$.
As projections of the faces of the envelope ${{\mathcal E}}(V)$ the cones in such a fan can encode an arbitrary digraph on $d$ nodes.
The maximal cell in Figure \[figure:infinite\_convex\_hull\] is the intersection of the sectors $S_3({{(0,1,0)}^\top})$, $S_2({{(0,0,1)}^\top})$ and $S_1({{(0,\infty,\infty)}^\top})$. On the other hand, it is the projection of the face of the envelope ${{\mathcal E}}(V)$ corresponding to the graph on three nodes with the arcs $(1,3),(2,2),(3,1)$ for the matrix $V$ from Example \[example:envelope\_with\_infty\].
The recession cone of this face is given by the graph in Figure \[figure:cone\_graph\_face\_envelope.tex\]. It has the strong components $1\times3$ and $23\times12$. Hence, a minimal generator of the pointed part of the cone is ${{(0,1,1;1,1,0)}^\top}$ by Proposition \[prop:rays\_of\_cone\]. This projects to the ray generated as the positive span of ${{(0,1,1)}^\top}$ which is indeed contained in the tropical cone $\operatorname{tcone}(V)$.
[![Bipartite graph for the face projecting to the maximal cell in Figure \[figure:infinite\_convex\_hull\][]{data-label="figure:cone_graph_face_envelope.tex"}](ex_envelope_face_ray "fig:")]{}
Clearly, we can also project the envelope ${{\mathcal E}}(V)$ onto the $[n]$ coordinates of the lower shore. This yields a covector decomposition of ${{\mathbb R}}^n$ induced by the $d$ rows of the matrix $V$. Applying Theorem \[thm:envelope\] to the transpose ${{V}^\top}$ gives an isomorphism between the envelope faces without any isolated node in $[d]$ and the cells in the covector decomposition of ${{\mathbb R}}^n$ induced by the rows of $V$.
Therefore, the cells whose covector graphs do not have any isolated node in their covector graphs project affinely isomorphic to ${{\mathbb R}}^d$ as well as to ${{\mathbb R}}^n$. This entails an isomorphism between the covector decompositions of $\operatorname{tcone}(V)$ and $\operatorname{tcone}({{V}^\top})$.
\[prop:char\_type\_graph\] Let $G$ be a subgraph of $[d]\times[n]$. Then the following statements are equivalent.
1. \[item:point\_env\] There is a point $(y,z) \in {{\mathcal E}}(V)=Q(W)$ for which the inequality corresponding to $(i,j) \in \Gamma(W)$ is attained with equality if and only if $(i,j) \in G$.
2. \[item:matching\]
1. For every pair of subsets $D \subseteq [d]$ and $N \subseteq [n]$ with $|D| = |N|$, every perfect matching of $G$ restricted to $D {\sqcup}N$ is a minimal matching of the complete graph $D\times N$ with the weights given by the corresponding submatrix of $V$;
2. if there are more minimal perfect matchings in $D\times N$ then each of them is contained in $G$.
3. \[item:cycles\]
1. The graph $\Gamma(W\#G)$ does not have any negative cycle, and
2. every arc of $\Gamma(W)$ in $\Gamma(W\#G)$ that is contained in a cycle of weight zero is contained in $G$.
To conclude \[item:matching\] let $D \subseteq [d]$ and $N \subseteq [n]$ with $|D| = |N|$ so that there is a perfect matching $\mathcal{M}_0$ in $D\times N \cap G$. Let $\mathcal{M}_1$ be any other perfect matching in $D\times N$. Then considering the corresponding inequalities and equations implies after summing up and reordering $$\sum_{(i,j) \in \mathcal{M}_0} v_{ij} \ = \ \sum_{i \in D} y_i - \sum_{j \in N} z_j \ \leq \ \sum_{(i,j) \in \mathcal{M}_1} v_{ij} \enspace .$$ Therefore, $\mathcal{M}_0$ is a minimal perfect matching. Furthermore, if $\mathcal{M}_1$ is also a minimal perfect matching, then equality follows in the former inequality. That implies the equations $y_i - z_j = v_{ij}$ for every $(i,j) \in \mathcal{M}_1$. Hence, every arc in $\mathcal{M}_1$ has to be contained in $G$.
We now want to show that this implies \[item:cycles\]. For this, we consider a non-positive cycle in $\Gamma(W\#G)$ with vertex set $D {\sqcup}N$. Let $A_W$ be the set of arcs directed from $[d]$ to $[n]$ and $A_G$ the set of arcs directed from $[n]$ to $[d]$. Since $\Gamma(W\#G)$ is bipartite, this implies $|D| = |N| = |A_W| = |A_G|$ and the arc sets $A_W$ and $A_G$ define perfect matchings in $D\times N$.
By definition of $\Gamma(W\#G)$ we obtain for the weight of the cycle $$\sum_{(i,j) \in A_W} v_{ij} + \sum_{(j,i) \in A_G} (-v_{ij})\ \leq\ 0 \qquad \text{ or, equivalently, } \qquad \sum_{(i,j) \in A_W} v_{ij}\ \leq\ \sum_{(j,i) \in A_G} v_{ij} \enspace .$$ If the inequality is strict, this contradicts the minimality of the matching via \[item:matching\]. If the cycle has weight zero and the inequality becomes an equality, this implies that $A_W$ also represents a minimal perfect matching. With \[item:matching\] every arc in $A_W$ is also in $G$ then.
The final goal is to lead \[item:cycles\] back to \[item:point\_env\]. If $\Gamma(W\#G)$ does not contain a negative cycle, the weighted digraph polyhedron $Q(W\#G)$ is not empty. Therefore, there is $(y,z)$ in the interior of the face $Q(W\#G)
\subseteq {{\mathbb R}}^d\times{{\mathbb R}}^n$. Let $(i,j)$ be some arc of $\Gamma(W)$. If the equality $y_i - z_j = v_{ij}$ holds, Lemma \[lemma:zero\_cycles\]\[lemma:item:zero\_cycles\] yields that there is a cycle of weight zero containing the arc $(i,j)$. With \[item:cycles\] we obtain $(i,j) \in G$. On the other hand, for $(i,j) \in G$, the graph $\Gamma(W\#G)$ contains the cycle $(i,j,i)$ of weight zero, and the claim follows.
Together with Proposition \[prop:projection\_face\_sector\] this also gives a characterization for the covector graphs which are contained in the tropical cone $\operatorname{tcone}(V)$. Furthermore, we obtain a corollary concerning the dimension of a cell.
\[coro:dim\_type\] If $G \subseteq {{\rm B}}(V)$ is a covector graph for $V$, the dimension of $F_G(W)$ and thus of $X_G(V)$ equals the number of weak components of $G$.
By property \[item:cycles\] of Proposition \[prop:char\_type\_graph\] two nodes in $[d] {\sqcup}[n]$ are connected by a path in $G$ if and only if they are in a cycle of weight zero in $\Gamma(W\#G)$. By Lemma \[lemma:zero\_cycles\]\[lemma:item:zero\_cycles\] these cycles exactly define the equality partition of $F_G(W)$. Finally, Lemma \[lemma:dim equality graph\] connects this to the dimension. Further, Proposition \[prop:projection\_face\_sector\] shows the equality for $F_G(W)$ and $X_G(V)$.
The envelope of $V$ is the set of points $(y,z)$ satisfying $$y_i - z_j \ \leq \ v_{ij} \quad \text{ for } (i,j) \in {{\rm B}}\, .$$ Substituting $z_j$ by $-z_j$ yields $$\label{eq:max_cover_lp}
y_i + z_j \ \leq \ v_{ij} \quad \text{ for } (i,j) \in {{\rm B}}\enspace ,$$ which is the form of the envelope in [@DevelinSturmfels:2004]. Maximizing the coordinate sum over the polyhedron defined in is dual to finding a minimum weight matching by Egerváry’s Theorem [@Schrijver:CO:A Theorem 17.1]. This gives rise to a primal-dual algorithm for computing matchings and vertex covers; the method is explained in detail in [@PapadiCO Theorem 11.1]. A partial matching of minimal weight in a subgraph can be expanded by growing so-called ‘Hungarian trees’, which are shortest path trees in a modified graph. The partial matchings, which encode tight inequalities in the dual description, are collected in the equality subgraphs. By Proposition \[prop:char\_type\_graph\] one can deduce that these equality subgraphs are exactly the covector graphs of the dual points $(y,z)$.
Tropical half-spaces
--------------------
The sectors $S_i(u)$ with $u_i\neq\infty$ from Lemma \[lem:sectors\], which are responsible for the combinatorial properties of $\min$-tropical point configurations, are precisely the (closures of the) complements of the $\max$-tropical hyperplane with apex $u$. The same combinatorial objects also control systems of tropical linear inequalities. To see this it is convenient to switch to $\max$ as the tropical addition now.
Let $c \in {{\mathbb T}}_{\min}^d$ and let $I$ be a non-empty proper subset of $[d]$, i.e., $I\neq\emptyset$ and $I\neq[d]$. Then the set $\bigcup_{\ell \in I} S_{\ell}(c)$ is a *$\max$-tropical half-space* with *apex* $c$. This is exactly the set of points in ${{\mathbb R}}^d$ which satisfies the homogeneous $\max$-tropical linear inequality $$\max_{\ell \in [d] \setminus I} (-c_{\ell} + x_{\ell}) \ \leq \ \max_{\ell \in I} (-c_{\ell} + x_{\ell}) \enspace .$$ Since here we allow for $\infty$ as a coordinate in $c$ this definition is more general than the one in [@MJ:2005]. Notice that $-c$ is an element of ${{\mathbb T}}_{\max}^d$ and that the halfspaces are defined over the $\max$-tropical semiring. By [@GaubertKatz:11 Theorem 1] each tropical cone is the intersection of finitely many tropical half-spaces and conversely; the Proposition 3.3 [@MJ:2005] and thus also the proof of [@MJ:2005 Theorem 3.6] is false. In [@Butkovic:10 §7.6] it is shown that the solution set of any system of max-tropical linear equalities is finitely generated. Since $u\leq v$ holds if and only if $\max(u,v)=v$, i.e., since in the tropical setting studying systems of linear equalities amounts to the same as studying systems of linear inequalities, that result is essentially equivalent to [@GaubertKatz:11 Theorem 1].
\[rem:wdp\_trop\_ineq\] Let $W$ be a $k{\times}k$-matrix. Each defining inequality of the weighted digraph polyhedron $Q(W)$ can be rewritten as $$x_i - w_{ij}\ \leq \ x_j \qquad \text{ for each arc $(i,j)$ in $\Gamma(W)$} \enspace .$$ Fixing $j$ and varying $i$ then yields $$\max_{i \in [k]}(x_i - w_{ij}) \ \leq \ x_j \qquad \text{ for each } j \in [k] \enspace .$$ Looking at all $j$ simultaneously we obtain the equation $$(-{{W}^\top}) \odot_{\max} x \ \leq \ x \enspace .$$ of column vectors. This means that each weighted digraph polyhedron is a max-tropical cone. In [@Butkovic:10 §1.6.2 and §2] a vector $x$ satisfying the inequality above is called a ‘subeigenvector’ of the matrix $-{{W}^\top}$.
We now want to introduce notation for inequality descriptions of tropical cones which is suitable for our combinatorial approach. Let $V=(v_{ij}) \in {{\mathbb T}}_{\min}^{d{\times}n}$ and let $\Psi$ be a subgraph of the complete bipartite graph $[d]\times[n]$ with arcs directed from $[d]$ to $[n]$. We define $$\label{eq:section_complex}
{\operatorname{thalf}}(V,\Psi) \ = \ \bigcap_{j \in [n]} \bigcup_{(i,j) \in \Psi} S_{i}(v^{(j)}) \enspace .$$ That is, ${\operatorname{thalf}}(V,\Psi)$ comprises those points $x \in {{\mathbb R}}^d$ which satisfy the homogeneous $\max$-tropical linear inequalities $$\max_{i \in [d],\, (i,j) \not\in \Psi}(-v_{ij} + x_i) \ \leq \ \max_{i \in [d],\, (i,j) \in \Psi}(-v_{ij} + x_i)$$ for each $j\in[n]$. In our notation the columns of the matrix $V$ collect the apices of the tropical half-spaces, and the graph $\Psi$ lists the sectors per half-space. In [@Butkovic:10 §7] exterior descriptions of tropical cones like are discussed under the name ‘two-sided max-linear systems’. To phrase our results below it is convenient to introduce two sets of subgraphs of $[d]\times[n]$, both of which depend on $\Psi$. We let $$\begin{aligned}
{{\mathcal G}}_\Psi \ &= \ {\left\{\left.G \subseteq \Psi\vphantom{\text{every node in } [n] \text{ has degree } 1 \text{ in } G}\ \right|\ \text{every node in } [n] \text{ has degree } 1 \text{ in } G\vphantom{G \subseteq \Psi}\right\}} \qquad \text{and}\\
{{\mathcal H}}_\Psi \ &= \ {\left\{\left.H \subseteq [d]\times[n]\vphantom{\text{every node in } [n] \text{ has degree } \geq 1 \text{ in }
\Psi\cap H}\ \right|\ \text{every node in } [n] \text{ has degree } \geq 1 \text{ in }
\Psi\cap H\vphantom{H \subseteq [d]\times[n]}\right\}} \enspace ,
\end{aligned}$$ which gives the following.
\[prop:graphs\_types\_cells\] For each graph $H\in{{\mathcal H}}_\Psi$ the cell $X_H$, which may be empty, is contained in ${\operatorname{thalf}}(V,\Psi)$. Further, ${{\mathcal G}}_\Psi \subseteq {{\mathcal H}}_\Psi$, and we have $${\operatorname{thalf}}(V,\Psi) \ = \ \bigcup_{G \in {{\mathcal G}}_\Psi} \bigcap_{(i,j) \in G} S_{i}(v^{(j)}) \ = \ \bigcup_{H \in {{\mathcal H}}_\Psi} \bigcap_{(i,j) \in H} S_{i}(v^{(j)}) \enspace .$$
Here the first equality is obtained by reordering the intersections and unions in the Definition . For the second equality notice that ${{\mathcal G}}_\Psi \subseteq {{\mathcal H}}_\Psi$. Since for every graph $H \in {{\mathcal H}}_\Psi$ there is a graph $G \in {{\mathcal G}}_\Psi$ so that $X_H(V) \subseteq X_G(V)$ the claim follows.
The preceding proposition says that a cell $X_G(V)=\bigcap_{(i,j)\in G} S_i(v^{(j)})$ in the covector decomposition ${{\mathcal T}(V)}$ of ${{\mathbb R}}^d$ with covector graph $G \subseteq [d]\times[n]$ is contained in the $\max$-tropical cone ${\operatorname{thalf}}(V,\Psi)$ if and only if no node in $[n]$ is isolated in the intersection of $G$ and $\Psi$. Moreover, ${\operatorname{thalf}}(V,\Psi)$ is a union of cells. In this way the Proposition \[prop:graphs\_types\_cells\] can be seen as some kind of a dual version of [@DevelinSturmfels:2004 Theorem 15], which is a key structural result in tropical convexity.
The covector decomposition of ${\operatorname{thalf}}(V,\Psi)$ induced by the columns of $V$ is dual to a subcomplex of the regular subdivision of $\Delta_{d-1}\times\Delta_{n-1}$ with weights given by $V$.
The apices ${{(0,1,1)}^\top}$ and ${{(0,2,1)}^\top}$ induce the cell decomposition depicted in Figure \[figure:not\_max\_cell\]. Every node in the bottom shore in the graph $G$ to the right has degree $1$. Hence, it is the kind of graph contained in ${{\mathcal G}}_{\Psi}$ for some appropriate $\Psi$ (for example $G$ itself). However, the corresponding cell is not full-dimensional since the apices are not in general position. Indeed, the covector graph of this cell is obtained from $G$ by adding the arcs $(3,1)$ and $(1,2)$.
[![The left figure shows the cell decomposition induced by two apices which are not in general position. The right figure depicts a graph $G$ corresponding to the black marked cell $X_G$ on the left[]{data-label="figure:not_max_cell"}](ex_not_max_cell1 "fig:")]{} [![The left figure shows the cell decomposition induced by two apices which are not in general position. The right figure depicts a graph $G$ corresponding to the black marked cell $X_G$ on the left[]{data-label="figure:not_max_cell"}](ex_not_max_cell2 "fig:")]{}
\[rem:tangent\_digraph\] The *tangent digraph*, defined in [@ABGJ-Simplex:A §3.1], describes the local combinatorics at a cell $C$ of ${\operatorname{thalf}}(V,\Psi)$. This is related to the above as follows. Deleting all nodes in $[n]$ (and incident arcs) for which all incident arcs are contained in $\Psi$ in the covector graph ${{\rm T}(C)}$ and forgetting about the orientation yields the *tangent graph* $\operatorname{TG}(C)$ of [@ABGJ-Simplex:A §3.1]. By taking the orientation into account and reversing every arc in $\operatorname{TG}(C)$ which is not in $\operatorname{TG}(C) \cap \Psi$ from the bottom shore $[n]$ (corresponding to the hyperplane apices) to the top shore $[d]$ (corresponding to the coordinate directions) we obtain the tangent digraph.
Proposition \[prop:graphs\_types\_cells\] implies that the $\max$-tropical cone ${\operatorname{thalf}}(V,\Psi)$ is compatible with the covector decomposition of ${{\mathbb R}}^d$ induced by $V$. Thus it makes sense to talk about the *covector decomposition* of a $\max$-tropical cone with respect to a fixed system of defining tropical half-spaces. This is the polyhedral decomposition formed by the cells which happen to lie in the tropical cone. A tropical cone is *pure* if each cell in its covector decomposition which is maximal with respect to inclusion shares the same dimension. While the covector decomposition does depend on the choice of the defining inequalities, pureness does not.
The *tropical determinant* of a square matrix $W=(w_{ij})\in{{\mathbb T}}_{\min}^{k{\times}k}$ is $$\label{eq:tdet}
\begin{aligned}
\operatorname{tdet}W \ &= \ \bigoplus_{\sigma\in\operatorname{Sym}(k)} \bigodot_{i\in[k]} w_{i,\sigma(i)} \\
&= \ \min_{\sigma\in\operatorname{Sym}(k)} (w_{1,\sigma(1)}+w_{2,\sigma(2)}+\dots+w_{k,\sigma(k)}) \enspace ,
\end{aligned}$$ which is the same as the solution to a minimum weight bipartite matching problem in the complete bipartite graph $[k]\times[k]$. The tropical determinant *vanishes* if the minimum in equals $\infty$ or if it is attained at least twice. In [@Butkovic:10 §6.2.1] a square matrix whose tropical determinant does not vanish is called ‘strongly regular’. A not necessarily square matrix is *tropically generic* if the tropical determinant of no square submatrix vanishes. A finite set of points is in *tropically general position* if any matrix whose columns (or rows) represent those points is tropically generic. Develin and Yu conjectured that a tropical cone is pure and full-dimensional if and only if it has a half-space description in which the apices of these half-spaces are in general position [@DevelinYu:2007 Conjecture 2.11]. The next result confirms one of the two implications.
\[thm:pure\] Let $V$ and $\Psi$ be as before. If $V$ is tropically generic with respect to the tropical semiring ${{\mathbb T}}_{\min}$ then the $\max$-tropical cone ${\operatorname{thalf}}(V,\Psi)$ is pure and full-dimensional.
As in Proposition \[prop:graphs\_types\_cells\] we consider the graph class ${{\mathcal G}}_\Psi$. If we can show that each ordinary polyhedron $X_G(V) = \bigcap_{(i,j) \in G} S_{i}(v^{(j)})$ for $G \in {{\mathcal G}}_\Psi$ is either full-dimensional or empty then the claim follows. Proposition \[prop:projection\_face\_sector\] implies that $X_G(V)$ is the projection of the weighted digraph polyhedron $Q(W\#G)$, which is a face of ${{\mathcal E}}(V)=Q(W)$. Assume that $Q(W\#G)$ is feasible. We have to show that $X_G(V)$ is full-dimensional, i.e., it suffices to show that $\dim Q(W\#G)=d$.
In view of Proposition \[prop:char\_type\_graph\] together with Corollary \[coro:dim\_type\] this will follow if we can show that no two nodes in $[n]$ are contained in a cycle of weight zero in $\Gamma(W\#G)$. Aiming at an indirect argument we suppose that such a cycle exists. Let $D {\sqcup}N$ be the vertex set of the zero cycle $(d_1,n_1,d_2,n_2,\dots,d_1)$. We have $|D| = |N|$. Then the arcs $(d_1,n_1),(d_2,n_2),\dots$ form a perfect matching ${{\mathcal M}}$ in $D\times N$ whose weight $\sum_i v_{d_i,n_i}$ is minimal by Proposition \[prop:char\_type\_graph\]. The complementary arcs $(n_1,d_2),(n_2,d_3),\dots$ of the cycle yield a second matching whose weight is the same as the weight of ${{\mathcal M}}$ since the total weight of the cycle is zero. This entails that the minimum $$\min_{\sigma} \sum_{i \in D} v_{i \sigma(i)} \enspace ,$$ where $\sigma$ ranges over all bijections from $D$ to $N$, is attained at least twice for the submatrix of $V$ indexed by $D\times N$. Hence, the apices are not in general position, and this is the desired contradiction.
Since the matrix $V$ is tropically generic it is immediate that $\operatorname{tcone}(V)$ has at least one full-dimensional cell; e.g., see [@Butkovic:10 Theorem 6.2.18] or [@DevelinSturmfels:2004 Proposition 24]. Yet, in general $\operatorname{tcone}(V)$ is not pure; see Example \[ex:type\_decomposition\]. The following shows that the reverse direction of Theorem \[thm:pure\] does not hold.
\[exmp:pure\] For $$V \ = \ \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 3 & 2 & 1 & \infty & \infty \\ 2 & 2 & \infty & 1 & 3 \end{pmatrix}$$ and $\Psi$ as in Figure \[figure:feasible\_graph\] we are interested in the $\max$-tropical cone $C={\operatorname{thalf}}(V,\Psi)$. Now $C$ is pure, but the first two columns, ${{(0,3,2)}^\top}$ and ${{(0,2,2)}^\top}$, of the matrix $V$ are not in general position with respect to $\min$. Notice that each one of the apices of the three remaining tropical half-spaces can be moved without changing the feasible set $C$. However, the first two tropical half-spaces are *essential* in the sense that they occur in any exterior description of $C$.
[![The graph $\Psi$ for the $\max$-tropical cone $C={\operatorname{thalf}}(V,\Psi)$ from Example \[exmp:pure\] and Figure \[figure:pure\_counterexample\][]{data-label="figure:feasible_graph"}](ex_feasible_graph "fig:")]{}
[![The pure $\max$-tropical cone $C$ from Example \[exmp:pure\]. The apices of any $\max$-tropical half-space description are not in general position with respect to $\min$[]{data-label="figure:pure_counterexample"}](ex_pure_counter "fig:")]{}
A related conjecture from the same paper [@DevelinYu:2007 Conjecture 2.10] was recently resolved by Allamigeon and Katz [@AllamigeonKatz:1408.6176].
Polytropes {#sec:polytropes}
----------
A *polytrope* is a tropical cone $P=\operatorname{tcone}(V)$ for $V\in{{\mathbb R}}^{d\times n}$, i.e., with a generating matrix with finite coefficients, which is also convex in the ordinary sense. In that case $d$ generators suffice [@DevelinSturmfels:2004 Proposition 18] and [@JoswigKulas:2010 Theorem 7]. Therefore we may assume that $n=d$. From this we obtain $\operatorname{tcone}(V)=Q(V)=Q(V^*)$ in view of Remark \[rem:wdp\_min\_cone\], and thus any polytrope is a weighted digraph polyhedron; see also [@JoswigKulas:2010 Proposition 10]. Yet another argument for the same goes through Theorem \[thm:envelope\] and Lemma \[lemma: face of wdp\]. This is slightly more general as it takes $\infty$ coefficients into account. Moreover, the covector decomposition of $P$ induced by the square matrix $V$ has a single cell. Its projection to the tropical projective torus ${{\mathbb R}}^d/{{\mathbb R}}{\mathbf 1}$ is bounded, namely the polytrope $P$ itself. The latter also gives a max-tropical exterior description. The polytropes are exactly the ‘alcoved polytopes of type A’ of Lam and Postnikov [@LamPostnikov:2007]. The weighted digraph polyhedra form the natural generalization to polyhedra which are not necessarily bounded. We sum up our discussion in the following statement.
Let $V\in{{\mathbb T}}_{\min}^{d\times n}$ such that the $\min$-tropical cone $\operatorname{tcone}(V)$ is also convex in the ordinary sense. Then there is a $d{\times}d$-matrix $U$ such that $\operatorname{tcone}(V)=Q(U)$ is a weighted digraph polyhedron.
In the context of proving a hardness result on the vertex-enumeration of polyhedra given in terms of inequalities Khachiyan and al. [@Khachiyan:2008] study the *circulation polytope* of the digraph $\Gamma$, which is the set of all points $u\in{{\mathbb R}}^{\Gamma}$ satisfying $$\begin{aligned}
\sum_{j: (i,j) \in \Gamma}u_{ij} - \sum_{\ell: (\ell, i) \in \Gamma}u_{\ell i} \ &= \ 0 \quad \text{ for all } i \in [k] \\
\sum_{(i,j) \in \Gamma} u_{ij} \ &= \ 1 \\
0 \ &\leq \ u_{ij} \quad \text{ for all } (i,j) \in \Gamma \enspace .\end{aligned}$$ The support set ${\{{(i,j) \in \Gamma}\,|\,{u_{ij} \neq 0}\}}$ of a vertex of the circulation polytope defines a cycle in $\Gamma$. Hence, by Lemma \[lemma:feasible\], minimizing the weight function $\gamma(W)$ over the circulation polytope yields a certificate for the feasibility of $Q(W)$. Tran uses this approach to characterize the feasibility of polytropes in terms of ordinary inequalities [@Tran:2013 §3].
Covector decompositions of tropical projective spaces {#sec:projective}
-----------------------------------------------------
The *tropical projective space* ${\mathbb{TP}_{\min}}^{d-1}$ is defined as the quotient of ${{\mathbb T}}_{\min}^d\setminus\{{{(\infty,\infty,\dots,\infty)}^\top}\}$ modulo ${{\mathbb R}}{\mathbf 1}$. That is, its points are equivalence classes of vectors with coefficients in ${{\mathbb T}}_{\min}={{\mathbb R}}\cup\{\infty\}$ with at least one finite entry, up to differences by a real constant; see [@Mikhalkin:2006 Example 3.10]. The tropical projective space ${\mathbb{TP}_{\min}}^{d-1}$ is a natural compactification of the tropical projective torus ${{\mathbb R}}^d/{{\mathbb R}}{\mathbf 1}$. It is easy to see that the pair $({\mathbb{TP}_{\min}}^{d-1},{{\mathbb R}}^d/{{\mathbb R}}{\mathbf 1})$ is homeomorphic to the pair of a $(d{-}1)$-simplex and its interior.
We assume that $V\in{{\mathbb T}}_{\min}^{d\times n}$ has no column identically $\infty$. Then $V$ gives rise to a configuration of $n$ labeled points in ${\mathbb{TP}_{\min}}^{d-1}$. The covector decomposition ${{\mathcal T}(V)}$ of ${{\mathbb R}}^d$ does not change if we add a real constant to the entries in any column. So it is an invariant of that point configuration, and, moreover, ${{\mathcal T}(V)}$ induces a covector decomposition of the tropical projective torus ${{\mathbb R}}^d/{{\mathbb R}}{\mathbf 1}$. Yet it makes sense to study tropical convexity and tropical cones also within the compactification ${\mathbb{TP}_{\min}}^{d-1}$. Our goal is to describe a decomposition of the tropical projective space into cells. Let $Z$ be a proper subset of $[d]$. We consider the matrix obtained by removing from $V$ all columns $j$ for which there is an $i \in Z$ with $v_{ij}\not=\infty$. Each row of the resulting matrix with a label in $Z$ has only $\infty$ as coefficients. Removing these rows yields yet another matrix, which we denote as $V(Z)$. Now this matrix induces a covector decomposition of the *boundary stratum* $${\mathbb{TP}_{\min}}^{d-1}(Z) \ = \ {\left\{\left.(p_1,p_2,\dots,p_d)\in{\mathbb{TP}_{\min}}^{d-1}\vphantom{p_i=\infty \text{ if and only if } i\in Z}\ \right|\ p_i=\infty \text{ if and only if } i\in Z\vphantom{(p_1,p_2,\dots,p_d)\in{\mathbb{TP}_{\min}}^{d-1}}\right\}} \enspace,$$ which is a copy of the tropical projective torus of dimension $d-1-|Z|$. In particular, we have ${\mathbb{TP}_{\min}}^{d-1}(\emptyset)={{\mathbb R}}^d/{{\mathbb R}}{\mathbf 1}$. Notice that for the induced covector decomposition we keep the original labels of the columns and the rows.
For $K \subseteq [d]$ let $b^{(K)}$ be the vector in ${{\mathbb T}}_{\min}^d$ with $$b_i^{(K)} \ = \
\begin{cases}
0 & \text{ for } i \in [d] \setminus K \\
\infty & \text{ for } i \in K
\end{cases} \enspace .$$ Consider $u \in {{\mathbb T}}_{\min}^d$ and let $\operatorname{supp}(u)={\{{i\in[d]}\,|\,{u_i\neq\infty}\}}$ be the support of $u$. Then the recession cone of the weighted digraph polyhedron $S_i(u)$ is given by the graph on $[d]$ where the nodes in $[d] \setminus \operatorname{supp}(u)$ are isolated and there are arcs from the nodes in $\operatorname{supp}(u) \setminus \{ i \}$ to $i$, see Equation . The supports of the rays of $S_i(u)$ are given by the sets in $${{\mathcal K}}\ = \ {\left\{\left.A \cup B\vphantom{A \in {{\mathcal A}}, B \in {{\mathcal B}}}\ \right|\ A \in {{\mathcal A}}, B \in {{\mathcal B}}\vphantom{A \cup B}\right\}}$$ where $${{\mathcal A}}= \bigl\{ \emptyset \cup {\left\{\left.M \cup \{i\}\vphantom{M \subseteq \operatorname{supp}(u)}\ \right|\ M \subseteq \operatorname{supp}(u)\vphantom{M \cup \{i\}}\right\}} \bigr\} \quad \text{and} \quad {{\mathcal B}}= \bigl\{M \subseteq [d]\setminus \operatorname{supp}(u)\bigr\} \enspace .$$ Here, the sets in ${{\mathcal A}}$ correspond to the faces of the pointed part of the recession cone of $S_i(u)$ described by Theorem \[thm:partition\]. The set ${{\mathcal B}}$ encodes rays arising from the lineality space of $S_i(u)$ which was characterized in Lemma \[lemma:recession\_cone\].
So, it is natural to define $${\overline{S_{i}(u)}} \ = \ S_i(u) \cup \bigcup_{K \in {{\mathcal K}}} \left(b^{(K)} + S_i(u)\right) \enspace$$ where the ‘$+$’-operator denotes elementwise ordinary addition of $b^{(K)}$ and the set $S_i(u)$.
In the following we will frequently identify subsets of $({{\mathbb R}}\cup\{\infty\})^d$ with their images modulo ${{\mathbb R}}/{\mathbf 1}$. In particular, we will typically view ${\overline{S_{i}(u)}}$ with $u_i\neq\infty$ as a subset of ${\mathbb{TP}_{\min}}^{d-1}$.
\[lem:sectors\_boundary\] The set ${\overline{S_{i}(u)}}$ for $u_i\neq\infty$ is the compactification of the sector $S_i(u)$ in ${\mathbb{TP}_{\min}}^{d-1}$.
Consider a cell $X_G$ in ${{\mathcal T}(V)}$ which contains a ray with support $Z$. Let $M$ be the index set of the columns of $V$ with $v_{ij} = \infty$ for all $i \in Z$ and $j \in M$. Construct the submatrix $Y$ of $V$ indexed by $([d] \setminus Z) \times M$ and the graph $H$ as the restriction of $G$ to the node set $([d] \setminus Z) {\sqcup}M$.
\[lemma:cell\_covector\_boundary\] The cell decomposition of ${\mathbb{TP}_{\min}}(Z)^{d-1}$ induced by $Y$ contains the cell $X_G(V) + b^{(Z)}$ which is given by the covector graph $H$ in the decomposition of ${{\mathbb R}}^{([d] \setminus Z)}$ by $Y$. Furthermore, we obtain the alternative description $$b^{(Z)} + X_G(V) \ = \ b^{(Z)} + \bigcap_{(i,j)\in G} {\overline{S_{i}(v^{(j)})}} \enspace .$$
The second claim is merely a reformulation with the definition of ${\overline{S_{i}(u)}}$.
The first claim follows if we show that $$\label{eq:proj_boundary}
\pi_Z(X_G(V)) \ = \ X_{H}(Y)$$ where $\pi_Z$ is the projection onto the coordinates in $[d] \setminus Z$. Since any ray is generated by the minimal generators of the pointed part of the recession cone and the generators of the lineality space, at first we assume that $Z$ is the support of a minimal generator of the pointed part of the recession cone. Setting $D'' = Z$ in Corollary \[coro:inf\_env\] yields that every shortest path is already defined on $([d]\setminus Z) {\sqcup}M$. Furthermore, the support of a generator of the lineality space is given by a weak component by Lemma \[lemma:recession\_cone\] what implies the same statement about the shortest paths for those generators.
Summarizing, equation follows with Lemma \[lemma:projection\_wdp\].
\[thm:covector\_decomposition\] The union of the covector decompositions induced by the matrices $V(Z)$ where $Z$ ranges over all proper subsets of $[d]$ yields a piecewise linear decomposition of ${\mathbb{TP}_{\min}}^{d-1}$.
If the graph ${{\rm B}}(V)$ is weakly connected, then by Lemma \[lemma:recession\_cone\] the intersection poset generated by the sets ${\{{{\overline{S_{i}(u)}}}\,|\,{u_i\neq\infty}\}}$ contains a $0$-dimensional cell, whence that piecewise linear decomposition of ${\mathbb{TP}_{\min}}^{d-1}$ is a cell complex.
By definition as the common refinement of polyhedral complexes the covector decomposition of ${{\mathbb R}}^d/{{\mathbb R}}{\mathbf 1}$ induced by $V$ is a polyhedral complex. The bounded cells are polytopes and therefore homeomorphic to closed balls. We need to check that the topology works out right for those cells which are unbounded in ${{\mathbb R}}^d/{{\mathbb R}}{\mathbf 1}$. This is gotten from an induction on $d$ as follows. In the base case $d=1$ there is nothing to show since the tropical projective torus ${{\mathbb R}}^1/{{\mathbb R}}{\mathbf 1}$ is a single point. For $d\geq 2$, by induction, we may assume that the covector decomposition induced on the closure $${\left\{\left.(p_1,p_2,\dots,p_d)\in{\mathbb{TP}_{\min}}^{d-1}\vphantom{p_i=\infty \text{ if } i\in Z}\ \right|\ p_i=\infty \text{ if } i\in Z\vphantom{(p_1,p_2,\dots,p_d)\in{\mathbb{TP}_{\min}}^{d-1}}\right\}} \enspace,$$ of ${\mathbb{TP}_{\min}}^{d-1}(Z)$ yields a cell decomposition if $Z$ is not empty. Now consider $Z = \emptyset$ and let $X_G(V)$ be an unbounded cell with covector $G=(G_1,G_2,\dots,G_d)$. By Lemma \[lemma:cell\_covector\_boundary\], the closure of $X_G(V)$ in ${\mathbb{TP}_{\min}}^{d-1}$ is the union of $X_G(V)$ with all the cells $X_H(V(Z'))$ where $Z'$ ranges over the supports of the rays contained in $X_G(V)$. Here $H$ is the covector which $G$ induces on ${\mathbb{TP}_{\min}}^{d-1}(Z')$ by omitting those $G_i$ with $i \in Z'$; this union is homeomorphic with a ball. The same argument also shows that intersections of cells are unions of cells.
By construction one can apply Lemma \[lemma:sectors\_tropical\_hull\] also to the cells in the boundary of the tropical projective space to check for containment in $\operatorname{tcone}(V)$. Consider $z \in {{\mathbb T}}^d$ and let $\operatorname{supp}(z) = {\{{i \in [d]}\,|\,{z_i \neq \infty}\}}$ be its support.
\[coro:sectors\_projective\_hull\] The point $z$ is contained in $\operatorname{tcone}(V)$ if and only if for every $i \in \operatorname{supp}(z)$ there is an index $s \in [n]$ with $z \in {\overline{S_{i}(v^{(s)})}}$ and $\operatorname{supp}(v^{(s)})\subseteq\operatorname{supp}(z)$. A point $z \in {{\mathbb T}}^d$ is contained in $\operatorname{tcone}(V)$ if and only if for every $i \in [d]$ there is an index $s \in [n]$ with $z \in {\overline{S_{i}(v^{(s)})}}$.
Let $$V' \ = \ \begin{pmatrix} 0 & 0 & 0 & 0 & \infty & \infty \\ 1 & 1 & \infty & \infty & 0 & \infty \\ 0 & 2 &
\infty & \infty & \infty & 0 \end{pmatrix} \enspace ,$$ where $d=3$ and $n=6$. The third and fourth columns of $V'$ are the same. Notice that the first three columns correspond to the matrix $V$ from Example \[example:envelope\_with\_infty\]. With $Z=\{1\}$ we obtain the matrix $$\BAtablenotesfalse
\begin{blockarray}{*3{>{\scriptstyle}{c}}}
\begin{block}{\Left{$V'(Z)=$}{(}cc)>{\scriptstyle}c}
0 & \infty & 2 \\
\infty & 0 & 3 \\
\end{block}
5 & 6 &
\end{blockarray} \enspace ,$$ where we keep the original row and column labels. The one-dimensional tropical projective torus $B={\mathbb{TP}_{\min}}^2(\{1\})$ is trivially subdivided; its covector reads $(\bullet,5,6)$. The union of the $1$-dimensional ball $B$ and the unbounded cell in ${{\mathbb R}}^3/{{\mathbb R}}{\mathbf 1}$ with covector $(1234,-,-)$ yields the $2$-dimensional cell with covector $(1234,5,6)$ in the covector decomposition of ${\mathbb{TP}_{\min}}^2$ induced by $V'$; see Figure \[fig:signed\_cells\] and compare with Figure \[figure:tropical\_types\].
Notice that, while the tropical projective torus works for min and max alike, the definition of the tropical projective space does depend on the choice of the tropical addition.
Arrangements of tropical halfspaces
-----------------------------------
So far we associated with a matrix $V\in{{\mathbb T}}_{\min}^{d\times n}$ the covector decompositions of ${{\mathbb R}}^d$ and ${\mathbb{TP}_{\min}}^{d-1}$, respectively, and Theorem \[thm:envelope\] describes the min-tropical cone $\operatorname{tcone}(V)$ as a union of their cells. Choose a subgraph $\Psi$ of the complete bipartite graph $[d]\times[n]$ (with arcs directed from $[d]$ to $[n]$) as in . This gives rise to the max-tropical cone ${\operatorname{thalf}}(V,\Psi) = \bigcap_{j \in
[n]} \bigcup_{(i,j) \in \Psi} S_{i}(v^{(j)})$, which again is a union of cells from the same covector decomposition. Here we want to describe yet another cell decomposition of ${{\mathbb R}}^d$ (or ${\mathbb{TP}_{\min}}^{d-1}$), which was introduced in [@ABGJ-Simplex:A §3.2].
For this, we introduce the max-tropical cone with boundary $${\overline{\operatorname{thalf}}}(V,\Psi) \ = \ \bigcap_{j \in [n]} \bigcup_{(i,j) \in \Psi} {\overline{S_{i}(v^{(j)})}} \enspace .$$
For a vector $\epsilon=\{\pm\}^n$ of $n$ signs we consider the directed bipartite graph $${\Psi_{\epsilon}} \ = \ {\left\{\left.(i,j)\in[d]\times[n]\vphantom{\bigl((i,j)\in\Psi \text{ and } \epsilon_j=+\bigr) \text{ or }
\bigl((i,j)\not\in\Psi \text{ and } \epsilon_j=-\bigr)}\ \right|\ \bigl((i,j)\in\Psi \text{ and } \epsilon_j=+\bigr) \text{ or }
\bigl((i,j)\not\in\Psi \text{ and } \epsilon_j=-\bigr)\vphantom{(i,j)\in[d]\times[n]}\right\}} \enspace .$$
The construction of ${\Psi_{\epsilon}}$ from $\Psi$ amounts to taking the complementary arcs incident to each node $j \in [n]$ with $\epsilon_j = -$. We call the max-tropical cone ${\overline{\operatorname{thalf}}}(V,{\Psi_{\epsilon}})$ the *inversion* of ${\overline{\operatorname{thalf}}}(V,\Psi)$ with respect to $\epsilon$. As a subset of ${\mathbb{TP}_{\min}}^{d-1}$ the inversion may be empty or not. In the latter case ${\overline{\operatorname{thalf}}}(V,\Psi_\epsilon)$ is the *signed cell* with respect to $V$, $\Psi$ and $\epsilon$. Each generic point, i.e., a point which does not lie on any of the max-tropical hyperplanes whose apices are columns of $V$, is contained in a unique signed cell. The trivial inversion with respect to $\epsilon={+}{+}\cdots{+}$ is the tropical cone ${\overline{\operatorname{thalf}}}(V,\Psi)$ itself. To denote cells in the boundary we use the symbol $\bullet$ at the component corresponding to an apex to mark if the cell is in a common boundary stratum with this apex. Each signed cell is a union of cells of the covector decomposition. So Theorem \[thm:covector\_decomposition\] together with Proposition \[prop:graphs\_types\_cells\] entails the following.
\[cor:signed\_cell\_decomposition\] The signed cells ${\overline{\operatorname{thalf}}}(V,{\Psi_{\epsilon}}) / {{\mathbb R}}{\mathbf 1}$, where $\epsilon$ ranges over all choices of sign vectors, generate a piecewise linear decomposition of ${\mathbb{TP}_{\min}}^{d-1}$.
Furthermore, a cell with graph $G$ in the covector decomposition of ${\mathbb{TP}_{\min}}^{d-1}$ by $V$ is contained in a cell ${\overline{\operatorname{thalf}}}(V,{\Psi_{\epsilon}})$ if and only if $\Psi_{\epsilon} \cap G$ has no isolated node.
The decomposition into signed cells is a tropical analogue of the decomposition into polyhedral cells defined by an ordinary affine hyperplane arrangement. As in Theorem \[thm:covector\_decomposition\] that piecewise linear decomposition is a cell complex, provided that ${{\rm B}}(V)$ is weakly connected.
Figure \[fig:signed\_cells\] shows the signed cell decomposition of ${\mathbb{TP}_{\min}}^2$ induced by the matrix $V$ from Example \[ex:type\_decomposition\] with the extra columns ${{(\infty,0,\infty)}^\top}$ and ${{(\infty,\infty,0)}^\top}$ and the directed bipartite graph $\Psi\subset\{1,2,3\}\times\{1,2,3,4,5\}$ with the six directed edges $(1,1),(2,1),(3,2),(1,3),(2,4),(3,5)$. The six signed cells correspond to the sign vectors ${+}{+}{+}{+}{+}{+}$, ${-}{+}{+}{+}{+}{+}$ and ${+}{-}{+}{+}{+}{+}$. The remaining $29$ inversions are empty. Finally, the three inversions ${-}{+}{\bullet}{\bullet}{\bullet}{+}$, ${+}{-}{\bullet}{\bullet}{+}{\bullet}$ and ${+}{-}{+}{+}{\bullet}{\bullet}$ form a decomposition of the boundary of the tropical projective plane.
Concluding remarks
==================
Tropical point configurations, or rather the dual tropical hyperplane arrangements, were generalized to ‘tropical oriented matroids’ by Ardila and Develin [@ArdilaDevelin:2009]. Horn showed that the latter are equivalent to subdivisions of a product of simplices which are not necessarily regular [@Horn:2012]. The tangent digraph discussed in Remark \[rem:tangent\_digraph\] also makes sense in the tropical oriented matroid setting. That graph is the crucial combinatorial device for the pivoting operation in the tropical simplex algorithm [@ABGJ-Simplex:A].
Give an oriented matroid version of the tropical simplex algorithm.
It is worth noting that the axioms for tropical oriented matroids given in [@ArdilaDevelin:2009] generalize the combinatorics of tropical convexity with finite coordinates only.
Generalize the axioms of tropical oriented matroids to cover point configurations or hyperplane arrangements in the tropical projective space.
In view of Theorem \[thm:envelope\] and the results in [@Horn:2012] this might be related not necessarily regular subdivisions of subpolytopes of products of simplices.
How are the signed cell decompositions related to tropical oriented matroids?
Acknowledgment {#acknowledgment .unnumbered}
==============
We are indebted to Xavier Allamigeon, Federico Ardila, Peter Butkovič, Veit Wiechert and two anonymous referees for several valuable hints.
|
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author:
- '[R. Salvaterra]{}, [M. Della Valle, [S. Campana$^{~1}$]{}, [G. Chincarini$^{~1}$]{}, [S. Covino$^{~1}$]{}, [P. D’Avanzo$^{~5 ~1}$]{}, [A. Fern[á]{}ndez-Soto]{}, [C. Guidorzi]{}, [F. Mannucci]{}, [R. Margutti$^{~5 ~1}$]{}, [C.C. Th[ö]{}ne$^{~1}$]{}, [L.A. Antonelli]{}]{}, [S.D. Barthelmy]{}, [M. De Pasquale ]{}, [V. D’Elia$^{~9}$]{}, [F. Fiore$^{~9}$]{}, [D. Fugazza$^{~1}$]{}, [L.K. Hunt$^{~8}$]{}, [E. Maiorano]{}, [S. Marinoni]{}, [F.E. Marshall$^{~10}$]{}, [E. Molinari$^{~13~1}$]{} [J. Nousek]{}, [E. Pian]{}, [J.L. Racusin$^{~15}$]{}, [L. Stella$^{~9}$]{}, [L. Amati$^{~12}$]{}, [G. Andreuzzi$^{~13}$]{}, [G. Cusumano]{}, [E.E. Fenimore]{}, [P. Ferrero]{}, [P. Giommi]{}, [D. Guetta$^{~9}$]{}, [S.T. Holland$^{~10}$]{}, [K. Hurley]{}, [G.L. Israel$^{~9}$]{}, [J. Mao$^{~1}$]{}, [C.B. Markwardt$^{~10~23}$]{}, [N. Masetti$^{~12}$]{}, [C. Pagani$^{~15}$]{}, [E. Palazzi$^{~12}$]{}, [D.M. Palmer$^{~18}$]{}, [S. Piranomonte$^{~9}$]{}, [G. Tagliaferri$^{~1}$]{}, [V. Testa$^{~9}$]{}'
title: 'GRB 090423 at a redshift of $z\simeq 8.1$'
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[**Gamma-ray bursts (GRBSs) are produced by rare types of massive stellar explosions. Their rapidly fading afterglows are often bright enough at optical wavelengths, that they are detectable up to cosmological distances. Hirtheto, the highest known redshift for a GRB was $z=6.7$ (ref. 1), for GRB 080913, and for a galaxy was $z=6.96$ (ref. 2). Here we report observations of GRB 090423 and the near-infrared spectroscopic measurement of its redshift $z=8.1^{+0.1}_{-0.3}$. This burst happened when the Universe was only $\sim 4\%$ of its current age[@Komatsu09]. Its properties are similar to those of GRBs observed at low/intermediate redshifts, suggesting that the mechanisms and progenitors that gave rise to this burst about 600 million years after the Big Bang are not markedly different from those producing GRBs $\sim 10$ billion years later.**]{}
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GRB 090423 was detected by NASA’s Swift satellite on 23 April 2009 at 07:55:19 UT as a double-peaked burst of duration $T_{90}=10.3\pm 1.1$ s. As observed by Swift’s Burst Alert Telescope (BAT)[@Palmer09], it had a 15–150 keV fluence $F=(5.9\pm0.4)\times 10^{-7}$ erg cm$^{-2}$ and a peak energy $E_p=48_{-5}^{+6}$ keV (errors at $90\%$ confidence level). Its X-ray afterglow was identified by Swift’s X-ray Telescope (XRT), which began observations 73 s after the BAT trigger[@Stratta09]. A prominent flare was detected at $t\sim 170$ s in the X-ray light curve, which shows that a typical ’steep decay/plateau/normal decay’ behaviour (Fig. 1). Swift’s UltraViolet Optical Telescope (UVOT) did not detect a counterpart even though it started settled explosures only 77 s after the trigger[@Pasquale09]. A 2$\mu$m counterpart was detected with the United Kingdom Infra-Red Telescope (UKIRT, Hawaii) 20 min after the trigger[@Tanvir09]. Evidence that this burst occurred at high redshift, was given by the multi-band imager Gamma-Ray Burst Optical/Near-Infrared Detector (GROND, Chile) multiband imager (from $g$’ band to $K$ band), which indicated a photometric redshift of $z = 8.0^{+0.4}_{-0.8}$ (ref. 7).
We used the 3.6m Telescopio Nazionale Galileo (TNG, La Palma) with the Near Infrared Camera Spectrometer (NICS) and the Amici prism to obtain a low-resolution ($R\approx50$) spectrum of GRB 090423 $\sim 14$ hrs after the trigger. NICS/Amici is an ideal instrument to detect spectral breaks in the continuum of faint objects because of its high efficiency and wide simultaneous spectral coverage (0.8-2.4 $\mu$m). The spectrum (Fig. 2) reveals a clear break at a wavelength of 1.1 $\mu$m (ref. 8). We derive a spectroscopic redshift for the GRB of $z = 8.1^{+0.1}_{-0.3}$ (ref. 9; see Supplementary Information, section 3), interpreting the break as Lyman-$\alpha$ absorption in the intergalactic medium. No other significant absorption features were detected. This result is consistent, within the errors, with the measurement reported in ref. 7.
At $z\sim 8.1$, GRB 090423 has a prompt-emission rest-frame duration of only $T_{90,rf}=1.13\pm0.12$ s in the redshifted 15-150 keV energy band, an isotropic equivalent energy $E_{iso}=1.0\pm0.3\times 10^{53}$ erg in the redshifted 8-1000 keV energy band[@Kienlin09] and a peak energy $E_{p,rf}=437\pm55$ keV. The short duration and the high peak energy are consistent both with the distribution of long bursts, linked to massive stellar collapse, and with the population of short bursts, thought to arise from the merger of binary compact stars[@Meszaros; @Zhang]. Although the analysis of the spectral lag between the high- and low-energy channels in the BAT band is inconclusive about the classification of GRB 090423, the high $E_{iso}$ argues in favor of a long GRB. The fact that GRB 090423 matches the $E_{iso}-E_{p,rf}$ correlation of long GRBs within 0.5$\sigma$ further supports this classification[@Amati08] (Supplementary Fig. 2).
The rest-frame $\gamma-$ray and X-ray light curve of GRB 090423 is remarkably akin to those of long GRBs at low, intermediate and high redshifts (Fig. 1), suggesting similar physics and interaction with the circumburst medium. The near-infrared light curve of GRB 090423 $\sim 15$h after the trigger shows a temporal decay with a power-law index of $\alpha_0\sim 0.5$, which is markedly different from the decay observed at X-ray energies during the same time interval, which has a power-law index of $\alpha_{X,2}\sim
1.3$ (Supplementary Fig. 3 and Supplementary Information, section 2). As for other lower-redshift GRBs, this behaviour is difficult to reconcile with standard afterglow models, although the sampling of the near-infrared light curve is too sparse for any firm conclusion to be drawn.
The spectral energy distribution of near-infrared afterglow is well fitted by a power-law with an index of $\beta = 0.4^{+0.2}_{-1.4}$ and an equivalent interstella extinction of E($B-V$)$<0.15$, assuming dust reddening consistent with the Small Magellanic Cloud[@Soto09]. On the other hand, the analysis of the XRT data in the time interval 3900s–21568s suggests the presence of intrinsic absorption (in excess of the Galactic value) with an equivalent hydrogen column density of $N_H(z)=6.8^{+5.6}_{-5.3}\times 10^{22}$ cm$^{-2}$ (90% confidence level; Supplementary Information, section 1). The low value of the dust extinction coupled with a relatively high value of $N_H$ suggests that GRB 090423 originates from a region with low dust content relative to those of low-$z$ GRBs[@Schady07], but one similar to that of the high-$z$ GRB 050904, for which $z=6.3$ (ref. 15). Because the absorbing medium must be thin from the point of view of “Thomson” scattering, the metallicity of the circumburst medium can be constrained to be $>4$% of the solar value, $\Zsun$. The implication is that previous supernova explosions have already enriched the host galaxy of GRB 090423 to more than the critical metallicity, $Z\sim
10^{-4}\;\Zsun$ (ref. 16) that prevents the formation of very massive stars (population III stars). Therefore the progenitor of GRB 090423 should belong to a second stellar generation. Its explosion injected fresh metals into the interstellar medium, further contributing to the enrichment of its host galaxy. Its existence empirically supports the cosmological models[@Springel05; @Nagamine06] in which stars and galaxies, already enriched by metals, are in place only $\sim 600$ million years after the Big Bang. Long GRBs are mostly associated with star forming dwarf galaxies, which are thought to be the dominant population of galaxies in the early Universe[@Choudhury08]. The fact that GRB 090423 appears to have exploded in an environment similar to that of low-$z$ GRB hosts[@Fruchter] is in agreement with this.
The occurrence of a GRB at $z\sim 8$ has important implications for the cosmic history of these objects[@Lamb; @Guetta05; @Bromm06; @SC07]. In a first, simple approach, we can assume that GRBs trace the cosmic star formation history, given the well-known link of the long GRBs and the deaths of massive stars[@WB], and that GRBs are well described by a universal luminosity function. However, under these assumptions the expected number of bursts at $z\ge 8$ with an observed photon peak flux larger than or equal to that of GRB 090423 is extremely low: $\sim 4\times 10^{-4}$ in $\sim 4$ yrs of Swift operation (Supplementary Fig. 6 and Supplementary Information, section 4). Hence, one or both of the above assumptions may be oversimplification[@SC07; @RS08]. The detection of a very high-$z$ burst such as GRB 090423 could be accomodated if the GRB luminosity function were shifted towards higher luminosity according to $(1+z)^\delta$ with $\delta\simgt 1.5$ or if the GRB formation rate were strongly enhanced in galaxies with $Z\simlt
0.2\Zsun$. The requirement for evolution may be mitigate if we assume a very high star formation rate at $z>8$. However, we note that the need for evolution is strongly supported by both the large number of Swift detections at $z>2.5$ (ref. 24) and the number of bursts with peak luminosities in excess of $10^{53}$ erg s$^{-1}$ (ref. 26). A possible explanation is that high-redshift galaxies are characterized by a top-heavy (bottom-light) stellar initial mass function with a higher incidence of massive stars than in the local Universe[@chary], providing an enhanced number of GRB progenitors. Such objects could be the main agents responsible for completing the reionization of the Universe[@Bolton07; @Choudhury08; @Furlanetto09; @Stiavelli].
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Stiavelli, M. [*From First Light to Reionization: The End of the Dark Ages.*]{} Vch Verlagsgesellschaft Mbh. (2009).
We acknowledge the TNG staff for useful support during ToO observations, in particular A. Fiorenzano, N. Sacchi, A.G. de Gurtubai Escudero. We thank A. Ferrara for useful discussions. This research was supported by the Agenzia Spaziale Italiana, the Ministero dell’Universit[à]{} e della Ricerca (MUR), the Ministero degli Affari Esteri, NASA, and the National Science Foundation (NSF).
All authors made contributions to this paper. This took the form of direct analysis of the Swift data (SC, GC, CG, RM, SDB, MDP, FEM, JN, JLR, GC, EEF, PG, STH, JM, CBM, CP, DMP), analysis of the TNG and photometric data (MdV, SC, PDA, AFS, CCT, LAA, FM, VE, FF, DF, LKH, EM, EM, SM), management of optical follow-up (PDA, LAA, VDE, EM, SM, GA, PF, GLI, NM, EP, SP, GT, VT), interpretation of the GRB properties (RS, MdV, SC, GC, SC, PDA, AFS, CG, RM, CCT, LA, EP, LS, KH), and modeling of the GRB luminosity function (RS, MdV, SC, GC, CG, DG GT). Additionally, all authors have made contributions through their major involvement in the programmes from which the data derives, and in contributions to the interpretation, content and discussion presented here.
The authors declare that they have no competing financial interests.
and requests for materials should be addressed to R.S. (e-mail: salvaterra@mib.infn.it).
[**Supplementary Information**]{}
[**This material presents technical details to support the discussion in the main paper. We discuss here the Swift data analysis, the analysis of the photometric data, the details of the analysis of the TNG spectroscopic data and the modelling of the GRB redshift distribution.**]{}
[**1. Swift data analysis**]{}
Swift-BAT triggered on GRB 090423 at 07:55:19.35 UT on 23 April 2009. BAT data were analysed using the HEASOFT software package (version 6.6.2) with the Swift Calibration Database (CALDB) version BAT(20090130). Background-subtracted light curves in different energy channels, energy spectra and corresponding response functions were derived from the BAT event file as processed with the BAT software tool [batgrbproduct]{}, by using the mask-weighting technique for the BAT refined position$^4$, and by using standard and BAT-dedicated software tools. The mask-weighted light curve (Supplementary Figure 1) showed a couple of overlapping peaks starting at $T_0-2$ s, peaking at $T_0+4$ s, and ending at $T_0+15$ s. The estimated duration, $T_{90}$, was $10.3\pm1.1$ s for the mask-weighted light curve in the 15-150 keV band. $T_{90}$ is defined as the duration 90% of the total prompt $\gamma$-ray fluence in the observer frame (i.e. the interval from 5% to 95% of the total fluence) and is estimated using the [battblocks]{} software tool. Noteworthy is the light curve of the hardest channel, from 100 to 150 keV, showing a very weak signal as compared with those of the other energy channels. This reflects the spectral softness of this GRB, as shown also by the total energy spectrum. The latter was accumulated from -0.7 to 11.7 s and is fitted with a cut-off power law, $N(E)\sim E^{-\Gamma}
\exp[(2-\Gamma)E/E_p]$, with the best-fit value for the peak energy, $E_p=48_{-5}^{+6}$ keV, and a photon index $\Gamma=0.6_{-0.6}^{+0.5}$. This value of $E_p$ is fully consistent with that determined by fitting the Fermi/Gamma-ray Burst Monitor spectrum$^{10}$ with the canonical “Band” function[@Band93]. The corresponding total fluence in the 15-150 keV energy band is $(5.9\pm
0.4)\times 10^{-7}$ erg cm$^{-2}$. The 1-s peak photon flux measured from 3.5 s in the 15-150 keV band is $1.7\pm0.2$ ph s$^{-1}$ cm$^{-2}$. Uncertainties are given at 90% confidence. At $z=8.1$, GRB 090423 is found to be consistent with the $E_{\rm p,rf}$ – $E_{iso}$ correlation$^{13}$ within 0.5$\sigma$ (Supplementary Figure 2). We note that, even considering the measured peak energy as obtained by a fit of the Fermi/GBM data with a cut-off power-law spectrum, i.e. $E_p = 82\pm 15$ keV and thus $E_{p,rf} = 746\pm 137$ keV (ref. $^{10}$), GRB 090423 would still be consistent within 2$\sigma$ with the $E_{p,rf}-E_{iso}$ correlation.
The BAT light curve shown in Supplementary Figure 3 is the mask-weighted curve extracted between 15 and 150 keV, binned so as to ensure S/N$>2$ with a minimum binning time of 0.512 s. Extrapolation of the BAT flux down to the 0.3-10 keV band was performed by assuming the above spectral model. We note that the remarkable X-ray flare detected by XRT is seen in the BAT data as well. We note that assuming that the flare is still part of the prompt emission[@Zhang09], the total duration of the prompt phase in the source rest frame might be $\sim 20$s, similar to other long GRBs.
The XRT observations began 73 s after the trigger: up to $\sim 300$ s the signal was dominated by a flare. As in many other GRBs, the light-curve then flattened to a shallow decay phase which could be well modelled by a power-law with index $\alpha_{X,1}=0.13\pm0.11$. At $t\sim 4500$ s the X-ray afterglow steepened to $\alpha_{X,2}=1.3\pm0.1$ (errors at 68% confidence level). The flare was modelled by a standard profile[@Norris05]: this is characterised by a $1/e$ rise-time $t_{rise}=29.1\pm3.6$ s; $1/e$ decay-time $t_{decay}= 65.5\pm3.6$ s; $1/e$ width of $\Delta t=94.6\pm 7.3$ s, while the asymmetry parameter is $k=0.38\pm0.03$. This implies a variability measure $\Delta t/t_{peak}=0.66$ and a brightness contrast $\Delta$Flux/Flux around 25. While the flare parameters are defined following ref. [@Norris05], the reported uncertainties are worked out by using the entire covariance matrix. At the redshift of the burst, the flare has an energy $E_{iso}=3.6\times 10^{51}$ erg in the redshifted 0.3-10 keV band of XRT, comparable to the energy released during the prompt emission of other GRBs.
To evaluate the intrinsic column density absorbing the GRB 090423 spectrum, we extracted data in the 3900–21568 s time interval (observer frame). This interval was selected in order to avoid the bright X-ray flare whose variable spectrum might alter the fit and in order to have sufficient signal in the extraction region which we define as a count rate of more than 0.01 counts s$^{-1 }$. The resulting 7984 s exposure contains 680 counts in the range between 0.3-10 keV. The ancillary response file (arf) was created with the task [xrtmkarf]{} (within [heasoft]{} v.6.2.2) using the relevant exposure file and the latest v.11 reponse matrix function (rmf). The spectrum was binned to 20 counts per bin in order to assure a reasonable $\chi^2$ statistic.
We fit the X-ray spectrum with a composite absorption model consisting of a Galactic contribution and an intrinsic absorption fixed to z=8.1 using the [tbabs]{} model within the XSPEC (v12.5.0aa) package. We left the Galactic value free to vary in the $2.9-3.2\times 10^{20}$ cm$^{-2}$ range (based on the absorption maps by [@Dickey90] and [@Kalberla04]). The X-ray continuum was modeled with a power law, as is customary for the afterglow spectra of GRBs. The overall fit is good with a reduced $\chi^2_{\rm red}=1.12$ (28 degrees of freedom, corresponding to a null hypothesis probability of $30\%$). The resulting power law photon index is $\Gamma_X=1.97^{+0.15}_{-0.16}$. For the intrinsic column density, we get a value of $N_H(z)=6.8^{+5.6}_{-5.3}\times10^{22}$ cm$^{-2}$ (90% confidence level), among the highest of all Swift GRBs[@Evans09]. The results refer to a solar composition and metallicity. Assuming that the medium is not Thomson thick, a lower limit of the metallicity can be obtained by $N_H(z)\simlt (1/\sigma_T) (Z/Z_\odot)^{-1}$, where $\sigma_T$ is the Thomson cross-section (e.g. ref. [@Campana07]). We find $Z>0.043\;Z_\odot$. A lower limit on the value of $N_H(z)>6\times 10^{21}$ cm$^{-2}$ is found at 95% confidence level corresponding to a lower limit on the metallicity of $Z>0.004\;Z_\odot$.
[**2. Analysis of the photometric data**]{}
We analyzed all the available photometric data$^{7}$ by using the [/it zphrem]{} code\[Soto03\], in order to determine the photometric redshift and spectral properties of the afterglow. Our code fits a model of functional form $f_\nu \propto
\nu^{-\beta} t^{-\alpha_O}$, including dust extinction (by an SMC-type extinction law) as a free parameter.
We find that the data are best fit within the time range $4.2 \times 103 <
t < 6.6 \times 104$ s) with a model characterised by a temporal decay with a power law index $\alpha_O = 0.50 \pm 0.05$ (we quote hereafter 95% confidence intervals). The dust content is constrained to be $E(B-V)<
0.15$, and the photometric redshift is $z_{\rm phot} = 8.3 \pm 0.3$, consistent with the spectroscopic results. We caution that the rest-frame wavelength observed extends only out to 2500Å, and only the three $JHK$ filters do indeed measure any flux redwards of Lyman-$\alpha$. That is the reason why the spectral index is only loosely constrained and its error is asymmetric ($\beta = 0.4^{+0.2}_{-1.4}$), although its relatively blue color still enables us to put stringent limits to the possible dust content in the afterglow environment. Supplementary Figure 4 shows the projections of the $(z, \alpha_O, \beta, E(B-V))$ four-dimensional confidence intervals on the different bidimensional planes.
Extending the analysis to the whole available temporal window ($2.5 \times
10^2 < t < 1.4 \times 10^6$ s) renders impossible to find a good fit with a single temporal power-law, because of the different decay regimes that the afterglow goes through.
[**3. Analysis of the TNG spectroscopic data**]{}
We observed the afterglow of GRB090423 with the near-IR camera and spectrograph NICS[@Baffa01] on the Italian 3.6m Telescopio Nazionale Galileo (TNG) at La Palma. We used the the lowest spectral resolution mode, offered by the Amici prism[@Oliva00]. This prism provides a simultaneous spectral coverage over a wide wavelength range, between 0.8 and 2.4 $\mu$m, and has a high efficiency. It yields a constant spectral resolution $R \approx 50$ over the whole wavelength range. These characteristics make the instrument especially well-suited for studying the spectral distribution of faint objects.
We obtained 128 minutes of on-target spectroscopy. The afterglow was positioned in the slit using as reference a nearby star approximately 30 arcseconds away (at J2000 coordinates 09:55:35.31, +18:09:03.9). We used a dithering mosaic of 8 cycles, each including two coadds of single 120s exposures, repeated 4 times. The mean time of our observations was Apr 23.98, approximately 15.5 hours after the burst detection. The 2-dimensional spectrum is shown in Supplementary Figure 5.
Standard reduction tasks for NIR spectroscopy were performed independently by four different groups in our team, all of them reaching consistent results. Wavelength calibration was obtained by using a standard calibration table provided by the TNG and matching the deep telluric absorption bands. This method allows for wavelength calibrations better than 0.005 $\mu$m at 1.1 $\mu$m, and its contribution ($\Delta z= \pm$0.04) to the final error budget on redshift is negligible.
Relative flux calibration was performed by using the observed spectral shape of the reference star. Its optical (SDSS) and near-IR (2MASS) colors are consistent with those of an M3-III star. The absolute calibration of the spectrum was obtained from the comparison with the simultaneous photometric measurements obtained by GROND (H=19.94 (Vega), ref. $^7$). We estimate the slit losses to be less than 30%.
The observed flux is compatible with zero below a wavelength of 1.1 $\mu$m, while a significant flux ($>99\%$ confidence level) is measured redwards of this limit. Assuming that this is due to hydrogen absorption by a virtually completely thick Lyman-$\alpha$ forest, then the redshift at which the GRB occurred is $z= 8.1^{+0.1}_{-0.3}$. The quoted error includes the uncertainties on the wavelength calibration and on the estimate of the break position. This value makes GRB090423 the most distant object spectroscopically identified to date. By using a standard cosmology with $\Omega_\Lambda = 0.73$, $\Omega_M = 0.27, H_0=71\, {\rm km s}^{-1}
{\rm Mpc}^{-1}$, we find that GRB 090423 was detected at a lookback time of greater than 13 Gyrs.
We tentatively identified two absorption features at 1.3 and 2.2 $\mu$m. These would be consistent with blends of Si IV and Fe II at 1400Å and 2400Å, $z=8.1$ rest-frame, respectively. The detection, however, has a low confidence level due to the low S/N of the spectrum.
[**4. Modelling the GRB redshift distribution**]{}
We compute the probability of detecting of GRB 090423 in three different scenarios for the formation and cosmic evolution of long GRBs: (i) no evolution model, where GRBs follow the cosmic star formation and their luminosity function (LF) is constant in redshift; (ii) luminosity evolution model, where GRBs follow the cosmic star formation but the LF varies with redshift; (iii) density evolution model, where GRBs form preferentially in low–metallicity environments. In the first two cases, the GRB formation rate is simply proportional to the global cosmic star formation rate as computed by [@Hopkins]. For the luminosity evolution model, the typical burst luminosity is assumed to increase with redshift as $(1+z)^\delta$. Finally, for the density evolution case, the GRB formation rate is obtained by convolving the observed SFR with the fraction of galaxies at redshift $z$ with metallicity below $Z_{th}$ using the expression computed by [@Langer06]. In this scenario, the LF is assumed to be constant.
The computation works as follows (see also $^{21,22,23,24,26,43}$). The observed photon flux, $P$, in the energy band $E_{\rm min}<E<E_{\rm max}$, emitted by an isotropically radiating source at redshift $z$ is
$$P=\frac{(1+z)\int^{(1+z)E_{\rm max}}_{(1+z)E_{\rm min}} S(E) dE}{4\pi d_L^2(z)},$$
where $S(E)$ is the differential rest–frame photon luminosity of the source, and $d_L(z)$ is the luminosity distance. To describe the typical burst spectrum we adopt the functional form proposed by [@Band93], i.e. a broken power–law with a low–energy spectral index $\alpha$, a high–energy spectral index $\beta$, and a break energy $E_b=(\alpha-\beta)E_p/(2+\alpha)$, with $\alpha=-1$ and $\beta=-2.25$ (ref. [@Preece00]). In order to broadly estimate the peak energy of the spectrum, $E_p$, for a given isotropic–equivalent peak luminosity, $L=\int^{10000\,\rm{keV}}_{1\,\rm{keV}} E S(E)dE$, we assumed the validity of the correlation between $E_p$ and $L$ (ref. [@Yonetoku04]).
Given a normalized GRB LF, $\phi(L)$, the observed rate of bursts with $P_1<P<P_2$ is
$$\frac{dN}{dt}(P_1<P<P_2)=\int_0^{\infty} dz \frac{dV(z)}{dz}
\frac{\Delta \Omega_s}{4\pi} \frac{\Psi_{\rm GRB}(z)}{1+z} \int^{L(P_2,z)}_{L(P_1,z)} dL^\prime \phi(L^\prime),$$
where $dV(z)/dz$ is the comoving volume element, $\Delta \Omega_s$ is the solid angle covered on the sky by the survey, and the factor $(1+z)^{-1}$ accounts for cosmological time dilation. $\Psi_{\rm GRB}(z)$ is the comoving burst formation rate and the GRB LF is described by a power law with an exponential cut–off at low luminosities[@PM01], i.e. $\phi(L) \propto
(L/L_{\rm cut})^{-\xi} \exp (-L_{\rm cut}/L)$.
For the three scenarios, we optimize the model free parameters (GRB formation efficiency, burst typical luminosity at $z=0$ and the power index $\xi$ of the LF) by fitting the differential number counts observed by BATSE (see ref. $^{24,26}$ for a detailed description of the models and of the analysis). We find that it is always possible to find a good agreement between models and data. Moreover, we can reproduce also the differential peak flux count distribution observed by Swift in the 15-150 keV band without changing the best fit parameters. On the basis of these results, we compute the probability to detect with Swift a GRB at $z\ge 8$ with photon flux $P$. The results are plotted in Supplementary Figure 6 (top panels) together with the cumulative number of GRBs at $z\ge 8$ expected to be detected by Swift in one year of observations (bottom panels). From the plot it is clear that the no evolution model fails to account for the observation of GRB 090423, since only $\sim 4\times 10^{-4}$ GRBs are expected to be detected at $z\ge 8$ in $\sim 4$ years of Swift observations. Evolutionary models (both in luminosity or in density) can easily account for the discovery of GRB 090423. We note that the results confirm the need for cosmic evolution in the GRB luminosity function and/or in the GRB density obtained by recent analysis of the whole Swift GRB dataset. Indeed, both the large number of $z\ge 2.5$ bursts$^{24}$ and the number of bright (i.e. with peak luminosity $L\ge 10^{53}$ erg s$^{-1}$) bursts$^{26}$ strongly require the existence of evolution.
Moreover, we want to stress here that our conclusions are conservative. First of all, many biases can hampered the detection of GRB at very high redshift. Indeed, a few very high-$z$ bursts may be hidden among the large sample of Swift bursts that lack of an optical detection. Thus, the discovery of a single event at $z>8$ in 4.5 yrs of Swift operation can be treated as a lower limit on the real number of high-$z$ detection. Moreover, our choice of the GRB LF is also conservative, since the existence of large population of faint GRBs (i.e. for an LF with a more gentle decline or a rise in the faint end) would lead to a decrease of the expeceted number of GRBs at $z>8$ strenghtening our conclusions.
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[**Supplementary Fig. 1.**]{} [ Four channels and combined 0.512 s mask-weighted light curve. The light curve of the 100-150 keV energy channel shows a weak signal, because of the soft spectrum; the corresponding integration time is 2.048 s. Errors are at 1$\sigma$ level.]{}
[**Supplementary Fig. 2.** ]{} [ Position of GRB 090423 in the $E_{\rm p,rf}$ – $E_{iso}$ plane based on Swift/BAT$^{4}$ and Fermi/GBM$^{10}$ (fitted with the Band function) results. The lines show the best–fit power–law and the $\pm 2\sigma$ region of the correlation as derived by $^{13}$. Also shown are the 70 GRBs included in the sample analyzed in that work (errors on individual bursts are at 1$\sigma$ level). Given that short GRBs do not follow the correlation$^{13}$, this evidence supports the hypothesis that, despite its cosmological rest–frame duration of $\sim 1.3$ s, GRB090423 belongs to the long GRB class.]{}
[**Supplementary Fig. 3.**]{} [ Light curve of GRB 090423 as observed by Swift/BAT (red crosses), Swift/XRT (blue plus) and in the NIR (cyan points). Errors on fluxes are at 1$\sigma$ level and horizontal bars refer to the integration time interval. The XRT 0.3–10 keV light-curve, starting at 73 s after the burst, shows a prominent flare at $t \sim 170$ s (also detected by BAT), and a flat phase ($\alpha_{X,1}=0.13 \pm 0.11$) followed by a rather typical decay (starting at $t=4513\pm491$ s) with power-law index $\alpha_{X,2}=1.3\pm0.1$. Available photometric data are plotted in the K band (AB magnitude) by transforming the fluxes, when the observations have been taken in a different filter, using a power law with $\beta = 0.4$, as estimated from the NIR spectral energy distribution. A small displacement in time for contemporary data in different bands is applied in order to increase the visibility. The NIR light curve is consistent with a plateau phase ($t\sim 10^2-10^3$ s) followed by a decay with $\alpha_{O}\sim 0.5$ ($t\sim 10^3-10^5$ s). This decay phase is shallower than the X-ray decay in the same time interval. Triangles at $t\sim 10^5$ s report NIR upper limits as obtained by our second epoch TNG observation with the NICS camera in the Y and J band and by GROND in the JHK band. These limits are consistent with the temporal decay observed by XRT.]{}
[**Supplementary Fig. 4.** ]{} [ Analysis of available photometric data for GRB090423 in the interval 70 min $< t <$ 1100 min. The code fits a model function with temporal index $\alpha_O$ and spectral index $\beta$, dust extinction $E(B-V)$, and redshift $z$. The different panels show the projection of the four-dimensional confidence intervals on the different two-dimensional planes of interest. The best-fit is marked by the black dot, with the red, cyan, and green contours defining respectively the 68%, 95%, and 99.5% confidence areas. The apparently bimodal distribution in the $\beta$ direction is an artifact of the parameter space discretization.]{}
[**Supplementary Fig. 5.** ]{} [ The spectrum has been taken by $\sim 14$ hrs from the trigger. The spectrum of the nearby reference star is also shown.]{}
[**Supplementary Fig. 6.**]{} [ Top panels: probability for a GRB with peak photon flux $P$ to be detected by Swift at $z\ge 8$. Luminosity evolution models are shown in the left panel, where shaded area refers to a typical burst luminosity increasing as $L_{\rm cut}\propto (1+z)^\delta$ with $\delta=1.5-3$. Density evolution models are shown in the right panel, where shaded area refers to a metallicity threshold for GRB formation $Z_{th}=0.02-0.2\;Z_\odot$ (the lower bound refers to the higher $Z_{th}$). In both panels, the dashed line shows the no evolution case. The red point marks the position of GRB 090423. We note that the point represents a lower limit on the number of detection at $z>8$ since a few very high-$z$ bursts may be hidden among those bursts that lack of an optical detection. Bottom panels: cumulative number of GRBs at $z>8$ to be detected by Swift with photon flux larger than $P$ in one year of Swift observations.]{}
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abstract: 'We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length $\leq k$ are all zero into a link with vanishing Milnor invariants of length $\leq 2k+1$, and we provide formulae for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self $\Delta$-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link $L$ is link-homotopic to the unlink if and only if a link $L$ with a single component Whitehead doubled is self $\Delta$-equivalent to the unlink.'
address:
- |
Institut Fourier\
Université Grenoble 1\
100 rue des Maths - BP 74\
38402 St Martin d’Hères , France
- |
Tokyo Gakugei University\
Department of Mathematics\
Koganeishi\
Tokyo 184-8501, Japan
author:
- 'Jean-Baptiste Meilhan'
- Akira Yasuhara
title: Whitehead double and Milnor invariants
---
[^1]
Introduction
============
In this paper, we consider the operation of Whitehead double, more generally of Whitehead $n$-double, on a component of a link, and we study the behavior of Milnor invariants under this operation. Milnor invariants $\overline{\mu}_L(I)$ of an $m$-component link $L$, where $I=i_1i_2...i_k$ with $1\le i_j\le m$, can be thought of as some sort of higher order linking number” of the link. See Section \[milnor\] for a definition.
A typical example is the Whitehead link, which is a Whitehead double of the Hopf link. The linking number of the Hopf link (which coincides with Milnor invariant $\overline{\mu}(12)$) is $\pm 1$, whereas the Whitehead link has linking number $0$. On the other hand, the Whitehead link has some nontrivial higher order Milnor invariants: its Sato-Levine invariant for instance, which is equal to $-\overline{\mu}(1122)$, is $\pm 1$. Our main result, stated below, generalizes this observation.
Let $K$ be a component of a link $L$ in $S^3$, regarded as $h(\{{\bf0}\}\times S^1)$ for some embedding $h:D^2\times S^1\rightarrow S^3\setminus(L\setminus K)$, such that $K$ and $h((0,1)\times S^1)$ have linking number zero. Let $n$ be a (nonzero) integer. Consider in the solid torus $T=D^2\times S^1$ the knot $\mathcal{W}_n$ depicted in Figure \[wh\]. The knot $h(\mathcal{W}_n)$ is called the *Whitehead $n$-double of $K$*, and it is denoted by $W_n(K)$.
![The knot $\mathcal{W}_n$ for $n<0$ and $n>0$ respectively.[]{data-label="wh"}](wh.eps)
Given an $m$-component link $L=K_1\cup...\cup K_m$ in $S^3$, we denote by $W^i_n(L)$ the link $\left(L\setminus K_{i}\right)\cup W_n(K_{i})$ obtained by Whitehead $n$-double on the $i^{th}$ component of $L$. Note that the case $n=\pm1$ coincides with the usual notion of (positive or negative) Whitehead double.
\[double\] Let $L$ be an $m$-component link in $S^3$, and let $n(\neq 0)$ be an integer. If all Milnor invariants $\overline{\mu}_L(Ji)$ of $L$ of length $|Ji|\leq k$ are zero $(k\ge 1)$, then all Milnor invariants $\overline{\mu}_{W^i_{n}(L)}(Ii)$ of $W^i_{n}(L)$ of length $|Ii|\leq2k+1$ are zero. Moreover, if $\overline{\mu}_L(Pi)\ne 0$, $\overline{\mu}_L(Qi)\ne 0$ with $P=p_1 p_2 ... p_k,~Q=q_1 q_2 ... q_k$ $($possibly $P=Q)$ such that $p_j\ne i,~q_j\ne i$ for all $1\le j\le k$, then we have the following formulae for the first non-vanishing Milnor invariants of $W^i_{n}(L)$ $$\left\{ \begin{array}{l}
\overline{\mu}_{W^i_{n}(L)}(PiQi)=2n\overline{\mu}_L(Pi)\overline{\mu}_L(Qi), \\
\overline{\mu}_{W^i_{n}(L)}(PQii)=-n\overline{\mu}_L(Pi)\overline{\mu}_L(Qi).
\end{array} \right.$$
\[satolevine\] In the case of a $2$-component link, the formulae given in Theorem \[double\] for the first nonvanishing Milnor invariants of $W^i_n(L)$ provide, as an immediate corollary, a generalization of a result of Shibuya and the second author [@SY] as follows:\
Let $L=K_1\cup K_2$ in $S^3$. Let $n\ne 0$ be an integer, and let $W_n(L)$ be obtained by Whitehead $n$-double on a component of $L$. Then the Sato-Levine invariant $\beta_2$ of $W_n(L)$ satisfies $$\beta_2(W_n(L)) = n\left(lk(K_1,K_2)\right)^2.$$ (Note that the Sato-Levine invariant of $W_n(L)$ is well-defined, as Theorem \[double\] ensures that the link has zero linking number.)
Recall that two links are *link-homotopic* if they are related by a sequence of ambient isotopies and *self crossing changes*, which are crossing changes involving two strands of the same component, see the left-hand side of Figure \[delta\]. In particular, a link is called *link-homotopically trivial* if it is link-homotopic to the unlink. It has long been known that Milnor invariants with no repeating indices are invariants of link-homotopy [@Milnor2]. Like crossing change, the $\Delta$-move is an unknotting operation [@MN]. Here we consider the notion of *self $\Delta$-move* for links, which is a local move as illustrated in the right-hand side of Figure \[delta\] involving three strands of the same component. Two links are [*self $\Delta$-equivalent*]{} if they are related by a finite sequence of ambient isotopies and self $\Delta$-moves. Self $\Delta$-equivalence is a generalized link-homotopy, i.e., self $\Delta$-equivalence implies link-homotopy. The self $\Delta$-equivalence was introduced by Shibuya [@Shi; @Shi1], and was subsequently studied by various authors [@FY2; @NO; @NS; @NSY; @SY2; @SY; @yasuhara]. A link is *self $\Delta$-trivial* if it is self $\Delta$-equivalent to the unlink.
![A crossing change and a $\Delta$-move[]{data-label="delta"}](delta.eps)
The following is a consequence of our main result.
\[self\] Let $L$ be an $m$-component link in $S^3$ which is not link-homotopically trivial. Then, for any $n(\neq 0)$ and $i~(1\le i\le m)$, $W^i_n(L)$ is not self $\Delta$-trivial.
Recall now that a link $L$ is Brunnian if all (proper) sublinks of $L$ are trivial. The next result shows that the converse of Corollary \[self\] also holds for Brunnian links.
\[br\] Let $L$ be an $m$-component Brunnian link in $S^3$. Let $n(\neq 0)$ and $i~(1\le i\le m)$ be integers. Then $L$ is link-homotopically trivial if and only if $W^i_n(L)$ is self $\Delta$-trivial.
Observe that an $m$-component Brunnian link always has vanishing Milnor invariants of length $\le m-1$ since these are Milnor invariants of sublinks of a Brunnian link, which are trivial links. So Theorem \[double\] implies that all Milnor invariants of $W^i_{n}(L)$ of length $\le 2m-1$ are zero for any choice of $1\le i\le m$ and $n(\neq 0)$. In other words, for $m$-component Brunnian links, Whitehead doubling kills all Milnor invariants of length $\le 2m-1$. It follows from a more general result (stated and proved in Section \[sat\]) that an additional Whitehead doubling, on either the same or another component of the link, actually kills *all* Milnor invariants, as the resulting link is always a boundary link, see Corollary \[boundary\].
The rest of the paper is organized as follows. In Section \[milnor\] we recall the definition of Milnor invariants and prove Theorem \[double\]. In Section \[sde\] we prove the two statements relating Whitehead doubling and self $\Delta$-equivalence, namely Corollary \[self\] and Theorem \[br\]. In Section \[sat\] we consider more general satellite constructions, involving a knot which is null-homologous in the solid torus. When applied twice to a Brunnian link, such a construction always yields a boundary link.
Milnor invariants {#milnor}
=================
J. Milnor defined in [@Milnor; @Milnor2] a family of invariants of oriented, ordered links in $S^3$, known as Milnor’s $\overline{\mu}$-invariants.
Given an $m$-component link $L$ in $S^3$, denote by $\pi(L)$ the fundamental group of $S^3\setminus L$, and by $\pi_q(L)$ the $q^{th}$ subgroup of the lower central series of $\pi(L)$. We have a presentation of $\pi(L)/ \pi_q(L)$ with $m$ generators, given by a meridian $\alpha_i$ of the $i^{th}$ component of $L$. So for $1\le i\le m$, the longitude $l_i$ of the $i^{th}$ component of $L$ is expressed modulo $\pi_q(L)$ as a word in the $\alpha_i$’s (abusing notations, we still denote this word by $l_i$).
The *Magnus expansion* $E(l_i)$ of $l_i$ is the formal power series in non-commuting variables $X_1,...,X_m$ obtained by substituting $1+X_j$ for $\alpha_j$ and $1-X_j+X_j^2-X_j^3+\cdots$ for $\alpha_j^{-1}$, $1\le j\le m$.
Let $I=i_1 i_2 ...i_{k-1} j$ be a multi-index (i.e., a sequence of possibly repeating indices) among $\{1,...,m\}$. Denote by $\mu_L(I)$ the coefficient of $X_{i_1}\cdots X_{i_{k-1}}$ in the Magnus expansion $E(l_j)$. *Milnor invariant* $\overline{\mu}_L(I)$ is the residue class of $\mu_L(I)$ modulo the greatest common divisor of all $\mu_L(J)$ such that $J$ is obtained from $I$ by removing at least one index, and permutating the remaining indices cyclically. We call $|I|=k$ the *length* of Milnor invariant $\overline{\mu}_L(I)$.
The indeterminacy comes from the choice of the meridians $\alpha_i$ or, equivalently, from the indeterminacy of representing the link as the closure of a string link [@HL].
Without loss of generality, we may suppose that $i=m$. We give the proof of the case $n<0$. The case $n>0$ is strictly similar and we omit it.
We denote by $\alpha_1$,...,$\alpha_{m-1}$, $\alpha_m$ and $a$ meridians of $K_1$,...,$K_{m-1}$, $K_m$ and $W_{n}(K_m)$ respectively, such that $\alpha_1,...,\alpha_{m}$ generate $\pi(L)/\pi_q(L)$ and $\alpha_1,...,\alpha_{m-1},a$ generate $\pi(W^m_{n}(L))/\pi_q(W^m_{n}(L))$.
The Magnus expansion of the longitude $l_m\in \pi(L)/\pi_q(L)$ of $K_m$, written as a word in $\alpha_1,...,\alpha_m$, has the form $$E(l_m)=1+\sum{\mu}_L(i_1...i_r,m)X_{i_1}...X_{i_r} = 1+f(X_1,...,X_m),$$ where $E(\alpha_i)=1+X_i$ for all $1\le i\le m$.
Now consider the Whitehead $n$-double of $K_m$, and consider $2n+1$ elements $a_0$, $a_1$, ... , $a_{2n}$ of $S^3\setminus W^m_{n}(L)$ as represented in Figure \[W1\]. Let $\phi(l_m)=l$, where $\phi: \pi(L)/\pi_q(L) \rightarrow \pi(W^m_{n}(L))/\pi_q(W^m_{n}(L))$ is the natural map that maps $\alpha_i$ to itself ($1\le i\le m-1$) and maps $\alpha_m$ to $a_{2n}^{-1}a$. (Abusing notation, we still denote by $a_i$, $0\le i\le 2n$, the corresponding elements in $\pi(W^m_{n}(L))/\pi_q(W^m_{n}(L))$.)
It follows from repeated uses of Wirtinger relations that $$\left\{ \begin{array}{ll}
a_0=l^{-1}al, & \\
a_{2r}=R^raR^{-r}, & \textrm{for all $r\ge 1$} \\
a_{2r+1}=R^raR^{-(r+1)}, & \textrm{for all $r\ge 0$}
\end{array} \right.$$ where $R=al^{-1}a^{-1}l$. In particular we have that $$\phi(\alpha_m)=a_{2n}^{-1}a= R^n a^{-1}R^{-n}a.$$ Let $E(a)=1+X$ denote the Magnus expansion of $a$. Observe that $$\begin{array}{rrcl}
& E(R)=E(al^{-1}a^{-1}l)&=&(1+X)E(l^{-1})(1-X)E(l)+{\mathcal O}_X(2)\\
& &=&1+X-E(l^{-1})XE(l)+{\mathcal O}_X(2),
\\
\textrm{and } & & & \\
& E(R^{-1})=E(l^{-1}ala^{-1})&=&E(l^{-1})(1+X)E(l)(1-X)+{\mathcal O}_X(2)\\
& &=&1-X+E(l^{-1})XE(l)+{\mathcal O}_X(2),
\end{array}$$ where ${\mathcal O}_X(2)$ denotes terms which contain $X$ at least $2$ times. So we have
$$\begin{aligned}
E(\phi(\alpha_m))&=&(1+X-E(l^{-1})XE(l))^n(1-X)\\
&&\times(1-X+E(l^{-1})XE(l))^n(1+X)+{\mathcal O}_X(2)\\
&=&(1+nX-nE(l^{-1})XE(l))(1-X)\\
&&\times(1-nX+nE(l^{-1})XE(l))(1+X)+{\mathcal O}_X(2),\\
&=&1+{\mathcal O}_X(2).\end{aligned}$$
This implies that $$E(l)=1+f(X_1,...,X_{m-1},{\mathcal O}_X(2))=1+f_1(X_1,...,X_{m-1})+f_2(X_1,...,X_{m-1},X),$$ where $$f_1(X_1,...,X_{m-1})=f(X_1,...,X_{m-1},0)\in {\mathcal O}(k)$$ and $$f_2(X_1,...,X_{m-1},X)=f(X_1,...,X_{m-1},{\mathcal O}_X(2))-f_1(X_1,...,X_{m-1})
\in {\mathcal O}(k+1),$$ and ${\mathcal O}(u)$ denotes terms of degree at least $u$ (the degree of a monomial in the $X_j$ is simply defined by the sum of the powers). Similarly we have $$E(l^{-1})=1+g(X_1,...,X_{m-1},{\mathcal O}_X(2))=1+g_1(X_1,...,X_{m-1})+g_2(X_1,...,X_{m-1},X),$$ where $g_1(X_1,...,X_{m-1})\in {\mathcal O}(k)$ and $g_2(X_1,...,X_{m-1},X)\in {\mathcal O}(k+1)$.
Let $f_1,f_2,~g_1,~g_2$ denote $f_1(X_1,...,X_{m-1})$, $f_2(X_1,...,X_{m-1},X)$, $g_1(X_1,...,X_{m-1})$, $g_2(X_1,...,X_{m-1},X)$ respectively, and set $f=f_1+f_2$ and $g=g_1+g_2$. Set $E(a^{-1})=1-X+X^2-X^3+\cdots=1+Y$. Note that $(1+f)(1+g)=(1+g)(1+f)=1$ and $(1+X)(1+Y)=(1+Y)(1+X)=1$, hence $f+g=-fg=-gf\in {\mathcal O}(2k)$ and $X+Y=-XY=-YX$. One can check, by induction, that $$\left\{ \begin{array}{l}
E(R^n)=1+n(gY-Xf+XgY+gYf)+{\mathcal O}(2k+2), \\
E(R^{-n})=1+n(Xf-gY+XfY+gXf)+{\mathcal O}(2k+2), \\
E((a^{-1}R)^n)=(1+Y)^n+(1+Y)^nf - f(1+Y)^n+n(gYf- fgY)+{\mathcal O}(2k+2).
\end{array} \right.$$
Since the preferred longitude $L_m$ of $W^m_{n}(K_m)$ is presented in $\pi(W^m_{n}(L))/\pi_q(W^m_{n}(L))$ by the word $$L_m = la^{-1}a_2^{-1}...a_{2n-2}^{-1}l^{-1}a_{2n-1}^{-1}a_{2n-3}^{-1}a_{3}^{-1}a_{1}^{-1}a^{2n}
= l(a^{-1}R)^{n}R^{-n}l^{-1}R^{n}a^{n},$$ we have $$\begin{aligned}
E(L_m)&=&(1+f)\big[(1+Y)^n+(1+Y)^nf - f(1+Y)^n+n(gYf- fgY)\big]\\
& &\times\big[1+n(Xf-gY+XfY+gXf) \big](1+g)\\%\big[1+n(gY-Xf+XgY+gYf) \big](1+X)^n\\
& &\times\big[1+n(gY-Xf+XgY+gYf) \big](1+X)^n\\%+{\mathcal O}(2k+2)\\
&=&\big[(1+Y)^n + n(2fXf - f^2X - Xf^2)\big](1+X)^n+{\mathcal O}(2k+2)\\
&=&1+n\left(2fXf - ffX - Xff\right)+{\mathcal O}(2k+2)\end{aligned}$$
Because $f\in {\mathcal O}(k)$, the first non-trivial terms in the Magnus expansion $E(L_m)$ are of degree $2k+1$. It follows that all Milnor invariants $\overline{\mu}_{W^m_{n}(L)}(Im)$ of length $|Im|\leq 2k+1$ of $W^m_{n}(L)$ are zero.
Moreover, we actually have $$E(L_m)=1+n\left(2f_1Xf_1 - f_1f_1X - Xf_1f_1\right)+{\mathcal O}(2k+2).$$ So if $\overline{\mu}_L(Pm)\ne 0$, $\overline{\mu}_L(Qm)\ne 0$ for some multi-indices $P=p_1...p_k$, $Q=q_1...q_k~(P\ne Q)$ with $p_j\ne m$, $q_j\ne m$ for all $1\le j\le k$, then $$f_1=\overline{\mu}_L(Pm)X_{p_1}...X_{p_k}+\overline{\mu}_L(Qm)X_{q_1}...X_{q_k}+{\mathcal O}(k),$$ and it follows from the above formula that $$\begin{aligned}
E(L_m)&=&1+2n\overline{\mu}_L(Pm)\overline{\mu}_L(Pm) X_{p_1}\cdots X_{p_k}XX_{p_1}\cdots X_{p_k}\\
&&+2n\overline{\mu}_L(Pm)\overline{\mu}_L(Qm) X_{p_1}\cdots X_{p_k}XX_{q_1}\cdots X_{q_k} \\
&&+2n\overline{\mu}_L(Qm)\overline{\mu}_L(Pm) X_{q_1}\cdots X_{q_k}XX_{p_1}\cdots X_{p_k}\\
&&+2n\overline{\mu}_L(Qm)\overline{\mu}_L(Qm) X_{q_1}\cdots X_{q_k}XX_{q_1}\cdots X_{q_k}\\
&&-n\overline{\mu}_L(Pm)\overline{\mu}_L(Pm) X_{p_1}\cdots X_{p_k}X_{p_1}\cdots X_{p_k}X \\
&&-n\overline{\mu}_L(Pm)\overline{\mu}_L(Qm) X_{p_1}\cdots X_{p_k}X_{q_1}\cdots X_{q_k}X \\
&&-n\overline{\mu}_L(Qm)\overline{\mu}_L(Pm) X_{q_1}\cdots X_{q_k}X_{p_1}\cdots X_{p_k}X \\
&&-n\overline{\mu}_L(Qm)\overline{\mu}_L(Qm) X_{q_1}\cdots X_{q_k}X_{q_1}\cdots X_{q_k}X \\
&&-n\overline{\mu}_L(Pm)\overline{\mu}_L(Pm) XX_{p_1}\cdots X_{p_k}X_{p_1}\cdots X_{p_k} \\
&&-n\overline{\mu}_L(Pm)\overline{\mu}_L(Qm) XX_{p_1}\cdots X_{p_k}X_{q_1}\cdots X_{q_k} \\
&&-n\overline{\mu}_L(Qm)\overline{\mu}_L(Pm) XX_{q_1}\cdots X_{q_k}X_{p_1}\cdots X_{p_k}\\
&&-n\overline{\mu}_L(Qm)\overline{\mu}_L(Qm) XX_{q_1}\cdots X_{q_k}X_{q_1}\cdots X_{q_k}
+ {\mathcal O}(2k+1)\end{aligned}$$ which implies the desired formulae for the first nonvanishing Milnor invariants of $W^m_{n}(L)$.
One may wonder what happens when we consider, in the definition of a Whitehead $n$-double, an odd number $2p+1$ of half-twists in place of $n$ full twists. For a link $L$, denote by $W^i_{odd}(L)$ any link obtained by such a satellite construction with an odd number of half-twists on the $i^{th}$ component of $L$. Then we can prove the following: If all Milnor invariants of $L$ with length $\le k$ vanish, then for any multi-index $Ii$ with $|Ii|\le k+1$, $\overline{\mu}_{W^i_{odd}(L)}(Ii)=2^{r_i+1}\overline{\mu}_L(Ii)$, where $r_i$ is the number of times that the index $i$ appears in $I$.
On self $\Delta$-equivalence {#sde}
============================
In this section we provide the proofs for Corollary \[self\] and Theorem \[br\].
We need the following additional notation. Given a multi-index $I$, we denote by $r(I)$ the maximum number of times that any index appears in $I$. For example, $r(1123)=2$ and $r(1233212)=3$.
Let $L$ be an $m$-component link which is not link-homotopically trivial. Then by [@Milnor] there exists some multi-index $I=i_1...i_p$ with $r(I)=1$ such that $\overline{\mu}_L(I)\ne 0$ and $\overline{\mu}_L(J)=0$ for all multi-index $J$ with length $|J|<|I|$ and $r(J)=1$.
Let $n(\neq 0)$ and $i~(1\leq i\leq m)$ be integers. If $I$ does not contain $i$, then $\overline{\mu}_{W^i_n(L)}(I)=\overline{\mu}_L(I)\neq 0$. So $W^i_n(L)$ is not link-homotopically trivial. Hence $W^i_n(L)$ is not self $\Delta$-trivial. Suppose that $I$ contains $i$. By cyclic symmetry” ([@Milnor2 Theorem 6]), we may assume that $i_p=i$. By Theorem \[double\], the link $W^i_n(L)$ thus satisfies $\overline{\mu}_{W^i_n(L)}(M)\neq 0$ for some multi-index $M$ with $r(M)\le 2$. Since Milnor invariants with $r\leq 2$ are self $\Delta$-equivalence invariants [@FY], $W^i_n(L)$ is not self $\Delta$-trivial.
\[prbr\] Let $L$ be an $m$-component Brunnian link. Let $n(\neq 0)$ and $i~(1\le i\le m)$ be integers. By Corollary \[self\] we already know that $L$ is link-homotopically trivial if $W^i_n(L)$ is self $\Delta$-trivial. Let us prove that the converse is also true.
The link $L$ being Brunnian, $\overline{\mu}_L(I)=0$ if $I$ does not contain an index in $ \{ 1,...,m \} $. Moreover, if $L$ is link-homotopically trivial, then $\overline{\mu}_L(I)=0$ for any $I$ with $r(I)=1$. In particular $\overline{\mu}_L(I)=0$ for all $|I|\le m$, and by Theorem \[double\] the link $W^i_n(L)$ thus satisfies $\overline{\mu}_{W^i_n(L)}(I)=0$ for all $|I|\le 2m+1$. This implies that $\overline{\mu}_{W^i_n(L)}(I)=0$ for any multi-index $I$ with $r(I)\le 2$. By [@yasuhara Corollary 1.5], we have that $W^i_n(L)$ is self $\Delta$-trivial.
From Brunnian links to boundary links {#sat}
=====================================
Boundary links from satellite construction
------------------------------------------
In this section we consider a more general satellite construction. Let $L=K_1\cup...\cup K_m$ be an $m$-component link in $S^3$, and let $h_i:D^2\times S^1\rightarrow S^3$ be an embedding such that $h_i(\{{\bf 0}\}\times S^1)$ is the $i^{th}$ component $K_i$ of $L$ (as in the introduction, we assume that $K_i$ and $h((0,1)\times S^1)$ have linking number zero). Now, instead of the knot $\mathcal{W}_n$ depicted in Figure \[wh\], consider in the solid torus $T=D^2\times S^1$ a fixed knot $\mathcal{K}$ which is null-homologous in $T$. Denote by $W^i_{\mathcal{K}}(L)$ the link $\left(L\setminus K_{i}\right)\cup h_i(\mathcal{K})$. We have the following result.
\[thsat\] Let $L=K_1\cup...\cup K_m$ be an $m$-component link in $S^3$, and let $\mathcal{K}$, $\mathcal{K}'$ be two null-homologous knots in the solid torus $T$. Then
1. If $L\setminus K_i$ is a boundary link, then $W^i_{\mathcal{K}}(W^i_{\mathcal{K}'}(L))$ is a boundary link.
2. If $L\setminus (K_i\cup K_j)$ is a boundary link and $K_i\cup K_j$ is null-homotopic in $S^3\setminus\left(L\setminus (K_i\cup K_j)\right)$, then $W^i_{\mathcal{K}}(W^j_{\mathcal{K}'}(L))$ is a boundary link.
Note that in particular a Brunnian link $L$ always satisfies the conditions in (i) and (ii). It follows that a link obtained from a Brunnian link by taking twice Whitehead double (on either the same or another component of the link) kills *all* Milnor invariants.
\[boundary\] Let $L$ be an $m$-component Brunnian link in $S^3$. Let $p,q ~(pq\neq 0)$ and $i,j\in \{ 1,...,m \} $ $($possibly equal$)$ be integers. Then the link $W^{i,j}_{p,q}(L)$, obtained by respectively Whitehead $p$-double and Whitehead $q$-double on the $i^{th}$ and $j^{th}$ components of $L$, is a boundary link.
Figure \[ex\] below illustrates this result in the case of the Borromean rings $B$.
![The boundary links $W^{1,1}_{-4,2}(B)$ and $W^{1,2}_{-4,2}(B)$.[]{data-label="ex"}](B.eps)
Proof of Theorem \[thsat\]
--------------------------
Before proving Theorem \[thsat\], we will introduce the notion of band presentation of a link.
Let $L_i=\gamma_{i0}\cup \gamma_{i1}\cup \gamma_{i2}\cup... \cup \gamma_{ip_i}$ be a link as illustrated in Figure \[canonical\]. Let $L_1\cup...\cup L_m$ be a split union of the links $L_1$,...,$L_m$, and let $\Delta=\bigcup \Delta_{ij}$ be a disjoint union of disks $\Delta_{ij}$ ($1\le i\le m$ ; $1\le j\le p_i$) such that $\partial \Delta_{ij}=\gamma_{ij}$ and $\Delta_{ij}\cap \left(\bigcup_{k} \gamma_{k0}\right)=\Delta_{ij}\cap \gamma_{i0}$ consists of a single point. It is known [@suzuki] that an $m$-component link $L$ in a $3$-manifold $M$ which is null-homotopic in $M$ can be expressed as a band sum of $L_1\cup...\cup L_m$, which is contained in a $3$-ball in $M$, along mutually disjoint bands $b_{ij}$ ($1\le i\le m$ ; $1\le j\le p_i$), disjoint from $\mathrm{int}\Delta$, such that $b_{ij}$ connect $\gamma_{ij}$ and $\left(\bigcup_{k} \gamma_{k0}\right)$. [^2] This presentation is called a *band presentation* of $L$, and $L_1\cup...\cup L_m$ is called the *base link*.
[(i)]{} We may suppose that $i=m$ without loss of generality. Since $K_1\cup...\cup K_{m-1}$ is a boundary link, it bounds a disjoint union of surfaces $E=E_1\cup...\cup E_{m-1}$. Denote by $W_{\mathcal{K}'}(K_m)$ the $m^{th}$ component of $W^m_{\mathcal{K}'}(L)$. Since $W_{\mathcal{K}'}(K_m)$ is null-homologous in $h_m(D^2\times S^1)$, it is null-homotopic in $S^3\setminus (L\setminus K_m)$. Hence there is a band presentation of $W_{\mathcal{K}'}(K_m)$ such that the base link is disjoint from $E$ and such that the intersections of each band and $E$ are ribbon singularities. So $W_{\mathcal{K}'}(K_m)\cap E$ is a union of copies of $S^0$, which are the endpoints of these ribbon singularities. By tubing the surfaces $E_i$ suitably at these endpoints, we obtain a union of mutually disjoint surfaces $F_1$,...,$F_{m-1}$ such that $F_i=\partial K_i$ and $F_i\cap W_{\mathcal{K}'}(K_m)=\emptyset$ for all $1\le i\le m-1$. Since the $m^{th}$ component of $W^m_{\mathcal{K}}(W^m_{\mathcal{K}'}(L))$ bounds a Seifert surface $F_m$ in a regular neighborhood of $W_{\mathcal{K}'}(K_m)$, it follows that the components of $W^m_{\mathcal{K}}(W^m_{\mathcal{K}'}(L))$ bound $m$ mutually disjoint Seifert surfaces $F_1$,...,$F_m$.
[(ii)]{} We may suppose that $i=m-1$ and $j=m$ without loss of generality. $K_1\cup...\cup K_{m-2}$ being a boundary link, it bounds a disjoint union of surfaces $E=E_1\cup...\cup E_{m-2}$. Since $K_{m-1}\cup K_m$ is null-homotopic in $S^3\setminus (K_1\cup...\cup K_{m-2})$, there is a band presentation of $K_{m-1}\cup K_m$ such that the base link is disjoint from $E$ and such that the intersections of each band and $E$ are ribbon singularities. By tubing the surfaces $E_i$ suitably at the endpoints of theses singularities, we obtain a union of mutually disjoint surfaces $F_1$,...,$F_{m-2}$ such that $F_i=\partial K_i$ and $F_i\cap (K_{m-1}\cup K_m)=\emptyset$ for all $1\le i\le m-2$. Since the $(m-1)^{th}$ and $m^{th}$ components of $W^{m-1}_{\mathcal{K}}(W^m_{\mathcal{K}'}(L))$ bound a disjoint union $F_{m-1}\cup F_m$ of Seifert surfaces in a regular neighborhood of $K_{m-1}\cup K_m$, it follows that the components of $W^{m-1}_{\mathcal{K}}(W^m_{\mathcal{K}'}(L))$ bound $m$ mutually disjoint Seifert surfaces $F_1$,...,$F_m$.
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[^1]: The second author is partially supported by a Grant-in-Aid for Scientific Research (C) ($\#$20540065) of the Japan Society for the Promotion of Science.
[^2]: The result is given in [@suzuki] for *knots* in $S^3$, but it can be easily extended to the link case.
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abstract: 'We explain that, like the usual [Padé]{} approximants, the barycentric [Padé]{} approximants proposed recently by Brezinski and Redivo-Zaglia can diverge. More precisely, we show that for every polynomial ${{P}\!\left(z\right)}$ there exists a function ${{g}\!\left(z\right)} = \sum_{n = 0}^\infty c_n z^n$, with $c_n$ arbitrarily small, such that the sequence of barycentric [Padé]{} approximants of ${{f}\!\left(z\right)} = {{P}\!\left(z\right)} + {{g}\!\left(z\right)}$ do not converge uniformly in any subset of ${\mathds{C}}{}$ with a non-empty interior.'
author:
- 'Walter F. Mascarenhas [^1]'
title: 'The divergence of the barycentric [Padé]{} interpolants [^2] '
---
intro\_2 main\_proof\_2 easy\_lemmas\_2 proof\_of\_lemmas\_2 references\_2
[^1]: Instituto de Matemática e Estatística, Universidade de São Paulo, Cidade Universitária, Rua do Matão 1010, São Paulo SP, Brazil. CEP 05508-090 Tel.: +55-11-3091 5411, Fax: +55-11-3091 6134, walter.mascarenhas@gmail.com
[^2]: This work was supported by grant \#2013/10916-2, São Paulo Research Foundation (FAPESP).
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---
abstract: |
The flux of cosmic ray hadrons at the atmospheric depth of 820 g/cm$^2$ has been measured by means of the EAS-TOP hadron calorimeter (Campo Imperatore, National Gran Sasso Laboratories, 2005 m a.s.l.).\
The hadron spectrum is well described by a single power law :\
$S_{h}(E_{h}) = (2.25 \pm 0.21 \pm 0.34^{sys}) \times 10^{-7}
(\frac{E_{h}}{1000})^{(-2.79 \pm 0.05)}$ m$^{-2}$s$^{-1}$sr$^{-1}$GeV$^{-1}$\
over the energy range 30 GeV $\div$ 30 TeV. The procedure and the accuracy of the measurement are discussed.\
.8 cm (\*) Corresponding author.
The primary proton spectrum is derived from the data by using the CORSIKA/QGSJET code to compute the local hadron flux as a function of the primary proton spectrum and to calculate and subtract the heavy nuclei contribution (basing on direct measurements). Over a wide energy range $E_{0}= 0.5 \div 50$ TeV its best fit is given by a single power law :\
$S(E_{0}) = (9.8 \pm 1.1 \pm 1.6^{sys}) \times 10^{-5}
(\frac{E_{0}}{1000})^{(-2.80 \pm 0.06)}$ m$^{-2}$ s$^{-1}$ sr$^{-1}$ GeV$^{-1}$.\
The validity of the CORSIKA/QGSJET code for such application has been checked using the EAS-TOP and KASCADE experimental data by reproducing the ratio of the measured hadron fluxes at the two experimental depths (820 and 1030 g cm$^{-2}$ respectively) at better than $10 \%$ in the considered energy range.
address:
- 'Istituto di Fisica dello Spazio Interplanetario, CNR,Torino, Italy'
- 'Istituto Nazionale di Fisica Nucleare,Torino, Italy'
- 'Istituto Nazionale di Fisica Nucleare,Bologna, Italy'
- 'Laboratori Nazionali del Gran Sasso, INFN, Assergi (AQ), Italy'
- 'Dipartimento di Fisica dell’ Università, L’ Aquila, Italy'
- 'Dipartimento di Fisica Generale dell’ Università,Torino, Italy'
- 'Dipartimento di Scienze Fisiche dell’ Università and INFN, Napoli, Italy'
author:
- 'M. Aglietta'
- 'B. Alessandro'
- 'P. Antonioli'
- 'F. Arneodo'
- 'L. Bergamasco'
- 'M. Bertaina'
- 'C. Castagnoli'
- 'A. Castellina$^{*,}$'
- 'A. Chiavassa'
- 'G. Cini Castagnoli'
- 'B. D’Ettorre Piazzoli'
- 'G. Di Sciascio'
- 'W. Fulgione'
- 'P. Galeotti'
- 'P. L. Ghia'
- 'M. Iacovacci'
- 'G. Mannocchi'
- 'C. Morello'
- 'G. Navarra'
- 'L. Riccati'
- 'O. Saavedra'
- 'G. C. Trinchero'
- 'S. Valchierotti'
- 'P. Vallania'
- 'S. Vernetto'
- 'C. Vigorito'
title: 'Measurement of the cosmic ray hadron spectrum up to 30 TeV at mountain altitude: the primary proton spectrum.'
---
The EAS-TOP Collaboration
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**[PACS: 96.40.Pq, 96.40.De, 29.40.Vj.]{}**
Cosmic Rays. Hadrons. Primary protons. High energy calorimetry.
Introduction
============
The spectrum of hadrons detected at different atmospheric altitudes retains significant information about the energy/nucleon spectrum of primary cosmic rays, which is dominated by the lightest component, i.e. the proton one. Its measurement has been carried on in the past, both at sea level [@dig; @bro; @bar; @fic; @ash; @cow] and at mountain altitude [@ame; @can; @ren; @sio; @ada], using different experimental techniques, like emulsion chambers, magnetic spectrometers and calorimeters.\
The knowledge of the primary proton spectrum is of main relevance for the understanding of the cosmic rays acceleration mechanisms and of the propagation processes in the Galaxy. Moreover, the proton component is mainly responsible for the uncorrelated particle production in the atmosphere: any uncertainties on the proton spectrum reflect in an uncertainty in the calculation of the secondary particle fluxes ($\pi$ and $K$) and thus, for example, on the knowledge of the atmospheric muon and neutrino fluxes. A precise knowledge of such spectra is of particular importance to interpret the observational data from muon and neutrino detectors deep underground [@cas].\
The measurement of the primary proton spectrum has been performed by means of satellite and balloon borne experiments [@boe; @men; @bel; @san; @web; @alc; @swo; @buc; @seo; @rya; @zat; @iva; @asa; @apa] and indirectly derived by using ground based detectors [@ino; @ame1; @sak; @aha]. In the region of tens of TeV, however, direct measurements lack statistics and moreover their energy determinations are not calorimetric and depend on the interaction parameters and their fluctuations. The data inferred from hadron measurements at ground level can therefore provide significant new information.\
On the other side, the derivation of the information on the primary cosmic ray spectrum from hadron measurements, as well as the comparison of the results from different experiments, relies on the use of simulation tools describing the interaction and propagation of primary cosmic rays in the atmosphere. The response of such hadron interaction models has therefore to be verified, especially considering that the recorded hadrons are the results of large fluctuations with respect to the average behavior.\
The EAS-TOP Extensive Air Shower array was located at Campo Imperatore, 2005 m a.s.l., above the underground Gran Sasso Laboratories, with the aim of studying the cosmic ray spectrum in the energy range $10^{13} \div 10^{16}$ eV through the detection of the different air shower components.\
In this paper, we present and discuss the results obtained in the study of the uncorrelated hadrons by means of the calorimeter of EAS-TOP, namely:\
a) the measurement of the hadron flux in the energy range 30 GeV $\div$ 30 TeV;\
b) the derivation of the primary proton energy spectrum in the range 0.5 $\div$ 50 TeV;\
c) the check of the propagation and interaction code (CORSIKA/QGSJET) used for the interpretation of the data.
**The detector and the hadron trigger**
=======================================
The Muon and Hadron Detector of EAS-TOP is a 144 m$^2$ calorimeter [@nim] made of nine layers, each composed by a 13 cm iron slab absorber (except for the uppermost plane which is unshielded), and three planes of limited streamer tubes, for a total depth of 818.5 g cm$^{-2}$.\
Two of the streamer tube layers (with 100 $\mu$m wire diameter at 4650 V voltage) act as tracking devices, and are read by a two-dimensional system based on the anode wires and external orthogonal pick up strips. The third one, which data are used for the present analysis, operates in saturated proportional mode (the wire diameter being 50 $\mu$m, and the HV at 2900 V) for hadron calorimetry and EAS core studies. The signal charges are collected by a matrix of 840 (40$\times$38) cm$^2$ pads placed above the tubes; the pad signals are transferred to charge integrating ADCs with 15 bit dynamic range. The pad read-out is converted to the equivalent number of vertical particles by means of periodical calibration runs based on single muon triggers (pressure and temperature dependence of the induced charge being corrected for).\
Different sets of scintillators are placed in the apparatus for different aims; in particular, of the six ones lodged below the second absorber layer, three are used for hadronic trigger purposes. Each scintillator, of dimensions (80$\times$80) cm$^2$, is centered on a pad, viewed by two identical photomultipliers operating in coincidence and discriminated at the level of 30 m.i.p., corresponding to the energy loss of a 30 GeV proton incident on the calorimeter. They provide the “local hadron trigger”, which generates the read-out of the whole detector.\
For each scintillator a “tower” is defined, as the stack of $3 \times 3$ pads of the 8 internal layers centered on the scintillator itself. The detector and its operation are fully described in ref. [@nim]; a scheme of a “tower” is shown in Fig.\[fi:mhdsk\].
Hadron selection, acceptance and energy calibration
===================================================
An event, recorded following the “local hadron trigger”, is accepted as a hadron if: a) the cascade crosses at least three consecutive internal layers of the calorimeter, including the one positioned immediately below the triggering scintillator, and b) the maximum energy release is recorded on the central pad of each plane of the corresponding “tower”. This allows the selection of hadrons with energy above 30 GeV, and the definition of the angular acceptance.\
The check of hadron selection, the detection efficiency, effective area, angular acceptance, and energy calibration have been obtained by means of simulations of the detector response based on the GEANT code [@mcgea] (with FLUKA option for hadronic cascades), including the full description of the apparatus. Protons at fixed primary energy and zenith angle have been generated and analyzed with the same procedure as the experimental data.\
Particular care has been put in the modelling of the chamber behavior in the saturated proportional mode; the saturation in the collected charge has been studied in detail and included in the simulation, as fully discussed in [@nim]. The modelling of the chamber response to large particle densities has been checked at the 50 GeV $e^+$ beam at CERN-PS, using a detector built by chambers with the same characteristics, read out and filling gas mixture as the ones operating on site, but with length reduced to 3 m. Lead was used as absorber in front of the chambers in order to reach maximum particle density [@nim]. The chamber response was tested and found to agree with the model inside 2$\%$ up to particle densities $\rho_{ch} \simeq$ 300 particles/cm$^2$, corresponding to a 50 GeV electromagnetic shower after 4 cm of lead absorber. For iron absorber and the calorimeter geometry such particle density corresponds to hadrons with energy $E_{h} \simeq$ 650 GeV.\
Such response, introduced in the quoted simulation, provides transition curves that can be directly compared with the experimental data. As shown in Fig.\[fi:longi\], the difference between the two curves is always less than 10 $\%$ even at shower maximum, where the particle density is the highest, thus showing that the chamber behavior and saturation are well described at least up to 5 TeV (i.e. at particles densities at which the chamber response could not be directly tested).
The verification of the hadron selection procedure has been performed by comparing the shapes of the longitudinal developments for individual events with the expectations from the simulated ones (the agreement on the average transition curves having been shown in Fig.\[fi:longi\]). For layers 1-7 (i.e. the ones shielded by more than two iron slabs, see Fig.\[fi:mhdsk\]) the experimental and simulated $N_{l}/N_{tot}$ distributions (i.e. the ratio between the equivalent particle number recorded in each layer and the total one in the tower) are in agreement inside the statistical errors ($\simeq 10\%$). For layer 8 (shielded by a single iron slab), the contamination from the accompanying shower adds an excess of 15$\%$ of $N_{tot}$ in 16$\%$ of the events, independent on $N_{tot}$. The effect does not alter the hadron selection and the spectrum measurement beyond the systematic effects discussed in the following.\
For the described triggering conditions, the effective area $A_{eff}(E_{h},\theta)$ was determined using the same simulation code and taking into account the inefficiency of the trigger scintillators due to the 30 m.i.p. threshold. Such area includes the detection efficiency, which, concerning energy, rises above 65 $\%$ at $E_{h} \simeq$ 130 GeV for vertical incidence inside the geometrical area of the central pad. As regards zenith angle, the efficiency at 30$^{\circ}$ is about 10 $\%$ of the vertical one. The selection condition therefore introduces a cut in the angular acceptance such that 90$\%$ of the events are found inside 22$^{\circ}$ from the vertical direction.\
The effective area of each “tower” is shown in Fig.\[fi:acce\] for 4 different zenith angles.
The hadron energy is inferred from the total charge induced on the 8 shielded layers of the defined “tower” (more than 95 $\%$ of the shower particles at all energies are contained inside a 20 cm radius from the hadron position).\
The conversion curve from the total number of particles induced in the “tower” ($N_{tot}$) to the primary hadron energy is shown in Fig.\[fi:neconv\]. The energy resolution is $\sigma(E_{h})/E_{h}
= 15 \%$ at 1 TeV, worsening to 25$\%$ at 5 TeV due to leakage losses and to 30$\%$ at 30 GeV due to sampling losses. The dependence of the total number of particles on the hadron zenith angle is less than 3$\%$ up to 30$^{\circ}$; the difference in the conversion curve between protons and pions impinging on the calorimeter is less than 2$\%$.
The possibility that the triggering and selection procedure includes more hadrons has been studied by means of a simulation of cascades in the atmosphere through CORSIKA/QGSJET. It results that such hadron pile-up effect, even at the highest energies ($E_{h}>$3 TeV), does not alter the average energy determination of more than 6$\%$. As a test, to evaluate possible contaminations from the accompanying shower particles, the particle-energy conversion curve has also been obtained using the total charge induced on the five innermost planes only. No statistically significant difference was found in the hadron fluxes obtained in the two cases over the considered energy range.
**The hadron flux** {#sec:hadflu}
===================
About one million triggers were recorded in T=14760 hours of effective live time used in the present analysis; 40832 of them survived the selection criteria and were classified as hadrons.\
The measured number of events in each energy bin for the flux $S(E,\theta)$ is : $$\begin{aligned}
n_{ev}^{meas}(E_{h} \div E_{h}+\Delta E_{h}) = \int_{0}^{\Omega}
\int_{E_{h}}^{E_{h} + \Delta E_{h}} S(E,\theta)T A_{eff}(E,\theta)
d\Omega dE\end{aligned}$$ The hadron flux at zenithal angle $\theta$ can be approximated as: $$S(E,\theta) = S(E) exp \big[-\frac{x(\theta)-x}{\Lambda(E)} \big]
\label{eq:lamflu}$$ where S(E) is the flux in vertical direction and $x(\theta)$ is the atmospheric depth along $\theta$. The attenuation length $\Lambda(E)$ has been derived using the CORSIKA [@corsi] code to simulate the interactions and propagation of primary protons in air. In fact the hadron flux in the atmosphere includes both residual primaries and secondaries; at the EAS-TOP atmospheric depth, their ratio rises from $\simeq$0.7 at 500 GeV to $\simeq$1.4 at 5 TeV. Therefore the obtained values of $\Lambda(E)$ represent the full evolution of such mixture: $\Lambda(E) \simeq$ 114 g/cm$^2$ for QGSJET [@qgs], and $\simeq$ 131 g/cm$^2$ for HDPM [@hdpm], for the EAS-TOP altitude and range of zenith angles.\
Assuming a power law spectrum ($\gamma$ = 2.7) inside each energy bin, the mean value $\overline{E}_{h}$ is obtained, the corresponding flux being $S(\overline{E}_{h})=S(E)(\frac{\overline{E}_{h}}{E})^{-\gamma}$ (a change of $\Delta \gamma=0.1$ in the spectral slope does not produce any appreciable difference in the resulting flux).\
The vertical flux is thus : $$\begin{aligned}
S(\overline{E}_{h}) = \frac{n_{ev}^{meas}(E_{h} \div E_{h}+\Delta
E_{h})} {2 \pi T \overline{E}_{h}^{\gamma}\int \int E^{-\gamma}
exp \big[-\frac{x(\theta)-x}{\Lambda(E)} \big] A_{eff}(E,\theta)
sin\theta d\theta dE} \label{eq:eqflu}\end{aligned}$$ The recorded number of events and the experimental hadron fluxes at the atmospheric depth of 820 g/cm$^2$ are listed in Tab.\[ta:tabflu\] with the corresponding statistical uncertainties.
---------------------- ------------------ ------------------ ----------------- ----------------------------------------- ----------------------------
[**Mean Energy** ]{} [$\bf{E_{0}}$]{} [$\bf{E_{1}}$]{} [**Hadron**]{} [$\bf{S_{had}}$ ]{} [$\bf{\sigma(S_{had})}$]{}
[**(GeV)** ]{} [**(GeV)** ]{} [**(GeV)**]{} [**numbers**]{} [$\bf{( m^{2} \ s \ sr \ GeV)^{-1}}$]{} [$\bf{(m^{2} \ s \ sr \
GeV)^{-1}}$]{}
41 32 56 10222 $.12 \ 10^{-2}$ $.12 \ 10^{-4}$
73 56 100 12875 $.27 \ 10^{-3}$ $.24 \ 10^{-5}$
129 100 178 9506 $.60 \ 10^{-4}$ $.63 \ 10^{-6}$
229 178 316 4930 $.14 \ 10^{-4}$ $.21 \ 10^{-6}$
408 316 562 2174 $.29 \ 10^{-5}$ $.66 \ 10^{-7}$
726 562 1000 802 $.47 \ 10^{-6}$ $.18 \ 10^{-7}$
1290 1000 1778 299 $.92 \ 10^{-7}$ $.56 \ 10^{-8}$
2295 1778 3162 119 $.17 \ 10^{-7}$ $.16 \ 10^{-8}$
4081 3162 5623 44 $.26 \ 10^{-8}$ $.46 \ 10^{-9}$
7257 5623 10000 23 $.84 \ 10^{-9}$ $.18 \ 10^{-9}$
12904 10000 17783 12 $.14 \ 10^{-9}$ $.49 \ 10^{-10}$
22945 17783 31623 3 $.55 \ 10^{-10}$ $.39 \ 10^{-10}$
---------------------- ------------------ ------------------ ----------------- ----------------------------------------- ----------------------------
: *[The measured hadron flux at 820 g/cm$^2$. The given uncertainties are the statistical ones.]{}*[]{data-label="ta:tabflu"}
The following sources of systematic uncertainties have to be considered:
- The uncertainty in the evaluation of the effective area, $\frac{\delta A_{eff}}{A_{eff}} \simeq 12 \%$ at all energies.
- The uncertainty in the hadron angular distribution, which reflects in the evaluation of the attenuation length $\Lambda(E)$. A comparison between two different models (QGSJET and HDPM) in the CORSIKA frame shows that the differences in $\Lambda(E)$ reflect in a flux uncertainty $\frac{\delta S}{S} \simeq 5 \%$.
- The uncertainty in the energy assignment to each single hadron, for a spectral slope $\gamma \simeq$ 2.7, results in a flux uncertainty $\frac{\delta S}{S} \simeq 7 \%$. This value reaches 10$\%$ at the highest energies, as shown by the comparison of the measured and simulated longitudinal shower profiles.
- An uncertainty $\frac{\delta S}{S} \simeq 15 \%$ on the flux, due to the different behavior and efficiency of the triggering scintillators and to the different calibrations and stability of the corresponding “towers”.
A total systematic energy dependent uncertainty $\frac{\delta
S}{S} \simeq 15 \%$ is obtained from the first 3 items. To this, the $15 \%$ constant systematic uncertainty due to item 4 has to be added.\
The hadron flux is fitted by a power law from 30 GeV up to 30 TeV as\
$$S_{h}(E_{h}) = (2.25 \pm 0.21) \times 10^{-7}
(\frac{E_{h}}{1000})^{(-2.79 \pm 0.05)} \ \ \ \
m^{-2}s^{-1}sr^{-1}GeV^{-1} \label{eq:flufit}$$ with $\chi^2$ = 0.91 and is shown in Fig.\[fi:hadflu\].\
In the fitting procedure (and in the plot), the energy dependent systematic uncertainties have been included; the $15 \%$ energy independent systematic effect has to be added.\
The hadron flux is compatible, within the errors, with a single power law. This has been tested by performing the same fit in independent narrower energy ranges, the resulting slopes being shown in Fig.\[fi:slopes\].
**The primary proton spectrum**
===============================
The primary proton spectrum is derived from the data by:\
a) checking the hadron propagation code in the atmosphere;\
b) subtracting from the measured hadron spectrum the contribution of heavy primaries;\
c) minimizing the difference between the measured and the expected hadron fluxes on the basis of different primary proton spectra.
a\) The region of interest coincides with the energy range in which QGSJET (the hadronic interaction model used to describe the cosmic ray interaction and propagation in the atmosphere) has been directly checked against accelerator data [@engel; @knapp], both concerning the leading particle and the secondary production physics. Its reliability to reproduce the present data has been checked by comparing its predictions to the measured ratio of hadron fluxes at sea level (KASCADE [@casca], 1030 g/cm$^2$) and mountain altitude (EAS-TOP, 820 g/cm$^2$). Primary protons and helium nuclei were generated in quasi vertical direction $\theta
\leq 5^{\circ}$, with energy spectra according to JACEE and RUNJOB [@asa; @apa] and the expected hadron fluxes at each observation level were calculated. As shown in Fig.\[fi:ratio\], the expected ratio does not depend on the differences between such primary spectra, and it is compatible with the measured one within the statistical uncertainties, the comparison leading to a $\chi^2$=1.2/d.o.f. On average, the model reproduces the experimental ratio at better than 10$\%$.\
We remind that the general features of the model relevant for the calculation of the hadron flux (and therefore object of the test) are the combination of the total cross section and inelasticity concerning the contribution of the surviving primaries, and the very forward production cross section for the contribution of the secondaries. We therefore conclude that QGSJET, as implemented in CORSIKA, can be reliably applied in the considered energy range in the description of the uncorrelated hadron fluxes at different atmospheric depths and therefore can be applied between the top of the atmosphere and the EAS-TOP observation level, thus allowing to derive the primary nucleon flux from the present measurement.
b\) The contribution to the hadron flux from helium primaries has been evaluated using their spectrum as directly measured by the balloon experiments. In order to derive the systematic uncertainties of the procedure, both the RUNJOB ($\gamma_{He}$ = 2.80) and JACEE ($\gamma_{He}$ = 2.68) data have been used and their contribution subtracted from the experimental hadron flux. At $E_{h} \simeq$ 1 TeV, such contribution is $15 \%$ and $29 \%$ from RUNJOB and JACEE respectively; the heavier nuclei one is less than 10 $\%$.
c)The primary proton spectrum is obtained as the one minimizing the difference between the measured hadron spectrum (after subtraction of the Helium contribution by means of the afore described procedure) and the expected one from simulated proton primaries. Extensive simulations have been carried on, generating primary protons in quasi vertical direction ($\theta < 5^o$), with energy extracted on power law spectra with slope varying between 2.5 and 3.2. The number of simulated events is such that the number of hadrons in each energy bin be much higher than the experimentally collected one. Most of the contribution to each hadron energy bin comes from different primary energy regions; hadrons of energies in different ranges, e.g. $E_{h}$ = 0.1 $\div$ 0.2, 0.2 $\div$ 0.5, 0.5 $\div$ 1, 1 $\div$ 2, 2 $\div$ 5, $\geq$ 5 TeV are produced by primaries with median energy $E_{MED} \simeq$ 0.5, 2, 4, 10, 20, 55 TeV respectively. The data thus allow to get information on the primary proton spectrum in the range 0.5 $\div$ 50 TeV.\
Assuming a primary spectrum of the power law form $S(E_{0})=S_{0}E_{0}^{-\gamma}$, the normalization factor $S_{0}$ and the slope $\gamma$ have been obtained minimizing the differences between the calculated and the measured number of hadrons in each energy bin. The minimizations have been carried on by taking into account both the statistical and the energy dependent systematic uncertainties in the hadron flux.\
The data are well described by power law spectra in the energy range 0.5 $\div$ 50 TeV, with best fits, for the case of subtraction of the RUNJOB Helium spectrum:
$S(E_{0}) = (1.05 \pm 0.16) \times 10^{-4}(\frac{E_{0}}{1000})^{(-2.80 \pm 0.05)}$ m$^{-2}$ s$^{-1}$sr$^{-1}$ GeV$^{-1}$
and for the case of subtraction of the JACEE Helium spectrum:
$S(E_{0}) = (0.91 \pm 0.15) \times 10^{-4}(\frac{E_{0}}{1000})^{(-2.80 \pm 0.06)}$ m$^{-2}$ s$^{-1}$sr$^{-1}$ GeV$^{-1}$
Including the 7$\%$ uncertainty in the helium contribution and the 15$\%$ constant systematic uncertainty on the measured hadron flux into a global systematic error term, the result can be summarized as follows :
$S(E_{0}) = (9.8 \pm 1.1 \pm 1.6^{sys}) \times 10^{-5}(\frac{E_{0}}{1000})^{(-2.80 \pm 0.06)}$ m$^{-2}$s$^{-1}$sr$^{-1}$GeV$^{-1}$ (5)
The obtained proton spectrum is shown in Figs.\[fi:prflu\] and \[fi:prflup\]. The full area and the shaded lines (in the two figures respectively) include the systematic and statistical uncertainties of the measurement.
**Conclusions**
===============
The hadron flux has been measured over a wide energy range (30 GeV$\div$30 TeV) by means of the EAS-TOP hadron calorimeter at the atmospheric depth of 820 g/cm$^{2}$. The spectrum is well described by a single power law in the whole range :\
$S(E_{h}) = (2.25 \pm 0.21 \pm 0.34^{sys}) \times 10^{-7} (\frac{E
_{h}}{1000})^{(-2.79 \pm 0.05)}$ m$^{-2}$ s$^{-1}$ sr$^{-1}$ GeV$^{-1}$.\
Taking into account the contamination from heavier nuclei, on the basis of direct measurements, the primary proton spectrum is obtained between 0.5 and 50 TeV and is found to be compatible with a single slope power law:\
$S(E_{0}) = (9.8 \pm 1.1 \pm 1.6^{sys}) \times 10^{-5}
(\frac{E_{0}}{1000})^{(-2.80 \pm 0.06)}$ m$^{-2}$ s$^{-1}$ sr$^{-1}$ GeV$^{-1}$.\
A systematic uncertainty of about 7 $\%$ due to the uncertainty in the helium flux is included. The data match very well with the direct measurements over a wide energy range, usually not available to a single experiment, where direct measurements become statistically poor.\
The reliability of the CORSIKA/QGSJET interaction and propagation code, which is used to propagate the hadrons in the atmosphere and to compute the heavy nuclei contribution, is directly checked in this energy range by comparison with accelerator data and, concerning the direct application to the present measurement, through its capability to reproduce the ratio of hadron fluxes as measured at two different atmospheric depths by EAS-TOP and KASCADE, at 820 and 1030 g/cm$^{2}$ respectively.
Acknowledgements
================
The cooperation of the Direction and Staff of the Gran Sasso National Laboratories, as well as the technical assistance of C.Barattia, R.Bertoni, G.Giuliani and G.Pirali are gratefully acknowledged.\
The comments of an unknown referee have contributed to improve and clarify the text.
[99]{}
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---
abstract: |
One major branch of saliency object detection methods is diffusion-based which construct a graph model on a given image and diffuse seed saliency values to the whole graph by a diffusion matrix. While their performance is sensitive to specific feature spaces and scales used for the diffusion matrix definition, little work has been published to systematically promote the robustness and accuracy of salient object detection under the generic mechanism of diffusion.
In this work, we firstly present a novel view of the working mechanism of the diffusion process based on mathematical analysis, which reveals that the diffusion process is actually computing the similarity of nodes with respect to the seeds based on diffusion maps. Following this analysis, we propose super diffusion, a novel inclusive learning-based framework for salient object detection, which makes the optimum and robust performance by integrating a large pool of feature spaces, scales and even features originally computed for non-diffusion-based salient object detection. A closed-form solution of the optimal parameters for the integration is determined through supervised learning.
At the local level, we propose to promote each individual diffusion before the integration. Our mathematical analysis reveals the close relationship between saliency diffusion and spectral clustering. Based on this, we propose to re-synthesize each individual diffusion matrix from the most discriminative eigenvectors and the constant eigenvector (for saliency normalization).
The proposed framework is implemented and experimented on prevalently used benchmark datasets, consistently leading to state-of-the-art performance.
author:
- 'Peng Jiang, Zhiyi Pan, Nuno Vasconcelos, Baoquan Chen, and Jingliang Peng, [^1][^2] [^3]'
bibliography:
- 'tip.bib'
title: Super Diffusion for Salient Object Detection
---
[JIANG : Super Diffusion for Salient Object Detection]{}
Saliency detection, diffusion, spectral clustering.
Introduction
============
he aim of saliency detection is to identify the most salient pixels or regions in a digital image which attract humans’ first visual attention. Results of saliency detection can be applied to other computer vision tasks such as image resizing, thumbnailing, image segmentation and object detection. Due to its importance, saliency detection has received intensive research attention resulting in many recently proposed algorithms.
In the field of saliency detection, two branches have developed, which are visual saliency detection [@v1; @v4; @v5; @v7; @v8; @v10; @v12; @v13; @v14; @v15; @v16; @v17] and salient object detection [@s1; @s2; @s3; @s4; @s5; @s7; @s9; @s10; @s11; @s12; @s14; @s15; @s16; @s17; @s18; @s19; @s20; @s21; @s22; @s23; @s24; @s28; @grab; @EQCUTS; @GP; @SP; @RW]. While the former tries to predict where the human eyes focus on, the latter aims to detect the whole salient object in an image. Saliency in both branches can be computed in a bottom-up fashion using low-level features [@v1; @v4; @v7; @v8; @v13; @v14; @v15; @v16; @v17; @s1; @s2; @s5; @s7; @s11; @s12; @s14; @s19; @s21; @s22; @s23; @s24; @s28; @GP; @grab; @EQCUTS; @SP; @RW], in a top-down fashion by training with certain samples driven by specific tasks [@s16; @v12; @s4; @v5; @s18; @s17; @v10; @s20], or in a way of combining both low-level and high-level features [@s3; @s9; @s10; @s15]. In this work, we focus on salient object detection and utilize both high-level training and low-level features.
Salient object detection algorithms usually generate bounding boxes, binary foreground and background segmentation, or saliency maps which indicate the saliency likelihood of each pixel. Over the past several years, contrast-based methods [@s1; @s2; @s7; @s9] significantly promote the benchmark of salient object detection. However, these methods usually miss small local salient regions or bring some outliers such that the resultant saliency maps tend to be nonuniform. To tackle these problems, diffusion-based methods [@s11; @s12; @s18; @s22; @grab; @EQCUTS; @GP; @SP; @RW] use diffusion matrices to propagate saliency information of seeds to the whole salient object. While most of them focus on constructing good graph structures, generating good seed vectors and/or controlling the diffusion process, they have not yet made sufficient effort in analyzing the fundamental working mechanism of the diffusion process and accordingly addressing the inherent problems with the diffusion-based approaches.
The existent diffusion-based methods more or less follow a restricted framework, , a specific diffusion matrix is defined in specific feature space and scale based on a specific graph structure, usually with the seed saliency vector computed according to specific color-space heuristics. As a result, they usually lack in extensibility and robustness. This has motivated our search in this work for an inclusive and extensible diffusion-based framework that incorporates a large pool of feature spaces, scales, and seeds for robust performance. Major contributions of this work reside in the following aspects.
- **Novel interpretation of the diffusion mechanism**. Through eigen-analysis of the diffusion matrix, we find that: 1) the saliency of a node (called focus node) is equal to a weighted sum of all the seed saliency values, with the weights determined by the similarity in diffusion map between the focus node and each seed node, and 2) since the diffusion map is formed by the eigenvectors and eigenvalues of the diffusion matrix, the process of saliency diffusion has a close relationship with spectral clustering. This novel interpretation provides the foundation for the novel framework and methods proposed in this work.
- **Super diffusion framework for salient object detection**. We propose an inclusive and extensible framework, named super diffusion, for salient object detection, which computes the optimal diffusion matrix by exploiting a pool of feature spaces, scales and even saliency features originally computed for non-diffusion-based salient object detection. The parameters for the optimal integration are derived in a closed-form solution through supervised learning. This contrasts with traditional diffusion-based methods that define the diffusion matrices and seeds with high specificity, compromising the robustness of performance.
- **Local refinement of saliency diffusion**. We propose to promote each individual saliency diffusion scheme prior to its integration into the overall super diffusion framework. Based on the close relationship between saliency diffusion and spectral clustering, the promotion is achieved by re-synthesizing an individual diffusion matrix from the most discriminative eigenvectors and the constant eigenvector (for saliency normalization). In addition, we propose efficient and effective ways to compute seed vectors based on background and foreground priors.
It should be noted that an initial version of this work was published as a conference paper [@GP], which has been extended to this journal version mainly in the following aspects: 1) proposal of the super diffusion framework with full-length explanation, 2) significantly extended experiments to evaluate the proposed framework in concrete implementation, and 3) more comprehensive coverage and analysis of related works.
Related Works
=============
Diffusion-based salient object detection methods(, [@s11; @s12; @s18; @grab; @GP; @SP; @RW; @EQCUTS]) share the same main formula: $$\begin{aligned}
\mathbf{y} = \mathbf{A}^{-1}\mathbf{s},
\end{aligned}
\label{eq:eq1}$$ where $\mathbf{A}^{-1}$ is the diffusion matrix (also called ranking matrix or propagation matrix), $\mathbf{s}$ is the seed vector (diffusion seed), and $\mathbf{y}$ is the final saliency vector to be computed. Here $\mathbf{s}$ usually contains preliminary saliency information of a portion of nodes, that is to say, usually $\mathbf{s}$ is not complete and we need to propagate the partial saliency information in $\mathbf{s}$ to the whole salient region based on a graph structure to obtain the final saliency map, $\mathbf{y}$ [@s18]. The diffusion matrix $\mathbf{A}^{-1}$ is designed to fulfill this task. The existent methods mostly focus on how to construct the graph structure, how to generate the seed and/or how to control the diffusion process. Accordingly, we review them based on their approaches to the three sub-problems, respectively, in the following sub-sections.
Graph Construction
------------------
A diffusion-based salient object detection algorithm needs to firstly construct a graph structure on a given image for the definition of diffusion matrix. Specifically, it segments the given image into $N$ superpixels first by an algorithm such as SLIC [@a3] or ERS [@ERS], and then constructs a graph $G = (V,E)$ with superpixels as nodes $v_i$, $1 \leq i \leq N$, and undirected links between node pairs $(v_i, v_j)$ as edges $e_{ij}$, $1 \leq i, j \leq N$, to define the adjacency. Note that superpixels but not pixels are usually used as nodes for efficiency and stability considerations.
Straightforwardly, two nodes are connected by an edge in the graph if they are contiguous in the image. In order to capture relationship between nodes farther on the image, some works [@s11; @s12; @s18; @grab; @GP; @RW] connect a node to not only its directly contiguous neighbors, but also its 2-hop and even up to 5-hop neighbors [@EQCUTS]. Besides, some works [@s11; @s12; @s18; @grab; @GP; @SP; @RW] make a close-loop graph by connecting the nodes at the four borders of the image to each other. As a result, the distance between two nodes close to two different borders will be shortened by a path through borders. Work [@grab] connects each node to all the nodes at the four borders to increase the connectivity of the graph, which provides certain robustness to noise.
The weight $w_{ij}$ of the edge $e_{ij}$ which encodes the similarity between linked nodes usually is defined as $$\begin{aligned}
w_{ij} = e^{-\frac{\|v_{i}-v_{j}\|_2}{\sigma^2}}
\end{aligned}
\label{eq:resp}$$ where $v_{i}$ and $v_{j}$ represent the mean feature value of two nodes respectively, and $\sigma$ is a scale parameter that controls the strength of the weight. All the mentioned diffusion-based methods use the CIE LAB color space as feature space. Finally, the affinity matrix is defined as $\mathbf{W} = [w_{ij} ]_{N{\times}N}$ with $w_{ij}$ computed by Eq. \[eq:resp\] if $i=j$ or edge $e_{ij}$ exists in the graph and assigned 0 otherwise; the degree matrix is defined as $\mathbf{D} = diag\{d_{11},...,d_{NN}\}$, where $d_{ii} = \sum_j{w_{ij}}$.
Diffusion Matrix and Seed Computation
-------------------------------------
Different algorithms derive diffusion matrices and seed vectors in different ways. Works [@s11; @grab; @RW] use inverse Laplacian matrix $\mathbf{L}^{-1}$ as the diffusion matrix. Correspondingly, the formula of saliency diffusion is $$\begin{aligned}
\mathbf{y} = &\mathbf{L}^{-1}\mathbf{s}
\end{aligned}
\label{eq:eq2}$$ where $\mathbf{L}=\mathbf{D}-\mathbf{W}$. Works [@s18; @SP] use inverse normalized Laplacian matrix $\mathbf{L_{rw}}^{-1}$ as the diffusion matrix which normalizes weights by degrees of nodes when computing similarity. Correspondingly, the formula of saliency diffusion is $$\begin{aligned}
\mathbf{y} = &\mathbf{L_{rw}}^{-1}\mathbf{s}
\end{aligned}
\label{eq:eq3}$$ where $\mathbf{L_{rw}}=(\mathbf{I}-\mathbf{D}^{-1}\mathbf{W})=\mathbf{D}^{-1}(\mathbf{D}-\mathbf{W})$.
While works [@s11; @grab; @RW; @SP] use binary background and foreground indication vectors as the seed vectors in two stages, respectively, work [@s18] computes $\mathbf{s}$ by combining hundreds of saliency features $\mathbf{F}$ with learned weight $\mathbf{w}$ ($\mathbf{s}=\mathbf{F}\mathbf{w}$).
Different from the above-mentioned methods, work [@s12] duplicates the superpixels around the image borders as the virtual background absorbing nodes, and sets the inner nodes as transient nodes. Then, the entry of seed vector $s_i=1$ if node $v_i$ is a transient node and $s_i=0$ otherwise. Correspondingly, the formula of saliency diffusion is $$\begin{aligned}
\mathbf{y} = &(\mathbf{I}-\mathbf{P})^{-1}\mathbf{s}=\mathbf{L_{rw}}^{-1}\mathbf{s}
\end{aligned}
\label{eq:eq4}$$ where $\mathbf{P}=\mathbf{D}^{-1}\mathbf{W}$ and $\mathbf{P}$ is called transition matrix. Note that Eq. \[eq:eq4\] is derived from but not identical to the original formula in reference [@s12] and the derivation process is described in Appendix \[sect:appendix\].
In general, the existing diffusion-based salient object detection methods derive their diffusion matrices from the basic form of Laplacian matrix. As a result, their performance is restricted by the Laplacian matrix that makes the performance sensitive to the scale parameter and the feature space used for the matrix construction.
Diffusion Process Control
-------------------------
Applying Eq. \[eq:eq1\] for once to complete the salient object detection task may not produce satisfactory results, as the seed saliency information may diffuse to the non-salient region or may not diffuse to the whole salient region. One common way to control the diffusion process is applying multi-stage diffusion instead of one-stage diffusion. Works [@s11; @grab; @SP; @RW] diffuse to estimate a non-saliency map using the background prior, and reverses and thresholds the map to get the most salient seed nodes at the first stage; they conduct another pass of diffusion at the second stage with the seed saliency estimated at the first stage. Work [@SP] further divides each pass of diffusion into a sequence of steps that, instead of computing saliency of all nodes at once, estimates saliency of a subset of nodes as selected according to certain rules. Though effective to a certain extent, these approaches lack in theoretical support and may not be robust in general.
To summarize, researchers have devised good ways to construct the graph structures, the diffusion matrices and the seed vectors exploiting effective heuristics and priors. In this work, we step further to explore novel views of the fundamental diffusion mechanism and, accordingly, make systematic promotion of the diffusion-based salient object detection performance.
Diffusion Re-Interpreted {#sect:diffusionmap}
========================
As discussed before, a diffusion-based salient objection detection algorithm (, [@s11; @s12; @s18; @grab; @SP; @RW; @EQCUTS]) usually defines the diffusion matrix by a certain form of the Laplacian matrix, denoted by $\mathbf{A}$, which usually is positive semi-definite. Thus, $\mathbf{A}$ can be decomposed as $\mathbf{A}={\mathbf{U}}{\mathbf{\Lambda}}{\mathbf{U}}^{T}$ where $\mathbf{\Lambda}$ is a diagonal matrix formed from the eigenvalues $\lambda_{l}$, $l=1,2,\ldots,N$, and the columns of $\mathbf{U}$ are the corresponding eigenvectors $\mathbf{u}_{l}$, $l=1,2,\ldots,N$. According to spectral decomposition theories, each element, $\tilde a(i,j)$, of $\mathbf{A}^{-1}$ can then be expressed as $$\begin{aligned}
\tilde a(i,j)=\sum_{l=1}^N \lambda_{l}^{-1} \mathbf{u}_{l}(i)\mathbf{u}_{l}(j).
\end{aligned}$$ and each entry, $y_i$, of $\mathbf{y}$ as $$\begin{aligned}
\mathbf{y}_i &= \sum_{j=1}^N \mathbf{s}_j \sum_{l=1}^N \lambda_{l}^{-1} \mathbf{u}_{l}(i)\mathbf{u}_{l}(j) \\
&= \sum_{j=1}^N \mathbf{s}_j\left<\mathbf{\Psi}_{i}, \mathbf{\Psi}_{j}\right>,
\end{aligned}
\label{eq:eq6}$$ $$\begin{aligned}
\mathbf{\Psi}_{i} = [\lambda_{1}^{-\frac{1}{2}}\mathbf{u}_{1}(i),...,\lambda_{N}^{-\frac{1}{2}}\mathbf{u}_{N}(i)]
\end{aligned}
\label{eq:eq7}$$ where $\left<\cdot,\cdot\right>$ is the inner product operation. According to the reference [@a4], $\mathbf{\Psi}_{i}$ is called diffusion map (diffusion map at time $t=-\frac{1}{2}$ to be more exactly) at the $i$-th data point (node).
Based on Eq.s \[eq:eq6\] and \[eq:eq7\], we make a novel interpretation of the working mechanism of diffusion-based salient object detection: the saliency of a node (called focus node) is determined by all the seed saliency values in the form of weighted sum, with each weight determined by diffusion map similarity (measured by inner product) between the corresponding seed node and the focus node. In other words, seed nodes having more diffusion map similarity to the focus node will influence more on the focus node’s saliency. In a nutshell, diffusion maps are key functional elements for the diffusion.
According to Eq.s \[eq:eq6\] and \[eq:eq7\], nodes with similar (, distinct) diffusion maps tend to obtain similar (, distinct) saliency values. Therefore, the process of saliency diffusion is closely related to the clustering of the nodes based on their diffusion maps. Further, diffusion maps are derived from the eigenvalues and eigenvectors of the diffusion matrix, , we form a matrix by putting the weighted eigenvectors in columns and each row of the matrix gives one node’s diffusion map (see Eq. \[eq:eq7\]). As such, the diffusion-map-based clustering is almost identical in form to the standard spectral clustering of the nodes [@a1; @a2].
According to [@a1; @a2; @a9], the spectral clustering performance tends to be sensitive to the scale parameter $\sigma$ and the feature space used for computing the Laplacian matrix (see Eq. \[eq:resp\]), and only a subset of the eigenvectors are the most discriminative while the rest are less discriminative or even cause confusions to the clustering. Due to the close relationship between spectral clustering and saliency diffusion, we foresee that the limitations of the spectral clustering also limit the performance of the saliency diffusion. As such, we address these limitations in this work to fundamentally promote the performance of saliency diffusion.
Super Diffusion
===============
Generic Framework {#sect:integration}
-----------------
We propose a framework that systematically integrates diffusion maps originally derived from various diffusion matrices and seed vectors originally derived by various heuristics, so as to get rid of the sensitiveness of traditional diffusion-based salient object detection methods to specific feature spaces, scales and heuristics. We call this framework *super diffusion*.
For the $i$-th, $1 \leq i \leq N$, node, we may integrate various diffusion maps by defining
$$\begin{aligned}
\hat{\mathbf{\Psi}}_{i} = [\alpha_1\mathbf{\Psi}_{i}^1, \alpha_2\mathbf{\Psi}_{i}^2,\cdots,\alpha_{M}\mathbf{\Psi}_{i}^M]
\end{aligned}
\label{eq:diffmap}$$
where $\mathbf{\Psi}_{i}^j$, $1 \leq j \leq M$, is a diffusion map computed from diffusion matrix $\mathbf{A}_j^{-1}$ at the $i$-th node, and $\alpha_j$ is its weight to be determined. Correspondingly, we may formulate a matrix
$$\begin{aligned}
\mathbf{A}_I^{-1} &= \mathbf{U}_I \mathbf{\Lambda}_I \mathbf{U}_I^T \\
&= \mathbf{U}_I\left[
\begin{array}{ccc}
\alpha_{1}\mathbf{\Lambda}_1^{-1}&\cdots&\cdots\\
\vdots&\ddots&\vdots\\
\cdots&\cdots&\alpha_{M}\mathbf{\Lambda}_{M}^{-1}\\
\end{array}
\right]\mathbf{U}_I^T,
\end{aligned}
\label{eq:lapinte}$$
where $\mathbf{U}_I = \left[\mathbf{U}_1, \cdots, \mathbf{U}_{M} \right]$, $\mathbf{\Lambda}_I$ is a diagonal matrix and $\mathbf{A}_j={\mathbf{U}_j}{\mathbf{\Lambda}_j}{\mathbf{U}_{j}^{T}}$ is the eigen decomposition of $\mathbf{A}_j$. Applying $\mathbf{A}_I^{-1}$ to a seed vector, $\mathbf{s}$, we obtain the saliency vector $\mathbf{y} = \mathbf{A}_I^{-1}\mathbf{s}$ with its entry, $y_i$, $1 \leq i \leq N$, expressed as $y_i = \sum_{j=1}^N s_j\left<\hat{\mathbf{\Psi}}_{i}, \hat{\mathbf{\Psi}}_{j}\right>$.
Further, we may integrate various seed vectors, $\mathbf{s}^1,\mathbf{s}^2,\cdots,\mathbf{s}^K$, by $$\begin{aligned}
\mathbf{s}_I &= [\mathbf{s}^1,\mathbf{s}^2,\cdots,\mathbf{s}^K][\beta_1,\beta_2,\cdots,\beta_K]^T
\end{aligned}
\label{eq:superseed}$$ with $\beta_k$, $1 \leq k \leq K$, being the weights to be determined.
Applying $\mathbf{A}_I^{-1}$ to $\mathbf{s}_I$, we obtain the saliency vector, $$\begin{aligned}
\mathbf{y} &= \mathbf{A}_I^{-1}\mathbf{s}_I\\
&= \sum_{i=1}^{M}\sum_{j=1}^{K} \alpha_i \beta_j \mathbf{U}_i \mathbf{\Lambda}_i^{-1} \mathbf{U}_i^T \mathbf{s}^j \\
&= \sum_{i=1}^{M}\sum_{j=1}^{K} \alpha_i \beta_j \mathbf{A}_i^{-1} \mathbf{s}^j\\
&= \sum_{i=1}^{M}\sum_{j=1}^{K} \alpha_i \beta_j \mathbf{y}^{i,j}\\
&= \mathbf{H} \mathbf{w}^T,
\end{aligned}
\label{eq:newdiffu}$$ where $\mathbf{A}_i^{-1}=\mathbf{U}_i\mathbf{\Lambda}_i^{-1}\mathbf{U}_i^T$, $\mathbf{y}^{i,j}=\mathbf{A}_i^{-1} \mathbf{s}^j$, $\mathbf{H}=[\mathbf{y}^{1,1}, \cdots, \mathbf{y}^{1,K}, \mathbf{y}^{2,1}, \cdots, \mathbf{y}^{2,K}, \cdots, \mathbf{y}^{M,1}, \cdots, \mathbf{y}^{M,K}]$, and $\mathbf{w}=[\alpha_1 \beta_1, \cdots, \alpha_1 \beta_K, \alpha_2 \beta_1, \cdots, \alpha_2 \beta_K, \cdots, \alpha_M \beta_1, \cdots, \\\alpha_M \beta_K]$. With $\mathbf{A}_i^{-1}$, $1 \leq i \leq M$, and $\mathbf{s}^j$, $1 \leq j \leq K$, given, the variables of this system are $\alpha_i$, $1 \leq i \leq M$, and $\beta_j$, $1 \leq j \leq K$. In other words, the degree of freedom (DOF) for our solution is $M+K$. In order to increase the room for optimization, we increase the DOF to $M \times K$ by replacing $\mathbf{w}$ in Eq. \[eq:newdiffu\] with $\mathbf{w}=[w_1, w_2, \cdots, w_{M \times K}]$ and solving for $w_i$, $1 \leq i \leq M \times K$, instead.
We determine the weighting vector, $\mathbf{w}$, by supervised learning from a training set of $L$ samples, with the loss function defined as $$\begin{aligned}
J &= \sum_{i=1}^L(\mathbf{y}(i)-\mathbf{y_{gt}}(i))^2\\
&= \sum_{i=1}^L(\mathbf{H}(i)\mathbf{w}^T-\mathbf{y_{gt}}(i))^2,
\end{aligned}
\label{eq:loss}$$ where $\mathbf{y}(i)$, $\mathbf{y_{gt}}(i)$ and $\mathbf{H}(i)$ are the computed saliency vector, the ground-truth binary saliency vector and the $\mathbf{H}$ matrix for the $i$-th training sample, respectively. As $J$ is convex, the optimal $\mathbf{w}$ has a closed-form expression of $$\begin{aligned}
\mathbf{w} &= \frac{\sum_{i=1}^L \mathbf{H}(i)^{T}\mathbf{y_{gt}}(i)}{\sum_{i=1}^L \mathbf{H}(i)^T \mathbf{H}(i)}.
\end{aligned}
\label{eq:w}$$
Local Refinement {#sect:measures}
----------------
While the proposed framework in Sec. \[sect:integration\] promotes the robustness by optimally integrating various diffusion matrices and seeds, each individual diffusion matrix on its own may be optimized as well.
As discussed in Sec. \[sect:diffusionmap\], only a subset of $\mathbf{A}$’s eigenvectors are the most discriminative. Thus, in order to increase the discriminative power of the diffusion maps associated with each specific $\mathbf{A}_i$, $1 \leq i \leq M$, in Sec. \[sect:integration\], we are motivated to keep only the most discriminative while discarding the rest of its eigenvectors. Specifically, we refine each individual $\mathbf{A}_i^{-1}$ by re-synthesizing it from $\mathbf{A}_i$’s most discriminative eigenvectors followed by a normalization step, as detailed in the following subsections. We call this process local refinement for short.
In practice, we first refine each individual diffusion matrix, $\mathbf{A}_i^{-1}$, and then use the refined diffusion matrices to compute all the saliency values in matrix $\mathbf{H}$ in Eq. \[eq:newdiffu\] and $\mathbf{H}(i)$ in Eq.s \[eq:loss\] and \[eq:w\]. Regarding the choice of $\mathbf{A}_i$, $1 \leq i \leq M$, we use a slightly modified $\mathbf{L_{rw}}$, $\tilde{\mathbf{L_{rw}}}$(Sec. \[sect:constant\]), as the basic form and define a series of diffusion matrices by varying the feature space and scale parameter when computing the edge weights (Eq. \[eq:resp\]). Our choice is motivated by the fact that $\mathbf{L_{rw}}$ often leads to better intra-cluster coherency and clustering consistency than $\mathbf{L}$ for spectral clustering ( [@a2]).
### Constant Eigenvector {#sect:constant}
The eigenvalues, $\lambda_l$, and eigenvectors, $\mathbf{u}_l$, $1 \leq l \leq N$, of $\mathbf{L_{rw}}$ (the same for $\mathbf{L}$) are ordered such that $0=\lambda_1 \leq \lambda_2 \leq \ldots \leq \lambda_N$ with $\mathbf{u}_1=\mathbf{1}$ [@a1]. Some works (, [@s11]) avoid zero eigenvalues by approximately setting $\tilde{\mathbf{L_{rw}}}=\mathbf{D}^{-1}(\mathbf{D}-0.99W)$ such that $\tilde{\mathbf{L_{rw}}}$ is always invertible. Assuming $\tilde{\lambda}_l$ and $\tilde{\mathbf{u}_l}$, $1 \leq l \leq N$, are the corresponding eigenvalues and eigenvectors of $\tilde{\mathbf{L}_{rw}}=\mathbf{D}^{-1}(\mathbf{D}-0.99\mathbf{W})$, it can be proven that $\tilde{\mathbf{u}_l}=\mathbf{u}_l$ and $\tilde{\lambda}_l=0.99\lambda_l+0.01$. Thus, $0.01=\tilde{\lambda}_1 \leq \tilde{\lambda}_2 \leq \ldots \leq \tilde{\lambda}_N$ with $\tilde{\mathbf{u}}_1=\mathbf{1}$.
The constant eigenvector $\tilde{\mathbf{u}}_1$ contains no discriminative information. Thus, we discard it and re-synthesize the diffusion matrix, as done in our early conference version [@GP] of this work. But novelly, we reuse the constant eigenvector later in Sec. \[sect:normalization\] for normalization of saliency.
### Eigengap
In each diffusion matrix, except $\mathbf{u}_1$ that is a constant vector, the more $\mathbf{u}_l$ ($l \in [2,N]$) is to the front of the ordered array, the more indicative it usually is for the clustering. For instance, we visualize in Fig. \[fig:fig2\] a leading portion (excluding $\mathbf{u}_1$) of the ordered array of eigenvectors for each of four sample images. From Fig. \[fig:fig2\], we see that, for each sample image, the first few eigenvectors well indicate node clusters while the later ones often convey less or even confusing information about the clustering. The key is how to determine the exact cutting point before which the eigenvectors should be kept and after which discarded.
In practice, $\mathbf{L_{rw}}$ (the same for $\mathbf{L}$) often exhibits an eigengap, , a few of its eigenvalues before the eigengap are much smaller than the rest. Specifically, we denote the eigengap as $r$ and define it as $$\begin{aligned}
r&=\underset{l}{\operatorname{argmax}}|\Delta{\Upsilon}_l|,\\
\Delta{\Upsilon}_l& = \lambda_l-\lambda_{l-1}, ~ l=2,\ldots,N.
\end{aligned}
\label{eq:eigen}$$ Usually, Eq. \[eq:eigen\] is called eigengap heuristic. According to [@a2], some leading eigenvectors (except $u_1$) before the eigengap are usually good cluster indicators which can capture the data cluster information with good accuracy (as observed in Fig. \[fig:fig2\]), meanwhile the location of the eigengap often indicates the right number of data clusters. Further, the larger the difference between the two successive eigenvalues at the eigengap is, the more important the leading eigenvectors are, since $u_l$ is weighted by $\lambda_l^{-\frac{1}{2}}$ in diffusion map $\mathbf{\Psi}$ (Eq. \[eq:eq7\]). Ideally, the eigenvalues before the eigengap are close to zero while the rest are much larger, which means that the leading eigenvectors (except $u_1$) will dominate the behavior of the diffusion map.
With the eigengap identified, we then keep only the eigenvectors prior to the eigengap, which are usually the most discriminative ones for the task of node clustering. It may sometimes happen that $r=2$ according to Eq. \[eq:eigen\], meaning that all the eigenvectors will be filtered out. In this case, we assume the position of the second largest $|\Delta{\Upsilon}_l|$ as the eigengap.
### Discriminability {#sect:discriminability}
In some cases, an eigenvector may only distinguish a tiny region from the background, , $\mathbf{u}_{5}$ , $\mathbf{u}_{6}$ in the second row and $\mathbf{u}_{6}$ in the last row of Fig. \[fig:fig2\]. Usually, these tiny regions are less likely to be the salient regions we search for. Besides, these tiny regions often have been captured by other leading eigenvectors as well. Therefore, such eigenvectors usually have low discriminability and may even worsen the final results by overemphasizing tiny regions.Therefore, we evaluate the discriminability of eigenvector $\mathbf{u}_{l}$ by its variance $var(\mathbf{u}_{l})$, and filter out eigenvectors with variance values below a threshold, $v$.
### Normalization {#sect:normalization}
After the above local enhancement operations, each original diffusion matrix $\mathbf{A}_i^{-1}$ becomes $\bar{\mathbf{A}}_i^{-1}= \bar{\mathbf{U}}_i \bar{\mathbf{\Lambda}_i}^{-1}\bar{\mathbf{U}}_i^T$ ($1 \leq i \leq M$) in Eq. \[eq:newdiffu\]. Immediately, we may compute $\mathbf{\bar{H}}(i)$ of the $i$-th ($1 \leq i \leq L$) training sample using its enhanced diffusion matrices to replace $\mathbf{H}(i)$ in Eq. \[eq:w\] and obtain $\mathbf{w}$. However, this usually is problematic as the saliency vectors computed on different samples and/or by different matrix-seed combinations often exhibit inconsistent ranges of componential values. Therefore, in order to derive an optimal $\mathbf{w}$ of generic applicability, we first normalize the saliency vector of each sample computed by each matrix-seed combination, as explained below.
On each specific image sample, for each matrix-seed combination $\left( \bar{\mathbf{A}_{i}}^{-1}, \mathbf{s}^j \right)$, $1 \leq i \leq M$, $1 \leq j \leq K$, we need to normalize the saliency vector $\bar{\mathbf{y}}^{i,j} = \bar{\mathbf{A}}_{i}^{-1} \mathbf{s}^j = \bar{\mathbf{U}}_i \bar{\mathbf{\Lambda}}_i^{-1} \bar{\mathbf{U}}_i^T \mathbf{s}^j$ to range its componential values to $[0,1]$. It is commonly known that a vector $\mathbf{x}$ whose componential values extend a range of $[p,q]$ may be normalized by $$\begin{aligned}
\hat{\mathbf{x}}&=b\mathbf{1}+\frac{\mathbf{x}}{a},\\
b&=\frac{p}{p-q},\\
a&=-(p-q).
\end{aligned}
\label{eq:ab}$$ Similarly, we normalize $\bar{\mathbf{y}}^{i,j}$ with a componential value range of $[p,q]$ by
$$\begin{aligned}
\hat{\mathbf{y}}^{i,j}&=[\mathbf{u}_1, \mathbf{\bar{U}}_i]\left[
\begin{array}{cc}
\lambda'{_1}^{-1}&\mathbf{0}^{T}\\
\mathbf{0}&\frac{1}{\hat{a}}\bar{\mathbf{\Lambda}}_i^{-1}\\
\end{array}
\right][\mathbf{u}_1, \bar{\mathbf{U}}_i]^T \mathbf{s}^j\\
&=\lambda'{_1}^{-1}\mathbf{u}_1\mathbf{u}_1^{T}\mathbf{s}^j + \frac{1}{\hat{a}}\bar{\mathbf{U}}_i\bar{\mathbf{\Lambda}}_i^{-1}\bar{\mathbf{U}}_i^T\mathbf{s}^j\\
&=\lambda'{_1}^{-1}\mathbf{u}_1\mathbf{u}_1^{T}\mathbf{s}^j + \frac{\bar{\mathbf{y}}^{i,j}}{\hat{a}},
\end{aligned}
\label{eq:norm}$$
where $\mathbf{u}_1$ is the constant vector and $\lambda'{_1}$ and $\hat{a}$ are scalars to be determined. By comparing Eq.s \[eq:ab\] and \[eq:norm\], we set $\hat{a}=a$ and $\lambda'{_1}^{-1}\mathbf{u}_1\mathbf{u}_1^{T}\mathbf{s}^j=b\mathbf{1}$ for the normalization. Equivalently, we have $\hat{a}=-(p-q)$ and $\lambda'{_1}=\sum_{i=1}^{N}\mathbf{{s}}^j(i)(p-q)/p$.
In essence, the above normalization process refines $\bar{\mathbf{A}}_i^{-1}$ to $$\begin{aligned}
\hat{\mathbf{A}}_{i,j}^{-1}&=\hat{\mathbf{U}}_i\hat{\mathbf{\Lambda}}_i^{-1}\hat{\mathbf{U}}_i^T\\
&=[\mathbf{u}_1, \bar{\mathbf{U}}_i]\left[
\begin{array}{cc}
\lambda'{_1}^{-1}&\mathbf{0}^{T}\\
\mathbf{0}&\frac{1}{\hat{a}}\bar{\mathbf{\Lambda}}_i^{-1}\\
\end{array}
\right][\mathbf{u}_1, \bar{\mathbf{U}}_i]^T
\end{aligned}
\label{eq:final}$$ which is used as the final diffusion matrix for $\mathbf{s}^j$ on the specific image sample.
Using the finally refined diffusion matrices, we compute $\hat{\mathbf{H}}(i)$ for the $i$-th ($1 \leq i \leq L$) training sample to replace $\mathbf{H}(i)$ in Eq. \[eq:w\] and finally obtain the solution of $\mathbf{w}$.
Choice of Seeds {#sect:choiceofseeds}
---------------
We utilize the foreground and background prior to design two kinds of seed vectors and use them as $\mathbf{s}^j$, $1 \leq j \leq K$ in Eq.s \[eq:superseed\] and \[eq:newdiffu\].
Firstly, we assume that nodes closer to the center of image are more salient, and initialize a sequence of Gaussian-filter-like images (with different variances) to compute the first kind of seed vectors, as people usually put salient objects in the central foreground area when taking a photo.
Secondly, we assume that nodes located at the border of image are the least salient, and compute the time that other non-border nodes random walk to them to form the seed vector. Nodes that take more time to reach the border nodes are more salient. Note that the transition matrix of random walk can also be derived from the highly discriminative diffusion matrices, $\bar{\mathbf{A}}_i^{-1}$, $1 \leq i \leq M$, as explained in Appendix \[sect:appendix\] with an in-depth analysis of the working mechanism of this proposed seed vector construction method.
The foreground and background prior leads to not only good accuracy of seed value estimation, but also high time-efficiency as it avoids an extra pass of color-based preliminary saliency search.
Implementation Details
----------------------
When constructing the graph, in order to utilize the cross-node correlation in a broader range, we connect not only nodes that are directly adjacent, but also those that are two hops apart. Furthermore, we connect the nodes at the four borders of an image to each other to make a close-loop graph.
The main training steps of the proposed salient object detection algorithm are summarized in Algorithm \[alg:alg2\]. As for testing, given an input image $I$, we conduct the superpixel segmentation and graph construction on it and compute its $\hat{\mathbf{H}}$, following the same initialization and local refinement steps in Alg. \[alg:alg2\], and apply the learned weight $\mathbf{w}$ to $\hat{\mathbf{H}}$ to obtain $\hat{\mathbf{y}}=\hat{\mathbf{H}}\mathbf{w}^T$. Finally, we obtain the saliency map $S$ by assigning the value of $\hat{\mathbf{y}}_{i}$ to the corresponding node $v_i$, $1 \leq i \leq N$.
\[!t\]
\
(a) A list of training images, $[I_1, \cdots, I_L]$,\
(b) A list of scale parameters, $[\sigma_1, \cdots, \sigma_m]$,\
(c) A list of feature spaces, $[f_1, \cdots, f_n$\],\
(d) A list of seed computing methods, $[c_1, \cdots, c_K]$,\
**Initialization:**\
Segment each training image into $N$ superpixels, use the superpixels as nodes, connect border nodes to each other and connect nodes that are one or two hops away to construct a graph $G$.\
**Local refinement:** For each training image in (a),\
Compute $\mathbf{A}_i=\mathbf{D}_i^{-1}(\mathbf{D}_i - 0.99\mathbf{W}_i)$ and its eigenvectors $\mathbf{U}_i$ and eigenvalues $\mathbf{\Lambda}_i$ for each setting $i$ in combination of (b)(c); For each $\mathbf{A}_i$, discard the constant eigenvector, the eigenvectors after the eigengap or with low discriminability to get $\bar{\mathbf{A}}_i^{-1}$, $\bar{\mathbf{U}}_i$, $\bar{\mathbf{\Lambda}}_i$ and $\bar{\mathbf{H}}$ by local refinement operations described in Sec.s \[sect:constant\] to \[sect:discriminability\]; For each $\bar{\mathbf{A}}_i^{-1}$ and each seed computation method, $c_j$, in (d),\
1. Compute the seed vector, $\mathbf{s}^j$;\
2. Re-add the constant eigenvector with an updated eigenvalue, and scale $\bar{\mathbf{\Lambda}}_i$ to normalize $\bar{\mathbf{y}}^{i,j}$ by Eq. \[eq:norm\];\
3. Correspondingly, re-synthesize $\bar{\mathbf{A}}_i^{-1}$ to get the final diffusion matrix $\hat{\mathbf{A}}_{i,j}^{-1}$ by Eq. \[eq:final\];\
Integrate all $\hat{\mathbf{y}}^{i,j}$ to get $\hat{\mathbf{H}}$.\
**Global optimization:** With $\hat{\mathbf{H}}(i)$, $1 \leq i \leq L$, for all the training images computed,\
Substitute $\hat{\mathbf{H}}(i)$ for $\mathbf{H}(i)$ in Eq. \[eq:w\] to compute the optimal weight $\mathbf{w}$. Weight $\mathbf{w}$.
\[alg:alg2\]
Saliency Features as Diffusion Maps {#sect:drfi}
-----------------------------------
Most diffusion-based salient object detection methods (, [@s11; @s12; @s18; @grab; @GP; @SP; @RW; @EQCUTS]) rely on raw color features, , they use the mean color vectors of two linked nodes to compute the edge weight (Eq. \[eq:resp\]) and, correspondingly, the affinity matrix and the diffusion matrix. However, the raw color features may sometimes not be well indicative of the saliency. As such, more saliency features have been devised and used by non-diffusion-based salient object detection methods. In particular, the work [@s16] effectively integrates hundreds of saliency features for the task of salient object detection. This has motivated us to integrate more saliency features seamlessly into our super diffusion framework.
By our interpretation of the diffusion mechanism (Sec. \[sect:diffusionmap\]), diffusion maps play a key role in saliency computation and nodes with similar (, dissimilar) diffusion maps tend to be assigned similar (, dissimilar) saliency values. Therefore, good diffusion maps themselves should be discriminative which are similar for nodes of similar factual saliency and dissimilar otherwise. As saliency features discriminate salient from non-salient nodes, we use them to construct discriminative maps at the nodes to imitate the diffusion process. We call them diffusion maps as well for the convenience of description.
We denote the $Z$ saliency features by $\mathbf{g}^1, \mathbf{g}^2, \ldots, \mathbf{g}^Z$ with each $\mathbf{g}^i$, $i \in [1, Z]$, being an $N$-dimensional vector containing the corresponding feature values of the nodes. Then we construct a diffusion map for each node by $$\begin{aligned}
\mathbf{\Psi}'_i = [\mathbf{1}(i), \mathbf{g}^1(i), \mathbf{g}^2(i), \cdots, \mathbf{g}^Z(i)].
\end{aligned}
\label{eq:ftpsi}$$ Incorporating $\mathbf{\Psi}'_i$ into Eq. \[eq:diffmap\], we update $\hat{\mathbf{\Psi}}_{i}$ to $$\begin{aligned}
\hat{\mathbf{\Psi}}_{i} = [\alpha_1\mathbf{\Psi}_{i}^1, \alpha_2\mathbf{\Psi}_{i}^2,\cdots,\alpha_{M}\mathbf{\Psi}_{i}^M, \alpha_{M+1}\mathbf{\Psi}_{i}^{M+1}]
\end{aligned}
\label{eq:diffmapft}$$ with $\mathbf{\Psi}_{i}^{M+1} = \mathbf{\Psi}'_i$. Correspondingly, we make $\mathbf{A}_{M+1}=\mathbf{U}_{M+1}\mathbf{\Lambda}_{M+1}\mathbf{U}_{M+1}^{T}$ with $\mathbf{U}_{M+1} = [\mathbf{1}, \mathbf{g}^1, \mathbf{g}^2, \cdots, \mathbf{g}^Z]$ and $\mathbf{\Lambda}_{M+1}=diag\{1, 1, \cdots, 1\}$, and update $\mathbf{A}_I^{-1}$ in Eq. \[eq:lapinte\] to $$\begin{aligned}
\mathbf{A}_I^{-1} &= \mathbf{U}_I \mathbf{\Lambda}_I \mathbf{U}_I^T \\
&= \mathbf{U}_I\left[
\begin{array}{ccc}
\alpha_{1}\mathbf{\Lambda}_1^{-1}&\cdots&\cdots\\
\vdots&\ddots&\vdots\\
\cdots&\cdots&\alpha_{M+1}\mathbf{\Lambda}_{M+1}^{-1}\\
\end{array}
\right]\mathbf{U}_I^T,
\end{aligned}
\label{eq:lapinteft}$$ where $\mathbf{U}_I=\left[\mathbf{U}_1, \cdots, \mathbf{U}_{M+1} \right]$. Further, we update $\mathbf{H}$ and $\mathbf{w}$ by $\mathbf{H}=[\mathbf{y}^{1,1}, \cdots, \mathbf{y}^{1,K}, \mathbf{y}^{2,1}, \cdots, \mathbf{y}^{2,K}, \cdots, \mathbf{y}^{M+1,1}, \cdots, \mathbf{y}^{M+1,K}]$ and $\mathbf{w}=[w_1, w_2, \cdots, w_{(M+1) \times K}]$ for Eq.s \[eq:newdiffu\], \[eq:loss\] and \[eq:w\].
Finally, for the training and the testing, the procedures described in the previous sections are still conducted except that the steps in Sec.s \[sect:constant\]–\[sect:discriminability\] are not applied on any $\mathbf{A}_{M+1}$ as it is not a common graph-based diffusion matrix. But still, the normalization step in Sec. \[sect:normalization\] is conducted for each matrix-seed combination, $\left( \mathbf{A}_{M+1}, \mathbf{s}^j \right)$, $1 \leq j \leq K$, on each specific image sample.
Experiments and Analysis {#sect:experiments}
========================
Datasets and Evaluation Methods
-------------------------------
Our experiments are conducted on three datasets: the MSRA10K dataset [@s2; @s23] with $10K$ images, the DUT-OMRON dataset [@s11] with $5K$ images and the ECSSD dataset [@s14] with $1K$ images. Each image in these datasets is associated with a human-labeled ground truth.
In order to study the performance of our final super diffusion method, we adopt prevalently used evaluation protocols including precision-recall (PR) curves [@s1], F-measure score which is a weighted harmonic mean of precision and recall [@s1], mean overlap rate (MOR) score [@s15] and area under ROC curve (AUC) score [@s18], as described in Sec. \[sect:salientobject\]. Further, to analyze how much the local enhancement operations benefit our method, we propose to measure the quality of a diffusion matrix by visual saliency promotion and constrained optimal seed efficiency (COSE), as detailed in Sec. \[sect:promotion\] and Sec. \[sect:OSE\], respectively. Finally, in Sec. \[sect:components\], we give an ablation study of all the global and local enhancement operations, to show the effects of different steps in Alg. \[alg:alg2\].
[@c@ @c@]{}
Experimental Settings
---------------------
We choose $11$ different settings for the scale parameter $\sigma$, $\sigma^2 \in{[10, 11,\cdots,20]}$, and 3 different color spaces, $Lab$, $RGB$ and $HSV$, for the feature space, which leads to $11\times3=33$ different diffusion matrices, , $M=33$ for Eq.s \[eq:diffmap\], \[eq:lapinte\] and \[eq:newdiffu\]. We set $v=300$ as the threshold to filter out eigenvectors of low discriminability in Sec. \[sect:discriminability\]. For the first kind of seed vectors, we take the Gaussian variance from {0.5, 1, 2}. We integrate the saliency features of the work [@s16] into our super diffusion framework (Sec. \[sect:drfi\]). For each dataset, we use one half of the images as training samples, and the other half for testing and evaluation. In Sec. \[sect:promotion\] and Sec. \[sect:OSE\], in order to avoid zero eigenvalues, we approximately set $\tilde{\mathbf{L_{rw}}}=\mathbf{D}^{-1}(\mathbf{D}-0.99\mathbf{W})$ and $\tilde{\mathbf{L}}=\mathbf{D}-0.99\mathbf{W}$ when comparing diffusion matrices, as done in the reference [@s11]. However, our each diffusion matrix $\hat{\mathbf{A}}_{j}$ is directly re-synthesized from $\mathbf{L_{rw}}=\mathbf{D}^{-1}(\mathbf{D}-\mathbf{W})$ by the local refinement.
To comprehensively report the effectiveness of our proposed local refinement operations in Sec. \[sect:measures\], in Sec. \[sect:promotion\] and Sec. \[sect:OSE\], we design two experiments to compare the diffusion results with and without the local refinement. In Sec. \[sect:salientobject\] and Sec. \[sect:components\], we further demonstrate how much our method gets promoted after global enhancement by integration of diffusions.
Promotion of Visual Saliency {#sect:promotion}
----------------------------
Visual saliency detection predicts human fixation locations in an image, which are often indicative of salient objects around. Therefore, we use the detected visual saliency as the seed information, and conduct diffusion on it to detect the salient object region in an image. In other words, we promote a visual saliency detection algorithm by diffusion for the task of salient object detection.
In this experiment, we use the results of nine visual saliency detection methods (, IT [@v1], AIM [@v12], GB [@v4], SR [@v13], SUN [@v14], SeR [@v15], SIM [@v16], SS [@v7] and COV [@v17]) on the MSRA10K dataset as the seed vectors, respectively, and compare the saliency detection results before and after diffusion. For the diffusion, we test three matrices including $\hat{\mathbf{A}}_{1}^{-1}$, $\tilde{\mathbf{L}}^{-1}$ and $\tilde{\mathbf{L_{rw}}}^{-1}$, which are all computed in $Lab$ feature space with $\sigma^2=10$. It’s worth noting that $\hat{\mathbf{A}}_{1}^{-1}$ is only one of our locally refined diffusion matrices (without normalization yet) before the integration.
The PR curves of the nine visual saliency detection methods before and after diffusion by $\hat{\mathbf{A}}_{1}^{-1}$, $\tilde{\mathbf{L}}^{-1}$ and $\tilde{\mathbf{L_{rw}}}^{-1}$ are plotted in Fig. \[fig:boost\](a), (b) and (c), respectively.
Remarkably, as shown in Fig. \[fig:boost\], previous visual saliency detection methods which usually can not highlight the whole salient object all get significantly boosted after diffusion with any of $\hat{\mathbf{A}}_{1}^{-1}$, $\tilde{\mathbf{L}}^{-1}$ and $\tilde{\mathbf{L_{rw}}}^{-1}$. The promotion is so significant that some promoted methods even outperform some state-of-the-art salient objection detection methods, as observed by comparing Fig. \[fig:boost\] and Fig. \[fig:main\]. This means that, with a good diffusion matrix, we can fill the performance gap between two branches of saliency detection methods.
Comparing Fig.s \[fig:boost\](a), \[fig:boost\](b) and \[fig:boost\](c), we observe that $\hat{\mathbf{A}}_{1}^{-1}$ leads to more significant performance promotion and more consistent promoted performance than $\tilde{\mathbf{L}}^{-1}$ and $\tilde{\mathbf{L_{rw}}}^{-1}$, demonstrating higher effectiveness and robustness of the refined diffusion matrix, $\hat{\mathbf{A}}_{1}^{-1}$, in visual saliency promotion.
{width="\linewidth"}
Constrained Optimal Seed Efficiency {#sect:OSE}
-----------------------------------
We prefer a diffusion matrix to use as little query information or, equally, as few non-zero seed values to derive as close saliency to the ground truth as possible. Correspondingly, for a diffusion matrix, we measure the constrained optimal saliency detection accuracy it may achieve at each non-zero seed value budget, leading to a constrained optimal seed efficiency curve, as detailed below.
Given the ground truth $\mathbf{GT}$ and the diffusion matrix $\mathbf{A}^{-1}$, we hope to find the optimal seed vector, $\mathbf{s}$, that minimizes the residual, $\mathbf{res}$, computed by $$\begin{aligned}
\mathbf{res}=\mathbf{GT}- \mathbf{A}^{-1}\mathbf{s}.
\end{aligned}
\label{eq:eq10}$$ Aiming to reduce the number of non-zero values in $\mathbf{s}$, we turn the residual minimization to a sparse recovery problem, to solve which we adapt the algorithm of orthogonal matching pursuit (OMP) [@a8], as described in Alg. \[alg:alg3\].
As shown in Alg. \[alg:alg3\], we adapt the residual computation to $\tilde{\mathbf{res}}=\mathbf{GT}-bin(\mathbf{A}^{-1}\mathbf{s})$ in Step $4$, where $bin$ is the binarization operation since $\mathbf{GT}$ is binary; we multiply a factor $\mathbf{GT}(j)$ in Step $1$ to ensure that the non-zero seed values are selected from only the salient region; we solve the nonnegative least-squares problem in Step $3$ to ensure nonnegative elements of $\mathbf{s}$. The adapted OMP will stop when $\|\tilde{\mathbf{res}}\|_2$ is below a threshold, $c$, or the nonnegative seed values at the salient region are all selected, as shown in Step $5$. We see that the optimization process in Alg. \[alg:alg3\] is constrained, , the seeds are selected from only the salient region, the optimization is conducted in a greedy fashion and so forth. Although the saliency detection performance of these resultant seed vectors provides a good reference for our diffusion matrix evaluation, it should be noted that their optimal performance is constrained but not absolute.
In order to obtain the constrained optimal seed efficiency curve over the full range of nonnegative seed value budget, we set $c=0$ in Alg. \[alg:alg3\] and, at the $i$-th ($0 \leq i \leq 100$) iteration, we compute and record the pair of nonnegative seed percentage, $r_i$, and saliency detection accuracy, $a_i$, according to the following formulae: $$\begin{aligned}
r_i&=\frac{100\times\|\mathbf{s}\|_0}{\|\mathbf{GT}\|_0}\%,\\
a_i&=\frac{\|\mathbf{GT}\|_2-\|\tilde{\mathbf{res}}\|_2}{\|\mathbf{GT}\|_2}.
\end{aligned}
\label{eq:accuracy}$$ Based on these ($r_i$, $a_i$) pairs, we can plot the OSE curve of $\mathbf{A}^{-1}$ on an image.
We substitute $\hat{\mathbf{A}}_{1}^{-1}$, $\tilde{\mathbf{L}}^{-1}$ and $\tilde{\mathbf{L}_{rw}}^{-1}$ in last section into Eq. \[eq:eq10\] for $\mathbf{A}^{-1}$, respectively. For each diffusion matrix, we plot the average OSE curve over all the images in the MSRA10K dataset, as shown in Fig. \[fig:boost.spres\]. From Fig. \[fig:boost.spres\], we observe that the constrained optimal seed efficiency rises sharply at the beginning and levels off at around the nonnegative seed percentage of $30\%$, that $\hat{\mathbf{A}}_{1}^{-1}$ exhibits significantly higher average constrained optimal seed efficiency than $\tilde{\mathbf{L}}^{-1}$ and $\tilde{\mathbf{L_{rw}}}^{-1}$, and that there is an inherent performance ceiling for each diffusion matrix while $\mathbf{\hat{A}_{1}}^{-1}$ has the highest one. According to the last observation, it appears that the performance of diffusion-based saliency detection is fundamentally determined by the diffusion matrix, again emphasizing the importance in constructing a good diffusion matrix.
\[!t\]
Dictionary$(\mathbf{A}^{-1}_{N{\times}N})$, Signal$(\mathbf{GT}_{N{\times}1})$ and Stop criterion$(c)$\
Coefficient vector$(\mathbf{s}_{N{\times}1})$ and Residual$(\mathbf{res})$\
**Initialization:** $\mathbf{res}=\mathbf{GT}$, $Inds=\emptyset$,\
$FgInds=\underset{i}{\operatorname{arg}}{\{\mathbf{GT}(i)=1\}}$\
**Iteration:** $ind=\underset{j}{\operatorname{argmax}}\{|\left<\mathbf{res},\mathbf{A}^{-1}(:,j)\right>|\cdot \mathbf{GT}(j)\}$, $j \in FgInds$;\
$Inds=Inds \cup ind$, $FgInds=FgInds \setminus ind$; $\mathbf{s}(Inds)=\underset{\mathbf{\tilde{s}}\geq0}{\operatorname{argmin}}\|\mathbf{GT}-\mathbf{A}^{-1}(:,Inds)\mathbf{\tilde{s}}\|_2$; $\tilde{\mathbf{res}}=\mathbf{GT}- bin(\mathbf{A}^{-1}\mathbf{s})$,\
**Go to** 1;
\[alg:alg3\]
Dataset Protocol
----------- ----------- --------- --------- --------- --------- --------- --------- --------- --------- --
Precision 0.80289 0.8532 0.88492 0.87807 0.8586 0.86753
Recall 0.67817 0.752 0.75455 0.73813 0.71551 0.78882
MSRA10K F-measure 0.7702 0.85399 0.85504 0.82357 0.83908 0.84637
AUC 0.94111 0.94379 0.95074 0.95888 0.93264 0.96358
Overlap 0.57652 0.69254 0.69386 0.65398 0.65576 0.71011
Precision 0.66047 0.76865 0.74891 0.76924 0.7376 0.74345
Recall 0.52427 0.64498 0.65227 0.64544 0.53912 0.68775
ECSSD F-measure 0.62311 0.73608 0.72219 0.70027 0.72547 0.73279
AUC 0.87643 0.89127 0.91113 0.9154 0.88534 0.91663
Overlap 0.39517 0.52335 0.53065 0.51352 0.45799 0.53146
Precision 0.4784 0.54751 0.53419 0.52623 0.51532 0.54588
Recall 0.63225 0.62502 0.67161 0.68049 0.57653 0.65185
DUT-OMRON F-measure 0.50686 0.57087 0.57189 0.56208 0.55072 0.54627
AUC 0.88716 0.85276 0.88691 0.8607 0.86891 0.88407
Overlap 0.34133 0.42156 0.40828 0.39755 0.41029 0.40978
[@c@ @c@]{}
\[fig:main\]
Salient Object Detection {#sect:salientobject}
------------------------
We experimentally compare our methods (Ours and Ours(ND)) with eight other recently proposed ones including PCA [@s31], GMR [@s11], MC [@s12], DSR [@s28], HS [@s14], GP [@GP] and DRFI [@s16] on salient object detection. When evaluating these methods, we either use the results from the original authors (when available) or run our own implementations. Among these methods, GMR, MC, and GP are the diffusion-based methods which lead to outstanding performance, and DRFI is the approach that integrates hundreds of saliency features and yields top performance on the saliency benchmark study [@a10][^4]. Ours(ND) (, Ours) is our super diffusion method without (, with) the saliency features of DRFI [@s16] integrated.
We plot the PR curves of all the nine methods on MSRA10K dataset, ECSSD dataset and DUT-OMRON dataset in Fig.s \[fig:main.msra\], \[fig:main.ecssd\] and \[fig:main.dut\], respectively. Further, we provide the performance statistics on the five prevalent protocols for all the methods on the three datasets in Tab. \[tab:table1\]. From both Fig. \[fig:main\] and Tab. \[tab:table1\], we clearly observe that Ours(ND) outperforms the other diffusion-based methods and, after integrating the saliency feature of DRFI, Ours yields the top performance.
For visual comparison, we show in Fig. \[fig:final\] the saliency object detection results by the benchmark methods and our methods on several images in MSRA10K. From Fig. \[fig:final\], we observe clearly that Ours produces much closer results to the ground truth than the others. It is worth noting that, although Ours(ND) and DRFI both miss some salient regions, after integrating the saliency features of DRFI into the super diffusion framework, Ours successfully highlights most of the salient regions.
Effects of Building Steps {#sect:components}
-------------------------
In this section, we demonstrate the incremental effects of building steps in the proposed global and local enhancement operations (Sec. \[sect:integration\], Sec. \[sect:measures\] and Sec. \[sect:drfi\]), as detailed below.
For each test image, we may obtain eight PR curves, $S0$ to $S7$. We start from its $S0$ and progressively obtain $S1$ to $S7$ when the constant eigenvector is discarded, the eigenvectors after the eigengap are filtered out, the discriminability weighting is conducted, diffusion maps derived from multiple color spaces are integrated, diffusion maps derived from multiple scales are integrated, multiple diffusion seeds are integrated and saliency features are imported as diffusion maps, respectively. Experimenting on the whole MSRA10K dataset [@s2; @s23], we obtain the average PR curves for $S0$ to $S7$, as plotted in Fig. \[fig:main.steps\].
From Fig. \[fig:main.steps\], we observe that all the local and global enhancement operations consistently improve the performance, and the introduction of saliency features as diffusion maps leads to the top performance.
Conclusions
===========
In this work, we have proposed a super diffusion framework that systematically integrates various diffusion matrices, saliency features and seed vectors into a generalized diffusion system for salient object detection. To the best of our knowledge, this is the first framework of this kind ever published. The whole framework is theoretically based on our novel re-interpretation of the working mechanism of diffusion-based salient object detection, , diffusion maps are core functional elements and the diffusion process is closely related to spectral clustering in general. It takes a learning-based approach and provides a closed-form best solution to the global weighting for the integration. At the local level, it refines each diffusion matrix by getting rid of less discriminating eigenvectors, normalizes each specific saliency vector, and even incorporates discriminative saliency features as diffusion maps. As a result, the proposed framework produces a highly robust salient object detection scheme, yielding the state-of-the-art performance.
It is worthwhile to emphasize that the proposed super diffusion framework is open and extensible. Besides those employed in this work, it may integrate any other diffusion matrices, saliency features and/or seed vectors as well into the system specifically trained for any application with specific criterion in saliency object detection. In particular, it would be interesting to integrate various CNN-learned saliency features (, [@deep1; @deep2].) into the proposed framework and examine the performance promotion for specific applications. This is being planned for our future work.
{#sect:appendix}
In this appendix, we give the proof of Eq. \[eq:eq4\] and clarify the working mechanism of the second kind of seed vectors proposed in Sec. \[sect:choiceofseeds\].
Proof of Eq. \[eq:eq4\]
-----------------------
The work [@s12] duplicates the superpixels around the image borders as virtual background absorbing nodes, and sets the inner nodes as transient nodes, thus constructing an Absorbing Markov Chain. It computes the absorbed time for each node as its saliency value. In Eq.s 1 and 8 in the work [@s12], it formulates the transition matrix as $$\begin{aligned}
&\mathbf{P} = \mathbf{D}^{-1}\mathbf{W}=
\begin{pmatrix}
\mathbf{Q} ~~~ \mathbf{R}\\
\mathbf{0} ~~~ \mathbf{I}
\end{pmatrix},~ \\
\end{aligned}
\label{eq:first}$$ where the first $m$ nodes are transient nodes and the last $N-m$ nodes are absorbing nodes, $\mathbf{Q}\in[0, 1]^{m \times m}$ contains the transition probabilities between any pair of transient nodes, while $\mathbf{R}\in[0, 1]^{m \times (N-m)}$ contains the probabilities of moving from any transient node to any absorbing node. $0$ is the $(N-m) \times m$ zero matrix and $\mathbf{I}$ is the $(N-m) \times (N-m)$ identity matrix. According to Eq. 2 in the work [@s12], the absorbed time for $m$ transient nodes is $$\begin{aligned}
&\mathbf{y^*} = (\mathbf{I}-\mathbf{Q})^{-1}\mathbf{c},\\
\end{aligned}
\label{eq:N}$$ where $\mathbf{c}$ is a $m$ dimensional column vector all of whose elements are $1$.
In our derivation, we extend Eq. \[eq:N\] to $$\begin{aligned}
&\mathbf{y^*} = (\mathbf{I}-\mathbf{Q})^{-1}\mathbf{c} = (\mathbf{Q}^0+\mathbf{Q}^1+\mathbf{Q}^2+\ldots)\mathbf{c}.\\
\end{aligned}
\label{eq:N1}$$ and compute the $n$-th power of $\mathbf{P}$ as $$\begin{aligned}
&\mathbf{P}^n=
\begin{pmatrix}
&\mathbf{Q}^n~~~&(\mathbf{Q}^0+\mathbf{Q}^1+\ldots+\mathbf{Q}^{n-1})\mathbf{R}\\
&\mathbf{0}~~~&\mathbf{I}\\
\end{pmatrix}.
\end{aligned}
\label{eq:P}$$ As the absorbed time for absorbing nodes is $0$, we define the absorbed time for all the nodes as $\mathbf{y} =
\begin{pmatrix}
\mathbf{y^*}\\
\mathbf{0}
\end{pmatrix}
$. From Eq.s \[eq:N1\], \[eq:P\] and \[eq:first\], we have $$\begin{aligned}
\mathbf{y} &= (\mathbf{P}^0+\mathbf{P}^1+\mathbf{P}^2+\ldots)\mathbf{x} = (\mathbf{1}-\mathbf{P})^{-1}\mathbf{x}\\&=(\mathbf{D}^{-1}(\mathbf{D}-\mathbf{W}))^{-1}\mathbf{x}=\mathbf{L_{rw}}^{-1}\mathbf{x},\\
\end{aligned}
\label{eq:aeq5}$$ where $\mathbf{x}= \begin{pmatrix}
\mathbf{c}\\
\mathbf{0}
\end{pmatrix}$. This completes the proof of Eq. \[eq:eq4\].
Further, based on our re-interpolation of the diffusion (*ref.* Sec. \[sect:diffusionmap\]), $$\begin{aligned}
\mathbf{y}_i & = \sum_{j=1}^N \mathbf{x}_j\left<\mathbf{\Psi_{{L_{rw}}_i}}, \mathbf{\Psi_{{L_{rw}}_j}}\right>\\
& = \sum_{j=1}^m\left<\mathbf{\Psi_{{L_{rw}}_i}}, \mathbf{\Psi_{{L_{rw}}_j}}\right>,
\end{aligned}
\label{eq:abs_re}$$ meaning that the absorbed time of each node is equal to the sum of the inner products of its diffusion map with those of all the $m$ non-border nodes on the Absorbing Markov Chain.
The Second Kind of Seed Vectors
-------------------------------
In effect, after connecting all the nodes at the four borders of the image, we have constructed a graph similar to the Absorbing Markov Chain. For every node at the border, it connects with all $bn$ border nodes (including itself) and only $bm$ non-border nodes ($bm \ll bn$), meaning that once a random walk reaches a border node, it will less likely escape from the border node set. Therefore, we may assume that all the non-border nodes are transient nodes and all the border nodes are background absorbing nodes.
Accordingly, we compute the absorbed time of all nodes in an image to form a seed vector of the second kind, That is, following Eq. \[eq:abs\_re\], we have $$\mathbf{e}^i(j) = \sum_{k=1}^d \left<\bar{\mathbf{\Psi}}_{j}^i, \bar{\mathbf{\Psi}}_{k}^i\right>
\label{eq:eq11-0}$$ or, equivalently, $$\mathbf{e}^i = \bar{\mathbf{A}}_i^{-1}\mathbf{z}
\label{eq:eq11}$$ where $\mathbf{z}(k) = 1$ if $v_k$ is a non-border node and $\mathbf{z}(k) = 0$ otherwise.
[Peng Jiang]{} received the BS and PhD degrees in computer science and technology from Shandong University, China, in 2010 and 2016, respectively. Currently, He is a Lecturer with the School of Qilu Transportation, Shandong University, China. His research interests include computer vision and machine learning.
[Zhiyi Pan]{} received the BS degree from Shandong University, China, in 2018. He is currently pursuing the MS degree in computer science and technology with Shandong University, China. His research interests include computer vision and machine learning.
[Nuno Vasconcelos]{} received his PhD from the Massachusetts Institute of Technology in 2000. From 2000 to 2002, he was a member of the research staff at the Compaq Cambridge Research Laboratory. In 2003, he joined the Department of Electrical and Computer Engineering at the University of California, San Diego, where he is the head of the Statistical Visual Computing Laboratory. His work spans various areas, including computer vision, machine learning, signal processing, and multimedia systems.
[Baoquan Chen]{} is a Chair Professor of Peking University, where he is the Executive Director of the Center on Frontiers of Computing Studies. Prior to the current post, he was the Dean of School of Computer Science and Technology at Shandong University, and the founding director of the Visual Computing Research Center, Shenzhen Institute of Advanced Technology (SIAT), Chinese Academy of Sciences (2008-2013), and a faculty member at Computer Science and Engineering at the University of Minnesota at Twin Cities (2000-2008). His research interests generally lie in computer graphics, visualization, and human-computer interaction, focusing specifically on large-scale city modeling, simulation and visualization. He has published more than 100 papers in international journals and conferences, including two dozens or so papers in SIGGRAPH and SIGGRAPH Asia. Chen received an MS in Electronic Engineering from Tsinghua University, Beijing (1994), and a second MS (1997) and then PhD (1999) in Computer Science from the State University of New York at Stony Brook. Chen is the recipient of the Microsoft Innovation Excellence Program 2002, the NSF CAREER award 2003, McKnight Land-Grant Professorship for 2004, IEEE Visualization Best Paper Award 2005, and NSFC “Outstanding Young Researcher” program in 2010. Chen served as conference co-chair of IEEE Visualization 2005, and served as the conference chair of SIGGRAPH Asia 2014.
[Jingliang Peng]{} received the PhD degree in electrical engineering from the University of Southern California in 2006, the BS and MS degrees in computer science from Peking University in 1997 and 2000, respectively. Currently, he is a professor at the School of Software, Shandong University, China. His research interest mainly resides in digital geometry processing and digital image/video analysis.
[^1]: This work was partially funded by the National Natural Science Foundation of China (NSFC Grants No.s 61702301, 61472223 and 61872398) and the China Postdoctoral Science Foundation (Grant No. 2017M612272).
[^2]: P. Jiang, Z. Pan and J. Peng are with Shandong University, China. (E-mails:{sdujump, jingliap, panzhiyi1996}@gmail.com). N. Vasconcelos is with University of California, San Diego, USA. (E-mail: nvasconcelos@ucsd.edu). B. Chen is with Peking University, China. (E-mail: baoquan.chen@gmail.com)
[^3]: J. Peng is the corresponding author.
[^4]: We have noted that recently proposed CNN based methods such as [@deep1; @deep2] achieved the best performance on their own report. However, these methods are hard to re-implement and saliency benchmark study [@a10] does not list these works either.
|
---
abstract: 'We investigate the response to temperature of a well-known colloid-polymer mixture. At room temperature, the critical value of the second virial coefficient of the effective interaction for the Asakura-Oosawa model predicts the onset of gelation with remarkable accuracy. Upon cooling the system, the effective attractions between colloids induced by polymer depletion are reduced, because the polymer radius of gyration is decreases as the $\theta$-temperature is approached. Paradoxically, this raises the effective temperature, leading to “melting” of colloidal gels. We find the Asakura-Oosawa model of effective colloid interactions with a simple description of the polymer temperature response provides a quantitative description of the fluid-gel transition. Further we present evidence for enhancement of crystallisation rates near the metastable critical point.'
address:
- 'School of Chemistry, University of Bristol, Bristol, BS8 1TS, UK.'
- 'H.H. Wills Physics Laboratory, University of Bristol, Bristol, BS8 1TL, UK.'
- 'School of Chemistry, University of Bristol, Bristol, BS8 1TS, UK.'
- 'Received April 20th, 2012'
author:
- 'Shelley L. Taylor'
- Robert Evans
- 'C. Patrick Royall'
title: 'Temperature as an external field for colloid-polymer mixtures : “quenching” by heating and “melting” by cooling'
---
Introduction
============
Colloid-polymer mixtures occupy a special place in soft matter physics [@poon2002; @lekkerkerker]. The introduction of non-adsorbing polymer introduces an effective attraction between the colloids whose strength and range can be tuned by altering the concentration and molecular weight, respectively, of the polymer [@lekkerkerker; @asakura1954; @asakura1958; @vincent1972; @vrij1976]. The existence of this entropy driven ‘depletion’ attraction opens up a vast swathe of behaviour inaccessible to colloidal systems with purely repulsive interactions, such that colloid-polymer mixtures may be regarded as true “model atomic systems” [@poon2002]. The best known examples include liquid-gas phase separation [@poon2002; @lekkerkerker] but there are also phenomena not seen in atomic systems such as gelation [@poon2002] and re-entrant glassy dynamics at high density [@pham2002]. Real-space analysis at the particle level has enabled direct observation of crystallisation [@dehoog2001], and behaviour related to liquid-gas phase separation such as capillary wave fluctuations at (colloidal) liquid-gas interfaces [@aarts2004] and fluid critical phenomena [@royall2007c]. Moreover colloid-polymer mixtures with a relatively short-ranged attractive interaction can (crudely) model proteins and may exhibit two-step crystal nucleation phenomena [@savage2009], to which we return below.
Theoretical treatments of colloid-polymer mixtures are based largely on the Asakura-Oosawa-Vrij (AO) model [@asakura1954; @asakura1958; @vrij1976] which treats the colloid-colloid interaction as that of hard-spheres (HS) and the polymer-polymer interaction as ideal, i.e. the polymer coils are assumed to be perfectly interpenetrating spheres. However, the polymer spheres have an excluded volume (hard) interaction with the HS colloids. This model binary (AO) mixture provides the simplest, zeroth-order description of the real mixture. It is characterized by the size ratio $q=\sigma_p/\sigma$, where sigma is the colloid diameter and $\sigma_p$ is the polymer sphere diameter. The colloid-polymer interaction is infinite for separations $r<(\sigma+\sigma_p)/2$. From simulation and theoretical studies it is well-known that for sufficiently large size ratios, $q \gtrsim 0.3$, the AO model exhibits phase separation into a colloid-rich (liquid) and a colloid-poor (gas) phase at sufficiently high polymer volume fractions. For smaller size ratios this phase transition becomes metastable w.r.t. the fluid-crystal transition[@gast1983; @lekkerkerker1992; @dijkstra1999]. The same trend in phase behaviour (with $\sigma_p$ set equal to twice the radius of gyration of the non-adsorbing polymer) is found in experimental studies[@ilett1995; @poon2002; @lekkerkerker]. In addition to predicting purely entropy driven fluid-fluid phase separation the AO model exhibits the elegant feature that for size ratios $q<(2/\sqrt 3-1)=0.1547$ the degrees of freedom of the ideal polymer can be integrated out exactly and the binary mixture maps formally to a one-component system of colloids described by an effective Hamiltonian containing only one and two-body (pair) contributions [@gast1983; @dijkstra1999]. The former contribution plays no role in determining phase equilibrium or structure, for a uniform (bulk) fluid, but does determine the total pressure and compressibility [@dijkstra2000]. The pair interaction between the HS colloids, given by integrating out the ideal polymer, is the standard AO potential:
$$\beta u_{AO}(r)=
\cases{
\infty & for $r<\sigma$\\
- \phi_p \frac{(1+q)^3}{q^3} \\
\times \left[ 1-\frac{3r}{2(1+q)\sigma} +\frac{r^{3}}{2(1+q)^{3}\sigma^{3}} \right] & for $\sigma< r < \sigma+\sigma_p$ \\
0 & for $r < \sigma+\sigma_p$ \\
}
\label{eqAO}$$
$\beta$ is $1/k_{B}T$ where $k_{B}$ is Boltzmann’s constant and $T$ is temperature. Since the analysis is performed in the semi-grand ensemble the polymer volume fraction in the reservoir $ \phi_p = \pi \sigma_p^3 z_p / 6 $ appears in Eq. (vref[eqAO]{}).The polymer fugacity $z_{p}$ is equal to the number density $\rho_p$ of ideal polymers in the reservoir at the given chemical potential $\mu_p$. As noted already, in relating the AO model to experiment, one usually sets $\sigma_p = 2 R_G$ where $R_G$ is the polymer radius of gyration. Hitherto, most work on colloidal-polymer mixtures was carried out at a fixed temperature, typically around 25 $^{\circ}$C. *Effective* temperature is varied by changing the interaction strength. The effective temperature is inversely proportional to the depth of the attractive well of the interaction potential in (Eq. \[eqAO\]), and is therefore fixed for a given polymer reservoir density. Scanning a phase diagram then requires preparation of a considerable number of different samples. Conversely, in molecular systems, interactions are usually constant over the (broad) temperature range of interest, one sample is prepared and temperature is used as a control parameter.
In our present study we note that colloid-polymer mixtures can respond to temperature in an intriguing and counter-intuitive manner. The effective temperature in Eq. (\[eqAO\]) is set by $z_{p}$ which in turn is equal to the polymer number density for ideal polymers. Real systems approximate this behaviour very well [@royall2007jcp]. Thus the primary response of the system to temperature is given by the response of the polymer depletant, since the polymer-polymer interactions are weak and colloid-colloid interactions are athermal (hard sphere). Now, close to (but above) their theta temperature $T^\theta$, polymers expand ($R_G$ increases) upon heating (Fig. \[figRgT\]) . This expansion has two effects : firstly, the polymer-colloid size ratio $q$ increases, thereby increasing the range of attraction, and secondly, the polymer reservoir volume fraction $\phi_p=\pi \rho_p \sigma_p^3 / 6$ also increases. Since the well-depth $- \beta u_{AO}(\sigma)= \pi \rho_p \sigma^3 q^2 (1+\frac{2}{3}q)/4$ this means the effective temperature falls strongly for a modest increase in polymer size which leads to a paradoxical result, namely raising the temperature of a colloid-polymer mixture near $T^\theta$ brings about a strong *effective cooling*. Although this effect has been exploited to drive phase transitions in mixtures of colloidal rods and polymers [@alsayed2004], these temperature-dependent depletion interactions have received relatively little attention. This is in contrast to other means of controlling the attractive interactions between colloids in-situ, such as the critical Casimir effect[@hertlein2008; @guo2008; @bonn2009] and multiaxial electric fields [@elsner2009]. We note that in-situ control of attractive interactions, combined with particle-resolved studies, has the power to provide much new insight into a variety of phenomena, including phase transitions [@book].
Here we make a quantitative experimental investigation, at the single-particle level, of the effect of temperature on an already well-studied colloid-polymer mixture. The elucidation of our results requires theoretical underpinning and we shall base this on the AO model described above. Specifically we investigate a mixture where the size ratio is about 0.2 but varies by 10 percent or so on changing temperature. The size ratio is such that the fluid-fluid transition is metastable w.r.t. the fluid-solid transition and therefore we consider out of equilibrium phenomena associated with metastable states. A recent simulation study [@fortini2008] provides a helpful framework for placing our results in context. These authors study the effective one-component AO model, where the pair-potential is given by Eq(1), for $q=0.15$ which is in the regime where the mapping to the effective one-component description using only a pair potential is exact. By changing the polymer reservoir density , equivalent to changing the depth of the attractive potential well, they determine both equilibrium and out-of equilibrium ‘phase diagrams’. More specifically, using Monte Carlo and Brownian dynamics, they investigate crystal nucleation and the onset of gelation in the vicinity of the metastable fluid-fluid binodal. They present convincing evidence that crystallization is enhanced by the binodal. We tackle the same issues in our experiments on a real colloid-polymer mixture, seeking to ascertain what role proximity to the binodal plays in forming gels and in determining crystal nucleation rates.
It is well-known that in experiments equilibrium is often not reached and in particular gelation can occur. This phenomenon has been linked to spinodal decomposition associated with colloidal gas-liquid condensation [@verhaegh1997; @lu2008]; gels are supposed to form within the metastable fluid-fluid spinodal. It follows that knowledge of the critical point is important in predicting where gelation might occur [@lu2008].
The connection between critical density fluctuations and crystal nucleation rates in short-ranged attractive systems, where the gas-liquid critical point is metastable with respect to crystallisation, has received considerable attention since it was elucidated by Ten Wolde and Frenkel [@tenWolde1997]. Critical fluctuations are expected to enhance the nucleation rate and may be responsible for the strong temperature dependence of nucleation rates found in globular proteins [@galkin2000; @vekilov2010]. A two-step nucleation process is envisaged where nuclei preferentially form in fluctuations of high density since the surface tension between the nucleus and the surrounding fluid is smaller. The reduction in free energy barrier to nucleation associated with such density fluctuations has been measured in a 2D depletion system [@savage2009]. Here we investigate crystallisation in the neighbourhood of the metastable critical point.
This paper is organised as follows. In section \[sectionMethods\] we first introduce the experimental system, and discuss the response of the polymer component to temperature. We then discuss relating experiment and theory in terms of the AO model. In section \[sectionResults\] we present results for *(i)* the room-temperature phase diagram, *(ii)* crystallisation around the metastable critical point and *(iii)* the response of the system to temperature. We conclude in section \[sectionConclusions\].
Methods {#sectionMethods}
=======
Experimental {#sectionExperimental}
------------
![(color online) The radius of gyration $R_G$ of polystyrene. This is a fit, Eq. (\[eqExpansion\]), to experimental data [@berry1966]. []{data-label="figRgT"}](figRgT){width="90mm"}
Our experimental system is based on polymethyl methacrylate (PMMA) colloids. The colloid diameter $\sigma=1080$ nm with polydispersity 4.6%, as determined from static light scattering. The polystyrene polymer used has a molecular weight $M_{w}=8.5\times10^{6}$, which corresponds to a radius of gyration of $R_{G}^{\theta}=95$ nm under $\theta$ conditions [@vincent1990]. This leads to a polymer-colloid size ratio of $q(T=T^\theta)=0.176$. The colloids and polymer are dispersed in a solution of *cis* decalin, where we find $T^\theta$ of polystyrene is $10^{\circ}$C . We image this system at the single particle level with a Leica SP5 confocal microscope. To this microscope we have fitted a temperature stage which uses a Peltier chip to cool from room temperature ($25{}^{\circ}$C) to the $\theta$-temperature and below. Note that our experimental system is not density-matched. The gravitational length $\lambda_{g}=k_{B}T/(mg)=1.96\sigma$, where $m$ is the buoyant mass of the colloid and $g$ is the acceleration due to gravity. Sedimentation therefore becomes an issue at long times, limiting our experimental timescales to about one hour. For studies of crystallisation, we orient the sample capillaries perpendicular to gravity, mitigating its effect. We note that for this system a disordered layer forms on the capillary walls which inhibits heterogenous nucleation.
We estimate the effect of temperature on the radiius of gyration of the polymer as shown in Fig. \[figRgT\]. We use the following expression
$$R_{G}(T)=R_{G}^{\theta} \left[\sqrt{2} \left(1-\exp \left( \frac{T^\theta-T}{\tau} \right) \right) +1 \right]
\label{eqExpansion}$$
for $T\geq T^{\theta}$ which closely matches experimental data over the relevant temperature range [@berry1966]. Here the parameter $\tau=20^\circ$ C.
Comparison with theory {#sectionMapping}
----------------------
As mentioned in the introduction we choose to interpret our experimental results within the framework of the simple AO model. A key ingredient is locating, in the AO phase diagram, the (metastable) fluid-fluid binodal for size- ratios $q\sim0.2$. There are computer simulation results for the binodal and its critical point for $q=0.1$ [@dijkstra1999; @ashton2011] and for $q=0.15$ [@fortini2008]. Clearly in both cases $q<0.1547$ so the mapping to an effective one-component fluid that is described by only the pair potential Eq. (\[eqAO\]) is exact. Although our experimental systems have $q\sim0.2$, within the context of AO we can expect three-body contributions to the effective Hamilitonian to play only a very small role. In order to ascertain the phase behaviour of the AO model it is tempting to turn to the free-volume theory [@lekkerkerker1992; @dijkstra1999] which yields simple recipes for calculating both fluid-solid and fluid-fluid phase equilibria for the binary AO mixture. This approximation is fairly successful for size- ratios $\gtrsim 0.4$. However, for $q=0.1$ free-volume theory provides a reasonably accurate description of fluid-solid coexistence [@dijkstra1999] but is quantitatively poor at describing the metastable fluid-fluid coexistence. Specifically it predicts a critical value of $\phi_p$ that is in reasonable agreement with simulation but a critical value of $\phi_c\sim 0.57$ that is unphysically large [@dijkstra1999].
In Fig. \[figPdRoom\] we plot the fluid-fluid spinodal for $q=0.214$, the value that corresponds to the experimental system at room temperature $T=25^\circ$ $C$, calculated from free-volume theory using the analytical expression derived by Schmidt *et al.* [@schmidt2002]:
$$\phi_{P}=\frac{\theta_{1}^{4}\theta_{2}/\phi_{c}}{\alpha\left(12\theta_{1}^{3}+15q\theta_{1}^{2}\theta_{2}+6q^2\theta_{1}\theta_{2}^{2}+q^3\theta_{2}^{3}\right)}
\label{schmidtSpin}$$
where $\theta_{1}=(1-\phi_{c})$ and $\theta_{2}=(1+2\phi_{c})$ and $\alpha$ is the free volume fraction [@lekkerkerker1992; @schmidt2002].
One finds that the critical point is at about $\phi_c =0.40, \phi_p=0.35$. Once again the critical colloid fraction appears rather high. We shall argue that a more accurate value is $\phi_c \sim 0.27$.
Clearly a more reliable prescription is required to estimate the critical point and therefore the location of the binodal. For models like the present, where attractive interactions are short-ranged (sticky), Vliegenthart and Lekkerkerker [@vliegenthart2000] and Noro and Frenkel [@noro2000] argued that a useful estimate of the critical temperature (or interaction strength) could be obtained by considering the reduced, with respect to HS, second virial coefficient given by
$$B_{2}^{*}=\frac{3}{\sigma^3}\intop_{0}^{\infty}drr^{2}\left[1-
\exp\left(-\beta u(r)\right)\right]
\label{eqB2}$$
where $u(r)$ is the pair potential. These authors proposed that for a wide class of model fluids $B_2^* \sim -1.5$ at the critical temperature. Later Largo and Wilding [@largo2006] carried out simulations for effective (depletion) potentials calculated for additive binary HS systems with size-ratios $q=0.1$ and $0.05$. For these small ratios the effective pair potentials are similar to the AO potential \[Eq. \[eqAO\]\] but with an additional repulsive barrier.
For all the potentials they considered, Largo and Wilding found that the value of $B_2^*$ at criticality obtained from simulation was very close to $B_2^{*AHS} =-1.207$, i.e. the value reported for the adhesive hard sphere (AHS) model at its critical point [@miller2003]. Ashton applied the same criterion for the AO potential \[Eq. (\[eqAO\])\] with $q=0.1$ [@ashton2011]. He found that his simulation result for the critical reservoir fraction $\phi_p$ was $0.249$, very close to the value $0.248$ given by the $B_2^{*AHS}$ criterion. We also considered the simulation results of Fortini et.al. [@fortini2008] for the AO model with $q=0.15$. In this case the critical value of $\phi_p$ is $\sim 0.316$ which is again close to the value $0.313$ from the $B_2^{*AHS}$ criterion. Since the size-ratios we consider are not vastly larger than those considered above, we chose to estimate the critical value of $\phi_p$ by calculating $B_2^*$ for the AO potential \[Eq. (\[eqAO\])\] and employing the $B_2^{*AHS}$ criterion. In order to estimate the critical colloid fraction we used the mapping to the square-well potential proposed by Noro and Frenkel [@noro2000] to obtain an effective range. For the three temperatures, i.e. the three $q$ values, that we consider the estimate of the critical colloid fraction is about $0.27$ which happens to be equal to the AHS value [@miller2003].
It is important to note that the simulations of the AO model, and of other models with short-ranged attraction, report broad, in $\phi_c$, gas-liquid coexistence curves extending to large values of $\phi_c$ and it is clear that extracting an accurate value for the critical colloid fraction can be difficult. Our resulting estimates of the AO model critical points are shown as open squares in Fig. \[figPdRoom\] and Fig. \[figPdTemper\] where we also sketch putative spinodals. Since our estimates are based on empirical recipes we remark that using the slightly higher value of $-1.174$ for the critical value of $B_2^{*AHS}$ ,reported by Largo et al. [@largo2008], makes no discernible difference in our plots. Note also that their revised value of the critical packing fraction for AHS is $0.29$, only slightly bigger than the earlier result for AHS. Moreover were we to employ the original estimate for the critical value of $B_2^*$, namely $-1.5$, we find this results in only minor changes (about 4%) to the critical value of $\phi_p$, for the size-ratios relevant for our systems.
Results {#sectionResults}
=======
![(color online) Phase diagram at room temperature in the colloid volume fraction $ \phi_c $ and reservoir polymer number density $\phi_p$ plane. Symbols are experimental data; circles are fluid (F), triangles are gels and squares are crystals (X). Shaded area marked x$^*$ denotes samples which crystallised on the experimental timescale: hatched squares were intially gels, and hatched circles were initially fluids. The data are compared with theoretical predictions for the AO model with $q=0.214$ from free volume theory for fluid-solid coexistence (solid lines) [@lekkerkerker1992] and for the liquid-gas spinodal (short-dashed line) (Eq. \[schmidtSpin\]). The unfilled square is the AO critical point according to our $B_2^{*AHS}$ criterion and the long dashed line is a sketch of the accompanying spinodal. (a)-(c) are confocal microscopy images of a gel (a), fluid (b) and (hard sphere) crystal (c) at states in the phase diagram shown in the main panel. Bars=20 $\mu m$. []{data-label="figPdRoom"}](figPdRoom){width="140mm"}
Room-temperature behaviour {#sectionRoom}
--------------------------
We begin our presentation of results with the phase diagram at ambient conditions as shown in Fig. \[figPdRoom\]. Throughout we work in the colloid volume fraction $\phi_{c}$ and reservoir polymer number density $\rho_{p}$ plane. We calculate $\rho_{p}$ following the free-volume prescription for the AO model [@lekkerkerker1992] : $\rho_{p}=\rho_{p}^{exp}/\alpha$, where $\rho_{p}^{exp}$ is the experimental value for the polymer number density and $\alpha$ is the free-volume fraction entering Eq. (\[schmidtSpin\]). In the free-volume approximation, $\alpha$ depends on $q$ and $\phi_c$ only [@dijkstra1999]. For a size-ratio $q \sim 0.2$ and colloid volume fractions up to $\phi_c \sim 0.4$ we expect this approximation to be accurate. Our choice of representation is motivated by two considerations : firstly, polymer number density is conserved during heating and cooling (while polymer “volume fraction” emphatically is not), and secondly, the reservoir representation permits easy visual comparison with theoretical spinodal lines and critical points.
We calculate theoretical phase boundaries as outlined in Section \[sectionMapping\]: fluid-solid coexistence is determined using free volume theory [@lekkerkerker1992] and the metastable fluid-fluid critical point is estimated using the $B_2^{*AHS}$ criterion. Experimentally, along the hard sphere line $\phi_p=0$ ($x$-axis in Fig. \[figPdRoom\]) we find hard sphere crystallisation for $\phi_{c}>0.54$. However, upon addition of polymer, we found fluid states around the AO fluid-solid phase boundary and gelation at higher polymer concentration. Only around the estimated AO critical point was a pocket of states found which crystallised on the experimental timescale of 6 days. This is the shaded region in Fig. \[figPdRoom\].
We already remarked that the fluid-gelation boundary has been identified with the fluid-fluid spinodal [@verhaegh1997; @lu2008]. In Fig. \[figPdRoom\] we plot the spinodal calculated from free-volume theory , i.e. Eq. (\[schmidtSpin\]), and a sketch of where the spinodal should be located for the AO model, based on our $B_2^{*AHS}$ criterion for the critical point. As mentioned earlier, free-volume theory grossly overestimates the colloid critical fraction for this size-ratio $q =0.214$ thus in making comparison with experiment it is appropriate to focus on the $B_2^{*AHS}$ result. We observe that the states identified as gels (triangles) all lie within the (putative) AO spinodal. Below the estimated AO fluid-fluid critical point we find only fluid states. We may conclude that the AO critical point provides an excellent indicator of the location of the experimental transition between fluid and gel states at this temperature.
Critical enhancement of crystallisation {#sectionXtal}
---------------------------------------
![(color online) Crystallisation in colloid-polymer mixtures at room temperature. (a) immediately after preparation, (b) after 1 day. Bars=$20$ $\mu m$. (c) Crystallisation times in units of the Brownian time $\tau_B$ as a function of polymer number density . Vertical line is the critical polymer number density estimated for the AO model, with $q=0.214$, from the $B_2^{*AHS}$ criterion. Dashed line is a guide to the eye.[]{data-label="figXtal"}](figXtal){width="140mm"}
Crystallisation of colloid-polymer mixures is described in Fig. \[figXtal\]. The crystals formed in the shaded region of Fig. \[figPdRoom\] are markedly different from those formed in the absence of polymer in that the fluid is at a much lower colloid packing fraction. Crystallisation times vary from 1 to 6 days. We observe crystallisation only in a small “pocket” around the critical polymer number density $\rho_p^c \sigma^3 \sim 72$.
Some samples which crystallised were fluids prior to freezing (hatched circles in Fig. \[figPdRoom\]), while some were gels (hatched squares in Fig. \[figPdRoom\]). That crystallisation is found to occur in the neighbourhood of the (metastable) critical point predicted by our $B_2^{*AHS}$ criterion is remarkable and we explore this aspect further in Fig. \[figXtal\](c) where we plot the crystallisation time as a function of polymer concentration. The crystallisation time $\tau_X$ is the time at which more than 50 % of the sample had crystallised. As seen in Fig. \[figXtal\](b), crystallisation is rather clear. The unit of time employed is the Brownian time, defined as $\tau_B=\pi \eta \sigma^3 / k_B T$, where $\eta$ is the viscosity and is equal to $0.42$ $s$ for this system. We find that in the (metastable) one-phase fluid region, moving further from criticality, the crystallisation time increases rapidly and takes values outside the experimental time-window. This is consistent with the two step nucleation scenario of ten Wolde and Frenkel [@tenWolde1997].
Response to temperature {#sectionTemper}
-----------------------
![(color online) The fluid-gel transition at different temperatures. (a-c) shows a typical experiment where the transition temperature is found by “quenching” a colloid-polymer mixture by heating. Here $\phi_c=0.107$ and $\rho_p=116 \sigma^{-3}$. (a) $8.5^\circ$C, fluid, (b) $10.2^\circ$C condensing, (c) $11.8^\circ$C, gel. The transition is then identified as around $10^\circ$C. In the main panel, experimental data indicate fluid-gel transitions at $10^\circ$C (blue squares), $11^\circ$C (turquoise up triangles), $12^\circ$C (cyan down triangles) and $16^\circ$C (green circles). As in Fig. \[figPdRoom\], AO free volume theory fluid-crystal coexistence lines are solid, and critical points (unfilled squares) follow our $B_2^{*AHS}$ criterion with accompanying sketched spinodal (long dashed lines). These are shown for room temperature ($25^\circ$C), $15^{\circ}$C and $10^{\circ}$C as indicated, corresponding to $q=0.214$, $0.197$ and $0.176$, respectively. Bars=20 $\mu m$.[]{data-label="figPdTemper"}](figPdTemper){width="140mm"}
We now consider the effect of temperature on our system. The results are given in Fig. \[figPdTemper\]. Using the temperature stage we cool the system to around $10^{\circ}$C. The images in Figs. \[figPdTemper\](a-c) show the effect of then gently heating the system. A metastable fluid (a) condenses (b) and finally forms a gel (c). The phase diagrams shown in the main panel pertain to the AO model and are obtained using the same prescriptions as in Fig. \[figPdRoom\]. It is assumed that the only effect of temperature $T$ is to change the radius of gyration according to Eq. \[eqExpansion\] and we calculate the size ratio using $q=2R_G(T)/\sigma$. We find $q= 0.214$, $0.197$ and $0.176$ for $T= 25$ $^\circ$C, $15$ $^\circ$C and $10$ $^\circ$C, respectively. Within the context of the free volume approximation the fluid-solid coexistence lines in the $\phi_c-\phi_p$ plane change little over this range of $q$ and we fixed these to be the lines for $q=0.18$. It is the scaling with $(\sigma/\sigma_p)^3$, from the polymer volume fraction to the polymer reservoir density, that gives rise to the variation shown in the figure. Although the experimental data show considerable scatter, which we attribute predominantly to sedimentation effects, there is reasonable overall agreement with the theoretical predictions. Specifically we find that a transition from a fluid to a gel as illustrated in Fig. \[figPdTemper\] (a-c) occurs at temperatures broadly consistent with the location of the spinodals as predicted by our $B_2^{*AHS}$ criterion. It appears that the assumption of ideal polymer behaviour is a reasonable first step to treating the temperature response of colloid-polymer mixtures.
Conclusions {#sectionConclusions}
===========
We have examined the room-temperature behaviour of a colloid-polymer mixture and its response to temperature quenches. To the best of our knowledge, these are the first particle-resolved studies of the latter. At ambient conditions, the fluid-gel transition is well described by an estimate of the spinodal based on a second virial coefficient criterion for the effective one-component Asakura-Oosawa model. Although free-volume theory provides a reasonable description of fluid-crystal coexistence, we emphasize that this approximation predicts a spinodal which lies at unphysically large colloid volume fractions.
The response to temperature is consistent with our assumptions that the polymers can be treated as ideal, and that the radius of gyration follows a simple fit to experimental results [@berry1966] \[Eq. (\[eqExpansion\])\]. This opens the way to 3D particle-resolved studies of a variety of phenomena related to systems with attractive interactions which are tuneable *in-situ*.
We find crystallisation on observable timescales close to the metastable critical point predicted by our $B_2^{*AHS}$ criterion. Assuming the mapping we have carried out is accurate, we find crystallisation in the metastable one-phase region, on a timescale which increases further from criticality, in addition to the metastable two-phase region. This contrasts with results from Brownian dynamics simulations where crystallisation was found only in the metastable gas-liquid two-phase region [@fortini2008; @royall2012]. We attribute this to the very mch longer timescale accessed in the experiments.
Concerning the lack of crystallisation in the fluid-crystal coexistence regions, a question arises in the apparent discrepancy between our results and Ilett *et al.* [@ilett1995], who found a closer agreement with the prediction of the fluid-crystal boundary from free volume theory in a comparable system (with size ratio $q=0.08$). However in their case, the colloid diameter was $\sigma\approx400$ nm, compared to $\sigma\approx1080$ nm here . This has drastic consequences for the dynamics of the system, as the time for a colloid to freely diffuse over its own diameter scales with the cube of the particle size. Thus the effective timescales are around 20 times less in this work. Typical crystallisation times were around 6 hours in their case. This corresponds to 120 hours for our systems, which is far beyond the experimental limits imposed by sedimentation of around one hour. Observations such as this underline the challenges for self-assembly in this size range and serve to emphasise the very dependence of timescales in these systems upon colloid size.
**Acknowledgments** SLT and CPR acknowledge the Royal Society for financial support and EPSRC grant code EP/H022333/1 for provision of the confocal microscope used in this work. Gregory N. Smith is kindly acknowledged for preliminary experiments. It is a pleasure to thank Daan Frenkel and Richard Sear for many stimulating discussions.
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---
abstract: 'The paper describes a time- and angle-resolved photoemission apparatus consisting of a hemispherical analyzer and a pulsed laser source. We demonstrate 1.48-eV pump and 5.90-eV probe measurements at the $\geq$10.5-meV and $\geq$240-fs resolutions by use of fairly monochromatic 170-fs pulses delivered from a regeneratively amplified Ti:sapphire laser system operating typically at 250 kHz. The apparatus is capable to resolve the optically filled superconducting peak in the unoccupied states of a cuprate superconductor, Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. A dataset recorded on Bi(111) surface is also presented. Technical descriptions include the followings: A simple procedure to fine-tune the spatio-temporal overlap of the pump-and-probe beams and their diameters; achieving a long-term stability of the system that enables a normalization-free dataset acquisition; changing the repetition rate by utilizing acoustic optical modulator and frequency-division circuit.'
author:
- 'Y. Ishida'
- 'T. Togashi'
- 'K. Yamamoto'
- 'M. Tanaka'
- 'T. Kiss'
- 'T. Otsu'
- 'Y. Kobayashi'
- 'S. Shin'
title: 'Time-resolved photoemission apparatus achieving sub-20-meV energy resolution and high stability'
---
Introduction {#Intro}
============
Angle-resolved photoemission spectroscopy (ARPES) is a powerful method to investigate the electronic structures of matter. In ARPES, monochromatic light is shined on a crystal. Photoelectrons emitted therefrom has a pattern in kinetic energy and emission angle that replicates the electronic band dispersions. Continuous development is being pursued in ARPES driven by the demands for investigating the fine electronic structures of complex materials such as superconducting copper oxides and heavy-fermion compounds [@RMP_ARPES]. Electron analyzers, both time-of-flight and hemispherical deflection types, have improved their energy-and-angular resolutions and throughputs by virtue of the multi-channel detection and precise control in the electron-lens ($e$-lens) [@Scienta_MultiChannel94; @Wannberg09; @ScientaTOF13]. Light source for photoemission has also been developing considerably. There is a trend to utilize lasers, as they can deliver bright, coherent, polarized, monochromatic, or ultra-short stroboscopic light. After the pioneering implementations of pulsed lasers into photoemission spectroscopy [@Haight_InP; @Haight_RSI88; @Haight_RSI94; @Haight_SurfSciRep], investigations of ultrafast and out-of-equilibrium phenomena became possible in the femtoseconds [@PetekOgawa; @Bauer_PRL01; @Perfetti_TaS2; @Perfetti_Bi2212; @Schmitt; @Bovensiepen_Rev; @Gedik_Science] and into the attoseconds [@Cavalieri; @PhotoDelay; @Krausz_Rev]. Meanwhile, 70-$\mu$eV energy resolution was achieved by utilizing the monochromaticity of the lasers [@Okazaki].
In an ultrafast pump-and-probe method employing a pulsed laser source, a laser pulse is split into two. One pulse (pump) initiates dynamics in the sample and the other pulse (probe) snapshots the non-equilibrated state. By varying the difference in the optical-path lengths of the pump and probe pulses, the recovery from the impact can be followed in the time domain. In the case for pump-and-probe-type time-resolved photoemission spectroscopy (TRPES), the photon energy of the probe pulse has to exceed the work function of the sample typically being $\sim$5 eV.
In TRPES, there are two methods, besides using free-electron lasers [@PRL10_TaS2_Flash; @NewJPhys_FLASH12], for generating probe pulses in the deep-to-extreme ultraviolet region; namely, high-harmonic generation in gaseous media [@Corkum; @Kulander; @Heinzmann_RSI01; @Bauer_Rev] and frequency up-conversion by using nonlinear crystals [@Lisowski04; @Carpene_RSI09; @Perfetti_RSI12; @Lanzara_TrPES_RSI12]. The former has the capability to generate high-energy pulses, which opens access to full valence band [@Miaja-Avila], shallow core levels [@Heinzmann_GaAs; @Ishizaka_TaS2], and wide momentum ($k$) space [@TiS2e_Nature11]. Achieving better time resolution is also straightforward in this method [@Krausz_Rev], because the pulse duration shortens for higher harmonics. However, a drawback is in the energy resolution: Due to the uncertainty principle, monochromaticity and short duration of a pulse are incompatible. 90-meV energy resolution is demonstrated by using the 35.6-eV high harmonics at the overall time resolution of 125 fs [@RSI13_HHG_Frietsch]. The latter method is limited to generating pulses up to $\sim$6 eV, because the nonlinear crystals lose transparency at higher energies. Nonetheless, it is suited for measurements aiming at high energy resolution and high signal-to-noise ratio (S/N) in the dataset. The latter owes to the fact that the probing pulses can be up-converted from moderately strong $\lesssim$1-$\mu$J pulses, and hence, at a high repetition rate of $>$100 kHz. Sub-30-meV time-resolved ARPES (TARPES) studies have been accessing into the carrier dynamics across the Fermi level ($E_F$) [@Lanzara_TrARPES_NPhys; @Lanzara_Science; @Sobota_PRL12], while sub-20-meV measurements also became feasible [@Ishida_HOPG; @KimPRL].
This paper describes the specification of a TARPES apparatus consisting of a hemispherical analyzer and a pulsed laser source. The design concept was to achieve a high energy resolution capable for detecting ultrafast responses of the fine electronic structures of complex materials. To this end, we adopted a laser source delivering fairly monochromatic, and hence, not too short pulses of 170-fs duration at a typical repetition rate of 250 kHz; namely, a regeneratively amplified Ti:sapphire laser system. For the generation of the probe pulses, we used nonlinear crystals.
We also describe technical points for the sake of TARPES to be a precise, convenient, and hence, a versatile method for materials investigation. First of all, compared to ARPES, TARPES has another dimension of time in the dataset. In addition, the flux of the pulsed probe has to be reduced in order to minimize the space-charge broadening effect [@PasslackAeschlimann_SpaceCharge; @Zhou_SpaceMirror]. These characteristics of TARPES make the acquisition time of the dataset longer than that in ARPES, in general. It was therefore a mandatory to achieve a long-term stability of the apparatus in order to obtain a high S/N dataset recorded under the reduced probing flux. Second, two beams, pump and probe, have to be manipulated in TARPES, which is contrasted to the single-beam manipulation in ARPES. Moreover, the manipulation has to be done in both space and time at the focal point of the $e$-lens in the ultra-high vacuum. We sought for simple methods that allow precise manipulation and characterization of the pump and probe beams.
The paper is organized as follows. After the present introduction (Sec. \[Intro\]), we describe the TARPES setup in Sec. \[Setup\]. In this section, we first sketch the TARPES system (\[layout\]), and then describe the procedures to improve the time resolution (\[TimeResolution\]) and to align the pump and probe beams to the focal point of the $e$-lens by use of some devices (\[align\]). A method to cross-correlate the pump and probe pulses in space and time is explicated in \[Graphite\], in which we use pump-and-probe photoemission signal of graphite [@Ishida_HOPG]. Description on the space-charge effect and stability of the TARPES system are presented in \[SpaceCharge\] and \[Stability\], respectively. In \[DelayScan\], we describe a unique data acquisition procedure in TARPES that becomes effective after achieving the long-term stability of the system. An optional device that varies the repetition rate is explicated in \[RepRate\]. In Sec. \[Performance\], we describe the performance of the TARPES system by presenting the dataset recorded on a cuprate superconductor in a high-energy resolution mode (\[Bi2212\]) and on Bi(111) surface during a prolonged acquisition (\[Bi\]). Finally, we conclude the paper in Sec. \[Conclusions\].
Experimental setup {#Setup}
==================
TARPES layout {#layout}
-------------
The general layout of the TARPES setup is sketched in Figure \[fig\_setup\](a). A Ti:sapphire laser system (Coherent RegA9000) repetitively generates 170-fs laser pulses with center photon energy at 1.48 eV (840 nm). The repetition rate $\Omega$ is tunable in the 60-300 kHz range, and set typically to 250 kHz. Using type I non-linear second-harmonic generation at two $\beta$-BaB$_2$O$_4$ (BBO) crystals, a portion of the laser pulse is up-converted into a 5.90 eV (210 nm) probing pulse. The first and second BBOs are 0.3- and 0.1-mm thick, respectively. The pump and probe beams are split by a harmonic separator (HS) located after the first BBO. A translational delay stage varies the optical-path length of the pump beam line. The pump and probe beams are finally joined at a dichroic mirror (DM) just before entering the ultra-high vacuum chamber (base pressure $<$5$\times$10$^{-11}$ Torr) through a CaF$_2$ window. Photoelectrons are collected by a hemispherical analyzer (VG Scienta R4000) equipped with an $e$-lens and a multi-channel-plate (MCP) detector.
![\[fig\_setup\] TARPES layout. (a) Schematic view of the TARPES system consisting of a pulsed laser source, pump and probe beam lines, and a hemispherical electron analyzer. The laser system consists of a Ti:sapphire regenerative amplifier (Coherent RegA 9000) seeded by a mode-locked Ti:sapphire oscillator lasing at 840 nm (Coherent Mira). Ti:sapphire crystals of the amplifier and oscillator are pumped by a Nd:YVO$_4$ laser system (Coherent Verdi18) that delivers 532-nm beam up to 18 W. (b) A sensor device simultaneously visualizing the infrared pump and ultraviolet probe beams. Snapshots taken before (b1) and after (b2) overlapping the pump and probe beams. ](Fig1.eps){width="8cm"}
The intensity of the pump beam is controlled by a combination of a polarizer (POL) and a $\lambda$/2 wave plate (WP1). By rotating the wave plate, we vary the amount of the polarization component that is passed though by the subsequent polarizer. The probe beam intensity is controlled through a similar mechanism: A $\lambda$/2 wave plate (WP2) regulates the polarization component that is passed through by subsequent Breuster prism (BP1). The wave plates WP1 and WP2 are mounted on rotary stages that are controlled from a personal computer (PC).
Improving the time resolution {#TimeResolution}
-----------------------------
A light pulse elongates after traveling through optical elements and air that act as dispersive media. The group-velocity dispersion (GVD) has to be either minimized or compensated in order to increase the time resolution in TARPES. GVD for a $\sim$200-fs pulse becomes pronounced at wavelength $\lambda$=200-400 nm. Particular care thus has to be taken in the probe beam line.
We inserted a prism-pair compressor [@PrismCompressor] in the probe beam line \[Figure \[fig\_setup\](a)\]: A pair of Breuster prisms, BP1 and BP2, are positioned between the first and second BBOs at the angle of minimum deviation. By installing the prism-pair compressor, the time resolution of TARPES improved from $\sim$400 to $\gtrsim$240 fs, as described later in Sec. \[Graphite\]. We have also kept the optical-path length of the 210 nm probe in air (from the second BBO to the CaF$_2$ window) to be less than 80 cm, because the GVD becomes pronounced when $\lambda$ approaches the absorption edge of air at $\sim$205 nm. GVD in the pump beam line was negligibly small, which was confirmed by measuring the duration of the pump pulse just before the CaF$_2$ window by a scanning-type autocorrelator (A$\cdot$P$\cdot$E Mini).
Optimization of the prism-pair compressor is done by shifting BP2 into the beam through a two-step process. First, the amount of the BP2 insertion is determined by maximizing the photoemission count rate, which is a good measure of the up-conversion efficiency at the second BBO, and hence the shortness of the pulse. In the second step, we maximize the contrast of the pump-and-probe signal of graphite in the unoccupied side (see later in Sec. \[Graphite\]). This ensures that the pump duration is optimized for the TARPES measurements. The time resolution of $<$300 fs was easily achieved just by undergoing the first step.
Alignment of the pump and probe beams {#align}
-------------------------------------
In a pump-and-probe method, the pump and probe beams are usually overlapped on the sample with the pump beam size slightly larger than that of the probe. In the case for TARPES using a hemispherical analyzer, another stringent condition exists: The spatial overlap of the pump and probe beams has to be done at the focal point of the $e$-lens, which is located $\sim$35 mm from the $e$-lens aperture in the ultra-high vacuum chamber. We here describe a procedure to direct the pump and probe beams to the focal point of the $e$-lens at the desired beam sizes. The key items are a sensor device \[Figure \[fig\_setup\](b)\], an XYZ stage \[Figure \[fig\_setup\](a)\], a microscope [@Muro], and a pin-hole device (Figure \[fig\_Pinhole\]).
First, we coarsely align the pump and probe beams co-linearly before the CaF$_2$ window in air. The alignment can be done conveniently by using a sensor device that visualizes the infrared (IR) and ultraviolet (UV) beams simultaneously. The sensor device is composed of an IR sensing card attached with UV fluorescence powder by a double-sided adhesive tape, as shown in Figure \[fig\_setup\](b). The UV probe glows on the surface of the tape, while the IR pump penetrates through the tape and glows on the card. Coarse focusing of the pump and probe beams is also done at this point by varying the translational stages of the f1000 and f700 lenses that focus the pump and probe beams, respectively.
Next, we shift the focal point of the $e$-lens into the coarsely-aligned beam. This can be done because the entire analyzer chamber is on a XYZ stage that is separated from the laser table. The alignment using the XYZ stage is done by centering and sharpening the photoelectron image on the MCP detector. The use of a long-working-distance microscope employing K2/S lens of Infinity Photo-Optical Co. [@Muro] is helpful to know the beam position in the analyzer chamber. The microscope is viewed by a CCD camera, which has sensitivity to the pump beam in the IR. Therefore, after the co-linear alignment in air, as described previously, the pump beam seen through the CCD on sample is also the locus of the probe beam. The advantage of using the XYZ stage is that the alignment can be done without losing the co-linearity of the pump-and-probe beam. Without the XYZ stage, pump and probe beams have to be moved one by one, during which the locus of the invisible UV probe beam is lost in the CCD image.
![\[fig\_Pinhole\] Setup around the sample. A layered BSTS sample [@KimPRL] to be cleaved by a Scotch-tape method is on a copper plate that is screwed to the cryogenic manipulator by M2 titan bolts. Graphite (already cleaved) is also attached next to the sample and is used as a reference. A pin hole device is also installed. $E_F$ is calibrated by recording the Fermi cutoff of evaporated Au in electrical contact to the sample and the analyzer. ](Fig2.eps){width="5.5cm"}
We next optimize the beam sizes by utilizing a pin hole drilled on a Cu plate attached to the sample mount; see Figure \[fig\_Pinhole\]. The advantage of the method presented below is that the tuning and estimation of the beam sizes are done at the focal point of the $e$-lens.
The probe beam size is estimated as follows: Similarly to doing photoemission spectroscopy on a small sample, we search the pin hole of 200 $\mu$m in diameter by monitoring the photoelectron count rate, which becomes minimal when the pin hole is properly aligned to the focal point of the $e$-lens. Finiteness in the beam size, or the tail of the Gaussian-like beam, generates photoelectrons from the copper plate surrounding the hole. From the amount of the loss of the photoelectron count rate on the 200-$\mu$m pin hole, we can estimate the beam size of the probe. Tightening or loosening of the focus on the pin hole is done by monitoring the loss of the count rate from the pin-hole device.
After the above procedure, the following three are spatially matched; the focal point of the $e$-lens, the pin hole, and the properly-sized probing beam. Under this configuration, we finally fine-tune the alignment of the pump beam and its size. This is done by maximizing the pump-beam intensity $I$ passing through the pin hole. The intensity $I$ is monitored by a power meter set after the viewing port behind the sample; see Figure \[fig\_setup\]. Note, the standard glass viewing port transmits the pump beam but blocks the probe beam; the latter can be seen on the glass as a florescence spot, which is also useful for visualizing the probe beam in the ultraviolet. The pump beam size is estimated from $I/I_0$, where $I_0$ is the intensity of the pump beam, or the read of the power meter when the pin-hole device is fully retracted from the optical path. By tuning the f700 focusing lens, and hence $I$, we optimize the pump beam size on the pin hole, which is identical to the focal point of the $e$-lens.
Cross correlating the pump and probe pulses by using graphite {#Graphite}
-------------------------------------------------------------
We find highly-oriented pyrolytic graphite (HOPG) as a useful cross correlator of the pump and probe pulses [@Ishida_HOPG]. Calibration of the origin of the pump-and-probe delay $t_0$, fine-tuning of the spatial overlap of the pump and probe beams, and estimation of the time resolution $\varDelta t$ are conveniently done by using the pump-and-probe photoemission signal of graphite at the focal point of the $e$-lens. Graphite can be easily cleaved and the exposed surface is stable over weeks in the vacuum chamber. These characteristics also facilitate us to use graphite as a reference sample. We cleave graphite by attaching Scotch tape on surface and peel it off in the ultra-high vacuum at room temperature; see Figure \[fig\_Pinhole\]: To our experience, Scotch tape is compatible to ultra-high vacuum unless baked.
![\[fig\_HOPG\] Pump-and-probe signal of graphite as a cross correlator. (a) Real-time photoemission signal displayed on the PC screen. White band in the MCP image is the photoemission signal dispersed horizontally in energy. When the spatio-temporal overlap of the pump and probe pulses is small (a1), photoelectron signal appears only in the occupied side. When the overlap becomes large (a2), the signal also appears in the unoccupied side. Fine tuning of the spatial overlap of the pump beam to the probe beam is done by maximizing the signal in the unoccupied side. (b) Determination of $t_0$ and time resolution $\varDelta t$. Pump-and-probe photoelectron signal around 7-eV kinetic energy, or 1.1 eV above $E_F$, is mapped during the delay-stage scan around $t_0$ (left panel). Right panel shows the integrated photoelectron intensity as a function of the delay-stage distance, or delay time. Gaussian fitting gives the locus of $t_0$ and $\varDelta t$=237$\pm$6 fs. ](Fig3.eps){width="8.0cm"}
When pumped by a 1.48-eV pulse, graphite exhibits photoemission signal in the unoccupied side that is strong enough to be observed in real time. Figure \[fig\_HOPG\](a) shows the PC screens displaying the photoemission image on the MCP detector. When the temporal overlap of the pump and probe pulses are small, the photoemission signal appears only in the occupied side (a1). As the delay stage is scanned, photoemission signal in the unoccupied side appears when the temporal overlap becomes large (a2). Final fine tuning of the spatial overlap of the pump beam to the probe beam is also done at this point by maximizing the signal in the unoccupied side around $t_0$. Pulse compression by the prism pair (the second step described in Sec. \[TimeResolution\]) is also done by monitoring the contrast between the on-$t_0$ (a2) and off-$t_0$ (a1) images.
The time resolution and origin of the delay are determined by using a resolution-limited fast response of graphite that appears at $\sim$1 eV above the Fermi level [@Ishida_HOPG]. The left panel in Figure \[fig\_HOPG\](b) shows the map of the photoemission signal from graphite around 7-eV kinetic energy, or 1.1 eV above the Fermi level, during the delay-stage scan. Signals of photoelectrons appear when the temporal overlap of the pump and probe pulses becomes large. Integrated intensity in the kinetic-energy region \[6.92, 7.08 eV\] is shown in the right panel in Figure \[fig\_HOPG\](b). By fitting the profile with a Gaussian, we obtain the time resolution and the locus of $t_0$: In the case shown in Figure \[fig\_HOPG\](b), full width at half maximum (FWHM) of the Gaussian is 240 fs, which is the typical time resolution after the tuning. The acquisition time was 10 min for the dataset presented in Figure \[fig\_HOPG\](b). The repetitive scanning of the delay stage was done (described in Sec. \[DelayScan\]) in a fixed kinetic-energy mode with the pass energy ($E_{\mbox{\it{pass}}}$) and slit width set to 5 eV and 0.8 mm, respectively.
Space-charge effect and its reduction {#SpaceCharge}
-------------------------------------
In TARPES implemented by the pump-and-probe method, the flux of the probe beam cannot be increased as much as desired because of the space-charge effect [@PasslackAeschlimann_SpaceCharge; @Zhou_SpaceMirror]. A too intense probing pulse generates a bunch of photoelectrons that repel each other through the Coulomb effect; thereby distorting the energy-and-angular distribution of the photoelectrons; see the schematic shown in Figure \[fig\_SpaceCharge\]. The space-charge effect has to be minimized by reducing the probe flux when fine structures such as superconducting gaps and small surface photo-voltage-induced shift of the spectra are of interest.
![\[fig\_SpaceCharge\] Space-charge broadening. (a) Fermi cutoff of Au recorded at various flux of the probe beam. (b) Shift of the Fermi cutoff of Au plotted as a function of the photoemission count rate, which is a measure of the probe flux. (c) Valence-band spectra of SmB$_6$ recorded at various probe flux. The schematic figure shows that a too intense probing pulse generates a bunch of photoelectrons that repel each other through the Coulomb effect. The spectra shown in (a) and (c) are arbitrarily normalized in intensity. ](Fig4.eps){width="8cm"}
Figure \[fig\_SpaceCharge\](a) shows the Fermi cutoff of gold (Au) recorded at various flux of the probe beam. As the flux is increased, the leading edge of the spectrum is shifted into higher kinetic energy due to the space-charge effect. The locus of the leading edge is plotted in Figure \[fig\_SpaceCharge\](b) as a function of the photoemission count rate, which is a good measure of the flux of the probe beam. For reducing the leading edge shift from 4 meV to $<$1 meV, the intensity of the probe has to be reduced for one order of magnitude.
The magnitude of the space-charge effect depends on materials. Therefore, the flux of the probe has to be determined for each sample before the measurement. The case for SmB$_6$ is shown in Figure \[fig\_SpaceCharge\](c), in which valence-band spectra recorded at various probing flux are displayed. As the flux is increased, the leading edge is shifted into higher kinetic energy. In addition, the spectral shape is deformed. These are due to the space-charge effect, while some non-linearity of the MCP to the photoemission count rate may have some effect at high count rates. The space-charge effect is judged to be negligibly small for the spectrum recorded at the lowest flux shown in Figure \[fig\_SpaceCharge\](c), because the locus of the leading edge and the spectral shape hardly changed from that recorded at the second lowest flux; see the spectra recorded at the count rates of 5.6 and 10 in arbitrary units. The unchanged spectral shape also ensures that the linearity of the MCP is fairly good when the photoemission count rate is sufficiently reduced.
Stabilizing the TARPES system {#Stability}
-----------------------------
In order to achieve good S/N in the TARPES dataset, the apparatus has to operate stably during the prolonged acquisition time. Particular care had to be taken not only in the laser power but also in the the sample position during the acquisition, as we describe below. The stability of the system was also important to realize a unique TARPES dataset acquisition scheme, which we describe in Sec. \[DelayScan\].
![\[fig\_stable\] Stability of the TARPES system. (a) A snapshot during TARPES measurement at low temperatures. Mechanical supporting is done around the He return line (left bottom) where ice is developing. The vinyl sheet housing the laser source is seen in the back side. Seen in the front side is the vacuum chamber and hemispherical analyzer, which are on the XYZ stage \[Figure \[fig\_setup\](a)\]. (b) Logging of the laser power and temperature. The laser reflected from POL in the pump beam line \[Figure \[fig\_setup\](a)\] was monitored by a pyrometer-type power meter. Temperature was simultaneously monitored near the oscillator in the vinyl house. The minimal step of the reads of the digital power meter and thermometer were 1 mW and 0.1$^{\circ}$C, respectively. ](Fig5.eps){width="8cm"}
The laser system is in a temperature-controlled environment \[Figure \[fig\_stable\](a)\]. It is housed by a vinyl sheet that prevents the system from being subjected to the flow of air; the vinyl house is in a room whose temperature is stabilized by an air conditioner equipped with a PID-controlled heater. Figure \[fig\_stable\](b) shows the temperature monitored on the laser table in the vinyl house. During the $\sim$12-hours acquisition, the read was stable at 26.8$^{\circ}$C, while fluctuations to 26.9$^{\circ}$C sometimes occurred. The laser intensity monitored simultaneously was virtually stable at 449 mW, as shown in Figure \[fig\_stable\](b).
The sample position should not move during the data acquisition. This becomes of particular importance not only because of the prolonged acquisition time, but also because the focal point of the $e$-lens becomes tight when analyzing the low-kinetic-energy photoelectrons generated by the 5.90 eV probe, which is rather low in photon energy for a photoemission light source. In fact, when the working distance (distance between the sample and the aperture of the $e$-lens) is varied for 100 $\mu$m, slight changes can be seen in the spectral shape.
Mechanical tightening of the cryostat that holds the sample is found to be effective; see Figure \[fig\_stable\](a). Here, not only the cryostat but also the return line of the vaporized He gas is also supported mechanically. After properly supporting the cryostat, the sample position does not move for 100 $\mu$m over at least 12 hours, as judged from the spectral shape. This is also true even when the He return line gains the weight of ice during the low-temperature measurement. Ice developing on the He return line can be seen in Figure \[fig\_stable\](a).
The flow rate of liquid He should also be kept as constant as possible. We keep the pressure in the liquid He vessel to 0.5 MPa by using a pressure regulator attached to the vessel. $T$=10 and 3.5 K can be maintained stably for at least 7 and 3 days, respectively, when using a 250-L liquid He vessel. We do not pump out the He gas from the return line in order to avoid the vibration of the pump to be transmitted to the sample at the focal point of the $e$-lens.
Data acquisition through repetitive scanning of the delay stage {#DelayScan}
---------------------------------------------------------------
The delay stage driver and the ARPES data acquisition software are linked. Therefore, a TARPES dataset can be acquired in a automated scan. After achieving the long-term stability of the TARPES system (Sec. \[Stability\]), the following dataset-acquisition scheme became very effective: A TARPES sequence is designed such that the delay stage repetitively scans over the specified delay points. In this way, we can keep the same S/N in each TARPES image recorded at different delay points. The major advantage is that the intensity of the raw image recorded at each delay points is normalized to the acquisition time. That is, the spectral intensity recorded at different delays can be compared directly without any normalizations; see Sec. \[Bi\]. In general, when comparing the intensity of the ARPES data recorded at different conditions, there would be concerns such as the difference in the surface conditions, slight change in the sample position due to the thermal expansion of the cryostat, and so on. Therefore, when comparing the intensity of the two datasets recorded at different conditions, some kind of intensity normalization is usually undertaken.
Varying the repetition rate {#RepRate}
---------------------------
In a pump-and-probe measurement employing a mode-locked pulsed laser source, the sample is repetitively impinged by the pump pulses. When the pumped sample does not fully recover before the next pump arrives, the response accumulates to result in a so-called periodic steady state. One of the examples relevant to TRPES is a semiconductor surface exhibiting surface photo-voltage (SPV) effect [@Marsi; @Kamada]. SPV occurs as a result of the pump-induced variation of surface band bending developed on the edge of semiconductors. The duration of SPV usually exceeds micro-seconds, so that pump-induced shifts of the spectrum are observed at $t$$<$0 when the interval time of the pulses is shorter than the SPV duration. When investigating a periodic steady state in a pump-and-probe method, interests may appear in its interval-time dependence.
As an option, we can insert a device that varies the interval time, or the repetition rate. This is done by using an acoustic optical modulator (AOM) drove by 80-MHz carrier frequency (CF). When CF is applied to the AOM crystal (TeO$_2$), acoustic wave is generated which acts as a grating that diffracts out the laser passing through the crystal. By using an AOM driver composed of frequency division and timing microchips, we can synchronically gate the AOM to the amplifier output at a repetition rate of $\Omega/2^n$ ($n$=0, 1, ..., 6; $n$ can be set manually). The AOM device can be inserted just after the amplifier, and the diffracted pulses are sent into the pump and probe beam lines, so that TARPES is done at the $\Omega/2^n$ repetition rate.
The repetition rate dependency of the pump-induced shift of SmB$_6$ spectra, which we attribute to the SPV effect, is presented elsewhere [@SmB6].
Performance {#Performance}
===========
TARPES of Bi2212: Observation of the superconducting peak in the unoccupied side {#Bi2212}
--------------------------------------------------------------------------------
We present TARPES data of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212), a copper oxide superconductor that has a critical temperature $T_c$ of $\sim$92 K [@Mochiku]. The electronic structure of Bi2212 is quasi-two dimensional and has strong anisotropy in $k$ space [@RMP_ARPES]. The superconducting gap closes in the nodal direction (0,0)-($\pi$,$\pi$), since Bi2212 is a $d$-wave superconductor. In the anti-nodal direction, or around ($\pi$,0) in $k$ space, there is a pseudo-gap region, which is characterized by the persistence of a gap even above $T_c$ [@Norman_Nature98] and by a particle-hole asymmetric electronic structure around $E_F$ [@Hashimoto_NPhys10]. Between the nodal and anti-nodal directions, a particle-hole symmetric structure across $E_F$ is found below $T_c$, namely the Bogoliubov bands in the occupied and unoccupied sides separated by a superconducting gap [@WSLee_nature]. This near-nodal region recently gained renewed interest, whether the superconducting gap disappears at $T_c$ similarly to that in the BCS theory of superconductivity [@WSLee_nature] or whether it persists slightly above $T_c$, which may be in conformity to the precursor pairing picture [@Dessau_NaturePhys12; @Dessau_PRB]. For a recent review that includes the above discussions in the near-nodal region, see Ref. [@NPhys_ARPES].
![\[fig\_Bi2212\] TARPES of Bi2212. (a) ARPES image of Au around the Fermi cutoff recorded at 10 K. (b) Fermi edge of Au. A Gaussian convolved Fermi-Dirac distribution function is overlaid. The EDC was obtained by integrating the angular region of the ARPES image indicated in (a). (c) Quasi-particle dispersions of Bi2212 recorded before the pump at 10 K ($t$=-1.0 ps; c1), without pump at 70 K (c2), and after the pump at 10 K ($t$=2.6 ps; c3). (d) EDCs of Bi2212. EDCs of (d1)-(d3) were obtained from the corresponding images of (c1)-(c3), with the integration done in the angular region indicated in (c1). ](Fig6.eps){width="8cm"}
A high energy resolution is a prerequisite to TARPES when the superconducting gaps in the near-nodal regions become of interest, because the gap size becomes diminishingly small on approach to the node. We set the TARPES spectrometer in a high-energy resolution mode (slit width of 0.5 mm and $E_{\mbox{\it{pass}}}$ of 2 eV) and investigated the pump-induced changes of the quasi-particle dispersion of Bi2212 in the near-nodal region.
We first present in Figure \[fig\_Bi2212\](a) the ARPES image of Au around the Fermi cutoff recorded at $T$=10 K. The spectra were recorded prior to the Bi2212 measurement, and were used for energy calibration. We had set the probe intensity so that the space-charge induced shift was much less than 1 meV. The corresponding energy distribution curve (EDC) is displayed in Figure \[fig\_Bi2212\](b), which is the integral of the ARPES image in the angular region indicated by arrows in Figure \[fig\_Bi2212\](a). By fitting the EDC to a Fermi-Dirac distribution function convolved with a Gaussian, the energy resolution (FWHM of the Gaussian) was estimated to be 10.5 meV.
Figure \[fig\_Bi2212\](c1) displays the TARPES image of Bi2212 at $t$=-1.0 ps along a near-nodal cut in $k$ space indicated in the inset of (d). The sample was held at $T$=10 K, and the pump density per pulse was set to $p$=14 $\mu$J/cm$^{-2}$. The particle-hole symmetry across $E_F$ in this cut was confirmed by heating the sample up to 70 K and confirming the thermally populated superconducting peak in the unoccupied side; see the quasi-particle dispersion recorded at $T$=70 K in Figure \[fig\_Bi2212\](c2) and the corresponding EDC shown in (d2), which exhibits a peak at $\sim$10 meV above $E_F$. The TARPES image recorded at $t\,=$2.6 ps is shown in Figure \[fig\_Bi2212\](c3). A shoulder or a peak is observed at $\sim$10 meV in the corresponding EDC \[\[fig\_Bi2212\](d3)\], which is attributed to the pump-induced population of the superconducting peak in the upper Bogoliubov band. Such features were not reported in the previous TARPES studies done at the energy resolution of $>$20 meV [@Perfetti_Bi2212; @Lanzara_TrARPES_NPhys; @Bovensiepen; @Lanzara_Science; @Lanzara_NCom]. We thus succeed in the direct observation of the unoccupied side of the superconducting gap by TARPES in the high energy resolution mode.
All the Bi2212 data presented in Figure \[fig\_Bi2212\](c) were recorded after reducing the space-charge-induced shift to $<$1 meV, and their energy axes were calibrated by using the Fermi cutoff of Au \[Figure \[fig\_Bi2212\](a)\] as done in usual ARPES data analyses. It is warned in the literature that photoemission spectroscopy utilizing intense pulsed light source can suffer from the Fermi level referencing problem [@PasslackAeschlimann_SpaceCharge; @Zhou_SpaceMirror]: Extrinsic energy shift and broadening due to the space-charge and possible mirror-charge effects can occur depending on the photon flux as well as the sample configuration such as the tilting angle. The Fermi level referencing using Au presented herein is judged to be reliable, because the calibrated Fermi level nicely falls into the internal energy reference of the sample; see Figures \[fig\_Bi2212\](d2) and \[fig\_Bi2212\](d3), in which the Fermi level is nicely located in-between the superconducting peaks in the occupied and unoccupied side. It is thus demonstrated that the Fermi level referencing to a metal, as done in usual ARPES studies, is equally safe in TARPES, if the photon flux of the probe beam is sufficiently reduced.
TARPES of Bi thin film grown on HOPG {#Bi}
------------------------------------
We show a dataset of TARPES recorded by scanning the delay stage repetitively; see Sec. \[DelayScan\]. The measurement was performed on a thin film of bismuth (Bi) grown on HOPG. Vacuum-evaporated Bi on cleaved HOPG is found to form micro-crystals with the 111 face oriented normal to surface. Because HOPG consists of layers of micron-sized crystallite sheets that have random in-plane orientation, the 111 surface of the Bi micro-crystals are also randomly oriented in the basal plane; see later. Details of the film growth and characterization will be presented elsewhere [@Ishida_YbFiber].
![\[fig\_Bi\] TARPES of micro-crystalline Bi thin film grown on HOPG. (a) Band dispersion recorded at $t$$\leq$-0.6 ps, namely, before the arrival of the pump pulse (left) and its second derivative in energy (right). (b) TARPES images. Top panels show band dispersions, and bottom panels show difference to the averaged image before pumped. (c) Intensity variations along the V-shaped band in the unoccupied side. Inset shows the magnified view of the V-shaped band recorded at $t$=0.13 ps and the integration windows A-C along the band. The intensity scales adopted in the ARPES, TARPES, and difference images presented in (a) and (b) are in common arbitrary units. ](Fig7.eps){width="8.6cm"}
The left panel of Figure \[fig\_Bi\](a) shows an average over 12 TARPES images recorded at $t$$\leq$-0.6 ps during a sequence composed of 58 delay points. Since the probing is done before the arrival of the pump pulse, band dispersions in the occupied side are observed as in usual ARPES. Characteristic electronic structures of the 111-oriented Bi thin film are nicely resolved such as the Rashba-split surface states dispersing at $>$-0.15 eV [@YuKorteev] and quantized states occurring around -0.3$\pm$0.1 eV [@Hirahara_PRL06; @Hirahara_PRB07]. The latter is clearly observed in the second-derivative image shown in the right panel, in which the color scale highlights the negative curvature in energy of the original image, and hence, the band dispersions. Some of the electronic structures are more clearly resolved than those reported in the literature, such as the crossing point of the Rashba-split branches at $\bar{\varGamma}$ [@YuKorteev; @Ast]. The clear view in the ARPES images presented herein not only owes to the high energy resolution of 17 meV adopted in the measurement (slit width and $E_{\mbox{\it{pass}}}$ were set to 0.8 mm and 2 eV, respectively) but also to the magnified view in $k$ space around $\bar{\varGamma}$ when probed by the low excitation energy of 5.9 eV. Since the probe beam illuminates a large number of micron-sized Bi(111) surface that have random in-plane orientation, the band dispersions that warp hexagonally in going away from $\bar{\varGamma}$ are all projected on the TARPES images. For example, the dispersions of the Rashba-split surface states along $\bar{\varGamma}$-$\bar{M}$ and $\bar{\varGamma}$-$\bar{K}$ directions, the former contributing to a lobe-shaped Fermi surface [@YuKorteev; @Ast], are simultaneously observed in the ARPES images shown in Figure \[fig\_Bi\](a).
The upper and lower panels of Figure \[fig\_Bi\](b) respectively display selected TARPES images and their difference to the averaged image before pumped. Note, the difference was taken without any data processing such as the intensity normalization, because the intensity of the raw data is already normalized to the acquisition time; thanks to the repetitive delay-stage scanning (Sec. \[DelayScan\]) and stability of the TARPES system during the data acquisition of $\sim$12 hours (Sec. \[Stability\]). On arrival of the pump ($t$=0 and 0.13 ps), spectral intensity is spread not only into the unoccupied side of the Rashba-split surface states but also into a V-shaped band dispersing at $>$0.4 eV. After the pump-induced filling of the unoccupied states, recovery dynamics follows over 10 ps. The recovery of the spectral intensity along the V-shaped band is shown in Figure \[fig\_Bi\](c). Here, the integrated intensities in the frames A-C indicated in the inset to Figure \[fig\_Bi\](c) are plotted as functions of the delay time. The recovery of the intensity slows on approach to the bottom of the V-shaped band, which indicates that the pump-induced carriers are running down along the V-shaped band during the recovery.
The above results not only show the efficacy of the normalization-free data-acquisition for capturing the pump-induced dynamics, but also demonstrate the capability of the apparatus for revealing the band dispersions in the unoccupied side at the sub-20-meV resolution. That is, the bands above $E_F$ can be investigated at the energy resolution comparable to those of standard ARPES done at modern synchrotron facilities. TARPES results on a topological insulator Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$ (BSTS) are presented elsewhere [@KimPRL], in which the dispersion of the topological surface states traversing from the valence band to the conduction band is fully visualized at the energy resolution of 15 meV. The unoccupied side up to $\sim$1 eV above $E_F$ can be nicely revealed for some materials such as topological insulators [@Sobota_PRL12; @Gedik_Science], Bi [@Perfetti_Bi], and semiconductors [@Haight_InP; @Azuma]; all of which may be categorized as low-carrier-density materials. On the other hand, in the case of metallic materials, the unoccupied states are less visible by TARPES, presumably because the pump pulse is efficiently reflected by the surface. As a result, the pump-induced changes are small and are confined within a narrow energy region around $E_F$ but not up to $\sim$1 eV above $E_F$, which may be the case for Bi2212 shown in Figure \[fig\_Bi2212\].
Conclusions {#Conclusions}
===========
We described a 1.48-eV pump and 5.90-eV probe TARPES apparatus operating at $\geq$10.5-meV and $\geq$240-fs resolutions. Capability of the apparatus was demonstrated by presenting the datasets recorded on Bi2212 and 111-face-oriented micro-crystalline Bi thin film. In the TARPES of Bi2212, we resolved a pump-induced superconducting peak in the unoccupied side at the high-energy-resolution settings of 10.5 meV. In the TARPES of a Bi thin film, carrier dynamics as well as band dispersions above $E_F$ were revealed. A normalization-free dataset acquisition was demonstrated in the latter, which became possible after achieving a long-term stability of the TARPES system. Investigations of the variations in the TARPES intensities, such as pump-induced spectral-weight transfers in correlated materials and pump-induced changes in the photoemission matrix element would thus become possible without any post-processing in the dataset.
We also described simple methods to align and characterize the pump and probe beams and to determine the time resolution by using graphite. After all, daily inspection of the beam alignment before TARPES measurements is done with a $\lesssim$15-min procedure: First, we coarsely check the pump-and-probe overlap in air by using the sensor device (Sec. \[align\]), and then check the spatio-temporal overlap and time resolution by using the pump-and-probe signal of graphite (Sec. \[Graphite\]) which is attached next to the sample to be measured (Figure \[fig\_Pinhole\]). When concerns still remain in the beam alignment, we check it by utilizing the pin-hole device (Sec. \[align\]), which takes not more than 5 min. We hope that the descriptions would facilitate TARPES to be a versatile tool for investigating the electronic structures and dynamics of matter.
The authors acknowledge T. Mochiku, S. Nakane, and K. Hirata for providing high-quality Bi2212 single crystals; F. Iga and T. Takabatake for providing high-quality SmB$_6$ single crystals; K. Yaji for preparing a calibrated Bi evaporator; Y. Ozawa for realizing the pinhole device; M. Endo, and T. Nakamura for developing the AOM driver; and T. Saitoh for help in the TARPES measurements. This work was supported by Photon and Quantum Basic Research Coordinated Development Program from MEXT and JSPS KAKENHI, Grant Nos. 23740256 and 26800165.
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abstract: 'The search for sharp features in the gamma-ray spectrum is a promising approach to identify a signal from dark matter annihilations over the astrophysical backgrounds. In this paper we investigate the generation of gamma-ray lines and internal bremsstrahlung signals in a toy model where the dark matter particle is a real scalar that couples to a lepton and an exotic fermion via a Yukawa coupling. We show that the Fermi-LAT and H.E.S.S. searches for line-like spectral features severely constrain regions of the parameter space where the scalar dark matter is thermally produced. Finally, we also discuss the complementarity of the searches for sharp spectral features with other indirect dark matter searches, as well as with direct and collider searches.'
author:
- Alejandro Ibarra
- Takashi Toma
- Maximilian Totzauer
- Sebastian Wild
title: 'Sharp Gamma-ray Spectral Features from Scalar Dark Matter Annihilations'
---
Introduction
============
Multiple astrophysical and cosmological observations have demonstrated that a significant fraction of the matter content of the Universe is in the form of new particles not included in the Standard Model, but belonging to the so-called dark sector (see [@Bertone:2004pz; @Bergstrom:2000pn] for reviews). The dark matter (DM) particles were presumably produced during the very early stages of the Universe and must have a relic abundance today $\Omega_{\rm DM} h^2\approx0.12$ [@Ade:2013zuv]. Among the various production mechanisms proposed, the freeze-out mechanism stands among the most appealing and predictive ones. In this framework, the dark matter particles were in thermal equilibrium with the Standard Model particles at very early times, but went out of equilibrium when the temperature reached a value $\sim m_{\rm DM}/25$. Below this temperature, the expansion rate became larger than the annihilation rate and therefore the number density of dark matter particles per comoving volume remained practically constant until today, the value being inversely proportional to their annihilation cross section into Standard Model particles.
The annihilations that lead to the freeze-out of dark matter particles in the early Universe presumably continue today, at a much smaller rate, in regions with high dark matter density, such as in galactic centers. There exists then the possibility of testing the freeze-out mechanism if the flux of energetic particles produced in the annihilations is detected at the Earth. Unfortunately, the expected flux from annihilations is typically much smaller than the background fluxes from astrophysical processes, which makes a potential signal difficult to disentangle from the still poorly understood backgrounds.
A promising strategy to identify a dark matter signal is the search for sharp gamma-ray spectral features, such as gamma-ray lines [@Srednicki:1985sf; @Rudaz:1986db; @Bergstrom:1988fp], internal electromagnetic bremsstrahlung [@Bergstrom:1989jr; @Flores:1989ru; @Bringmann:2007nk] or gamma-ray boxes [@Ibarra:2012dw]. Most dark matter models predict rather faint sharp spectral features, however, the predicted signatures are qualitatively very different to the ones expected from known astrophysical processes, thus allowing a very efficient background subtraction. As a result, searches for sharp gamma-ray spectral features provide limits on the model parameters which are competitive, and sometimes better, than those from other approaches to indirect dark matter detection.
Recent works have thoroughly investigated the generation of sharp gamma-ray spectral features in simplified models, as well as the complementarity of the searches for spectral features with other search strategies, in scenarios where the dark matter particle is a Majorana fermion that couples to a Standard Model fermion via a Yukawa coupling [@Bringmann:2012vr; @Garny:2013ama; @Chang:2013oia; @An:2013xka; @Bai:2013iqa; @DiFranzo:2013vra; @Kopp:2014tsa; @Garny:2014waa], or in the inert doublet dark matter model [@Gustafsson:2007pc; @LopezHonorez:2006gr; @Garcia-Cely:2013zga].
In this paper, we investigate the generation of sharp gamma-ray spectral features in the toy model of real scalar dark matter considered in ref. [@Toma:2013bka; @Giacchino:2013bta], where the Standard Model is extended with a real singlet scalar $\chi$, candidate for dark matter, and an exotic vector-like fermion $\psi$, which mediates the interactions with the Standard Model fermions. In this model, the stability of the dark matter particle is ensured by imposing a discrete $\mathbb{Z}_2$ symmetry, under which $\chi$ and $\psi$ are odd while the Standard Model particles are even. We assume for simplicity that the new sector only couples to a right-handed lepton of one generation, $f_R=e_R$, $\mu_R$ or $\tau_R$, in order to suppress potential contributions to lepton flavor violating processes such as $\mu\rightarrow e \gamma$. Under these simplifying assumptions the interaction Lagrangian of the dark matter particle with the Standard Model particles reads: $$-{\cal L}_{\rm int}=\frac{\lambda}{2}\chi^2(H^\dagger H)+(y \chi \bar \psi f_R+{\rm h.c.})$$ where $H$ is the Standard Model Higgs doublet.
This model has the peculiarity that the cross section for the tree-level two-to-two annihilation process $\chi \chi\rightarrow f\bar f$, which sets the relic abundance over large regions of the parameter space, is d-wave suppressed in the limit $m_f\rightarrow 0$. On the other hand, the processes generating gamma-ray lines at the one loop level $\chi\chi\rightarrow \gamma \gamma,\gamma Z$ or internal bremsstrahlung $\chi\chi\rightarrow f\bar f \gamma$ proceed in the s-wave. Therefore, for values of the parameters leading to the correct relic abundance, the expected indirect detection signals are relatively large compared to other models and, under some conditions, at the reach of present instruments [@Toma:2013bka; @Giacchino:2013bta].
The paper is organized as follows. In Section \[sec:gamma-ray\] we present the result for the cross sections and we discuss the relative strength of both signals. In Section \[sec:constraints\] we present constraints on the model from perturbativity, thermal production, direct detection, indirect detection with charged cosmic rays and collider experiments. In Section \[sec:numerics\] we present a numerical analysis showing the complementarity of all these constraints, under the assumption that the dark matter particle was thermally produced. Finally, in Section \[sec:conclusions\] we present our conclusions.
Sharp gamma-ray spectral features from scalar dark matter annihilations {#sec:gamma-ray}
=======================================================================
The gamma-ray flux generated by the annihilation of scalar dark matter particles receives several contributions. In this paper we will concentrate on the generation of sharp gamma-ray spectral features, which, if observed, would constitute a strong hint for dark matter annihilations. We will neglect, however, the gamma-ray emission generated by the inverse Compton scattering of the electrons/positrons produced in the annihilation on the interstellar radiation field, and will only briefly discuss the gamma-rays produced by the decay and hadronization of Higgs or gauge bosons since they do not generate sharp features in the gamma-ray spectrum.
The scalar $\chi$ does not have tree-level electromagnetic interactions. Nevertheless, annihilations into one or two photons are possible via higher order effects mediated by the Standard Model fermion $f$ and the exotic fermion $\psi$, which do carry electroweak charges. More specifically, the interaction Lagrangian with photons of the charged fermions of the model reads: $$\mathcal{L}_{\mathrm{QED}}=eA_{\mu}\overline{f}\gamma^{\mu}f
+eA_{\mu}\overline{\psi}\gamma^{\mu}\psi,$$ while the interaction Lagrangian with the $Z$ boson reads: $$\mathcal{L}_{Z}=
- e \tan\theta_W Z_{\mu}\overline{f_R}\gamma^{\mu} f_R
-e \tan\theta_W Z_{\mu}\overline{\psi}\gamma^{\mu}\psi.$$
The annihilations $\chi\chi\rightarrow \gamma\gamma$ and $\gamma Z$ are generated at the one loop level, through the diagrams shown in Fig. \[fig:box\]. In the Milky Way center, dark matter particles are expected to be very non-relativistic, $v\approx 10^{-3}$, thus generating a monoenergetic photon in the annihilation process. In the limit of zero relative velocity, the transition amplitude for the annihilation $\chi\chi\rightarrow \gamma\gamma$ can be cast as $$i\mathcal{M}_{\gamma\gamma}=-\frac{i\alpha_{\mathrm{em}}y^2}{\pi}
\epsilon_{\mu}^*(k_1)\epsilon_{\nu}^*(k_2) g^{\mu\nu} \mathcal{A}_{\gamma\gamma},
\label{eq:MGammaGamma}$$ as required by gauge invariance. Here, $\epsilon_\mu(k)$ is the polarization vector of the photon, $\alpha_{\mathrm{em}}$ is the electromagnetic fine structure constant, defined as $\alpha_{\mathrm{em}}\equiv e^2/(4\pi)$ and $\mathcal{A}_{\gamma\gamma}$ is a form factor. The explicit expression for the form factor $\mathcal{A}_{\gamma\gamma}$ at the one loop level is rather complicated and is reported in the Appendix. The form factor greatly simplifies in the limit $m_f\to0$ and reads[^1]: $$\mathcal{A}_{\gamma\gamma}=
2+\mathrm{Li}_2\left(\frac{1}{\mu}\right)-\mathrm{Li}_2\left(-\frac{1}{\mu}\right)
-2\mu\,\mathrm{arcsin}^2\left(\frac{1}{\sqrt{\mu}}\right),
\label{eq:loop-f}$$ with $\mu\equiv m_\psi^2/m_\chi^2$. [^2] Finally, the annihilation cross section for $\chi\chi\rightarrow \gamma\gamma$ is given by $$\sigma{v}_{\gamma\gamma}=\frac{y^4\alpha_{\mathrm{em}}^2}
{32\pi^3 m_\chi^2}\left|\mathcal{A}_{\gamma\gamma}\right|^2.
\label{eq:sigmatotal2gamma}$$
On the other hand, the transition amplitude for $\chi\chi\to\gamma Z$ can be cast, in the zero velocity limit, as $$i\mathcal{M}_{\gamma Z}=\frac{i y^2\alpha_{\mathrm{em}}\tan\theta_W}{\pi}
\epsilon_{\mu}^*(k_1)\epsilon_{\nu}^*(k_2)g^{\mu\nu} \mathcal{A}_{\gamma
Z} ,
\label{eq:MGammaZ}$$ where $\epsilon_\mu(k_1)$, $\epsilon_\nu(k_2)$ are the polarization vectors of the $Z$ boson and the photon, respectively, and $\mathcal{A}_{\gamma Z}$ is the corresponding form factor, which is reported in the Appendix. The cross section is in this case given by $$\sigma{v}_{\gamma Z}=\frac{y^4 \alpha_{\mathrm{em}}^2\tan^2\theta_W}{16\pi^3
m_\chi^2}
\left(1-\frac{m_Z^2}{4m_\chi^2}\right)\left|\mathcal{A}_{\gamma Z}\right|^2.
\label{eq:sigmatotalZgamma}$$
On the other hand, the two-to-three annihilation into a fermion-antifermion pair with the associated emission of a photon results from Feynman diagrams shown in Fig. \[fig:vib\], where the photon can be attached to either of the charged fermions. The differential cross section for this process reads:
$$\frac{d\sigma{v}_{f\overline{f}\gamma}}{dx}=
\frac{y^4\alpha_{\mathrm{em}}}{4\pi^2 m_\chi^2}\left(1-x\right)
\left[
\frac{2x}{\left(\mu+1\right)\left(\mu+1-2x\right)}
-\frac{x}{\left(\mu+1-x\right)^2}
-\frac{\left(\mu+1\right)\left(\mu+1-2x\right)}{2\left(\mu+1-x\right)^3}
\log\left(\frac{\mu+1}{\mu+1-2x}\right)
\right],$$
with $x=E_\gamma/m_\chi$, while the total cross section is given by $$\sigma{v}_{f\overline{f}\gamma}=
\frac{y^4\alpha_{\mathrm{em}}}{8\pi^2 m_\chi^2}
\left[
\left(\mu+1\right)\left\{\frac{\pi^2}{6}-\log^2\left(\frac{\mu+1}{2\mu}\right)
-2\mathrm{Li}_2\left(\frac{\mu+1}{2\mu}\right)\right\}
+\frac{4\mu+3}{\mu+1}+\frac{\left(4\mu+1\right)\left(\mu-1\right)}{2\mu}
\log\left(\frac{\mu-1}{\mu+1}\right)
\right].
\label{eq:sigmatotal2to3}$$
As it is well known, the gamma-ray spectrum for the two-to-three process displays a sharp peak close to the kinematical endpoint of the spectrum, which becomes more and more prominent as $\mu\rightarrow 1$ [@Bringmann:2007nk].
The relative importance of the one loop processes $\chi \chi\rightarrow \gamma V$, with $V=\gamma, Z$, and the three body process $\chi \chi\rightarrow f\bar f \gamma$ is determined by $\mu\equiv m_\psi^2/m_\chi^2$ and by $m_V^2/m_\chi^2$. This dependence is explicitly shown in Fig. \[fig:ratio\], where we have taken $m_\chi=500$ GeV for definiteness. For $\mu=1$ the cross section for the two-to-three process is $1.6 \times 10^3$ ($2.9 \times 10^3$) times larger than the cross section for $\gamma\gamma$ ($\gamma Z$). As $\mu$ increases, $|\mathcal{A}_{\gamma \gamma}|$ decreases and eventually changes sign at $\mu-1 \simeq 8.0 \times 10^{-3}$. Accordingly, at this point the $\gamma \gamma$ cross section vanishes. For larger mass splittings, the relative importance of the one loop processes increases and they become the dominant process when $\mu\gtrsim 10$ ($\gtrsim 19$). A similar behavior arises in scenarios with Majorana dark matter particles coupling to leptons via a Yukawa coupling with a scalar field, as discussed in ref. [@Garny:2013ama].
![Ratio between the cross sections of the one-loop induced two-to-two annihilation into $\gamma\gamma$ (or $\gamma Z$) and the two-to-three annihilation into $f\bar f\gamma$, as a function of the parameter $\mu\equiv m_\psi^2/m_\chi^2$. For the plot it was assumed $m_\chi = 500$ GeV.[]{data-label="fig:ratio"}](./CrossSectionRatios)
 \
 
The gamma-ray flux generated by the two-to-three and the one loop annihilation processes depends on the cross sections in those channels. The dependence of these cross sections on the different parameters of the model is shown in Fig. \[fig:vib\_vs\_loop\], where we have fixed in all the plots the Yukawa coupling $y$ to 1. Approximate analytic expressions for the cross sections when $\mu\gg 1$ are $$\begin{aligned}
\langle \sigma v \rangle_{f \bar{f} \gamma} &\simeq 3.0 \times 10^{-26} \, \text{cm}^3\text{s}^{-1} \frac{y^4}{\mu^4} \left( \frac{100 \,\text{GeV}}{m_\chi} \right)^2 \nonumber\\
\langle \sigma v \rangle_{\gamma \gamma} &\simeq 1.3 \times 10^{-28} \, \text{cm}^3\text{s}^{-1} \frac{y^4}{\mu^2} \left( \frac{100 \,\text{GeV}}{m_\chi} \right)^2 \,, \nonumber\\
\langle \sigma v \rangle_{\gamma Z} &\simeq 7.6 \times 10^{-29} \, \text{cm}^3\text{s}^{-1} \frac{y^4}{\mu^2} \left( \frac{100 \,\text{GeV}}{m_\chi} \right)^2 \,,\end{aligned}$$ from where it is apparent the different dependence of the two-to-three and the loop-induced two-to-two processes with $\mu$.
As mentioned above, for (moderately) small mass splittings $m_\psi/m_\chi$ between the fermionic mediator and the scalar dark matter particle, the two-to-three process has the largest cross section, while for $\mu\gtrsim 10$ the one loop processes dominate the annihilation. This is illustrated in Fig. \[fig:vib\_vs\_loop\], top plots, where we show the annihilation cross sections for the relevant processes as a function of the dark matter mass for $\mu=1.1$ (left plot) and for $\mu=25$ (right plot). It is important to note that the values of the cross section when the loop processes dominate are fairly small. However, this does not imply that the one loop processes can be neglected in this model. As we will show below, gamma-ray lines can dominate the total gamma-ray energy spectrum, even for smaller values of $\mu$, due to the sharpness of the gamma-ray line signal compared to the internal bremsstrahlung signal and the fairly good energy resolution of present gamma-ray telescopes, $\Delta E\sim 10\%$.
We further investigate the dependence of the cross sections with the parameters of the model in Fig. \[fig:vib\_vs\_loop\], bottom plots, where we show the cross sections as a function of the degeneracy parameter $\mu$ for two choices of the dark matter mass, $m_\chi=150$ GeV (left plot) and $m_\chi=500$ GeV (right plot). Again, and as apparent from the plots, for choices of parameters where the one loop processes dominate, the resulting cross section is rather small.
   
The total energy spectrum of the channels leading to sharp spectral features reads: $$\frac{dN_\gamma}{dx}=
\frac{1}{\langle\sigma{v}\rangle}\left[
\frac{d\langle\sigma{v}\rangle_{f\overline{f}\gamma}}{dx}+
2\frac{d\langle\sigma{v}\rangle_{\gamma\gamma}}{dx}+
\frac{d\langle\sigma{v}\rangle_{Z\gamma}}{dx}\right] \,.$$ To investigate the relative strength of the gamma-ray line and the internal bremsstrahlung features, we have calculated the gamma-ray energy spectrum for various choices of $\mu$ assuming a Gaussian energy resolution of $10\%$; the result is shown in Fig. \[fig:spectrum\] for $\mu=1.1$ (top left), 4 (top right), 9 (bottom left) and 25 (bottom right). It is interesting that the gamma-ray lines give a significant contribution to the spectrum for $\mu\sim 4$ and totally dominate the high energy part of the spectrum when $\mu\gtrsim 9$, despite the smaller cross section. Therefore, in the search for gamma-ray spectral features the contribution from gamma-ray lines ought not to be neglected. The relative importance of the gamma-ray lines will increase with the next generation of gamma-ray telescopes, such as GAMMA-400 [@Galper:2012fp] and DAMPE [@DAMPE], that aim to an energy resolution of $\sim 1\%$ at $E_\gamma>10~\mathrm{GeV}$.
Complementary constraints {#sec:constraints}
=========================
The possibility of observing a spectral feature in the gamma-ray sky arising from scalar dark matter annihilations is subject to a series of theoretical and observational constraints. In this section, we discuss each of these constraints individually, while we will study their complementarity in section \[sec:numerics\], along with the results of the searches for gamma-ray spectral features.
Perturbativity
--------------
Demanding perturbativity of the model translates into an upper bound on the Yukawa coupling $y$. We will use the common condition $y<4 \pi$ as a conservative upper limit, however, and since the perturbative calculation of the annihilation and scattering rates is performed as an expansion in $\alpha_y\equiv y^2/(4\pi)$, we will also use the condition $y<\sqrt{4\pi}$.
Relic density
-------------
The freeze-out of the dark matter particles from the thermal bath, and accordingly the dark matter relic density, is determined by the following annihilation processes: [*i)*]{} the two-to-two annihilation $\chi\chi\rightarrow f\bar f$ through the Yukawa coupling $y$, [*ii)*]{} the higher order processes $\chi \chi \rightarrow f \bar{f} V$ via $t$-channel exchange of the heavy fermion $\psi$ and the one loop annihilations $\chi \chi \rightarrow V V'$ (with $V,V'=\gamma,Z$), [*iii)*]{} the two-to-two annihilation $\chi \chi \rightarrow h \rightarrow X X'$ via the Higgs portal interactions, where $X$, $X'$ are Standard Model particles.
We note that the two-to-three annihilation channels $\chi \chi \rightarrow f \bar{f} V$ (and to a lesser extent also the one loop annihilations into a pair of gauge bosons) can contribute significantly to the total cross section at the time of freeze-out, due to the d-wave suppression of the annihilation into $f \bar{f}$. Furthermore, for mass ratios $m_\psi/m_\chi \lesssim 1.2$, the relic density is not set by the annihilation channels listed above, but by other (co-)annihilation processes, the most relevant ones being $\psi \psi \rightarrow f f, \chi \psi \rightarrow f \gamma$ and $\psi \bar{\psi} \rightarrow F \bar{F}$, where $F$ can be any Standard Model fermion. In particular, the latter process is present even for $y = \lambda = 0$. Consequently, for a given mass ratio $m_\psi/m_\chi$, there is a lower bound on the dark matter mass $m_\chi$, below which these annihilation processes are sufficiently strong to suppress the relic density to values smaller than the observed one.
![Yukawa coupling $y$ leading to the observed dark matter relic density, for $\lambda=0$, and for different values of the mass ratio $m_\psi/m_\chi$.[]{data-label="fig:yThermal"}](./fThermalScalarDM)
We fully take into account all relevant annihilation and coannihilation channels by adapting the micrOMEGAs code [@Belanger:2013oya] for the solution of the Boltzmann equation, taking $\Omega_{\rm DM} h^2 \simeq 0.12$ as measured by Planck [@Ade:2013zuv]. In this way, also the increase of the annihilation cross section close to the Higgs resonance is included, which suppresses the relic density for $m_\chi \simeq m_h/2$ and values slightly below. For $y = 0$, our results agree reasonably well with [@Cline:2013gha]. In Fig. \[fig:yThermal\] we show the Yukawa coupling $y$ leading to the observed dark matter relic density, for $\lambda=0$ and different choices of the mass ratio $m_\psi/m_\chi$. Setting the scalar coupling $\lambda$ to non-zero values leads to a decrease of $y_\text{thermal}$ with respect to the values shown in Fig. \[fig:yThermal\]. This will be further discussed in section \[sec:numerics\].
Direct Detection
----------------
The dark matter particle $\chi$ can scatter off nuclei via $t$-channel Higgs exchange with quarks[^3]. The corresponding spin independent scattering cross section with nucleons is given by $$\sigma_{\text{SI}}=\frac{\lambda^2 f_N^2}{4\pi} \frac{\mu_N^2 m_N^2}{m_h^4 m_\chi^2} \,,$$ where $\mu_N=m_N m_\chi / \left(m_N + m_\chi\right)$ is the DM-nucleon reduced mass, and $f_N=\sum_q f_q$ is the Higgs-nucleon coupling. The latter is subject to significant nuclear uncertainties, in particular it is sensitive to the strangeness content $f_s$ of protons and neutrons. In the following we use the value $f_N=0.345$, as determined in the recent study [@Cline:2013gha]. Furthermore, we fix the Higgs mass to be $m_h=125$ GeV.
The present upper limit from the LUX experiment [@Akerib:2013tjd] on the spin independent cross section can be used to set a limit on the scalar coupling $\lambda$ as a function of the dark matter mass $m_\chi$, independently of the values of $y$ and $m_\psi$. This limit will be improved in the future by the XENON1T experiment [@Aprile:2012zx], which will have a sensitivity to $\sigma_{\text{SI}}$ which is approximately a factor 100 better than the current XENON100 experiment [@Aprile:2012nq]. The present LUX limit, as well as an estimate on the projected sensitivity to $\lambda$ of the XENON1T experiment, are shown in Fig. \[fig:LamdaConstraintsDD\] as a solid and a dashed line, respectively.
![Upper limit on the scalar coupling $\lambda$ as a function of the dark matter mass derived from the LUX data (solid line) as well as the prospected sensitivity of the XENON1T experiment (dashed line).[]{data-label="fig:LamdaConstraintsDD"}](./LambdaConstraintsDD)
Indirect detection
------------------
In addition to the sharp gamma-ray spectral features, the scalar dark matter model we consider in this paper leads to other potentially observable signals in indirect detection experiments, such as exotic contributions to the antimatter fluxes or the continuum gamma-ray flux. We will consider first the scenario where the Higgs portal coupling is set to zero and will discuss later on the impact of a sizable $\lambda$ on the fluxes.
The annihilation channel $f \bar{f} \gamma$ is necessarily accompanied by the process $f \bar{f} Z$, leading to the production of antiprotons via the decay and hadronization of the $Z$ boson; this has been studied in detail in the context of Majorana dark matter [@Kachelriess:2009zy; @Ciafaloni:2011sa; @Garny:2011cj; @Garny:2011ii]. In the case of the scalar dark matter model at hand, we find for the differential cross section the expression $$\begin{aligned}
&\frac{\text{d}\sigma{v}_{f\overline{f}Z}}{\text{d}x \,\text{d}z}=
\frac{y^4\alpha_\mathrm{em}\tan^2\theta_W}
{\pi^2m_\chi^2\left(1-\mu-2z\right)^2\left(3+\mu-2x-2z\right)^2} \times \nonumber\\
&\left\{(1-x)\left[x^2-2x(1-z)+2(1-z)^2\right]+\frac{\xi}{4}(x^2-2x+2)\right\},\end{aligned}$$ where $x=E_Z/m_\chi$, $z=E_f/m_\chi$ and $\xi=m_Z^2/m_\chi^2$. Note that, while the differential cross section for $f \bar{f} \gamma$ in the case of scalar DM is simply a factor of 8 larger than for Majorana DM, this expression for $f \bar{f} Z$ has a different functional dependence on $x$ and $y$ compared to the corresponding process for Majorana DM (see e.g. [@Garny:2011ii] for the latter). The one loop annihilations into $\gamma Z$ and $ZZ$ also contribute to the antiproton flux, however, these channels are only relevant when $m_\psi/m_\chi \gtrsim 3$. In this regime, the resulting antiproton flux is too small to be observed, therefore, the one loop annihilation channels will be neglected in our analysis.
We use CalcHEP [@Pukhov:1999gg; @Pukhov:2004ca] interfaced with PYTHIA [@Sjostrand:2007gs] to produce the corresponding injection spectrum of antiprotons. The propagation to the Earth is implemented by using the standard two-zone diffusion model, neglecting energy losses and reacceleration; hereby, we use the same setup as in [@Garny:2011ii] to which we refer the reader for details. We employ the MIN, MED and MAX propagation parameters from [@Donato:2003xg] in order to bracket the uncertainty arising from the different possible parameters of the propagation model; furthermore, the analysis is done assuming a NFW halo profile, although this choice does not affect the limits significantly. Finally, solar modulation is included by means of the force-field approximation [@1967ApJ149L115G] with $\Phi_{F} = 500$ MeV. We then compare the sum of the propagated antiproton spectrum and the expected spallation background, which we take from [@Bringmann:2006im], with the PAMELA $\bar{p}/p$ data [@Adriani:2010rc]. We perform a $\chi^2$-test at $95 \%$ C.L. in order to obtain the corresponding constraints on the Yukawa coupling $y$. The resulting limit is shown in Fig \[fig:antimatter\_constraints\] as a function of the dark matter mass for $m_\chi/m_\chi=1.01$ (left plot), 1.1 (middle plot) and 2 (right plot). We also show for comparison the values of the Yukawa coupling $y$ leading to the observed relic abundance via thermal freeze-out for any values of $m_\chi$ and $m_\psi$. As apparent from the plots, present experiments are not sensitive enough to detect the antiprotons produced in the annihilations of scalar dark matter particles.
  
Besides antiprotons, also electrons and positrons can be produced in the annihilation processes. However, in contrast to the case of antiprotons, the observed positron fraction $e^+/\left(e^- + e^+ \right)$ is not compatible with the expectation from purely secondary positron production by spallation of cosmic rays. While the reason for the unexpected rise of the positron fraction as observed by PAMELA [@Adriani:2008zr], Fermi-LAT [@FermiLAT:2011ab] and AMS-02 [@Aguilar:2013qda] remains unclear, the extremely precise data from AMS-02 can be used for setting strong limits on the annihilation cross section of dark matter into leptonic final states, notably when $m_{\text{DM}} \lesssim 300$ GeV [@Bergstrom:2013jra; @Ibarra:2013zia].
In our scenario, for $f=e^-$ and $m_\psi/m_\chi \simeq 1$, the injection spectrum of the positrons arising in the annihilation process $\chi \chi \rightarrow e^- e^+ \gamma$ exhibits a pronounced spectral feature towards the kinematical end point, which makes it separable from the smooth background, even after taking into account propagation effects. In the following, we use the limits on $\langle \sigma v \rangle_{e^- e^+ \gamma}$ derived in [@Bergstrom:2013jra], and convert them into limits on the Yukawa coupling $y$, neglecting the subdominant process $\chi \chi \rightarrow e^- e^+ Z$. It is important to note that the limit in [@Bergstrom:2013jra] was derived under the assumption $m_\psi \equiv m_\chi$, leading to the above-mentioned sharp spectral feature in the positron spectrum. Up to mass ratios $m_\psi/m_\chi \lesssim 1.2$, the spectrum of the positrons does not change significantly and our procedure of obtaining the corresponding limits on the Yukawa coupling is applicable; for larger values of $m_\psi/m_\chi$, the positron spectrum becomes broader, making it harder to distinguish from the background. The limits for $\mu=1.01$ and 1.1 are shown, respectively, in the left and middle plots of Fig. \[fig:antimatter\_constraints\] and, as for the antiprotons, are well above the value required by the freeze-out mechanism.
![Maximally allowed annihilation cross sections for the Higgs-mediated processes, imposing the upper limit on $\lambda$ from the LUX experiment, together with bounds from PAMELA and Fermi-LAT.[]{data-label="fig:IndirectConstraintsMaxLambda"}](./indirect_constraints_max_lambda_by_LUX)
For $\lambda > 0$, new annihilation channels mediated by the Standard Model Higgs open up, the most relevant ones being $\chi \chi \rightarrow W^+ W^-$, $\chi \chi \rightarrow ZZ$ and $\chi \chi \rightarrow hh$. In the limit $v\rightarrow 0$, the annihilation cross sections read
$$\begin{aligned}
\langle \sigma v \rangle_{ZZ} &= \frac{\lambda^2 \left( 4 m_\chi^4 - 4 m_\chi^2 m_Z^2 + 3 m_Z^4 \right) \sqrt{m_\chi^2-m_Z^2}}{16 \pi m_\chi^3 \left( m_h^2-4 m_\chi^2\right)^2 } \,, \nonumber\\
\langle \sigma v \rangle_{W^+ W^-} &= \frac{\lambda^2 \left( 4 m_\chi^4 - 4 m_\chi^2 m_W^2 + 3 m_W^4 \right) \sqrt{m_\chi^2-m_W^2}}{8 \pi m_\chi^3 \left( m_h^2-4 m_\chi^2\right)^2 } \,, \nonumber\\
\langle \sigma v \rangle_{hh} &= \frac{\sqrt{m_\chi^2 -m_h^2}}{16 \pi m_\chi^3 \left( 8 m_\chi^4 -6m_\chi^2 m_h^2+m_h^4\right)^2} \times \nonumber\\
&\hspace{-0.5cm}\lambda^2 \left( 4 m_\chi^4 +4 \lambda m_\chi^2 v_{\text{EW}}^2-m_h^2 \left[ m_h^2+\lambda v_{\text{EW}}^2 \right] \right)^2 \,,\end{aligned}$$
with $v_{\text{EW}} = 246$ GeV being the vacuum expectation value of the Standard Model Higgs. These annihilation channels contribute to the antiproton flux, and also lead to a featureless gamma-ray spectrum. In Fig. \[fig:IndirectConstraintsMaxLambda\], we show the annihilation cross sections into these final states, fixing $\lambda$ to be the upper limit deduced from the LUX experiment (as shown in Fig. \[fig:LamdaConstraintsDD\]). When comparing to the upper bounds derived in [@Fornengo:2013xda] from the PAMELA data on the antiproton-to-proton fraction, as well as the latest Fermi-LAT limits obtained by searches for gamma-ray emission from dwarf galaxies [@Ackermann:2013yva], we find that these limits from indirect detection are always superseded by the upper limit on $\lambda$ from the LUX experiment. Note that for simplicity, we only show the PAMELA and Fermi-LAT bounds for the annihilation channel $\chi \chi \rightarrow b \bar{b}$; the constraints for the relevant annihilation channels listed above are all comparable [@Fornengo:2013xda; @Ackermann:2013yva], leading to the same conclusion.
 
Collider Constraints
--------------------
The scalar dark matter model can be probed at colliders, most notably at the LHC. A full analysis of the collider constraints is beyond the scope of this work, and we focus on the Drell-Yan production of a pair of heavy fermions $\psi\overline{\psi}$. These particles in turn both decay into the dark matter particle $\chi$ and the Standard Model lepton $f$ (or $\bar{f}$), leading to a charged lepton pair plus missing energy. This signal is similar to the slepton search at LHC, however the production cross section for the charged fermion is about one order of magnitude larger than for a charged scalar [@Bai:2014osa], leading to tighter constraints.
We calculate the production cross section $pp\to\psi\overline{\psi}$ using CalcHEP [@Pukhov:1999gg; @Pukhov:2004ca], and compare it with the upper bound of the production cross section along with the analysis in ref. [@Chang:2014tea]. The computed production cross section depends only the heavy fermion mass $m_\psi$. The experimental data is taken from ATLAS with the integrated luminosity of $20.3~\mathrm{fb^{-1}}$ and $\sqrt{s}=8~\mathrm{TeV}$ [@TheATLAScollaboration:2013hha]. Additionally, we estimate a bound from the LEP experiment, based on a search for the right-handed selectron in the minimal supersymmetric Standard Model, with data collected in the energy range $\sqrt{s}=183-208~\mathrm{GeV}$ [@LEP_bound]. This bound does not exactly apply to the model discussed in this work, and hence only serves as an estimation.
Constraints for thermally produced scalar dark matter {#sec:numerics}
=====================================================
We present in this section the constraints on the parameter space of the scalar dark matter model from the negative searches for gamma-ray spectral features, assuming that the dark matter density was generated via thermal freeze-out, as well as the complementarity of these constraints with those discussed in section \[sec:constraints\] from direct detection and collider experiments. Let us discuss separately the scenario where the scalar coupling $\lambda$ is small and where it is sizable.
Small scalar coupling $\lambda$
-------------------------------
 
Let us discuss first the case where the coupling $\lambda$ between the dark matter particle $\chi$ and the Standard Model Higgs doublet is very small, concretely $\lambda\lesssim 10^{-3}$. In this limit, and as argued in section \[sec:constraints\], the Higgs portal interaction is too suppressed to lead to observable signatures in the direct detection experiments LUX and XENON1T or in the indirect detection experiments AMS-02 or PAMELA. In this case, the requirement of reproducing the observed dark matter abundance via thermal production fixes the Yukawa coupling of the model as a function of $m_\chi$ and $m_\psi$.
We show in Fig. \[fig:sigmav\_thermal\] the annihilation cross sections into $f \bar{f} \gamma$ and $\gamma \gamma$ for $\lambda=0$ and for $m_\psi/m_\chi=1.1$ (left plot) and 3 (right plot) as a function of $m_\chi$ in the scalar dark matter scenario. These cross sections can be compared to the searches for sharp spectral features in the gamma-ray sky performed by the Fermi-LAT [@Ackermann:2013uma] and H.E.S.S. [@Abramowski:2013ax] collaborations. Upper limits on the combined annihilation cross section $\langle \sigma v \rangle_{f \bar{f} \gamma} + 2 \langle \sigma v \rangle_{\gamma \gamma}$ were derived in [@Garny:2013ama], employing the spectrum of gamma-rays originating from a region close to the Galactic Center, for a model of Majorana dark matter which couples to a heavy scalar and a right-handed Standard Model fermion $f_R$ via a Yukawa coupling. Since the normalized energy spectrum of gamma-rays from the two-to-three processes is identical for the cases of scalar and Majorana dark matter, we adopt those limits for our analysis. These constraints are shown in Fig. \[fig:sigmav\_thermal\], together with the prospected sensitivity of CTA [@Bernlohr:2012we], which we also take from [@Garny:2013ama].[^4] It follows from Fig. \[fig:sigmav\_thermal\] that the Fermi-LAT and H.E.S.S. observations are currently probing regions of the parameter space where the scalar dark matter particle is thermally produced, as first mentioned in [@Toma:2013bka; @Giacchino:2013bta]. We also show for comparison the corresponding thermal annihilation cross sections for the analogous model involving a Majorana dark matter candidate, as studied in e.g. [@Garny:2013ama], and which is not expected to produce any observable signal in current- or next-generation experiments searching for gamma-ray spectral features, unless a boost factor of the Galactic gamma-ray signal is invoked [@Garny:2013ama].
The complementarity of the various search strategies is illustrated in Fig. \[fig:lambda\_zero\], where we show the excluded regions in the parameter space of the scalar dark matter model, spanned by the mass ratio $m_\psi/m_\chi$ and dark matter mass $m_\chi$ (assuming thermal production). In the left panel, the blue shaded regions enclosed by the solid lines are excluded by Fermi-LAT or H.E.S.S., while the red shaded regions surrounded by dashed lines are excluded by collider searches. The dark gray shaded regions in the right upper part of each plot are ruled out by the conservative perturbativity requirement $y< 4\pi$ while the dotted black line shows the points with $y=\sqrt{4\pi}$. Lastly, the gray shaded regions in the lower left corners of the parameter space correspond to choices of parameters where the measured dark matter abundance cannot be generated via thermal freeze-out, due to very efficient coannihilations. It follows from the figures that the parameter space for thermally produced scalar dark matter particles is considerably constrained by the requirement of perturbativity of the theory, by collider searches and by the searches for sharp gamma-ray spectral features. In the right panel of Fig. \[fig:lambda\_zero\], we show the reach of CTA on the scalar dark matter model, assuming again thermal production. Remarkably, CTA has good prospects to probe practically the whole parameter space, in particular values of the dark matter mass which are inaccessible to present and projected collider searches.
Sizable scalar coupling $\lambda$
---------------------------------
If the coupling $\lambda$ is sizable, the direct detection signals can be significantly enhanced, as discussed in section \[sec:constraints\]. Moreover, the Higgs portal coupling opens new additional annihilation channels at the time of freeze-out. Therefore, and in order to reproduce to observed dark matter abundance, the Yukawa coupling $y$ must be smaller compared to the case $\lambda=0$. Accordingly, the intensity of the gamma-ray spectral features is expected to be smaller in this case than when $\lambda=0$.
 \
 
We show in Fig. \[fig:lambda\_finite\] the parameter space of the scalar dark matter model, spanned by $m_\psi/m_\chi$ and $m_\chi$, for the value of the Yukawa coupling $y$ leading to the observed dark matter abundance for a fixed value of the coupling $\lambda=0.03$ (upper panels) or 0.1 (lower panels). Compared to Fig. \[fig:lambda\_zero\], which assumed $\lambda=0$, we also include in the plots the choices of parameters excluded by the LUX experiment (left panels) and the projected sensitivity of the XENON1T experiment (right panels), shown as a hatched region. Moreover, depending on the value of $\lambda$, there is a range of dark matter masses $m_\chi$ for which the Higgs-mediated annihilation channels are sufficiently strong to suppress the relic density below its observed value, independently of the Yukawa coupling $y$. This is indicated in Fig. \[fig:lambda\_finite\] by the (nearly) vertical gray shaded strips. In particular, it is impossible to obtain the observed relic density for dark matter masses around $m_\chi \simeq m_h/2 \simeq 63$ GeV, even for very small values of $\lambda$, due to the resonantly enhanced annihilation processes mediated by the exchange of a Higgs particle in the s-channel.
It follows from Fig. \[fig:lambda\_finite\] an interesting complementarity between direct detection and collider constraints on the one hand, and searches for gamma-ray spectral features on the other hand: the LUX experiment and the LHC constrain the model for dark matter masses below $\simeq 100 - 200$ GeV (depending on the value of $\lambda$), while the Fermi-LAT and H.E.S.S. provide the strongest constraints for larger dark matter masses. Interestingly, some regions of the parameter space have been probed both by direct and indirect searches (and in some cases also by collider searches). The non observation of a dark matter signal then allows to more robustly exclude that part of the parameter space, in spite of the astrophysical uncertainties that plague the calculation of the direct and indirect detection rates. Conversely, some regions of the parameter space will be probed in the next years both by the XENON1T experiment and by CTA, thus opening the exciting possibility of observing dark matter signals in more than one experiment.
Summary and Conclusions {#sec:conclusions}
=======================
We have studied the generation of sharp gamma-ray spectral features in a toy model consisting in a scalar particle as dark matter candidate, that couples to a heavy exotic vector-like fermion and a Standard Model fermion via a Yukawa coupling. More specifically, we have calculated the cross section for the processes generating gamma-ray lines at the one loop level and generating line-like features through the two-to-three annihilation into two Standard Model fermions with the associated emission of a gauge boson. We have showed that the cross section for the two-to-three process is larger than for the loop annihilation when the fermionic mediator is degenerate in mass with the dark matter particle, the importance of the latter increasing as the ratio between the mediator mass and the dark matter mass becomes larger and larger.
We have also calculated the expected intensity of the sharp spectral features for a dark matter population produced via the freeze-out mechanism. In this model, the annihilation cross section into a fermion-antifermion pair proceeds in the d-wave, therefore reproducing the correct relic abundance via thermal production requires a rather large Yukawa coupling, which in turn translates into relatively intense indirect detection signals. In fact, we find that large parts of the parameter space are already excluded by the Fermi-LAT and H.E.S.S. searches for line-like features. We have also investigated the limits on the model from other indirect dark matter searches, direct searches and collider searches, and we have discussed the complementarity of these limits with those from the non-observation of sharp features in the gamma-ray sky.
Note Added {#note-added .unnumbered}
==========
During the completion of this work, we learned about an analysis of gamma-ray spectral features in the scalar dark matter model [@Giacchino2014]. Their results agree with ours, in the aspects where our analyses overlap.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Federica Giacchino, Laura Lopez-Honorez and Michel Tytgat for communications and for sharing with us the results of [@Giacchino2014]. We also thank Celine Boehm and Stefan Vogl for useful discussions. TT acknowledges support from the European ITN project (FP7-PEOPLE-2011-ITN, PITN-GA-2011-289442-INVISIBLES). AI, MT and SW were partially supported by the DFG cluster of excellence “Origin and Structure of the Universe”, and SW further acknowledges support from the TUM Graduate School and the Studienstiftung des Deutschen Volkes.
Appendix A {#appendix-a .unnumbered}
==========
In this appendix we provide the complete expressions for the form factors $\mathcal{A}_{\gamma \gamma}$ and $\mathcal{A}_{\gamma Z}$ (see Eq. (\[eq:MGammaGamma\]) and (\[eq:MGammaZ\]) for its definitions)[^5]. We work exclusively in the limit $v \rightarrow 0$, i.e. we only keep the (dominant) s-wave term of the annihilation cross sections.
For $\mathcal{A}_{\gamma \gamma}$, we find $$\begin{aligned}
\mathcal{A}_{\gamma\gamma} \, &= 2 + m_\chi^2 \,\bigg\{\nonumber\\
&\frac{1-\mu-\epsilon}{1+\mu-\epsilon}
\frac{2\epsilon}{\mu-\epsilon}
C_0\left(-m_\chi^2,m_\chi^2,0;m_f^2,m_\psi^2,m_f^2\right)\nonumber\\
+&\frac{1-\epsilon-\mu}{1+\epsilon-\mu}
\frac{2\mu}{\epsilon-\mu}
C_0\left(-m_\chi^2,m_\chi^2,0;m_\psi^2,m_f^2,m_\psi^2\right)\nonumber\\
+&\frac{4\epsilon\left(1-\epsilon\right)}{1+\mu-\epsilon}
C_0\left(4 m_\chi^2 ,0,0;m_f^2,m_f^2,m_f^2\right)\nonumber\\
+&\frac{4\mu\left(1-\mu\right)}{1+\epsilon-\mu}
C_0\left(4m_\chi^2 ,0,0;m_\psi^2,m_\psi^2,m_\psi^2\right)\bigg\}.
\label{eq:exact_gamma}\end{aligned}$$
where $\epsilon = m_f^2/m_\chi^2$ and $\mu = m_\psi^2/m_\chi^2$. $C_0$ is the scalar three-point Passarino-Veltman integral [@Passarino:1978jh] defined by $$\begin{aligned}
&C_0\left(p_1^2,\left(p_1-p_2\right)^2,p_2^2;m_1^2,m_2^2,m_3^2\right)
\nonumber\\
&=
\int\frac{d^d\ell}{i\pi^2}
\frac{1}{\ell^2-m_1^2}\frac{1}{\left(\ell+p_1\right)^2-m_2^2}
\frac{1}{\left(\ell+p_2\right)^2-m_3^2}.\end{aligned}$$
As already mentioned in the main text, our result differs from the one reported in [@Tulin:2012uq]. We have checked that our full expression of $i \mathcal{M}_{\gamma \gamma}$ satisfies the Ward identity, and we also cross-checked the imaginary part of $\mathcal{A}_{\gamma \gamma}$ with the one deduced from the optical theorem [@Asano:2012zv]. In the limit $\epsilon\to0$, one can explicitly evaluate all Passarino-Veltman functions in eq. (\[eq:exact\_gamma\]), leading to the result given in eq. (\[eq:loop-f\]).
For the $\chi\chi\to \gamma Z$ process, we find
$$\begin{aligned}
\mathcal{A}_{\gamma Z}\!\!&\,=\,&\!\!
2-\frac{\xi}{4-\xi}B_0\left(m_Z^2;m_f^2,m_f^2\right)
-\frac{\xi}{4-\xi}B_0\left(m_Z^2;m_\psi^2,m_\psi^2\right)\nonumber\\
&&
+\frac{2\xi\left(1+\mu+\epsilon\right)}
{\left(4-\xi\right)\left(1+\mu-\epsilon\right)\left(1+\epsilon-\mu\right)}
\left[1-\frac{1-\mu+\epsilon}{1+\mu+\epsilon}\frac{\epsilon}{2}\right]
B_0\left(m_\chi^2;m_f^2,m_\psi^2\right)\nonumber\\
&&
-\frac{\epsilon}{1+\mu-\epsilon}\frac{\xi}{4-\xi}
B_0\left(4m_\chi^2;m_f^2,m_f^2\right)
-\frac{2\mu}{1+\epsilon-\mu}\frac{\xi}{4-\xi}
B_0\left(4m_\chi^2;m_\psi^2,m_\psi^2\right)\nonumber\\
&&
+m_\chi^2\left\{
\frac{\epsilon}{2}\frac{4-4\epsilon-\xi}{1+\mu-\epsilon}
C_0\left(m_Z^2,4m_\chi^2,0;m_f^2,m_f^2,m_f^2\right)
+\mu\frac{4-4\mu-\xi}{1+\epsilon-\mu}
C_0\left(m_Z^2,4m_\chi^2,0;m_\psi^2,m_\psi^2,m_\psi^2\right)\right.\nonumber\\
&&
+\frac{\epsilon}{2}\left[\frac{\left(4+\xi\right)\left(-2+2\mu+2\epsilon+\xi\right)}
{\left(1+\mu-\epsilon\right)\left(4\epsilon-4\mu+\xi\right)}
+\frac{1}{2}\frac{4-4\epsilon-\xi}{1+\mu-\epsilon}
\right]
C_0\left(-m_\chi^2+\frac{m_Z^2}{2},m_\chi^2,0;m_f^2,m_\psi^2,m_f^2\right)\nonumber\\
&&
+\frac{\mu}{2}\left[\frac{\left(4+\xi\right)\left(-2+2\epsilon+2\mu+\xi\right)}
{\left(1+\epsilon-\mu\right)\left(4\mu-4\epsilon+\xi\right)}
-\frac{8\epsilon}{4\mu-4\epsilon+\xi}
\right]
C_0\left(-m_\chi^2+\frac{m_Z^2}{2},m_\chi^2,0;m_\psi^2,m_f^2,m_\psi^2\right)\nonumber\\
&&
+\left[
\frac{2\xi\left(1+\mu\right)+\epsilon\left(4\mu-\xi\right)}{4\left(1+\mu-\epsilon\right)}
-\frac{4\left(1+\mu\right)}{4-\xi}+\frac{4\mu\left(1+\mu-2\epsilon\right)}{4\mu-4\epsilon+\xi}
\right]C_0\left(-m_\chi^2+\frac{m_Z^2}{2},m_\chi^2,m_Z^2;m_f^2,m_\psi^2,m_f^2\right)\nonumber\\
&&
+\left.\left[
\frac{2\mu\left(1-\mu+3\epsilon\right)+\xi\left(1+\epsilon\right)}
{2\left(1+\epsilon-\mu\right)}
-\frac{4\left(1+\mu+\epsilon\right)}{4-\xi}
+\frac{4\epsilon\left(1+3\mu+\epsilon\right)}{4\epsilon-4\mu+\xi}
\right]
C_0\left(-m_\chi^2+\frac{m_Z^2}{2},m_\chi^2,m_Z^2;m_\psi^2,m_f^2,m_\psi^2\right)
\right\},\nonumber\\\end{aligned}$$
where $B_0$ is defined by $$B_0\left(p_1^2;m_1^2,m_2^2\right)=
\int\frac{d^d\ell}{i\pi^2}
\frac{1}{\ell^2-m_1^2}\frac{1}{\left(\ell+p_1\right)^2-m_2^2}.$$
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[^1]: In [@Boehm:2006gu], the $\gamma \gamma$ amplitude was calculated for a scenario of MeV dark matter, but for the case of couplings to both left- and righthanded fermions, to leading order in $1/m_\psi$.
[^2]: Our result differs from the one reported in [@Bertone:2009cb; @Tulin:2012uq]. In particular, we obtain a finite amplitude when $\mu\rightarrow 1$.
[^3]: In principle, also scattering processes involving the Yukawa coupling to the SM lepton $f$ are possible; however, the corresponding scattering cross section is heavily suppressed as the first non-zero contribution arises from two-loop diagrams [@Kopp:2009et].
[^4]: Strictly, the limits for the Majorana case can only be applied to the scalar case when the fermionic mediator and the dark matter particle are close in mass, $m_\psi/m_\chi \lesssim 2-3$, namely when the two-to-three annihilations into $f \bar{f} \gamma$ dominate over the one loop annihilations into two photons. For larger mass ratios, $m_\psi/m_\chi \gtrsim 3$, the relative importance of the two-to-three and one loop annihilations is different for Majorana and for scalar dark matter and the limits for the former cannot be straightforwardly applied for the latter. Due to the mild dependence of the upper limits on $\langle \sigma v \rangle$ on the mass ratio [@Garny:2013ama], we estimate that for $m_\psi/m_\chi>3$ the actual upper limits for the scalar dark matter model deviate from these results by at most a factor of two.
[^5]: We have used FeynCalc [@Mertig:1990an] and LoopTools [@Hahn:1998yk] for parts of the computations.
|
---
date: |
Akash Goel, Reevu Maity, Pratim Roy, Tapobrata Sarkar [^1] 0.4cm [*Department of Physics,\
Indian Institute of Technology,\
Kanpur 208016,\
India*]{}
title: Tidal Forces in Naked Singularity Backgrounds
---
Introduction
============
Einstein’s theory of General Relativity (GR) [@Weinberg] is the most successful theory of gravity till date. An intriguing prediction of the theory is the existence of mathematical singularities, which have been a major focus of research in gravity over the last century. Indeed, black holes, which are singular solutions of the Einstein’s equations predicted by GR, form a basic ingredient in our understanding of the universe as it is generally believed that the centre of supermassive galaxies are candidate black holes. There are also other singular solutions of GR called naked singularities that are not as well understood, but their existence cannot be ruled out. Possible distinctions between black holes and naked singularities have been of great interest in the recent past. To substantiate and carry forward such analysis is the purpose of this paper.
Recall that singularities in GR arise in gravitational collapse processes, see, e.g [@Poisson]. Indeed, In the absence of pressure, a spherically symmetric distribution of matter will collapse under its own gravity. This scenario is often repeated in more realistic situations including effects of pressure. The end stage of a generic collapse process is singular, and results in a black hole or a naked singularity [@Joshi1], depending on initial conditions. While black holes are space-time singularities in GR that are covered by an event horizon, naked singularities do not have such a cover. The cosmic censorship conjecture originally proposed by Penrose more than four decades back states that nature does not allow naked singularities. However, a complete proof for this conjecture is lacking till date, and several research groups have directed their attention to the formation of naked singularities, their stability, and in particular distinguishing features between naked singularities and black holes. This last points assumes significance in predicting observational differences, if any, between black holes and naked singularities. Such analyses has been carried out for example in the context of accretion discs [@Joshi2] and gravitational lensing [@Virbhadra], [@tapo1]. In this work, we focus on another important aspect of gravity, namely tidal forces, and the purpose of this paper is to investigate the difference in the nature of these forces in naked singularity backgrounds, compared to black hole cases. The motivation for this line of approach is that such differences might be significant to be observable. In particular, tidal effects on neutron stars [@Shapiro] in black hole backgrounds have been extensively studied in the literature and one of our aims is to quantify the differences between these vis a vis naked singularity backgrounds, which might be relevant in realistic situations.
Tidal forces are manifestations of non-local gravitational interactions. Consider, for example, a celestial object moving under the gravity of a more massive object. Non local gravity might cause this object to disintegrate. Viewed in the Newtonian context, this is easy to visualize. Consider for example a massive star of mass $M$ and radius $R$ that has a satellite made of incompressible matter, of mass $M_s$ and radius $r_s$, both objects assumed to be spherical, with their centres at a distance $d$ apart. An estimate of the tidal force can be obtained by computing the gravitational forces due to the star at the (near) end of the satellite and its centre. Assuming that $r_s/R \ll 1$ and equating this with the self gravitational force of the satellite predicts that the satellite will disintegrate if the dimensionless ratio $(M_s/M)(d/R_s)^3$ is less than a critical value, this value depending on the spin angular velocity of the satellite. Then, this ratio translates into the fact that tidal disruptions occur if the radial separation between the star and the satellite is less than a critical value, known as the Roche limit.
While tidal forces are easy to understand in the Newtonian context when the stellar mass is made of incompressible fluids, they may be substantially difficult to compute in the framework of GR, especially if one considers the fluid dynamics of the stellar object. The importance of this latter fact has been recognized of late, and by now a large body of literature is available, which considers the nature of the fluid star that is being tidally disrupted. Appropriate coordinates to deal with the problem in the framework of GR are the Fermi normal coordinates [@Manasse]. Here, one first sets up a locally flat system of coordinates along a geodesic, and then computes the tidal force taking into account the internal fluid dynamics of the star, and hence obtain the Roche limit. The actual computation procedure is non-trivial, and with little analytical handle available, one usually resorts to numerical analysis, along with a number of simplifying assumptions. This method was first elaborated upon in the work of Ishii et al [@ishii], where tidal effects on objects in circular geodesics around a Kerr black hole were computed, upto fourth order in the Fermi normal coordinates. This built upon the work of Fishbone [@Fishbone], and to the best of our knowledge, was the first GR computation of tidal effects taking into account the equilibrium fluid dynamics of the stellar mass. The fluid star was treated in a Newtonian approximation, wherein it was possible to superpose a tidal potential calculated in Fermi normal coordinates on the star’s Newtonian potential, and a numerical routine was used to extract various physical quantities related to the tidal force.
![Critical Mass For Circular geodesics in the Kerr geometry for different values of $a$, the rotation parameter. The singularity at $r=0$ is naked for $a>1$.[]{data-label="Kerr"}](kerrcircf.eps)
The purpose of this paper is to generalize and extend the analysis of [@ishii] to naked singularity backgrounds. A simple possibility for example is to consider tidal forces in the Kerr background considered in [@ishii], and extend it to the regime where the singularity is naked. This might happen for example, when a black hole accretes angular momentum by capturing rapidly spinning stellar objects. High spin black holes are known to exist in nature, and studies involving the spin of the Kerr black hole lesser than, but close to its mass, have appeared in [@Liu]. Here we investigate the situation in which the spin of the black hole exceeds its mass.
We will, for most of this paper, adopt geometrized units in which the Newton’s constant and the speed of light is set to unity. Then, the Kerr solution given in Boyer Lindquist coordinates reads $$\begin{aligned}
ds^2 &=& -\left(1-\frac{2Mr}{\rho^2}\right)dt^2 + \frac{\rho^2}{\Delta}dr^2 + \frac{(r^2 + a^2)^2 - \Delta a^2{\rm sin}^2\theta}{\rho^2}{\rm sin}^2\theta d\phi^2 \nonumber\\
&+& \rho^2 d\theta^2 -\frac{4Mra{\rm sin}^2\theta}{\rho^2}dtd\phi~,\end{aligned}$$ with $\rho^2 = r^2 + a^2{\rm cos}^2\theta$, and $\Delta = r^2 + a^2 - 2Mr$. Here, $M$ is the ADM mass and $a$, the angular momentum per unit mass of the source is a parameter that denotes the spin of the black hole. For $a > M$, the singularity at $r=0$ is naked, and there is no event horizon. We will also set the ADM mass of the singularity, $M=1$ and the radius of the fluid star to $0.5$, a choice made consistently in all examples considered in this paper. Fig.(\[Kerr\]) (obtained by methods illustrated in the later sections where we have used the Fermi normal coordinates upto second order) shows the variation of the critical mass below which a star in a circular geodesic orbit is tidally disrupted, as a function of the rotation parameter (circular orbits in the Kerr naked singularity background and their stability has been studied extensively in [@Stuchlik]. We use the result of that paper that all circular orbits in the Kerr naked singularity background are stable). Here we have chosen the radius of the orbit of the star as $r=8$ (in geometrized units), and used a polytropic equation of state for the star (as given later in Eq.(\[poly\])). The solid red, dot dashed blue and dotted green curves correspond to different values of the polytropic index ($0.5$, $1$, and $1.5$ respectively). The Kerr singularity is naked for $a>1$. We see from the figure that there is no special behavior of the critical mass near the region $a=1$, and that the tidal force decreases with increasing angular momentum of the source. This latter fact implies that for the Kerr naked singularity background, tidal effects are smaller than those in Kerr black hole backgrounds. That this is not generically the case is one of the main results of this paper.
Two things need to be kept in mind at this stage. First of all, we are using a vacuum solution of GR, which is an good approximation in the above example, with $r=8$, i.e far from the central singularity. More realistic situations arise when there is diffused matter (possibly dark matter) around the central singularity, and this will be considered later in this paper. Secondly, as can be seen from Fig.(\[Kerr\]), the change in the critical mass as a function of the rotation parameter is small. Hence, there is little hope to distinguish between a Kerr black hole and a Kerr naked singularity from the point of view of tidal disruptions. As we will see later in this paper, other naked singularity backgrounds offer a better scenario, and that one could in principle obtain effects of tidal disruption that are orders of magnitude apart from corresponding black hole situations.
This paper is organized as follows. In the next section, we will first summarize the main ingredients used for our calculations and the various space-times considered in this work. This involves three main steps. First we set up the Fermi normal coordinates for the naked singularity backgrounds. Next, the hydrostatic equations for a fluid star at equilibrium are set up, and in the final step, a numerical routine is elaborated upon, for both circular and radial geodesic motions. The last two analyses discussed here essentially follow the work of Ishii et al [@ishii]. Towards the end of section 2, we will lay down all the approximations that we have made, and also comment on some of existing important results in the area. Then, in section 3, we present our main numerical results. Section 4 ends this paper with discussions and directions for future research.
Computation of Tidal Effects in Naked Singularity Backgrounds
=============================================================
We remind the reader that throughout this paper, we adopt geometrized units and set $G=c=1$. Units will be explicitly restored when we discuss some examples towards the end of section 3. We will also use $T_{(ij)k}$ to denote the part of the tensor $T$ symmetric in indices $i$ and $j$. Latin and Greek indices are used to denote spatial and space-time coordinates respectively. To begin with, we will illustrate the setup of the Fermi normal coordinates, following the work of [@Manasse].
Metric in Fermi Normal Coordinates
----------------------------------
In order to compute the tidal potential observed by a local inertial observer, we set up a locally flat system of coordinates on an arbitrary timelike geodesic, $G$. This would provide a way for a freely falling observer to report the effects of graviational field gradients in local experiments. We follow the standard procedure for constructing Fermi normal coordinates by setting up a tetrad basis at a point on the geodesic as the origin of our new coordinates, $\hat{e}^{\mu'}_{\nu}|_{P_0} $ where unprimed indices denote the Fermi normal coordinates. We parallely transport the tetrad basis along $G$ and in particular choose $\boldsymbol{\hat{e}}_0(\tau)$ to be the tangent vector along $G$ which is, by definition, parallel transported. We hence obtain the tetrad basis on the entire geodesic as a function of the proper time. The Fermi normal coordinates, $x^{\alpha}$ of a point $P'$ in the neighbourhood of the geodesic are then specified. We define $x^0=\tau$ and $x^i$ corresponds to the point along the unique spacelike geodesic, $G'$ at proper distance $s$ with tangent vector at $P(\tau)$ given by direction cosines $x^i/s$ in the tetrad basis, $\boldsymbol{\hat{e}}_i$.
In order to demonstrate the differences in tidal effects, we will consider general static, spherically symmetric spacetimes to cover a broad class of physical scenarios. We present an explicit computation of the Fermi normal coordinates along circular and radial geodesics for these. The class of metrics that we consider in this paper are given by $$\mbox{d}s^2=-A(r)\mbox{d}t^2+B(r)\mbox{d}r^2+C(r)\mbox{d}\Omega^2~,$$ where the coefficients are positive functions of the radial coordinate and $\mbox{d}\Omega^2=\mbox{d}\theta^2+\sin^2\theta\mbox{d}\phi^2$ is the metric on the unit two sphere.
The metric possesses two cyclic coordinates, $t$ and $\phi$ corresponding to the conserved quantities $$E=A(r)\dot{t},~~~L=C(r)\dot{\phi}~,$$ where $E$ and $L$ are the energy and angular momentum per unit mass of the test particle. From the remaining geodesic equations, we reduce the problem to the equivalent one-dimensional problem with effective potential $$V(r)=\frac{1}{B(r)}\left[1+\frac{L^2}{C(r)}-\frac{E^2}{A{r}}\right]~.$$ Imposing $V(r)=V'(r)=0$ for circular orbits gives $$E=\frac{A(r)\sqrt{C'(r)}}{\sqrt{A(r)C'(r)-C(r)A'(r)}},~~~
L=\frac{ C(r)\sqrt{A'(r)}} {\sqrt{A(r)C'(r)-C(r)A'(r)}}~.$$ We now set up the Fermi normal basis which must satisfy the parallel transport conditions $$\boldsymbol{\hat{e}}_{\mu}.\boldsymbol{\hat{e}}_{\nu}=\eta_{\mu\nu},~~~
\nabla_{\alpha'}\left( \hat{e}^{\beta'}_{\mu } \right) \hat{e}^{\alpha'}_0=0~,
\label{eq:ptr}$$ and $\boldsymbol{\hat{e}}_0$ is the tangent vector to the geodesic. This gives us the tetrad $$\begin{aligned}
\hat{e}^{\alpha'}_0 &=&\left( \frac{E}{A(r)},0,0,\frac{L}{C(r)} \right) \nonumber\\
\hat{e}^{\alpha'}_1 &=&\left( -\frac{L\sin(\Omega\tau)}{\sqrt{A(r)C(r)}},\frac{\cos(\Omega\tau)}{\sqrt{B(r)}},0, -\frac{E\sin(\Omega\tau)}{\sqrt{A(r)C(r)}} \right) \nonumber\\
\hat{e}^{\alpha'}_2 &=&\left( 0,0,\frac{1}{\sqrt{C(r)}},0 \right)\nonumber\\
\hat{e}^{\alpha'}_3 &=&\left( \frac{L\cos(\Omega\tau)}{\sqrt{A(r)C(r)}},\frac{\sin(\Omega\tau)}{\sqrt{B(r)}},0, \frac{E\cos(\Omega\tau)}{\sqrt{A(r)C(r)}} \right)~.\end{aligned}$$ Here, the constant angular velocity of motion is $$\Omega=D(r)\dot{\phi}=\frac{1}{2}\sqrt{\frac{A'(r)C'(r)}{A(r)B(r)C(r)}}~,$$ where we have defined $$D(r) =\frac{1}{2}\sqrt{\frac{C'(r)\left[A(r)C'(r)-A'(r)C(r)\right] }{A(r)B(r)C(r) }}~.$$ In case of radial geodesics, proceeding similarly, we obtain $$\dot{t}=\frac{E}{A(r)}~,~~~
\dot{r}=\sqrt{\frac{E^2-A(r)}{A(r)B(r)}}~.$$ We set up the tetrad satisfying Eq.(\[eq:ptr\]) using the geodesic equations in this case
$$\begin{aligned}
\hat{e}^{\alpha'}_0 &=&\left( \dot{t},\dot{r},0,0 \right) \nonumber\\
\hat{e}^{\alpha'}_1 &=&\left( \dot{r}\sqrt{\frac{B(r)}{A(r)}},\dot{t}\sqrt{\frac{A(r)}{B(r)}},0,0 \right) \nonumber\\
\hat{e}^{\alpha'}_2 &=&\left( 0,0,\frac{1}{\sqrt{C(r)}},0 \right) \nonumber\\
\hat{e}^{\alpha'}_3 &=&\left( 0,0,0,\frac{1}{\sqrt{C(r)}\sin\theta} \right)~.\end{aligned}$$
Calculating the Tidal Tensor
----------------------------
Once the tetrad is constructed, we obtain the components of the curvature tensor in the tetrad basis $$R_{\alpha\beta\gamma\delta}=\hat{e}^{\mu'}_{\alpha}\hat{e}^{\nu'}_{\beta}
\hat{e}^{\rho'}_{\gamma}\hat{e}^{\kappa'}_{\delta} R_{\mu'\nu'\rho'\kappa'}~.$$ The metric of an observer in the Fermi normal coordinate system can be expanded as $$g_{\mu\nu}=\eta_{\mu\nu}+\frac{1}{2}g_{\mu\nu,ij}x^ix^j+\frac{1}{6}g_{\mu\nu,ijk}x^ix^jx^k+O(x^4)~.$$ The expressions for the derivatives of the metric in the Fermi normal basis were computed in [@Manasse] and [@ishii]. The tidal potential, $\phi_t$ associated with the metric can then be expanded as $$\phi_t = -\frac{1}{2}(g_{00}+1)~ = \frac{1}{2}R_{0i0j}x^ix^j+\frac{1}{6}R_{0(i|0|j;k)}x^ix^jx^k+O(x^4)~.
\label{phit}$$
In case of the circular geodesic, the non-vanishing derivatives of the metric needed for the computation of the tidal tensor at leading order are $$\begin{aligned}
g_{00,11}&=&\frac{CA'^2C'-AC'^2A'-2ACC'A''+2ACA'C''}{4ABC(AC'-A'C)}\nonumber\\
g_{00,22}&=&\frac{A'}{A'C-AC'},~~g_{00,33} = -\frac{A'C'}{4ABC}~.\end{aligned}$$
For radial geodesic, the energy of the test particle $E$ is a free parameter. This parameter cannot however assume large values, as we discuss in subsection 2.5. The corresponding non-zero quantities in this case are $$\begin{aligned}
g_{00,11}&=&\frac{A^2[-2ABA''+A'(BA'+AB')]}{4A^4B^2}\nonumber\\
g_{00,22}&=&g_{00,33} = -\frac{E^2BCA'C'+A(E^2-1)(BC'^2+C(B'C'-2BC''))}{4A^2B^2C^2}~.\end{aligned}$$
In our analysis, we will focus on the following class of static, spherically symmetric metrics to compare the black hole and naked singularity cases-
- [*Janis Newmann Winicour (JNW) spacetime*]{} [@JNW1],[@JNW2]- This is a solution to the Einstein-Klein-Gordon system of equations. The spacetime consists of a singularity that is globally naked at $r=B$. $$\mbox{d}s^2=-\left(1-\frac{B}{r}\right)^{\nu}\mbox{d}t^2+\left(1-\frac{B}{r}\right)^{-\nu}\mbox{d}r^2+r^2\left(1-\frac{B}{r}\right)^{1-\nu}\mbox{d}\Omega^2~.
\label{JNWmetric}$$ This spacetime is sourced by the scalar field with magnitude $q$ $$\psi=\frac{q}{B\sqrt{4\pi}}\ln\left(1-\frac{B}{r}\right)~.$$ Here, $B$ is a parameter related to the ADM mass by $B=2\sqrt{q^2+M^2}$ and $\nu=2M/B$ lies between 0 and 1. Note that the case $\nu=1$ corresponds to the Schwarzschild metric which is a black hole solution. For a fixed value of the ADM mass, moving away from $\nu = 1$ takes us deep into the naked singularity regime. Note however that there is a tradeoff here, namely that smaller values of $\nu$ implies larger values of $B$, which in turn means that the solution is valid from a higher value of the radial coordinate $r$. Also, we would be mostly interested in stable circular orbits, and the criteria for such orbits are well established. We simply quote the result that defining the variable $$r_{\pm} = \frac{B}{2}\left( 1 + 3\nu \pm \sqrt{5\nu^2 - 1}\right)~,$$ it can be checked that for $0 < \nu < 1/\sqrt{5}$, stable circular orbits exits for all values of the radius (greater than $B$). For $\nu$ lying between $1/\sqrt{5}$ and $1/2$, stable circular orbits are possible for radii less than $r_-$ and greater than $r_+$, while for $\nu$ greater than $1/2$, such orbits are possible for all values of the radius greater than $r_+$. In our analysis on the JNW spacetimes, we have chosen values of the radii that correspond to stable circular orbits.
- [*Bertrand spacetimes (BSTs)*]{} [@Perlick],[@dbs2]- These are seeded by matter that can be given an effective two-fluid description and are a candidate for galactic dark matter. We look at the type II BST $$\mbox{d}s^2=-\frac{1}{D+\frac{\alpha}{r}}\mbox{d}t^2+\frac{1}{\beta^2}\mbox{d}r^2+r^2\mbox{d}\Omega^2~.
\label{BSTmetric}$$ Here, $\beta$ is a rational. The parameters $\alpha$ and $\beta$ can be used to provide a phenomenological estimate for the total dark matter mass of the galaxy (in good agreement of known results for low surface brightness galaxies) as [@dbs1] $$M=\frac{\alpha R_g^2}{2(\alpha+DR_g)^2}~,
\label{massBST}$$ where a good estimate for the size of the galaxy is $R_g=\alpha/D$. It can be checked that here we have a central singularity at $r=0$, which is naked. For computations performed with BSTs, we have set $M=1$, which implies, via Eq.(\[massBST\]) that $\alpha = 8D^2$. This will be understood in sequel. It can be checked that for BSTs, all circular orbits are stable.
- [*Joshi Malafarina Narayan (JMN) spacetime*]{} [@JMN1],[@JMN2] - describes a class of geometries that result from the gravitational collapse of a matter cloud with regular initial conditions into an astympotically static equilibrium configuration containing a central naked singularity at $r=0$ $$\mbox{d}s^2=-(1-M_0)\left(\frac{r}{R_0}\right)^{\frac{M_0}{1-M_0}}\mbox{d}t^2+\frac{1}{1-M_0}\mbox{d}r^2+r^2\mbox{d}\Omega^2~.
\label{JMNmetric}$$ This metric also matches smoothly to a Schwarzschild exterior at $r=R_0$ with mass $M=M_0R_0/2$. The system possesses stable circular orbits for $M_0\leq 2/3$.
While the JNW, BST and JMN space-times discussed above contain naked singularities, towards the end of this paper, we will also briefly comment on the nature of tidal forces in the interior Schwarzschild solution. This describes the metric of a fluid of constant density that is matched with an external Schwarzschild solution of mass $M$, and reads $$\mbox{d}s^2=-\left( \frac{3}{2}\sqrt{1-\frac{2M}{R_0}} - \frac{1}{2}\sqrt{1-\frac{2Mr^2}{R_0^3}} \right)^2\mbox{d}t^2+
\left(1-\frac{2Mr^2}{R_0^3}\right)^{-1}\mbox{d}r^2+r^2\mbox{d}\Omega^2~.
\label{IntSchmetric}$$ The matching radius here is $r=R_0$.
Hydrostatic equations for Computing Equilibrium Configurations
--------------------------------------------------------------
We will now review a formulation to compute the equilibrium configurations under gravitational tidal forces arising from the potential gradients in a curved spacetime geometry. The discussion in this and the next subsection essentially follows [@ishii], and we closely follow the notation of that paper. We assume that the radius of the star [^2] is much smaller than the radius of its orbit so that in Fermi normal coordinates, the self gravity of the star is described by Newtonian gravity, with an additional tidal potential due to the curved background. In such a case, the fluid star obeys the hydrodynamic equation $$\label{euler}
\rho\frac{\partial v_i}{\partial \tau} + \rho v^j \frac{\partial v_i}{\partial x^j}=-\frac{\partial P}{\partial x^i}-
\rho\frac{\partial (\phi+\phi_{t})}{\partial x^i}+\rho\left[ v_j\left( \frac{\partial A_j}{\partial x^i} -\frac{\partial A_i}
{\partial x^j} \right) -\frac{\partial A_j}{\partial \tau} \right]~.$$ This equation is analogous to the Euler equation with the tidal potential $\phi_{t}$ superposed with the Newtonian potential $\phi$ and an additional term associated with the gravitomagnetic force described by the vector potential $A_k=\frac{2}{3}R_{k(ij)0}x^ix^j$. Here, $v^i=dx^i/d\tau$ is the fluid three-velocity and $P$ is the fluid pressure. The Newtonian potential produced due to the mass of the star obeys the Poisson equation $\Delta \phi=4\pi\rho$.
As is common in the literature, we assume a polytropic equation of state for the star given by $$P=\kappa \rho^{1+\frac{1}{n}}~,
\label{poly}$$ where $\kappa$ and $n$ are polytropic constants. The equation of continuity, along with that for hydrostatic equilibrium then leads to the Lane-Emden equation $$\frac{1}{\xi^2}\frac{d}{d\xi}\left( \xi^2\frac{d\theta}{d\xi} \right)+\theta^n=0~,$$ where $\xi$ is a dimensionless radial parameter obtained by scaling the radius, and $\rho=\rho_0\theta^n$, where $\rho_c$ is the central density. In our computations, this will be used to fix the stellar radius in terms of the Lane-Emden coordinate at the stellar surface $\xi_0$ $$R=\left(\ \frac{(n+1)\kappa\rho_c^{(1-n)/n}}{4\pi} \right)^{1/2}\xi_0~.$$ We will consider the values of the polytropic index $n=$ $0.5$, $1$ and $1.5$ for which it can be checked that $\xi_0 =$ $2.75$, $\pi$ and $3.65$ respectively.
We first consider the case of circular geodesics. Following [@ishii], we assume a co-rotational velocity field for the fluid star with the velocity field $$v^i=\left[ -{x^3-x_c\sin(\Omega\tau)},0,{x^1-x_c\cos(\Omega\tau)} \right]~,$$ where $x_c$ is a small correction constant. In order to simplify numerical computations, we go to a frame where the Euler equation is independent of $\tau$. This is achieved by choosing the frame, $\tilde{x}^i$ rotating at angular velocity $\Omega$ about $x^2$ axis. The hydrostatic equation corresponding to the first integral of the Euler equation in this frame is then given as $$\label{int}
\frac{\Omega^2}{2}\left[(\tilde{x}^1-2x_c)^2+(\tilde{x}^3)^2\right] = \kappa(n+1)\rho^{\frac{1}{n}} + \phi +\phi_{t} + \phi_{m} + C~.$$ Here $\phi_m$ is the contribution due to the gravitomagnetic force and we have used the polytropic equation.
In case of radial geodesics, [^3] we take the velocity field to be irrotational, and hence the gradient of a velocity potential, $$v_i+A_i=\frac{\partial\psi}{\partial x^i}~.$$ The first integral for is then given as $$\label{eulerint}
-\frac{\partial\psi}{\partial\tau}-\frac{1}{2}\delta_{ij}\frac{\partial\psi}{\partial x^i}\frac{\partial\psi}{\partial x^j}=
\kappa(n+1)\rho^{\frac{1}{n}} + \phi +\phi_{t} -\frac{\partial\psi}{\partial x^j}A^j + C~.$$ Here, $\psi$ is determined by solving the equation of continuity, $$\label{continuity}
\rho\Delta\psi+\delta_{ij}\frac{\partial\psi}{\partial x^i}\frac{\partial\rho}{\partial x^j}=0~.$$ These, along with the Poisson equation are the governing equations for the radial case.
Numerical Procedure
-------------------
For our numerical procedure, we again closely follow [@ishii]. We first consider the circular case. Here the governing equations are the Poisson equation and Eq., and these are to be solved iteratively. In order to achieve a convergence in the iteration, we switch to a dimensionless coordinate, $\tilde{x}^i= h q^i$. We correspondingly define the rescaled potentials, $\bar{\phi}=\phi/h^2$, $\bar{\phi_t}=\phi_t/h^2$ so that our system of equations reduces to $$\Delta_h \bar{\phi}=4\pi\rho~,$$ $$\label{euler2}
\frac{\Omega^2}{2}h^2\left[(q^1-2q_c)^2+(q^3)^2\right] = \kappa(n+1)\rho^{\frac{1}{n}} + h^2(\bar{\phi} +\bar{\phi}_{t} + \bar{\phi}_{m}) + C~.$$ This system contains three primary constants, namely $h$, $q_c$ and $C$. The remaining constants are fixed by choosing units so that the Newtonian mass of the central body is unity or by choosing the metric parameters in these units. The equilibrium configurations will be computed for different values of $\rho_c$. The three primary constants will have to be fixed at each step of the iteration. As in [@ishii], we solve the Poisson equation in Cartesian coordinates on a uniform grid of size (2$N$+1, $N$+1, 2$N$+1) in order to cover the region $-L\leq q^1\leq L$, $0\leq q^2\leq L$, $-L\leq q^3\leq L$. Here, we have assumed reflection symmetry with respect to the $q^2=0$ plane. We typically choose $N=50$ and the grid spacing to be $q_s/40$, where $q_s$ is the coordinate at the stellar surface along the $q^1$ axis. We use a Fortran subroutine to solve the fourth-order finite difference approximations to the elliptic partial difference equations. We implement a numerical algorithm that is summarized as :
- We take a trial density distribution and use the cubic grid-based Poisson solver to solve the Poisson equation. We impose Neumann boundary conditions on the $q^2=0$ plane to achieve reflection symmetry along the $q^2$ axis and Dirichlet boundary conditions elsewhere, namely, $$\phi\to -\frac{1}{r}\int \rho\,d^3r + O(r^{-3})~.$$ The distribution $\phi$ is obtained which is used at the next step.
- The constants $h$, $q_c$, $C$ are determined from the Euler equation by imposing constraints on the density profile. We match the central density, $\rho(0,0,0)=\rho_c$ and require that $\partial\rho/\partial q^1=0$ at the center. Also, we require the density to vanish at the stellar surface, $\rho(q_s,0,0)=0$. The set of constraints when substituted in can be solved to determine the values of the constants. From these values, the new distribution $\rho(q^i)$ is computed again using .
- The new density distribution is truncated at the first instance where it goes to zero and is again used as a source in the Poisson equation. The process is repeated iteratively until it converges to the equilibrium distribution and the density profile becomes stationary upto a numerical tolerance value.
- We compute several equilibrium configurations for decreasing $\rho_c$ and track the quantity $\chi=\partial \rho/\partial q^1$ at the stellar surface ($q_s$, 0, 0). The critical value of $\rho_c$ corresponding to the tidal disruption limit i.e. the Roche limit is obtained when $\chi$ goes abruptly from a positive value to zero. Beyond this point, the density distribution becomes flat, signalling tidal disruption. For positive $\chi$, the star is in a stable configuration.
We can adopt a similar procedure for radial geodesics where there is an extra function $\psi$ that has to be determined. This can be handled using the additional constraint imposed by the continuity equation. We proceed as follows. As before, a trial density function is used to solve the Poisson equation and obtain $\phi$. This initial density distribution is also used to solve the elliptic partial differential Eq. using a three dimensional Cartesian PDE solver based on multigrid iteration. Once $\phi$ and $\psi$ are known, we use Eq. to determine the contants $h$ and $C$. This is done by imposing two constraints, namely $\rho=\rho_c$ at the center and the density must vanish at the stellar surface. One can now use the values of these cosnstants in Eq. to determine $\rho$ on the entire grid. This new density distribution in again fed into the Poisson and continuity equations and the iterative procedure is used as previously to compute the equilibrium configuration, and hence the Roche limit.
Approximations, Limitations and Related Issues
----------------------------------------------
Before we move on to present our results, we point out the approximations that we will make, and the limitations of our analysis and possible caveats that need to be kept in mind.
Since we are mainly interested in demonstrating the difference in tidal forces between naked singularities and black hole backgrounds, it suffices to work up to second order in the tidal potential. Fourth order effects can be included in our analysis, but these, apart from being small, will not qualitatively change our results. A more refined analysis of the scenarios presented here including higher order corrections to the metric in Fermi normal coordinates is left for the future. Further, the contribution of the gravitomagnetic term in the hydrodynamic equations is only significant at fourth order in the expansion and so, these terms can be neglected at our level of approximation. Also, for the radial geodesics, for simplicity, we have chosen the velocity potential to be zero to give the irrotational field. In the rest of the paper, we will proceed with these approximations.
There are a few caveats that we need to keep in mind. First, note that we work in a probe approximation, where the effects of the star back reacting on the metric is ignored. Effects of back-reaction are somewhat intractable in the present formalism and pose a formidable challenge. Here we proceed with the assumption that the background metric is fixed.
Second, in case of radial geodesics, our analysis for the Roche radius is valid under the approximation of instantaneous equilibrium. In general, this radius will be less (or equivalently, the critical mass will be smaller) than that calculated from the numerical simulation by a small correction equal to the distance travelled by the star in the delay period during which the star attains the equilibrium configuration. The approximation is valid in the limit that the distance travelled in the time scale during which the star attains equilibrium is smaller than the distance over which the tidal force changes appreciably. This condition is most naturally satisfied for low energy scales. We will, for radial geodesics, set $E = 10^{-2}$ and it is assumed that at our level of approximation, the tidal force is constant over the range of validity of the Fermi normal expansion.
Next, we note that a general tidal model can also result in a weak tidal encounter between a star orbiting in a massive background. Such encounters can result in a mass loss from the star, as discussed in details in [@cheng]. These effects on white dwarfs are seen to increase the likelihood of tidal disruption as a function of time. In this work, we will however restrict ourselves to conditions for complete tidal stripping for both radial and circular orbital motion.
Finally, although not presented here, the present analysis can be used to build upon the work of [@kesden] to calculate rates for tidal disruption events for supermassive black holes capturing not only the relativistic treatment of tidal disruption but also the hydrodynamics of the orbiting star. The method is summarized as follows. Monte Carlo simulations are performed with an appropriate distribution of the free variables in the geodesic equations for a general orbit. These equations, in Fermi normal coordinates, are integrated upto the pericenter. In our case, we can run the numerical code to determine if the star is directly captured or is tidally disrupted at this point by computing the equilibrium hydrodynamic configuration. This should give us an estimate for the tidal disruption rate of the simulated orbits. One can then study the effects of varying different metric parameters and astronomical data from flares associated with these tidal disruption events can be used to provide insight into the nature of the background singularity. In principle, this should improve upon the work of [@kesden], but such an analysis is left for the future.
Results and Analysis
====================
In this section, we will present the results of our numerical analysis. We remind the reader that unless mentioned otherwise, in all examples below, we have set the radius of the star to be $0.5$, and the mass of the background to unity. We start with circular geodesics in various space-times.
![Critical Mass for circular geodesics in BSTs for different $\beta$, with $D=10^5$, $n=0.5$[]{data-label="bstcirc"}](jnwcircf.eps)
![Critical Mass for circular geodesics in BSTs for different $\beta$, with $D=10^5$, $n=0.5$[]{data-label="bstcirc"}](bstcircf.eps)
In Fig.(\[jnwcirc\]), we show the critical mass as a function of the radial distance for JNW space-times, with the polytropic index (of Eq.(\[poly\])) $n=0.5$. The solid red, dotted green and dot-dashed blue curves correspond to the values $\nu =$ $0.01$, $0.1$ and $0.6$ respectively, in Eq.(\[JNWmetric\]). For comparison, we have shown with the dashed black curve the corresponding result for the Schwarzschild black hole ($\nu = 1$ in Eq.(\[JNWmetric\])). Clearly, the effect of this naked singularity background is seen to increase the tidal force on the star. This is our first observation : if, at a given radius, stellar objects above the critical mass predicted from the Schwarzschild black hole are seen to exist, they might point towards a naked singularity, rather than one with a horizon.
In Fig.(\[bstcirc\]), the computation is repeated for the BST naked singularity background of Eq.(\[BSTmetric\]). In this figure, the solid red, the dotted green and the dot-dashed blue curves correspond to choosing the values $\beta$ of Eq.(\[BSTmetric\]) as $0.2$, $0.5$ and $0.8$, respectively. We see here that increase in the value of $\beta$ corresponds to higher tidal forces. This is again indicative of the fact that for similar central masses, the BST naked singularity predicts higher tidal disruption limits. If observational indications of this fact are found in future experiments, BSTs can possibly be used as models to understand such a scenario.
![Leading Order Tidal Potential for Circular BST, with the chosen values $D=10^5$, $\beta=0.8$, $r=8$.[]{data-label="bsttidpot"}](bsttidpot2.eps)
![Leading Order Tidal Potential for Circular BST, with the chosen values $D=10^5$, $\beta=0.8$, $r=8$.[]{data-label="bsttidpot"}](bsttidpotf.eps)
![Density variation near equilibrium for BST (Circular Orbit), with $D=10^5$, $\beta=0.8$, $n=1.5$[]{data-label="bstden2"}](bstdens0-5f.eps)
![Density variation near equilibrium for BST (Circular Orbit), with $D=10^5$, $\beta=0.8$, $n=1.5$[]{data-label="bstden2"}](bstdens1-5f.eps)
![Critical mass for the radial geodesics in the JMN background for different values of the radius of the star $R_0$ with $M_0$ = 0.1[]{data-label="radjmn"}](radjnwf.eps)
![Critical mass for the radial geodesics in the JMN background for different values of the radius of the star $R_0$ with $M_0$ = 0.1[]{data-label="radjmn"}](radjmn1f.eps)
Next, we come to the tidal potentials, computed from Eq.(\[phit\]). We show this for two examples. Figs.(\[bsttidpot2\]) and (\[bsttidpot\]) shows the variation of the second order tidal potential along the three axes in the Fermi normal coordinates, for $r=3$ and $r=8$. Although the plots have been shown for BST, we have checked that the nature of the tidal potential is similar for other metrics, for circular geodesics. The gradient of the plot is a measure of the tidal force. The given plots indicate that the potential is confining along the $q^2$ and $q^3$ axes and hence the resulting equilibrium configuration will have major axis along $q^1$. Note that the leading order tidal potential has a symmetry about the origin along all the axes. This symmetry is generically broken in small amounts when higher order terms in the tidal potential are taken into account. We observe from Figures (\[bsttidpot2\]) and (\[bsttidpot\]) that as $r$ is decreased, the tidal potential becomes steeper, indicating higher tidal effects. For the Kerr case, we observed that the potential is less steep along the $q^1$ and $q^2$ axes for a higher value of $a$ indicating that that the tidal forces weaken as the effect of spin is taken into account.
Metric Parameters $M_{crit}$
--------------- ----------------------- ----------------------
Schwarzschild $M=1$ $2.5 \times 10^{-3}$
JNW $\nu=0.8$ $2.8 \times 10^{-3}$
JNW $\nu=0.6$ $3.5 \times 10^{-3}$
JNW $\nu=0.4$ $6.1 \times 10^{-3}$
BST $D=10^3$, $\beta=0.1$ $5.7 \times 10^{-4}$
BST $D=10^3$, $\beta=0.6$ $1.9 \times 10^{-3}$
BST $D=10^3$, $\beta=0.8$ $5.4 \times 10^{-3}$
BST $D=10^4$, $\beta=0.3$ $2.8 \times 10^{-3}$
JMN $M_0=0.2$ $1.1 \times 10^{-3}$
JMN $M_0=0.08$ $4.4 \times 10^{-4}$
JMN $M_0=0.02$ $1.1 \times 10^{-4}$
: Comparative Table For Circular Geodesics for $r=8$
\[tab1\]
It is also interesting to look at the behavior of the density variation of the star near equilibrium. Figs.(\[bstden1\]) and (\[bstden2\]) show the typical variation of the equilibrium density distribution at the Roche limit in the second order tidal potential along the $q^1$ axis for different values of $q^2$, where we have chosen the polytropic index to be $0.5$ and $1.5$, respectively, as a function of the coordinate $q^1$. In Fig.(\[bstden1\]), this is shown for $q^2=q^3=0$ (solid red), $q^2=0,q^3=3R/8$ (dashed green), $q^2=0,q^3 = R/2$ (dot-dashed blue) and $q^2=0, q^3=5R/8$ (sparse-dotted magenta). In Fig.(\[bstden2\]), the same values of $q^2$ and $q^3$ are used for the solid red and the dashed green curve. In this figure, the dot-dashed blue curve corresponds to $q^2=0, q^3=R/4$.
The density in equilibrium is set to zero at $q^1=0.8$ along the major axis on the grid corresponding to $R_0$. As expected, the density along the $q^2$ direction goes to zero faster than in the $q^1$ direction. We observe that the density is symmetric with respect to the $q^1=0$ plane. This symmetry is generically broken slightly when higher order terms in the tidal potential are taken into consideration. For smaller values of the polytropic index, the pressure, in accordance with the stiffer polytropic equation of state, has a stronger density dependence. This is reflected in the fact the equilibrium configuration is more compact with higher surface density for $n=0.5$ whereas for $n=1.5$, the density goes to zero smoothly at the surface due to lower pressures.
Finally, in Figs.(\[radjnwn\]) and (\[radjmn\]), we show the critical mass versus radial distance for the JNW and JMN singularities. In Fig.(\[radjnwn\]), we show the variation of the critical mass for different values of the polytropic index, with the solid red, dotted green and dot-dashed blue curves corresponding to $n=0.5$, $1$ and $1.5$ respectively. In Fig.(\[radjmn\]), this is shown as a function of the radius of the star, with the solid red, dashed green and dot-dashed blue lines corresponding to this radius being $0.5$, $0.75$ and $1$, respectively.
We now tabulate some numerical data on the various cases that we have discussed. In table (\[tab1\]), we have provided such data for circular geodesics in the Schwarzschild, JNW, BST and JMN metrics, with the mass of the singularity taken to be unity in all the cases. One can see from the table that at this value of the radial distance, the tidal force can be of similar orders of magnitude for the Schwarzschild black hole, and the JNW and the BST naked singularity backgrounds.
Metric Parameters $M_{crit}$
--------------- ----------------------- ----------------------
Schwarzschild $M=1$ $1.5 \times 10^{-3}$
JNW $\nu=0.8$ $1.9 \times 10^{-3}$
JNW $\nu=0.6$ $2.1 \times 10^{-3}$
JNW $\nu=0.4$ $4.7 \times 10^{-3}$
BST $D=10^3$, $\beta=0.1$ $1.4 \times 10^{-4}$
BST $D=10^3$, $\beta=0.6$ $3.4 \times 10^{-3}$
BST $D=10^3$, $\beta=0.8$ $3.9 \times 10^{-3}$
BST $D=10^4$, $\beta=0.3$ $1.3 \times 10^{-2}$
JMN $M_0=0.2$ $5.4 \times 10^{-4}$
JMN $M_0=0.08$ $2.7 \times 10^{-4}$
JMN $M_0=0.02$ $8.1 \times 10^{-5}$
: Comparative Table For Radial Geodesics for $r=8$, $E=0.01$
\[tab2\]
Specifically, tidal forces are seen to be stronger as one decreases the value of $\nu$ in the JNW background. For example, at $\nu = 0.4$, the critical mass allowed by this background is about three times that in a same-mass Schwarzschild background. This would imply that if a stellar object of higher critical mass than predicted by a Schwarzschild analysis is found, this might indicate that one needs to look closer at the nature of the central singularity. On the other hand, the JMN background indicates that stellar objects having masses much smaller (by even two orders of magnitude) than those predicted by a Schwarzschild analysis might exist in stable circular orbits. Although direct observational evidence for such objects might be extremely difficult in practise, nonetheless our simple minded analysis indicates useful bounds on the masses of stellar objects in circular orbits of a given radius. Of course, we have confined our analysis to circular orbits, whereas actual orbits of stellar objects might be highly elliptical. We however expect that qualitative features of our analysis will remain unchanged in such situations also, though this requires a more detailed analysis.
As a consequence of our analysis, we note that for large values of $r$, the differences in the magnitude of the tidal force between the Schwarzschild and the JNW naked singularity background increases appreciably. For example, for circular geodesics at $r=12$, we find that whereas $M_{\rm crit} = 9.8 \times 10^{-4}$ for the Schwarzschild case, it is $5.6 \times 10^{-3}$ for the JNW naked singularity with $\nu = 0.2$. For circular geodesics at $r=21$, the Schwarzschild critical mass is $8.6 \times 10^{-3}$, while the JNW background in this case with $\nu = 0.1$ predicts a critical mass $5.5 \times 10^{-4}$, almost an order of magnitude difference. This latter set of numbers translate into interesting realistic ones as we now illustrate. Take the mass of the central singularity as $1.8 \times 10^3 M_{\odot}$, and $R_0=14~{\rm km}$ (which correspond to $R_0 = 5\times 10^{-3}$ in natural units). Then, $r=21$ in natural units leads to $r = 5.6\times 10^4~{\rm km}$. The Schwarzschild background in this case predicts a critical mass $M_{\rm crit}=1.0M_{\odot}$, while a JNW background with $\nu = 0.1$ predicts $M_{\rm crit} = 15.7 M_{\odot}$, where we have assumed $n=1$ in the polytropic equation of state of the neutron star. Given a typical neutron star mass $\sim 1.5 M_{\odot}$, we see that the neutron star is tidally disrupted for the naked singularity background but not for the Schwarzschild background at this radius.
It is also worthwhile to mention that while the discussion of the previous paragraph focused on the JNW naked singularity, the result assumes significance given the fact that similar conclusions can be reached for BSTs as well. Indeed, from Fig.(\[bstcirc\]), we see indications that the critical mass increases sharply as the value of the parameter $\beta$ in Eq.(\[BSTmetric\]) is increased. It is thus expected that appropriately tuning the parameters of a BST, one can in principle obtain values of the critical mass substantially larger than what is obtained in Schwarzschild backgrounds. On the other hand, as Table (\[tab1\]) indicates, JMN backgrounds typically show lower tidal effects compared to the Schwarzschild cases.
Table (\[tab2\]) summarizes the results for the radial geodesics, which have to be understood along with the limitations discussed in subsection 2.5. We see that these are qualitatively similar to those for the circular geodesics. In this table, we have considered a small energy of the star. Increase in this leads to a higher difference in magnitude for the critical mass for the naked singularity background compared to the Schwarzschild case. However, the assumption that the star has travelled a distance that is small compared to the scale over which the tidal forces change will not be valid for high energies of the star, and thus higher values of energy will probably not be very trustable in our framework.
Before we end, let us briefly comment on the nature of tidal forces in the interior Schwarzschild solution of Eq.(\[IntSchmetric\]). We considered radial geodesics in this geometry, with energy $E=0.01$. Here, we have taken a matching radius $r=20$. Our results in this case suggest that the critical mass reaches a saturation value as the radial distance of the stellar object decreases. However, we were able to achieve good numerical convergence in this case only for a limited number of parameter values. This case is of interest as this exemplifies motion in a diffused matter background by a stellar object (which is assumed not to back react on the background) and merits further understanding, which we defer for a later study.
Discussions and Conclusions
===========================
In this paper, we have examined the nature of tidal forces in a class of non-rotating naked singularity backgrounds on stellar objects in radial and circular geodesic motion. The purpose of this work was to understand theoretical differences in the nature of these forces, keeping in mind observational aspects. Broadly, this paper establishes the magnitudes of these forces for three different naked singularity backgrounds. Our main conclusion here is that tidal forces can be significantly different for these cases when compared to the Schwarzschild background. If in experiments can indicate numerical bounds on the masses of objects that can be tidally disrupted by a singularity, our results might be effective in modelling such situations, to indicate the nature of the underlying singularity.
In this paper, we have confined ourselves to a second order expansion of the tidal potential in Fermi normal coordinates. The analysis of [@ishii], which is based on a fourth order expansion of the potential is numerically more accurate. However, in this paper our main focus was comparing the magnitudes of the tidal force in different backgrounds. Inclusion of higher order corrections, although important, will not qualitatively affect the results of the present analysis.
Also, our analysis is simple minded, has a number of constraints which we have extensively discussed. Nevertheless, we believe that the present analysis complements the work of [@ishii] and sets the stage for a deeper question, namely if tidal disruptions can be an effective indicator of the central singularity in galaxies or galaxy clusters. As an immediate future application, we hope to build upon the results of this paper to understand rates of tidal disruption events, while taking into account the fluid nature of the star. This work is in progress. Further, as is well known, a realistic neutron star has a superfluid character, and its fluid dynamics might be different from what is assumed in this work. How this will affect tidal forces on neutron stars might be an important issue to understand.\
[**Note added :**]{} A preliminary version of the FORTRAN code used for the computations in this paper is available upon request.
[99]{} Steven Weinberg, “Gravitation And Cosmology,” John Wiley & Sons(1972). E. Poisson, “A Relativist’s Toolkit : The Mathematics of Black-Hole Mechanics,” Cambridge University Press, Cambridge, U.K (2004). P. S. Joshi, D. Malafarina and R. Narayan, Class. Quant. Grav. [**28**]{}, 235018 (2011). P. S. Joshi, D. Malafarina and R. Narayan, Class. Quant. Grav. [**31**]{}, 015002 (2014). K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D [**65**]{}, 103004 (2002). D. Dey, K. Bhattacharya and T. Sarkar, Phys. Rev. D [**88**]{}, 083532 (2013). S. L. Shapiro and S. A. Teukolsky, “Black Holes, White Dwarfs, and Neutron Stars,” Wiley Interscience (New York), 1983. F. K. Manasse and C. W. Misner, Journal of Math. Phys. [**4**]{} 735 (1963). M. Ishii, M. Shibata, and Y. Mino, Physical Review [**D71**]{} 044017 (2005). L. G. Fishbone, Astrophys. J. [**185**]{} 43 (1973). Y. T. Liu, Z. B. Etienne, and S. L. Shapiro, Phys. Rev. [**D80**]{} 121503(R) (2009). Z. Stuchlik, Bull. Astron. Inst. Czechosl. [**31**]{}, 129 (1980). A. I. Janis, E. T. Newman, J. Winicour, Phys. Rev. Lett. [**20**]{}, 878 (1968), M. Wyman, Phys. Rev. D [**24**]{}, 839 (1981). K. S. Virbhadra, Int J Mod Phys A [**12**]{} 4831 (1997). V. Perlick, Class. Quantum Grav. [**9**]{}, 1009 (1992). D. Dey, K. Bhattacharya and T. Sarkar, Phys. Rev. D [**88**]{}, 083532 (2013). D. Dey, K. Bhattacharya and T. Sarkar, Phys. Rev. D [**87**]{}, no. 10, 103505 (2013). P. S. Joshi, D. Malafarina, R. Narayan, Class. Quant. Grav. [**28**]{}, 235018 (2011). S. Sahu, M. Patil, D. Narasimha, P. S. Joshi, Phys. Rev. D [**86**]{}, 063010 (2012). R. M. Cheng and C. R. Evans, Phys. Rev. D [**87**]{} 10410 (2013). M. Kesden, Phys. Rev. D [**85**]{} 024037 (2012).
[^1]: E-mail: akagoel, reevu, proy, tapo @iitk.ac.in
[^2]: By “radius” of a stellar object, we really mean the length of its major axis. This slight abuse of notation should be kept in mind.
[^3]: There is an assumption of instantaneous equilibrium here. See the discussion in subsection 2.5.
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---
abstract: 'The 2002 status of unification models for extragalactic radio sources is examined, with particular emphasis on the dust properties of these objects.'
address: 'Kapteyn Astronomical Institute, P.O.Box 800, NL–9700 AV Groningen'
author:
- 'Peter Barthel,'
- Ilse van Bemmel
bibliography:
- '/ber1/users/bemmel/Tex/BibTex/ivb.bib'
title: 'Radio galaxies: unification and dust properties'
---
radio galaxies; unification; dust; torus
Introduction
============
Unification of active galaxies combines the mechanisms of relativistic beaming in radio jets (when present – the radio-loud objects) with anisotropy due to dust shadowing or non-spherical optically thick emission; key parameter is the aspect angle. These geometric effects, in interplay with source evolution – youth, adulthood, seniority, as well as duty cycle – provide the full framework in which we seek to explain the active galaxy populations.
In case of Seyfert galaxies, an optically thick torus was invoked in the mid 1980’s to explain the apparent differences (strength of continuum and emission line radiation) between Seyfert’s of Types 1 and 2. As for the radio-loud population of radio galaxies and quasars such tori were postulated in combination with relativistic effects in their radio jets. The model whereby quasars and BLLac objects are favourably oriented radio galaxies has drawn considerable interest: the unification church still has many members. The review articles by [@ant93] and [@up95] provide excellent accounts of these models. Increased attention has been paid in the last decade to the subject of life-time evolution of active galaxies and AGN: some of that work will be dealt with here, while [@bir02] provide an excellent full account. For a nice up-to-date summary of the general aspects of unification studies we refer to the Proceedings of the Elba workshop “Issues in Unification of AGN” [@elba02].
In this short review we will discuss new evidence from the past decade in favour or against unification of active galaxies. Furthermore, we will examine the infrared properties of active galaxies, in order to gain insight in the physical behaviour of the dusty toroid.
Status of radio-loud unification models
=======================================
Aspect being the key parameter in geometric unification, how do we know the aspect angle to or the inclination of an active galactic nucleus? Here radio-loudness helps because the fractional radio core strength (radio core strength normalized with total radio or extended radio luminosity) – obviously taken from kpc-resolution radio images – provides a reasonably good orientation indicator. As pointed out by [@wb95], the optical core luminosity normalized with the total radio luminosity provides an additional, improved orientation indicator. In addition, radio-loud objects provide us with a radio (jet) axis: radio images permit determination of the projected source axis so only the inclination angle w.r.t. this jet axis is unknown. While radio-quiet objects occasionally display optical cone emission yielding the optical axis, that axis is generally less well constrained.
As such, unification models for radio-loud objects are further developed and encompass more seemingly different classes. Within the framework of this radio-loud meeting we will concentrate on the radio-loud populations, but mention results for radio-quiet objects where relevant.
FRI unification
---------------
The model whereby Fanaroff & Riley Class I radio galaxies at small inclination manifest themselves as BLLacertae objects has gained considerable support from HST observations of the former. Ultraviolet/optical cores in FRI host galaxies were found to correlate with their radio cores arguing for a common, beamed synchrotron origin [@cap02]. No evidence for slower milli-arcsec scale jets in comparison to FRII radio galaxies was found: FRI jets – or their relativistic spines – must slow down from the parsec to the kiloparsec scale [e.g., @giov01]. Dust disks are often observed in FRI hosts galaxies: their optical depth is much lower as compared to the opaque circumnuclear tori postulated in FRII radio galaxies [@chiab02]. Added to the apparent absence of broad line emission in FRI radio galaxies [at least: luminous BLR – see e.g., @corb00], this may mark an important distinction between the Fanaroff & Riley classes, but the distinction is blurred [e.g., @br01]. Further evidence is provided by the far-infrared SEDs: FRI’s are generally less far-infrared-bright than FRII’s at comparable radio luminosity [@heck94], by a moderate factor of $\sim$4. Note that the broadband optical photometric host properties differ little or nothing [@ledlow02] and that the masses of the central black holes have no connection to radio luminosity whatsoever [e.g., @woo02] – it is most likely the accretion mechanism itself that will determine the FR nature of the radio galaxy.
Backyard FRI Centaurus A was studied in quite some detail and found to conform to the unification model [e.g., @capi00; @chia01]. [@why02] did however point out differences between archetypal FRI’s Centaurus A and Virgo A (M87).
FRII unification
----------------
[@pdb93] found substantial supportive evidence for the favourable orientation of all radio-loud quasars; most of this evidence has become stronger. This is however [*not*]{} equivalent with the picture whereby all FRII radio galaxies contain a QSO hidden from direct view. In fact, the evidence for the existence of a population of FRII pure-radio-galaxies – without a big blue bump – has grown. Such a population of optically “dull” FRII radio galaxies, in which the nuclear accretion activity is currently switched off or at a low level, together with the astrophysically attractive model of a receding torus (e.g., Simpson in these Proceedings) may well account for the reported number density and linear size incompatibilities between FRII radio galaxies and quasars.
Investigations into possible orientation invariants, à la [@hes96] and [@bak97] continue to be important. Relatively weak \[OIII\] emission may be due to obscuration but also to a weak ionizing spectrum [e.g., @tadh98]. With regard to the invariance issue, the mid- and far-infrared emission remains controversial. It is still not clear to what extent this emission is suitable to test and/or constrain unification models [e.g., @ivb00]; see Sect. 5. The multi-component, partly anisotropic nature of the far-infrared radiation nevertheless does not provide major inconsistencies with unification models; it is however likely that a cool dust component, related to host star-formation activity plays a significant additional role.
Beautiful radiation cones have been reported, in objects varying from the nearest Seyfert galaxies to high redshift radio galaxies. These provide clear evidence for anisotropic nuclear radiation fields, with an added component of jet driven star-formation (of difficult to determine magnitude). The strength of the latter is – not surprisingly – related to the size of the radio source [e.g., @best00]. However, in the powerful radio galaxy Cygnus A there is solid evidence that the ionization cones seen in optical images are generated by an outflow, driven by the radiation pressure of the central quasar [@ivb_phd].
As for the host galaxies, the HST studies carried out by the Edinburgh group have yielded strong support for the FRII unification, and also the K($z$) behaviour provided consistency – see the contribution by McLure in these Proceedings. HST and ground-based studies of radio galaxy hosts by de Vries and collaborators provide in addition consistency with evolutionary models for the growth of radio sources; see Sect. 3.
Obscured AGN are required by the X-ray background: this was already pointed out in the late 1980’s and the evidence is still strong [e.g., @comas95]. X-ray spectra will soon yield the cosmologically evolving obscuration, including the contribution from highly obscured AGN [Fabian, in @elba02].
Spectropolarimetry provides the tool to [*proof*]{} unification. Following up on the beautiful early work by Miller, Antonucci c.s., both the Caltech and the Hatfield group obtained these proofs, for an as yet small number of objects. Polarization and obscuration/reddening appear to go hand-in-hand, cf. the models [e.g., @young96; @mhc99].
Observations of the archetypal FRII narrow-line radio galaxy Cygnus A were wonderfully revealing: its radiation cones were imaged with HST [e.g., @jack98; @tadh99] whereas the hidden BLR was detected with spectropolarimetry [@ogle97]. We stress however that ultraluminous backyard radio source Cygnus A harbours a QSO of only moderate strength!
Effects of source evolution
===========================
Considerable effort was spent to study the nature of compact radio sources. These objects, of the Gigahertz-Peaked Spectrum (GPS) and Compact Steep-Spectrum (CSS) classes, are now thought to represent the progenitors of the large classical doubles. Whereas the latter display radio structure of supergalactic ($\gea 100$kpc) dimensions, the former are subgalactic – typically a few tens of kpc for the CSS and a few tens of pc for the GPS class. See also the contribution by Snellen in these Proceedings. Noteworthy is the determination of the spectral ages of several CSS objects [@murgia99] which appear in agreement with their postulated youth. Cold HI gas has been detected in several CSS radio galaxies (e.g., the contribution by Vermeulen in these Proceedings); the CSS quasar class displays pronounced associated CIV absorption [@baker02]. Combined with the fact that some CSS radio galaxies and quasars radiate unusually strong far-infrared emission this has been postulated to imply a young evolutionary stage with strong star-formation [@baker02]. Well-known CSS quasar 3C48 provides an excellent example [@can00]. It is likely that the radio jets, trying to find their way through the circumnuclear ISM play an important role [e.g., @odea02]. True (proper motion) expansion velocities, of order 100 km/sec, for compact radio galaxies have been measured, arguing for their very young ages [e.g., @iza99] and a nice case of a reborn GPS in a large double lobed radio galaxy was recently reported [@mari02]. Radio source number density data require an increasing expansion speed and/or decreasing radio luminosity with age – to tie this down, the evolutionary models for extragalactic radio sources are currently being investigated with larger samples. Overall, the global host galaxy properties of compact and large scale radio galaxies do not show inconsistencies with the evolutionary models [@wimdev00].
Support for the occurrence of recurrent nuclear activity (duty cycle ?) is slowly accumulating [e.g., @arno00]. Given the general occurrence of massive black holes in luminous galaxies [e.g., @mago98], such repetitive activity triggered by repetitive fueling is not unexpected. A multi-wavelength approach including deep optical imaging seems at order.
Broad-line radio galaxies make up an important subset of the radio galaxy population. Their nature is most likely composite: the class encompasses low-luminosity QSRs, possibly with different torus opening angles as compared to high luminosity QSRs, as well as radio galaxies with somewhat transparent, porous tori [@tadh98; @jdt00]. [@ivb01] point out that BLRGs are characterized by the absence of star-formation: one explanation could be that some BLRG represent old, dying radio galaxies. The absence of optical synchrotron (jet) components in some BLRGs [@chiab02] may be consistent with that view.
Very dusty sources
==================
Intriguing recent development is the detection in X-rays of active nuclei in seemingly non-active galaxies, apart from their dusty starburst nature: NGC4945 [e.g., @guai00] and NGC6240 [@iwa98]. Such highly obscured AGN are important in the possible evolutionary connection between starburst galaxies and AGN and may be important contributors to the X-ray background – ongoing Chandra and XMM-Newton investigations will undoubtedly shed light on these issues.
Noting that dust obscuration lies at the heart of unification models, we proceed by reviewing the status of torus models.
Modelling the toroid in AGN
===========================
Introduction
------------
Key to the unification model is the obscuring torus, creating an angle and wavelength dependent anisotropy in active galaxies. The soft X-ray and UV emission from the central engine is reprocessed to infrared wavelengths by the dust in the torus. In 1983 IRAS was the first satellite to detect infrared emission from the relatively distant active galaxies. Soon after, several groups started to model this reprocessing using radiative transfer codes. Among the first to present their models are [@pk92], hereafter PK92. A few years later they are followed by [@gra94], hereafter GD94 and [@efst95], hereafter ER95, and recently models have appeared by [@nenk02] and [@ivb03], hereafter BD03. The ISO satellite provided a wealth of additional data, however, only for the Seyfert galaxies these allow a good constraint of the models. For the radio-loud AGN population only few objects have a well defined broad-band SED, e.g. Cygnus A.
The early models: PK92, GD94 and ER95
-------------------------------------
To ease the radiative transfer calculations, all groups assume azimuthal symmetry, but the actual torus geometries differ among the different groups. PK92 define a so-called pill-box geometry, where the thickness of the torus is constant with radius, and the inner walls are perpendicular to the plane of the toroid. GD94 use a similar geometry, but with the possibility of having a conical hole in the center, instead of perpendicular walls. They also allow for different dust mixtures to be present in the torus. ER95 use three different geometries: a conical disk, where the thickness of the disk increases linearly with radius, an anisotropic sphere, and a pillbox with a conical hole. In all their models, the torus has a constant inner radius, causing the central opening to be circular (the opposite of the GD94 central cavity). All these ’early’ models assume Galactic dust properties.
The silicate problem
--------------------
All three groups predict significant 10[$\mu$m]{} silicate emission in type 1 active galaxies, which is not observed. The more powerful radio-loud population still lacks proper spectra, but the absence of 10[$\mu$m]{} emission is well established by the ISO spectrometers for the Seyfert type AGN. GD94 postulate depletion of small grains by shocks in order to explain the lack of 10[$\mu$m]{} emission in type 1’s. Only one of the ER95 pill-box model does not predict 10[$\mu$m]{} emission.
This leads several groups to the conclusion that AGNs do not contain standard Galactic dust. The grain size distribution might differ significantly from the Galactic distribution, in the sense that AGN toroid dust is dominated by larger grains. This is first recognized by [@laor93], and several groups have brought forth explanations for this. From UV spectra it becomes clear that also the 2200Å absorption is shallower than expected in many AGN, consistent with the lack of small grains [@maio01_2].
Small grains are easily destroyed in the strong radiation field of an active nucleus. However, they should survive in the regions shielded by the torus. Several reasons have been presented in literature for the lack of small grains in AGN: most recently a clumping theory is presented by [@maio01_1]. The grains are swept up by the radiation pressure, and clump together to form larger grains.
The width problem
-----------------
The early models encounter a second problem in explaining the full width of the broad-band SED observed in all types of active galaxies. Although the results from the models are generally broader than a single grey body, due to the temperature gradient in the dust, most of them are not yet broad enough. PK92 recognize this problem, and both GD94 and ER95 have tried to improve, but did not succeed.
A second component?
-------------------
To deal with these issues, recent models have appeared in which the dust is no longer smooth, but clumpy [@nenk02], or has a non-Galactic grain size distribution (BD03). Both sets of models can produce the full observed width of the infrared SED of active galaxies, and both manage to circumvent significant 10[$\mu$m]{} emission in models at small inclinations.
However, observational evidence is mounting that not all the infrared emission in active galaxies arises from the compact torus [see e.g., @spi02; @prie01]. There must be a significant contribution from a second dust component, which seems to be related to star-formation in the host galaxy, i.e. at scales much larger than the radius at which the AGN can influence the dust. So far, all papers have ignored this possibility, except BD03. Such a second component may well provide a solution to the width problem in the early models. BD03 provide colour-colour diagrams for their models, showing that the observed 25–60 and 60–100[$\mu$m]{} colours are not well fitted by single component torus models, and demand an additional dust component.
This secondary dust component might also be responsible for the lack of silicate emission observed in type 1 AGN; if the torus does produce 10[$\mu$m]{} emission but this passes through a colder layer of dust, the resulting spectrum shows no 10[$\mu$m]{} emission, or even a shallow absorption. However, this would require a large scale, spherical dust screen that covers all viewing angles to the nuclear regions. In the case of NGC1068 there is no evidence of large scale silicate absorption: the nucleus seems to be the main source of the absorption. However, in this object the star-formation provides only a marginal fraction of the total infrared luminosity [@mar03], whereas it is thought to contribute at least 50% to the far-infrared emission in powerful radio sources.
For a proper understanding of dust in the nuclear regions of AGN the relative contributions of large and small scale dust need to be assessed. Decomposition will result in a better understanding of the behaviour of dusty tori in active galaxies, but also of the properties of the dust associated with star-formation in the host galaxies. Although the current generation of models is roughly consistent with observations, many important issues need to be solved. For this purpose, deep SIRTF low resolution spectroscopy data will be most welcome.
Conclusions
===========
Good progress has been made during the past decade regarding aspect angle unification of radio-loud objects. The basic picture is correct, but the distinction between various classes appears somewhat blurred. The actual torus, big blue bump, and jet set-up, connected to source power and lifetime evolution, is not yet understood. The infrared might provide valuable clues, but we first need to understand how the various components contribute to the overall SED.
[*Acknowledgements*]{}\
We acknowledge expert reading by healthy Ski Antonucci noting in passing that he fell ill just before the Workshop, leading the SOC to request the present reviewers to take over Ski’s review talk at 48h notice ..... The authors realize that this review is far from complete and does no justice to many active workers in the field. Apologies to those outside the review cone ....
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abstract: 'We compare the structures of model atmospheres and synthetic spectra calculated using different line lists for TiO and water vapor. We discuss the effects of different line list combinations on the model strutures and spectra for both dwarf and giant stars. It is shown that recent improvements result in significantly improved spectra, in particular in the optical where TiO bands are important. The water vapor dominated near-IR region remains problematic as the current water line lists do not yet completely reproduce the shapes of the observed spectra. We find that the AMES TiO list provides more opacity in most bands and that the new, smaller oscillator strengths lead to systematically cooler temperatures for early type M dwarfs than previous models. These effects combine and will help to siginificantly improve the fits of models to observations in the optical as well as result in improved synthetic photometry of M stars. We show that the [@davis86] -values for the $\delta$ and $\varphi$ bands of TiO best reproduce the observed (V-I) color indices.'
author:
- France Allard
- 'Peter H. Hauschildt'
- David Schwenke
bibliography:
- 'yeti.bib'
- 'opacity.bib'
- 'mdwarf.bib'
- 'radtran.bib'
- 'general.bib'
- 'opacity-fa.bib'
- 'local.bib'
title: 'TiO and H$_2$O absorption lines in cool stellar atmospheres'
---
[*ApJ, in press. Also available at\
ftp://calvin.physast.uga.edu/pub/preprints*]{}
Introduction
============
Over the past decade, model atmospheres and synthetic spectra for late-type stars have improved hand-in-hand with higher quality opacities. In 1994, quality lists of transitions of the water vapor molecule based on ab initio molecular calculations became available [@UCL; @schryb94; @H2OJorg] which allowed the computation of the first Direct Opacity Sampling (hereafter dOS) model atmospheres for late-type dwarfs [@h2olet] and brown dwarfs [@gl229b], later to become the NextGen models described in [@ng-hot]. Showing a more physical description of their main opacities, the NextGen dOS model atmospheres promised a better description of the Spectral Energy Distribution (hereafter SED) of cool stars. And this appeared to be verified for the infrared SED of M dwarfs [@Jones96; @leg96; @araa].
But despite these fundamental improvements, the NextGen models have failed to match adequately several of the optical (spectroscopic and photometric) properties of late type dwarfs and giants. In fact, the dOS models [@B95; @h2olet; @araa; @ng-hot] provided a worse fit to the optical SED of lower main sequence dwarfs than previous models based on simplified molecular opacities [@MDpap hereafter, AH95]. The models could not reproduce the tight relation formed by M dwarfs in the V-R versus R-I two-color diagram, indicating a systematically shallower slope of the optical SED (defined by TiO absorption) then observed in these stars. A systematic flux excess in the spectral region sampled by the V bandpass ($0.4-0.65 \m$m) was noted as well in dwarfs as in red giants. [@BCAH98] observed that this excess in the NextGen dwarf models translated into lower main sequence isochrones deviating progressively to the blue (by up to 1.0 magnitude!) in (M$_V$,V-I) color-magnitude diagrams, for masses lower than about 0.5 M$_\odot$ (T$_{\rm eff}\leq 3800\,$K). [@BCAH97] examined a variety of globular clusters and showed that this departure of the models decreased in amplitude with decreasing metallicity. The problem seemed therefore confirmed to be caused by a lack of opacity of an oxygen compound[^1].
Only three independent models of the TiO molecule and corresponding lists of transitions were available so far to the construction of model atmospheres. The first model was constructed over two decades ago by [@coll75] and was restricted by the computational limitations of the time. The Collins line list was intended to model the extended atmospheres of red giants and did not include high energy and otherwise weak transitions important by their number in the hotter environments of red dwarf atmospheres. It also neglected the red $\epsilon$ system of TiO. [@TiOJorg] extended Collins’ work to TiO isotopic transitions, included the $\epsilon$ system from revised molecular rotational constants, and adopted the laboratory oscillator strengths of [@davis86]. It is therefore understandable that the resulting limited list of transitions caused shortcomings in the NextGen model atmospheres. The second TiO list was constructed by [@cdrom15] and is used in his ATLAS9-12 atmospheres. The third model was constructed by [@Plez92] using also the [@davis86] oscillator strengths, and is used in his version of the MARCS atmosphere code. All three independent models yielded the visual flux excess in different proportions. [@Plez98] suggested that the missing opacity is due to missing TiO band systems in current lists, and added the TiO a-f system at 0.5 $\m$m to his list. However, this helped him only partially to resolve the V-band flux excess problem.
Recently, [@Langhoff97] constructed a new model of the TiO molecule and published new lifetimes and oscillator strengths that improved significantly upon the 1986 values of Davis et al.[@Valenti97; @Plez98]. [@ames-tio] has subsequently computed a corresponding list of transitions complete to the high energies and therefore more suitable for general model atmosphere applications. In this paper we present the results of including this new TiO line list as well as the new list in the construction of model atmospheres and synthetic spectra for late-type dwarfs and red giants.
Model calculations
==================
We have calculated the models presented in this paper using version 10.3 of our general model atmosphere code . Details of the code and the general input physics are discussed in [@ng-hot] and references cited therein. The models for M giants were calculated with the same setup, however, they employ spherical geometry (including spherically symmetric radiative transfer). For giant models with low gravities ($\logg \le 3.5$), this can be an important effect for the correct calculation of the structure of the model atmosphere and the synthetic spectrum [@epscma; @betacma]. The main difference between the models presented in [@ng-hot] and the models presented here is the use of the new AMES line lists for [@ames-water-new] and TiO [@ames-tio], but we have also adjusted the empirical oscillator strengths of VO and CaH absorption bands to respect their strength relative to TiO bands (note that VO and CaH absorption is still treated in the Just Overlapping Line Approximation due to lack of adequate line data). Our combined molecular line list includes about 500 million molecular lines. These lines are treated with a dOS technique where each line has its individual Voigt (for strong lines) or Gauss (weak lines) line profile (in the standard OS method tables of precomputed opacities are used). They are selected for every model from the master line list to the beginning of each model iteration to account for changes in the model structure, see [@ng-hot] for details. This procedure selects about 215 million molecular lines for a typical giant model with $\Teff\approx 3000\K$ and about 130 million molecular lines for a dwarf model with the same effective temperature. Therefore, we generally use the parallelized version of phoenix [@parapap; @parapap2; @jcam] to perform the calculation efficiently on parallel supercomputers. Details of the TiO and H$_2$O lists are given in the next subsections.
Water lines
-----------
The effects of water lines on the M dwarfs SED was discussed in [@h2olet]. For the work presented here, we have replaced the UCL water vapor line-list [@UCL; @schryb94 hereafter: ] used in [@ng-hot] with the AMES water line-list [@ames-water-new hereafter: ]. This list includes about 307 million lines of water vapor. For the calculations shown in this paper we have used H$_2 ^{16}$O and neglected other, much less abundant, isotopes of this molecule.
The water vapor opacity is governed by the completeness of the line list used, but also by the adopted atomization energy. The partition function of the molecule cancels out in the final absorption coefficient, after we have multiplied cross-sections by number densities. But since water is an important chemical equilibrium specie, errors in the partition function can affect indirectly the model structure and spectra. The AH95 models were based on the [@ludwig] hot flames water cross-sections in the form of straight means, and used the JANAF partition function for water vapor [@irwin88]. The NextGen models where, on the other hand, computed with the line list and a partition function computed from the levels. We note that the partition function is practically identical to JANAF values, while the value is smaller than JANAF for temperatures above 3000K, possibly due to the energy levels missing in the data. We have therefore adopted for this and later work the JANAF partition function. We use an atomization energy of $9.5119\,$eV from [@irwin88] for all models since AH95.
TiO lines
---------
The main point of this paper is the comparison of the model structure and the synthetic spectra obtained by using the list of TiO lines from [@TiOJorg hereafter: ] and the new list of TiO lines from [@ames-tio hereafter, ]. The list includes a total of about 172 million lines, about 44.6 million of these are for the most abundant isotope $~^{48}$TiO and about 32 million lines for each of the remaining 4 isotopes $^{46,47,49,50}$TiO). But beyond the completeness of the line-list, two more considerations affects the overall opacity produced by TiO, and explain systematic differences between model versions and by different authors: the atomization energy ($D_0^0$) determines the number density of TiO, and the TiO band oscillator strengths[^2] have been derived from sunspot observations [@davis86 hereafter: DLP86], laboratory experiments [@TiOfel95 hereafter: HNC95], as well as from ab initio calculations [@Langhoff97 hereafter L97]. derived astrophysical values by fitting the optical SEDs of red giants, using an atomization energy of $7.76\,$eV. He quoted that reducing this value by $0.3\,$eV would increase his by a factor 2.5. The most recent estimate of D$_0^0$ for TiO is now $6.92\,$eV, which suggests that the B90 values are underestimated by as much as a factor 7! We summarize in table \[feltab\] the various sources of oscillator strengths available for TiO.
The models of [@allardphd] used values from B90 together with the straight-mean TiO opacities by [@coll75] and [@tio74], and assuming an atomization energy of $6.87\,$eV. The first comparison of these models to the SED of M dwarfs [@kirk93] revealed the inadequacy of this combination of parameters for TiO which produced far too weak optical opacities. We have therefore, since the AH95 model series, employed the updated value of $6.92\,$eV, together with the larger laboratory -values of DLP86. These two modifications combined to significantly increased the strength of TiO opacities in the models, bringing the AH95 and later the dOS NextGen models in improved agreement with the SED of M dwarfs. Any differences in the predictions of the AH95 and NextGen models are therefore purely due to the opacity technique (Straight-Mean versus dOS) and to the completeness of the line-list used. The incompleteness of the line-list allows photons to escape between absorption bands [see e.g. @Valenti97], and thus leads to systematically and increasingly (with higher $\Teff$) bluer optical colors (V-I) than observed [@solar-evol; @clusterpap]. For the current models we therefore explore the use of the more complete line-list, and the yet larger theoretical -values of L97.
Results
=======
We have calculated a number of model atmospheres using either the or the list of TiO lines and using either or as source of the lines. All the other input physics is the same for both sets of models. All models have been fully converged with their respective set of parameters. Note that these models have been constructed for the purpose of this paper only and not to model individual stars and thus do not include dust formation and opacities which is important in atmosphere models with effective temperatures below about $2500\K$, such models are presented in a subsequent paper. In the following, we will discuss the results for the dwarf and giant models separately. The baseline for our comparisons are the NextGen models [@ng-hot] for the dwarfs and the NG-giant models [@ng-giants] for the giants.
Effects of different TiO line lists
-----------------------------------
The models discussed in this section were all calculated using to isolate the effects of different TiO line lists on the model spectra and structures.
### M dwarf models
In Figs. \[dwarfs-blue\] and \[dwarfs-red\] we show a comparison of model spectra calculated with (full curves) and with (dotted curves) (both using our adopted -set, as quoted in table \[feltab\]) for several effective temperatures. The gravity ($\logg=5.0$) and abundances (solar) were selected to be representative of M dwarfs in the solar neighborhood. In both figures, the resolution of the synthetic spectra was reduced by boxcar smoothing to $20\ang$. At high effective temperatures, the two sets of models are nearly identical due to reduced importance of TiO absorption. At very low $\Teff$ the two line lists apparently agree very well since only the lowest levels of TiO remain populated. It is essentially between $\Teff \approx 2000\K$ and $\approx 3500\K$ that the largest completeness and quality effects of the TiO line lists are seen.
Fig. \[tio29\] indicates the location of each TiO band system for a 2900K model. From this it becomes clear that the addition of a-f transitions, which depress the continuum from 0.4 to 0.5$\m$m, is one of the largest improvements brought by the list to our models. We note that the entire optical regime from 0.4 to 0.75$\m$m shows generally [**more**]{} opacity in the models then using the [@TiOJorg] (hereafter: J94) line list. The $\epsilon$ bands at 0.82 to 0.88$\m$m have a more precise shape in the list, and come out stronger as well. This is a result of the completeness of the list which also removes flux excess escaping between the troughs of the bands. We note however that some regions, such as in the $\gamma$ band near 0.78$\m$m, show less opacity in the new models.
The main effect of the new on spectroscopic and photometric $\Teff$ estimates will however be dominated by the change we make to the oscillator strengths. The L97 values being generally smaller than the DLP86 values adopted by J94, models of early-type M dwarfs using the new setup should predict systematically lower effective temperatures then did prior models (NextGen, AH95, etc., see also Fig.\[TiO3500\] below). And beyond the enhanced completeness of the list to high temperature transitions, the need for a cooler model should also contribute to making the TiO bands fit better a given star i.e. larger bands with less flux escaping from deeper, hotter atmospheric regions between them.
We could have opted to use the HNC95 laboratory -values as did [@Plez98], but since the L97 ab initio values agree quite well, we decided to keep these, except for the $\delta$ and $\varphi$ band systems. The reason the oscillator strengths for the $\delta$ and $\varphi$ bands are less accurate is that it is very hard to get a good description of the b state, which is the upper state in both bands. For the $\delta$ system, L97 derives an oscillator strength which is, as opposed to all other bands, twice as large as the DLP86 value. And the $\delta$ and $\varphi$ -values cannot be corroborated by recent experimental values. Such a strong $\delta$ band system would be difficult to bring in agreement with M dwarfs observations. Indeed, prior models have all shown a gradually increasing departure to the blue of the main sequence in M$_V$ vs $V-I$ diagrams [@baraffe95; @solar-evol]. Such departure is significantly improved using the new TiO list if one keeps a weak $\delta$ band as indicate preliminary results of evolution models to be published separately (see also Fig. \[VRI\] below). We have therefore adopted to keep the DLP86 oscillator strength values for the reddest two TiO bands until new laboratory experiments can either confirm or infirm the L97 predictions. The summary of our adopted set of oscillator strengths for TiO is presented in Table \[feltab\].
### M giant models
The results for the giant models are similar to the results for the dwarfs. Figures \[giants-blue\] and \[giants-red\] show synthetic spectra for 3 representative giant models with the indicated effective temperatures. The models have in common the parameters $\logg=0.5$, $M=5\Msun$ and solar abundances. The differences between the (full curves) and (dotted curves) models are somewhat larger for giants than for the dwarfs in the blue spectral region due to an increased sensitivity to the added a-f system opacities in the line list. It is however somewhat less pronounced in the red spectral region where TiO bands are weaker in giants. The “spikes” that are apparent in the spectrum with $\Teff=3000\K$ are absent in the models. These spikes were one of our major problems in fitting observed spectra of giants. For larger $\Teff$ the differences between the spectra diminishes quickly as TiO becomes less important in the giants. This happens at lower effective temperatures compared to the dwarfs because of the lower pressures in giant atmosphere which results in smaller partial pressure of molecules as compared to dwarfs.
### Model Structures
A comparison of the model structures for both dwarf and giant model reveals only very small differences between structures calculated with and . We plot the differences in electron temperature as well as the relative differences between the and models for dwarfs and giants in Figs. \[dwarf-structure\] and \[giant-structure\], respectively. The changes are generally very small, only in the per-cent range for the gas pressures and about $10\K$ maximum difference between the electron temperatures for the dwarf models. For the giant models the differences are somewhat larger. The changes in the opacity averages are generally small but largest for the Rosseland mean opacity in the outer layers of the giant models. The temperatures are higher in model for both the giant and the dwarf models, however, the gas pressures are lower in the dwarf model but higher in the giant model. Overall the changes are modestly small, indicating that the detailed effects of the TiO line lists do not have a large effect on the model structure itself.
Effects of different water vapor line lists
-------------------------------------------
In figure \[mt-compare\] we show the effects of different water line lists on the synthetic spectra for M dwarfs. All models shown in the graph otherwise use the same line-lists ( was used for the TiO lines). In overall, we can see that the changes in the water vapor line lists are of larger amplitude than the changes in the completeness of the TiO line list. The models calculated with show a totally different shape of the $1.4\,\mu$m band, both weaker and wider then predicted by the model. The completeness of the new line list to high temperatures helps block more flux escaping from deeper, hotter layers of the models around $1.6\,\mu$m, and $2.2\,\mu$m. This promises a much better description of observations in general.
We also find important changes of the model structure as shown in Fig.\[mt-structures\]. The differences of the electron temperatures can reach $100\K$, the gas pressures can differ by $20\,$% and the opacity averages, in particular the Rosseland mean, can differ by close to $60\,$%. These effects are much larger in the outskirts of the atmosphere than the changes in the structures caused by different TiO line lists (see Fig. \[dwarf-structure\]). As a result, the use of the line list also affects the optical spectra, causing weaker TiO bands than obtained with the line list. Models of early-type M dwarfs based on the line list therefore systematically predict yet lower effective temperatures for a given star.
Combined effects
----------------
In Figs. \[TiO3500\] and \[H2O3500\] we display a comparison between NextGen models (which use and ) and models that use the and . The wavelength range of important filters and band identifications for TiO are given on the figures. The TiO bands in the “AMES-atmosphere” are considerably weaker than those of the NextGen model spectrum as a result of the smaller oscillator strengths used, and the structural effects. On the other hand, the water bands are stronger in the AMES-atmosphere than in the NextGen model. This model has a relatively high temperature, thus the higher energy levels of the water molecule are relatively more important than for models with lower $\teff$ (however, the concentration of water molecules is reduced so the overall water opacity is smaller than in the cooler models).
To better judge of the impact of these opacity changes on the overall SED of M dwarfs in general, we have computed synthetic photometry as described in AH95 for three sets of models: (1) the NextGen grid based on J94 and (2) the AMES grid based on and opacities, and (3) the AMES-MT grid based on and opacities. The results are compared to a photometric sample of M dwarfs [@leggett92] in Figs. \[VRI\], \[JHK\], and \[IJK\]. Since M dwarfs form a tight sequence in optical VRI two-colors diagram despite the age and metallicity scatter of the sample (see AH95), this diagram imposes a strong constraint on model atmospheres. We find that models based on opacities are systematically redder in $V-R$ and $V-I$ than models based on the J94 line list. The new models agree much better with observations and the new TiO data removes most of the discrepancy shown by the NextGen models in the lower main-sequence. Small remaining discrepancies may be attributed to the JOLA handling of VO and CaH which tend to overestimate slightly their opacities in the present models. [@lhs1070] already studied the low resolution HST/FOS spectra of an M6 dwarf (LHS1070A) and found the AMES-MT models indeed agree quite well both with the observed SED and absolute fluxes within errors on the parallax of the system. They however noticed that some “continuum” flux excess remains important in the visual part of the SED (0.45 to $0.65\,\mu$m). However, there is no [*a priori*]{} reason to assume that the a-f oscillator strengths are inaccurate, and these remaining problems could be related to other effects on the model structure.
The use of seems also to bring some improvements to the modeling of near-infrared colors. Fig. \[JHK\] shows that the late-type dwarfs can be better reproduced by the new water opacities than by the line list. However this diagram is sensitive to both gravity (lower gravity models loop lower) and metallicity, which makes it difficult to constrain the models on the adequacy of the water opacities used with them. [@leg98] and [@leg99] have already used the AMES models in their analysis of M dwarfs and brown dwarfs, and found an excellent general agreement of the predicted near-infrared SED with observations. However these analyses used the models to derive the parameters of the studied stars and brown dwarfs based on fits to the near-infrared SED or photometry, and could not make an independent statement on the quality of the water line list.
M dwarfs form again a sequence in the mixed-colors IJK diagram (Fig. \[IJK\]), although less tightly than in the VRI diagram. Unresolved binary stars produce K-band flux excess and lie below the sequence. H$_2$ pressure-induced opacities depress the K-band flux of metal-depleted dwarfs so that they systematically lie above the sequence. But, as opposed to the JHK diagram, this one is not particularly sensitive to gravity in M dwarfs which allows a sequence to be defined. Models should pass therefore through the bulk of early-type M dwarfs at $J-K=0.8$, and follow a relatively $J-K$-insensitive sequence towards late-type dwarfs. We find that models based on the opacities lie, as did the AH95 models before them, 0.2 magnitude in $J-K$ to the blue of the observed sequence! And our tests show that this result is independent of the TiO opacities used. The NextGen models already reproduced perfectly the location of lower main-sequence stars in this diagram. And AMES-MT models computed using the and line lists behave adequately both in the optical and infrared. Why? Perhaps the new water vapor line list is still not complete enough to high temperatures and lacks opacity in the $J$-bandpass i.e. around $1.3\,\mu$m? Or would it have too much opacity in the $K$-window i.e. around $2.2\,\mu$m? Until these questions can be answered, we hope that the two main grids of models we have computed (AMES and AMES-MT) will allow independent detailed confrontations to observations of cool stars that will locate more precisely the source of the problem (e.g. Leinert et al., in preparation).
Summary and Conclusions
=======================
A long standing problem with M dwarf models was that prior TiO line lists were incomplete to high temperatures. The use of “straight means” (AH95 models) helped by the coarseness of the treatment to block flux which otherwise escapes between lines in the incomplete list. But these models also blocked too much flux in most cases, and were only appropriate in late-type M dwarfs when TiO bands are already very strong. Clearly, a more complete line list was needed to model stars from the onset of TiO formation to its gradual disappearance from the gas phase in brown dwarfs. The list now serves beautifully this purpose. We find that the list provides more opacity in most bands and suppress adequately flux between bands. The new, smaller oscillator strength values also play an important role in systematically assigning cooler models (at least for early type M dwarfs) to a given star, this way contributing to broader bands and lesser inter-band flux as well. These effects combine and should resolve most of the previously observed discrepancy between models and observations in the optical SED and photometry of M stars. [@lhs1070] note however that flux excess remains substantial in the visual spectrum, suggesting some further incompleteness or inaccuracies of the new TiO in the a-f system.
In order to better reproduce the observed (V-I) color indices, we had to retain in the present models the [@davis86] -values for the two reddest band systems: $\delta$ and $\varphi$. For these two band systems, the theoretical estimates of [@Langhoff97] predicts an unexpectedly large -value ratio, while no laboratory estimate [@TiOfel95] are available to corroborate this. And we find, as did [@AP98] in red giants for the $\delta$ band, that models based upon the DLP86 -values for these two bands reproduce adequately their observed depths in M dwarfs.
The introduction of the opacities bring solid improvements of the near-infrared SED of late-type dwarfs, but fails as the AH95 models did to reproduce adequately the $J-K$ colors of hotter stars. Water vapor is a more important factor for the structure of the atmosphere than TiO because its overall opacity is larger and its lines are closer to the peak of the SED than the TiO bands, so the flux blocking effect of water vapor is more important for the temperature structure than that of TiO opacities for these low temperatures. Schwenke and collaborators at NASA AMES are preparing a new dipole moment function for H$_2$O, which may change the high temperature high overtone water bands, and help resolve this discrepancy in the near future.
Until a revised version of the line list becomes available, we have therefore generated two sets of model atmospheres for cool stars which allow to investigate these issues: the AMES grid based on the new TiO and H2O opacities, and the AMES-MT grid which rely on the and opacities.
We thank David Alexander for helpful discussions and the referee, U.G. J[ø]{}rgensen, for his very helpful comments. This work was supported in part by grants from CNRS and INSU, NSF grant AST-9720704, NASA ATP grant NAG 5-3018 and LTSA grant NAG 5-3619 to the University of Georgia, and NASA LTSA grant NAG5-3435 to Wichita State University. This work was supported in part by the Pôle Scientifique de Modélisation Numérique at ENS-Lyon. Some of the calculations presented in this paper were performed on the IBM SP2 of CNUSC, the SGI Origin 2000 of the UGA UCNS, and on the IBM SP2 of the San Diego Supercomputer Center (SDSC) with support from the National Science Foundation, and on the Cray T3E of the NERSC with support from the DoE. We thank all these institutions for a generous allocation of computer time.
Tables
======
[\*[8]{}[r]{}]{} system & lam0 & B90 & J94\* & DLP86 & L97 & AP98 &adopted\
alpha &$5170.7$ & $0.10$ & $0.17$ & $0.106$ & $0.105$ &$ 0.106$ & $0.105$\
beta &$5605.2$ & $0.15$ & $0.28$ & $0.125$ & $0.176$ &$ 0.125$ & $0.176$\
gamma’ &$6192.5$ & $0.08$ & $0.14$ &$0.0935$ & $0.108$ &$0.0935$ & $0.108$\
gamma &$7095.8$ & $0.09$ & $0.15$ &$0.0786$ & $0.092$ &$0.0786$ & $0.092$\
epsilon &$8407.6$ &$0.0024$ &$0.014$ &$<0.006$ & $0.002$ &$0.0023$ & $0.002$\
delta &$8870.9$ & $0.02$ &$0.048$ & – & $0.096$ &$ 0.048$ & $0.048$\
phi &$11044.8$ & $0.02$ &$0.052$ & – & $0.018$ &$0.0178$ & $0.052$\
\* laboratory values determined by HNC95
B90: [@brett90]\
DLP86: [@davis86]\
J94: [@TiOJorg]\
HNC95: [@TiOfel95]\
L97: [@Langhoff97]\
AP98: [@AP98]\
Figures
=======
[^1]: Hydride absorption bands only get stronger relative to the continuum with decreasing metallicity over the range covered by the globular clusters studied in [@BCAH97] in the optical spectral range.
[^2]: $\fel= f_{\nu'\nu''}/q_{\nu'\nu''}$, where the $f$’s are the oscillator strengths and the $q$’s are the Franck-Condon factors of the transition $\nu'\nu''$
|
---
abstract: 'We calculate the two-loop vertex function for the crossed topology, and for arbitrary masses and external momenta. We derive a double integral representation, suitable for a numerical evaluation by a Gaussian quadrature. Real and imaginary parts of the diagram can be calculated separately.'
---
-2cm
[MZ-TH/96-30]{}\
[hep-ph/9610285]{}\
[September 1996]{}\
[**New representation of the two-loop crossed vertex function**]{}
[Alexander Frink, Ulrich Kilian, Dirk Kreimer[^1]]{}\
PACS numbers: 02.70.+d, 12.38.Bx, 11.20.Dj
Introduction
============
In the recent past many efforts have been made to calculate two-loop Feynman diagrams with masses, e.g. [@dirk1; @dirk2; @newrepplanar; @tarasov; @kato]. Though now very effective methods exist for the general mass case of the two-loop two point function [@xloops; @baub], these methods need considerable extensions or modifications to cope with two-loop three-point functions in general. The planar topology has been discussed extensively, e.g. in [@dirk2; @newrepplanar; @fujimoto92; @fleischer93]. For the other important topology — the crossed topology — so far two methods have been presented.
The method presented in [@kato] is based on Feynman parameters and uses high dimensional Monte Carlo integration, resulting in extensive CPU usage and slow convergence when high accuracy is needed.
Taylor expansion, analytic continuation and Pad[é]{} approximations, as presented in [@tarasov], gives results with very high accuracy, but the method is, at this stage, still restricted to one kinematical variable.
Our approach is based on [@dirk2; @newrepplanar], where it gave excellent results for the planar two-loop three point function. The calculation of the crossed topology is similar to the planar case. Again we succeed by considering orthogonal and parallel space variables. From this starting point, a four-fold integration is immediately obtained. Still following the lines of [@newrepplanar] we use Euler transformations for subsequent integrations. The difference between the planar and the non-planar topology is apparent in the increasing number of different cases which have to be considered for the latter, while, fortunately, the conceptual frame remains unaffected. Once more we end up with a two-fold integral over a finite region, solely involving dilogarithms and related functions. This integral representation allows for a Gaussian quadrature, and is thus perfectly suitable for practical purposes [@FKKR].
Calculation
===========
Our task is to calculate the two-loop crossed vertex function, as shown in fig.\[fig:crossedfig\]. We restrict ourselves to the scalar integral, which is given by $$\label{eq:theintegral}
V(q_1,q_2) = \int\!\! d^4l \int\!\! d^4k \frac{1}{P_1 P_2 P_3 P_4 P_5 P_6}
{\, ,}$$ where $q=q_1+q_2$ is the momentum of the incoming particle and $q_1$ and $q_2$ are the momenta of the outgoing particles. Further, $l$ and $k$ are the internal loop momenta, and $P_i$, $i=1 \dots 6$ are the propagators of the internal particles, as labeled in fig.\[fig:crossedfig\]. We assume that all external particles are massive and have time-like momenta. The massless limit is approached smoothly as long as no on-shell singularities occur.
The integral will be splitted into parallel and orthogonal space integrations. The parallel space is two-dimensional here, hence leaving two dimensions for the orthogonal space. The integral is convergent in $D=4$ dimensions, and there is no need to use a regularization scheme.
Since the integral is Lorentz invariant, the splitting into parallel and orthogonal space variables is equivalent to the choice of a Lorentz frame. For time-like external particles, we choose the rest frame of the incoming particle, and assume that outgoing particles are moving along the $x$-axis.
Explicitly, the four-momenta are chosen as $$\begin{aligned}
q^\mu & = & (e_1+e_2,0,0,0)
\nonumber\\
q_1^\mu & = & (e_1,q_z,0,0)
\nonumber\\
q_2^\mu & = & (e_2,-q_z,0,0)
\nonumber\\
l^\mu & = & (l_0,l_1,\vec{l}_\perp)
\nonumber\\
k^\mu & = & (k_0,k_1,\vec{k}_\perp)
\label{eq:momenta}\end{aligned}$$ with $q_z$, $e_1$ and $e_2$ positive and $q_z<e_1$, $q_z<e_2$.
The propagators $P_i$ of the internal particles can be written down using the flow of loop momenta as shown in fig.\[fig:crossedfig\] $$\begin{aligned}
P_1 & = & (l+k-q_1)^2-m_1^2+i\eta
\nonumber\\
P_2 & = & (l+k+q_2)^2-m_2^2+i\eta
\nonumber\\
P_3 & = & (l-q_1)^2-m_3^2+i\eta
\nonumber\\
P_4 & = & (k+q_2)^2-m_4^2+i\eta
\nonumber\\
P_5 & = & l^2-m_5^2+i\eta
\nonumber\\
P_6 & = & k^2-m_6^2+i\eta {\, .}\end{aligned}$$ A small imaginary part $-i\eta$ with $\eta > 0$ is assigned to the squared masses of the internal particles and is chosen equal for all particles for convenience. Hence differences of propagators have a vanishing imaginary part, which we utilize in the following. The choice of different (small, positive) $\eta_i$’s for the propagator would not change our final result, but is technically more cumbersome.
Using the explicit representation of the momenta in eq.(\[eq:momenta\]), a shift of the loop momenta with a trivial Jacobian according to $$\begin{aligned}
l_1 & \rightarrow & l_1 + q_z
\nonumber\\
l_0 & \rightarrow & l_0 + l_1
\nonumber\\
k_0 & \rightarrow & k_0 + k_1 {\, ,}\label{eq:shifts} \end{aligned}$$ achieves that the propagators are linear in $l_1$ and $k_1$ and hence can be written as $$\begin{aligned}
P_1 & = & (l_0+k_0-e_1)^2+2 (l_0+k_0-e_1)(l_1+k_1)
-l_\perp^2-k_\perp^2-2 l_\perp k_\perp z-m_1^2+i\eta
\nonumber\\
P_2 & = & (l_0+k_0+e_2)^2+2 (l_0+k_0+e_2)(l_1+k_1)
-l_\perp^2-k_\perp^2-2 l_\perp k_\perp z-m_2^2+i\eta
\nonumber\\
P_3 & = & (l_0-e_1)^2+2 (l_0-e_1)l_1-l_\perp^2-m_3^2+i\eta
\nonumber\\
P_4 & = & (k_0+e_2+q_z)^2+2 (k_0+e_2+q_z)(k_1-q_z)-k_\perp^2-m_4^2+i\eta
\nonumber\\
P_5 & = & (l_0-q_z)^2+2 (l_0-q_z)(l_1+q_z)-l_\perp^2-m_5^2+i\eta
\nonumber\\
P_6 & = & k_0^2+2 k_0 k_1 -k_\perp^2-m_6^2+i\eta {\, ,}\end{aligned}$$ where $z$ is the cosine of the angle between $\vec{l}_\perp$ and $\vec{k}_\perp$.
The volume element in the integral in eq.(\[eq:theintegral\]) is given by (cf. [@newrepplanar]) $$d^4l \, d^4k = \frac{1}{2} dl_0 \, dk_0 \, dl_1 \, dk_1 \, ds \, dt \,
d\alpha \, \frac{dz}{\sqrt{1-z^2}}$$ with $s \equiv l_\perp^2$, $t \equiv k_\perp^2$ and the angle $\alpha$ describing the absolute position of $\vec{l}_\perp$ and $\vec{k}_\perp$. The integration over $\alpha$ gives a trivial factor $2\pi$.
An important difference between the crossed and the planar two-loop vertex function is that now necessarily two (instead of one) propagators depend on $z$, which, in our notation, are $P_1$ and $P_2$. This is a result of the fact that in the planar case the loop momenta can be arranged to flow through only one common propagator, which is not possible for the crossed topology.
But after applying a partial fraction decomposition to the integrand $$\frac{1}{P_1 P_2 P_3 P_4 P_5 P_6} =
\frac{1}{(P_2-P_1) P_1 P_3 P_4 P_5 P_6}
+\frac{1}{(P_1-P_2) P_2 P_3 P_4 P_5 P_6}
\label{eq:parfrac1}$$ the $z$-dependence in $P_1-P_2$ drops out and the $z$-integration can be performed as in the planar case: $$\int\limits_{-1}^1 \frac{dz}{\sqrt{1-z^2}} \frac{1}{A_k+B_k z+i\eta}
= \frac{\pi}{\sqrt[c]{A_k^2-B_k^2+2 A_k i\eta}}, \quad k=1,2$$ where $P_1=A_1+B_1 z+i\eta$ for the first term in eq. (\[eq:parfrac1\]) and $P_2=A_2+B_2 z+i\eta$ in the second, respectively. $\sqrt[c]{z}$ denotes the square root of a complex number with a cut along the positive real axis, whereas $\sqrt{x}$ is the usual square root of a positive real number (if $x$ has a small imaginary part of any sign, it can be ignored).
The next step is the integration over $l_1$ and $k_1$ using Cauchy’s theorem for both of them. It is important to notice that, as a result of the partial fraction decomposition in eq. (\[eq:parfrac1\]), in the difference $P_1-P_2$ the imaginary part of the masses $-i\eta$, which has been chosen to be equal for all masses, drops out. Therefore not all poles in $l_1$ and $k_1$ lie in the upper or lower complex half-plane, but some also on the real axis. The integral can be made meaningful if interpreted as a principal value integral.[^2] Then one has to use the modified Cauchy’s theorem [@henrici1]: $$\begin{aligned}
{{\mbox{P$\!$.V$\!$.}}\!\int}\limits_{-\infty}^{\infty} f(z) \, dz
& = & 2 i \pi \sum_i^n {\mbox{Res}}(f(z)) \bigg|_{z=z_i, \,\mbox{\scriptsize Im}(z_i) {\!\!\begin{array}{c} \mbox{\tiny $<$} \\[-2.1ex]
\mbox{\tiny $>$} \end{array}\!\!}0}\nonumber\\
& & + i \pi \sum_j^m {\mbox{Res}}(f(z)) \bigg|_{z=z_j, \,\mbox{\scriptsize Im}(z_j)=0},\end{aligned}$$ where the $z_i$ are the $n$ poles of $f(z)$ inside the closed integration contour, whereas $z_j$ are $m$ poles along the integration path. Poles on the path contribute only with half the weight compared to poles inside the path. For the cases we are confronted with it is guaranteed that the function tends to zero sufficiently fast for large $l_1$ and $k_1$, hence the theorem is applicable.
The integration path has to be closed either in the upper or in the lower complex half-plane, depending on the position of the cuts of the square root $\sqrt[c]{A^2-B^2+2 A i\eta}$. Similarly to the planar case, the position of the cuts is determined by the sign of $l_0+k_0-e_1$ for the first term in eq.(\[eq:parfrac1\]) and by the sign of $l_0+k_0+e_2$ for the second term.
Let us concentrate on the first term of the partial fraction decomposition, the second can be handled analogously. Assume $l_0+k_0-e_1<0$, so that we have to close the contour of the $l_1$ integration in the lower half-plane. Then, some propagators have poles inside the path: $P_3$ iff $l_0-e_1>0$ and $P_5$ iff $l_0-q_z>0$. Additionally, $P_2-P_1$ always contributes with half its residue, since it has a pole on the real axis. In the following we will call residues, which involve only $P_3 \dots P_6$, [*complex*]{} contributions, because all poles lie in the upper or lower complex half-plane, whereas residues involving $P_1-P_2$ are called [*real*]{} contributions, because the pole lies on the real axis.
For the $k_1$ integration, we close the contour in the same half-plane as for the $l_1$ integration, depending on the sign of $l_0+k_0-e_1$. This is not necessary for the $P_2-P_1$ contribution, where the square root becomes independent of $k_1$, but it is done for consistency. Let us first assume that we had a $l_1$-pole in $P_3$. Then we have poles in $k_1$ from $P_4$ iff $k_0+e_2+q_z>0$ and $P_6$ iff $k_0>0$, and a pole in $P_2-P_1$, iff $l_0-e_1<0$. But the last case is a contradiction to $l_0-e_1>0$ above, so it does not contribute.
Analogously, if we had a $l_1$-pole in $P_5$, we have the same constraints from $P_4$ and $P_6$, but $l_0-q_z<0$ from $P_2-P_1$, which is again a contradiction.
The last case, a $l_1$-pole in $P_2-P_1$, gives contributions from all other propagators, namely $P_3$ iff $l_0-e_1<0$, $P_4$ iff $k_0+e_2+q_z>0$, $P_5$ iff $l_0-q_z<0$ and $P_6$ iff $k_0>0$.
If $l_0+k_0-e_1>0$, the relation operators have to be reversed in all inequalities. Out of the [*complex*]{} combinations, only the pairs $(P_3,P_4)$, $(P_5,P_4)$ and $(P_5,P_6)$ contribute in triangular regions in the $(k_0,l_0)$ plane if $l_0+k_0-e_1<0$ (see fig.\[fig:complcontr1\]). If $l_0+k_0-e_1>0$, no terms contribute. All [*real*]{} residues contribute for $l_0+k_0-e_1<0$ as well as for $l_0+k_0-e_1>0$. The areas are not triangles, but unbound, as can be seen in fig.\[fig:realcontr1\].
The second term of the partial fraction decomposition also gives us three contributing [*complex*]{} residues (all for $l_0+k_0+e_2>0$, fig.\[fig:complcontr2\]) and four [*real*]{} residues, which have the same constraints as for the first term, but with $l_0+k_0-e_1 {\begin{array}{c} < \\[-1.7ex] > \end{array}}0$ replaced by $l_0+k_0+e_2 {\begin{array}{c} < \\[-1.7ex] > \end{array}}0$ (fig.\[fig:realcontr2\]).
If we compare the [*real*]{} contributions from both terms of the partial fraction decomposition eq.(\[eq:parfrac1\]), we notice that the terms for two corresponding residues (e.g. $(P_2-P_1, P_3)$ and $(P_1-P_2,P_3)$) are equal, except for an overall minus sign. As a consequence they cancel in the area where both contribute together. This is everywhere the case except in the strip $-e_2 < l_0+k_0 < e_1$ (see fig.\[fig:realcontr3\]). Furthermore it can be shown that all four [*real*]{} contributions from fig.\[fig:realcontr3\] add up to zero where they all contribute together. This is the case everywhere outside the finite area shown in fig.\[fig:realcontr4\], which is also the joined area of all triangles in figs.\[fig:complcontr1\] and \[fig:complcontr2\] from the [*complex*]{} residues.
After the residue integrations we end up with a four-fold integral representation which is similar in nature, but slightly more complicated in its technical appearance than the planar case. It can be written in the general form $$\begin{aligned}
V(q_1,q_2) & = & \sum_j {\int\!\!\!\!\int}\limits_{A_j} \!\! dl_0 \, dk_0
\int\limits_0^\infty\!\! ds
\int\limits_0^\infty\!\! dt \: C(k_0,l_0)
\nonumber\\
& & \hspace{-1.0cm}
\frac{1}{{\tilde{a}}_1 t+{\tilde{b}}_1+{\tilde{c}}_1 s}
\frac{1}{{\tilde{a}}_2 t+{\tilde{b}}_2+{\tilde{c}}_2 s}
\frac{1}{{\tilde{a}}_3 t+{\tilde{b}}_3+{\tilde{c}}_3 s}
\frac{1}{\sqrt[c]{(a t+b+c s)^2-4 s t}} {\, .}\label{eq:4fold}\end{aligned}$$ The $j$-sum runs over all areas $A_j$ depicted in figs.\[fig:complcontr1\] – \[fig:realcontr2\]. The coefficients $C(k_0,l_0)$, ${\tilde{a}}_i$, ${\tilde{b}}_i$, ${\tilde{c}}_i$ ($i=1 \dots 3$), $a$, $b$ and $c$ are in general dependent on $j$, $l_0$ and $k_0$. ${\tilde{b}}_i$ and $b$ have an infinitesimal imaginary part, all other coefficients are real. Some coefficients are vanishing: for the [*complex*]{} residues, ${\tilde{a}}_1$ and ${\tilde{c}}_3$ are zero, and we have one pure pole in $s$, one in $t$ and a mixed $(s,t)$ pole. For the [*real*]{} residues, either ${\tilde{c}}_3=0$ or ${\tilde{a}}_3=0$, i.e. we have two mixed poles and either one pure pole in $t$ or in $s$. In the latter case we can relabel $s {\displaystyle\leftrightarrow} t$, together with exchanging ${\tilde{a}}_i \leftrightarrow {\tilde{c}}_i$ and $a \leftrightarrow c$, so that we always have exactly two poles in $s$, either pure or mixed, and $c_3=0$. In the planar case we always had one pure $s$ pole and two pure $t$ poles, with mixed poles altogether absent.
To proceed further with the $s$ and $t$ integrations, we apply a partial fraction decomposition in $s$, followed by a similar partial fraction decomposition in $t$. $$\begin{aligned}
\lefteqn{
\frac{1}{{\tilde{a}}_1 t+{\tilde{b}}_1+{\tilde{c}}_1 s}
\frac{1}{{\tilde{a}}_2 t+{\tilde{b}}_2+{\tilde{c}}_2 s}
\frac{1}{{\tilde{a}}_3 t+{\tilde{b}}_3}
} & &
\nonumber\\
& = & C'
\left(
\frac{1}{t+\bar{t}_{02}} -
\frac{1}{t+\bar{t}_{01}}
\right)
\left(
\frac{1}{s+\bar{s}_{02}(t)} -
\frac{1}{s+\bar{s}_{01}(t)}
\right)\end{aligned}$$ with $$\begin{aligned}
C' & = & \frac{1}{{\tilde{c}}_2({\tilde{a}}_1{\tilde{b}}_3-{\tilde{b}}_1{\tilde{c}}_3)-
{\tilde{c}}_1({\tilde{a}}_2{\tilde{b}}_3-{\tilde{b}}_2{\tilde{c}}_3)}
\nonumber\\
\bar{t}_{01} & = & \frac{{\tilde{b}}_3}{{\tilde{c}}_3}
\nonumber\\
\bar{t}_{02} & = & \frac{{\tilde{c}}_2{\tilde{b}}_1-{\tilde{c}}_1{\tilde{b}}_2}
{{\tilde{c}}_2{\tilde{a}}_1-{\tilde{c}}_1{\tilde{a}}_2} {\, .}\nonumber\\
\bar{s}_{01}(t) & = & \frac{{\tilde{a}}_1 t+{\tilde{b}}_1}{{\tilde{c}}_1}
\nonumber\\
\bar{s}_{02}(t) & = & \frac{{\tilde{a}}_2 t+{\tilde{b}}_2}{{\tilde{c}}_2} {\, .}\end{aligned}$$ Thus we have to calculate four integrals of the form $$V' = \int\limits_0^\infty \frac{dt}{t+\bar{t}_0}
\int\limits_0^\infty \frac{ds}{s+\bar{s}_0(t)}
\frac{1}{\sqrt[c]{(a t+b+c s)^2 - 4st}}$$ with $s_0(t)$ being either a linear function of $t$ or a constant. Now we have to split these into real and imaginary part. It can be shown that $b$ always has a positive small imaginary part, whereas the sign of the imaginary part of $\bar{s}_0$ and $\bar{t}_0$ is a function of $l_0$ and $k_0$, so that in general $\bar{s}_0=s_0 \pm i\eta$ and $\bar{t}_0=t_0 \pm i\eta$. In contrast, in the planar case the imaginary part was always negative.
Now there are three possible sources for an imaginary part of $V'$: either $s_0$, $t_0$ or the argument of the square root can become negative. In the first two cases the imaginary part can be extracted using $$\label{eq:pvplusipidelta}
\lim_{\eta \to 0} \int\limits_0^\infty \frac{dx}{x+x_0 \pm i\eta} f(x)
= {{\mbox{P$\!$.V$\!$.}}\!\int}\limits_0^\infty \frac{dx}{x+x_0} f(x)
\mp i \pi \int\limits_0^\infty \delta(x+x_0) f(x) dx {\, .}$$ To analyze the contribution from the square root, we have to distinguish between [*complex*]{} and [*real*]{} residues. For the [*complex*]{} case, the area in the $s-t$ plane, where the argument is negative, is an ellipse. For a given positive $t$, the $s$ values on the border of the ellipse can be calculated as $$\label{eq:sigmas}
\sigma_{1/2} =
\frac{1}{c^2}\left[ \sqrt{t} \pm \sqrt{t(1-a c)-b c} \right] ^2 {\, ,}$$ which has real and positive solutions for $0 < t < bc/(1-ac)$ if $b>0$, since $1-ac<0$, cf. fig. \[fig:ellipse\]. For the [*real*]{} residues, $a=-1$ and $c=-1$, therefore the area where the argument becomes negative is unbounded with a parabola as its boundary (fig. \[fig:parabel\]).
Now we can do the $s$-integration with the aid of an Euler’s change of variables [@fichtenholz] and obtain $$V' = \frac{1}{|c|}\int\limits_0^\infty \frac{1}{t+t_0}
\left( f(s_0,\sigma_i) + i k(t) g(s_0,\sigma_i) \right) \, dt$$ with $$\label{eq:sint1}
f(s_0,\sigma_i) = \left\{
\begin{array}{lll}
\frac{2}{\sqrt{-R}}
\arctan\left(\frac{s_0\pm{\sqrt{\sigma_1\sigma_2}}}{\sqrt{-R}}\right)
& \mbox{if} \quad \sigma_1 < -s_0 < \sigma_2
& \mbox{(i)}
\\
-\frac{1}{\sqrt{R}}
\log\left|\frac{s_0\pm{\sqrt{\sigma_1\sigma_2}}+\sqrt{R}}
{s_0\pm{\sqrt{\sigma_1\sigma_2}}-\sqrt{R}}
\right|
& \mbox{else}
& \mbox{(ii)}
\end{array}\right.$$ and $$\label{eq:sint2}
g(s_0,\sigma_i) = \left\{
\begin{array}{lll}
\frac{i}{\sqrt{-R}}
& \mbox{if} \quad \sigma_1 < -s_0 < \sigma_2
& \mbox{(i)}
\\
\frac{1}{\sqrt{R}}
& \mbox{else}
& \mbox{(ii)}
\end{array}\right.$$ where we note that $s_0$ and $\sigma_i$ are functions of $t$ and further $R={s_0^2+s_0(\sigma_1+\sigma_2)+\sigma_1\sigma_2}$ and $k=-2,-1,0,1$ or 2, depending on the relative magnitudes of $s_0$ and $\sigma_{1/2}$ and the sign of the imaginary part of $s_0$. One has to choose $+{\sqrt{\sigma_1\sigma_2}}$ in eqs.(\[eq:sint1\]) and (\[eq:sint2\]) if $t>-b/a$, and $-{\sqrt{\sigma_1\sigma_2}}$ else. The condition $\sigma_1 < -s_0 < \sigma_2$ is equivalent to $R<0$.
The $t$-intervals where $k$ is constant, together with the corresponding $k$ values, can be calculated by solving $s_0=0$ and $s_0=-\sigma_i$ for $t$. With another Euler’s change of variables, using eq.(\[eq:pvplusipidelta\]) and exploiting the relevant properties of the functions $\log(x)$ and $\arctan(x)$, the $t$-integration leads to expressions of the form $${{\mbox{P$\!$.V$\!$.}}\!\int}\limits_{x_1}^{x_2} \frac{1}{x^2+px+q}
\quad\mbox{from} \;\; g(s_0,\sigma_i) {\, ,}$$ $${{\mbox{P$\!$.V$\!$.}}\!\int}\limits_{x_1}^{x_2} \frac{\arctan(rx+s)}{x^2+px+q}
\quad\mbox{from}\; f(s_0,\sigma_i),\; \mbox{case (i)}$$ and $${{\mbox{P$\!$.V$\!$.}}\!\int}\limits_{x_1}^{x_2} \frac{\log|x^2+rx+s|}{x^2+px+q}
\quad\mbox{from}\; f(s_0,\sigma_i),\; \mbox{case (ii)} {\, ,}$$ which can be expressed in terms of logarithms, arcus-tangens, dilogarithms and Clausen’s functions [@lewin], as it was the case for the planar topology. The full result is, as expected, a lengthy expression in terms of these special functions, which we cannot list here. We rather follow the philosophy of [@newrepplanar] and present examples in the next section.
We stress that to obtain stable numerical results in the two dimensional integral over $l_0$ and $k_0$ it is very important to add all contributions ([*real*]{} and [*complex*]{}) to the same $l_0$ and $k_0$, because eq.(\[eq:parfrac1\]) introduces some artificial divergences which cancel in the sum.
As expected, the numerical integrand for the crossed topology is of the same nature as for the planar case, but involves more terms and different cases. This naturally increases the amount of CPU time needed to obtain the requested accuracy: as a thumb rule, we found that the crossed topology demands 5-10 times more time compared to the planar case.
The threshold behaviour can be examined with the four-fold integral representation eq.(\[eq:4fold\]). The crossed vertex function has three two-particle thresholds and six three-particle thresholds. As stated above a possible source for an imaginary part is a negative argument of the square root. A necessary condition for this to happen is $b>0$ for some $l_0$, $k_0$ inside the integration region. It can be shown that each three-particle threshold corresponds to one of the six $b$ coefficients of the [*complex*]{} triangles [@xloops]. The other $b$ coefficients belonging to the [*real*]{} regions are identical and correspond to the two-particle threshold $q^2>(m_1+m_2)^2$. The remaining two-particle thresholds $q_1^2>(m_3+m_5)^2$ and $q_2^2>(m_4+m_6)^2$ can be identified with the coefficients $\bar{s}_{01}$ (when ${\tilde{a}}_1 = 0$) and $\bar{t}_{01}$ respectively which produce an imaginary part when $\bar{s}_{01}$ and $\bar{t}_{01}$ become negative. A possible four-particle threshold $q^2>(m_3+m_4+m_5+m_6)^2$ disconnects the graph into three parts and is therefore a combination of the two two-particle thresholds $q_1^2>(m_3+m_5)^2$ and $q_2^2>(m_4+m_6)^2$.
Examples
========
Only few analytical and/or numerical results are known for the crossed vertex function to compare with. Besides the symmetries with respect to internal masses and external momenta, some limiting cases can be checked. In the case of zero momentum transfer the crossed vertex function reduces to the master two-point topology [@dirk1; @baub] with a squared propagator. [fig.\[fig:exmaster\]]{} shows these limits for some arbitrary masses ($m_1=2$, $m_2=3$, $m_3=1$, $m_4=1.5$, $m_5=0.6$ and $m_6=0.2$), $q^2=1$ and either $q_1 \to 0$ or $q_2 \to 0$. Additionally, for vanishing $q^2$ we obtain the vacuum bubble calculated in [@davydtausk].
For the case of all internal masses vanishing an analytical formula in known [@davyd2]. This limit is approached smoothly in [fig.\[fig:zeromass\]]{} for all masses going to zero simultaneously and $e_1=4$, $e_2=3$ and $q_z=1$.
In the limit of the momenta of the outgoing particles on the light-cone ($q_1^2=q_2^2=0$) and all internal masses equal this diagram can be calculated with the small momentum expansion technique [@tarasov]. A comparison for real and imaginary part is show in [fig.\[fig:tarasov\]]{}.
We can obtain this limit easily by transforming our parallel space coordinates to light cone coordinates, which does not interfere with our subsequent steps.
As a last example in [fig.\[fig:kato\]]{} we show the decay $\rm Z \to t \bar{t}$ with the exchange of two Z bosons. This diagram has been calculated in [@kato] by a five-dimensional numerical integration over the Feynman parameters.
Conclusions {#conclusions .unnumbered}
===========
In this paper, we demonstrated the calculation of the two-loop vertex function for the crossed topology. We outlined how to achieve the reduction to a manageable two-fold integral representation, and verified its correctness by comparison with the literature, wherever data were available. Further, our integral representation was used in [@FKKR], with results in perfect agreement with the expectations.
Our results complement the results for the planar topology in [@newrepplanar]. Together, the scalar two-loop three-point function is now available in $D=4$ dimensions for all topologies, all masses, and arbitrary external momenta. Some degenerated topologies were already given in [@FKKR], obtained by similar methods. Such degenerated cases typically also appear when one confronts tensor integrals. For the future, we plan to incorporate such cases in the package XLOOPS [@xloops]. Also, code which implements the results presented here will be incorporated there.
To our knowledge, no other method is at this stage able to deliver reliable results for the massive two-loop vertex function in such generality and accuracy.
Acknowledgements {#acknowledgements .unnumbered}
================
We like to thank David Broadhurst, Jochem Fleischer, Bernd Kniehl, Kurt Riesselmann, Karl Schilcher and Volodya Smirnov for interesting discussions and support, and the participants and organizers of AIHENP96 (Lausanne, September 1996) for a stimulating workshop. D.K. thanks the DFG for support. This work was supported in part by HUCAM grant CHRX-CT94-0579.
[99]{} D. Kreimer. [*Phys. Lett.*]{} [**B273**]{} (1991) 277. D. Kreimer. [*Phys. Lett.*]{} [**B292**]{} (1992) 341. A. Czarnecki, U. Kilian and D. Kreimer, [*Nucl. Phys.*]{}[**B433**]{} (1995) 259. O.V. Tarasov. [*An algorithm for small momentum expansion of Feynman diagrams*]{}, publ. in the proceedings of AIHENP95: [*New Computing Techniques in Physics Research IV*]{}, Eds. B. Denby, D. Perret-Gallix, World Scientific (1995). J. Fujimoto, Y. Shimizu, K. Kato and T. Kaneko, [*Numerical approach to two-loop three point functions with masses*]{}, publ. in the proceedings of AIHENP95: [*New Computing Techniques in Physics Research IV*]{}, Eds. B. Denby, D. Perret-Gallix, World Scientific (1995). L.Brücher, J.Franzkowski, A.Frink, D.Kreimer, contributed talks given at [*AIHENP96*]{}, Lausanne, September 1996, to appear. See also the homepage\
http://dipmza.physik.uni-mainz.de/${}^\sim$Bruecher/xloops.html S. Bauberger and M. Böhm. [*Nucl. Phys.*]{} [**B445**]{} (1995) 25. J. Fujimoto, Y. Shimizu, K. Kato and Y. Oyanagi, [*Numerical approach to two-loop integrals*]{}, publ. in the proceedings of the workshop on [*HEP and QFT*]{}, Sochi (1992), in [*Phys.Atom.Nucl.*]{}[**56**]{}. J. Fleischer and O.V. Tarasov. [*Z.Phys.C64*]{} (1994) 413. A.Frink, B.Kniehl, D.Kreimer, K.Riesselmann. [*Phys.Rev.*]{}[**D54**]{} (1996) 4548. P. Henrici. [*Applied and computational complex analysis, Vol.I.*]{} John Wiley & Sons, New York, 1974. G.M. Fichtenholz. [*Integral and differential calculus, Vol. II.*]{} Fizmatgiz, Moscow, 1959. N.I. Ussyukina and A.I. Davydychev. [*Phys. Lett.*]{} [**B298**]{} (1993) 363. N.I. Ussyukina and A.I. Davydychev. [*Phys. Lett.*]{} [**B332**]{} (1994) 159. L. Lewin. [*Polylogarithms and associated functions.*]{} North Holland, New York, 1981. A.I. Davydychev and J.B. Tausk. [*Nucl. Phys.*]{} [**B397**]{} (1993) 123.
[^1]: email: [*author*]{}@dipmza.physik.uni-mainz.de
[^2]: One can avoid principal value integrals by choosing different imaginary parts $\eta_i$ for the propagators, as mentioned earlier. After having convinced ourselves that the results remain unchanged, we prefer to follow the route outlined here, for purely technical reasons.
|
---
abstract: 'We study excess noise in a quantum conductor in the presence of constant voltage and alternating external field. Due to a two particle interference effect caused by Fermi correlations the noise is sensitive to the phase of the time dependent transmission amplitude. We compute spectral density and show that at $T=0$ the noise has singular dependence on the [dc]{} voltage $V$ and the [ac]{} frequency $\Omega$ with cusplike singularities at integer $eV/\hbar\Omega$. For a metallic loop with an alternating flux the phase sensitivity leads to an oscillating dependence of the strengths of the cusps on the flux amplitude.'
address:
- |
Universit[" a]{}t zu K[" o]{}ln,\
Institut f[" u]{}r Theoretische Physik,\
Z[" u]{}lpicher Str. 77, D-5000 K[" o]{}ln 41, Germany
- |
Massachusetts Institute of Technology,\
12-112, Department of Physics,\
77 Massachusetts Ave., Cambridge, MA 02139
author:
- 'G.B. Lesovik[@*]'
- 'L.S.Levitov[@**]'
title: |
Noise in an [ac]{} biased junction.\
Non-stationary Aharonov-Bohm effect.
---
There is a variety of phenomena related with the quantum coherence of transport in small conductors[@1]: weak localization, Aharonov-Bohm effect with the flux quantum $hc/2e$, universal conductance fluctuations, etc. Each of these effects can also be seen in the spectrum of noise, equilibrium or non-equilibrium. The equilibrium noise is simply proportional to conductance according to the fluctuation-dissipation theorem. The non-equilibrium noise in coherent conductors is expressed through eigenvalues of the scattering matrix[@2; @3; @4], and, therefore, is also related with the conductance, though in a less trivial way. For that reason all the coherence phenomena are present in the non-equilibrium noise as well. However, for a better understanding of transport in small conductors it is interesting to analyze the converse line of thinking and to look for coherence effects that are present in the noise but are absent in the conductance. Such effects, if they exist, are genuinely many-particle[@5], otherwise they would show up in the conductance. As long as we are talking about non-interacting fermions it is only statistics that can produce such coherence. A purpose of this letter is to describe an effect caused by two-particle statistical correlations that leads to phase sensitivity of a two-particle observable, i.e., of electric noise, but does not affect one-particle observables, e.g., conductance. The phase sensitivity manifests itself in an oscillating dependence on the amplitude of an [ac]{} flux, in many aspects similar to the A-B effect. However, it will occur in a single-connected conducting loop, i.e., in the geometry where the normal A-B effect is absent.
Let us specify which coherence effects we are going to study. In simple words, when an electron is scattered inside a conductor its wavepacket splits into two portions, forward and backward, presenting a choice to the electron to be either transmitted or reflected with the probabilities $D$ and $1-D$. Part of this picture of the wavepacket splitting, involving the relation of $D$ with the conductance[@6; @7] and of $D(1-D)$ with the noise[@2; @3; @4; @8] is well understood. However, there is another part, quite unusual, related with the behavior of current fluctuations in the time domain. Recently, we studied the distribution of the charge transmitted through a resistor during fixed interval of time[@9]. We found that the distribution is very close to the binomial, which means that the attempts to have electron transmitted are highly correlated in time. (Were the sequence of the attempts perfectly periodic the distribution would be exactly binomial.) The origin of the correlation is the Pauli principle that forbids passing of electrons through the resistor simultaneously. The attempts follow almost periodically, spaced by the interval $h/eV$. Because the periodicity is not perfect it does not affect the average current, but shows up in its second moment, i.e., noise, leading[@10] at zero temperature to a sharp edge of the spectral density of excess noise $S_\omega$ near $\omega_0= eV/\hbar$: $S_\omega = 2{e^2\over\pi
}D(1-D)\hbar (\omega_0 - |\omega |)$ for $|\omega | < \omega_0$, $0$ otherwise. (Excess noise is the difference of the actual noise and the equilibrium $S_\omega=2{e^2\over\pi }D\hbar\omega\
{\rm coth}(\hbar\omega/2T)$.) The corresponding current-current correlation in time is ${\langle}{\langle}j(t)j(t+{\tau}){\rangle}{\rangle}= {2e^2
\over {\pi}^2}D(1-D) {\rm sin}^2({\omega}_0{\tau}/2)/{\tau}^2$, oscillating with the period $2\pi/\omega_0$ and decaying.
Having realized that the frequency $eV/\hbar$ is characteristic for the time correlation of the attempts one has to think of a simple experimental situation were the presence of this frequency could be studied. It is natural to consider a system driven both [dc]{} and [ac]{}, and to look for the effects of commensurability of $\Omega$ and $eV/\hbar$, where $\Omega$ is the frequency of the [ac]{} bias and $V$ is the [dc]{} voltage. In this letter we study such a system and demonstrate that due to the [ac]{} bias the singularity at $\omega
=\omega_0$ can be shifted down to zero frequency thus making it easier to observe. Below, we compute the noise in a model resistor in the presence of combined [dc]{}-[ac]{} bias and find that the low frequency noise power $S_0$ has singularities at $eV=n\hbar\Omega$, when the “internal” frequency of the problem $eV/\hbar$ is a multiple of the external frequency $\Omega$. We find that $\partial S_0/\partial V$ is a stepwise function of $V$ that rises in positive steps at $V_n=n\hbar\Omega/e$. Another interesting observation is that the heights of the steps of $\partial S_0/\partial V$ are [*phase sensitive*]{}, i.e., they depend on the phase of the transmission amplitude in an oscillating way resembling A-B effect. The phase sensitivity of the noise should be opposed to the pure [dc]{} situation where only the probabilities of transmission and reflection enter the expression for the noise, which makes the noise power insensitive to the phase picked by the wavefunction across the system. In the simplest situation when the [ac]{} bias is supplied by alternating flux threading the current loop, $\Phi (t)=\Phi_a\sin (\Omega t)$, the heights of the steps in $\partial S_0/\partial V$ are proportional to the squares of the Bessel functions $J_n^2(2\pi\Phi_a/\Phi_0)$, where $\Phi_0=h c/e$. Let us note that we are not talking about the trivial effect of the e.m.f. $-\partial\Phi /c
\partial t$ induced in the circuit by the alternating flux. The effect in the noise will persist in the quasistatic limit $|\partial\Phi/c\partial t|\ll V$ when the [ac]{} component of the current vanishes.
Let us start with recalling general facts about scattering off an oscillating potential. We consider a model one dimensional system where electrons are scattered by alternating scalar and vector potentials $U(x,t)$, $A(x,t)$ localized in the interval $[-d,d]$, $U(x,t)=A(x,t)=0$ for $|x|>d$. As a function of time they are periodic: $U(x,t)=\sum_{m=-\infty}^{\infty} U_m(x)\exp(-im\Omega t)$, where $U_0(x)$ is the static part of the potential, and the other harmonics $U_m(x)$, $m\ne0$ describe [ac]{} bias. (Expression for $A(x,t)$ is similar.) The [dc]{} bias is expressed in the framework of the Landauer model as the difference of the population of the right and the left scattering states. An important difference is that in our case the states describe [*inelastic*]{} scattering because an electron can gain several quanta $\hbar \Omega$ while passing through the region $[-d,d]$. It will be useful to have the states expressed through the amplitudes of transmission and reflection: $${\psi}_{L,k}(x,t)=\left\{ \begin{array}{ll}
e^{-iEt+ikx}
+\sum\limits_n B_{L,n}e^{-iE_nt-ik_nx} & x<-d \\
\sum\limits_n A_{L,n}e^{-iE_nt+ik_nx} & x>d
\end{array} \right. ,$$ $${\psi}_{R,k}(x,t)=\left\{ \begin{array}{ll}
\sum\limits_n A_{R,n}e^{-iE_nt-ik_nx} & x<-d \\
e^{-iEt-ikx}+\sum\limits_n B_{R,n}e^{-iE_nt+ik_nx} & x>d
\end{array} \right. ,$$ where the amplitudes $A_{L(R),n},B_{L(R),n}$ are time independent. Here $E=\hbar^2k^2/2m$, $E_n=E+n\hbar\Omega$, and $k_n$ are defined by $\hbar^2k_n^2=2mE_n$. The states (1) are solutions of the Schr[ö]{}dinger equation\
$$E\psi (x,t)=[{1\over 2}(-i{\partial\over\partial x }-{e\over
c}A(x,t))^2+U(x,t)]\psi (x,t)\ .$$ They can be used as a basis to study transport through the system the same way it is done for the static barrier. The amplitudes $A_{L(R),n}, B_{L(R),n}$ satisfy unitarity relation, $$\sum_{n,n',\alpha,\alpha'} \delta (E_n-E'_{n'})
{\big (} \bar A_{\alpha,n}(E) A_{\alpha',n'}(E')+
\bar B_{\alpha,n}(E) B_{\alpha',n'}(E'){\big )}=
\delta(E-E') \delta _{\alpha \alpha'}$$ that one obtains by the standard reasoning about conservation of current.
The operator of electric current, $\hat j(x,t)=-ie\hat \psi^+(x,t)\nabla\hat \psi(x,t)$ is written in terms of second-quantized electrons, ${\hat {\psi}}(x,t)= {\hat {\psi}}_L(x,t)+{\hat {\psi}}_R(x,t)$, $\hat \psi_L(x,t)= \sum_k \psi_{L,k}(x,t) {\hat a}_k$, $\hat \psi_R(x,t)=
\sum_k \psi_{R,k}(x,t) {\hat b}_k$, where $a_k$ and $b_k$ are canonical Fermi operators corresponding to the states (1) coming out of the reservoirs, the left and the right respectively. It is straightforward to compute the mean value $I(t)=\langle\hat j(x,t)\rangle$, where the brackets $\langle...\rangle$ stand for averaging with the density matrix $\rho$ of the reservoirs. As usual, we assume absence of correlations in the reservoirs, $\hat \rho =\hat
\rho_L \otimes \hat \rho_R$, which physically means that after having been scattered into a reservoir electrons have enough time to relax to the equilibrium before they return. Below we assume equilibrium Fermi distributions $\rho_{L,R}=
n(E-E_F\pm eV/2)$. One obtains a generalized Landauer formula: $$I(t)=\sum_{m=-\infty}^{\infty} I_m\exp(-im\Omega t)\ ,$$ where it is straightforward to write the coefficients $I_m$ in terms of the scattering amplitudes $A_{L(R),n}$ and $B_{L(R),n}$. Expr.(2) describes both steady current and generation of harmonics in the presence of the [ac]{} bias.
Quite similarly one can obtain an expression through $A_{L(R),n}$, $B_{L(R),n}$ for the noise. Noise is related with the correlation function $S(t_1,t_2)=\langle\!\langle [\hat j(x,t_1), \hat j(x,t_2)]_+
\rangle\!\rangle$. In the usual [dc]{} situation $S(t_1,t_2)=S(t_1-t_2)$ and its Fourier transform gives spectral density $S_\omega=\langle\!\langle j_\omega j_{-\omega}\rangle\!\rangle$ of the noise. In our [ac]{} case the situation is somewhat more complex because the spectral density $S_\omega$ does not provide a complete description of the noise. Indeed, $S(t_1,t_2)$ will depend now separately on $t_1$ and $t_2$, not only on the difference $t_1-t_2$. However, it satisfies $S(t_1,t_2)= S(t_1+2\pi/\Omega,
t_2+ 2\pi/\Omega)$ resulting from the periodicity of the [ac]{} bias. For the Fourier components $\hat j_\omega$ it means that the average $\langle\!\langle j_\omega j_{\omega'} \rangle \!
\rangle$ does not vanish whenever $\omega+\omega'=m\Omega$, where $m$ is any integer. Thus, in addition one gets generalized spectral densities $S_{\omega,m}=\langle\!\langle j_\omega j_{m
\Omega -\omega} \rangle \! \rangle$, an integer parameter family of functions. Among them there is the ’ordinary’ $m=0$ spectral density $S_\omega = S_{\omega,0}=\langle\!\langle j_\omega
j_{-\omega} \rangle \! \rangle$, the one easiest to access experimentally. In what follows we concentrate on it and do not study other $S_{\omega,m}$, $m\ne 0$.
To compute the noise one has to average the product of two current operators over the distribution in the reservoirs. Evaluation of the average is similar to Refs.[@2; @3; @4], so we do not need to repeat it here. General expression simplifies quite substantially in the practically interesting limit of $t_f$, the time of flight through the barrier $U(x,t)$ being much shorter than $2\pi/\Omega$ and $\hbar/eV$. The point is that $\hbar
/t_f$ defines the characteristic scale of energy dependence of the scattering amplitudes, so the condition $t_f\Omega \ll 1$, $t_feV \ll \hbar$ enables one to neglect the energy dependence of $A_{L(R),n}$, $B_{L(R),n}$ in the interesting energy domain $E_F \pm {\rm max}[eV, \hbar \Omega ]$. We also assume $E_F \gg {\rm max}[eV, \hbar \Omega]$, which allows to neglect the difference of $k_n/m$ and $k/m$, the velocities of scattered and incident states, and set $k_n/m=v_F$. It should be remarked that the physical picture we discuss below is not really dependent on any of these assumptions, they only make our expressions more compact. The more general case of arbitrary relation between $\hbar /t_f$, $E_F$, $eV$, $\hbar \Omega$ presents no difficulty.
With the above assumptions made it becomes convenient to use Fourier transform of the amplitudes $A$ and $B$. Let us define $A_\alpha
(t)=\sum_n A_{\alpha,n} \exp ( -i n \Omega t)$, $\alpha=L,R$. Similarly we introduce $B_\alpha (t)$, and rewrite Expr.(1) as $${\psi}_{L,k}(x,t)=\left\{ \begin{array}{ll}
e^{ikx}+B_L(t+x/v_F)e^{-ikx} & x<-d \\
A_L(t-x/v_F)e^{ikx} & x>d
\end{array} \right. ,$$ $${\psi}_{R,k}(x,t)=\left\{ \begin{array}{ll}
A_R(t+x/v_F)e^{-ikx} & x<-d \\
e^{-ikx}+B_R(t-x/v_F)e^{ikx} & x>d
\end{array}\right. .$$ (To obtain (3) from (1) we substitute $k_n=k+n\Omega /v_F$ in the phase shifts $e^{ik_nx}$ and then do the sum over $n$.) The amplitudes $A_{L(R)}(t)$, $B_{L(R)}(t)$ have clear meaning of the transmission and reflection amplitudes at given instant of time for a slowly varying potential. The retarded time $t-|x|/v_F$ in Expr.(3) accounts for the finite speed of motion after scattering. The unitarity relation now takes the form $$|A_{L(R)}(t)|^2+|B_{L(R)}(t)|^2=1,\
\bar A_L(t)B_R(t)+\bar B_L(t) A_R(t)=0\ .$$ To clarify the character of the simplification thus achieved let us remark that with Expr.(3) the formula (2) for the current $I(t)$ becomes just $I(t)=2{e^2\over h}|A(t)|^2eV$ which means that the current ’adiabatically’ follows time variation of the transparency of the barrier according to the Landauer formula. Now we shall compute noise and find that, unlike $I(t)$, it is not reduced to anything trivially related with the static limit. Let us write the average of two currents $\langle\!\langle\hat j(t_1)\hat j(t_2)\rangle\!\rangle=$ $${2e^2\over h^2}\sum_{E,E'}e^{-i(E_k-E_{k'})(t_1-t_2)}
{\big [} |A(t_1)A(t_2)|^2{\big (}n_L(E')(1-n_L(E))+
n_R(E')(1-n_R(E)){\big )}$$ $$+\bar B(t_1)A(t_1) \bar A(t_2)B(t_2) n_R(E') (1-n_L(E))+
\bar A(t_1)B(t_1) \bar B(t_2)A(t_2) n_L(E') (1-n_R(E)){\big ]} .$$ To compute $S_\omega$ we have to do Fourier transform and substitute Fermi distributions $n_{L(R)}(E)$. Explicit calculation yields the result $$S_{\omega}=
{2e^2\over \pi} \sum\limits_n 2 N_0(\omega-n\Omega)
|(|A|^2)_n|^2 + N_1(\omega ,n\Omega+eV) |(A\bar B)_n|^2,$$ where $$N_0(x)=\int (n(E-x)+n(E+x))(1-n(E))dE= x{\rm coth}(x/2T)\ ,$$ $$N_1(x,y)=N_0(x+y)+N_0(x-y)={x \sinh (x/2T) - y \sinh (y/2T)\over
\cosh (x/2T) - \cosh (y/2T)}\ ,$$ and $(...)_n$ denotes Fourier components, e.g., $(A\bar B)_n={\Omega
\over 2\pi}\int A(t)\bar B(t) e^{in\Omega t}dt$. Expr.(4) describes the noise as function of $eV,\ \Omega,\ \omega$ and $T$. The behavior is simplest at $T=0$ when $N_0(x)=|x|$, $N_1(x,y)=|x+y|+|x-y|$. Given by Expr.(4) as a weighted sum of terms like $|n\Omega + eV \pm\omega|$, $|\omega-n\Omega|$ the noise $S_{\omega}$ will then depend on $V,\ \Omega,\ \omega$ in a piecewise linear way, changing from one slope to another when $n\Omega+eV\pm\omega$ or $\omega-n\Omega$ equals $0$. This condition defines the locations where $S_{\omega}$ has singularities. They are cusps, sharp at $T=0$ and rounded on the scale $T$ at $T>0$.
With the general Expr.(4) one can explore the noise in all possible limiting situations that one obtains for different combinations of $eV,\ \Omega,\ \omega$ and $T$. Particularly interesting for us will be the case $T=0$, $\omega =0$ corresponding to the noise $S_0 =\langle\!\langle j_\omega j_{-\omega}\rangle\!
\rangle_{\omega \rightarrow 0}$ measured at low frequency. Let us remark here that setting $\omega =0$ means only that $\omega$ is small compared to the parameters $eV$ and $\Omega$ that define the width of the frequency band of the excess noise. Such $\omega$ may still be much higher than the band width for other sources of noise, e.g., the $1/f$. Let us concentrate on the dependence of $S_0$ on $V$. It is a piecewise linear function which is easiest to characterize by its derivative, $$\partial S_0/\partial V={2e^3\over \pi }\sum\limits_n \lambda_n
\theta (eV-n\hbar\Omega),$$ where $\lambda_n=|(\bar A_L B_R)_n|^2$ and $\theta (x)=1$ for $x>0$, $-1$ otherwise. The function $\partial S_0/\partial V$ rises in positive steps at all $V_n=\hbar \Omega n/e$ (see Fig. \[figure1\]), the property that can be alternatively formulated as convexity of $S_0(V)$ in $V$.
The meaning of the singularities in $S_0(V)$ was clarified recently in a study of the statistics of transmitted charge [@IL]. The generating function of the charge distribution was expressed through the single-particle scattering matrix, and it was found that the distribution arises from Bernoulli statistics (i.e., it is a generalized binomial distribution). The frequencies of attempts were given as function of $V$ and $\Omega$. The probabilities of outcomes of a single attempt were found in terms of many-particle scattering amplitudes, and it was shown that they change at the thresholds $V_n=n\hbar\Omega/e$ in a discontinuous way due to statistical correlation in the outgoing channels of the scattering. The discontinuity manifests itself in the second moment of the distribution that corresponds to the noise $S_0(V)$ discussed above.
There is an interesting and simple example where one can explicitly evaluate the heights of the steps. Let us consider a junction with ideal leads bent into a loop of length $L$ (see inset of Fig. \[figure1\]) and placed into an external magnetic field varying with time. In this problem the junction is the only source of scattering. For simplicity let us assume that only one scattering channel is involved and that the junction is symmetric, $A_L=A_R=A$, $B_L=B_R=B$. The [ac]{} bias is supplied by the alternating flux of the magnetic field through the loop, $\Phi (t)=\Phi_a\sin (\Omega t)$. Also let us suppose that the magnetic field is quasistatic, i.e., the time of flight through the system, $t_f=L/v_F$ is much shorter than $2\pi/\Omega$, that makes it possible to introduce the time dependent amplitudes $A_{L(R)}(t),\ B_{L(R)}(t)$ as it was discussed above. As is common, in such a situation the vector potential can be treated semiclassically, and one can write the wavefunction as $\psi (x,t)= \exp ({ie\over\hbar
c}\int_{-\infty}^{x} A(x')dx') \psi_0 (x,t)$, where $x$ is the coordinate along the lead and $\psi_0 (x,t)$ is found by solving the Schr[ö]{}dinger equation in the absence of the magnetic field. Thus all the dependence on the magnetic flux can be accumulated in the phase of the transmission amplitude, $$A_{R(L)}(t)=\exp (\pm i\Phi (t)/\Phi_0) A$, $B_{R(L)}(t)=B,$$ where $\Phi_0=h c/e$ is single electron flux quantum. Since $|A(t)|^2=D=const$ the current is time independent: $I={2e^2\over h}DV$. According to Expr.(4) $S_{\omega}$ is written through the Fourier components of $\bar A_L(t) B_R(t)$ in this case given by the Bessel functions: $(\bar A_L
B_R)_n=J_n(2\pi\Phi_a/\Phi_0)AB$. Thus we find $$S_{\omega}={2e^2\over \pi} {\big [}
2N_0(\omega )D^2+ \sum\limits_n N_1(\omega ,n\Omega+eV)
D(1-D)J_n^2(2\pi\Phi_a/\Phi_0){\big ]}.$$ The heights $\lambda_n$ of the steps in $\partial
S_0/\partial V$ are then given by $$\lambda_n=
D(1-D) J_n^2(2\pi\Phi_a/\Phi_0).$$ They oscillate as function of $\Phi_a / \Phi_0$ and vanish at the nodes of Bessel functions.
Exprs.(6),(7) illustrate one important feature of the noise in the [ac]{} biased system, the sensitivity to the [*phase*]{} of the transmission amplitude $A$. By varying the amplitude $\Phi_a$ of the alternating flux one can make $\lambda_n$ vanish separately for each harmonic $n\Omega$ of the [ac]{} frequency. This should be compared with the case of the [dc]{} bias where the noise is expressed only through $|A|^2$ and thus cannot be phase dependent. We call the oscillating dependence (6),(7) [*non-stationary Aharonov-Bohm effect*]{}. To compare it with the usual A-B effect let us recall that the latter is observed as an oscillation of the [dc]{} conductance under variation of flux in the situation when one has interference of transmission amplitudes corresponding to different classical trajectories of a quantum system, e.g., in a conductor with multiply connected leads forming one or several closed loops. The [dc]{} A-B effect cannot be observed in the single path geometry like Fig.1. Alternatively, the non-stationary A-B effect appears as a result of interference of the right and left scattering states travelling in the opposite directions along same path and having energies shifted by $n\Omega$. It is clear from our discussion that such interference does not contribute to the [ac]{} conductance but is important for the noise and, therefore, one obtains the non-stationary A-B effect in the noise even in the topologically trivial situation of Fig.1.
One can derive a sum rule: $$\sum\limits_n \lambda_n={\Omega\over 2\pi}
\int_0^{2\pi/\Omega} D(t)(1-D(t))dt \ ,$$ where $D(t)=|A(t)|^2$. For $\lambda_n$ given by Expr.(7) it follows from the definition of the Bessel functions. In the general case of Expr.(5) the sum rule is obtained by applying Plancherel’s formula to Fourier components of $A(t)\bar B(t)$. The sum rule clarifies the relation of our problem with the previous calculation[@2; @3; @4] of the noise in the pure [dc]{} case for which the result does not depend on the phase of $A(t)$. When the limit is taken $\Omega\rightarrow 0$, $V=const$, the steps in $\partial S_0/\partial V$ do not vanish but just move closer to zero, thus effectively condensing then all together in a single step at $V=0$. The height of this step is not phase sensitive and is simply given by the expression (8) for the [dc]{} noise averaged over the period $2\pi/\Omega$.
It is worth mentioning that our results for $S_\omega$ are quite general. Indeed, it is clear after what have been said that the singularities at $V=n\hbar\Omega/e$ are only due to the sharp edge of the Fermi distribution, and not related with any specific geometry assumed for the junction. Because of that the phenomenon should be displayed by any coherent conductor, provided that the main source of inelastic scattering is the [ac]{} potential. The reason is that an elastic scattering, if any, can smear the Fermi distribution of momenta but it will not affect the sharpnes of the step in the energies distribution, and our effect is sensitive only to the latter. The same remark applies to the oscillating dependence of the singularities on the amplitude of the [ac]{} signal.
Let us briefly discuss a generalization of the system shown in Fig.1 where the loop is not an ideal lead but a real metallic wire with disorder, i.e., instead of one scatterer there are now many of them uniformly distributed over the bulk of the wire. Most interesting is the case of a purely coherent conductor for which the energy relaxation time $\tau_E$ and the phase breaking time $\tau_\phi$ are much longer than the flight time $t_f$. (One can estimate $t_f\approx \hbar/E_c$, where $E_c$ is Thouless’ energy $\hbar D/L^2$.) In such a system transport is described by channels of the scattering matrix with transmission coefficients $T_m$ assigned to each channel[@7]. In the [dc]{} case the noise can be written[@4] in terms of $T_m$ as $S_0={2e^2\over \pi} \sum_m T_m(1-T_m)eV$. In the presence of the alternating flux the extension of our formalism can be carried out easily and one obtains expressions similar to (6) and (7), with $D^2$ and $D(1-D)$ replaced by $\sum_m T_m^2$ and $\sum_m T_m (1-T_m)$ respectively. However, the limitations under which the result is valid, $eV\ll\hbar/t_f$, $\Omega\ll 1/t_f$, are now slightly more stringent than for Exprs.(6),(7) because the flight time $t_f$ is longer.
A more fundamental limitation to the general validity of our calculation is in the assumption that the flux threads only the phase coherent part of the conductor. It would certainly be of interest to better understand the opposite limit when the [ac]{} voltage increases smoothly over a distance much larger than the phase breaking length $L_\phi=\sqrt {\tau_\phi/D}$.
To summarize, we studied current and noise in a conductor driven by [dc]{} and [ac]{} and we expressed them through time-dependent one particle scattering amplitudes. In the quasistatic limit of short time of flight through the conductor the current is given by the Landauer formula with time-dependent transmission coefficient, i.e., by a trivial generalization of the static case. The situation with the noise is quite different because of the two-particle interference. Spectral density of the noise $S_\omega$ depends on the scattering amplitudes in such a way that the phases do not drop out, and this leads to a non-stationary Aharonov-Bohm effect. Because of the way the Fermi statistics affects the two-particle interference the noise measured at $T=0$ is singular at $\omega =\pm eV/\hbar
+m\Omega$, where $m$ is any integer. To illustrate the phase sensitivity of the noise we consider a conducting metallic loop in which the [ac]{} signal is supplied by an oscillating magnetic flux. Because of the sensitivity to the phase of transmission amplitude the strengths of the singularities in the noise display oscillatory dependence on the amplitude of the [ac]{} flux given by squares of the Bessel functions.
We are indebted to D. E. Khmelnitskii for drawing our attention to the problem of harmonic generation in coherent conductors, and to J. Hajdu for illuminating discussions.\
Research of L. L. is partly supported by Alfred Sloan fellowship. The work of G. L. is performed within the research program of the Sonderforshungsbereich 341, Köln-Aachen-Jülich.
also the Institute for Solid State Physics, Chernogolovka 142432, Moscow district, Russia also at the L. D. Landau Institute for Theoretical Physics, Moscow 117334, Russia P. A. Lee, T. V. Ramakrishnan, Rev. Mod. Phys. [**57**]{}, p.287 (1985),\
B. L. Altshuler, P. A. Lee, Phys. Today (December), p.2 (1988) G. B. Lesovik, JETP Letters, [**49**]{}, p.594 (1989) B. Yurke and G. P. Kochanski, Phys. Rev. [**41**]{}, p.8184 (1989) M. B[ü]{}ttiker, Phys. Rev. Lett., [**65**]{}, p.2901 (1990),\
M. B[ü]{}ttiker, in: Granular Nanoelectronics, D. K. Ferry (ed.), Plenum Press, New York (1991) M. Büttiker, Phys. Rev. Lett., [**68**]{}, p.843 (1992) R. Landauer, in: Localization, Interaction and Transport Phenomena, eds. B. Kramer, G. Bergmann and Y. Bruynsraede (Springer, Heidelberg, 1985) Vol.[**61**]{}, p.38;\
see also a review by R. Landauer in: W. van Haeringen and D. Lenstra (eds.), Analogies in Optics and Micro Electronics, pp.243-257, Kluwer Academic Publishers (1990) M. B[ü]{}ttiker, Phys. Rev. Lett., [**57**]{}, p.1761 (1986) Th. Martin and R. Landauer, Phys. Rev. [**B45**]{}, 1742 (1992) L. S. Levitov, G. B. Lesovik, JETP Letters [**58**]{} (3), p.230, (1993) S.-R. E. Yang, Solid State Commun. [**81**]{}, p. 375 (1992) D. Ivanov, L. S. Levitov, JETP Letters [**58**]{}(6), p.461 (1993)
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author:
- 'D. Shalybkov'
- 'G. Rüdiger'
title: 'Stability of density-stratified viscous Taylor-Couette flows'
---
Introduction
============
The flow pattern between concentric rotating cylinders with a stable axial density stratification was firstly studied by Thorpe (1966) who concluded that stable stratification stabilizes the flow. The further experimental and theoretical studies by Boubnov, Gledzer & Hopfinger (1995) and Caton, Janiaud & Hopfinger (2000) confirmed the stabilizing role of the density-stratification and showed that i) the critical Reynolds number depend on the buoyancy frequency (or Brunt-Väisälä frequency) of the fluid and ii) the stratification reduces the vertical extension of the Taylor vortices. The computational results of Hua, Le Gentil & Orlandi (1997) have indeed reproduced the experiment results.
The common feature of these studies is that the outer cylinder is at rest and flow is unstable after the Rayleigh condition for inviscid flow (which was extended to stratified fluids by Ooyama 1966), i.e. ( R\^4\^2) <0 \[ray\] where $\Om$ is the angular velocity of the flow.
Recently, Molemaker, McWilliams & Yavneh (2001) and Yavneh, McWilliams & Molemaker (2001) found <0 \[mri\] as the sufficient condition for (nonaxisymmetric) instability. The condition (\[mri\]) is identical with the condition for magnetorotational instability of Taylor-Couette flow (Velikhov 1959). These results have been derived by a linear stability analysis for inviscid flow. The numerical results of Yavneh, McWilliams & Molemaker (2001) demonstrate the existence of the hydrodynamic instability also for finite viscosity.
An instability of density-stratified Taylor-Couette flow beyond the Rayleigh line $\hat\mu=\hat\eta^2$ for nonaxisymmetric disturbances have also been found experimentally (Withjack & Chen 1974). With their wide gap container ($\hat\eta=0.2$) they found for $\hat \mu>0$ the stability curve [*crossing*]{} the classical Rayleigh line. The observed instability was reported as [*nonaxisymmetric*]{}. The resulting experimental stability line, however, is very steep for positive $\hat\mu$ (see their Fig. 8) and does never cross the line $\hat\mu=\hat\eta$.
For real viscous flows there are very illustrative results by Yavneh, McWilliams & Molemaker (2001). In the present paper a more comprehensive study of such flows is given. The governing equations and the restrictions of the used Boussinesq approximation are discussed in Sect. 2 while the numerical results are presented in Sect. \[res\]. Summary and final discussion are given in Sect. \[disc\].
The existence of an instability in Taylor-Couette flows with a stable radial rotation law and with a stable $z$-stratification of the density is a surprise in the light of the Solberg-Høiland criterion (see Rüdiger, Arlt & Shalybkov 2002). The necessary condition for stability reads = [g\_z R]{}, \[eg\] with ${\mbox{\boldmath $g$}}$ is the external force acceleration. Any [*conservative*]{} force is a particular solution of (\[eg\]). Without external forces this relation is thus always fulfilled. If – as it is in accretion disks – the gravity balances the pressure and the centrifugal force, then Eq. (\[eg\]) is automatically fulfilled. Note that after the Poincare theorem for rotating media with potential force and $\Om=\Om(R)$ both the density and the pressure can be written as functions of the generalized potential so that (\[eg\]) is always fulfilled. Generally, the magnetic field is [*not*]{} conservative and can never fulfill the condition (\[eg\]). This is the basic explanation for the existence of the magnetorotational instability (MRI) driven by (weak) magnetic fields.
Equation (\[eg\]) has been derived by means of the short-wave approximation $m<R/\delta R$. For waves which are large in radial directions this condition might easily be fulfilled only for $m=0$. It is, therefore, important to break the short-wave approximation in order to probe also the nonaxisymmetric modes. For Kepler flows (with uniform gravity acceleration) this has recently been done by Dubrulle et al. (2004). With Boussinesq approximation and direct numerical simulation for viscous flow they find all stratified flows with negative ${\rm d}\Om/{\rm d}R$ unstable against nonaxisymmetric disturbances. According to their results a critical Froude number (as defined by Eq. \[Fr\]) exists below which the flow is stable. Whether the Boussinesq approximation can be used for too small Froude numbers seems still to be an open question.
Equations and basic state
=========================
In cylindric coordinates ($R$, $\phi$, $z$) the equations of incompressible stratified fluid with uniform dynamic viscosity, $\mu$, are\
&& - + ,\
\
&& - + ,\
\
\[sys\] where ( )u\_R=u\_R+ +u\_z \[unabla\] and u\_R=+ ++. $\rho$ is the density, $P$ is the pressure, $g$ is the gravity, $\nu=\mu/\rho$ is the kinematic viscosity.[^1] The equation which describes the evolution of the density fluctuation moving in the general density field is + ( )=0. \[density\] We have to formulate the basic state with prescribed velocity profile ${\mbox{\boldmath $u$}}=(0,R\Om(R),0)$ and given density vertical stratification $\rho=\rho(z)$. The system (\[sys\]) takes the form\
\[sysb\] The last equation defines the angular velocity =a+, \[Om\] where $a$ and $b$ are two constants related to the boundary values of the angular velocity, $\Om_{\rm in}$, $\Om_{\rm out}$, of the inner cylinder with radius, $R_{\rm in}$, and the outer cylinder with radius, $R_{\rm out}$. It follows a=\_[in]{}, b=\_[in]{}R\_[in]{}\^2, \[ab\] with =, =. Differentiating the first equation of the system (\[sysb\]) by $z$ and the second equation by $R$, subtracting each other and using the supposed profiles of density and angular velocity one gets R\^2 =0. \[cond\] After this relation the density can depend only on the vertical coordinate $z$ in the absence of rotation ($\Om=0$) and the angular velocity can only depend on radius in the absence of the vertical density stratification (${\rm d}\rho/{\rm d}z=0$). The supposed profiles of the angular velocity, $\Om=\Om(R)$, and the density $\rho=\rho(z)$ are, therefore, not self-consistent. Thus, we must admit more general profile for the density $\rho=\rho(R,z)$ even though the initial stratification for the resting fluid is only vertical. In this case, the condition (\[cond\]) takes the form R\^2 +g =0. \[cond1\] The fluid transforms under the centrifugal force from the pure vertical stratification at the initial state to mixed (vertical and radial) stratification under the influence of the rotation strongly complicating the problem.
For real experiments the initial (without rotation) vertical stratification is small $|d{\textrm{log}}\rho/d
{\textrm{log}}z \ll 1 |$ as is the ratio of centrifugal acceleration to the vertical gravitation acceleration || 1, \[conda\] so that after (\[cond1\]) the radial stratification is also small. Let us therefore consider the case of a weak stratification =\_0+\_1(R,z), \_1 \_0, \[smrho\] where $\rho_0$ is the uniform background density and (\[cond1\]) is fulfilled in zero-order. The perturbed state of the flow is described by\
where $|P_1/P_0| \ll 1$ and $u_R$, $u_\phi$, $u_z$, $P$ and $\rho$ are the perturbations. Linearizing the system (\[sys\]) and selecting only the terms of the largest order we have\
&& + \_0 ,\
\
&& + \_0 ,\
\
\
\[sysl\] with uniform $\nu_0=\mu /\rho_0$. The first-order terms are left in the mass conservation equation due to vanishing of the zero-order terms.
Due to (\[conda\]) we can neglect $\partial P_0/\partial R$ in the first equation and $\partial\rho_1/\partial R$ in the fourth arising from radial stratification (they will be $|R^2\Om /g|$ times smaller than terms arising from the vertical stratifications) and the system takes exactly the Boussinesq form\
&& + \_0 ,\
\
&& + \_0 ,\
\
\
\[sysbo\] where $N$ is the vertical buoyancy frequency with N\^2=-. Suppose that the linear vertical density stratification $\partial \rho_1/\partial z={\textrm{const}}$ and thus $N^2$ is a constant, too. Then the coefficients of the system (\[sysbo\]) only depend on the radial coordinate and we can use a normal mode expansion of the solution ${F}={F}(R){\textrm{exp}}({\textrm{i}}(m\phi+kz+\omega t))$ where $F$ represents any of the disturbed quantities.
Let $D=R_{\rm out} - R_{\rm in}$ be the gap between the cylinders. We use $R_0=(R_{\rm in}D)^{1/2}$ as the unit of length, the velocity $\Om_{\rm in} R_0$ as the unit of the perturbed velocity, $\Om_{\rm in}$ as the unit of $\omega$, $N$ and $\Om$. Using the same symbols for normalized quantities and redefining $\rho$ as the dimensionless density $\rho g/\rho_0R_0\Om_{\rm in}^2$ we finally find\
&& -(+m) u\_R +2u\_-=0,\
\
&& -(+m) u\_-P -(R\^2 )=0,\
\
&& -(+m) u\_z -kP-=0,\
\[sysf\] and + +u\_+ku\_z=0 \[sysf1\] with the Reynolds number =.
The standard no-slip boundary conditions used at the inner and outer cylinder, i.e. u\_R=u\_=u\_z=0, \[bound\] complete the classical eigenvalue problem. The same numerical method as in our previous papers about the Taylor-Couette problem (see e.g. Rüdiger & Shalybkov 2002) is used. Here we use a small negative imaginary part of $\omega$ to avoid problems with the corotation point $\omega=m\Om$ for $m>0$. Thus, the calculated critical Reynolds numbers are not for the marginally stable state but for slightly unstable state. To be sure that the calculated unstable state can be realized in experiments we checked the existence of the transition from stable to unstable state for several arbitrary points.
The code has been tested by computing for $m=1$ the critical Reynolds number for the run 2 of Withjack & Chen (1974, their Tab. 1) with the experimental value 196.2 (with our normalizations) and our computed result 200.6 which we accepted to be in sufficiently good accordance.
Results {#res}
=======
The imaginary parts of $\omega$, $\Im(\omega)$, decrease with increasing Reynolds number. The Reynolds numbers above which the imaginary part of $\omega$ is smaller than some fixed value depend on the vertical wave number. They have a minimum at a certain wave number for fixed other parameters. This minimum value is called the critical Reynolds number.
In Fig. \[comp\] we compare the calculated marginal stability line (i.e. $\Im(\omega)=0$) for axisymmetric disturbances with experimental values by Boubnov, Gledzer & Hopfinger (1995). The agreement is rather good except the small values of the Froude number $${\rm Fr}=\frac{\Om_{\rm in}}{N}.
\label{Fr}$$ This disagreement may indicate the violation of the Boussinesq approximation. For Kepler disks we find ${\rm Fr}\simeq 0.5$ (Dubrulle et al. 2004). The unstratified fluids possess infinite Froude number.
The dependence of the critical Reynolds numbers on $\hat\mu$ is given by Fig. \[remu\] for a narrow gap and in Fig. \[remu2\] for a wide gap. The exact line of marginal stability is plotted only for $m=0$. The axisymmetric disturbances are unstable only for $\hat\mu < \hat\eta^2$ in accordance to the Rayleigh condition (\[ray\]). For $m>0$ the slightly unstable lines with $\Im(\omega)=-10^{-3}$ are given.
The nonaxisymmetric disturbances are unstable also beyond the Rayleigh line (plotted as solid in the figures). The higher the $m$, however, the more the corresponding instability line approaches the Rayleigh line. Note, therefore, that the ‘stratorotational instability’ (SRI, Dubrulle et al. 2004) only produces low-$m$ modes. This is an indication that indeed within the short-wave approximation (high-$m$) it does not exist (see Rüdiger, Arlt & Shalybkov 2002).
For nonstratified Taylor-Couette flows the nonaxisymmetric modes are only the most unstable disturbances for counter-rotating cylinders (e.g. Drazin & Reid 1981). The nonaxisymmetric instability of the stratified Taylor-Couette flow beyond the Rayleigh line ($\hat\mu > \hat\eta^2$) leads to the existence of some critical value, $\hat\mu_c$, beyond which the nonaxisymmetric disturbances are the most unstable. Our results show that $\hat \mu_c \sim 0.27$ and almost independent of the Froude number for $\hat\eta=0.78$ (small gap) and $\hat\mu_c < 0$ for $\hat\eta=0.3$ (wide gap). It is possible that for some $\hat\eta$ the nonaxisymmetric disturbances are the most unstable for all values of $\hat\mu$ corresponding to unstable flows.
There is another observation with the Figs. \[remu\] and \[remu2\]. Approaching the line $\hat\mu = \hat\eta$ the instability lines become more and more steep. We did not find any solution for $\hat\mu > \hat\eta$. If this is true one has to apply rather steep rotational profiles (not so steep as for nonstratified fluids but also not so weak as for the MRI) to find the modes of the linear SRI. This result seems to be of relevant for the discussion of the stability or instability of Kepler disks.
For the narrow gap ($\hat\eta=0.78$) the critical Reynolds numbers of the nonaxisymmetric modes only slightly depend on $m$. The same is true for the critical vertical wave number and the pattern speed $\Re({\omega})/m$ (Fig. \[kom\]). The vertical wave number only weakly depends on $\hat\mu$ and the values $\Re{(\omega)}/m$ linearly run with $\hat\mu$. The situation is changed, however, for the wide gap ($\hat\eta=0.3$). All parameters now strongly depend on both $m$ and $\hat\mu$ (Fig. \[kom03\]). The trend for the vertical wave numbers is opposite for $m=3$ to those for $m=1$ and $m=2$.
The vertical wave numbers for both containers are of order 10 for $m=1$. With our normalization the vertical extent of the Taylor vortices is given by = . With the mentioned value of $k$ it is order of 0.1 for both the small-gap case and the wide-gap case. The cell becomes thus rather flat. For nonstratified TC-flows one finds $\delta z \simeq R_{\rm out} - R_{\rm in}$ while the cells under the influence of an axial magnetic field become more and more prolate. The stratification generally reduces the height of the Taylor vortices.
Unlike to the nonstratified Taylor-Couette flow the $\Re(\omega)$ is not zero for stratified Taylor-Couette even for axisymmetric disturbances ($m=0$). The onset of instability is thus oscillatory (‘overstability’). The question is whether a critical Froude number exists corresponding to the transition from stationary solutions to oscillating solutions? The answer is No. One cannot fulfill Eq. (\[sysf\]) for marginal stability ($\Im(\omega)=0$) without a finite real part of $\omega$ for $N^2 \ne 0$. The axially stratified Taylor-Couette flow bifurcates from the purely azimuthal flow through a direct Hopf bifurcation to a wavy regime. Depending on the value of $\hat \mu$ and $\hat \eta$ this new regime can be either oscillating and axisymmetric or nonaxisymmetric and azimuthally drifting (see Figs. \[remu\], \[remu2\]). For both our containers the pattern speeds are negative for positive $m$. The drift of the spirals is thus always in the direction of the cylinder rotation.
Experiments have really demonstrated the oscillating onset of the axisymmetric instability (e.g. Caton, Janiaud & Hopfinger 2000). It would be interesting to design experiments with either rotating outer cylinder or wider gap to probe the bifurcation from the overstable oscillating axisymmetric flow pattern to the spiral nonaxisymmetric flow pattern.
As an example for the container with the narrow gap in Fig. \[eign\] the velocity eigenfunctions are presented for $m=1$ and for $\hat\mu=0.7$ exceeding the value of $\hat\eta^2$. The functions are smooth enough and do not suggest that the instability must be explained as a boundary effect.
The Reynolds number, the vertical wave number and the real part of $\omega$ for the transition from positive to negative imaginary part of $\omega$ are given in Fig. \[trans\], i.e. for the transition from negative to positive growth rates. We find a continuous transition across the marginal stability line. It should thus be possible to realize the transition from stable to unstable flows in experiments. The vertical wave number and real part of $\omega$ are hardly influenced by the transition but, not surprisingly, the Reynolds numbers has a remarkably clear trend.
Discussion {#disc}
==========
It is shown that the Boussinesq approximation yields nonaxisymmetric disturbances with low $m$ of the stratified Taylor-Couette flow as unstable even beyond the Rayleigh line $\hat\mu > \hat\eta^2$. Our results, however, also show that the critical Reynolds numbers are extremely increasing approaching the line $\hat\mu=\hat\eta$ so that as the condition for instability now the relation <\[eta\] seems to appear rather than $\hat\mu < 1$ according to Yavneh, McWilliams & Molemaker (2001) and Molemaker, McWilliams & Yavneh (2001). It is challenging to interprete the line $\hat\mu=\hat\eta$ with the (galactic) rotation profile $u_\phi=$const in the same sense as to interprete the line $\hat\mu=\hat\eta^2$ with the rotation law for uniform specific angular momentum $R^2\Om=$ const. Below we shall consider the line $\hat\mu=\hat\eta^{1.5}$ as concerning the Kepler flow.[^2] Dubrulle et al. (2004) for their model of rotating plane Couette flow have found unstable solutions also beyond the line $\hat\mu=\hat\eta$ but also in these computations the rotation profile must be steeper than $R^{-2/3}$ (their Fig. 7).
The SRI also leads to the situation that nonaxisymmetric disturbances can be the most unstable modes not only for counterrotating cylinders but also for corotating cylinders. The characteristic values of $\hat\mu$ where the nonaxisymmetric disturbances are the most unstable ones strongly depend on the gap width. For $\hat\eta=0.3$ all positive $\hat\mu$ are concerned (see Fig. \[remu2\]). It cannot be excluded that the nonaxisymmetric disturbances are the most unstable ones for all $\hat\mu$ for $\hat\eta$ smaller than some critical value.
These results were obtained with the Boussinesq approximation so that two restrictions remain. The vertical density stratification should be weak enough and the rotation should be so slow that the centrifugal acceleration can be neglected in total. If one of these conditions is violated the Boussinesq approximation cannot be used and the situation becomes much more complicated. The disagreement between the calculated and the observed critical Reynolds numbers for small Fr (see Fig. \[comp\]) may already indicate the violation of the Boussinesq approximation for strong stratifications.
We have shown that for not too small negative ${\rm d}\Om/{\rm d}R$ Taylor-Couette flows with vertical density stratification become unstable against nonaxisymmetric disturbance with $m=1$ even if they are stable without density stratification. Kepler flows seem to be concerned by this phenomenon. In the Figs. \[remu\], \[remu2\] the dotted lines represent the limit $\hat\mu=\hat\eta^{1.5}$ which might mimic the radial shear in Kepler disks. In both Figures we find a critical Reynolds number of only about 500 for the lowest ($m=1$) mode. This is indeed a rather small number whose meaning, however, should not be overestimated. Approaching the line $\hat\mu=\hat\eta$ also these values become more and more large. Even more important is the finding that in magnetohydrodynamic Taylor-Couette experiments the MRI only needs [*magnetic*]{} Reynolds numbers of O(10). This leeds to hydrodynamic Reynolds numbers exceeding O($10{^6}$) only for experiments with liquid metals in terrestrial laboratories. For hot plasma with magnetic Prandtl numbers of order 10 (Noguchi & Pariev 2003) the necessary hydrodynamic Reynolds number is also only O(1) or even smaller!
The existence of the SRI might be important for astrophysical applications. As suggested first by Richard & Zahn (1999) one should not forget (in particular for protoplanetary disks) to probe hydrodynamical instabilities as the source for the necessary turbulence in accretion disks. In partial confirmation of results of Dubrulle et al. (2004) we have here demonstrated that for the rotation laws of Taylor-Couette flows under the presence of vertical density gradients indeed linear hydrodynamic instabilities for low $m$ exist. The properties of these modes are described above. Whether they are important for the accretion disk physics is still an open question. Note that the density stratification in accretion disks completely vanishes in the equatorial region and that the unstable modes discussed above would more or less lead to a nonaxisymmetric structure of the disk rather than to turbulence. The next step in this direction must include the numerical simulation for well-designed but simplified [*global*]{} accretion disk models.
D.S. thanks for financial support from the AIP visiting program. His work was also partly supported by RFBR (grant 03-02-17522). GR thanks J.-P. Zahn for detailed discussions about the subject of this paper.
Boubnov, B.M., Gledzer, E.B., & Hopfinger, E.J. 1995, J. Fluid Mech., 292, 333
Caton, F., Janiaud, B., & Hopfinger, E.J. 2000, J. Fluid Mech., 419, 93
Dubrulle, B., Marie, L., Normand, Ch. et al. 2004, A&A (subm.)
Drazin, P.G., & Raid, W.H. 1981, Hydrodynamic stability, Cambridge Univ. Press
Hua, B.L., Le Gentil, S., & Orlandi, P. 1997, Phys. Fluids, 9, 365
Molemaker, M.J., McWilliams, J.C., & Yavneh, I. 2001, Phys. Rev. Lett., 86, 5270
Noguchi, K., & Pariev, V. 2003, astro-ph/0309340
Ooyama, K. 1966, J. Atmos. Sci., 23, 43
Richard, D., & Zahn, J.-P. 1999, A&A, 347, 734
Rüdiger, G., Arlt, R., & Shalybkov, D. 2002, A&A, 391, 781
Rüdiger, G., & Shalybkov, D. 2002, Phys. Rev. E, 66, 016307
Thorpe, S.A. 1966, Notes on 1966 Summer Geophysical Fluid Dynamics (Woods Hole Oceanographic Institute, Woods Hole, MA 1966) p. 80
Velikhov, E.P. 1959, Sov. Phys. JETP, 9, 995
Withjack, E.M., & Chen, C.F. 1974, J. Fluid Mech., 66, 725
Yavneh, I., McWilliams, J.C., & Molemaker, M.J. 2001, J. Fluid Mech., 448, 1
[^1]: the density diffusion term in the mass conservation equation is neglected (see e.g. Caton, Janiaud & Hopfinger 2000)
[^2]: note, however, the general difference of the Kepler rotation law $\Om\propto R^{-1.5}$ and the Taylor-Couette rotation law (\[Om\])
|
---
abstract: 'Aim of the QUAX (QUaerere AXion) proposal is to exploit the interaction of cosmological axions with the spin of electrons in a magnetized sample. Their effect is equivalent to the application of an oscillating rf field with frequency and amplitude which are fixed by axion mass and coupling constant, respectively. The rf receiver module of the QUAX detector consists of magnetized samples with the Larmor resonance frequency tuned to the axion mass by a polarizing static magnetic field. The interaction of electrons with the axion-equivalent rf field produces oscillations in the total magnetization of the samples. To amplify such a tiny field, a pump field at the same frequency is applied in a direction orthogonal to the polarizing field. The induced oscillatory magnetization along the polarizing field is measured by a SQUID amplifier operated at its quantum noise level.'
address: |
$^1$ Laboratori Nazionali di Legnaro, Viale dell’Università 2, 35020 Legnaro (Italy)\
$^2$ INFN and Dipartimento di Fisica e Astronomia, Via Marzolo 8, 35131 Padova (Italy)\
$^3$ School of Physics and Astronomy, University of Birmingham, West Midlands B15 2TT (UK)
author:
- 'G Ruoso$^1$, A Lombardi$^1$, A Ortolan$^1$, R Pengo$^1$, C Braggio$^2$, G Carugno$^2$, C S Gallo$^2$, C C Speake$^3$'
title: 'The QUAX proposal: a search of galactic axion with magnetic materials'
---
Introduction
============
An outstanding result of modern cosmology is that a significant fraction of the universe is made of dark matter. However, the nature of such component is still unknown, apart its gravitational interaction with ordinary baryonic matter. A favored candidate for dark matter is the axion: a new particle introduced by Peccei and Quinn to solve the strong CP problem [@PQ], i.e. the absence of CP violation in the strong interaction sector of the Standard Model. Axions have properties similar to a $\pi_0$ particle and have mass $m_a$ inversely proportional to the Peccei-Quinn symmetry breaking scale $f_a$. For certain ranges of $f_a$ and $m_a$ (typically with masses ranging from $\mu$eV to meV), large quantities of axions may had been produced in the early Universe that could be able to account for a large portion of the cold dark matter forming today galactic halos. Axions have extremely small coupling to normal matter and radiation, but they can be converted into detectable photons by means of the inverse Primakoff effect as shown by Sikivie [@Sikivie]. The idea of Sikivie has been exploited by the several experiments [@MelissinosAX; @Florida], of which the most recent is ADMX [@ADMX]. The latter experiment is still running, and for the moment it has been capable of exploring the axion model for masses of a few $\mu$eV.
The QUAX (QUaerere AXion) proposal explores in details the ideas of Krauss [@Krauss1985], Barbieri et al [@Barbieri1989], and Kolokolov and Vorobyev [@Kolo1991]. These authors proposed to study the interaction of the cosmological axion with the spin of fermions (electrons or nucleons). In fact, due to the motion of the Solar System through the galactic halo, the Earth is effectively moving through the cold dark matter cloud surrounding the Galaxy and an observer on Earth will see such axions as a wind. In particular, the effect of the axion wind on a magnetized material can be described as an effective oscillating rf field with frequency determined by $m_a$, and amplitude related to $f_a$. Thus, a possible detector for the axion wind can be a magnetized sample with Larmor resonance frequency tuned to the axion mass by means of an external polarizing static field (e.g. 0.6 T for 17 GHz, corresponding to a 70 $\mu$eV axion mass). The interaction with the axion effective field will drive the total magnetization of the sample, and so producing oscillations in the magnetization that, in principle, can be detected. In order to optimize the detection, a pump rf field is applied in a direction orthogonal to the polarizing field. Due to the non linearity of Bloch equations in magnetized materials, a pump field applied to a suitable magnetized sample, (e.g. a small Yttrium Iron Garnet (YIG) sphere) can amplify the equivalent rf field generated by the axion wind. The induced change in the magnetic flux along the polarizing field is then fed to a SQUID magnetometer through a superconducting transformer. It is worth noticing that the magnetized sample must be cooled to ultra-cryogenic temperature to avoid fluctuations of the magnetization due to the thermal bath.
The QUAX collaboration is exploring the experimental feasibility of the proposed detection scheme, starting at a precise value of the axion mass of about 70 $\mu$eV. The QUAX R&D activities are conducted at the Laboratori Nazionali di Legnaro (LNL) of the Istituto Nazionale di Fisica Nucleare (INFN), and funded in the framework of the research call [*What Next*]{} of INFN. We are building a prototype detector to study its sensitivity, intrinsic and technical noises, and demonstrate the possibility to reach the signal level corresponding to the axion coupling constant as predicted by the theoretical model.
Theoretical introduction
========================
Let’s start from the well known Lagrangian which describes the interaction of 1/2 spin particle with the axion field $a(x)$ $$\label{eq1}
L=\bar{\psi}(x)(i\hbar
\gamma^\mu\partial_\mu- mc)\psi(x) - ia(x) \bar{\psi}(x)g_p\gamma_5\psi(x)$$ where $\psi(x)$ and $a(x)$ are the spinor field of the fermion with mass $m$ and the axion field with dimension of momentum, respectively; here $\gamma^\mu$ are the 4 Dirac matrices, $\gamma^5=i \gamma^0\gamma^1\gamma^2\gamma^3$, and $a(x)$ is coupled to the matter by the pseudo-scalar coupling constant $g_p$. By obtaining the Euler-Lagrange equation and in the non relativistic limit, the time evolution of a 1/2 spin particles can be described by the usual Schroedinger equation $$\label{eqsh}
i\hbar\frac{\partial \varphi}{c\partial t}=\left[-\frac{\hbar^2}{2m}\nabla^2+g_sca- \frac{g_p\hbar }{2m}\boldsymbol{\sigma}\cdot \boldsymbol{\nabla}a \right]\varphi\ ,$$ where the most relevant interaction term $$-\frac{g_p\hbar }{2m}\boldsymbol{\sigma}\cdot \boldsymbol{\nabla}a \equiv -2 \frac{e \hbar }{2m}\boldsymbol{\sigma}\cdot\left(\frac{g_p}{2e} \right)\boldsymbol{\nabla}a$$ has the form of the interaction between the spin magnetic moment of a fermion ($-2 \frac{e \hbar }{2m}\boldsymbol{\sigma} = -\mu_B \boldsymbol{\sigma}$ with $\mu_B$ the Bohr magneton) and an effective magnetic field $B_a \equiv \frac{g_p}{2e} \boldsymbol{\nabla}a $ . It is worth noticing that the axion field couples to fermions as a pseudo-scalar field; in fact, $\boldsymbol{\nabla}a$ is a pseudo-vector (like the magnetic field) and $\boldsymbol{\sigma}\cdot \boldsymbol{\nabla} a$ is a true scalar interaction term.
Axions represent the best example of non-thermal dark matter candidate [@Turner]. The expected dark matter density is $\rho\simeq 300$ MeV/cm$^3$, and we will suppose that axion is the dominant component. As the axion mass should be in the range $ 10^{-6}{\rm eV} < m_a < 10^{-2} {\rm eV}$, we have $n_a\sim 3\times10^{12}\ (10^{-4}\ {\rm eV}/m_a)$ axions per cubic centimeter which is a remarkably high density number. The axion velocities $v$ are distributed according to a Maxwellian distribution (the non relativistic limit of the Bose-Einstein distribution), with a velocity dispersion $\sigma_v\approx 270$ km/sec. Due to Galaxy rotation and Earth motions in the Solar system, the rest frame of an Earth based laboratory is moving through the local axion cloud with a time varying velocity $\boldsymbol{v}_E= \boldsymbol{ v}_S+ \boldsymbol{ v}_O+ \boldsymbol{ v}_R$, where $\boldsymbol{ v}_S$ represents the Sun velocity in the galactic rest frame (magnitude 230 km/sec), $\boldsymbol{ v}_O$ is the Earth’s orbital velocity around the Sun (magnitude 29.8 km/sec), and $ \boldsymbol{ v}_R$ the Earth’s rotational velocity (magnitude 0.46 km/sec). The observed axion velocity is then $\boldsymbol{ v}_a = \boldsymbol{ v}- \boldsymbol{ v}_E$ . The effect of this motion is to broaden the Maxwell distribution, as well as to modulate it with a periodicity of one sidereal day and one sidereal year.
Axions kinetic energy is expected to be distributed with a mean relative to the rest mass $7 \times 10^{-7}$ and a dispersion about the mean $5.2 \times 10^{-7}$ [@turner1]. The inverse of this last number represent the natural figure of merit of the axion linewidth, $Q_a \simeq 1.9\times 10^6$. The mean De Broglie wavelength of an axion is $\lambda_d\simeq {h}/ ({m_a v_a}) \simeq 13.8 \left({10^{-4} {\rm eV}}/{m_a} \right)$ m, therefore $\lambda_d$ is much greater than the typical length of experimental apparatus, in our case the magnetized samples. Such theoretical and experimental aspects allow us to treat $a(x)$ as a classical field that interacts coherently with fermions with a mean value $a(x)=a_0 {\rm exp}[ {i ( p^0 c t-\boldsymbol{p}_E\cdot\boldsymbol{x} )/\hbar}]$ where $\boldsymbol{ p}_E = m_a \boldsymbol{ v}_E$, $p^0=\sqrt{m_a^2 c^4 + |\boldsymbol{ p}_E |^2 c^2} \approx m_a c^2 + |\boldsymbol{p}_E|^2 /(2 m_a)$ and $a_0$ is the field amplitude. The amplitude $a_0$ can be easily computed by equating the moment transported by this field per unit of volume (i.e. the associated energy momentum tensor $T^{0i}=a_0^2 p^0 p_E^i$) to the number of axion per unit volume times the average momentum (i.e. $n_a<p^i> =n_a p_E^i$), and it reads $a_0=\sqrt{({n_a\hbar^3 c})/({m_a c}})$. To calculate the effective magnetic field associated with the mean axion field we multiply the gradient $\boldsymbol{\nabla}a(x) =i (\boldsymbol{p}_E / \hbar)\ a(x)$ by $g_p/(2 e)$ and take the real part $$\mathbf{B}_a = \frac{g_p}{2 m \gamma} \left(\frac{n_a\hbar }{m_a c}\right)^{1/2}\, \boldsymbol{ p}_E \, \sin\left(\frac{p^0 c t-\boldsymbol{p}_E\cdot\boldsymbol{x}}{\hbar}\right)$$ where $\gamma\equiv g_L e/ (2 m)$ is the gyromagnetic ratio and $g_L=2$ is the Landé g-factor for elementary fermions. In the framework of DFSZ model of axions[@DFSZ; @DFSZ1], the value of the coupling constant $g_p$ with electrons can be calculated and expressed in terms of the $\pi^0$ mass and decay constant $m_{\pi^0}=135 \, {\rm MeV}$ and $f_{\pi^0} = 93 \, {\rm MeV}$ as $g_p \simeq ({m \, m_a})/({m_{\pi^0} f_{\pi^0}}) = 2.8 \times 10^{-11} ({m_a}/({1\ \rm{eV}})) $ where we have used $ m =0.5 \,{\rm MeV}$ for the electron mass.
Putting the magnetized samples in $\boldsymbol{ x} =\boldsymbol{ 0}$, we have that the equivalent oscillating rf field along the $\boldsymbol{ p}_E$ direction has a mean amplitude and central frequency $$B_a = 9.2 \cdot 10^{-23} \left(\frac{m_a}{10^{-4} {\rm eV}}\right) \,\,\,\, {\rm T}\ ,\,\,\,\,\, \frac{\omega_a}{2 \pi} = 24 \left(\frac{m_a}{10^{-4} {\rm eV}}\right) \,\,\,\, {\rm GHz} \,
\label{axionfield}$$ with a relative linewidth $\Delta \omega_a/\omega_a \simeq 5.2 \times 10^{-7}$.
Experimental scheme
===================
To detect an extremely small rf field we will exploit the Electron Spin Resonance (ESR) in a magnetized media. In particular, we have to recourse to the continuous wave ESR in a magnetic material using a coil oriented along the polarizing field direction. This method is known as LOngitudinal Detection scheme (LOD), and during the last decades it was investigated systematically, both theoretically and experimentally [@pescia].
A magnetized spherical sample of volume $V_s$ and magnetization $M_0$ is placed in the bore of a solenoid, which generates a static magnetic field $B_0$ (polarizing field). The value $B_0$ determines the Larmor frequency of the electrons, and so the axion mass under scrutiny, through the relation $$B_0 = \frac{\omega_L} {\gamma} = \frac{ m_a c^2}{\gamma \hbar } = 0.85 \left(\frac{m_a}{10^{-4} {\rm eV}}\right) \,\,\,\, {\rm T}.$$ Then, an additional radio frequency field $B_{1}$ is applied to the sample in a direction orthogonal to $\mathbf{B}_0$, that we suppose along the $z$ axis. As the radiofrequency field is in the $x-y$ plane, it can drive the magnetic resonance causing the electron spin to flip between the two Zeeman sublevels. We don’t need the quantum formalism as the rf field amplitude is sufficiently large to drive a macroscopic number of spins. The evolution of magnetization $\mathbf{ M}$ of the sample is described by the Bloch equations with dissipations and radiation damping [@bloom] $$\begin{aligned}
\label{bloch}
\frac{dM_x}{dt}&=&\gamma (\mathbf{M}\times\mathbf{ B})_x - \frac{M_x}{\tau_2} -\frac{M_x M_z}{M_0 \tau_r} \nonumber \\
\frac{dM_y}{dt}&=&\gamma (\mathbf{M}\times\mathbf{ B})_y - \frac{M_y}{\tau_2} - \frac{M_y M_z}{M_0\tau_r} \nonumber \\
\frac{dM_z}{dt}&=&\gamma (\mathbf{M}\times\mathbf{ B})_z - \frac{M_0- M_z}{\tau_1}- \frac{M_x^2+M_y^2}{M_0 \tau_r}\end{aligned}$$
where $M_0$ is the static magnetization directed along the $z$ axis, $\tau_r$ is the radiation damping time, $\tau_1$ and $\tau_2$ are the longitudinal (or spin-lattice) and transverse (or spin-spin) relaxation time, respectively. The non linear terms proportional to $\tau_r^{-1}$ were introduced by Bloom in 1957 to account for electromotive force induced in the rf coils of the driving circuit by magnetization changes without taking into account the dynamics of the rf coils. In this case $ \tau_r^{-1}= 2 \pi \zeta \gamma \mu_0 M_0 Q$, where the filling factor $\zeta$ and the quality factor $Q$ account for geometrical coupling of coil and magnetized sample and circuit dissipations, respectively.
In the higher frequency regime, which is our case, $\omega_L> \widehat{\omega}_L $, radiation damping is dominated by magnetic dipole emission in free space from the magnetized sample, and the relaxation time reduces to $ \tau_r^{-1}=1/(4 \pi) \frac{\omega_L^3}{c^3} \gamma \mu_0 M_0 V_s $. The steady state solutions of Eq.(\[bloch\]) in the presence of radiation damping and for various approximations are given in ref. [@augustine]. A new relaxation time is introduced $\tau^\star_2=1/(\tau_2^{-1} + \tau_r^{-1})$.
Let us suppose that the rf field is linearly polarized and a linear superposition of the equivalent field $\mathbf{ B}_a$, due to the axions, and a pump field $\mathbf{ B}_p$, with amplitude $B_p>>B_a$. We assume also that the axion field direction is parallel to the direction of $\mathbf{B}_a$. In addition, the axion and pump frequencies $\omega_a$ and $\omega_p$ are within the linewidth of the Larmor resonance of the magnetized sample, i.e. ($\omega_p-\omega_L)<1/\tau_2^\star$ and ($\omega_a-\omega_L)<1/\tau_2^\star$. In this way, the Larmor frequency of electrons is driven by the rf field $$\label{fields}
B_1 = B_p \cos \omega_p t + B_a \cos \omega_a t .$$
Due to the non-linearity of the Bloch’s equations, as a result of this [*amplitude modulated*]{} rf field, a variable magnetization is produced along the $z$-axis. In fact, the stationary solution for $M_z$ in Eq.(\[bloch\]) can be used even in quasi-stationary regimes [@augustine], provided that $|\omega_p-\omega_a|<<\omega_L$. At the beat note between the two field components we have $$\Delta m_z\equiv M_z-M_0 \simeq \frac{M_0}{4}\frac{\tau_2^\star}{ \tau_2} \ \gamma^2 \tau_1\tau_2^\star B_p B_a \cos \omega_D t\\$$ where we have defined the detection frequency $\omega_D=\omega_p-\omega_a$ and assumed $\omega_D<\min\{1/\tau_1,1/\tau_2^\star \} $.
Such low frequency oscillations of the magnetization can be detected with suitable devices, e.g. a SQUID amplifier. In this detection scheme we can choose the best detection frequency for increasing the signal to noise ratio, and we can use amplifiers that could not work at very high frequencies. Moreover, the effective rf field $B_a$ is down converted to a low frequency field with amplitude $B_D=\mu_0 \Delta m_z \equiv G_m B_a$, where $G_m$ is the transduction gain, and it is given by $$G_m= \mu_0 \frac{M_0}{4}\frac{\tau_2^\star}{\tau_2} \ \gamma^2 \tau_1\tau_2^\star B_p$$ If we assume that the relaxation times satisfy $\tau_1 \sim \tau_2$ and $\tau_r<\tau_2$, the transduction gain will depend only on $\tau_r$: $G_m \simeq \mu_0 M_0 \tau_r^2 \gamma^2 B_p \approx 1/(8 \pi^2) (\lambda_L^3/V_s) \gamma \tau_r B_p $, where $\lambda_L\equiv 2 \pi c /\omega_L $ is the wavelength corresponding to the Larmor frequency. Thus to obtain a gain $G_m>1$ in free space, the sample volume must satisfy the inequality $V_s< 1/(8 \pi^2)\lambda_L^3 \tau_r \gamma B_p$. On the other hand, the pump field amplitude must be far from saturation $\gamma^2 B_p^2 \tau_r \tau_1 <<1$, which implies $\gamma \tau_r B_p <<1$. This is the reason why we can’t get $G_m>1$ in free space with realistic sample volumes and pumping fields.
Inside an rf cavity, this problem could be solved but the thermal noise in the hybridized system formed by cavity and magnetized sample is much greater than the axion equivalent field. The solution is to detect axions by placing the sample in a waveguide with a cutoff frequency $\omega_c$ above the Larmor frequency of the sample. For instance, we can place the sample at a distance $\ell$ from the aperture of a rectangular waveguide of cross section $ab$. The lowest cut-off frequency for an open waveguide $(a>b)$ is given by (the waveguide behaves as a high pass filter): $\omega_c={1}/({4 \pi a \sqrt{\epsilon \mu}})$.
Due to boundary conditions, if the Larmor frequency is lower than $\omega_c$, the magnetic resonant mode cannot propagate inside the waveguide, but only evanescent waves can exist. We expect then a reduction of the radiation damping mechanism for the selected magnetic resonance. The advantage of this experimental configuration is twofold: i) relaxation times are no longer dominated by radiation damping but by the intrinsic relaxation times of the sample $\tau_1$ and $\tau_2$; and ii) the noise associated to thermal photons is greatly reduced. The main difference with the case of a microwave cavity is that we exploit the non-radiative field to pump energy in the sample by means of a rf coil placed around the sample itself.
Noise considerations
====================
We are interested in fluctuations of the magnetization of the sample. There are at least four sources of noise to consider: a) pump field noise; b) noise at the Axion frequency due to thermal photons that can be down converted after interaction with the pump field; c) magnetization noise generated by sample that can either be down converted by the pump or can act as a noise source in the low frequency readout; d) noise associated with the SQUID pickup at the readout frequency. A preliminary evaluation of these noise sources shows that the most challenging issue is related with the pump field noise. All the noises are in the process of being carefully evaluated.
In addition to these noise sources, we have to take into account any relaxation process that can be active at the readout frequency and can induce fluctuations in the z component of magnetization. The presence of such processes and their magnitude has to be directly measured on the magnetized sample.
References {#references .unnumbered}
==========
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|
---
abstract: 'The aim of this paper is to study the basic properties of the Thompson metric $d_T$ in the general case of a linear spaces $X$ ordered by a cone $K$. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T,$ and some results concerning the topology of $d_T,$ including a brief study of the $d_T$-convergence of monotone sequences. It is shown most results are true without any assumption of Archimedean-type property for $K$. One considers various completeness properties and one studies the relations between them. Since $d_T$ is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. Thompson metric $d_T$ and order-unit (semi)norms $|\cdot|_u$ are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although $d_T$ and $|\cdot|_u$ are only topological (and not metrical) equivalent on $K_u$, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.'
address:
- 'S. Cobzaş, Babes-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania'
- 'M.-D. Rus, Technical University of Cluj-Napoca, Cluj-Napoca, Romania'
author:
- 'S. Cobzaş and M.-D. Rus'
title: Normal cones and Thompson metric
---
Introduction
============
In his study on the foundation of geometry, Hilbert [@hilbert1895] introduced a metric in the Euclidean space, known now as the Hilbert projective metric. Birkhoff [@birkhoff57] realized that fixed point techniques for nonexpansive mappings with respect to the Hilbert projective metric can be applied to prove the Perron-Frobenius theorem on the existence of eigenvalues and eigenvectors of non-negative square matrices and of solutions to some integral equations with positive kernel. The result on the Perron-Frobenius theorem was also found independently by Samelson [@samelson57]. Birkhoff’s proof relied on some results from differential projective geometry, but Bushell [@bushell73a; @bushell73b] gave new and more accessible proofs to these results by using the Hilbert metric defined on cones, revitalizing the interest for this topic (for a recent account on Birkhoff’s definition of the Hilbert metric see the paper [@lem-nuss13], and for Perron-Frobenius theory, the book [@Lem-Nuss12]). A related partial metric on cones in Banach spaces was devised by Thompson [@thomp63], who proved the completeness of this metric (under the hypothesis of the normality of the cone), as well as some fixed point theorems for contractions with respect to it. It turned out that both these metrics are very useful in a variety of problems in various domains of mathematics and in applications to economy and other fields. Among these applications we mention those to fixed points for mixed monotone operators and other classes of operators on ordered vector spaces, see [@chen93; @chen99; @chen01a; @chen02; @RusM10; @rusm11]. Nussbaum alone, or in collaboration with other mathematicians, studied the limit sets of iterates of nonexpansive mappings with respect to Hilbert or Thompson metrics, the analog of Denjoy-Wolff theorem for iterates of holomorphic mappings, see [@lins07; @lins-nuss06; @lins-nuss08; @Nuss88; @Nuss89; @nuss07]. These metrics have also interesting applications to operator theory—to means for positive operators, [@iz-nakamura09; @nakamura09], and to isometries in spaces of operators on Hilbert space and in $C^*$-algebras, see [@hat-molnar14; @molnar09], and the papers quoted therein.
Good presentations of Hilbert and Thompson metrics are given in the monographs [@Hy-Is-Ras; @Nuss88; @Nuss89], and in the papers [@akian-nuss13; @lem-nuss13; @nuss-walsh04]. A more general approach—Hilbert and Thompson metrics on convex sets— is proposed in the papers [@bauer-bear69] and [@bear-weis67].
The aim of this paper, essentially based on the Ph. D. thesis [@RusM10], is to study the basic properties of the Thompson metric $d_T$ in the general case of a vector space $X$ ordered by a cone $K$. Since $d_T$ is defined in the context of a generic ordered vector space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in the present paper and, to the best of our knowledge, this has not been done yet.
For the convenience of the reader, we survey in Section \[S.ord-vs\] some notions and notations which will be used throughout and list, without proofs, the most important results that are assumed to be known. Since there is no a standard terminology in the theory of ordered vector spaces, the main purpose of this preliminary section is to provide a central point of reference for a unitary treatment of all of the topics in the rest of the paper. As possible we have given exact references to textbooks were these results can be found, [@Alipr; @BreckW; @Deimling; @Guo; @Jameson; @Peressini; @Schaef].
Section \[S.Thompson-metric\] is devoted to the definition and basic properties of the Thompson metric. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T.$ We close this section with some results concerning the topology of $d_T,$ including a brief study of the $d_T$-convergence of monotone sequences. Note that most of these results are true without the assumption of an Archimedean–type property for $K$.
We show that the Thompson metric $d_T$ and order–unit (semi)norms $|\cdot|_u$ are strongly related and share important properties (e.g., they are topologically equivalent), as both are defined in terms of the ordered linear structure.
Section \[S.Completeness\] is devoted to various kinds of completeness. It is shown that, although $d_T$ and $|\cdot|_u$ are only topologically, and not metrically, equivalent, we are able to prove that the completeness is a common feature. Also we study a special notion, called self–completeness, we prove that several completeness conditions are equivalent and that the Thompson metric on a sequentially complete normal cone $K$ in a locally convex space $X$ is complete.
In the last subsection we show that in the case when $X$ is a Banach space, the completeness of $K$ with respect to $d_T$ is also necessary for the normality of $K$. This is obtained as a consequence of a more general result (Theorem \[t2.complete-T-metric-B-sp\]) on the equivalence of several conditions to the completeness of $K$ with respect to $d_T$ .
Cones in vector spaces {#S.ord-vs}
======================
Ordered vector spaces
---------------------
A *preorder* on a set $Z$ is a reflexive and transitive relation $\,\le\,$ on $Z$. If the relation $\,\le\,$ is also antisymmetric then it is called an *order* on $Z$. If any two elements in $Z$ are comparable (i.e. at least one of the relations $x\le y$ or $y\le x$ holds), then one says that the order (or the preorder) $\le $ is *total*.
Since in what follows we shall be concerned only with real vector spaces, by a “vector space" we will understand always a “real vector space".
A nonempty subset $W$ of a vector space $X$ is called a *wedge* if $$\label{def.cone}
\begin{aligned}
&{\rm (C1)}\qquad W+W\subset W, \\
&{\rm (C2)}\qquad tW\subset W, \quad\mbox{for all } \; t\ge 0.
\end{aligned}$$
The wedge $W$ induces a preorder on $X$ given by $$\label{def.order}
x\le_W y \iff y-x\in W.$$
The notation $\,x<_W y\,$ means that $\,x\le_W y\,$ and $\,x\ne y.$ If there is no danger of confusion the subscripts will be omitted.
This preorder is compatible with the linear structure of $X$, that is $$\label{eq.lin.ord}
\begin{aligned}
&{\rm (i)}\qquad x\le y\Longrightarrow x+z\le y+z,\quad\mbox{and} \\
&{\rm (ii)}\qquad x\le y\Longrightarrow tx\le ty,
\end{aligned}$$ for all $x,y,z\in X$ and $t\in\mathbb{R}_+,$ where $\mathbb{R}_+=\{t\in \mathbb{R} : t\ge 0\}.$ This means that one can add inequalities $$x\le y\;\mbox{and}\; x'\le y'\Longrightarrow x+x'\le y+y',$$ and multiply by positive numbers $$x\le y \iff tx\le ty,$$ for all $x,x',y,y'\in X$ and $t>0.$ The multiplication by negative numbers reverses the inequalities $$\forall t<0,\;\; (x\le y \iff tx\ge ty).$$
As a consequence of this equivalence, a subset $A$ of $X$ is bounded above iff the set $-A$ is bounded below. Also $$\inf A =-\sup(-A)\quad \sup A=-\inf(-A).$$
It is obvious that the preorder $\,\le_W$ is total iff $\,X=W\cup(-W).$
It follows that in definitions (or hypotheses) we can ask only one order condition. For instance, if we ask that every bounded above subset of an ordered vector space has a supremum, then every bounded below subset will have an infimum, and consequently, every bounded subset has an infimum and a supremum. Similarly, if a linear preorder is upward directed, then it is automatically downward directed, too.
Obviously, the wedge $W$ agrees with the set of positive elements in $X$, $$W=X_+:=\{x\in X : 0\le_W x\}.$$
Conversely, if $\,\le\,$ is a preorder on a vector space $X$ satisfying (such a preorder is called a *linear preorder*), then $W=X_+$ is a wedge in $X$ and $\,\le\, = \,\le_W.$ Consequently, there is a perfect correspondence between linear preorders on a vector space $X$ and wedges in $X$ and so any property in an ordered vector space can be formulated in terms of the preorder or of the wedge.
A *cone* $K$ is a wedge satisfying the condition $$\label{def.pcone}
{\rm (C3)}\qquad K\cap(-K)=\{0\}.$$
This is equivalent to the fact that the induced preorder is antisymmetric, $$\label{eq.antiref}
x\le y\;\mbox{and}\; y\le x \Longrightarrow y=x,$$ for all $x,y\in X,$ that it is an order on $X$.
A pair $(X,K),$ where $K$ is a cone (or a wedge) in a vector space $X,$ is called an *ordered* (resp. *preodered*) *vector space*.
An *order interval* in an ordered vector space $(X,K)$ is a (possibly empty) set of the form $$[x;y]_o=\{z\in X : x\le z\le y\}=(x+K)\cap(y-K),$$ for some $x,y\in X.$ It is clear that an order interval $[x;y]_o$ is a convex subset of $X$ and that $$[x;y]_o=x+[0;y-x]_o.$$
The notation $[x;y]$ will be reserved to algebraic intervals: $[x;y]:=\{(1-t) x+t y : t\in [0;1]\}.$
A subset $A$ of $X$ is called *order–convex* (or *full*, or *saturated*) if $[x;y]_o\subset A$ for every $x,y\in A.$ Since the intersection of an arbitrary family of order–convex sets is order–convex, we can define the order–convex hull $[A]$ of a nonempty subset $A$ of $X$ as the intersection of all order–convex subsets of $X$ containing $A$, i.e. the smallest order–convex subset of $X$ containing $A$. It follows that $$\label{eq.o-cv-hull}
[A]=\bigcup\{[x;y]_o : x,y\in A\} =(A+K)\cap (A-K).$$
Obviously, $A$ is order–convex iff $A=[A].$
It is obvious that if $x\le y,$ then $[x;y]\subset [x;y]_o,$ but the reverse inclusion could not hold as the following example shows. Taking $X=\mathbb{R}^2$ with the coordinate order and $x=(0,0),\, y=(1,1),$ then $[x;y]_o$ equals the (full) square with the vertices $(0,0), \,(0,1),\, (1,1)$ and $(0,1),$ so it is larger than the segment $[x;y].$
We mention also the following result.
\[p.char-total-o-cv\] Let $(X,\le)$ be an ordered vector space. Then the order $\,\le\,$ is total iff every order–convex subset of $X$ is convex.
We shall consider now some algebraic-topological notions concerning the subsets of a vector space $X$. Let $A$ be a subset of $X$.
The subset $A$ is called:
*balanced* if $\lambda A\subset A$ for every $|\lambda|\le 1;$
*symmetric* if $-A=A;$
*absolutely convex* if it is convex and balanced;
*absorbing* if $\{t>0 : x\in tA\}\ne \emptyset$ for every $x\in X.$
The following equivalences are immediate: $$\begin{aligned}
A \;\mbox{is absolutely convex}\; \iff&\; \forall a,b \in A,\;\forall \alpha,\beta \in \mathbb{R},\; \mbox{with}\; |\alpha|+|\beta|=1,\quad \alpha a+\beta b\in A\\
\iff&\; \forall a,b \in A,\; \forall \alpha,\beta \in \mathbb{R},\;\mbox{with}\; |\alpha|+|\beta|\le 1,\quad \alpha a+\beta b\in A.
\end{aligned}$$
Notice that a balanced set is symmetric and a symmetric convex set containing 0 is balanced.
The following properties are easily seen.
\[p.full-hull\] Let $X$ be an ordered vector space and $A\subset X$ nonempty. Then
1. If $A$ is convex, then $[A]$ is also convex.
2. If $A$ is balanced, then $[A]$ is also balanced.
3. If $A$ is absolutely convex, then $[A]$ is also absolutely convex.
One says that $a$ is an *algebraic interior* point of $A$ if $$\label{def.a-int-pt}
\forall x\in X,\; \exists \delta >0,\; \mbox{such that}\; \forall \lambda\in [-\delta;\delta],\; a+\lambda x\in A.$$
The (possibly empty) set of all interior points of $A$, denoted by ${\operatorname{aint}}(A),$ is called the *algebraic interior* (or the *core*) of the set $A$. It is obvious that if $X$ is a TVS, then ${\operatorname{int}}(A)\subset{\operatorname{aint}}(A).$ where ${\operatorname{int}}(A)$ denotes the interior of the set $A$. In finite dimension we have equality, but the inclusion can be proper if $X$ is infinite dimensional.
A cone $K$ is called *solid* if ${\operatorname{int}}(K)\ne\emptyset.$
Zălinescu [@Zali] uses the notation $A^i$ for the algebraic interior and $^iA$ for the algebraic interior of $A$ with respect to its affine hull (called the relative algebraic interior). In his definition of an algebraic interior point of $A$ one asks that the conclusion of holds only for $\lambda \in [0;\delta],$ a condition equivalent to .
The set $A$ is called *lineally open* (or *algebraically open*) if $A={\operatorname{aint}}(A),$ and *lineally closed* if $X\setminus A$ is lineally open. This is equivalent to the fact that any line in $X$ meets $A$ in a closed subset of the line. The smallest lineally closed set containing a set $A$ is called the *lineal* (or *algebraic*) closure of $A$ and it is denoted by ${\operatorname{acl}}(A).$ Again, if $X$ is a TVS, then any closed subset of $X$ is lineally closed. The subset $A$ is called *lineally bounded* if the intersection with any line $D$ in $X$ is a bounded subset of $D$.
The terms “lineally open", “lineally closed", etc, are taken from Jameson [@Jameson].
\[re.a-int\] Similar to the topological case one can prove that $$\label{eq1.re.a-int}
a\in {\operatorname{aint}}(A),\; b\in A \;\mbox{and}\; \lambda\in [0;1)\;\Longrightarrow \; (1-\lambda) a+\lambda b\in {\operatorname{aint}}(A).$$
Consequently, if $A$ is convex then ${\operatorname{aint}}(A)$ is also convex.
If $K$ is a cone, then ${\operatorname{aint}}(K)\cup\{0\}$ is also a cone and $$\label{eq2.re.a-int}
{\operatorname{aint}}(K)+K\subset {\operatorname{aint}}(K).$$
We justify only the second assertion. Let $x\in {\operatorname{aint}}(K)$ and $y\in K.$ Then $$x+y=2\left(\frac12 x+\frac12 y\right)\in {\operatorname{aint}}(K).$$
Now we shall consider some further properties of linear orders. A linear order $\,\le\,$ on a vector space $X$ is called:
*Archimedean* if for every $x,y\in X,$ $$\label{def.Arch}
(\forall n\in\mathbb{N},\; nx\le y)\;\Longrightarrow \; x\le 0;$$
*almost Archimedean* if for every $x,y\in X,$ $$\label{def.a-Arch}
(\forall n\in\mathbb{N},\; -y\le nx\le y)\;\Longrightarrow \; x = 0;$$
The following four propositions are taken from Breckner [@BreckW] and Jameson [@Jameson]. In all of them $X$ will be a vector space and $\,\le\,$ a linear preorder on $X$ given by the wedge $W=X_+.$
\[p.char.Arch\] The following are equivalent.
1. The preorder $\le $ is Archimedean.
2. The wedge $W$ is lineally closed.
3. For every $x\in X$ and $\,y\in W,\; 0=\inf\{n^{-1}x : n\in\mathbb{N}\}.$
4. For every $\,x\in X$ and $y\in W,\; nx\le y,\,$ for all $\, n\in\mathbb{N},\,$ implies $\,x\le 0.$
5. For every $A\subset \mathbb{R}$ and $x,y\in X,\;\; y\le \lambda x$ for all $\lambda\in A,$ implies $y\le \mu x,$ where $\mu=\inf A.$
\[p.char.a-Arch\] The following are equivalent.
1. The preorder is almost-Archimedean.
2. ${\operatorname{acl}}(W)\,$ is a wedge.
3. Every order interval in $X$ is lineally bounded.
A wedge $W$ in $X$ is called *generating* if $X=W-W.$ The preorder $\,\le\,$ is called *upward* (*downward*) *directed* if for every $x,y\in X$ there is $z\in X$ such that $x\le z,\, y\le z$ (respectively, $x\ge z,\, y\ge z$). If the order is linear, then these two notions are equivalent, so we can say simply that $\,\le\,$ is directed.
\[p.char.gener-cone\] The following are equivalent.
1. The wedge $W$ is generating.
2. The order $\,\le\,$ is directed.
3. $\forall x\in X,\; \exists y\in W,\; x\le y. $
Let $(X,W)$ be a preordered vector space. An element $u\in W$ is called an *order unit* if the set $[-u;u]_o$ is absorbing. It is obvious that an order unit must be different of 0 (provided $X\ne\{0\}$).
\[p.char-o-unit\] Let $u\in W\setminus \{0\}.$ The following are equivalent.
1. The element $u$ is an order unit.
2. The order interval $[0;u]_o$ is absorbing.
3. The element $u$ belongs to the algebraic interior of $W.$
4. $[\mathbb{R} u]=X.$
Completeness in ordered vector spaces
-------------------------------------
An ordered vector space $X$ is called a vector lattice if any two elements $x,y\in X$ have a supremum, denoted by $x\vee y.$ It follows that they have also an infimum, denoted by $x\wedge y,$ and these properties extend to any finite subset of $X$. The ordered vector space $X$ is called *order complete* (or *Dedekind complete*) if every bounded from above subset of $X$ has a supremum and *order* $\sigma$-*complete* (or *Dedekind $\sigma$-complete*) if every bounded from above countable subset of $X$ has a supremum. The fact that every bounded above subset of $X$ has a supremum is equivalent to the fact that every bounded below subset of $X$ has an infimum. Indeed, if $A$ is bounded above, then $\sup\{y : y\; \mbox{is a lower bound for }\; A\} =\inf A.$
An ordered vector space $X$ is order complete iff for each pair $A,B$ of nonempty subsets of $X$ such that $A\le B$ there exists $z\in X$ with $A\le z\le B$.
This similarity with “Dedekind cuts" in $\mathbb{R}$ justifies the term *Dedekind complete* used by some authors. Here $A\le B $ means that $\, a\le b \,$ for all $(a,b)\in A\times B.$
The following results gives characterizations of these properties in terms of directed subsets.
\[p.Dedekind\] Let $X$ be a vector lattice.
1. The space $X$ is order complete iff every upward directed bounded above subset of $X$ has a supremum (equivalently, if every bounded above monotone net has a supremum).
2. The space $X$ is Dedekind $\sigma$-complete iff every upward directed bounded above countable subset of $X$ has a supremum (equivalently, if every bounded above monotone sequence has a supremum).
Ordered topological vector spaces (TVS)
---------------------------------------
In the case of an ordered TVS $(X,\tau)$ some connections between order and topology hold. In the following propositions $(X,\tau)$ will be a TVS with a preorder or an order, $\,\le\,$ generated by a wedge $W,$ or by a cone $K,$ respectively. We start by a simple result.
\[p1.order-tvs\] A wedge $W$ is closed iff the inequalities are preserved by limits, meaning that for all nets $(x_i : i\in I),\, (y_i : i\in I) $ in $X,$ $$\forall i\in I,\; x_i\le y_i \;\;\mbox{and}\;\; \lim_ix_i=x,\, \lim_iy_i=y \Longrightarrow \; x\le y.$$
Other results are contained in the following proposition.
\[p2.order-tvs\] Let $(X,\tau)$ be a TVS ordered by a $\tau$-closed cone $K$. Then
1. The topology $\tau$ is Hausdorff.
2. The cone $K$ is Archimedean.
3. The order intervals are $\tau$-closed.
4. If $(x_i:i\in I)$ is an increasing net which is $\tau$-convergent to $x\in X$, then $x=\sup_ix_i.$
5. Conversely, if the topology $\tau$ is Hausdorff, $\;{\operatorname{int}}(K)\ne\emptyset$ and $K$ is Archimedean, then $K$ is $\tau$-closed.
In what follows by an ordered TVS we shall understand a TVS ordered by a closed cone. Also, in an ordered TVS $(X,\tau,K)$ we have some parallel notions—with respect to topology and with respect to order. To make distinction between them, those referring to order will have the prefix “order–", as, for instance, “order–bounded", “order–complete", etc, while for those referring to topology we shall use the prefix “$\tau$-", or “topologically–", e.g., “$\tau$-bounded", “$\tau$-complete" (resp. “topologically–bounded", “topologically–complete"), etc.
Normal cones in TVS and in LCS (locally convex spaces)
------------------------------------------------------
Now we introduce a very important notion in the theory of ordered vector spaces. A cone $K$ in a TVS $(X,\tau)$ is called *normal* if there exists a neighborhood basis at 0 formed of order–convex sets.
The following characterizations are taken from [@BreckW] and [@Peressini].
\[t1.char-normal-cone\] Let $(X,\tau,K)$ be an ordered TVS. The following are equivalent.
1. The cone $K$ is normal.
2. There exists a basis $\mathcal B$ formed of order–convex balanced 0-neighborhoods.
3. There exists a basis $\mathcal B$ formed of balanced 0-neighborhoods such that for every $B\in \mathcal B,\; y\in B $ and $0\le x\le y$ implies $x\in B.$
4. There exists a basis $\mathcal B$ formed of balanced 0-neighborhoods such that for every $B\in \mathcal B,\; y\in B $ implies $[0;y]_o\subset B.$
5. There exists a basis $\mathcal B$ formed of balanced 0-neighborhoods and a number $\gamma >0$ such that for every $B\in \mathcal B,\; [B]\subset \gamma B.$
6. If $(x_i:i\in I)$ and $(y_i:i\in I)$ are two nets in $X$ such that $\forall i\in I,\; 0\le x_i\le y_i$ and $\lim_iy_i=0,$ then $\lim_ix_i=0.$
If further, $X$ is a LCS, then the fact that the cone $K$ is normal is equivalent to each of the conditions 2–5, where the term “balanced" is replaced with “absolutely convex".
Condition 6 can be replaced with the equivalent one:
If $(x_i:i\in I), \,(y_i:i\in I)$ and $(z_i:i\in I) $ are nets in $X$ such that $\forall i\in I,\; x_i\le z_i\le y_i$ and $\lim_ix_i=x=\lim_iy_i,$ then $\lim_iz_i=x.$
The normality implies the fact that the order–bounded sets are bounded.
\[p1.normal-cone-bd\] If $(X,\tau)$ is a TVS ordered by a normal cone, then every order–bounded subset of $X$ is $\tau$-bounded.
In the case of a normed space this condition characterizes the normality, see Theorem \[t4.char-normal-cone\] below. Also, it is clear that a subset $Z$ of an ordered vector space $X$ is order–bounded iff there exist $x,y\in X$ such that $Z\subset [x;y]_o.$
The existence of a normal solid cone in a TVS makes the topology normable.
\[p1.normal-c-TVS\] If a Hausdorff TVS $(X,\tau)$ contains a solid $\tau$-normal cone, then the topology $\tau$ is normable.
In order to give characterizations of normal cones in LCS we consider some properties of seminorms. Let $\gamma >0$. A seminorm $p$ on a vector space $X$ is called:
$\gamma$-*monotone* if $0\le x\le y\;\Longrightarrow\; p(x)\le \gamma p(y);$
$\gamma$-*absolutely monotone* if $-y\le x\le y\;\Longrightarrow\; p(x)\le \gamma p(y);$
$\gamma$-*normal* if $x\le z\le y\;\Longrightarrow\; p(z)\le \gamma\max\{p(x),p(y)\}.$
A 1-monotone seminorm is called *monotone*. Also a seminorm which is $\gamma$-monotone for some $\gamma>0$ is called sometimes semi-monotone (see [@Deimling]).
These properties can be characterized in terms of the Minkowski functional attached to an absorbing subset $A$ of a vector space $X$, given by $$\label{def.Mink-fc}
p_A(x)=\inf\{t>0 : x\in tA\},\quad (x\in X.)$$
It is well known that if the set $A$ is absolutely convex and absorbing, then $p_A$ is a seminorm on $X$ and $${\operatorname{aint}}(A)= \{x\in X : p_A(x)<1\}\subset A\subset \{x\in X : p_A(x)\le 1\}={\operatorname{acl}}(A).$$
\[p1.Mink-fc-monot\] Let $A$ be an absorbing absolutely convex subset of an ordered vector space $X$.
1. If $\,[A]\subset \gamma A,\,$ then the seminorm $p_A$ is $\gamma$-normal.
2. If $\,\forall y\in A,\; [0;y]\subset \gamma A,\,$ then the seminorm $p_A$ is $\gamma$-monotone.
3. If $\,\forall y\in A,\; [-y;y]\subset \gamma A,\,$ then the seminorm $p_A$ is $\gamma$-absolutely monotone.
Based on Theorem \[t1.char-normal-cone\] and Proposition \[p1.Mink-fc-monot\] one can give further characterizations of normal cones in LCS.
\[t2.char-normal-cone\] Let $(X,\tau)$ be a LCS ordered by a cone $K.$ The following are equivalent.
1. The cone $K$ is normal.
2. There exists $\gamma>0$ and a family of $\gamma$-normal seminorms generating the topology $\tau$ of $X$.
3. There exists $\gamma>0$ and a family of $\gamma$-monotone seminorms generating the topology $\tau$ of $X$.
4. There exists $\gamma>0$ and a family of $\gamma$-absolutely monotone seminorms generating the topology $\tau$ of $X$.
All the above equivalences hold also with $\gamma =1$ in all places.
Normal cones in normed spaces
-----------------------------
We shall consider now characterizations of normality in the case of normed spaces. For a normed space $(X,\|\cdot\|),$ let $B_X=\{x\in X : \|x\|\le 1\}$ be its closed unit ball and $S_X=\{x\in X : \|x\|= 1\}$ its unit sphere.
\[t4.char-normal-cone\] Let $K$ be a cone in a normed space $(X,\|\cdot\|).$ The following are equivalent.
1. The cone $K$ is normal.
2. There exists a monotone norm $\|\cdot\|_1$ on $X$ equivalent to the original norm $\|\cdot\|.$
3. For all sequences $(x_n),\, (y_n),\, (z_n)$ in $X$ such that $x_n\le z_n\le y_n,\,n\in\mathbb{N},$ the conditions $\lim_nx_n=x=\lim_ny_n$ imply $\lim_nz_n=x.$
4. The order–convex hull $[B_X]$ of the unit ball is bounded.
5. The order interval $[x;y]_o$ is bounded for every $x,y\in X.$
6. There exists $\delta > 0$ such that $\forall x,y\in K\cap S_X,\; \|x+y\|\ge\delta.$
7. There exists $\gamma > 0$ such that $\,\forall x,y\in K,\; \|x+y\|\ge\gamma\max\{\|x\|,\|y\|\}.$
8. There exists $\lambda > 0$ such that $\, \|x\|\le\lambda \|y\|,$ for all $\,x,y\in K\,$ with $x \le y.$
We notice also the following result, which can be obtained as a consequence of a result of T. Andô on ordered locally convex spaces (see [@Alipr Theorem 2.10]).
\[p.ord-B-sp\] Let $X$ be a Banach space ordered by a generating cone $X_+$ and $B_X$ its closed unit ball. Then $
\,(B_X \cap X_+)- (B_X \cap X_+)\,$ is a neighborhood of 0.
Completeness and order completeness in ordered TVS
--------------------------------------------------
The following notions are inspired by Cantor’s theorem on the convergence of bounded monotone sequences of real numbers.
Let $X$ be a Banach space ordered by a cone $K$. The cone $K$ is called:
*regular* if every increasing and order–bounded sequence in $X$ is convergent;
*fully regular* if every increasing and norm-bounded sequence in $X$ is convergent.
By Proposition \[p.Dedekind\] if $X$ is a regular normed lattice, then every countable subset of $X$ has a supremum.
These notions are related in the following way.
\[t.regular-cone\] If $X$ is a Banach space ordered by a cone $K$, then $$K\; \mbox{fully regular} \; \Longrightarrow\; K\; \mbox{regular} \; \Longrightarrow\; K\; \mbox{normal}.$$
If the Banach space $X$ is reflexive, then the reverse implications hold too, i.e. both implications become equivalences.
Some relations between completeness and order completeness in ordered topological vector spaces were obtained by Ng [@ng72], Wong [@wong72] (see also the book [@Wong-Ng]). Some questions about completeness in ordered metric spaces are discussed by Turinici [@turinici80].
Let $(X,\tau)$ be a TVS ordered by a cone $K$. One says that the space $X$ is
*fundamentally $\sigma$-order complete* if every increasing $\tau$-Cauchy sequence in $X$ has a supremum;
*monotonically sequentially complete* if every increasing $\tau$-Cauchy sequence in $X$ is convergent in $(X,\tau)$.
In the following propositions $(X,\tau)$ is a TVS ordered by a cone $K.$
The following result is obvious.
\[p1.o-complete\]
1. If $ X$ is sequentially complete, then $X$ is monotonically sequentially complete.
2. If $ X$ is monotonically sequentially complete, then $X$ is fundamentally $\sigma$-order complete.
3. If $ K $ is normal and generating, and $X$ is fundamentally $\sigma$-order complete, then $X$ is monotonically sequentially complete.
The following characterizations of these completeness conditions will be used in the study of the completeness with respect to the Thompson metric.
\[p2.o-complete\] The following conditions are equivalent.
1. $X$ is fundamentally $\sigma$-order complete.
2. Any decreasing Cauchy sequence in $X$ has an infimum.
3. Any increasing Cauchy sequence in $K$ has a supremum.
4. Any decreasing Cauchy sequence in $K$ has an infimum.
\[p3.o-complete\] The following conditions are equivalent.
1. $X$ is monotonically sequentially complete.
2. Any decreasing Cauchy sequence in $X$ has limit.
3. Any increasing Cauchy sequence in $K$ has limit.
4. Any decreasing Cauchy sequence in $K$ has limit.
\[p4.o-complete\] If $K$ is lineally solid, then the following conditions are equivalent.
1. $X $ is fundamentally $\sigma$-order complete.
2. Any increasing Cauchy sequence in ${\operatorname{aint}}(K)$ has a supremum.
3. Any decreasing Cauchy sequence in ${\operatorname{aint}}(K)$ has an infimum.
\[p5.o-complete\] If K is lineally solid, then the following conditions are equivalent.
1. $X$ is monotonically sequentially complete.
2. Any increasing $\tau$-Cauchy sequence in ${\operatorname{aint}}(K)$ has limit.
3. Any decreasing $\tau$-Cauchy sequence in ${\operatorname{aint}}(K)$ has limit.
The Thompson metric {#S.Thompson-metric}
===================
Definition and fundamental properties {#Ss.def-T-metric}
-------------------------------------
Let $X$ be a vector space and $K$ a cone in $X$. The relation $$\label{def.linked}
x\sim y \iff \exists \lambda,\mu >0,\;\; x\le \lambda y\;\mbox{and}\; y \le \mu x,$$ is an equivalence relation in $K$. One says that two elements $x,y\in K$ satisfying are *linked* and the equivalence classes are called *components*. The equivalence class of an element $x\in K $ will be denoted by $K(x).$
\[p1.equiv-cone\] Let $X$ be a vector space ordered by a cone $K$.
1. $K(0)=\{0\}$ and ${\operatorname{aint}}(K)$ is a component of $K$ if $K$ is lineally solid.
2. Every component $Q$ of $K$ is order–convex, convex, closed under addition and multiplication by positive scalars, that is $Q\cup\{0\}$ is an order–convex cone.
We justify only the assertion concerning ${\operatorname{aint}}K$, the others being trivial. If $x,y\in {\operatorname{aint}}K, $ then there exist $\alpha,\beta>0$ such that $x+t y\in K$ for all $t\in[-\alpha,\alpha]$ and $y+sx\in K$ for all $s\in[-\beta,\beta].$ It follows $y-\beta x\in K,$ i.e. $y\ge \beta x,$ and $x-\alpha y\in K,$ i.e. $x\ge \alpha y.$
For two linked elements $x,y\in K$ put $$\label{def1.t-metric}
\sigma(x,y)=\{s\ge 0 : e^{-s}x\le y\le e^sx\},$$ and let $$\label{def2.t-metric}
d_T(x,y)=\inf\sigma(x,y) .$$
\[re.extended-T-metric\] It is convenient to define $d_T$ for any pair of elements in $K$, by setting $d_T(x,y)=\infty$ for any $x,y$ not lying in the same component of $K$ which, by , is in concordance with the usual convention $\inf\emptyset =\infty.$ In this way, $d_T$ becomes an extended (or generalized) (semi)metric (in the sense of Jung [@jung69]) on $K$ and, for all $x,y\in K,\, x\sim y \iff d(x,y) < \infty.$ Though $d_T$ is not a usual (semi)metric on the whole cone, we will continue to call $d_T$ a metric. The Thompson metric is also called, by some authors, the part metric (of the cone $K$).
\[re2.T-metric\]It is obvious that the definition of $d(x,y)$ depends only on the ordering of the vector subspace spanned by $\{x,y\}$. This ensures that if $x$ and $y$ are seen as elements of some vector subspace $Y$ of $X$, then $d_T(x,y)$ is the same in $Y$ as in $X$ (assuming, of course, that $Y$ inherits the ordering from $ X$).
The initial approach of Thompson [@thomp63] was slightly different. He considered the set $$\label{def3.t-metric}
\alpha(x,y)= \{\lambda \ge 1 : x\le \lambda y\} .$$ and defined the distance between $x$ and $y$ by $$\label{def4.t-metric}
\delta(x,y)=\ln\left(\max\{\inf\alpha(x,y),\inf\alpha(y,x)\}\right).$$
The following proposition shows that the relations and yield the same function.
\[p1.T-metric\] For every $x,y\in K$ the following equality holds $$d_T(x,y)=\delta(x,y).$$
It suffices to prove the equality for two linked elements $x,y\in K.$ In this case let $$\alpha_1=\inf\alpha(x,y),\;\; \alpha_2=\inf\alpha(y,x)\;\; \mbox{and}\;\; \alpha=\max\{\alpha_1,\alpha_2\}.$$
Put also $$d=d_T(x,y)=\inf\sigma(x,y)\quad\mbox{and}\quad \delta=\delta(x,y)=\ln\alpha.$$
For $s\in\mathbb{R}$ let $\lambda =e^s.$ Then the following equivalences hold $$\label{eq2.p1.T-metric}\begin{aligned}
s\in\sigma(x,y)\iff& \lambda^{-1}x\le y\le\lambda x\\ \iff& x\le\lambda y\;\wedge \; y\le \lambda x \iff \lambda\in\alpha(x,y)\cap\alpha(y,x) .
\end{aligned}$$
Consequently $\lambda\ge\max\{\alpha_1,\alpha_2\}=\alpha$ and $s\ge \ln\alpha=\delta,$ for every $s\in \sigma(x,y),$ and so
$$\label{eq1.p1.T-metric}
d=\inf\sigma(x,y)\ge \delta.$$
To prove the reverse inequality, suppose that $\alpha_1\ge \alpha_2$ and let $\lambda>\alpha_1$. Then $\lambda\in\alpha(x,y)\cap\alpha(y,x)$ and the equivalences show that $s=\ln\lambda\in \sigma(x,y),$ so that $\ln\lambda \ge d.$ It follows $$\delta=\inf\{\ln\lambda : \lambda >\alpha_1\}\ge d ,$$ which together with yields $\delta=d.$
There is another metric defined on the components of $K$, namely the *Hilbert projective metric*, defined by $$\label{def.Hilb-metric}
d_H(x,y)=\ln\left(\inf\alpha(x,y)\cdot\inf\alpha(y,x)\}\right),$$ for any two linked elements $x,y$ of $K$.
The term projective comes from the fact that $d_H(x,y)=0$ iff $x=\lambda y$ for some $\lambda>0.$
The original Hilbert’s definition (see [@hilbert1895]) of the metric was the following. Consider an open bounded convex subset $\Omega$ of the Euclidean space $\mathbb{R}^n$. For two points $x,y \in \Omega$ let $\ell_{xy}$ denote the straight line through $x$ and $y$, and denote the points of intersection of $\ell_{xy}$ with the boundary $\partial\Omega$ of $\Omega$ by $x',y',$ where $x$ is between $x'$ and $y$, and $y$ is between $x$ and $y'$. For $x\ne y$ in $\Omega$ the Hilbert distance between $x$ and $y$ is defined by $$\label{def.Hilb-metric-R2}
\delta_H(x,y)=\ln\left(\frac{\|x'-y\|\cdot\|y'-x\|}{\|x'-x\|\cdot\|y'-y\|}\right)\,,$$ and $\delta_H(x, x) = 0$ for all $x\in\Omega,$ where $\|\cdot\|$ stands for the Euclidean norm in $\mathbb{R}^n.$ The metric space $(\Omega,\delta_H)$ is called the Hilbert geometry on $\Omega$. In this geometry there exists a triangle with non-colinear vertices such that the sum of the lengths of two sides equals the length of the third side. If $\Omega$ is the open unit disk, the Hilbert metric is exactly the Klein model of the hyperbolic plane.
The definition of Hilbert metric on cones in vector spaces was proposed by Bushell [@bushell73a] (see also [@bushell73b]).
As we shall consider only the Thompson metric, the subscript $T$ will be omitted, that is $d(\cdot,\cdot)$ will stand always for the Thompson metric.
In the following proposition we collect some properties of the set $\sigma(x,y).$
\[p2.T-metric\] Let $X$ be a vector space ordered by a cone $K$ and $x,y,z$ linked elements in $K$.
1. Symmetry:$\sigma(y,x)=\sigma(x,y).$
2. $(d(x,y);\infty)\subset \sigma(x,y)\subset [d(x,y);\infty).\;$ If the cone $K$ is Archimedean, then $d(x,y)\in\sigma(x,y),$ that is $\sigma(x,y)=[d(x,y);\infty)$.
3. $\sigma(x,y)+\sigma(y,z)\subset\sigma(x,z).$
1\. The symmetry follows from the definition of the set $\sigma(x,y).$
2\. The inclusion $(d(x,y);\infty)\subset \sigma(x,y)$ follows from the fact that $0<\lambda<\mu$ and $x\ge 0 $ implies $ \lambda x\le \mu x.$ The second inclusion follows from the fact that no $\lambda<d(x,y)$ belongs to $\sigma(x,y).$
Let $d=d(x,y)=\inf\sigma(x,y). $ Since an Archimedean cone is lineally closed and $y-e^{-s}x\in K$ for every $s>d$, it follows $y-e^{-d}x\in K.$ Similarly $e^{d}x-y\in K,$ showing that $d\in \sigma(x,y).$
3\. Let $s\in\sigma(x,y)$ and $t\in\sigma(y,z).$ Then $$e^{-s}x\le y\le e^sx\quad\mbox{and}\quad e^{-t}y\le z\le e^ty.$$
It follows $$e^{-(s+t)}x\le e^{-t}y\le z\quad\mbox{and} \quad z\le e^{t}y \le e^{s+t}x,$$ which shows that $s+t\in \sigma(x,z).$
Now it is easy to show that the function $d$ given by is an extended semimetric.
\[p3.T-metric\] Let $X$ be a vector space ordered by a cone $K$.
1. The function $d$ defined by is a semimetric on each component of $K$.
2. The function $d$ is a metric on each component of $K$ iff the order defined by the cone $K$ is almost Archimedean.
1\. The fact that $d$ is a semimetric follows from the properties of the sets $\sigma(x,y)$ mentioned in Proposition \[p2.T-metric\].
2\. Suppose now that the cone $K$ is almost Archimedean and $d(x,y)=0$ for two linked elements $x,y\in K.$ It follows $$\begin{aligned}
\forall s>0,\;\; e^{-s}x\le y\le e^sx \iff& \forall s>0,\;\; (e^{-s}-1)x\le y-x\le (e^s-1)x\\
\iff& \forall s>0,\;\; -\frac{e^{s}-1}{e^s}x\le y-x\le (e^s-1)x .
\end{aligned}$$
The inequality $ e^{-s}(e^{s}-1) \le e^s-1$ implies $-e^{-s}(e^{s}-1)x\ge -(e^s-1)x$. Consequently, $$\forall s>0,\;\; -(e^{s}-1)x\le y-x\le (e^s-1)x \iff \forall \lambda >0,\;\; -\lambda x\le y-x\le \lambda x.$$
Taking into account that $K$ is almost Archimedean it follows $y-x=0,$ that is $y=x.$
To prove the converse, suppose that $K$ is not almost Archimedean. Then there exists a line $D=\{x+\mu y : \mu\in\mathbb{R}\},\, $ with $\, y\ne 0,$ contained in $K$. If $x=0, $ then $\pm y\in K,$ that would imply $y=0, $ a contradiction.
Consequently $x\ne 0.$ Observe that in this case, for all $\mu\in\mathbb{R},$ $$\label{eq1.p3.T-metric}
d(x,x+\mu y)=0 ,$$ which shows that $d$ is not a metric. The equality is equivalent to $$\label{eq2.p3.T-metric}
\forall s>0,\;\; e^{-s}x\le x+\mu y\le e^sx.$$
The inclusion $D\subset K$ implies $x\pm \lambda y\in K$ for all $\lambda>0,$ and so $$-x\le \lambda y\le x,$$ for all $\lambda >0.$ Taking $\lambda =\mu (1-e^{-s})^{-1}$ the first inequality from above becomes $$-(1- e^{-s})x\le \mu y \iff e^{-s}x\le x+\mu y .$$
From the second inequality one obtains $$\mu y\le (1-e^{-s})x=e^{-s}(e^s-1)x\le (e^s-1)x,$$ which implies $$x+\mu y\le e^s x,$$ showing that the inequalities hold.
\[re1.T-metric\] By the triangle inequality, the equality implies that $d(u,v)=0$ for any two points $u,v$ on $D$, that is $$d(x+\lambda y,x+\mu y)=0,$$ for all $\lambda, \mu\in\mathbb{R}.$
If $X=\mathbb{R}^n$ and $K=\mathbb{R}^n_+$, then the components of $K$ are $\{0\},\, (0;\infty)\cdot e_i,\, 1\le i\le n,$ and ${\operatorname{aint}}(K) =\{x\in K : x_i>0,\, i=1,\dots,n\},$ while $d(x,y)=\max\{|\ln x_i-\ln y_i| : 1\le i\le n\},$ for any $x=(x_i)_{i=1}^n $ and $y=(y_i)_{i=1}^n $, with $x_i,y_i > 0,\, i=1,\dots,n$.
The following proposition contains some further properties of the sets $\sigma(x,y)$ and their corespondents for the Thompson metric.
\[p4.T-metric\] Let $X$ be a vector space ordered by a cone $K$.
1. For $x,y\in K$ and $\lambda,\mu >0$ $$\begin{aligned}
{\rm (i)}&\quad \sigma(\lambda x,\lambda y)= \sigma(x,y)\quad\mbox{and so}\quad d(\lambda x,\lambda y)= d(x,y);\\
{\rm (ii)}&\quad \sigma(\lambda x,\mu x)= [\big|\ln\big(\frac\lambda\mu\big)\big|;\infty)\quad\mbox{and so}\quad d(\lambda x,\mu x)= \big|\ln\big(\frac\lambda\mu\big)\big|;\\
{\rm (iii)}& \quad \mbox{If}\;\; \mu x\le y\le \lambda x, \quad\mbox{for some}\quad \lambda,\mu>0,\quad\mbox{then}\quad
d(x,y)\le\ln\max\{\mu^{-1},\lambda\}.
\end{aligned}$$
2. If $\sigma(x,y)\subset\sigma(x',y'),$ then $d(x,y)\ge d(x',y').$ The converse is true if the order is Archimedean. Also $$\label{eq1.p4.T-metric}
\max\{d(x,y),d(x',y')\}=\inf[\sigma(x,y)\cap \sigma(x',y')].$$
3. The following monotony inequalities hold $$\label{eq2.p4.T-metric}\begin{aligned}
{\rm(i)}\quad x\le x'\;\mbox{and}\; y'\le y\;&\Longrightarrow\; d(x',x'+y')\le d(x,x+y);\\
{\rm(ii)}\quad\;\; x\le x'\le y'\le y\;&\Longrightarrow\; d(x',y')\le d(x,y).
\end{aligned}$$
4. For all $x,y,x',y'\in K$ and $\lambda,\mu >0,$ $$\label{eq3.p4.T-metric}
d(\lambda x+\mu y,\lambda x'+\mu y')\le \max\{d(x,x'),d(y,y')\}.$$
1\. The equalities from (i) are obvious.
To prove (ii) suppose $\lambda >\mu.$ Then $e^{-s}x\le x\le \lambda\mu^{-1}x$ implies $ \mu e^{-s} x\le \lambda x,$ for every $s>0.$ Since $$\lambda\mu^{-1}x \le e^sx \iff s\ge \ln\big(\lambda\mu^{-1}\big),$$ it follows $\sigma(\lambda x,\mu x)=\big[\ln\big(\lambda\mu^{-1}\big);\infty)$ and $d(\lambda x,\mu x)= \ln\big(\lambda\mu^{-1}\big).$
To prove (iii) observe that $\mu x\le y$ is equivalent to $x\le \mu^{-1} y,$ that is $\mu^{-1}\in\alpha(x,y),$ and so $\mu^{-1}\ge \inf\alpha(x,y).$ Similarly, $y\le \lambda x$ is equivalent to $\lambda\in\alpha(y,x),$ implying $\lambda\ge\inf\alpha(x,y).$ It follows $$\ln\max\{\mu^{-1},\lambda\}\ge \ln(\max\{\inf\alpha(x,y),\inf\alpha(y,x)\}=d(x,y).$$
2\. The first implication is obvious. The converse follows from the fact that $\sigma(x,y)=[d(x,y);\infty)$ and $\sigma(x',y')=[d(x',y');\infty)$ if $K$ is Archimedean (Proposition \[p2.T-metric\].2).
The equality follows from the inclusions $$\begin{aligned}
&(d(x,y);\infty)\subset \sigma(x,y)\subset [d(x,y);\infty)\quad\mbox{and}\\
&(d(x',y');\infty)\subset \sigma(x',y')\subset [d(x',y');\infty).\end{aligned}$$
3\. The inequality (i) for the metric $d$ will follow from the inclusion $$\label{eq4.p4.T-metric}
\sigma(x,x+y)\subset \sigma(x',x'+y').$$
Let $s\in \sigma(x,x+y),$ that is $s>0$ and $$e^{-s}x\le x+y\le e^s x.$$
Then $$\begin{aligned}
e^{-s}x'\le& x'\le x'+y'\le x'+y=x+y +(x'-x)\\
\le&e^{s}x+ e^{s}(x'-x)=e^{s}x',\end{aligned}$$ showing that $s\in \sigma(x',x'+y').$
The inequality (ii) follows from (i) by taking $y:=y-x\ge y'-x'=:y'.$
4\. By 1.(i), $\,d(\lambda x,\lambda x')=d(x,x')$ and $d(\mu y ,\mu y')=d(y,y'),$ so that it is sufficient to show that $$\label{eq3b.p4.T-metric}
d(x+y,x'+y')\le \max\{d(x,x'),d(y,y')\}.$$
Taking into account and the assertion 2 of the proposition, the inequality will be a consequence of the inclusion $$\sigma(x,x')\cap \sigma(y,y')\subset \sigma(x+y,x'+y') .$$
But, if $s\in \sigma(x,x')\cap \sigma(y,y'),$ then $e^{-s}x\le x'\le e^{s}x $ and $e^{-s}y\le y'\le e^{s}y, $ which by addition yield $e^{-s}(x+y)\le x'+y'\le e^{s}(x + y),$ that is $s\in \sigma(x+y,x'+y').$
Based on these properties one obtains other properties of the Thompson metric.
\[t1.T-metric\] Let $X$ be a vector space ordered by a cone $K$.
1. The function $d$ is quasi-convex with respect to each of its argument, that is $$\label{eq1.q-cv}\begin{aligned}
&d((1-t) x+t y,v)\le\max\{d(x,v),d(y,v)\}\quad\mbox{and} \\
&d(u,(1-t) x+t y)\le\max\{d(u,x),d(u,y)\},
\end{aligned}$$ for all $x,y,u,v\in K$ and $t\in[0;1].$
2. The following convexity–type inequalities hold $$\label{eq2.q-cv}\begin{aligned}
&d((1-t)x+t y,v)\le\ln\left((1-t)e^{d(x,v)}+te^{d(y,v)}\right) , \\
&d(u,(1-t)x+t y)\le\ln\left((1-t)e^{d(u,x)}+te^{d(u,y)}\right),
\end{aligned}$$ for all $x,y,u,v\in K$ and $t\in[0;1],$ and $$\label{eq3.q-cv}\begin{aligned}
&d((1-t)x+t y,(1-s)x+s y)\le\ln\left(|s-t)e^{d(x,y)}+1-|s-t|\right),
\end{aligned}$$ for all $x,y \in K,\, x\sim y,\,$ and $s,t\in [0;1].$
1\. By and Proposition \[p4.T-metric\].1.(i), $$\begin{aligned}
d((1-t)x+t y,v) =& d((1-t)x+t y,(1-t)v+t v) \\\le& \max\{d((1-t)x,(1-t)v),d(t y,tv)\}= \max\{d(x,v),d(y,v)\},\end{aligned}$$ showing that the first inequality in holds. The second one follows by the symmetry of the metric $d.$
2\. For $s_1\in \sigma(x,v) $ and $s_2\in \sigma(y,v) $ put $s=\ln\left((1-t)e^{s_1}+ te^{s_2}\right). $ By a straightforward calculation it follows that $$\left((1-t)e^{s_1}+ te^{s_2}\right)\cdot\left((1-t)e^{-s_1}+ te^{-s_2}\right)=2t(1-t)(\cosh(s_1-s_2)-1)\ge 0,$$ which implies $$-s\le \ln\left((1-t)e^{-s_1}+ te^{-s_2}\right),$$ or, equivalently, $$e^{-s}\le (1-t)e^{-s_1}+ te^{-s_2}.$$
The above inequality and the inequalities $e^{-s_1}v\le x,\, e^{-s_2}v\le y$ imply $$e^{-s}v\le \left((1-t)e^{-s_1}+ te^{-s_2}\right) v\le (1-t)x+ty.$$
Similarly, the inequalities $x\le e^{s_1}v,\, y\le e^{s_2}v, $ and the definition of $s$ imply $$(1-t)x+ty\le \left((1-t)e^{-s_1}+ te^{-s_2}\right)v = e^s v.$$
It follows $s\in \sigma((1-t)x+ty,v)$ and so
$$d((1-t)x+ty,v)\le s = \ln\left((1-t)e^{-s_1}+ te^{-s_2}\right) ,$$ for all $s_1\in\sigma(x,v)$ and all $s_2\in \sigma(y,v).$ Passing to infimum with respect to $s_1$ and $s_2, $ one obtains the first inequality in . The second inequality follows by the symmetry of $d$.
It is obvious that holds for $s=t$, so we have to prove it only for $s\ne t.$ By symmetry it suffices to consider only the case $t>s.$ Putting $z_t=(1-t)x+t y$ and $z_s=(1-s)x+s y,$ it follows $z_s=(1-\frac st)x+\frac st z_t,$ so that, applying twice the inequality , $$d(x,z_t)\le \ln \left(1-t+te^{d(x,y)}\right),$$ and $$\begin{aligned}
d(z_s,z_t)\le& \ln\left((1-\frac st)e^{d(x,z_t)}+\frac st\right)\\
\le& \ln\left((1-\frac st)(1-t+te^{d(x,y)}) +\frac st\right)=\ln\left((t - s) e^{d(x,y)}+1-(t-s)\right).\end{aligned}$$
Recall that a metric space $(X,\rho)$ is called *metrically convex* if for every pair of distinct points $x,y\in X$ there exists a point $z\in X\setminus\{x,y\}$ such that $$\label{def.metric-cv}
\rho(x,y)=\rho(x,z)+\rho(z,y).$$
The following theorem, asserting that every component of $K$ is metrically convex with respect to the Thompson metric, is a slight extension of a result of Nussbaum [@Nuss88 Proposition 1.12].
\[t1.metric-cv\] Every component of $K$ is metrically convex with respect to the Thompson metric $d$. More exactly, for every pair of distinct points $x,y\in X$ and every $t\in(0;1)$ the point $$z=\frac{\sinh r(1-t)}{\sinh r}x+ \frac{\sinh rt}{\sinh r} y,$$ where $r=d(x,y)$, satisfies .
By the triangle inequality it suffices to show that $$\label{eq2.metric-cv}
r=d(x,y)\ge d(x,z)+d(z,y).$$
If $s\in\sigma(x,y),$ that is $e^{-s}x\le y\le e^sx,$ then $$\label{eq3.metric-cv}\begin{aligned}
&\left(\frac{\sinh r(1-t)}{\sinh r}+ \frac{\sinh rt}{\sinh r}e^{-s}\right)x\le z\le \left(\frac{\sinh r(1-t)}{\sinh r} + \frac{\sinh rt}{\sinh r} e^s\right)x .
\end{aligned}$$
Putting $$\mu(s)=\frac{\sinh r(1-t)}{\sinh r}+ \frac{\sinh rt}{\sinh r}e^{-s} \quad\mbox{and}\quad
\lambda(s)=\frac{\sinh r(1-t)}{\sinh r}+ \frac{\sinh rt}{\sinh r}e^{s},$$ the inequalities imply $$d(x,z)\le\ln(\max\{\mu(s)^{-1},\lambda(s)\}),$$ for all $s>r.$ Since the functions $\mu(s)^{-1}$ and $\lambda(s)$ are both continuous on $(0;\infty)$, it follows $$\label{eq4.metric-cv}
d(x,z)\le\ln(\max\{\mu(r)^{-1},\lambda(r)\}).$$
Taking into account the definition of the function $\sinh,$ a direct calculation shows that $\mu(r)^{-1}=\lambda(r)=e^{rt}$, and so the inequality becomes $$d(x,z)\le rt.$$
By symmetry $$d(z,y)\le r(1-t),$$ so that holds.
Order-unit seminorms {#Ss.u-semin}
--------------------
Suppose that $X$ is a vector space ordered by a cone $K$. For $u\in K\setminus\{0\}$ put $$\label{def.Xu}
X_u=\cup_{\lambda\ge 0}\lambda [-u;u]_o.$$
It is obvious that $X_u$ is a nontrivial subspace of $X\; (\mathbb{R} u\subset X_u),$ and that $[-u;u]_o$ is an absorbing absolutely convex subset of $X_u$ and so $u$ is a unit in the ordered vector space $(X_u,K_u)$, where $K_u$ is the cone in $X_u$ given by $$\label{def.Ku}
K_u=K\cap X_u\,,$$ or, equivalently, by $$\label{char.Ku}
K_u=\cup_{\lambda \ge 0}\lambda[0;u]_o\,.$$
The Minkowski functional $$\label{def.normu}
|x|_u=\inf\{\lambda > 0 : x\in\lambda[-u;u]_o\},$$ corresponding to the set $[-u;u]_o,$ is a seminorm on the space $X_u$ and $$\label{eq.u-sn}
|-u|_u=|u|_u =1.$$
For convenience, denote by the subscript $u$ the topological notions corresponding to the seminorm $|\cdot|_u$. Let also $B_u(x,r),\, B_u[x,r]$ be the open, respectively closed, ball with respect to $|\cdot|_u$. For $x\in X_u$ let $$\label{def.Mink}
\mathcal M_u(x)=\{\lambda > 0 : x\in\lambda[-u;u]_o\},$$ so that $$|x|_u=\inf\mathcal M_u(x).$$
Taking into account the convexity of $[-u;u]_o$ it follows that $$\label{eq-Mink-u-sn}
(|x|_u;\infty)\subset \mathcal M_u(x)\subset [|x|_u;\infty),$$ for every $x\in X_u.$
\[p1.Ku\] Let $u\in K\setminus\{0\}$ and $X_u, K_u, |\cdot|_u$ as above.
1. If $v\in K$ is linked to $u$, then $ X_u = X_v,\, K_u=K_v$ and the seminorms $|\cdot|_u, |\cdot|_v$ are equivalent. More exactly the following inequalities hold for all $x\in X_u$ $$\label{eq.p1.Ku-equiv}
|x|_u\le |v|_u |x|_v\quad\mbox{and}\quad |x|_v\le |u|_v |x|_u.$$
2. The Minkowski functional $|\cdot|_u$ is a norm on $X_u$ iff the cone $K_u$ is almost Archimedean.
3. The seminorm $|\cdot|_u$ is monotone: $x,y\in X_u$ and $0\le x\le y$ implies $|x|_u\le |y|_u.$
4. The cone $K_u$ is generating and normal in $X_u\, .$
5. For any $x\in X_u$ and $r>0,\; B_u(x,r)\subset x+r[-u;u]_o\subset B_u[x,r].$
6. The following equalities hold: $$\label{eq1.p1.Ku}
{\operatorname{aint}}(K_u) = K(u) = {\operatorname{int}}_u(K_u) .$$
7. The following are equivalent:
[(i)]{} $K_u$ is $|\cdot|_u$-closed;
[(ii)]{} $K_u$ is lineally closed;
[(iii)]{}$K_u$ is Archimedean.
In this case, $|x|_u\in\mathcal M_u(x)$ $($that is $\mathcal M_u(x)=[|x|_u;\infty))$ and $\,B_u[0,1]=[-u;u]_o.$
1\. If $v\sim u,$ then $v\in X_u$ and $u\in X_v$ which imply $X_u=X_v$ and $K_u=K_v.$ We have $$\label{eq1.p1.Ku-equiv}
\forall \alpha > |v|_u,\quad -\alpha u\le v\le \alpha u.$$
Let $x\in X_u$. If $\beta >0$ is such that $$\label{eq2.p1.Ku-equiv}
-\beta v\le x\le \beta v,$$ then $$\forall \alpha > |v|_u,\quad -\alpha\beta u \le x\le \alpha \beta u.$$
It follows $$|x|_u\le\alpha \beta,$$ for all $\beta>0$ for which is satisfied and for $\alpha >|v|_u,$ implying $|x|_u\le |v|_u |x|_v.$ The second inequality in follows by symmetry.
2\. It is known that the Minkowski functional corresponding to an absorbing absolutely convex subset $Z$ of a linear space $X$ is a norm iff the set $Z$ is radially bounded in $X$ (i.e. any ray from 0 intersects $Z$ in a bounded interval). Since a cone is almost Archimedean iff any order interval is lineally bounded (Proposition \[p.char.a-Arch\]), the equivalence follows.
3\. If $0\le x\le y,$ then $\mathcal M_u(y)\subset \mathcal M_u(x)$ and so $|y|_u=\inf\mathcal M_u(y)\ge \inf \mathcal M_u(x)=|x|_u.$
4\. The fact that $K_u$ is generating follows from definitions. The normality follows from the fact that the seminorm $|\cdot|_u$ is monotone and Theorem \[t4.char-normal-cone\].
5\. If $p$ is a seminorm corresponding to an absorbing absolutely convex subset $Z$ of a vector space $X$, then $$B_p(0,1)\subset Z\subset B_p[0,1],$$ which in our case yield $$B_u(0,1)\subset [-u;u]_o\subset B_u[0,1],$$ which, in their turn, imply the inclusions from 4.
6\. We shall prove the inclusions $$\label{eq2.p1.Ku}
{\operatorname{int}}_u(K_u)\subset{\operatorname{aint}}(K_u)\subset K(u)\subset {\operatorname{int}}_u(K_u) .$$
The first inclusion from above is a general property in topological vector spaces.
*The inclusion* $\, {\operatorname{aint}}(K_u)\subset K(u).$
For $x\in {\operatorname{aint}}(K_u)$ we have to prove the existence of $\alpha,\beta >0$ such that $$\alpha u\le x\le \beta u.$$
Since $x\in {\operatorname{aint}}(K_u)$ there exists $\alpha >0$ such that $x+tu\in K_u$ for all $t\in[-\alpha,\alpha]$ which implies $x-\alpha u\in K_u,$ that is $x\ge \alpha u.$
From and the fact that $x\in K_u$ follows the existence of $\beta >0$ such that $x\in\beta[0;u]_o,$ so that $x\le \beta u.$
*The inclusion* $\, K(u)\subset {\operatorname{int}}_u(K_u).$
If $x\in K(u),$ then there exist $\alpha,\beta>0$ such that $\alpha u\le x\le \beta u.$ But then $$B_u\big(x,\frac{\alpha}{2}\big)= x+B_u\big(0,\frac{\alpha}{2}\big)\subset x+\frac{\alpha}{2} [-u;u]_o \subset \left[\frac\alpha 2 u;\left(\beta+\frac\alpha 2\right)u\right]_o\subset K_u,$$ proving that $x$ is a $|\cdot|_u$-interior point of $K_u.$
7\. The implication (i) $\Rightarrow$ (ii) is a general property. By Proposition \[p.char.Arch\], (ii) $\iff$ (iii).
It remains to prove the implication (iii) $\Rightarrow$ (i).
Let $x\in X_u$ be a point in the $|\cdot|_u$-closure of $K_u.$ Then for every $n\in\mathbb{N}$ there exists $x_n\in K_u$ such that $|x_n-x|_u<\frac 1n.$ By the definition of the seminorm $\,|\cdot|_u, $ $$x_n-x\in \frac 1n [-u;u]_o,$$ so that $$-x\le -x+x_n\le \frac 1n\,u,$$ for all $n\in\mathbb{N}.$
By Proposition \[p.char.Arch\], this implies $-x\le 0,$ that is $x\ge 0,$ which means that $x\in K_u.$
Suppose now that the cone $K_u$ is Archimedean. For $x\in X_u\setminus\{0\}$ put $\alpha:=|x|_u>0.$ Then there exists a sequence $\alpha_n\searrow \alpha$ such that $x\in\alpha_n[-u;u]_o$ for all $n\in\mathbb{N},$ so that $$\frac{1}{\alpha_n}x+u\ge 0\quad \mbox{and}\quad -\frac{1}{\alpha_n}x+u\ge 0,$$ for all $n\in \mathbb{N}.$ Since the cone $K_u$ is lineally closed, it follows $$\frac{1}{\alpha}x+u\ge 0\quad \mbox{and}\quad -\frac{1}{\alpha}x+u\ge 0,$$ which means $x\in\alpha[-u;u]_o.$
By 4, $[-u;u]_o\subset B_u[0,1].$ If $x\in B[0,1]$ (i.e. $|x|_u\le 1$), then $\mathcal M_u(x)=[|x|_u;\infty),$ and so $$x\in |x|_u \,[-u;u]_o\subset [-u;u]_o.$$
The above construction corresponds to the one used in locally convex spaces. For a bounded absolutely convex subset $A$ of a locally convex space $(X,\tau)$ one considers the space $X_A$ generated by $A$, $$\label{def.XA}
X_A=\cup_{\lambda>0}\lambda A=\cup_{n=1}^\infty n A .$$
Then $A$ is an absolutely convex absorbing subset of $X_A$ and the attached Minkowski functional $$\label{def.pA}
p_A(x)=\inf\{\lambda >0 : x\in\lambda A\},\;\; x\in A,$$ is a norm on $X_A\,.$
\[t.B-disc\] Let $(X,\tau)$ be a Hausdorff locally convex space and $A$ a bounded absolutely convex subset of $X$.
1. The Minkowski functional $p_A$ is a norm on $X_A$ and the topology generated by $p_A$ is finer than that induced by $\tau $ (or, in other words, the embedding of $(X_A,p_A)$ in $(X,\tau)$ is continuous).
2. If, in addition, the set $A$ is sequentially complete with respect to $\tau,$ then $(X_A,p_A)$ is a Banach space. In particular, this is true if the space $X$ is sequentially complete.
In the case when $(X_A,p_A)$ is a Banach space one says that $A$ is a *Banach disc*. These spaces are used to prove that every locally convex space is an inductive limit of Banach spaces and to prove that weakly bounded subsets of a sequentially complete Hausdorff LCS are strongly bounded. (A subset of a $Y$ LCS $X$ is called *strongly bounded* if $$\sup\{|x^*(y)| : y\in Y, x^*\in M\}<\infty ,$$ for every weakly bounded subset $M$ of $X^*).$
For details concerning this topic, see the book [@Bonet §3.2], or [@Kothe §20.11].
In our case, the normality of $K$ guarantees the completeness of $(X_u,|\cdot|_u).$
\[t1.complete-Xu\] Let $(X,\tau)$ be a Hausdorff LCS ordered by a closed normal cone $K$ and $u\in K\setminus\{0\}.$
1. The functional $|\cdot|_u$ is a norm on $X_u$ and the topology generated by $|\cdot|_u$ on $X_u$ is finer than that induced by $\tau$ (or, equivalently, the embedding of $(X_u,|\cdot|_u)$ in $(X,\tau)$ is continuous).
2. If the space $X$ is sequentially complete, then $(X_u,|\cdot|_u)$ is a Banach space.
3. If $u$ is a unit in $(X,K),$ then $X_u=X.$ If $u\in{\operatorname{int}}(K),$ then the topology generated by $|\cdot|_u$ agrees with $\tau.$
By Theorem \[t2.char-normal-cone\], we can suppose that the topology $\tau$ is generated by a directed family $P$ of $\gamma$-monotone seminorms, for some $\gamma>0.$
1\. By Proposition \[p1.normal-cone-bd\], the set $[-u;u]_o$ is bounded and so $|\cdot|_u$ is a norm. We show that the embedding of $(X_u,|\cdot|_u)$ in $(X,P)$ is continuous.
Let $p\in P.$ The inequalities $-|x|_u u\le x\le |x|_u u$ imply $$0\le x + |x|_u u\le 2 |x|_u u\quad\mbox{and}\quad 0\le-x + |x|_u u\le 2 |x|_u u,$$ for all $x\in X_u$
By the $\gamma$-monotonicity of the seminorm $p$ these inequalities imply in their turn $$\begin{aligned}
2p(x)\le& p(x + |x|_u u) + p(x - |x|_u u)= p(x + |x|_u u) + p(-x + |x|_u\, u)\\
\le& 4\gamma |x|_u\, p(u).\end{aligned}$$
Consequently, for every $p\in P,$ $$\label{eq1.t1.complete-Xu}
p(x)\le 2 \gamma p(u) |x|_u ,$$ for all $x\in X_u,$ which shows that the embedding of $(X_u,|\cdot|_u)$ in $(X,\tau)$ is continuous.
2\. Suppose now that $(X,\tau)$ is sequentially complete and let $(x_n)$ be a $|\cdot|_u$-Cauchy sequence in $X_u$. By , $(x_n)$ is $p$-Cauchy for every $p\in P,$ so it is $\tau$-convergent to some $x\in X.$ By the Cauchy condition, for every $\varepsilon >0$ there is $n_0\in\mathbb{N}$ such that $|x_{n+k}-x_{n}|_u <\varepsilon,$ for all $n\ge n_0$ and all $k\in \mathbb{N}.$ By the definition of the functional $|\cdot|_u,$ it follows $$-\varepsilon u\le x_{n+k}-x_{n}\le \varepsilon u,$$ for all $n\ge n_0$ and all $k\in \mathbb{N}.$ Letting $k\to\infty,$ one obtains $$-\varepsilon u\le x-x_{n}\le \varepsilon u,$$ for all $n\ge n_0$, which implies $x\in X_u$ and $|x-x_n|_u\le \varepsilon, $ for all $n\ge n_0$. This shows that $x_n\xrightarrow{|\cdot|_u} x.$
3\. If $u$ is a unit in $(X,K)$, then the order interval $[-u;u]_o$ is absorbing, and so $X=\cup_{n=1}^\infty n\,[-u;u]_o=X_u.$
Suppose now that $u\in{\operatorname{int}}(K).$ Then $u$ is a unit in $(X,K)$, so that $X=X_u$ and, by 1, the topology $\tau_u$ generated by $|\cdot|_u$ is finer than $\tau,\; \tau\subset \tau_u.$
Since $u\in{\operatorname{int}}(K)$, there exists $p\in P$ and $r>0$ such that $B_p[u,r]\subset K.$ Let $x\in X,\, x\ne 0.$
If $p(x)=0$, then $u\pm tx\in B_p[u,r]\subset K$ for every $t>0,$ so that $-t^{-1}u\le x\le t^{-1}u$ for all $t>0,$ which implies $|x|_u=0,$ in contradiction to the fact that $|\cdot|_u$ is a norm on $X$.
Consequently, $p(x)>0$ and $u\pm p(x)^{-1}r x\in B_p[u,r]\subset K,$ that is $$-\frac{p(x)}{r}\, u\le x\le \frac{p(x)}{r}\, u,$$ and so $$|x|_u\le \frac{p(x)}{r}.$$
But then, $B_p[0,r]\subset B_{|\cdot|_u}[0,1]$, which implies $B_{|\cdot|_u}[0,1]\in \tau,$ and so $\tau_u\subset \tau.$
Incidentally, the proof of the third assertion of the above theorem gives a proof to Proposition \[p1.normal-c-TVS\].
The topology of the Thompson metric {#Ss.top-T-metric}
-----------------------------------
We shall examine some topological properties of the Thompson extended metric $d$. An extended metric $\rho$ on a set $Z$ defines a topology in the same way as a usual one, via balls. In fact all the properties reduces to the study of metric spaces formed by the components with respect to $\rho$. For instance, a sequence $(z_n)$ in $(Z,\rho)$ converges to some $z\in Z,$ iff there exists a component $Q$ with respect to $\rho$ and $n_0\in\mathbb{N}$ such that $ z\in Q,\, x_n\in Q$ for $n\ge n_0,$ and $\rho(z_n,z)\to 0$ as $n\to \infty,$ that is $(z_n)_{n\ge n_0}$ converges to $z$ in the metric space $(Q,\rho|_Q).$
The following results are immediate consequences of the definition.
\[p1.top-T-metric\] Let $X$ be a vector space ordered by a cone $K$.
1. The following inclusions hold $$\label{eq.p1.top-T-metric}
B_d(x,r)\subset[e^{-r}x;e^{r}x]_o\subset B_d[x,r].$$
If $K$ is Archimedean, then $ B_d[x,r]=[e^{-r}x;e^{r}x]_o.$
2. If $K$ is Archimedean, then the set $\,[x;\infty)_o:=\{z\in K : x\le z\}\,$ and the order interval $[x;y]_o$ are $d$-closed, for every $x\in K$ and $y\ge x.$
3. Let $x,y\in K$ with $x\le y.$ Then the order interval $[x;y]_o$ is $d$-bounded iff $x\sim y.$
1\. If $d(x,y)<r,$ then there exists $s,\, d(x,y)\le s<r,$ such that $y\in [e^{-s}x;e^{s}x]_o.$ Since $[e^{-s}x;e^{s}x]_o \subset [e^{-r}x;e^{r}x]_o,$ the first inclusion in follows. Obviously, $y\in[e^{-r}x;e^{r}x]_o$ implies $d(x,y)\le r.$
Suppose that $K$ is Archimedean and $d(x,y)=r.$ Let $t_n > r$ with $t_n\searrow r.\,$ Then $e^{-t_n}x\le y\le e^{t_n}x$ for all $n.$ Since $K$ is Archimedean, these inequalities imply $e^{-r}x\le y\le e^{r}x.$
2\. Let $z$ be in the $d$-closure of $[x;\infty)_o\,.$ Let $t_n>0,\, t_n\searrow 0.$ Then for every $n\in\mathbb{N}$ there exists $z_n\ge x$ such that $d(z,z_n)<t_n,$ implying $x\le z_n\le e^{t_n}z.$ The inequalities $x \le e^{t_n}z$ yield for $n\to \infty,\, x\le z,$ that is $z\in [x;\infty)_o.$
In a similar way one shows that $[0;x]_0$ is $d$-closed. But then, $[x;y]_o=[x;\infty)_o\cap [0;y]_o$ is also $d$-closed.
3\. If $[x;y]_o$ is bounded, then $d(x,y)<\infty,$ and so $x\sim y.$ Conversely, if $x\sim y,$ then there exist $\alpha,\beta>0$ such that $\alpha x\le y\le\beta x.$ Then, $x\le z\le y$ implies $x\le z\le y\le\beta x$, and so, by Proposition \[p4.T-metric\].1.(iii), $d(x,z)\le\ln \beta.$
\[p2.top-T-metric\] Let $X$ be a vector space ordered by a cone $K$.
1. The multiplication by scalars $\cdot:(0;\infty)\times K\to K$ and the addition $+:K\times K\to K$ are continuous with respect to the Thompson metric.
2. If $Q$ is a component of $K$, then the mapping $(\lambda,x,y)\mapsto (1-\lambda)x+\lambda y$ from $[0;1]\times Q^2$ to $Q$ is continuous with respect to the Thompson metric.
1\. Let $(\lambda_0,x_0)\in (0;\infty)\times K$. Appealing to Proposition \[p4.T-metric\].1 it follows $$\begin{aligned}
d(\lambda x,\lambda_0x_0)\le& d(\lambda x,\lambda_0x)+d(\lambda_0 x,\lambda_0x_0)\\
=&|\ln \lambda -\ln\lambda_0| +d(x,x_0) \to 0,\end{aligned}$$ if $\lambda\to \lambda_0$ and $x\xrightarrow{d}x_0$.
The continuity of the addition can be obtained from (with $\lambda =\mu =1$).
2\. Let $\lambda,\lambda_0\in[0;1]$ and $x,x_0,y,y_0\in Q.$ This time we shall appeal to the inequalities and to write $$\begin{aligned}
&d((1-\lambda) x+\lambda y,(1-\lambda_0) x_0+\lambda_0 y_0)\le\\&\le d((1-\lambda) x+\lambda y,(1-\lambda) x_0+\lambda y_0) +
d((1-\lambda) x_0+\lambda y_0,(1-\lambda_0) x_0+\lambda_0 y_0)\\
&\le\max\{d(x,x_0),d(y,y_0)\} +\ln\big(|\lambda-\lambda_0|e^{d(x_0,y_0)}+1-|\lambda-\lambda_0|\big)\to 0\end{aligned}$$ as $\lambda\to \lambda_0,\, x\xrightarrow{d}x_0$ and $y\xrightarrow{d}y_0$.
\[c.T-metric-connex\] Every component of $K$ is path connected with respect to the Thompson metric.
Follows from Proposition \[p2.top-T-metric\].2. If $x_0,x_1$ are in the same component, then $\varphi(t)=(1-t)x_0+t x_1,\, t\in [0;1],$ is a path connecting $x_0$ and $x_1.$
\[re.T-metric-connex\] The equivalence classes with respect to the equivalence $\sim$ are exactly the equivalence classes considered by Jung [@jung69] (the equivalence relation considered by Jung is $x \simeq y \iff d(x,y)<\infty,$ see Remark \[re.extended-T-metric\]). Since these classes are both open and closed, it follows that the components of $K$ with respect to $\sim$ are, in fact, the connected components of $K$ with respect to to the Thompson (extended) metric $d$.
In the following proposition we give a characterization of $d$-convergent monotone sequences.
\[p.T-converg-seq\] Let $X$ be a vector space ordered by an Archimedean cone $K$.
1. If $(x_n)$ is an increasing sequence in $K$, then $(x_n)$ is $d$-convergent to an $x\in K$ iff $$\begin{aligned}
{\rm(i)}\quad &\forall n\in \mathbb{N},\quad x_n\le x,\quad\mbox{and}\\
{\rm(ii)}\quad &\forall \lambda>1,\;\exists k\in \mathbb{N},\quad x\le\lambda x_k.
\end{aligned}$$ In this case, $x=\sup_nx_n$ and there exists $k\in\mathbb{N}$ such that $x_n\in K(x)$ for all $n\ge k.$
2. If $(x_n)$ is a decreasing sequence in $K$, then $(x_n)$ is $d$-convergent to an $x\in K$ iff $$\begin{aligned}
{\rm(i)}\quad &\forall n\in \mathbb{N},\quad x\le x_n,\quad\mbox{and}\\
{\rm(ii)}\quad &\forall \lambda\in(0;1),\;\exists k\in \mathbb{N},\quad x\ge\lambda x_k.
\end{aligned}$$ In this case, $x=\inf_nx_n$ and there exists $k\in\mathbb{N}$ such that $x_n\in K(x)$ for all $n\ge k.$
3. Let $(X,\tau)$ be a TVS ordered by a cone $K$. If $(x_n)$ is a $d$-Cauchy sequence in $K$ which is $\tau$-convergent to $x\in K$, then $x_n\xrightarrow{d}x.$
We shall prove only the assertion 1, the proof of 2 being similar.
Suppose that the condition (i) and (ii) hold and let $\varepsilon>0.$ Then $\lambda:=e^\varepsilon>1,$ so that, by (ii), there exists $k\in\mathbb{N}$ such that $x\le \lambda x_k=e^{\varepsilon} x_k.$ Taking into account the monotony of the sequence $(x_n)$ it follows that $$e^{-\varepsilon} x_n\le x_n\le x\le e^{\varepsilon} x_n,$$ for all $n\ge k$, which implies $d(x,x_n)\le\varepsilon$ for all $n\ge k,$ that is $x_n\xrightarrow{d}x$ as $n \to \infty.$
Conversely, suppose that $(x_n)$ is an increasing sequence in $K$ which is $d$-convergent to $x\in K.$ For $\lambda >1$ put $\varepsilon:=\ln\lambda>0.$ Then there exists $k\in\mathbb{N}$ such that $$\forall n\ge k,\;\; d(x,x_n)<\varepsilon,$$ which implies $$\forall n\ge k,\;\; e^{-\varepsilon} x_n \le x\le e^{\varepsilon} x_n.$$
By the second inequality above, $x\le \lambda x_k,$ which shows that (i) holds. Since $(x_n)$ is increasing the first inequality implies that for every $n\in\mathbb{N}$ $$e^{-\varepsilon} x_n \le x ,$$ for all $\varepsilon >0. $ By Proposition \[p.char.Arch\], the cone $K$ is lineally closed, so that the above inequality yields for $\varepsilon\searrow 0,\, x_n\le x$ for all $n\in\mathbb{N},$ that is (i) holds too.
It is clear that if $x_n\xrightarrow{d}x$, then there exists $k\in\mathbb{N}$ such that $d(x,x_n)\le 1<\infty,$ for all $n\ge k,$ which implies $x_n\in K(x)$ for all $n\ge k.$
It remains to show that $x=\sup_nx_n.$ Let $y$ be an upper bound for $(x_n)$. Then for every $n\in\mathbb{N},$ $$\label{eq.p.T-converg-seq}
x_n\le x_{n+k}\le y \iff x_{n+k}\in [x_n;y]_o,$$ for all $k\in \mathbb{N}.$ By Proposition \[p1.top-T-metric\].2 the interval $[x_n;y]_o$ is $d$-closed, so that, letting $k\to\infty$ in it follows $x\in[x_n;y]_o.$ The inequality $x\le y$ shows that $x=\sup_nx_n.$
3\. It follows that $(x_n)$ is eventually contained in a component $Q$ of $K,$ so we can suppose $x_n\in Q,\, n\in\mathbb{N}.$ Since $(x_n)$ is $d$-Cauchy, there exists $n_0$ such that $d(x_n,x_{n_0})<1$ for all $n\ge n_0.$ Then $e^{-1}x_{n_0}\le x_n\le e x_{n_0},$ for all $n\ge n_0.$ Letting $n\to \infty,$ one obtains $e^{-1}x_{n_0}\le x\le e x_{n_0},$ which shows that $x\sim x_{n_0},$ that is $x\in Q.$ Now for $\varepsilon >0$ there exists $n_\varepsilon\in\mathbb{N}$ such that $d(x_{n+k},x_{n})<\varepsilon$ for all $n\ge n_\varepsilon$ and all $k\in\mathbb{N}$. Then for every $n\ge n_\varepsilon,\,$ $e^{-\varepsilon}x_{n}\le x_{n+k}\le e^{\varepsilon} x_{n},$ for all $k\in\mathbb{N}.$ Letting $k\to \infty,$ one obtains $e^{-\varepsilon}x_{n}\le x\le e^{\varepsilon} x_{n},$ implying $d(x_n,x)\le\varepsilon$ for all $n\ge n_\varepsilon$, that is $x_n\xrightarrow{d} x.$
The Thompson metric and order–unit seminorms
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The main aim of this subsection is to show that the Thompson metric and the metric seminorms are equivalent on each component of $K$. We begin with some inequalities.
\[p1.T-metric-u-sn\] Let $X$ be a vector space ordered by a cone $K, u\in K\setminus\{0\}$ and $x,y\in K(u).$ The following relations hold.
1. $d(x,y)=\ln(\max\{|x|_y, |y|_x\}).$
2. $d(x,y)\ge|\ln|x|_u-\ln|y|_u|,$ or, equivalently,
[(ii)]{}$|x|_u\le e^{d(x,y)}|y|_u\quad\mbox{and}\quad |y|_u\le e^{d(x,y)}|x|_u.$
3. $e^{-d(x,u)}\le |x|_u\le e^{d(x,u)}.$
4. $|u|_x\le e^{d(x,y)}|u|_y .$
5. $d(x,y)\le \ln\big(1+|x-y|_u\cdot\max\{|u|_x,|u|_y\}\big).$
6. $\big(e^{d(x,y)}-1\big)\cdot\min\{|u|_x^{-1},|u|_y^{-1}\}\le |x-y|_u\le
\big(2e^{d(x,y)}+ e^{-d(x,y)}-1\big)\cdot \min\{|x|_u,|y|_u\}.$
7. $\big(1-e^{-d(x,u)}\big)\cdot\max\{|u|_x^{-1},|u|_y^{-1}\}\le |x-y|_u.$
8. $ |x-y|_x\ge 1-e^{-d(x,y)}$ .
1\. Recalling , it is easy to check that $$s\in\sigma(x,y) \iff e^s\in\mathcal M_x(y)\cap \mathcal M_y(x),$$ and so $$d(x,y)=\ln\big(\inf\big\{\mathcal M_x(y)\cap \mathcal M_y(x)\big\}\big)=\ln(\max\{|x|_y,|y|_x\}).$$
2\. By , $\,|x|_u\le|y|_u|x|_y.$ Taking into account 1, it follows $$d(x,y)\ge \ln|x|_y\ge \ln|x|_u-\ln|y|_u.$$
By symmetry, $\,d(x,y) \ge \ln|y|_u-\ln|x|_u,$ so that 2.(i) holds. It is obvious that (i) and (ii) are equivalent.
3\. Taking $y:= u$ in both the inequalities from 2.(ii), one obtains $$|x|_u\le e^{d(x,u)}|y|_u\quad\mbox{and}\quad 1\le e^{d(x,u)}|x|_u\,.$$
4\. By , $\,|u|_x\le|u|_y|y|_x$ and, by 3, $|y|_x\le e^{d(x,y)},$ hence $|u|_x\le e^{d(x,y)}|u|_y.$
5\. By and the triangle inequality $$\begin{aligned}
&|y|_x\le |x|_x+|x-y|_x\le 1+|x-y|_u|u|_x\, ,\quad \mbox{and}\\
&|x|_y\le |y|_y+|x-y|_y\le 1+|x-y|_u|u|_y\, ,\end{aligned}$$ so that $$\max\{|x|_y,|y|_x\}\le 1+|x-y|_u\cdot\max\{|u|_x,|u|_y\}\, .$$
The conclusion follows from 1.
6\. The inequality 6 can be rewritten as $|x-y|_u\max\{|u|_x,|u|_y\}\ge e^{d(x,y)}-1,$ so that $$|x-y|_u\ge \big(e^{d(x,y)}-1\big)\big[\max\{|u|_x,|u|_y\}\big]^{-1}=\big(e^{d(x,y)}-1\big)\cdot \min\{|u|_x^{-1},|u|_y^{-1}\} .$$
To prove the second inequality, take $s\in\sigma(x,y)$ arbitrary. Then $-(e^s-1)x\le x-y\le (1-e^{-s}s)x,$ so that $0\le x-y+(e^s-1)x\le (e^s-^{-s})x.$ The monotony of $|\cdot|_u$ and the triangle inequality imply $$|x-y|_u-(e^s-1)|x_u\le |x-y-(e^s-1) (e^s-e^{-s})x|_u\le (e^s-e^{-s})|x|_u,$$ so that $\, |x-y|_u\le \big(2e^s+e^{-s}-1\big)|x|_u. $ Since this holds for every $s\in \sigma(x,y)$ it follows $$|x-y|_u\le \big(2e^{d(x,y)}+ e^{-d(x,y)}-1\big) |x|_u .$$
By interchanging the roles of $x$ and $y$ in the above inequality, one obtains $$|x-y|_u\le \big(2e^{d(x,y)}+ e^{-d(x,y)}-1\big) |y|_u .$$
These two inequalities imply the second inequality in 6.
7\. By 4, $$|u|_x^{-1}\ge e^{-d(x,y)}|u|_y^{-1}\quad\mbox{and}\quad |u|_y^{-1}\ge e^{-d(x,y)}|u|_x^{-1},$$ so that $$\min\big\{|u|_x^{-1},|u|_y^{-1}\big\}\ge e^{-d(x,y)}\max\big\{|u|_x^{-1},|u|_y^{-1}\big\}.$$
The conclusion follows by 6.
8\. This can be obtained by taking $u:=x$ in 7.
\[t1.T-metric-u-sn\] Let $X$ be a vector space ordered by a cone $K$ and $ u\in K\setminus\{0\}$. Then the Thompson metric and the $u$-seminorm are topologically equivalent on $K(u).$
We have to show that $d$ and $|\cdot|_u$ have the same convergent sequences, that is $$x_n\xrightarrow{d}x \iff x_n\xrightarrow{|\cdot|_u} x ,$$ for any sequence $(x_n)$ in $K(u)$ and any $x\in K(u)$. But, by Proposition \[p1.Ku\].1, $$x_n\xrightarrow{|\cdot|_u} x\iff x_n\xrightarrow{|\cdot|_x} x,$$ hence we have to prove the equivalence $$\label{eq.t1.T-metric-u-sn}
x_n\xrightarrow{d}x \iff x_n\xrightarrow{|\cdot|_x} x .$$
Suppose that $x_n\xrightarrow{d}x .$ By Proposition \[p1.T-metric-u-sn\].6 $$|x_n-x|_x\le 2e^{d(x_n,x)}+ e^{-d(x_n,x)}-1\to 0\quad\mbox{as}\quad n\to \infty,$$ showing that $x_n\xrightarrow{|\cdot|_x} x .$
Conversely, if $x_n\xrightarrow{|\cdot|_x} x, $ then by Proposition \[p1.T-metric-u-sn\].8 , $$|x_n-x|_x\ge 1-e^{-d(x_n,x)},$$ which implies $d(x_n,x)\to 0.$
\[re.T-metric-u-semin\] The seminorm $|\cdot|_u$ and the metric $d$ are not metrically equivalent on $X_u.$ Take, for instance, $U:=[0;u]_o\cap K(u).$ Then $|x|_u\le 1$ for every $x\in U$. But $U$ is not $d$-bounded because $e^{-n}u$ belongs to $U$ for all $n\in \mathbb{N},$ and $d(x_n,u)=n\to\infty$ for $n\to \infty.$
\[c.T-metric-tau\] Let $K$ be a solid normal cone in a Hausdorff LCS $(X,\tau).$ Then the topology generated by $d$ on ${\operatorname{int}}K$ agrees with the restriction of $\tau $ to ${\operatorname{int}}K$.
Let $u\in {\operatorname{int}}K.$ By Theorem \[t1.complete-Xu\], $X_u=X$ and the topology generated by $|\cdot|_u$ agrees with $\tau,$ that is $|\cdot|_u$ is a norm on $X$ generating the topology $\tau.$ Since $K(u)={\operatorname{int}}K,$ Theorem \[t1.T-metric-u-sn\] implies that $d$ and $|\cdot|_u$ are topologically equivalent on $K(u).$
Completeness properties {#S.Completeness}
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Self-bounded sequences and self-complete sets in a cone
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Let $X$ be a vector space ordered by a cone $K$. A sequence $(x_n)$ in $K$ is called:
*self order–bounded from above* (or *upper self-bounded*) if for every $\lambda >1$ there exists $k\in\mathbb{N}$ such that $x_n\le\lambda x_k$ for all $n\ge k$.
*self order–bounded from below* (or *lower self-bounded*) if for every $\mu\in(0;1)$ there exists $k\in\mathbb{N}$ such that $x_n\ge\mu x_k$ for all $n\ge k$.
*self order–bounded* (or, simply, *self-bounded*) if is self order–bounded both from below and from above.
If the sequence $(x_n)$ is increasing, then it is self order–bounded from above iff for every $\lambda>1$ there exists $k\in\mathbb{N}$ such that $\lambda x_k$ is an upper bound for the sequence $(x_n)$.
Similarly, if the sequence $(x_n)$ is decreasing, then it is self order–bounded from below iff for every $\mu\in(0;1)$ there exists $k\in\mathbb{N}$ such that $\mu x_k$ is an lower bound for the sequence $(x_n)$.
The following propositions put in evidence some connections between self order bounded sequences and $d$-Cauchy sequences.
\[p1.s-bd-seq\] Let $X$ be a vector space ordered by a cone $K$.
1. Any $d$-Cauchy sequence in $K$ is self-bounded.
2. An increasing sequence in $K$ is upper self-bounded iff it is $d$-Cauchy.
3. A decreasing sequence in $K$ is lower self-bounded iff it is $d$-Cauchy.
1\. Let $(x_n)$ be a $d$-Cauchy sequence in $K$. If $\lambda>1,$ then for $\varepsilon:=\ln\lambda>0$ there exists $k\in \mathbb{N}$ such that $$e^{-\varepsilon}x_n\le x_k\ge e^{\varepsilon}x_n\iff \lambda^{-1}x_n\le x_k\ge \lambda x_n\,,$$ for all $n\ge k.$ Consequently $x_n\le \lambda x_k, $ for all $n\ge k,$ proving that $(x_n)$ is upper self–bounded.
The lower self-boundedness of $(x_n)$ is proved similarly, taking $\varepsilon'=-\ln \mu,$ for $\mu\in(0;1).$
2\. It suffices to prove that an increasing upper self-bounded sequence is $d$-Cauchy. For $\varepsilon >0$ let $\lambda=e^\varepsilon$ and $k\in\mathbb{N}$ such that $x_n\le\lambda x_k$ for all $n\ge k.$ It follows that $$e^{-\varepsilon}x_{m}\le x_m\le x_n\le \lambda x_k\le\lambda x_m=e^{-\varepsilon}x_m\, ,$$ for all $n\ge m\ge k$. Consequently, $d(x_n,x_m)\le \varepsilon,$ for all $n\ge m\ge k$, which shows that the sequence $(x_n)$ is $d$-Cauchy.
The proof of 3 is similar to the proof of 2, so we omit it.
\[p2.s-bd-seq\] Let $X$ be a vector space ordered by an Archimedean cone $K$ and $(x_n)$ an increasing sequence in $K$. The following statements are equivalent.
1. The sequence $(x_n)$ is $d$-convergent.
2. The sequence $(x_n)$ is $d$-Cauchy and has a supremum.
3. The sequence $(x_n)$ is upper self-bounded and has a supremum.
In the affirmative case $x_n\xrightarrow{d}\sup_nx_n.$
1$\Rightarrow$2 Follows from Proposition \[p.T-converg-seq\].
2$\Rightarrow$3. Follows from Proposition \[p1.s-bd-seq\].1
3$\Rightarrow$1. If $x=\sup_nx_n,$ then $x_n\le x$ for all $n\in\mathbb{N},$ showing that condition (i) from Proposition \[p.T-converg-seq\].1 holds. Now let $\lambda>1.$ Since $(x_n)$ is upper self-bounded there exists $k\in\mathbb{N}$ such that $x_n\le \lambda x_k$ for all $n\ge k,$ and so $x_n\le \lambda x_k$ for all $n\in\mathbb{N}$ (because $(x_n)$ is increasing). But then $x=\sup_nx_n\le \lambda x_k,$ which shows that condition (ii) of the same proposition is also fulfilled. Consequently $x_n\xrightarrow{d}x.$
The last assertion follows by the same proposition.
Similar equivalences, with similar proofs, hold for decreasing sequences.
\[p3.s-bd-seq\] Let $X$ be a vector space ordered by an Archimedean cone $K$ and $(x_n)$ a decreasing sequence in $K$. The following statements are equivalent.
1. The sequence $(x_n)$ is $d$-convergent.
2. The sequence $(x_n)$ is $d$-Cauchy and has an infimum.
3. The sequence $(x_n)$ is lower self-bounded and has an infimum.
In the affirmative case $x_n\xrightarrow{d}\inf_nx_n.$
The following proposition emphasizes a kind of duality between upper and lower self-bounded sequences. If $Y$ is a subset of an ordered set $X, $ then one denotes by $\sup\!|_Y A$ ($\inf\!|_Y A$) the supremum (resp. infimum) in $Y$ of a subset $A$ of $Y$. This may differ from the supremum (resp. infimum) of the set $A$ in $X$.
\[p4.s-bd-seq\] Let $X$ be a vector space ordered by a cone $K,$ $(x_n)$ an increasing, upper self-bounded sequence in $K$, and $(t_k)$ a decreasing sequence of real numbers, convergent to 1. Then there exists a subsequence $(x_{n_k})$ of $(x_n)$ such that the following conditions are satisfied.
1. The sequence $(y_k)$ given by $y_k=t_kx_{n_k},\, k\in\mathbb{N},\, $ is decreasing and lower self-bounded and $$\label{eq1.p4.s-bd-seq}
\forall n,k\in\mathbb{N},\quad x_n\le y_k.$$
2. If the cone $K$ is Archimedean, $x$ is an upper bound for $(x_n)$ and $y$ is a lower bound for $(y_k)$, then $y\le x.$
3. If the cone $K$ is Archimedean and $(x_n)$ lies in a vector subspace $Y$ of $X$, then the following statements are equivalent.
- $(x_n)$ has suppremum; $(y_k)$ has an infimum;
- $(x_n)$ has suppremum in $Y$; $(y_k)$ has an infimum in $Y$;
- there exists $x\in K$ such that $$\label{eq2.p4.s-bd-seq}
\forall n,k\in\mathbb{N},\quad x_n\le x\le y_k.$$
In the affirmative case $$\sup\{x_n : n\in \mathbb{N}\}= \sup\!|_Y\{x_n : n\in \mathbb{N}\}=\inf\{y_k : k\in \mathbb{N}\}= \inf\!|_Y\{y_k : k\in \mathbb{N}\}=x,$$ and $x_n\xrightarrow{d}x$ and $y_k\xrightarrow{d}x.$
1\. Since $(t_k)$ is decreasing, $\lambda_k:=t_k/t_{k+1}>1,\,k\in \mathbb{N}.$ The upper self-boundedness of the sequence $(x_n)$ implies the existence of $n_1\in\mathbb{N}$ such that $$\label{eq3.p4.s-bd-seq}
\forall n\ge n_1,\quad x_n\le\lambda_1x_{n_1}.$$
Since $(x_n)$ is increasing, the inequalities hold for all $n\in\mathbb{N}$. If $m_2\in \mathbb{N}$ is such that $x_n\le\lambda_2x_{m_2}$ for all $n\in\mathbb{N},$ then $n_2:=1+\max\{n_1,m_2\}>n_1$ and $x_n\le \lambda_2x_{n_2}$ for all $n\in\mathbb{N}.$ Continuing in this way one obtains a sequence of indices $n_1<n_2 < \dots$ such that $$\label{eq4.p4.s-bd-seq}
\quad x_n\le\lambda_kx_{n_k},$$ for all $ n\in \mathbb{N}$ and all $k\in \mathbb{N}.$
Let $y_k:=t_kx_{n_k},\, k\in\mathbb{N}.$ Putting $n=n_{k+1}$ in it follows $y_{k+1}\le y_k.$ By the same inequality $$x_n\le t_{k+1}x_n\le t_kx_{n_k}=y_k,$$ for all $n,k\in\mathbb{N}.$
Let now $\mu\in(0;1)$. Since $t_k\to 1$ there exists $k_0$ such that $t_{k_0}<\mu^{-1}.$ But then, by , $$\mu y_{k_0}\le t_{k_0}^{-1}y_{k_0}=x_{n_{k_0}}\le y_k,$$ for all $k\in\mathbb{N},$ proving that $(y_k)$ is lower self-bounded.
2\. Suppose that $x_n\le x,\, n\in\mathbb{N},$ and $y_k\ge y,\, n\in\mathbb{N}.$ Then, for all $k\in\mathbb{N},$ $$y\le y_k=t_kx_{n_k}\le t_k x\, .$$
Since $K$ is Archimedean and $t_k\to 1,$ the inequalities $y\le t_kx,\, k\in\mathbb{N},$ yield for $k\to \infty,\, y\le x.$
3\. The implications (a)$\Rightarrow$(b) and (c)$\Rightarrow$(d) are obvious.
Let as prove (b)$\Rightarrow$(d). Observe first that $y_k\in Y,\, k\in\mathbb{N}.$ Let $x=\sup_Y\{x_n : n\in\mathbb{N}\}.$ By $y_k$ is an upper bound for $(x_n),$ for every $k\in\mathbb{N}, $ so that $\,x\le y_k\,,$ for all $k\in\mathbb{N}.$ If $y\in Y$ is such that $y\le y_k$ for all $k\in\mathbb{N},$ then, by 2, $y\le x,$ proving that $x=\inf_Y\{y_k : k\in\mathbb{N}\}.$ On the way we have shown that $x_n\le x\le y_k$, for all $n,k\in\mathbb{N},$ that is the implication (b)$\Rightarrow$(e) holds too.
Similar reasonings show that (d)$\Rightarrow$(b), that is (b)$\iff$(d). The equivalence (a)$\iff$(c) can be proved in the same way (just let $Y:=X$).
Finally, let us show that (e)$\Rightarrow$(c). Assume that for some $x\in K,\, x_n\le x\le y_k$ for all $n,k\in\mathbb{N}.$ Suppose that $y\in K$ is such that $y\le y_k $ for all $k\in\mathbb{N}.$ Then, by 2, these inequalities imply $y\le x,$ showing that $x =\inf_ky_k.$ (Similar arguments show that $x=\sup_ny_n,$ that is (e)$\Rightarrow$(a)). The equivalence of the assertions from 3 is (over) proven.
The last assertions of the proposition follow from Propositions \[p2.s-bd-seq\] and \[p3.s-bd-seq\].
Similar results, with similar proofs, hold for decreasing lower self-bounded sequences.
\[p5.s-bd-seq\] Let $X$ be a vector space ordered by a cone $K,$ $(x_n)$ a decreasing, lower self-bounded sequence in $K$, and $(t_k)$ an increasing sequence of real numbers, convergent to 1. Then there exists a subsequence $(x_{n_k})$ of $(x_n)$ such that following conditions are satisfied.
1. The sequence $(y_k)$ given by $y_k=t_kx_{n_k},\, k\in\mathbb{N},\, $ is increasing and upper self-bounded and $$\label{eq1.p5.s-bd-seq}
\forall n,k\in\mathbb{N},\quad x_n\ge y_k.$$
2. If the cone $K$ is Archimedean, $x$ is a lower bound for $(x_n)$ and $y$ is an upper bound for $(y_k)$, then $y\ge x.$
3. If the cone $K$ is Archimedean and $(x_n)$ lies in a vector subspace $Y$ of $X$, then the following statements are equivalent.
- $(x_n)$ has an infimum; $(y_k)$ has a supremum;
- $(x_n)$ has an infimum in $Y$; $(y_k)$ has a supremum in $Y$;
- there exists $x\in K$ such that $$\label{eq2.p5.s-bd-seq}
\forall n,k\in\mathbb{N},\quad x_n\ge x\ge y_k.$$
In the affirmative case $$\inf\{x_n : n\in \mathbb{N}\}= \inf\!|_Y\{x_n : n\in \mathbb{N}\}=\sup\{y_k : k\in \mathbb{N}\}= \sup\!|_Y\{y_k : k\in \mathbb{N}\}=x,$$ and $x_n\xrightarrow{d}x$ and $y_k\xrightarrow{d}x.$
The following notions will play a crucial role in the study of completeness of the Thompson metric.
Let $X$ be a vector space ordered by a cone $K$. A nonempty subset $U$ of $K$ is called:
*self order–complete from above* (or *upper self-complete*) if every increasing self-bounded sequence $(x_n)$ in $U$ has a supremum and $\sup_n x_n \in U.$
*self order–complete from below* (or *lower self-complete*) if every decreasing self-bounded sequence $(x_n)$ in $U$ has an infimum and $\inf_nx_n \in U.$
*self order–complete* (or, simply, *self-complete*) if it is self order–complete both from below and from above.
If we do not require the supremum (resp. infimum) to be in $U$, then we say that $U$ is *quasi upper* (resp. *lower*) *self-complete*.
The duality results given in Propositions \[p4.s-bd-seq\] and \[p5.s-bd-seq\] have the following important consequence.
\[t1.s-complete\] Let $X$ be a vector space ordered by an Archimedean cone $K.$ If $U$ is an order–convex, strictly positively-homogeneous, nonempty subset of $K$, then all six completeness properties given in the above definitions are equivalent.
It is a simple observation that the stated equivalences hold if we show that self-completeness is implied by each of the conditions of quasi upper self-completeness and quasi lower self-completeness.
Assume that $U$ is quasi upper self-complete and show first that $U$ is upper self-complete.
Let $(x_n)$ be an increasing upper self-bounded sequence in $U.$ By hypothesis, there exists $x:=\sup_nx_n\in K.$ Also there exists $k\in\mathbb{N}$ such that $x_n\le 2 x_k$ for all $n\in\mathbb{N}.$ Consequently $x_k\le x\le 2x_k.$ Since $x_k$ and $2x_k$ belong to $U$ and $U$ is order–convex, $x\in U$, proving that $U$ is upper self-complete.
Let us show now that $U$ is lower self-complete too. Suppose that $(x_n)$ is a decreasing lower self-bounded sequence $(x_n)$ in $U$ and let $(t_k)$ be an increasing sequence of positive numbers which converges to 1 (e.g., $t_k=1-\frac{1}{2k})$. By Proposition \[p5.s-bd-seq\] there exists a subsequence $(x_{n_k})$ of $(x_n)$ such that the sequence $y_k:=t_kx_{n_k},\, k\in\mathbb{N},\,$ is increasing and upper self-bounded. Since we have shown that $U$ is upper self-complete, there exists $x:=\sup_ky_k\in U.$ By the last part of the same proposition, $\inf_nx_n=x\in U,$ proving that $U$ is lower self-complete.
When $U$ is quasi lower self-complete, the proof that $U$ is self-complete follows the same steps as before, using Proposition \[p4.s-bd-seq\] instead of Proposition \[p5.s-bd-seq\].
The following corollary shows that we can restrict to order–convex subspaces of $X$.
\[c1.s-complete\] Let $X$ be a vector space ordered by an Archimedean cone $K$ and $Y$ an order–convex vector subspace of $X$. If $U$ is an order–convex, strictly positively-homogeneous, nonempty subset of $Y\cap K\,$, then $U$ is self-complete in X iff $U$ is self-complete in $Y$.
For a lineally solid cone $K,$ the self-completeness is equivalent to the self-completeness of its algebraic interior.
\[p6.s-complete\] Let $X$ be a vector space ordered by an Archimedean cone $K$.
1. The cone $K$ is self-complete iff every component of $K$ is self-complete.
2. If, in addition, $K$ is lineally solid and ${\operatorname{aint}}(K)$ is self-complete then $K$ is self-complete.
1\. Suppose that $K$ is self-complete. Then any component $Q$ of $K$ is quasi upper self-complete. By Proposition \[p1.equiv-cone\], $Q$ satisfies the hypotheses of Theorem \[t1.s-complete\], so that it is self-complete.
Conversely, suppose that every component of $K$ is self-complete and let $(x_n)$ be an increasing upper self-bounded sequence in $K$. By Proposition \[p1.s-bd-seq\] the sequence $(x_n)$ is $d$-Cauchy, so there exists $k\in\mathbb{N}$ such that $d(x_k,x_n)\le 1 <\infty, $ for all $n\ge k,$ implying that the set $\{x_n : n\ge k\}$ is contained in a component $Q$ of $K$. By the self-completeness of $Q$ there exists $x:=\sup\{x_n : n\ge k\}\in Q.$ Since the sequence $(x_n)$ is increasing it follows $x=\sup_nx_n.$ Consequently, $K$ is upper self-complete and, by Theorem \[t1.s-complete\], self-complete.
2\. Let $(x_n)$ be an increasing, upper self-bounded sequence in $K$. Fix $x\in {\operatorname{aint}}(K).$ Then, by Remark \[re.a-int\], the sequence $y_n:=x_n+x,\, n\in\mathbb{N},$ is contained in ${\operatorname{aint}}(K)$ and it is obviously increasing and upper self-bounded. Consequently, $(y_n)$ has a supremum, $y:=\sup_ny_n\in {\operatorname{aint}}(K).$ But then there exists $\sup_nx_n=y-x.$ Therefore the cone $K$ is upper self-complete and, by Theorem \[t1.s-complete\], it si self-complete.
\[re.s-complete\] All the results proven so far can be restated into local versions, by replacing $X $ with $X_u$, hence $K$ with $K_u$ (where $u\in K \setminus\{0\}$). In this way, we can weaken the Archimedean condition by requiring only that $K_u$ is Archimedean. In this case, the conditions “has a supremum", respectively “has an infimum" must be understood with respect to $X_u$. Consequently, a subset $U$ of $K_u$ can be self-complete in $X_u$, but may be not self-complete in $X$ (yet, by Corollary \[c1.s-complete\], this cannot happen when $K$ is Archimedean). Note that the definition of the Thompson metric is not affected by this change (see Remark \[re2.T-metric\]).
Properties of monotone sequences with respect to order–unit seminorms {#Ss.seq-u-semin}
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In this subsection we shall examine the behavior of monotone sequences with respect to to order–unit seminorms, considered in Subsection \[Ss.u-semin\]. The results are analogous to those established in Subsection \[Ss.top-T-metric\] for the Thompson metric.
Throughout this subsection $X$ will be a vector space ordered by a cone $K,\, u\in K\setminus\{0\}$, and $X_u,K_u$ are as in Subsection \[Ss.u-semin\]. We shall assume also that the cone $K_u=X_u\cap K$ is Archimedean.
\[p1.seq-u-semin\] Let $(x_n)$ be an increasing sequence in $X_u$ and $x\in X_u\,.$
1. The sequence $(x_n)$ is $|\cdot|_u$-Cauchy iff $$\label{eq1.p1.seq-u-semin}
\forall \varepsilon>0,\; \exists k\in\mathbb{N},\;\forall n\in\mathbb{N},\quad x_n\le x_k+\varepsilon u.$$
2. The sequence $(x_n)$ is $|\cdot|_u$-convergent to $x\in X_u$ iff $$\begin{aligned}
{\rm(i)}\quad &\forall n\in\mathbb{N},\quad x_n\le x;\\
{\rm(ii)}\quad &\forall \varepsilon>0,\; \exists k\in\mathbb{N},\quad\mbox{such that}\quad x\le x_k+\varepsilon u.
\end{aligned}$$ In the affirmative case, $x=\sup\!|_{X_u}\{x_n :n\in\mathbb{N}\}.$ If $K$ is Archimedean, then $x=\sup_nx_n.$
3. The sequence $(x_n)$ is $|\cdot|_u$-convergent to $x\in X_u$ iff it is $|\cdot|_u$-Cauchy and has a supremum in $X_u$. In the affirmative case $x_n\xrightarrow{|\cdot|_u}\sup\!|_{X_u}\{x_n :n\in\mathbb{N}\}.$
1\. The sequence $(x_n)$ is $|\cdot|_u$-Cauchy iff $$\label{eq2.p1.seq-u-semin}
\forall \varepsilon>0,\; \exists k\in\mathbb{N},\; \forall n\ge m\ge k,\quad -\varepsilon u\le x_n-x_m\le\varepsilon u.$$
Suppose that $(x_n)$ is $|\cdot|_u$-Cauchy and for $\varepsilon>0$ let $k$ be given by the above condition. Taking $m=k$ in the right inequality, one obtains $ x_n\le x_k+\varepsilon u$ for all $n\ge k,$ and so for all $n\in \mathbb{N}.$
Suppose now that $(x_n)$ satisfies . For $\varepsilon >0$ let $k$ be chosen according to this condition. By the monotony of $(x_n),\, x_n-x_m\ge 0\ge-\varepsilon u,$ for all $n\ge m,$ and so the left inequality in is true. By and the monotony of $(x_n)$, $$x_n\le x_k+\varepsilon u\le x_m+\varepsilon u,$$ for all $m\ge k,$ so that the right inequality in holds too.
2\. We have $x_n\xrightarrow{|\cdot|_u}x $ iff $$\forall \varepsilon>0,\; \exists k\in\mathbb{N},\; \forall n\ge k,\quad -\varepsilon u\le x-x_n\le\varepsilon u.$$
The left one of the above inequalities implies $x_n\le x+\varepsilon u$ for all $n\ge k,$ and so, by the monotony of $(x_n)$, for all $n\in\mathbb{N}.$ Since $K_u$ is Archimedean, letting $\varepsilon\searrow 0$ it follows $x_n\le x,\, $ for every $n\in\mathbb{N}.$ The right inequality implies $x\le x_k+\varepsilon u.$
Conversely, suppose that (i) and (ii) hold. For $\varepsilon>0$ choose $k$ according to (ii). Then, by the monotony of $(x_n)$, $$x\le x_k+\varepsilon u\le x_n+\varepsilon u ,$$ and so $x-x_n\le \varepsilon u,$ for all $n\ge k.$ By (i), $x_n\le x\le x+\varepsilon u,$ and so $x-x_n\ge -\varepsilon u$ for all $n\in\mathbb{N}.$ Consequently, $-\varepsilon u\le x-x_n\le \varepsilon u $ for all $n\ge k.$
By Proposition \[p1.Ku\].7, the cone $K_u$ is $|\cdot|_u$-closed, and so, by Proposition \[p2.order-tvs\], $x_n\xrightarrow{|\cdot|_u}x $ implies $x=\sup\!|_{X_u}\{x_n : n\in\mathbb{N}\}.$
Suppose now that the cone $K$ is Archimedean and that $y\in X$ is such that $x_n\le y$ for all $n\in\mathbb{N}.$ By (ii) for every $\varepsilon>0$ there exists $k\in\mathbb{N}$ such that $x\le x_k+\varepsilon u,$ hence $x \le y+\varepsilon u.$ Letting $\varepsilon\searrow 0$ one obtains $x\le y,$ which proves that $x=\sup_nx_n.$
The direct implication in 3 follows from 1 and 2. Suppose that $(x_n)$ is $|\cdot|_u$-Cauchy and let $x=\sup\!|_{X_u}\{x_n :n\in\mathbb{N}\}.$ For $\varepsilon >0$ there exists $k\in\mathbb{N}$ such that $x_n\le x_k+\varepsilon u$ for all $n\in\mathbb{N},$ implying $x\le x_k+\varepsilon u.$ Taking into account 2, it follows $x_n\xrightarrow{|\cdot|_u}x. $
As before, similar results, with similar proofs, hold for decreasing sequences.
\[p2.seq-u-semin\] Let $(x_n)$ be a decreasing sequence in $X_u$ and $x\in X_u\,.$
1. The sequence $(x_n)$ is $|\cdot|_u$-Cauchy iff $$\label{eq1.p2.seq-u-semin}
\forall \varepsilon>0,\; \exists k\in\mathbb{N}, \;\forall n\in\mathbb{N},\quad x_n\ge x_k-\varepsilon u.$$
2. The sequence $(x_n)$ is $|\cdot|_u$-convergent to $x\in X_u$ iff $$\begin{aligned}
{\rm(i)}\quad &\forall n\in\mathbb{N},\quad x_n\ge x;\\
{\rm(ii)}\quad &\forall \varepsilon>0,\; \exists k\in\mathbb{N},\quad\mbox{such that}\quad x\ge x_k-\varepsilon u.
\end{aligned}$$ In the affirmative case, $x=\inf\!|_{X_u}\{x_n :n\in\mathbb{N}\}.$ If $K$ is Archimedean, then $x=\inf_nx_n.$
3. The sequence $(x_n)$ is $|\cdot|_u$-convergent to $x\in X_u$ iff it is $|\cdot|_u$-Cauchy and has an infimum in $X_u$. In the affirmative case $x_n\xrightarrow{|\cdot|_u}\inf\!|_{X_u}\{x_n :n\in\mathbb{N}\}.$
Now we consider the connection with self-bounded sequences.
\[p3.seq-u-semin\] Let $(x_n)$ be a $|\cdot|_u$-Cauchy sequence in $K_u.$
1. If there exists $\alpha>0$ and a subsequence $(x_{n_k})$ of $(x_n)$ such that $x_{n_k}\ge\alpha u,$ then $(x_n)$ is self-bounded.
2. If $(x_n)$ is increasing and there exists $n_0\in\mathbb{N}$ such that $x_{n_0}\in K(u),$ then then $(x_n)$ is self-bounded.
3. If $(X,\tau)$ is a TVS ordered by a cone $K$ and $(x_n)$ is a $|\cdot|_u$-Cauchy sequence in $X_u,\, \tau$-convergent to some $x\in X_u,$ then $x_n\xrightarrow{|\cdot|_u}x.$
1\. For $\lambda >1$ put $\varepsilon:=\alpha(\lambda-1).$ Since the sequence $(x_n)$ is $|\cdot|_u$-Cauchy, there exists $k\in\mathbb{N}$ such that $x_n\le x_m+\varepsilon u$ for all $n,m\ge k.$ Taking $m=n_k$ and $n\ge n_k (\ge k)$, it follows $$x_n\le x_{n_k}+(\lambda-1)\alpha u\le x_{n_k}+(\lambda-1)x_{n_k}=\lambda x_{n_k},$$ proving that $(x_n)$ is upper self-bounded.
The fact that $(x_n)$ is lower self-bounded, can be proved in a similar way, taking $\varepsilon:=\alpha(1-\mu)$ for $0<\mu<1.$
2\. If $x_{n_0}$ belongs to the component $K(u)$ of $K$, then $x_{n_0}\sim u,$ so there exists $\alpha>0$ such that $x_{n_0}\ge\alpha u.$ It follows $x_{n}\ge\alpha u,$ for all $n\ge n_0,$ and so the hypotheses of 1 are satisfied.
3\. For $\varepsilon>0$ let $n_\varepsilon\in\mathbb{N}$ be such that $|x_{n+k}-x_n|_u<\varepsilon$ for all $n\ge n_\varepsilon$ and all $k\in\mathbb{N}.$ By $\, \varepsilon\in \mathcal M_u(x_{n+k}-x_n),$ that is $-\varepsilon u\le x_{n+k}-x_n\le\varepsilon u,$ for every $n\ge n_\varepsilon$ and all $k\in\mathbb{N}.$ Letting $k\to \infty,$ one obtains $\, -\varepsilon u\le x-x_n\le \varepsilon u,$ implying $|x_n-x|_u\le \varepsilon, $ for all $n\ge n_\varepsilon.$ This shows that $x_n\xrightarrow{|\cdot|_u}x.$
The completeness results
------------------------
The following important result shows that the completeness of the Thompson metric $d$ on $K(u)$ and that of the $u$-norm on $X_u$ are equivalent when $K_u$ is Archimedean (by Remark \[re.T-metric-u-semin\] this result is nontrivial) and also reduces the completeness to the convergence of the monotone Cauchy sequences. We also show that the completeness of $d$ on $K(u)$ is equivalent to several order–completeness conditions in $X_u$.
The notation in the following theorem is that of Subsections \[Ss.def-T-metric\] and \[Ss.u-semin\].
\[t1.complete-T-m-u-sn\] Let $X$ be a vector space ordered by a cone $K$ and let $u\in K\setminus\{0\}$ be such that $K_u$ is Archimedean. Then the following assertions are equivalent.
1. $K(u)$ is $d$-complete.
2. $K(u)$ is self-complete in $X_u$.
3. $K_u$ is self-complete in $X_u$.
4. $(X_u,|\cdot|_u)$ is fundamentally $\sigma$-order complete.
5. $(X_u,|\cdot|_u)$ is monotonically sequentially complete.
6. $X_u$ is $|\cdot|_u$-complete.
If, in addition, $K$ is Archimedean, then the assertions 2 and 3 can be replaced by the stronger versions:
1. $K(u)$ is self-complete (in $X$).
2. $K_u$ is self-complete (in $ X$).
1$\Rightarrow$2. If $(x_n)$ is an increasing, upper self-bounded sequence in $K(u)$, then, by Proposition \[p1.s-bd-seq\].2, it is $d$-Cauchy, so that it is $d$-convergent to some $x\in K(u)$, and, by Theorem \[t1.T-metric-u-sn\], also $|\cdot|_u$-convergent to $x$. By Proposition \[p1.Ku\] the cone $K_u$ is $|\cdot|_u$-closed in $X_u,$ so that, by Proposition \[p2.order-tvs\].4, $\, x=\sup_nx_n.$ Consequently, $K(u)$ is upper self-complete in $X_u$. But then, by Proposition \[p1.equiv-cone\] and Theorem \[t1.s-complete\], self-complete in $X_u$.
2$\iff$3. By , $K(u)={\operatorname{aint}}(K_u),$ so that, by Proposition \[p6.s-complete\], $K_u$ is self-complete iff $K(u)$ is self-complete.
4$\iff$5. The implication 5$\Rightarrow$4 is trivial and 4$\Rightarrow$5 follows by Propositions \[p1.seq-u-semin\].3 and \[p2.seq-u-semin\].3.
2$\Rightarrow$4. Using again the fact that $K(u)={\operatorname{aint}}(K_u),$ it is sufficient to show that every increasing $|\cdot|_u$-Cauchy sequence in $K(u)$ has a supremum in $X_u$. Indeed, by Proposition \[p2.o-complete\] this is equivalent to the fact that the space $(X_u,|\cdot|_u)$ is fundamentally $\sigma$-order complete. By Proposition \[p1.Ku\].4, the cone $K_u$ is normal and generating, so that, by Proposition \[p1.o-complete\].3, it is monotonically sequentially complete.
But, by Proposition \[p3.seq-u-semin\].2, the sequence $(x_n)$ is self-bounded so it has a supremum in $X_u.$
5$\Rightarrow$6. Since a Cauchy sequence is convergent if has a convergent subsequence, it is sufficient to show that every sequence $(x_n)$ in $X_u$ satisfying $$\label{eq1.t1.complete-T-m-u-sn}
\forall n\in\mathbb{N}_0,\quad |x_{n+1}-x_n|_u\le \frac{1}{2^n},$$ is convergent in $(X_u,|\cdot|_u)$, where $\mathbb{N}_0=\mathbb{N}\cup\{0\}.$
The inequality implies $$\label{eq2.t1.complete-T-m-u-sn}
-\frac{1}{2^n} \, u\le x_{n+1}-x_n \le \frac{1}{2^n} \, u,$$ for all $n\in\mathbb{N}_0$. Writing for $0,1,\dots,n-1$ and adding the obtained inequalities, one obtains $$-\left(2-\frac{1}{2^{n-1}}\right)\le x_n-x_0\le \left(2-\frac{1}{2^{n-1}}\right),$$ for all $n\in\mathbb{N}.$ Putting $y_n:=x_n-x_0+\left(2-\frac{1}{2^{n-1}}\right)\,u$ it follows $$0\le y_n\le \left(4-\frac{1}{2^{n-2}}\right)\,u,\quad n\in\mathbb{N},$$ which proves that $y_n\in K_u$ for all $n\in\mathbb{N}.$ Also from $y_{n+1}-y_n=x_{n+1}-x_n+ \frac{1}{2^{n}}\,u$ and , one obtains $$0\le y_{n+1}-y_n\le \frac{1}{2^{n-1}}\,u,$$ implying $$0\le y_{n+k}-y_n\le \left(\frac{1}{2^{n-1}}+\frac{1}{2^{n}}+\dots+\frac{1}{2^{n+k-2}}\right)\,u<\frac{1}{2^{n-2}}\,u.$$
It follows that $(y_n)$ is an increasing $|\cdot|_u$-Cauchy sequence in $K_u,$ hence it is $|\cdot|_u$-convergent to some $y\in X_u$. But then $$x_n=y_n+x_0-\left(2-\frac{1}{2^{n-1}}\right)\,u$$ is $|\cdot|_u$-convergent to $y+x_0-2 u\in X_u.$
6$\Rightarrow$1. Again, to prove the completeness of $(K(u),d)$ it is sufficient to show that every sequence $(x_n)$ in $K(u)$ which satisfies $$\label{eq3.t1.complete-T-m-u-sn}
\forall n\in\mathbb{N}_0,\quad d(x_{n+1},x_n)\le \frac{1}{2^{n}},$$ is convergent in $(K(u),d)$. Let $s_0=d(x_0,u).$ Then, by the triangle inequality and applied successively, $$d(x_n,x_0)\le 1+\frac{1}{2}+\dots+\frac{1}{2^{n-1}}<2,$$ so that $$d(x_n,u)\le d(x_n,x_0)+ d(x_0,u)<2+s_0,$$ implying $$\label{eq4.t1.complete-T-m-u-sn}
e^{-(2+s_0)}u\le x_n \le e^{2+s_0}u$$ for all $n\in\mathbb{N}_0.$ The inequality implies $$x_{n+1}\le e^{1/2^n}x_n,$$ and $$x_n\le e^{1/2^n}x_{n+1},$$ so that, taking into account the second inequality in , one obtains the inequalities $$x_{n+1}-x_n\le \big(e^{1/2^n}-1\big)x_n\le \big(e^{1/2^n}-1\big)e^{2+s_0}u,$$ and $$x_{n+1}-x_n\ge -\big(e^{1/2^n}-1\big)x_n\ge -\big(e^{1/2^n}-1\big)e^{2+s_0}u,$$ which, in their turn, imply $$|x_{n+1}-x_n|_u\le \big(e^{1/2^n}-1\big)e^{2+s_0},$$ for all $n\in\mathbb{N}_0.$ The convergence of the series $\sum_n\big(e^{1/2^n}-1\big)e^{2+s_0}$[^1]and the above inequalities imply that the sequence $(x_n)$ is $|\cdot|_u$-Cauchy, and so it is $|\cdot|_u$-convergent to some $x\in X_u$. By Proposition \[p2.order-tvs\] the order intervals in $X_u$ are $|\cdot|_u$-closed and, by , $x_n\in \big[e^{-(2+s_0)}u;e^{2+s_0}u\big]_u$ it follows $x\in \big[e^{-(2+s_0)}u;e^{2+s_0}u\big]_u\subset K(u).$ Since, by Theorem \[t1.T-metric-u-sn\], $d$ and $|\cdot|_u$ are topologically equivalent on $K(u),$ it follows $x_n\xrightarrow{d}x.$
Combining Proposition \[p6.s-complete\] and Theorem \[t1.complete-T-m-u-sn\] one obtains the following corollaries.
\[c1.complete-T-m-u-sn\] If $K$ is Archimedean, then $d$ is complete iff $K $ is self-complete.
\[c2.complete-T-m-u-sn\] If $K$ is Archimedean and lineally solid, then the following conditions are equivalent.
1. The Thompson metric $d$ is complete.
2. The cone $K$ is self-complete.
3. The algebraic interior ${\operatorname{aint}}(K)$ of $K$ is self-complete.
4. The algebraic interior ${\operatorname{aint}}(K)$ of $K$ is $d$ -complete.
The completeness of the Thompson metric in LCS
----------------------------------------------
In this subsection we shall prove the completeness of the Thompson metric $d$ corresponding to a normal cone $K$ in a sequentially complete LCS $X$. In the case of a Banach space the completeness was proved by Thompson [@thomp63]. In the locally convex case we essentially follow [@Hy-Is-Ras].
Note that if $(X,\rho)$ is an extended metric space, then the completeness of $X$ means the completeness of every component of $X$. Indeed, if $(x_n)$ is a $d$-Cauchy sequence, then there exits $n_0\in\mathbb{N}$ such that $d(x_n,x_{n_0})\le 1, $ for all $n\ge n_0,$ implying that $x_n\in Q,$ for all $n\ge n_0,$ where $Q$ is the component of $X$ containing $x_{n_0}.$ Also if $x_n\xrightarrow{d}x,$ then there exists $n_1>n_0$ in $\mathbb{N}$ such that $d(x_n,x)\le 1$ for all $n\ge n_1,$ implying that the limit $x$ also belongs to $Q$.
\[t1.complete-T-metric\] Let $(X,\tau)$ be a locally convex space, $K$ a sequentially complete closed normal cone in $X$. Then each component of $K$ is a complete metric space with respect to the Thompson metric $d.$
By Theorem \[t2.char-normal-cone\] one can suppose that the topology $\tau$ is generated by a family $P$ of monotone seminorms.
We start by a lemma which is an adaptation of Lemma 2.3.ii in [@kraus-nuss93], proved for Banach spaces, to the locally convex case.
\[le.T-normal-cone\] Let $(X,\tau)$ be a Hausdorff LCS ordered by a closed normal cone $K$ and $d$ the Thompson metric corresponding to $K$. Supposing that $P$ is a directed family of monotone seminorms generating the topology $\tau,$ then for every $x,y\in K\setminus\{0\}$ and every $p\in P,$ the following inequality holds $$\label{eq1.le.T-normal-cone}
p(x-y)\le \big(2e^{d(x,y)}+ e^{-d(x,y)}-1\big)\cdot \min\{p(x),p(y)\}\,.$$
We can suppose $d(x,y)<\infty\,$ (i.e. $x\sim y$). By Proposition \[p2.order-tvs\] the cone $K$ is Archimedean, so that, by Proposition \[p2.T-metric\], $d(x,y)\in \sigma(x,y).$ Putting $\alpha =e^{d(x,y)},$ it follows $$\alpha^{-1}x\le y\le \alpha x,$$ so that $(\alpha-1)x\le x-y\le (1-\alpha^{-1})x,$ and so $$0\le (x-y)+(\alpha-1)x\le (\alpha-\alpha^{-1})x\, .$$
Let $p\in P.$ By the monotony of $p$ the above inequalities yield $$p(x-y)-(\alpha-1)p(x)\le p((x-y)+(\alpha-1)x)\le (\alpha-\alpha^{-1})p(x),$$ and so $$p(x-y) \le (2\alpha-\alpha^{-1}-1)p(x).$$
Interchanging the roles of $x$ and $y$ one obtains, $$p(x-y) \le (2\alpha-\alpha^{-1}-1)p(y),$$ showing that holds.
Let $(x_n)$ be a $d$-Cauchy sequence in a component $Q$ of $K$.
Observe first that the sequence $(x_n)$ is $\tau$-bounded, that is $p$-bounded for every $p\in P.$
Indeed, if $n_0\in\mathbb{N}$ is such that $d(x_n,x_{n_0})\le 1, $ for all $n\ge n_0,$ then $x_n\le e^{d(x_n,x_{n_0})}x_{n_0} \le e x_{n_0},$ for all $n\ge n_0.$ By the monotony of $p$, it follows $p(x_n)\le ep(x_{n_0})$ for all $n\ge n_0$ and every $p\in P.$ This fact and the inequality imply that $(x_n)$ is $p$-Cauchy for every $p\in P$, hence it is $P$-convergent to some $x\in X.$
If $n_0$ is as above, then the inequalities $e^{-1}x_{n_0}\le x_n\le e x_{n_0},$ valid for all $n\ge n_{0}$, yield for $n\to \infty,\; e^{-1}x_{n_0}\le x\le e x_{n_0},$ showing that $x\sim x_{n_0},$ that is $x\in Q.$
Since $(x_n)$ is $d$-Cauchy and $\tau$-convergent to $x$, Proposition \[p.T-converg-seq\].3 implies that $x_n\xrightarrow{d}x,$ proving the completeness of $(K,d)$.
The case of Banach spaces
-------------------------
We have seen in the previous subsection that the normality of a cone $K$ in a sequentially complete LCS $X$ is a sufficient condition for the completeness of $K$ with respect to the Thompson metric. In this subsection we show that, in the case when $X$ is a Banach space ordered by a cone $K$, the completeness of $d$ implies the normality of $K$. The proof will be based on the following result.
\[t2.complete-T-m-u-sn\] Let $(X,\|\cdot\|)$ be a Banach space ordered by a cone $K$ and $u\in K\setminus\{0\}$. Then the following assertions are equivalent.
1. The Thompson metric $d$ is complete on $K(u)$.
2. $(X_u,|\cdot|_u)$ is a Banach space.
3. The embedding of $(X_u,|\cdot|_u)$ into $(X,\|\cdot\|)$ is continuous.
4. The order interval $[0;x]_o$ is $\|\cdot\|$-bounded for every $x\in K(u).$
5. The order interval $[0;u]_o$ is $\|\cdot\|$-bounded.
6. Any sequence $(x_n)$ in $K(u)$ which is $d$-convergent to $x\in K(u)$ is also $\|\cdot\|$-convergent to $x$.
The equivalence 1$\iff$2 is in fact the equivalence 1$\iff$6 in Theorem \[t1.complete-T-m-u-sn\].
2 $\Rightarrow$3. Since both $(X,\|\cdot\|)$ and $(X_u,|\cdot|_u)$ are Banach spaces, by the closed graph theorem it suffices to show that the embedding mapping $I:X_u\to X, \, I(x)=x,\,$ has closed graph. This means that for every sequence $(x_n)$ in $X_u,\, x_n\xrightarrow{|\cdot|_u}x$ and $ x_n\xrightarrow{\|\cdot\|}y$ imply $y=x.$ Passing to limit for $n\to \infty$ with respect to $\|\cdot\|$ in the inequalities $$x_n-x+|x_n-x|_u u\ge 0\quad\mbox{and}\quad x_n-x+|x_n-x|_u u\ge 0\,,$$ and taking into account the fact that the cone $K$ is $\|\cdot\|$-closed, one obtains $$y-x\ge 0\quad\mbox{and}\quad x-y\ge 0,$$ that is $y=x.$
3 $\Rightarrow$4. By the continuity of the embedding, there exists $\gamma>0$ such that $\|x\|\le\gamma |x|_u$ for all $x\in X_u.$ By Proposition \[p1.Ku\] the norm $ |\cdot|_u$ is monotone, so that $0\le z\le x$ implies $\|z\|\le\gamma |z|_u \le\gamma |x|_u,$ for all $z\in[0;x]_o.$
The implication 4 $\Rightarrow$5 is obvious.
5 $\Rightarrow$3. Let $\gamma>0$ be such that $\|z\|\le \gamma$ for every $z\in[0;u]_u.$ For $x \ne 0$ in $X_u$, the inequalities $\,-|x|_u u\le x\le |x|_u u,$ imply $$\frac{x+|x|_uu}{2 |x|_u}\in [0;u]_o,$$ so that $\|x+|x|_uu\|\le 2 \gamma |x|_u \,.$
Hence, $$\|x\|- |x|_u\|u\|\le \|x+|x|_uu\| \le 2 \gamma |x|_u ,$$ and so $$\|x\|\le (2 \gamma+\|u\|)|x|_u),$$ for all $x\in X_u$, proving the continuity of the embedding of $(X_u,|\cdot|_u)$ into $(X,\|\cdot\|)$.
3 $\Rightarrow$2. Let $(x_n)$ be a $|\cdot|_u$-Cauchy sequence in $X_u$. The continuity of the embedding implies that it is $\|\cdot\|$-Cauchy and so, $\|\cdot\|$-convergent to some $x\in X.$ But then, by Proposition \[p3.seq-u-semin\].3, $(x_n)$ is $|\cdot|_u$-convergent to $x.$
The implication 3$\Rightarrow$6 follows by Theorem \[t1.T-metric-u-sn\].
6$\Rightarrow$3. Let $(x_n)$ be a sequence in $X_u$ which is $|\cdot|_u$-convergent to $x\in X_u.$ Then $(x_n)$ is $|\cdot|_u$-bounded, so there exists $\alpha>0$ such that $-\alpha u\le x_n\le\alpha u.$ It follows that $y_n:=x_n+(\alpha+1)u\in[u;(2\alpha+1)u]_o,$ and so $y_n\in K(u),\, n\in\mathbb{N},$ and $y_n\xrightarrow{|\cdot|_u}x+(\alpha+1)u.$ By Theorem \[t1.T-metric-u-sn\], $\,y_n\xrightarrow{d}x+(\alpha+1)u,$ so that, by hypothesis, $y_n\xrightarrow{\|\cdot\|}x+(\alpha+1)u.$ It follows $x_n\xrightarrow{\|\cdot\|}x ,$ proving the continuity of the embedding.
Now we present several conditions equivalent to the completeness of the Thompson metric.
\[t2.complete-T-metric-B-sp\] Let $(X,\|\cdot\|)$ be a Banach space ordered by a cone $K$. The following assertions are equivalent.
1. The Thompson metric $d$ is complete.
2. The cone $K$ is self-complete.
3. The cone $K$ is normal.
4. The norm topology on $K$ is weaker than the topology of $d$.
The equivalence 1$\iff$2 follows by Corollary \[c1.complete-T-m-u-sn\] (remind that, by Proposition \[p2.order-tvs\], the cone $K$ is Archimedean).
2$\iff$3. By Proposition \[p6.s-complete\], the cone $K$ is self-complete iff each component of $K$ is self-complete. By Theorem \[t2.complete-T-m-u-sn\], this happens exactly when the order interval $[0;x]_o$ is $\|\cdot\|$-bounded for every $x\in K,$ which is equivalent to the fact that the order intervals $[x;y]_o$ are $\|\cdot\|$-bounded for all $x,y\in K. $ By Theorem \[t4.char-normal-cone\] this is equivalent to the normality of $K$.
1$\iff$4. By Theorem \[t2.complete-T-m-u-sn\] the cone $K$ is $d$-complete iff the norm topology on each component of $K$ is weaker that the topology generated by $d$, and this is equivalent to 4.
By Theorem \[t2.complete-T-metric-B-sp\] in the case of an ordered Banach space the normality of the cone is both necessary and sufficient for the completeness of the Thompson metric. The proof, relying on Theorem \[t2.complete-T-m-u-sn\], uses the closed graph theorem and the fact that a cone in a Banach space is normal iff every order interval is norm bounded. As these results are not longer true in arbitrary LCS, we ask the following question.
**Problem.** *Characterize the class of LCS for which the normality of $K$ is also necessary for the completeness of the Thompson metric (or, at least, put in evidence a reasonably large class of such spaces).*
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[^1]: Follows from the inequality $e^{1/2^n}-1=\frac{1}{2^n}+\frac{1}{2!}\cdot\frac{1}{2^{2n}}+\dots<\frac{1}{2^n}(1+\frac{1}{2!}+\dots)= \frac{1}{2^n}(e-1)$.
|
---
abstract: 'This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with boundary. It is proved by applying a variational principle that the length of boundary components is uniquely determined by the combinatorial conformal factor. The combinatorial Yamabe flow is a gradient flow of a concave function. The long time behavior of the flow and the geometric meaning is investigated.'
address: 'School of Mathematics, University of Minnesota, Minneapolis, MN, 55455'
author:
- Ren Guo
title: Combinatorial Yamabe flow on hyperbolic surfaces with boundary
---
Introduction
============
Piecewise flat metrics
----------------------
In trying to develop the analogous piecewise linear conformal geometry, Luo studied the combinatorial Yamabe problem for piecewise flat metrics on triangulated surfaces [@Luo04]. We summarize a part of this work in the following. Suppose $\Sigma$ is a connected orientable closed surface with a triangulation $T$ so that $V,E,F$ are sets of all vertexes, edges and triangles in $T.$ We identify a vertex of $T$ with an index, an edge of $T$ with a pair of indexes and a triangle of $T$ with a triple of indexes. This means $V=\{1,2,...n\}, E=\{ij\ |\ i,j\in V\}$ and $F=\{ijk\ |\ i,j,k\in V\},$ where $n$ is the number of vertexes.
A piecewise flat metric on $(\Sigma, T)$ is identified with a vector indexed by the set of edges $E$. More precisely, it is an assignment to each edge a positive number such that the three numbers assigned to the three edges of a triangle satisfy the triangle inequality. Equipped with a piecewise flat metric, each triangle of $T$ can be realized as a Euclidean triangle and $\Sigma$ can be realized as a Euclidean polyhedral surface.
Let’s fix a piecewise flat metric on $(\Sigma, T)$ as $l^0\in \mathbb{R}_{>0}^{|E|}$. The assignment to the edge $ij$ is denoted by $l^0_{ij}.$ A *combinatorial conformal factor on $(\Sigma, T)$ is a vector $w=(w_1,w_2,...,w_n)\in \mathbb{R}^n$ which assigns each vertex $i\in V$ a number $w_i.$ (In [@Luo04], the notation $u_i$ is used, where $u_i=e^{w_i}$.) From a combinatorial conformal factor, we obtained a new vector $l\in \mathbb{R}_{>0}^{|E|}$ as follows: $$\begin{aligned}
\label{fml:e-conformal}
l_{ij}=e^{w_i+w_j}l^0_{ij}\end{aligned}$$ for each edge $ij\in E.$*
Let $\mathcal{W}_E$ be the space of combinatorial conformal factors $w$ such that each vector $l$ corresponding to a vector $w\in \mathcal{W}_E$ is indeed a piecewise flat metric. In other words, the triangle inequality holds for each triangle under the assignment $l.$ Obviously, $\mathcal{W}_E$ depends on the initial metric $l^0.$
For a vector $w\in \mathcal{W}_E,$ the vector $l$ corresponding to $w$ is a piecewise flat metric. Each triangle of $T$ is realized as a Euclidean triangle under the metric $l$. At a vertex $i,$ the curvature of the metric $l$ is defined as follows. Let $\alpha^i_{jk}$ be the inner angle of the triangle $ijk\in F$ between the edges $ij$ and $ik.$ Then $$K_i=2\pi-\sum_{ijk\in F} \alpha^i_{jk}$$ is the curvature at the vertex $i$.
This produces a map $$\begin{array}{ccccccc}
\psi_E: \mathcal{W}_E & \to & \mathbb{R}^n\\
(w_1,w_2,...,w_n) & \mapsto & (K_1,K_2,...,K_n)
\end{array}$$ sending a combinatorial conformal factor to the curvature.
The map $\psi_E$ is a local homeomorphism.
The theorem is proved by applying a variational principle. An local convex energy function is constructed using the derivative cosine law. And $\psi_E$ truns out to be the gradient of the energy function.
Motivated by establishing a discrete Uniformization Theorem, Luo introduced the combinatorial Yamabe flow
$$\begin{aligned}
\label{fml:eflow}
\left\{
\begin{array}{ccccccc}
\frac{dw_i(t)}{dt}=-K_i(t),\\
w_i(0)=0.
\end{array}
\right.\end{aligned}$$
The combinatorial Yamabe flow (\[fml:eflow\]) is the negative gradient flow of a locally convex function in terms of $w$. And $\sum_{i=1}^n K_i^2(t)$ is decreasing in time $t$.
Related work
------------
Motivated by the application in computer graphics, Springborn, Schröder, and Pinkall [@SSP08] considered this combinatorial conformal change of piecewise flat metrics (\[fml:e-conformal\]). They found an explicit formula of the energy function. Glickenstein [@Gli05a; @Gli05b] studied the combinatorial Yamabe flow on 3-dimensional piecewise flat manifolds relating to the ball packing metric of Cooper and Rivin [@CR96]. Recently Glickenstein [@Gli09] set the theory of combinatorial Yamabe flow of piecewise flat metric in a broader context including the theory of circle packing on surfaces. This combinatorial conformal change of metrics has appeared in physic literature [@RW84] and numerical analysis literature [@Ker96; @PC98]. We were informed by Luo in 2009 that Springborn considered the combinatorial conformal change of hyperbolic metric on a triangulated closed surface. He introduced the combinatorial conformal change as $$\begin{aligned}
\label{fml:h-conformal}
\sinh\frac{l_{ij}}{2}=e^{w_i+w_j}\sinh\frac{l^0_{ij}}{2}.\end{aligned}$$
Hyperbolic metrics
------------------
In this paper we study the combinatorial Yamabe flow on hyperbolic surfaces with geodesic boundary. Let $\Sigma$ be a connected orientable compact surface with $n$ boundary components. The set of boundary components is $B=\{1,2,...n\}$ where a boundary component is identified with an index.
A colored hexagon is a hexagon with three non-pairwise adjacent edges labeled by red and the opposite edges labeled by black. Take a finite disjoint union of colored hexagons and identify all red edges in pairs by homeomorphisms. The quotient is a compact surface with non-empty boundary together with an *ideal triangulation. The faces in the ideal triangulation are quotients of the hexagons. The quotients of red edges are called the edges of the ideal triangulation while the quotients of black edges are called the boundary arcs.*
It is well-known that each connected orientable compact surface $\Sigma$ of non-empty boundary and negative Euler characteristic admits an ideal triangulation.
Let $T$ be an ideal triangulation of $\Sigma$. Since an edge of $T$ connects two boundary components of $\Sigma$, an edge of $T$ is indentified with a pair of indexes. The set of edges of $T$ is $E=\{ij\ |\ i,j\in B\}.$ Since a face in $T$ is determined by its boundary arcs in three boundary components, a face of $T$ is indentified with a triple of indexes. The set of faces of $T$ is $F=\{ijk\ |\ i,j,k\in B\}.$
A hyperbolic metric on $(\Sigma,T)$ is identified with a vector indexed by the set of edges $E$. More precisely, it is an assignment to each edge a positive number. It is well-known that [@Bus92], for any three positive numbers, there exists a hyperbolic right-angled hexagon the length of whose three non-pairwise adjacent edges are the three numbers. Furthermore, the hexagon is unique up to isometry. Therefore, for vector in $\mathbb{R}_{>0}^{|E|}$, each face of $F$ can be realized as a unique hyperbolic right-angled hexagon and the surface $\Sigma$ can be realized as a hyperbolic surface with geodesic boundary.
Let’s fix a hyperbolic metric on $(\Sigma,T)$ as $l^0\in \mathbb{R}_{>0}^{|E|}$. The assignment to the edge $ij$ is denoted by $l^0_{ij}.$ A *combinatorial conformal factor on $(\Sigma, T)$ is a vector $w=(w_1,w_2,...,w_n)\in \mathbb{R}^n$ which assigns each boundary component $i\in B$ a number $w_i.$ From a combinatorial conformal factor, we obtained a new assignment $l\in \mathbb{R}^{|E|}$ as follows $$\begin{aligned}
\label{fml:hb-conformal}
\cosh\frac{l_{ij}}{2}=e^{w_i+w_j}\cosh\frac{l^0_{ij}}{2}\end{aligned}$$ for any edge $ij\in E$. This definition (\[fml:hb-conformal\]) is an analogue of Springborn’s definition of combinatorial conformal change of hyperbolic metrics on triangulated closed surfaces (\[fml:h-conformal\]).*
Denote by $\mathcal{W}$ the set of combinatorial conformal factors such that the corresponding assignment is positive, i.e., $l\in \mathbb{R}_{>0}^{|E|}$. Therefore, for $w\in\mathcal{W}$, we obtained a new hyperbolic metric on $(\Sigma,T)$. Each face of $T$ is a hyperbolic right-angled hexagon. Let $\theta^i_{jk}$ be the length of the boundary arc in the boundary component $i$ of the face $ijk\in F$. Then the length of the boundary component $i$ is $$B_i=\sum_{ijk\in F} \theta^i_{jk}.$$
This produces a map $$\begin{array}{ccccccc}
\psi: \mathcal{W} & \to & \mathbb{R}^n\\
(w_1,w_2,...,w_n) & \mapsto & (B_1,B_2,...,B_n)
\end{array}$$ sending a combinatorial conformal factor to the length of boundary components.
\[thm:homo\] The map $\psi$ is a homeomorphism.
This is a result of global rigidity while Theorem 1 is a result of local rigidity.
We also consider the combinatorial Yamabe flow in this situation
$$\begin{aligned}
\label{fml:hflow}
\left\{
\begin{array}{ccccccc}
\frac{dw_i(t)}{dt}=B_i(t),\\
w_i(0)=0.
\end{array}
\right.\end{aligned}$$
\[thm:flow\] The combinatorial Yamabe flow (\[fml:hflow\]) is the gradient flow of a concave function in terms of $w$. And $\sum_{i=1}^n B_i^2(t)$ is decreasing in time $t$.
We investigate the long time behavior of the flow.
\[thm:cusp\] The combinatorial Yamabe flow (\[fml:hflow\]) has a solution for $t\in[0,\infty)$. Along the flow (\[fml:hflow\]), any initial hyperbolic surface with geodesic boundary converges to a complete hyperbolic surface with cusps.
Variational principle
---------------------
The approach of variational principle of studying polyhedral surfaces was introduced by Colin de Verdiére [@CdV91] in his proof of Andreev-Thurston’s circle packing theorem. Since then, many works about variational principles on polyhedral surfaces have appeared. For example, see [@Bra92; @Riv94; @Lei02; @CL03; @BS04; @Luo06; @Guo07; @Spr08; @GL09; @Guo09] and others.
Organization of the paper
-------------------------
Theorem \[thm:homo\] is proved in section 2. Corollary \[thm:flow\] and Theorem \[thm:cusp\] are proved in section 3.
Homeomorphism
=============
Space of combinatorial conformal factors
----------------------------------------
Let $l^0\in \mathbb{R}_{>0}^{|E|}$ be a fixed hyperbolic metric on $(\Sigma,T)$. We investigate the space of combinatorial conformal factor such that the corresponding new assignment is a hyperbolic metric. For a face $ijk\in F,$ denote by $\mathcal{W}^{ijk}$ the space of vectors $(w_i,w_j,w_k)$ such that $l_{jk},l_{ki}$ and $l_{ij}$ are positive.
\[thm:space\] $\mathcal{W}^{ijk}$ is a convex polytope.
By definition (\[fml:hb-conformal\]) $\cosh\frac{l_{ij}}{2}=e^{w_i+w_j}\cosh\frac{l^0_{ij}}{2}$. The only requirement is $l_{ij}>0.$ Hence $$w_i+w_j>-\ln \cosh\frac{l^0_{ij}}{2}.$$ Similar inequalities hold for $w_j+w_k$ and $w_k+w_i$. Therefore $\mathcal{W}^{ijk}$ is the intersection of three half space.
\[thm:wspace\] The space $\mathcal{W}$ is a convex polytope.
$\mathcal{W}=\cap_{ijk\in F} \mathcal{W}^{ijk}.$
Energy function
---------------
Let’s focus on one face $ijk\in F.$ When $(w_i,w_j,w_k)\in \mathcal{W}^{ijk},$ from (\[fml:hb-conformal\]), we obtain a hyperbolic right-angled hexagon whose non-pairwise adjacent edges have length $l_{jk}, l_{ki}$ and $l_{ij}$. Let $\theta^i_{jk}, \theta^j_{ki}$ and $\theta^k_{ij}$ be the length of the hyperbolic arcs opposite to the edges $jk, ki$ and $ij$ of this hexagon. Therefore $\theta^i_{jk}, \theta^j_{ki}$ and $\theta^k_{ij}$ are functions of $w_i,w_j,w_k$.
\[thm:symmetry\] The Jacobian matrix of functions $\theta^i_{jk}, \theta^j_{ki}, \theta^k_{ij}$ in terms of $w_i,w_j,w_k$ is symmetric.
The cosine law for hyperbolic right-angled hexagon induces the derivative cosine law: $$\begin{gathered}
\left(
\begin{array}{ccc}
d\theta^i_{jk}\\
d\theta^j_{ki}\\
d\theta^k_{ij}
\end{array}\right)
=\frac{-1}{\sinh \theta^k_{ij} \sinh l_{ki}\sinh l_{jk}} \left(
\begin{array}{ccc}
\sinh l_{jk}&0&0 \\
0&\sinh l_{ki}&0 \\
0&0&\sinh l_{ij}
\end{array}
\right)\\
\left(
\begin{array}{ccc}
-1 & \cosh \theta^k_{ij} &\cosh \theta^j_{ki} \\
\cosh \theta^k_{ij}& -1 &\cosh \theta^i_{jk} \\
\cosh \theta^j_{ki}& \cosh \theta^i_{jk} &-1
\end{array}
\right) \left(
\begin{array}{ccc}
dl_{jk} \\
dl_{ki} \\
dl_{ij}
\end{array}\right).\end{gathered}$$
By differentiating the two sides of the equation (\[fml:hb-conformal\]), $\cosh\frac{l_{ij}}{2}=e^{w_i+w_j}\cosh\frac{l^0_{ij}}{2}$, we obtain $$dl_{ij}=\frac{2\sinh l_{ij}}{\cosh l_{ij}-1}(d w_i+d w_j).$$ Similar formulas hold for $d l_{jk}$ and $d l_{ki}$. Then we have
$$\begin{gathered}
\left(
\begin{array}{ccc}
d\theta^i_{jk}\\
d\theta^j_{ki}\\
d\theta^k_{ij}
\end{array}\right)
=\frac{-2}{\sinh \theta^k_{ij} \sinh l_{ki}\sinh l_{jk}} \left(
\begin{array}{ccc}
\sinh l_{jk} & 0 &0 \\
0 &\sinh l_{ki}&0 \\
0 &0 &\sinh l_{ij}
\end{array}
\right)\\
\left(
\begin{array}{ccc}
-1 & \cosh \theta^k_{ij} &\cosh \theta^j_{ki} \\
\cosh \theta^k_{ij}& -1 &\cosh \theta^i_{jk} \\
\cosh \theta^j_{ki}& \cosh \theta^i_{jk} &-1
\end{array}
\right)
\left(
\begin{array}{ccc}
\frac{\sinh l_{jk}}{\cosh l_{jk}-1} & 0 &0 \\
0 &\frac{\sinh l_{ki}}{\cosh l_{ki}-1}&0 \\
0 &0 &\frac{\sinh l_{ij}}{\cosh l_{ij}-1}
\end{array}
\right)\\
\left(
\begin{array}{ccc}
0 & 1 &1 \\
1 & 0 &1 \\
1 & 1 &0
\end{array}
\right)
\left(
\begin{array}{ccc}
d w_i \\
d w_j \\
d w_k
\end{array}\right).\end{gathered}$$
For simplicity of the notations, the above formula is written as $$\left(
\begin{array}{ccc}
d\theta^i_{jk}\\
d\theta^j_{ki}\\
d\theta^k_{ij}
\end{array}\right)
=\frac{-2}{\sinh \theta^k_{ij} \sinh l_{ki}\sinh l_{jk}}
M
\left(
\begin{array}{ccc}
d w_i \\
d w_j \\
d w_k
\end{array}\right)$$ where $M$ is a product of four matrixes. To prove the lemma, it is enough to show that the matrix $M$ is symmetric.
Represent $\cosh\theta^i_{jk}, \cosh\theta^j_{ki}, \cosh\theta^k_{ij}$ as functions of $\cosh l_{jk}, \cosh l_{ki}, \cosh l_{ij}$ using the cosine law. For simplicity of the notations, let $a:=\cosh l_{jk},b:=\cosh l_{ki},c:=\cosh l_{ij}$, Then we have $$M=
\left(
\begin{matrix}
\displaystyle \frac{c+ab}{b-1} + \frac{b+ac}{c-1}
& \displaystyle \frac{a+b-c+1}{c-1}
& \displaystyle \frac{a+c-b+1}{b-1} \\
\displaystyle\frac{a+b-c+1}{c-1}
&\displaystyle \frac{c+ab}{a-1} + \frac{a+bc}{c-1}
&\displaystyle \frac{b+c-a+1}{a-1} \\
\displaystyle \frac{a+c-b+1}{b-1}
&\displaystyle \frac{b+c-a+1}{a-1}
&\displaystyle \frac{b+ac}{a-1} + \frac{a+bc}{b-1}
\end{matrix}
\right).$$
\[thm:definite\] The Jacobian matrix of functions $\theta^i_{jk}, \theta^j_{ki}, \theta^k_{ij}$ in terms of $w_i,w_j,w_k$ is negative definite.
We only need to show that the matrix $M$ is positive definite. Let $M_{rs}$ be the entry of $M$ at $r-$th row and $s-$th column. First, $M_{11}>0.$ Second, $$\begin{aligned}
M_{11}-M_{12}= \frac{c+ab}{b-1} + a+1 >0,\\
M_{22}-M_{21}= \frac{c+ab}{a-1} + b+1 >0.\end{aligned}$$ Then $M_{11}M_{22}-M_{12}M_{21}>0.$ Third, $$\begin{gathered}
\det M=\sinh l_{jk} \sinh l_{ki}\sinh l_{ij}\cdot(\sinh \theta^i_{jk} \sinh \theta^j_{ki} \sinh l_{ij})^2 \\
\cdot \frac{\sinh l_{jk}}{\cosh l_{jk}-1} \frac{\sinh l_{ki}}{\cosh l_{ki}-1} \frac{\sinh l_{ij}}{\cosh l_{ij}-1}
\cdot 2>0. \end{gathered}$$
\[thm:function\] The differential 1-form $\theta^i_{jk} d w_i+ \theta^j_{ki} d w_j + \theta^k_{ij} d w_k$ is closed on $\mathcal{W}^{ijk}$. For any $c\in \mathcal{W}^{ijk},$ the integral $$\mathcal{E}(w_i,w_j,w_k)=\int_c^{(w_i,w_j,w_k)}(\theta^i_{jk} d w_i+ \theta^j_{ki} d w_j + \theta^k_{ij} d w_k)$$ is a strictly concave function on $\mathcal{W}^{ijk}$ satisfying
$$\begin{aligned}
\frac{\partial \mathcal{E}}{\partial w_i} =\theta^i_{jk},
\frac{\partial \mathcal{E}}{\partial w_j} =\theta^j_{ki},
\frac{\partial \mathcal{E}}{\partial w_k} =\theta^k_{ij}.\end{aligned}$$
The differential 1-form is closed due to Lemma \[thm:symmetry\]. Since $\mathcal{W}^{ijk}$ is connected and simply connected due to Lemma \[thm:space\], then the function $\mathcal{E}(w_i,w_j,w_k)$ is well defined, i.e., independent the path of integration. By Lemma \[thm:definite\], the Hessian matrix of $\mathcal{E}(w_i,w_j,w_k)$ is negative definite.
Homeomorphism
-------------
In this subsection we prove Theorem \[thm:homo\].The following two lemmas are needed.
The first one is well-known in analysis.
\[thm:convex\] Suppose $X$ is an open convex set in $\mathbb{R}^N$ and $f: X\to \mathbb{R}$ a smooth function. If the Hessian matrix of $f$ is positive definite for all $x \in X$, then the gradient $\nabla f:X\to\mathbb{R}^N$ is a smooth embedding.
\[thm:converge\] For a family of combinatorial conformal factor $w^{(m)}\in \mathcal{W}$, if $\lim_{m\to \infty}w_k^{(m)}=\infty$ for some $k$, then $\lim_{m\to\infty}B_k^{(m)}=0$ and the convergence is independent of the values of $\lim_{m\to\infty} w_r^{(m)}$ for $r\neq k.$
By definition (\[fml:hb-conformal\]), $$\cosh l_{jk}^{(m)}=e^{2w_j^{(m)}+2w_k^{(m)}}c_1-1,$$ $$\cosh l_{ki}^{(m)}=e^{2w_k^{(m)}+2w_i^{(m)}}c_2-1,$$ $$\cosh l_{ij}^{(m)}=e^{2w_i^{(m)}+2w_j^{(m)}}c_3-1,$$ where $c_1=2\cosh^2 \frac{l^0_{jk}}2, c_2=2\cosh^2 \frac{l^0_{ki}}2, c_3=2\cosh^2 \frac{l^0_{ij}}2.$ Then $$\begin{aligned}
&\ \lim_{m\to\infty} \cosh (\theta^k_{ij})^{(m)}\\
= &\ \lim_{m\to\infty}
\frac{\cosh l_{ij}^{(m)} + \cosh l_{jk}^{(m)}\cosh l_{ki}^{(m)}}{\sinh l_{jk}^{(m)}\sinh l_{ki}^{(m)}}\\
= &\ \lim_{m\to\infty}
\frac{e^{2w_i^{(m)}+2w_j^{(m)}}c_3-1 + (e^{2w_j^{(m)}+2w_k^{(m)}}c_1-1)(e^{2w_k^{(m)}+2w_i^{(m)}}c_2-1)}
{e^{w_j^{(m)}+w_k^{(m)}}\sqrt{e^{2w_j^{(m)}+2w_k^{(m)}}c^2_1-2c_1}\cdot
e^{w_k^{(m)}+w_i^{(m)}}\sqrt{e^{2w_k^{(m)}+2w_i^{(m)}}c^2_2-2c_2}}\\
= &\ \lim_{m\to\infty}
\frac{c_3-e^{-2w_i^{(m)}-2w_j^{(m)}}}
{e^{4w_k^{(m)}}\sqrt{(c^2_1-2c_1e^{-2w_j^{(m)}-2w_k^{(m)}})(c^2_2-2c_2e^{-2w_i^{(m)}-2w_k^{(m)}})}}\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \lim_{m\to\infty}
\frac{(c_1-e^{-2w_j^{(m)}-2w_k^{(m)}})(c_2-e^{-2w_i^{(m)}-2w_k^{(m)}})}
{\sqrt{(c^2_1-2c_1e^{-2w_j^{(m)}-2w_k^{(m)}})(c^2_2-2c_2e^{-2w_i^{(m)}-2w_k^{(m)}})}}\\
= &\ 0+1.\end{aligned}$$ Hence $\lim_{m\to\infty}(\theta^k_{ij})^{(m)}=0$ independent the values of $\lim_{m\to\infty} w_r^{(m)}$ for $r\neq k.$ Thus $\lim_{m\to\infty}B_k^{(m)}=\lim_{m\to\infty}\sum_{ijk\in F}(\theta^k_{ij})^{(m)}=0.$
Let $l^0\in \mathbb{R}^{|E|}_{>0}$ be a fixed hyperbolic metric on $(\Sigma,T).$ For any combinatorial conformal factor $w\in \mathcal{W}$, we obtain a new hyperbolic metric $l\in \mathbb{R}^{|E|}_{>0}$. By Corollary \[thm:function\], for each face $ijk\in F$, there is a function $\mathcal{E}(w_i,w_j,w_k).$ Define a function $\overline{\mathcal{E}}:\mathcal{W}\to \mathbb{R}$ by $$\overline{\mathcal{E}}(w_1,w_2,...,w_n)=\sum_{ijk \in F} \mathcal{E}(w_i, w_j, w_k)$$ where the sum is over all faces in $F$. By Corollary \[thm:function\], $\overline{\mathcal{E}}$ is strictly concave on $\mathcal{W}$ and $$\begin{aligned}
\label{fml:gradient}
\frac{\partial \overline{\mathcal{E}}}{\partial w_i}
=\sum_{ijk \in F}\frac{\partial \mathcal{E}(w_i, w_j, w_k)}{\partial w_i}
=\sum_{ijk \in F} \theta^i_{jk}
=B_i.\end{aligned}$$ That means the gradient of $\overline{\mathcal{E}}$ is exactly the map $\psi$ sending a combinatorial conformal factor $w$ to the corresponding length of boundary components. Thus $\psi$ is a smooth embedding due to Lemma \[thm:convex\].
To show that $\psi$ is a homeomorphism, we will prove that $\psi(\mathcal{W})$ is both open and closed in $\mathbb{R}^n_{>0}.$
Since $\psi$ is a smooth embedding, $\psi(\mathcal{W})$ is open in $\mathbb{R}^n_{>0}$.
To show that $\psi(\mathcal{W})$ is closed in $\mathbb{R}^n_{>0},$ take a sequence of combinatorial conformal factor $w^{(m)}$ in $\mathcal{W}$ such that $\lim_{m\to\infty} (B_1^{(m)},B_2^{(m)},...,B_n^{(m)})\in\mathbb{R}^n_{>0}.$ To prove the closeness, it is sufficient to show that there is a subsequence of $w^{(m)}$ whose limit is in $\mathcal{W}$.
Suppose otherwise, there is a subsequence, still denoted by $w^{(m)}$, so that its limit is on the boundary of $\mathcal{W}$. The first possibility is that there is some $k$ such that $\lim_{m\to\infty} w_k^{(m)}=\infty.$ By Lemma \[thm:converge\], $\lim_{m\to\infty}B_k^{(m)}=0$. This contradicts the assumption that $\lim_{m\to\infty} (B_1^{(m)},B_2^{(m)},...,B_n^{(m)})\in\mathbb{R}^n_{>0}.$
The second possibility is that $\lim_{m\to \infty}(w_i^{(m)}+w_j^{(m)})=-\ln \cosh\frac{l^0_{ij}}{2}$ for some edge $ij.$ That means $\lim_{m\to \infty}l_{ij}^{(m)}=0.$ For the face $ijk\in F$, we have $$\begin{aligned}
\lim_{m\to\infty}(\theta^i_{jk})^{(m)} =&\
\lim_{m\to\infty}
\frac{\cosh l_{jk}^{(m)} + \cosh l_{ik}^{(m)}\cosh l_{ij}^{(m)}}{\sinh l_{ik}^{(m)}\sinh l_{ij}^{(m)}}\\
\geq &\ \lim_{m\to\infty}
\frac{\cosh l_{ik}^{(m)}\cosh l_{ij}^{(m)}}{\sinh l_{ik}^{(m)}\sinh l_{ij}^{(m)}}\\
\geq &\ \lim_{m\to\infty}
\frac{\cosh l_{ij}^{(m)}}{\sinh l_{ij}^{(m)}}=\infty.\end{aligned}$$ Therefore $\lim_{m\to\infty}(\theta^i_{jk})^{(m)}=\infty$ and $\lim_{m\to\infty}B_i^{(m)}=\infty.$ This contradicts the assumption that $\lim_{m\to\infty} (B_1^{(m)},B_2^{(m)},...,B_n^{(m)})\in\mathbb{R}^n_{>0}.$
Flow
====
The combinatorial Yamabe flow (\[fml:hflow\]) is the gradient flow of the concave function $\overline{\mathcal{E}}(w_1,w_2,...,w_n)$ due to the equation (\[fml:gradient\]).
Since $$\begin{aligned}
\frac{d B_i(t)}{dt}=&\ \sum_{ijk\in F }\frac{d \theta^i_{jk}}{dt}\\
=&\ \sum_{ijk\in F } (\frac{d \theta^i_{jk}}{dw_i}\frac{dw_i}{dt}
+\frac{d \theta^i_{jk}}{dw_j}\frac{dw_j}{dt}
+\frac{d \theta^i_{jk}}{dw_k}\frac{dw_k}{dt})\\
=&\ \sum_{ijk\in F } (\frac{d \theta^i_{jk}}{dw_i}B_i
+\frac{d \theta^i_{jk}}{dw_j}B_j
+\frac{d \theta^i_{jk}}{dw_k}B_k),\end{aligned}$$ we have $$\begin{aligned}
&\ \frac12\frac{d}{dt}\sum_{i=1}^n B_i^2(t) \\
= &\ \sum_{i=1}^n B_i \frac{d B_i}{dt}\\
=&\ \sum_{ijk\in F }(\frac{d \theta^i_{jk}}{dw_i}B_i^2
+ \frac{d \theta^j_{ki}}{dw_j}B_j^2
+ \frac{d \theta^k_{ij}}{dw_k}B_k^2\\
&\ \ \ \ \ \ \ \ \ +(\frac{d \theta^j_{ki}}{dw_j}+\frac{d \theta^k_{ij}}{dw_k})B_jB_k
+(\frac{d \theta^k_{ij}}{dw_k}+\frac{d \theta^i_{jk}}{dw_i})B_kB_i
+(\frac{d \theta^i_{jk}}{dw_i}+\frac{d \theta^j_{ki}}{dw_j})B_iB_j)\\
=&\ \sum_{ijk\in F } (B_i,B_j,B_k)
\frac{\partial(\theta^i_{jk},\theta^j_{ki},\theta^k_{ij})}{\partial(w_i,w_j,w_k)} (B_i,B_j,B_k)^T.\end{aligned}$$ By Lemma \[thm:definite\], the Jacobian matrix $\frac{\partial(\theta^i_{jk},\theta^j_{ki},\theta^k_{ij})}{\partial(w_i,w_j,w_k)}$ is negative definite.
Hence $\frac12\frac{d}{dt}\sum_{i=1}^n B_i^2(t)<0.$ Therefore $\sum_{i=1}^n B_i^2(t)$ is decreasing in $t$.
Since $B_i(t)>0,$ then $w_i(t)>w_i(0)=0.$
For any $L<\infty,$ we claim that $\lim_{t\to L}w_i(t)<\infty.$
Otherwise, if $\lim_{t\to L}w_i(t)=\infty$ for some $L<\infty,$ then by Lemma \[thm:converge\], we see $\lim_{t\to L}B_i(t)=0.$ Therefore, for any $\epsilon>0,$ there exists some $\delta>0$ such that when $t\in(L-\epsilon,L),$ the inequalities $0<B_i(t)<\epsilon$ holds. Hence, by the flow (\[fml:hflow\]), $0<\frac{dw_i(t)}{dt}<\epsilon$ holds. Thus $w_i(0)<w_i(t)<\epsilon t< \epsilon L.$ This contradicts to the assumption that $\lim_{t\to L}w_i(t)=\infty$.
Hence the solution of the flow (\[fml:hflow\]) exists for all time $t\in[0,\infty).$
To obtain the geometric picture, we claim that $\lim_{t\to\infty}B_i(t)=0$ for each $1 \leq i\leq n$. There are two cases to consider.
First, if $\lim_{t\to\infty}w_i(t)=\infty,$ by Lemma \[thm:converge\], $\lim_{t\to\infty}B_i(t)=0$ holds.
Second, if $\lim_{t\to\infty}w_i(t)<\infty,$ we claim $\lim_{t\to\infty}B_i(t)=0$ still holds. Otherwise, $\lim_{t\to\infty}B_i(t)=a>0.$ Then, for any $\epsilon\in (0,a),$ there exists some $P>0$ such that when $t>P,$ the inequality $B_i(t)>a-\epsilon$ holds. Hence, by the flow (\[fml:hflow\]), $\frac{dw_i}{dt}>a-\epsilon$ holds. Therefore $w_i(t)>(a-\epsilon)t.$ This contradicts to the assumption $\lim_{t\to\infty}w_i(t)<\infty.$
Once $\lim_{t\to\infty}B_i(t)=0$ for each $1 \leq i\leq n$, we have $\lim_{t\to\infty}\theta^i_{jk}(t)=0$ for any face $ijk\in F.$ Hence, for any edge $ij,$ $$\begin{aligned}
\lim_{t\to\infty}l_{ij}(t) =&\
\lim_{t\to\infty}
\frac{\cosh \theta^k_{ij}(t) + \cosh \theta^i_{jk}(t)\cosh \theta^j_{ki}(t)}{\sinh \theta^i_{jk}(t)\sinh \theta^j_{ki}(t)}\\
\geq &\ \lim_{t\to\infty}
\frac{\cosh \theta^i_{jk}(t)\cosh \theta^j_{ki}(t)}{\sinh \theta^i_{jk}(t)\sinh \theta^j_{ki}(t)}\\
\geq &\ \lim_{t\to\infty}
\frac{\cosh \theta^i_{jk}(t)}{\sinh \theta^i_{jk}(t)}=\infty.\end{aligned}$$ Therefor each hyperbolic right-angled hexagon converges to a hyperbolic ideal triangle.
Acknowledgment {#acknowledgment .unnumbered}
==============
The author would like to thank Feng Luo for encouragement and helpful conversations.
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|
---
author:
- 'Isabel M. C. Salavessa'
title: |
On the Kähler angles of Submanifolds\
[To the memory of Giorgio Valli]{}
---
16.cm 23.cm -2.5cm 0.0cm 1.5cm
.6cm
\
\
[ We prove that under certain conditions on the mean curvature and on the Kähler angles, a compact submanifold $M$ of real dimension $2n$, immersed into a Kähler-Einstein manifold $N$ of complex dimension $2n$, must be either a complex or a Lagrangian submanifold of $N$, or have constant Kähler angle, depending on $n=1$, $n=2$, or $n\geq 3$, and the sign of the scalar curvature of $N$. These results generalize to non-minimal submanifolds some known results for minimal submanifolds. Our main tool is a Bochner-type tecnique involving a formula on the Laplacian of a symmetric function on the Kähler angles and the Weitzenböck formula for the Kähler form of $N$ restricted to $M$. ]{}\
Introduction
============
Let $(N,J,g)$ be a Kähler-Einstein manifold of complex dimension $2n$, complex structure $J$, Riemannian metric $g$, and $F:M^{2n}{\rightarrow}N^{2n}$ be an immersed submanifold $M$ of real dimension $2n$. We denote by $\omega(X,Y)=g(JX,Y)$ the Kähler form and by $R$ the scalar curvature of $N$, that is, the Ricci tensor of $N$ is given by $Ricci = Rg$. The cosine of the Kähler angles $\{\theta_{{\alpha}}\}_{1\leq {\alpha}\leq n}$ are the eigenvalues of ${F^{*}\omega}$. If the eigenvalues are all equal to 0 (resp. 1), $F$ is a Lagrangian (resp. complex) submanifold. A natural question is to ask if $N$ allows submanifolds with arbitrary given Kähler angles and mean curvature. An answer is that, the Kähler angles and the second fundamental form of $F$, and the Ricci tensor of $N$ are interrelated. Conditions on some of these geometric objects have implications for the other ones. There are obstructions to the existence of minimal Lagrangian submanifolds in a general Kähler manifold, but these obstructions do not occur in a Kähler-Einstein manifolds, where such submanifolds exist with abundance (\[Br\]). This is the reason we choose Kähler-Einstein manifolds as ambient spaces. An example how the sign of the scalar curvature of $N$ determines the Kähler angles is the fact that if $F$ is a totally geodesic immersion and $N$ is not Ricci-flat, then either $F$ has a complex direction, or $F$ is Lagrangian (\[S-V,1\]). A relation among the $\theta_{{\alpha}}$, ${{\mbox{\large $\nabla$}}_{\!\!}}dF$, and $R$ can be described through a formula on the Laplacian of a locally Lipschitz map $\kappa$, symmetric on the Kähler angles of $F$, where the Ricci tensor of $N$ and some components of the second fundamental form of $F$ appear. Such kind of formula was used for minimal immersions in \[W,1\] for $n=1$, and in \[S-V,1,2\] for $n\geq 2$.
A natural condition for $n\geq 2$ is to impose equality on the Kähler angles. Products of surfaces immersed with the same constant Kähler angle $\theta$ into Kähler-Einstein surfaces of the same scalar curvature $R$, give submanifolds immersed with constant equal Kähler angle $\theta$ into a Kähler-Einstein manifold of scalar curvature $R$. The slant submanifolds introduced and exhaustively studied by B-Y Chen (see e.g. \[Che,1,2\], \[Che-M\], \[Che-T,1,2\]) are just submanifolds with constant and equal Kähler angles. Examples are given in complex spaces form, some of them via Hopf’s fibration \[Che-T,1,2\]. A minimal 4-dimensional submanifold of a Calabi-Yau manifold of complex dimension 4, calibrated by a Cayley calibration, also called Cayley submanifold, is just the same as a minimal submanifold with equal Kähler angles (\[G\]). Existence theory of such submanifolds in ${l\!\!\! C}^{4}$, with given boundary data, is guaranteed by the theory of calibrations of Harvey and Lawson \[H-L\].
Submanifolds with equal Kähler angles have a role in 4 and 8 dimensional gauge theories. For example, each of such Cayley submanifolds in ${l\!\!\! C}^{4}$ carries a 21-dimensional family of (anti)-self-dual $SU(2)$ Yang-Mills fields \[H-L\]. Recentely, Tian \[T\] proved that blow-up loci of complex anti-self-dual instantons on Calabi-Yau 4-folds are Cayley cycles, which are, except for a set of 4-dimensional Hausdorff measure zero, a countable union of $C^1$ 4-dimensional Cayley submanifolds.
If $N$ is an hyper–Kähler manifold of complex dimension 4 and hyper-Kähler structure $(J_{x})_{x\in S^2}$, any submanifold of real dimension 4 that is $J_x$-complex for some $x\in S^2$, is a minimal submanifold with equal Kähler angles of each $(N,J_{y},g)$ (\[S-V,2\]), and the common Kähler angle is given by $\cos\theta(p)=\|(J_{y}X)^{\top}\|$, where $X$ is any unit vector of $T_{p}M$. A proof of this assertion is simply to remark that, if $\{X, J_xX, Y, J_x Y\}$ is an o.n. basis of $T_pM$, then the matrix of the Kähler form $\omega_y$ w.r.t. $J_y$, restricted to this basis, is just a multiple of a matrix in ${I\!\!R^{4}}$ that represents an orthogonal complex structure of ${I\!\!R^{4}}$, i.e. of the type $aI+bJ+cK$, where $I,J,K$ defines the usual hyper-Kähler structure of ${I\!\!R^{4}}$, and $a^2+b^2+c^2=1$. The square of this multiple is given by $\langle x,y\rangle^2+\langle J_y X,Y\rangle^2 +\langle J_{x\times y}X,Y\rangle^2=
\|(J_y X)^{\top}\|^2$. This example suggests us a way to build examples of (local) submanifolds with equal Kähler angles. Let $(N,I,g)$ be a Kähler manifold of complex dimension 4, and $U\subset N$ an open set where an orthornormal frame of the form $\{X_1,IX_1, X_2, IX_2, Y_1,IY_1, Y_2, IY_2 \}$ is defined. If for each $p\in U$, we identify $T_p N$ with ${I\!\!R^{4}}\times {I\!\!R^{4}}$, through this frame, we are defining a family of local $g$-orthogonal almost complex structures $ J_{x}= a i\times i+b j\times j+c k\times k$, for $x=(a,b,c) \in S^2$, where $i,j,k$ denotes de canonical hyper-Kähler structure of ${I\!\!R^{4}}$. Then any almost $J_x$-complex 4-dimensional submanifold $M$ is a submanifold with equal Kähler angles of the Kähler manifold $(N, I, g)$. It may not be minimal, because $J_x$ may not be a Kähler structure, or not even integrable.
Such a condition on the Kähler angles, turns out to be more restrictive for submanifolds of non Ricci-flat manifolds, or if $M$ is closed, that is, compact and orientable. A combination of the formula of $\triangle\kappa$ for minimal immersions with equal Kähler angles, with the Weitzenböck formula for ${F^{*}\omega}$, lead us in \[S-V,2\] to the conclusion that the Kähler angle must be constant, and in general it is either $0$ or $\frac{\pi}{2}$. Namely, we have:
Let $F:M^{2n}{\rightarrow}N^{2n}$ be a minimal immersion with equal Kähler angles.\
$(i)~($[**\[W,1\]**]{}$)$ If $n=1$, $M$ is closed, $R<0$, and $F$ has no complex points, then $F$ is Lagrangian.\
$(ii)~($[**\[S-V,2\], \[G\]**]{}$)$ If $n=2$ and $R\neq 0$, then $F$ is either a complex or a Lagrangian submanifold.\
$(iii)~($[**\[S-V,2\]**]{}$)$ If $n\geq 3$, $M$ is closed, and $R<0$, then $F$ is either a complex or a Lagrangian submanifold.\
$(iv)~($[**\[S-V,2\]**]{}$)$ If $n\geq 3$, $M$ is closed, $R=0$, then the common Kähler angle must be constant.
If $n=2$ and $R=0$ we cannot conclude the Kähler angle is constant. It is easy to find examples of minimal immersions with constant and non-constant equal Kähler angle, for the case of $M$ not compact and $N$ the Euclidean space. Namely, the most simple family of submanifolds with constant equal Kähler angle of $~{l\!\!\! C}^{2n}$ can be given by the vector subspaces defined by a linear map $F:{I\!\!R^{2n}}{\rightarrow}{l\!\!\! C}^{2n}\equiv ({I\!\!R^{2n}}\times {I\!\!R^{2n}}, J_{0})$, $F(X)=(X, a {J_{\omega}}X)$, where $a$ is any real number and ${J_{\omega}}$ is a $g_0$-orthogonal complex structure of ${I\!\!R^{2n}}$, and where $g_0$ is the Euclidean metric and $J_0(X,Y)=(-Y,X)$. These are totally geodesic submanifolds with constant equal Kähler angle $\cos\theta=\frac{2|a|}{1+a^2}$, and ${F^{*}\omega}(X,Y)= \cos\theta \,F^{*}\!g_{0}\,(\pm{J_{\omega}}X,Y)$, with $F^{*}\!g_{0}$ a ${J_{\omega}}$-hermitian euclidean metric. In (\[D-S\]) we have the following example of non-constant Kähler angle well away from $0$. The graph of the anti-$i$-holomorphic map $f:{I\!\!R^{4}}{\rightarrow}{I\!\!R^{4}}$ given by $f(x,y,z,w)=(u,v,-u,-v)$, where $$\begin{array}{l}
u(x,y,z,w)=\phi(x+z)\xi '(y+w),\\
v(x,y,z,w)=-\phi'(x+z)\xi(y+w)\\
\phi(t)=\sin t,~~~\xi(t)=\sinh t, \end{array}$$ defines a minimal complete submanifold of ${l\!\!\! C}^{4}$ with equal Kähler angles satisfying $$\cos\theta =\frac{2\sqrt{\cos^2(x+z)+\sinh^2(y+w)}}{1+4(\cos^2(x+z)+\sinh^2(y+w))}.$$ This graph has no complex points, for $0\leq \cos\theta\leq {{\mbox{\scriptsize $\frac{1}{2}$}}}$, and the set of Lagrangian points is a infinite discrete union of disjoint 2-planes, $${\cal L}=\bigcup_{-\infty\leq k\leq +\infty} span_{{I\!\!R^{}}}\{(1,0,-1,0), (0,1,0,-1)\} +(0,0,({{\mbox{\scriptsize $\frac{1}{2}$}}}+k)\pi,0).$$
In this paper we present a formula for $\triangle\kappa$, but now not assuming minimality of $F$, obtaining some extra terms involving the mean curvature $H$ of $F$. We will see that the above conclusions still hold for $F$ not minimal, but under certain weaker condition on the mean curvature of $F$. These conclusions show how rigid Kähler-Einstein manifolds are with respect to the Kähler angles and the mean curvature of a submanifold, leading to some non-existence of certain types of submanifolds, depending on the sign of the scalar curvature $R$ of $N$ and on the dimension $n$.
We summarize the main results of this paper:
Assume $n=2$, and $M$ is closed, $N$ is non Ricci-flat, and $F:M{\rightarrow}N$ is an immersion with equal Kähler angles, $\theta_{{\alpha}}=\theta ~\forall{\alpha}$. If $$R\,{F^{*}\omega}((JH)^{\top}, \nabla \sin^2\theta)\leq 0$$ then $F$ is either a complex or a Lagrangian submanifold. This is the case when $F$ has constant Kähler angle.
Let $n=2$, $R<0$, and $F:M{\rightarrow}N$ be a closed submanifold with parallel mean curvature and equal Kähler angles. If $\|H\|^2 \geq -\frac{R}{8}\sin^2\theta$, then $F$ is either a complex or a Lagrangian submanifold.
Assume $M$ is closed, $n\geq 3$, and $F:M{\rightarrow}N$ is an immersion with equal Kähler angles.\
$(A)$ If $R< 0$, and if $ \delta{F^{*}\omega}((JH)^{\top})\geq 0 $, then $F$ is either complex or Lagrangian.\
$(B)$ If $R=0$, and if $\delta{F^{*}\omega}((JH)^{\top})\geq 0 $, then the Kähler angle is constant.\
$(C)$ If $F$ has constant Kähler angle and $R\neq 0$, then $F$ is either complex or Lagrangian.
In case $n=1 $ we obtain:
If $M$ is a closed surface and $N$ is a non Ricci-flat Kähler-Einstein surface, then any immersion $F:M{\rightarrow}N$ either has complex or Lagrangian points. In particular, if $F$ has constant Kähler angle, then $F$ is either a complex or a Lagrangian submanifold.
This generalizes a result in \[M-U\], for compact surfaces immersed with constant Kähler angle (and so orientable, if not Lagrangian) into ${l\!\!\! C}\!I\!\!P^2$.\
For $M$ not necessarily compact we have the following proposition:
If $F:M{\rightarrow}N$ is an immersion with constant equal Kähler angle $\theta$ and with parallel mean curvature, then:\
(1) If $R=0$, $F$ is either Lagrangian or minimal.\
(2) If $R>0$, $F$ is either Lagrangian or complex.\
(3) If $R<0$, $F$ is either Lagrangian, or $\|H\|^2 = -\frac{\sin^2\theta}{4n}R.$\
(4) If $H=0$, then $R=0$ or $F$ is either Lagrangian or complex.
Note that (4) of the above proposition is an improvement of Theorem 1.3 of \[S-V,2\], for, compactness is not required now. We also observe that from Corollary 1.1, if $n=2$ and $M$ were closed, that later case of (3) implies as well $F$ to be complex or Lagrangian. Compactness of $M$ is a much more restrictive condition. In \[K-Z\] it is shown that, if $n=1$ and $N$ is a complex space form of constant holomorphic sectional curvature $4\rho$ and $M$ is a surface of non-zero parallel mean curvature and constant Kähler angle, then either $F$ is Lagrangian and $M$ is flat, or $\sin\theta=-\sqrt{
\frac{8}{9}}$, $\rho=-\frac{3}{4}\|H\|^{2}$ and $M$ has constant Gauss curvature $K=-\frac{\|H\|^2}{2}$. These values of $\theta$ and $\rho$ ($R=6\rho$) are according to our relation in (3) of Proposition 1.2. Chen in \[Che,2\] and \[Che-T,2\] shows explicitly all possible examples of such (non-compact) surfaces of the 2-dimensional complex hyperbolic spaces. In \[K-Z\] it is also shown all examples of surfaces immersed into $~{l\!\!\! C}\! I\!\!H^2$ with non-zero parallel mean curvature and non-constant Kähler angle. In case (1), if $F$ is not minimal, then $(JH)^{\top}$ defines a global nonzero parallel vector field on $M$ (see Proposition 3.6 of section 3).
Let $F$ be a closed surface immersed with parallel mean curvature into a non Ricci flat Kähler-Einstein surface . If $F$ has no complex points and if $\frac{{F^{*}\omega}}{Vol_{M}}\geq 0$ (or $\leq 0$) on all $M$, then $F$ is Lagrangian. If $F$ has no Lagrangian points, then $F$ is minimal.
Some formulas on the Kähler angles
==================================
On $M$ we take the induced metric $g_{M}=F^{*}g$, that we also denote by $\langle,\rangle$. We denote by ${{\mbox{\large $\nabla$}}_{\!\!}}$ both Levi-Civita connections of $M$ and $N$, and by ${{\mbox{\large $\nabla$}}_{\!\!X}}dF(Y)={{\mbox{\large $\nabla$}}_{\!\!}}dF(X,Y)$ the second fundamental form of $F$, a symmetric tensor on $M$ with values on the normal bundle $NM=(dF(TM))^{\bot}$ of $F$. The mean curvature of $F$ is given by $H=\frac{1}{2n}trace{{\mbox{\large $\nabla$}}_{\!\!}}dF$. At each point $p\in M$, let $\{X_{{\alpha}}, Y_{{\alpha}}\}_{1\leq {\alpha}\leq n}$ be a $g_{M}$-orthonormal basis of eigenvectors of ${F^{*}\omega}$. On that basis, ${F^{*}\omega}$ is a $2n\times 2n$ block matrix $$F^{*}\omega=\bigoplus_{0\leq {\alpha}\leq n} \left[ \begin{array}{cc}
0 & -\cos\theta_{{\alpha}} \\
\cos\theta_{{\alpha}} & 0 \end{array} \right],$$ where $\cos\theta_{1}\geq \cos\theta_{2}\geq \ldots \geq \cos\theta_{n}
\geq 0$, are the corresponding eigenvalues ordered in decreasing way. The angles $\{\theta_{{\alpha}}\}_{1\leq {\alpha}\leq n}$ are the Kähler angles of $F$ at $p$. We identify the two form ${F^{*}\omega}$ with the skew-symmetric operator of $T_{p}M$, $({F^{*}\omega})^{\sharp}:T_{p}M{\rightarrow}T_{p}M$, using the musical isomorphism with respect to $g_{M}$, that is, $g_{M}((F^{*}\omega)^{\sharp}(X),Y)$ $=F^{*}\omega(X,Y)$, and we take its polar decomposition, $(F^{*}\omega)^{\sharp} = |(F^{*}\omega)^{\sharp}| \, {J_{\omega}}$, where $J_{\omega}:T_{p}M{\rightarrow}T_{p}M$ is a partial isometry with the same kernel ${\cal K}_{\omega}$ as of $F^{*}w$, and where $|(F^{*}\omega)^{\sharp}|=\sqrt{-(F^{*}\omega)^{\sharp 2} }$. On ${\cal K}^{\bot}_{\omega}$, the orthogonal complement of ${{\cal K}_{\omega}}$ in $T_{p}M$, $J_{\omega}:{\cal K}^{\bot}_{\omega}{\rightarrow}{\cal K}^{\bot}_{\omega}$ defines a $g_{M}$-orthogonal complex structure. On a open set without complex directions, that is $\cos\theta_{{\alpha}}<1$ $\forall{\alpha}$, we consider the locally Lipschitz map $$\kappa=\sum_{1\leq {\alpha}\leq n}\log \left(\frac{1+\cos\theta_{{\alpha}}}{1-\cos\theta_{{\alpha}}}\right).$$ For each $0\leq k\leq n$, this map is smooth on the largest open set $\Omega^{0}_{2k}$, where ${F^{*}\omega}$ has constant rank $2k$. On a neighbourhood of a point $p_{0}\in
\Omega^{0}_{2k}$, we may take $\{X_{{\alpha}}, Y_{{\alpha}}\}_{1\leq {\alpha}\leq n}$ a smooth local $g_{M}$-orthonormal frame of $M$, with $Y_{{\alpha}}=J_{\omega}X_{{\alpha}}$ for ${\alpha}\leq k$, and where $\{X_{{\alpha}},Y_{{\alpha}}\}_{{\alpha}\geq k+1}$ is any $g_{M}$-orthonormal frame of ${\cal K}_{\omega}$. Moreover, we may assume that this frame diagonalizes $F^{*}\omega$ at $p_{0}$. Following the computations of the appendix in \[S-V,2\], without requiring now minimality, we see that the components of the mean curvature of $F$ appear three times in the formula for $\triangle \kappa$. Namely, when we compute $(5.9)$ and $(5.10)$ of \[S-V,2\], we get respectively, the extra terms $ig(\frac{n}{2}{{\mbox{\large $\nabla$}}_{\!\!\mu}}H, JdF({\bar{\mu}}))$ and $-ig(\frac{n}{2}{{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}H, JdF(\mu))$, and when we sum $\sum_{{\beta}}-R^M(\mu,{\bar{\beta}},{\beta},{\bar{\mu}})-R^M({\bar{\mu}},{\bar{\beta}},{\beta},\mu)$ we obtain the extra term $ng(H,{{\mbox{\large $\nabla$}}_{\!\!\mu}}dF({\bar{\mu}}))$. Then, we have to add in the final expression for $\sum_{{\beta}} Hess \tilde{g}_{\mu{\bar{\mu}}}({\beta},{\bar{\beta}})$ of Lemma 5.4 of \[S-V,2\] the expression $\sum_{{\beta}}ig(\frac{n}{2}{{\mbox{\large $\nabla$}}_{\!\!\mu}}H, JdF({\bar{\mu}}))
-ig(\frac{n}{2}{{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}H, JdF(\mu)) +\cos\theta_{\mu}ng(H,{{\mbox{\large $\nabla$}}_{\!\!\mu}}dF({\bar{\mu}}))$. Introducing these extra terms in the term $\sum_{{\beta},\mu}\frac{32}{\sin^2\theta_{\mu}}Hess
\tilde{g}_{\mu{\bar{\mu}}}({\beta},{\bar{\beta}})$ of $(5.7)$ of \[S-V,2\], we obtain our more general formula for $\triangle \kappa$:
For any immersion $F$, at a point $p_0$ on a open set where ${F^{*}\omega}$ has constant rank $2k$ and no complex directions, we have $$\begin{aligned}
\triangle \kappa
&=&4i\sum_{{\beta}} Ricci^{N}(JdF({{\beta}}),dF({{\bar{\beta}}}))\\[-2mm]
&&+\sum_{{\beta},\mu}\!\frac{32}{\sin^{2}\theta_{\mu}}Im {\mbox{\Large $($}}\!
R^{N}(dF({{\beta}}),dF({\mu}),dF({{\bar{\beta}}}),
JdF({{\bar{\mu}}})\!+\!i\cos\theta_{\mu}dF({\bar{\mu}}))\!{\mbox{\Large $)$}}{\nonumber}\\[-2mm]
&&-\sum_{{\beta},\mu,\rho}\!\!\frac{64(\cos\theta_{\mu}
\!+\!\cos\theta_{\rho})}{\sin^{2}\theta_{\mu}\sin^{2}\theta_{\rho}}
Re{\mbox{\large $($}}{g(\mbox{\large ${\nabla}$}_{\!\!{{\beta}}} dF({\mu}),JdF({{\bar{\rho}}}))}{g(\mbox{\large ${\nabla}$}_{\!\!{{\bar{\beta}}}} dF({\rho}),JdF({{\bar{\mu}}}))}\!{\mbox{\large $)$}}
{\nonumber}\\[-2mm]
&& +\sum_{{\beta},\mu,\rho}\!\!\!
\frac{32(\cos\theta_{\rho}\!\!-\!\cos\theta_{\mu})}
{\sin^{2}\theta_{\mu}\sin^{2}\theta_{\rho}}\;(|{g(\mbox{\large ${\nabla}$}_{\!\!{{\beta}}} dF({\mu}),JdF({\rho}))}|^{2}
\!\!+\!|{g(\mbox{\large ${\nabla}$}_{\!\!{{\bar{\beta}}}} dF({\mu}),JdF({\rho}))}|^{2}){\nonumber}\\[-1mm]
&&+\sum_{{\beta},\mu,\rho}\frac{32(\cos\theta_{\mu}+\cos\theta_{\rho})}
{\sin^{2}\theta_{\mu}}\,{\mbox{\LARGE $($}}
|\langle{{\mbox{\large $\nabla$}}_{\!\!{{\beta}}}}\mu,{\rho}\rangle|^{2} +
|\langle{{\mbox{\large $\nabla$}}_{\!\!{{\bar{\beta}}}}}\mu,{\rho}\rangle|^{2}{\mbox{\LARGE $)$}}{\nonumber}\\
&&+\!\!\sum_{\mu}\!\!\frac{8n}{\sin^{2}\theta_{\mu}}{\mbox{\LARGE $($}}ig{\mbox{\large $($}}
{{\mbox{\large $\nabla$}}_{\!\!\mu}}H,\!JdF({\bar{\mu}}){\mbox{\large $)$}}
\!-\!ig{\mbox{\large $($}}{{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}H,\!JdF(\mu){\mbox{\large $)$}}\!+\!2\cos\theta_{\mu}
g( H,{{\mbox{\large $\nabla$}}_{\!\!\mu}}dF({\bar{\mu}})){\mbox{\LARGE $)$}}{\nonumber}\end{aligned}$$ where $``{\alpha}"=Z_{{\alpha}}=\frac{X_{{\alpha}}-i Y_{{\alpha}}}{2}$ and $``\bar{{\alpha}}"=
\overline{Z_{{\alpha}}}$.
Projecting $JH$ on $dF(TM)$, we define a vector field $(JH)^{\top}$ on $M$, and we denote by $((JH)^{\top})^{\flat}$ the corresponding 1-form, $((JH)^{\top})^{\flat}(X)$ $=g_{M}((JH)^{\top},X)$ $=g(JH,dF(X))$. If $F$ is a Lagrangian immersion, the above formula on $\triangle\kappa$ leads to a well-known result:
[**(\[W,2\])**]{} If $F$ is a Lagrangian immersion, then $((JH)^{\top})^{\flat}$ is a closed 1-form on $M$.
A proof of this corollary will be given in section 3. The formula (2.1) is considerably simplified when $F$ is an immersion with equal Kähler angles. Now we recall the Weitzenböck formula for ${F^{*}\omega}$, that we used in \[S-V,2\] $$\frac{1}{2}\triangle \|{F^{*}\omega}\|^{2} =-\langle \triangle {F^{*}\omega},{F^{*}\omega}\rangle
+\|{{\mbox{\large $\nabla$}}_{\!\!}}{F^{*}\omega}\|^{2}+\langle S{F^{*}\omega}, {F^{*}\omega}\rangle,$$ where $\langle,\rangle$ denotes the Hilbert-Schmidt inner product for 2-forms, and $S$ is the Ricci operator of $\bigwedge^{2}T^{*}M$, and $\triangle = d\delta +\delta d$ is the the Laplacian operator on forms. ${F^{*}\omega}$ is a closed 2-form. If it is also co-closed, that is $\delta{F^{*}\omega}=0$, then it is harmonic. If $M$ is compact, $$\int_{M}\langle\triangle{F^{*}\omega}, {F^{*}\omega}\rangle Vol_{M}
= \int_{M}\|\delta{F^{*}\omega}\|^{2}Vol_{M}$$ We will use this formula when $F$ has equal Kähler angles.
Immersions with equal Kähler angles
===================================
In this section we recall some formulas for immersions with equal Kähler angles. $F$ is said to have equal Kähler angles, if all the angles are equal, $\theta_{{\alpha}}=\theta$ $\forall {\alpha}$. In this case, $({F^{*}\omega})^{\sharp}=\cos\theta {J_{\omega}}$, and ${J_{\omega}}$ is a smooth almost complex structure away from the set of Lagrangian points ${\cal L}=\{p\in M: \cos\theta(p)=0\}$. Let ${\cal L}^{0}$ denote the largest open set of ${\cal L}$, ${\cal C}=\{p\in M: \cos\theta(p)=1\}$ the set of complex points, and ${\cal C}^{0}$ its largest open set. Recall that $\cos^{2}\theta$ is smooth on all $M$, while $\cos\theta$ is only locally Lipschitz on $M$, but smooth on ${\cal L}^{0}\cup (M\sim {\cal L})$. For immersions with equal Kähler angles, any local frame of the form $\{X_{{\alpha}}, Y_{{\alpha}}={J_{\omega}}X_{{\alpha}}\}_{1\leq {\alpha}\leq n}$ diagonalizes ${F^{*}\omega}$ on the whole set where it is defined. We use the letters ${\alpha}, {\beta},\mu,\ldots$ to range on the set $\{1,\ldots, n\}$ and the letters $j,k,\ldots$ to range on $\{1,\ldots, 2n\}$. As in the previous section, we denote by $``{\alpha}"=Z_{{\alpha}}=\frac{X_{{\alpha}}-
i Y_{{\alpha}}}{2}$ and $``\bar{{\alpha}}"=\overline{Z_{{\alpha}}}=\frac{X_{{\alpha}}+
i Y_{{\alpha}}}{2}$, defining local frames on the complexifyied tangent space of $M$.
On tensors and forms we use the Hilbert-Shmidt inner product. We denote by $\delta$ the divergence operator on (vector valued) forms, and by $div_{M}$ the divergence operator on vector fields over $M$. The $(1,1)$-part of ${{\mbox{\large $\nabla$}}_{\!\!}}dF$ with respect to ${J_{\omega}}$, is given by $({{\mbox{\large $\nabla$}}_{\!\!}}dF)^{(1,1)}(X,Y)=$ ${{\mbox{\scriptsize $\frac{1}{2}$}}}({{\mbox{\large $\nabla$}}_{\!\!}}dF(X,Y) + {{\mbox{\large $\nabla$}}_{\!\!}}dF({J_{\omega}}X,{J_{\omega}}Y))$. This tensor is defined away from Lagrangian points, and it vanish on ${\cal C}^{0}$, for, on that set, $F$ is a complex submanifold of $N$, and ${J_{\omega}}$ is the induced complex structure.
[**\[S-V,2\]**]{} On $(M\sim{\cal L})\cup {\cal L}^{0}$, $$\begin{aligned}
\|{F^{*}\omega}\|^{2} & = & n\cos^{2}\theta\\
\|{{\mbox{\large $\nabla$}}_{\!\!}}{F^{*}\omega}\|^{2} &=& n\|\nabla\cos\theta\|^{2}+
\frac{1}{2}\cos^{2}\theta\|{{\mbox{\large $\nabla$}}_{\!\!}}{J_{\omega}}\|^{2}\\
\delta({F^{*}\omega})^{\sharp} &=& (\delta{F^{*}\omega})^{\sharp}
=(n-2){J_{\omega}}(\nabla\cos\theta) \\
\|\delta{F^{*}\omega}\|^{2} &=& (n-2)^{2}\|\nabla\cos\theta\|^{2}\\
\cos\theta\delta{J_{\omega}}&= &(n-1){J_{\omega}}(\nabla\cos\theta)\end{aligned}$$ and on $ {\mbox{\Large $($}}M\sim ({\cal L}\cup{\cal C}){\mbox{\Large $)$}}\cup {\cal L}^{0}
\cup {\cal C}^{0}$,\
$$(1\!-\!n)\nabla\sin^{2}\theta =
16\cos\theta\, Re{\mbox{\LARGE $($}} i\sum_{{\beta},\mu}
{\mbox{\Large $($}}{g(\mbox{\large ${\nabla}$}_{\!\!{{\bar{\mu}}}} dF({\mu}),JdF({{\beta}}))}\!-\!
{g(\mbox{\large ${\nabla}$}_{\!\!{{\bar{\mu}}}} dF({{\beta}}),JdF({\mu}))}{\mbox{\Large $)$}}{\bar{\beta}}{\mbox{\LARGE $)$}}.$$ In particular, for $n\neq 2$, ${J_{\omega}}(\nabla\cos\theta)$, $\|\nabla\cos\theta\|^{2}$, $\cos^{2}\theta\|{{\mbox{\large $\nabla$}}_{\!\!}}{J_{\omega}}\|^{2}$, and $\cos\theta\,\delta{J_{\omega}}$ can be smoothly extended to all $M$. Furthermore, for $n\geq 2$, there is a constant $C>0$ such that on $M$, $\|\nabla\sin^2\theta\|^2 \leq C
\cos^2\theta€~\sin^2\theta~ \|({{\mbox{\large $\nabla$}}_{\!\!}}dF)^{(1,1)}\|^2$.
The estimate on $\|\nabla\sin^2\theta\|^2$ given above follows from the expression on $(1-n)\nabla \sin^2\theta $ and the following explanation. From Schwarz inequality, $|g({{\mbox{\large $\nabla$}}_{\!\!X}}dF(Y),JdF(Z))|= |g({{\mbox{\large $\nabla$}}_{\!\!X}}dF(Y),\Phi(Z))|\leq
\|{{\mbox{\large $\nabla$}}_{\!\!X}}dF(Y)\|\, \|\Phi(Z)\|$, where $\Phi(Z)=(JdF(Z))^{\bot}$, and $(~)^{\bot}$ denotes the orthogonal projection onto the normal bundle. But (cf \[S-V,2\]) $JdF(Z) = \Phi(Z) + dF(({F^{*}\omega})^{\sharp}(Z))$. An elementary computation shows that $$\|\Phi(Z)\|^2 = g{\mbox{\Large $($}}JdF(Z)\!-\!dF(({F^{*}\omega})^{\sharp}(Z)),
JdF(Z)\!-\!dF(({F^{*}\omega})^{\sharp}(Z)){\mbox{\Large $)$}}
=\sin^2\theta\, \|Z\|^{2}$$ Obviously the formula on $\nabla\sin^2\theta$ as well the estimate on $\|\nabla\sin^2\theta\|^2$, are still valid on all complex and Lagrangian points, since those points are critical points for $ \sin^2\theta$, and at complex points $JdF(TM)\subset TM$ . Also
If $n=2$, ${F^{*}\omega}$ is an harmonic 2-form. If $n\neq 2$, ${F^{*}\omega}$ is co-closed iff $\theta$ is constant. For any $n\geq 2$, if $(M\sim {\cal L}, {J_{\omega}}, g_{M})$ is Kähler, then $\theta$ is constant and ${F^{*}\omega}$ is parallel.
Following chapter $4$ of \[S-V,2\] and using the new expression for $\triangle\kappa$ of Proposition $2.1$, with the extra terms involving the mean curvature $H$, and noting that now both $(4.4)$ and $(4.7)+(4.5)$ of \[S-V,2\] have extra terms involving $H$, we obtain:
Away from complex and Lagrangian points, $$\begin{aligned}
\lefteqn{
\triangle\kappa=}\\[-1mm]
&=&\!\!\!\! \cos\theta {\mbox{\Large $($}}\! -2nR
+\frac{32}{\sin^{2}\theta}\sum_{{\beta},\mu}\!\! R^{M}({\beta},\mu,{\bar{\beta}},{\bar{\mu}})
+\frac{1}{\sin^{2}\theta}\|{{\mbox{\large $\nabla$}}_{\!\!}}{J_{\omega}}\|^{2}
+\frac{8(n\!-\!1)}{\sin^{4}\theta}\|\nabla\cos\theta\|^{2}{\mbox{\Large $)$}}\\[-1mm]
&&-\frac{16n}{\sin^{4}\theta}\cos\theta\sum_{{\beta}}d\cos\theta{\mbox{\Large $($}}
i g (H, JdF({\beta})){\bar{\beta}}-ig( H, JdF({\bar{\beta}})){\beta}{\mbox{\Large $)$}}\\[-1mm]
&&+\frac{8n}{\sin^{2}\theta}\sum_{\mu}{\mbox{\Large $($}}
ig( {{\mbox{\large $\nabla$}}_{\!\!\mu}}H,JdF({\bar{\mu}}))-ig( {{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}H,JdF(\mu))
{\mbox{\Large $)$}}.\end{aligned}$$
Let us denote by ${{\mbox{\large $\nabla$}}^{\bot}_{\!\!}}$ the usual connection in the normal bundle, and denote by $(JH)^{\top}$ the vector field of $M$ given by $$g_{M} ((JH)^{\top}, X)= g(JH, dF(X))~~~~~\forall X\in TM.\\[3mm]$$
$\forall X,Y\in T_{p}M$, $$\begin{array}{lrcl}
(i)&g({{\mbox{\large $\nabla$}}_{\!\!X}}H,JdF(Y)) &=& -\langle
{{\mbox{\large $\nabla$}}_{\!\!X}}(JH)^{\top}, Y\rangle - g( H, J{{\mbox{\large $\nabla$}}_{\!\!X}}dF(Y))~~~~~~~~~~~~~~(\mbox{on}~M)\\
& &=& -g(H,{{\mbox{\large $\nabla$}}_{\!\!X}}dF (({F^{*}\omega})^{\sharp}(Y))) + g({{\mbox{\large $\nabla$}}^{\bot}_{\!\!X}}H,JdF(Y))~~~~~(\mbox{on}~M)
\\[4mm]
(ii)&({{\mbox{\scriptsize $\frac{1}{2}$}}}{J_{\omega}}( (JH)^{\top})&=&\sum_{{\beta}}i g(H,
JdF({\beta})){\bar{\beta}}-i g( H, JdF({\bar{\beta}})){\beta}~~~~~~~~~~(\mbox{on~}M\sim{\cal L})
\end{array}$$ $$\begin{aligned}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
(iii)~~~~~~~~~\lefteqn{\sum_{\mu}2i g( {{\mbox{\large $\nabla$}}_{\!\!\mu}}H,\!JdF({\bar{\mu}}))
-2ig( {{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}H,\!JdF(\mu))=}\\[-2mm]
&=&\sum_{\mu} 4 Im\langle {{\mbox{\large $\nabla$}}_{\!\!\mu}} (JH)^{\top}, {\bar{\mu}}\rangle=
-\sum_{\mu}2i d((JH)^{\top})^{\flat}(\mu, {\bar{\mu}})~~~~~~~(\mbox{on}~M)\\[-2mm]
&=& -2n\cos\theta\|H\|^2 - 4 \sum_{\mu} Im {\mbox{\Large $($}} g({{\mbox{\large $\nabla$}}^{\bot}_{\!\!\mu}}H,JdF({\bar{\mu}})){\mbox{\Large $)$}}~~~~~~~~~~~(\mbox{on}~M)\\[-1mm]
&=&- div_{M}{\mbox{\Large $($}}\!{J_{\omega}}((JH)^{\!\top}){\mbox{\Large $)$}}
+ \langle (JH)^{\!\top}\!,\delta{J_{\omega}}\rangle ~~~~~~~~~~~~~~~~~~~
(\mbox{on~}M\sim{\cal L}).\\[-3mm]\end{aligned}$$ (iv) $div_{M}((JH)^{\top}) = \sum_{\mu}-4 Re{\mbox{\Large $($}} g( {{\mbox{\large $\nabla$}}^{\bot}_{\!\!\mu}}H, JdF({\bar{\mu}})){\mbox{\Large $)$}}
~~~~~~~~~~~~~~~~~~(\mbox{on~}M)$.\
*Proof. *Assume that ${{\mbox{\large $\nabla$}}_{\!\!}}~Y(p)=0$. Then we have at the point $p$ $$\begin{aligned}
g( {{\mbox{\large $\nabla$}}_{\!\!X}} H, JdF(Y)) &=& d {\mbox{\Large $($}}g( H, JdF(Y)){\mbox{\Large $)$}} (X)
- g( H, {{\mbox{\large $\nabla$}}_{\!\!X}} (JdF(Y)))\\
&=& -d \langle (JH)^{\top}, Y\rangle (X)
- g( H, J {{\mbox{\large $\nabla$}}_{\!\!X}}dF(Y) ) \\
&=& - \langle {{\mbox{\large $\nabla$}}_{\!\!X}}(JH)^{\top}, Y \rangle
- g( H, J {{\mbox{\large $\nabla$}}_{\!\!X}}dF(Y)).\end{aligned}$$ On the other hand, from $JdF(Y)= dF( ({F^{*}\omega})^{\sharp}(Y)) + (JdF(Y))^{\bot}$, we get the second equality of $(i)$. For $p\in M\sim{\cal L}$, since ${J_{\omega}}{\beta}=i{\beta}$, and ${J_{\omega}}{\bar{\beta}}=-i{\bar{\beta}}$, $$\begin{aligned}
\lefteqn{\sum_{{\beta}}i g( H, JdF({\beta})) {\bar{\beta}}-i g( H, JdF({\bar{\beta}})) {\beta}=}\\[-2mm]
&&\begin{array}{rcl}
&=& \sum_{{\beta}}~g(H, JdF({J_{\omega}}{\beta})) {\bar{\beta}}+ g( H, JdF({J_{\omega}}{\bar{\beta}})) {\beta}\\[2mm]
&=&\sum_{{\beta}}~- g( JH, dF({J_{\omega}}{\beta})) {\bar{\beta}}- g( JH, dF({J_{\omega}}{\bar{\beta}})) {\beta}\\[2mm]
&=&\sum_{{\beta}}~-\langle (JH)^{\top}, {J_{\omega}}{\beta}\rangle {\bar{\beta}}-\langle (JH)^{\top},
{J_{\omega}}{\bar{\beta}}\rangle {\beta}\\[2mm]
&=&\sum_{{\beta}}~\langle {J_{\omega}}((JH)^{\top}), {\beta}\rangle {\bar{\beta}}+
\langle {J_{\omega}}((JH)^{\top}),
{\bar{\beta}}\rangle {\beta}\\[2mm]
& =&{{\mbox{\scriptsize $\frac{1}{2}$}}}{J_{\omega}}((JH)^{\top}),
\end{array}\end{aligned}$$ and $(ii)$ is proved. From the first equality of $(i)$, $$\begin{aligned}
\lefteqn{\sum_{\mu} i g({{\mbox{\large $\nabla$}}_{\!\!\mu}}H, JdF({\bar{\mu}}))
-i g( {{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}H, JdF(\mu))=}\\[-1mm]
&=& \sum_{\mu}-i\langle {{\mbox{\large $\nabla$}}_{\!\!\mu}}(JH)^{\top}, {\bar{\mu}}\rangle
+i\langle {{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}(JH)^{\top}, \mu\rangle=\sum_{\mu}2 Im {\mbox{\Large $($}}\langle {{\mbox{\large $\nabla$}}_{\!\!\mu}}(JH)^{\top}, {\bar{\mu}}\rangle{\mbox{\Large $)$}} \\[-1mm]
&=&\sum_{\mu}-id((JH)^{\top})^{\flat}(\mu,{\bar{\mu}}).\end{aligned}$$ On the other hand, from second equality of $(i)$ $$\begin{aligned}
\sum_{\mu} g({{\mbox{\large $\nabla$}}_{\!\!\mu}}H, JdF({\bar{\mu}}))&=&
\sum_{\mu}-g(H,{{\mbox{\large $\nabla$}}_{\!\!\mu}}dF(\cos\theta {J_{\omega}}({\bar{\mu}})))
+g({{\mbox{\large $\nabla$}}^{\bot}_{\!\!\mu}}H,JdF({\bar{\mu}}))\\
&=&\frac{ni}{2} \cos\theta\, g(H,H) +\sum_{\mu}g({{\mbox{\large $\nabla$}}^{\bot}_{\!\!\mu}}H,JdF({\bar{\mu}})).\end{aligned}$$ Hence $$\begin{aligned}
\lefteqn{\sum_{\mu} i g({{\mbox{\large $\nabla$}}_{\!\!\mu}}H, JdF({\bar{\mu}}))
-i g( {{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}H, JdF(\mu))=}\\[-1mm]
&=& -n\cos\theta\|H\|^2 -\sum_{\mu}2Im{\mbox{\Large $($}}
g({{\mbox{\large $\nabla$}}^{\bot}_{\!\!\mu}}H,JdF({\bar{\mu}})){\mbox{\Large $)$}}.\end{aligned}$$ Similarly, from $div_{M}((JH)^{\top})=\sum_{\mu} 2\langle
{{\mbox{\large $\nabla$}}_{\!\!\mu}}(JH)^{\top}, {\bar{\mu}}\rangle + 2\langle
{{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}(JH)^{\top}, \mu\rangle$ and $(i)$ we get $(iv)$.\
Finally, using the symmetry of ${{\mbox{\large $\nabla$}}_{\!\!}}dF$ and that $\langle {{\mbox{\large $\nabla$}}_{\!\!Z}}{J_{\omega}}(X), Y\rangle
=-\langle {{\mbox{\large $\nabla$}}_{\!\!Z}}{J_{\omega}}(Y), X\rangle$ (cf. \[S-V,2\]) $$\begin{aligned}
\lefteqn{\sum_{\mu} i g({{\mbox{\large $\nabla$}}_{\!\!\mu}}H, JdF({\bar{\mu}}))
-i g( {{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}H, JdF(\mu))=}\\
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\begin{array}{cl}
&= \sum_{\mu} \langle {{\mbox{\large $\nabla$}}_{\!\!\mu}}(JH)^{\top}, {J_{\omega}}({\bar{\mu}})\rangle
+\langle {{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}(JH)^{\top}, {J_{\omega}}(\mu)\rangle \\[2mm]
&= \sum_{\mu}
-\langle {J_{\omega}}({{\mbox{\large $\nabla$}}_{\!\!\mu}}(JH)^{\top}), {\bar{\mu}}\rangle
-\langle {J_{\omega}}({{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}(JH)^{\top}),\mu\rangle \\[2mm]
&= \sum_{\mu}\!\! -\langle{{\mbox{\large $\nabla$}}_{\!\!\mu}} ({J_{\omega}}(JH)^{\top}\!)
-\!{{\mbox{\large $\nabla$}}_{\!\!\mu}}{J_{\omega}}((JH)^{\top}\!), {\bar{\mu}}\rangle
-\langle {{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}({J_{\omega}}(JH)^{\top}\!)-{{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}{J_{\omega}}((JH)^{\top}\!),
\mu\rangle \\[2mm]
&=-{{\mbox{\scriptsize $\frac{1}{2}$}}}div_{M} ({J_{\omega}}(JH)^{\top}) +\sum_{\mu}\langle {{\mbox{\large $\nabla$}}_{\!\!\mu}}
{J_{\omega}}( (JH)^{\top}), {\bar{\mu}}\rangle + \langle {{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}
{J_{\omega}}( (JH)^{\top}), \mu\rangle\\[2mm]
&=-{{\mbox{\scriptsize $\frac{1}{2}$}}}div_{M} ({J_{\omega}}(JH)^{\top}) +\sum_{\mu} -\langle (JH)^{\top},
{{\mbox{\large $\nabla$}}_{\!\!\mu}}{J_{\omega}}( {\bar{\mu}})\rangle - \langle (JH)^{\top},
{{\mbox{\large $\nabla$}}_{\!\!{\bar{\mu}}}}{J_{\omega}}( \mu)\rangle\\[2mm]
&=-{{\mbox{\scriptsize $\frac{1}{2}$}}}div ({J_{\omega}}(JH)^{\top}) +\langle (JH)^{\top},
{{\mbox{\scriptsize $\frac{1}{2}$}}}\delta{J_{\omega}}\rangle.~~~~~~~~~~~{\mbox{~~~\boldmath $\Box$}}\end{array}\end{aligned}$$\
Using $div(fX)=fdiv(X)+df(X)$, with $f=\frac{1}{\sin^{2}\theta}$, and $X={J_{\omega}}( (JH)^{\top})$, and that $2\cos\theta d\cos\theta=
d\cos^{2}\theta=-d\sin^{2}\theta$, we obtain applying Lemma $3.1$ to Proposition 3.2**
Away from complex and Lagrangian points $$\begin{aligned}
\triangle\kappa &=& {{\mbox{\small $\cos$}}\theta {\mbox{\Large $($}} -2nR
+\frac{32}{\sin^{2}\theta}\sum_{{\beta},\mu} R^{M}({\beta},\mu,{\bar{\beta}},{\bar{\mu}})
+\frac{1}{\sin^{2}\theta}\|{{\mbox{\large $\nabla$}}_{\!\!}}{J_{\omega}}\|^{2}
+\frac{8(n-1)}{\sin^{4}\theta}\|\nabla\cos\theta\|^{2}~{\mbox{\Large $)$}}}\\
&&-div_{M}\left({J_{\omega}}{\mbox{\LARGE $($}}\frac{4n(JH)^{\top}}{\sin^{2}\theta}{\mbox{\LARGE $)$}}
\right) + g_{M}{\mbox{\LARGE $($}}\delta{J_{\omega}}, \frac{4n(JH)^{\top}}{\sin^{2}\theta}{\mbox{\LARGE $)$}}.\end{aligned}$$
If $n=1$ then $(M,{J_{\omega}},g)$ is a Kähler manifold (away from Lagrangian points), and so, $\delta {J_{\omega}}={{\mbox{\large $\nabla$}}_{\!\!}}{J_{\omega}}=0$. Obviously the curvature term on $M$ in the expression of $\triangle\kappa$ vanishes. Then, $\triangle\kappa$ reduces to:
If $n=1$, away from complex and Lagrangian points $$\triangle\kappa = -2R\cos\theta -4div_{M}\left(
{J_{\omega}}{\mbox{\LARGE $($}}\frac{(JH)^{\top}}{\sin^{2}\theta}{\mbox{\LARGE $)$}}.
\right)$$
Now we compute $\triangle\cos^2\theta$ from $\triangle\kappa$ of Proposition 3.3 and applying Proposition 3.1, following step by step the proof of Proposition 4.2 of \[S-V, 2\]. Recall that, if $F$ has equal Kähler angles at $p$, then, at $p$ (cf.\[S-V,2\]) $$\langle S{F^{*}\omega},{F^{*}\omega}\rangle =16\cos^{2}\theta\sum_{\rho,\mu}
R^{M}(\rho,\mu,{\bar{\rho}},{\bar{\mu}}),$$ where $S{F^{*}\omega}$ is the Ricci operator applied to ${F^{*}\omega}$, appearing in the Weitzenböck formula $(2.2)$. If $(M,{J_{\omega}},g_{M})$ is Kähler in a neighbourhood of $p$, then $\langle S{F^{*}\omega},{F^{*}\omega}\rangle=0$ at $p$.
Away from complex and Lagrangian points: $$\begin{aligned}
n\triangle\cos^{2}\theta &=& -2n\sin^{2}\theta\cos^{2}\theta R +
2\langle S{F^{*}\omega}, {F^{*}\omega}\rangle + 2\|{{\mbox{\large $\nabla$}}_{\!\!}}{F^{*}\omega}\|^{2}{\nonumber}\\
&& +4(n-2)\|\nabla \,|\sin\theta|~\|^{2}-4n \,div_{g_{M}}\left( ({F^{*}\omega})^{\sharp}((JH)^{\top})\right) {\nonumber}\\
&&-\frac{4n(2 +(n-4)\sin^2\theta)}{\sin^{2}\theta}\langle
\nabla\cos\theta, {J_{\omega}}( (JH)^{\top})\rangle\end{aligned}$$ The last term $(3.2)$ can be written, for $n=2$ as $$(3.2)= 8\,{F^{*}\omega}((JH)^{\top},\nabla \log\sin^2\theta)$$ and for $n\geq 3$, $$(3.2)= \frac{4n(2 +(n-4)\sin^2\theta)}{\sin^{2}\theta(n-2)}
\delta{F^{*}\omega}((JH)^{\top})$$
The expressions in (3.3) and (3.4) come from Proposition $3.1$ and the fact that $({F^{*}\omega})^{\sharp}=\cos\theta{J_{\omega}}$.\
*Remark 1. *Let ${\omega^{\bot}}=\omega_{| NM}$ be the restriction of the Kähler form $\omega$ to the normal vector bundle $NM$, and ${\omega^{\bot}}= |{\omega^{\bot}}|{J^{\bot}}$ be its polar decomposition, when we identify it with a skew-symmetric operator on the normal bundle, using the musical isomorphism. Let $\cos\sigma_{1}\geq \cos\sigma_{2}\geq \ldots\geq \cos\sigma_{n}\geq 0 $ be the eigenvalues of ${\omega^{\bot}}$. The $\sigma_{{\alpha}}$ are the Kähler angles of $NM$. If $\{U_{{\alpha}},V_{{\alpha}}\}$ is an orthonormal basis of eigenvectors of ${\omega^{\bot}}$ at $p$, then ${\omega^{\bot}}= \sum_{{\beta}}\cos\sigma_{{\beta}} U_{*}^{{\beta}}\wedge V_{*}^{{\beta}}$. For each $p$, $CD(F)= \bigoplus _{{\alpha}:~\cos\theta_{{\alpha}}=1}
span \{ X_{{\alpha}}, Y_{{\alpha}}\}$ defines the vector subspace of complex directions, or equivalently, the largest $J$-complex vector subspace contained in $T_{p}M$. Similarly we define $CD(NM)$, the largest $J$-complex subspace of $NM$ at $p$. Then $$\begin{array}{ccl}
{F^{*}\omega}&=& \omega_{|CD(F)} +\sum_{\cos\theta_{{\alpha}} <1
}\cos\theta_{{\alpha}} X_{*}^{{\alpha}}\wedge Y_{*}^{{\alpha}}\\[2mm]
{\omega^{\bot}}& =& \omega_{|CD(NM)} +
\sum_{\cos\sigma_{{\alpha}} <1}\cos\sigma_{{\alpha}} U_{*}^{{\alpha}}\wedge V_{*}^{{\alpha}}
\end{array}$$ We define the following morphisms between vector bundles of the same dimension $2n$, where $(~)^{\top}$ and $(~)^{\bot}$ denote the orthogonal projection onto $TM$ and $NM$ respectively, $$\begin{array}{cccccc}
\Phi : TM & {\rightarrow}& NM ~~~~~~~~~~~~~~~~~ \Xi : NM & {\rightarrow}& TM\\
X & {\rightarrow}& (JdF(X))^{\bot}~~~~~~~~~~~~~~ U & {\rightarrow}& (JU)^{\top}
\end{array}$$ Then $\Phi^{-1}(0)=CD(F)$, $\Xi^{-1}(0)=CD(NM)$. Note that $\forall X,Y\in TM$ and $\forall U,V\in NM$ $$\begin{array}{cl}
(JdF(X))^{\top}= dF(({F^{*}\omega})^{\sharp} (X))&~~~(JU)^{\bot}= {\omega^{\bot}}(U),\\
\Phi(X)= JdF(X) - dF(({F^{*}\omega})^{\sharp}(X))&~~~\Xi(U)= JU - {\omega^{\bot}}(U).\end{array}$$ A simple computation shows that, if $\cos\theta_{{\alpha}}\neq 1$, we may take $U_{{\alpha}}=\Phi(\frac{Y_{{\alpha}}}{\sin\theta_{{\alpha}}})$, and $V_{{\alpha}}=\Phi(\frac{X_{{\alpha}}}{\sin\theta_{{\alpha}}})$. Moreover, $CD(NM)=CD(F)^{\bot}\cap NM$ and $dim~CD(F)= dim~CD(NM)$. Then ${\omega^{\bot}}$ and ${F^{*}\omega}$ have the same eigenvalues, that is $NM$ and $F$ have the same Kähler angles. We also define $LD(F)=Ker~{F^{*}\omega}={\cal K}_{\omega}$, $LD(NM)=Ker~{\omega^{\bot}}$ the vector subspaces of Lagrangian directions of $F$ and $NM$ respectively. Then we have $J(LD(F))=LD(NM)$. Furthermore, ${J^{\bot}}\circ \Phi= -\Phi\circ {J_{\omega}}$, ${J_{\omega}}\circ \Xi = -\Xi\circ {J^{\bot}}$, $ -\Xi\circ \Phi = Id_{TM} + (({F^{*}\omega})^{\sharp})^{2}$, $ -\Phi\circ \Xi =Id_{NM} + ({\omega^{\bot}})^{2}$. Considering the Hilber-Smidt norms, $\|\Phi\|^{2}=\|\Xi\|^{2}=2\sum_{{\alpha}}\sin^{2}\theta_{{\alpha}}$. If $F$ has equal Kähler angles, $-\Xi\circ \Phi=\sin^{2}\theta Id_{TM}$, $-\Phi\circ \Xi=\sin^{2}\theta Id_{NM}$, and $$g(\Phi(X),\Phi(Y))=\sin^2\theta \langle X,Y\rangle~~~~~~
\langle\Xi(U),\Xi(V)\rangle =\sin^2\theta\, g(U,V).$$ If $F$ has equal Kähler angles, since $NM$ and $F$ have the same Kähler angles, we see that, at a point $p\in M$ such that $H\neq 0$, $(JH)^{\top}=0$ iff $ p$ is a complex point of $F$. We also note that, from lemma 3.1(iv), if $F$ has parallel mean curvature $(JH)^{\top}$ is divergence-free, or equivalentely, $((JH)^{\top})^{\flat}$ is co-closed.\
In \[S-V,2\] we have defined non-negative isotropic scalar curvature, as a less restrictive condition than non-negative isotropic sectional curvature of \[Mi-Mo\]. If such curvature condition on $M$ holds, then $\sum_{\rho,\mu}R^{M}(\rho,\mu,{\bar{\rho}},{\bar{\mu}})\geq 0$, where $\{\rho, {\bar{\rho}}\}_{1\leq \rho\leq n}$ is the complex basis of $T_{p}^c M$ defined by a basis of eigenvectors of ${F^{*}\omega}$. Hence, if $F$ has equal Kähler angles $\langle S{F^{*}\omega},{F^{*}\omega}\rangle\geq 0$. A simple application of the Weitzenbök formula (2.2) shows in next proposition, that such curvature condition on $M$, implies the angle must be constant. No minimality is required.**
([**\[S-V,2\]**]{}) Let $F$ be a non-Lagrangian immersion with equal Kähler angles of a compact orientable $M$ with non-negative isotropic scalar curvature into a Kähler manifold $N$. If $n=2$, $3$ or $4$, then $\theta$ is constant and $(M,{J_{\omega}},g_{M})$ is a Kähler manifold. For any $n\geq 1$ and $\theta$ constant, ${F^{*}\omega}$ is parallel, that is, $(M,{J_{\omega}},g_{M})$ is a Kähler manifold.
Finally, before we prove Corollary 2.1, we state a more general proposition. Let $F:M{\rightarrow}N$ be an immersion with equal Kähler angles, and let $M'=\{p\in M: H=0\}$ be the set of minimal points of $F$. On $M\sim {\cal C}$ a 1-form is defined $$\sigma=\frac{2n}{\sin^2\theta} ((JH)^{\top})^{\flat} +
\frac{\delta {F^{*}\omega}}{\sin^2\theta}$$ Following the proof of \[G\], but now neither requiring $n=2$ nor $\delta {F^{*}\omega}=0$, we obtain $$\begin{array}{lcl}
\sigma(X) &=& -trace~ \frac{1}{\sin^2\theta}
g({{\mbox{\large $\nabla$}}_{\!\!}}dF(\cdot, X), JdF(\cdot))\\[2mm]
d\sigma(X,Y) &=&
Ricci^{N}(JdF(X),dF(Y))=R{F^{*}\omega}(X,Y)\end{array}$$ We note that this form $\sigma$ is well known (see e.g \[Br\], \[Che-M\], \[W,2\]). Now we have:\
If $n=2$, or if $n\geq 2$ and $\theta$ is constant, then $\sigma=\frac{2n}{\sin^2\theta}((JH)^{\top})^{\flat}$ and does not vanish on $M\sim (M'\cup {\cal C})$. Moreover, if $R=0$, then $d\sigma=0$. Thus, if $\theta$ is constant $\neq 0$, $\sigma\in H^{1}(M,{I\!\!R^{}})$, and in particular, if $F$ has non-zero parallel mean curvature, and $R=0$, then $F$ is Lagrangian and $\sigma$ is a non-zero parallel 1-form on $M$.\
For any immersion with constant equal Kähler angles, the following equalities hold $$R\cos\theta\sin^2\theta = \sum_{{\beta}}2d ((JH)^{\top})^{\flat}(X_{{\beta}},Y_{{\beta}})=-4n\cos\theta\|H\|^2 -\sum_{\mu}8 Im{\mbox{\Large $($}} g({{\mbox{\large $\nabla$}}^{\bot}_{\!\!\mu}}H, JdF({\bar{\mu}})){\mbox{\Large $)$}},$$\
where $\{X_{{\alpha}},Y_{{\alpha}}\}$ is any basis of eigenvalues of ${F^{*}\omega}$.\
*Proof of Proposition 3.6 and Corollary 2.1. *We start by proving Corollary 2.1. For a Lagrangian immersion, the formula on $\triangle\kappa$ (valid on $\Omega^{0}_{0}$), reduces to $$0=\triangle\kappa = \sum_{\mu,{\beta}} 32 Im {\mbox{\Large $($}}
R^N (dF({\beta}), dF(\mu),dF({\bar{\beta}}), JdF({\bar{\mu}})){\mbox{\Large $)$}} -
\sum_{\mu}16n Im {\mbox{\Large $($}} g{\mbox{\large $($}}
{{\mbox{\large $\nabla$}}_{\!\!\mu}}H,JdF({\bar{\mu}}){\mbox{\large $)$}}{\mbox{\Large $)$}}.$$ Applying Codazzi equation to the curvature term and noting that $JdF(TM)$ is the orthogonal complement of $dF(TM)$, and that $\sum_{{\beta}}{{\mbox{\large $\nabla$}}_{\!\!\mu}}{{\mbox{\large $\nabla$}}_{\!\!}}dF ({\beta},{\bar{\beta}}) =\frac{n}{2}{{\mbox{\large $\nabla$}}^{\bot}_{\!\!\mu}}H$, we get\
$$0= \sum_{{\beta},\mu} Im ~{\mbox{\LARGE $($}} g{\mbox{\Large $($}} {{\mbox{\large $\nabla$}}_{\!\!{\beta}}}{{\mbox{\large $\nabla$}}_{\!\!}}dF (\mu,{\bar{\beta}}),
JdF({\bar{\mu}}){\mbox{\Large $)$}}{\mbox{\LARGE $)$}}.$$ Note that, since $F$ is Lagrangian, we can choose arbitrarily the orthonormal frame $X_{{\alpha}},Y_{{\alpha}}$. Then we may assume they have zero covariant derivative at a given point $p$. Since $F$ is a Lagrangian immersion $g({{\mbox{\large $\nabla$}}_{\!\!}}dF({\beta},{\bar{\mu}}),JdF(\mu))=g({{\mbox{\large $\nabla$}}_{\!\!}}dF({\bar{\mu}},\mu),JdF({\beta}))$ (see e.g \[S-V,2\]). Taking the derivative of this equality at the point $p$ in the direction ${\bar{\beta}}$ we obtain $$\begin{aligned}
\lefteqn{g{\mbox{\Large $($}} {{\mbox{\large $\nabla$}}_{\!\!{\bar{\beta}}}}{{\mbox{\large $\nabla$}}_{\!\!}}dF ({\beta},{\bar{\mu}}),JdF(\mu){\mbox{\Large $)$}}
+g{\mbox{\Large $($}} {{\mbox{\large $\nabla$}}_{\!\!}}dF ({\beta},{\bar{\mu}}),J{{\mbox{\large $\nabla$}}_{\!\!}}dF({\bar{\beta}}, \mu){\mbox{\Large $)$}}=}\\
&&g{\mbox{\Large $($}} {{\mbox{\large $\nabla$}}_{\!\!{\bar{\beta}}}}{{\mbox{\large $\nabla$}}_{\!\!}}dF ({\bar{\mu}},\mu),JdF({\beta}){\mbox{\Large $)$}}
+g{\mbox{\Large $($}} {{\mbox{\large $\nabla$}}_{\!\!}}dF ({\bar{\mu}},\mu),J{{\mbox{\large $\nabla$}}_{\!\!}}dF({\bar{\beta}}, {\beta}){\mbox{\Large $)$}}.\end{aligned}$$ Taking the summation on $\mu,{\beta}$ and the imaginary part, we obtain from (3.5) $$\sum_{{\beta}} Im~{\mbox{\Large $($}} g({{\mbox{\large $\nabla$}}_{\!\!{\bar{\beta}}}}H,JdF({\beta}) ){\mbox{\Large $)$}}=\sum_{{\beta}} Im~{\mbox{\Large $($}} g({{\mbox{\large $\nabla$}}^{\bot}_{\!\!{\bar{\beta}}}}H,JdF({\beta}) ){\mbox{\Large $)$}} =0.$$ From Lemma 3.1 we conclude, $${{\mbox{\scriptsize $\frac{1}{2}$}}}i\sum_{{\beta}}d((JH)^{\top})^{\flat}(X_{{\beta}},Y_{{\beta}})=
-\sum_{{\beta}}d((JH)^{\top})^{\flat}({\bar{\beta}},{\beta})=
\sum_{{\beta}}-2i Im g_{M} ({{\mbox{\large $\nabla$}}_{\!\!{\bar{\beta}}}}(JH)^{\top},{\beta})=
0.$$ From the arbitrarity of the orthonormal frame, we may interchange $X_1$ by $-X_{1}$, obtaining $d((JH)^{\top})^{\flat}(X_{1},Y_{1})=0$. Hence $d((JH)^{\top})^{\flat}=0$.\
Now we prove Proposition 3.6. The first part is an immediate conclusion from the expressions for $\sigma$, $d\sigma$, and the fact that, under the above assumptions, $\delta {F^{*}\omega}=0$ (see Corollary 3.1), besides the considerations on the zeroes of $(JH)^{\top}$ in the previous remark. The conclusion that $F$ is Lagrangian and $\sigma$ is parallel, under the assumption of non-zero parallel mean curvature and $R=0$, comes from the equalities stated in the proposition, which we prove now, and from Lemma 4.1 of next section . It is obviously true if $\cos\theta= 1$, that is for complex immersions, and it is true for $\cos\theta=0$, as we have seen above. Now, if $\cos\theta$ is constant and different from $0$ or $1$, from Proposition 3.3, $$\begin{aligned}
0=\triangle\kappa &=& \cos\theta {\mbox{\Large $($}} -2nR
+\frac{32}{\sin^{2}\theta}\sum_{{\beta},\mu} R^{M}({\beta},\mu,{\bar{\beta}},{\bar{\mu}})
+\frac{1}{\sin^{2}\theta}\|{{\mbox{\large $\nabla$}}_{\!\!}}{J_{\omega}}\|^{2}
~{\mbox{\Large $)$}}\\
&&-\frac{4n}{\sin^{2}\theta}div_{M}\left({J_{\omega}}{\mbox{\LARGE $($}}(JH)^{\top}{\mbox{\LARGE $)$}}
\right) +\frac{4n}{\sin^{2}\theta} g{\mbox{\LARGE $($}}\delta{J_{\omega}}, (JH)^{\top}{\mbox{\LARGE $)$}}.\end{aligned}$$ Since ${F^{*}\omega}$ is harmonic (see Corollary 3.1), Weitzenböck formula (2.2) with $\theta$ constant reduces to $$16\cos^2\theta \sum_{{\beta},\mu} R^{M}({\beta},\mu,{\bar{\beta}},{\bar{\mu}})
=\langle S{F^{*}\omega},{F^{*}\omega}\rangle=-\|{{\mbox{\large $\nabla$}}_{\!\!}}{F^{*}\omega}\|^{2}=-{{\mbox{\scriptsize $\frac{1}{2}$}}}\cos^2\theta
\|{{\mbox{\large $\nabla$}}_{\!\!}}{J_{\omega}}\|^2$$ Thus, from lemma 3.1 $$\begin{aligned}
{{\mbox{\scriptsize $\frac{1}{2}$}}}R\cos\theta\sin^2\theta&=&-div_{M}\left({J_{\omega}}{\mbox{\LARGE $($}}(JH)^{\top}{\mbox{\LARGE $)$}}
\right) +g_{M}{\mbox{\Large $($}}\delta{J_{\omega}}, (JH)^{\top}{\mbox{\LARGE $)$}}\\[-1mm]
&=& -2n\cos\theta \|H\|^2 -4\sum_{\mu} Im{\mbox{\Large $($}} g({{\mbox{\large $\nabla$}}^{\bot}_{\!\!\mu}} H,JdF({\bar{\mu}})){\mbox{\Large $)$}}.~~~~{\mbox{~~~\boldmath $\Box$}}\end{aligned}$$**
Proofs of the main results
==========================
*Proof of Proposition 1.1. *Assume ${\cal C}\cup {\cal L}=\emptyset$. Then the formula in Corollary 3.2 is valid on all $M$ with all maps involved smooth everywhere. By applying Stokes we get $\int_{M} R \cos\theta\, Vol_{M}=0$, where $\cos\theta>0$, which is impossible if $R\neq 0$.\
*Proof of Proposition 1.2. *Follows immediately from Proposition 3.6.\
*Proof of Theorem 1.4. *In case $n=1$, ${F^{*}\omega}$ is a multiple of the volume element of $M$, that is ${F^{*}\omega}=\cos\tilde{\theta}Vol_{M}$. This $\tilde{\theta}$ is the genuine definition of Kähler angle given by Chern and Wolfson \[Ch-W\]. Our is just $\cos\theta= |\cos\tilde{\theta}|$. While $\cos\tilde{\theta}$ is smooth on all $M$, $\cos\theta$ may not be $C^{1}$ at Lagrangian points. But we see that the formula $(3.1)$ is also valid on $M\sim {\cal L}\cup {\cal C}$ replacing $\cos\theta$ by $\cos\tilde{\theta}$ and the corresponding replacement of $\kappa$ by $\tilde{\kappa}$, and $\sin^2\theta$ by $\sin^2\tilde{\theta}$ and ${J_{\omega}}$ by $J_{M}$, the natural $g_{M}$-orthogonal complex structure on $M$, defining a Kähler structure. We denote this new formula by $(3.1)'$. Note that on $M\sim {\cal L}$, ${J_{\omega}}=\pm J_{M}$, the sign being $+$ or $-$ according to the sign of $\cos\tilde{\theta}$. Hence a change of the sign of $\cos\tilde{\theta}$ will give a change of sign on $\tilde{\kappa}$ and on ${J_{\omega}}$ (w.r.t. $J_{M}$). The formula $(3.1)'$ is in fact also valid on ${\cal L}^{0}$. To see this we use the following lemma, as an immediate consequence of Lemma $3.1~(i)$:******
If $F:M^{2n}{\rightarrow}N^{2n}$ is a submanifold with parallel mean curvature, then $(JH)^{\top}$ is a parallel vector field along ${\cal L}$, that is ${{\mbox{\large $\nabla$}}_{\!\!}}(JH)^{\top}(p)=0$ $\forall p\in {\cal L}$.
Now it follows that $div_{M}( J_{M}((JH)^{\top}) )=0$ on ${\cal L}$. Hence, the formula $(3.1)'$ on $\triangle \tilde{\kappa}$ is valid on ${\cal L}^0$, that is, at interior Lagrangian points. If we assume ${\cal C}=\emptyset$, then $(3.1)'$ is valid over all $M$, because now $\tilde{\kappa}$, $\cos\tilde{\theta}$, $J_{M}$, and $\sin^2\tilde{\theta}$ are smooth everywhere and ${\cal L}
\sim{\cal L}^{0}$ is a set of Lagrangian points with no interior. Integrating and using Stokes, $2R\int_{M}\cos\tilde{\theta}
=0$. Hence if $\cos\tilde{\theta}$ is non-negative or non-positive everywhere, and if $R\neq 0$, then $F$ is Lagrangian. If $F$ has no Lagrangian points, from Lemma 3.1 $(iii)$, since $\delta{J_{\omega}}=0$, $$div_{M}{\mbox{\Large $($}} {J_{\omega}}(JH)^{\top}{\mbox{\Large $)$}}=2\cos\theta\|H\|^2$$ is valid on $M$. Integration leads to $H=0$. .\
*Proof of Theorem 1.2. *If $n=2$, using (3.3) in the expression of $\triangle\cos^{2}\theta$ in Proposition 3.4, we get an expression that is smooth away from complex points, and valid at interior Lagrangian points, and hence on all $M\sim {\cal C}$. Then, following the same steps in the proofs of \[S-V,2\] chapter 4, combining the formulae for $\triangle\cos^2\theta$ of Proposition 3.4 and the Weitzenbök formula (2.2), and applying Proposition 3.1, we get, away from complex points $$sin^{2}\theta\cos^{2}\theta R =-2div_{M}(({F^{*}\omega})^{\sharp}((JH)^{\top}))
+ 2{F^{*}\omega}((JH)^{\top}, \nabla\log \sin^2\theta)$$ Set $P= sin^{2}\theta\cos^{2}\theta R +2 div_{M}(({F^{*}\omega})^{\sharp}((JH)^{\top}))$. This map is defined and smooth on all $M$ and vanishes on ${\cal C}^{0}$. If $R>0$ (resp. $R<0$), and under the assumption $(1.1)$, we have from (4.1) that $P\leq 0$ (resp. $\geq 0$) on $M\sim {\cal C}$. Since the remaining set ${\cal C}\sim {\cal C}^0$ is a set of empty interior, then $P\leq 0$ (resp. $\geq 0$) is valid on all $M$. In fact, from Proposition 3.1, $ |{F^{*}\omega}((JH)^{\top}, \nabla \sin^2\theta)|
\leq \sqrt{C}\cos^2 \theta\sin^2\theta\| H\|~\|({{\mbox{\large $\nabla$}}_{\!\!}}dF)^{(1,1)}\|$. Since $ ({{\mbox{\large $\nabla$}}_{\!\!}}dF)^{(1,1)}$ vanishes on ${\cal C}^{0}$, and so also on $
\overline{{\cal C}^{0}}$, we can smoothly extend to zero ${F^{*}\omega}((JH)^{\top}, \nabla \log\sin^2\theta)$ on $\overline{{\cal C}^0}$. This we can also get from (4.1). Moreover, such equation tells us we can smoothly extend the last term to all complex points, giving exactly the value $2div_{M}(({F^{*}\omega})^{\sharp}((JH)^{\top}))$ at those points. Integration of $P\leq 0$ (respectively $\geq 0$) and applying Stokes, we have $$\int_{M}\sin^2\theta\cos^2\theta R Vol_{M}\leq 0 \mbox{~~~~(resp.~~}\geq 0)$$ and conclude that $F$ is either complex or Lagrangian. \
*Proof of Corollary 1.1. *Instead of using Stokes on the term $div_{M}{\mbox{\Large $($}}({F^{*}\omega})^{\sharp} ( (JH)^{\top}) ){\mbox{\Large $)$}}$, to make it disapear as we did in the proof of theorem 1.2, we develop it into $$\begin{aligned}
div_{M}{\mbox{\Large $($}}({F^{*}\omega})^{\sharp} ( (JH)^{\top}) ){\mbox{\Large $)$}}&=&
div_{M}{\mbox{\Large $($}}\cos\theta{J_{\omega}}( (JH)^{\top}) ){\mbox{\Large $)$}}\\
&=& \cos\theta div_{M}{\mbox{\Large $($}}{J_{\omega}}( (JH)^{\top}) ){\mbox{\Large $)$}} +
d\cos\theta {\mbox{\Large $($}}{J_{\omega}}( (JH)^{\top}) ){\mbox{\Large $)$}},\end{aligned}$$ and use Lemma 3.1 to give, away from complex and Lagrangian points, $$\begin{aligned}
sin^{2}\theta\cos^{2}\theta R&=&-2\cos\theta div_{M}({J_{\omega}}((JH)^{\top}))
-2\langle {J_{\omega}}( (JH)^{\top}), \nabla \cos\theta \rangle\\
&&+2{F^{*}\omega}((JH)^{\top}, \nabla\log \sin^2\theta)\\
&=& -8\cos^2\theta\| H\|^2+2{F^{*}\omega}((JH)^{\top}, \nabla\log \sin^2\theta).\end{aligned}$$ Hence, away from complex and Lagrangian points $$\sin^4\theta \cos^2\theta R + 8 \sin^2\theta \cos^2\theta\| H\|^2
=2{F^{*}\omega}((JH)^{\top}, \nabla \sin^2\theta).$$ Obviously, this equality also holds at Lagrangian and complex points, for, those points are critical points for $\sin^2\theta$. The corollary now follows immediately from Theorem 1.2. \
*Proof of Theorem 1.3. *If $n\geq 3$ we set $$P= n\triangle\cos^2\theta +
4n\, div_{M}(({F^{*}\omega})^{\sharp}((JH)^{\top})) +2n\sin^2\theta\cos^2\theta R -2\|{{\mbox{\large $\nabla$}}_{\!\!}}{F^{*}\omega}\|^2 -2\langle S{F^{*}\omega}, {F^{*}\omega}\rangle.$$ This map is defined on all $M$ and is smooth. From Proposition (3.4) and using (3.4), on $M\sim {\cal C}$ $$P= \frac{4n(2+(n-4)\sin^2\theta)}{(n-2)\sin^{2}\theta}
\delta{F^{*}\omega}((JH)^{\top}) + 4(n-2)\|\nabla |\sin\theta|~\|^2$$ In $(A)$ and $(B)$, by assumption, $P\geq 0$ on $M\sim{\cal C}$, because for $n\geq 3$, $ (2 + (n-4)\sin^2\theta) \geq 0$. But on ${\cal C}^0$, $P=0$, for $(M,{J_{\omega}},g_{M})$ is a complex submanifold, and so, $(JH)^{\top}=0$ and $\langle S{F^{*}\omega}, {F^{*}\omega}\rangle =0$. Thus, $P\geq 0$ on all $M$. Integrating $P\geq 0$ on $M$ we obtain using Stokes, Weitzenböck formula (2.2), and (2.3) $$\int_{M}2n R\sin^2\theta\cos^2\theta Vol_{M}\geq \int_{M}2\|
\delta{F^{*}\omega}\|^2Vol_{M}.$$ Thus, if $R<0$ we conclude $F$ is either complex or Lagrangian, and if $R=0$ we conclude that $\delta{F^{*}\omega}=0$, which implies, by Corollary 3.1, that $\theta$ is constant. This last reasoning proves $(C)$ as well.\
*Remark 2. *In Theorem 1.3 we can replace the condition $\delta
{F^{*}\omega}((JH)^{\top})\geq 0$ by a weaker condition $$\delta{F^{*}\omega}((JH)^{\top})\geq -\frac{(n-2)^{2}}{4n(2+(n-4)\sin^2\theta)}
\|\nabla \cos^2\theta\|^2$$ to achieve the same conclusion. This condition is sufficient to obtain $P\geq 0$ in the above proof. Then we can obtain for $n\geq 3$ a corollary similar to Corollary 1.1, by requiring $$4n^2 \cos^2\theta\|H\|^2 + n \sin^2\theta\cos^2\theta R -(n-2)^2
\|\nabla \cos\theta\|^2\geq -2n\delta{F^{*}\omega}((JH)^{\top}).$$ [**[References]{}**]{}\
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[Centro de Física das Interacções Fundamentais,\
Instituto Superior Técnico,\
Edifício Ciência,\
Piso 3, 1049-001 LISBOA, Portugal;\
e-mail: isabel@cartan.ist.utl.pt]{}********
|
---
abstract: 'We extend the notion of conservativeness, given by Fredkin and Toffoli in 1982, to generic gates whose input and output lines may assume a finite number $d$ of truth values. A physical interpretation of conservativeness in terms of conservation of the energy associated to the data used during the computation is given. Moreover, we define *conservative computations*, and we show that they naturally induce a new NP–complete decision problem. Finally, we present a framework that can be used to explicit the movement of energy occurring during a computation, and we provide a quantum implementation of the primitives of such framework using creation and annihilation operators on the Hilbert space ${{\mathbb C}}^d$, where $d$ is the number of energy levels considered in the framework.'
author:
- |
[G. Cattaneo$^\ast$, G. Della Vedova$^\dagger$, A. Leporati$^\ast$, R. Leporini[^1]]{}\
${}^\ast$ Dipartimento di Informatica, Sistemistica e Comunicazione\
${}^\dagger$ Dipartimento di Statistica\
Università degli Studi di Milano – Bicocca\
Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy\
e-mail: { cattang, leporati, leporini }@disco.unimib.it\
gianluca.dellavedova@unimib.it
title: '**Towards a Theory of Conservative Computing**'
---
Introduction
============
Conservative logic has been introduced in [@fredkin-toffoli] as a mathematical model that allows one to describe computations which reflect some properties of microdynamical laws of Physics, such as reversibility and conservation of the internal energy of the physical system used to perform the computations. The model is based upon the so called Fredkin gate, a three–input/three–output Boolean gate originally introduced by Petri in [@petri], whose input/output map $\text{\sc FG} : {\{0,1\}^3 \rightarrow
\{0,1\}^3}$ associates any input triple $(x_1,x_2,x_3)$ with its corresponding output triple $(y_1,y_2,y_3)$ as follows: $$y_1 = x_1
\hspace{0.7cm}
y_2 = (\lnot x_1\land x_2) \lor (x_1\land x_3)
\hspace{0.7cm}
y_3 = (x_1 \land x_2) \lor (\lnot x_1 \land x_3)$$
The Fredkin gate is *functionally complete* for the Boolean logic: by fixing $x_3 = 0$ we get $y_3 = x_1 \land x_2$, whereas by fixing $x_2 = 1$ and $x_3 = 0$ we get $y_2 = \lnot x_1$. A useful point of view is that the Fredkin gate behaves as a *conditional switch*: that is, $\text{\sc FG}(1, x_2, x_3) = (1, x_3, x_2)$ and $\text{\sc FG}(0, x_2, x_3) = (0, x_2, x_3)$ for every $x_2, x_3 \in \{0,1\}$. In other words, the first input line can be considered as a control line whose value determines whether the input values $x_2$ and $x_3$ have to be exchanged or not.
According to [@fredkin-toffoli], *conservativeness* is usually modeled by the property that the output patterns of the involved gates are always a permutation of the patterns given in input. Let us stress that this does not mean that a fixed permutation is applied to every possible input pattern; on the contrary, the applied permutation depends on the input pattern. Here we just mention the fact that every permutation can be written as a composition of transpositions. Hence not only the Fredkin gate can be used to build an appropriate circuit to perform any given conservative computation (and thus it is universal also in this sense with respect to conservative computations), but it is also the most elementary conceivable operation that can be used to describe conservative computations. In this paper we will propose some analogous elementary operations with respect to our notion of conservativeness.
The Fredkin gate is also *reversible*, that is, [FG]{} is a bijective map on $\{0,1\}^3$. Notice that conservativeness and reversibility are two independent notions: a gate can satisfy both properties, only one of them, or none. Since every reversible gate computes a bijective map between input and output patterns, and every conservative gate produces permutations of its input patterns, it follows that they must necessarily have the same number of input and output lines.
In this paper we extend the notion of conservativeness to generic gates whose input and output lines may assume a finite number $d$ of truth values, and we derive some properties which are satisfied by conservative gates. By associating equispaced energy levels to the truth values, we show that our notion of conservativeness corresponds to the energy conservation principle applied to the data which are manipulated during the computation. Let us stress that we are *not* saying that the entire energy used to perform the computation is preserved, or that the computing device is a conservative physical system. In particular we do not consider the energy needed to transform the input values into output values, that is, the energy needed to *perform* the computation.
Successively we introduce the notion of *conservative computation*, based upon gates which are able to store some finite amount of energy and to reuse it during the computation. We show that the decision problem to determine whether a given computation can be performed in a conservative way through a gate which is able to store at most $C$ units of energy is NP–complete.
Finally, we introduce a framework that allows one to visualize the movement of energy occurring during a computation performed by a generic gate. The framework is based upon some primitive operators that conditionally move one unit of energy between any two given input/output lines of the gate. Using creation and annihilation operators on the Hilbert space ${{\mathbb C}}^d$, we show a quantum realization of these non–unitary conditional movement operators.
Conservativeness
================
Our notion of conservativeness, and the framework we will introduce, are based upon many–valued logics. These are extensions of the classical Boolean logic which are widely used to manage incomplete and/or uncertain knowledge. Different approaches to many–valued logics have been considered in literature: for an overview, see [@rescher; @rosser-turquette]. However, here we are not interested into the study of syntactical or algebraic aspects of many–valued logics; we just define some gates whose input and output lines may assume “intermediate” truth values, such as the gates defined in [@cattaneo-leporati-leporini-mv].
For every integer $d \ge 2$, we consider the finite set $L_d = \{ 0,
\frac{1}{d-1}, \frac{2}{d-1}, \ldots$, $\frac{d-2}{d-1}, 1\}$ of truth values; $0$ and $1$ denote falsity and truth, respectively, whereas the other values of $L_d$ indicate different degrees of indefiniteness. As usually found in literature, we will use $L_d$ both as a set of truth values and as a numerical set equipped with the standard order relation on rational numbers.
An $n$–input/$m$–output $d$–valued *function* (also called an $(n, m,
d)$–function for short) is a map $f: L_d^n \to L_d^m$. Analogously, an $(n, m, d)$–*gate* and an $(n, m, d)$–*circuit* are devices that compute $(n, m, d)$–functions. A gate is considered as a *primitive operation*, that is, it is assumed that a gate cannot be decomposed into simpler parts. On the other hand, a circuit is composed by *layers* of gates, where any two gates $G_1$ and $G_2$ of the same layer satisfy the property that no output line of $G_1$ is connected to any input line of $G_2$.
Let us consider the set ${\cal E}_d = \left\{ \varepsilon_0,
\varepsilon_{\frac{1}{d-1}}, \varepsilon_{\frac{2}{d-1}}, \ldots,
\varepsilon_{\frac{d-2}{d-1}}, \varepsilon_1 \right\} \subseteq {\mathbb{R}}$ of real values; for exposition convenience, we can think to such quantities as energy values. To each truth value $v \in L_d$ we associate the energy level $\varepsilon_v$; moreover, let us assume that the values of ${\cal E}_d$ are all positive, equispaced, and ordered according to the corresponding truth values: $0 < \varepsilon_0 < \varepsilon_{\frac{1}{d-1}} < \cdots <
\varepsilon_{\frac{d-2}{d-1}} < \varepsilon_1$. If we denote by $\delta$ the gap between two adjacent energy levels then the following holds: $$\varepsilon_v = \varepsilon_0 + \delta \, (d-1) \, v
\hspace{1.5cm}
\forall \, v \in L_d
\label{eq:linear-relation}$$ Notice that it is not required that $\varepsilon_0 = \delta$.
Now, let ${\underline{x}}= (x_1, \ldots, x_n) \in L_d^n$ be an input *pattern* for an $(n, m, d)$–gate. We define the *amount of energy associated to ${\underline{x}}$* as $E_n({\underline{x}}) =
\sum_{i=1}^n \varepsilon_{x_i}$, where $\varepsilon_{x_i} \in {\cal E}_d$ is the amount of energy associated to the $i$–th element $x_i$ of the input pattern. Let us remark that the map $E_n : L_d^n \to {\mathbb{R}}^+$ is indeed a family of mappings parameterized by $n$, the size of the input. Analogously, for an output pattern ${\underline{y}}\in L_d^m$ we define the associated amount of energy as $E_m({\underline{y}}) = \sum_{i=1}^m \varepsilon_{y_i}$. We can now define a conservative gate as follows.
An $(n, m , d)$–gate, described by the function $G: L_d^n \to L_d^m$, is *conservative* if the following condition holds: $$\forall \, {\underline{x}}\in L_d^n \qquad E_n({\underline{x}}) = E_m(G({\underline{x}}))
\label{eq:conservativegate}$$
Notice that it is not required that the gate has the same number of input and output lines, as it happens with the reversible and conservative gates considered in [@fredkin-toffoli; @cattaneo-leporati-leporini-mv; @cattaneo-leporati-leporini-qm].
Using relation , equation can also be written as: $$\frac{\varepsilon_0 n}{\delta (d-1)} + \sum_{i=1}^n x_i =
\frac{\varepsilon_0 m}{\delta (d-1)} + \sum_{j=1}^m y_j$$ Hence, when $n = m$ (as it happens, for example, with reversible gates) conservativeness reduces to the conservation of the sum of truth values given in input, as in weak conservativeness introduced in [@cattaneo-leporati-leporini-mv]. In the Boolean case this is equivalent to requiring that the number of $1$’s given in input is preserved, as in the original notion of conservativeness given in [@fredkin-toffoli].
An interesting remark is that conservativeness entails an upper and a lower bound to the ratio $\frac{m}{n}$ of the number of output lines versus the number of input lines of a gate. In fact, the maximum amount of energy that can be associated to an input pattern is $\sum_{i=1}^n \varepsilon_1 = n \, \varepsilon_1$, whereas the minimum amount of energy that can be associated to an output pattern is $\sum_{i=1}^m \varepsilon_0 = m \, \varepsilon_0$. Clearly, if it holds $n \, \varepsilon_1 < m \, \varepsilon_0$ then the gate cannot produce any output pattern in a conservative way. As a consequence, it must hold $\frac{m}{n} \le
\frac{\varepsilon_1}{\varepsilon_0}$. Analogously, if we consider the minimum amount of energy $n \, \varepsilon_0$ that can be associated to an input pattern ${\underline{x}}$ and the maximum amount of energy $m \, \varepsilon_1$ that can be associated to an output pattern ${\underline{y}}$, it clearly must hold $n \, \varepsilon_0 \le m \, \varepsilon_1$, that is $\frac{m}{n} \ge \frac{\varepsilon_0}{\varepsilon_1}$. Summarizing, we have the bounds $\frac{\varepsilon_0}{\varepsilon_1} \le
\frac{m}{n} \le \frac{\varepsilon_1}{\varepsilon_0}$, that is, for a conservative gate (or circuit) the number $m$ of output lines is constrained to grow linearly with respect to the number $n$ of input lines.
A natural question is whether we can compute all functions in a conservative way. Let us consider the Boolean case. Let $f: \{0,1\}^n \to \{0,1\}^m$ be a non necessarily conservative function, and let us define the following quantities: $$\begin{aligned}
&O_f = \max\left\{0, \max\limits_{{\underline{x}}\in \{0,1\}^n} \left\{E_m(f({\underline{x}})) -
E_n({\underline{x}})\right\}\right\} \\
&Z_f = \max\left\{0, \max\limits_{{\underline{x}}\in \{0,1\}^n} \left\{E_n({\underline{x}}) -
E_m(f({\underline{x}}))\right\}\right\}\end{aligned}$$ Informally, $O_f$ (resp., $Z_f$) is the maximum number of $1$’s (resp., $0$’s) in the output pattern that should be converted to $0$ (resp., $1$) in order to make the computation conservative. This means that if we use a gate $G_f$ with $n + O_f + Z_f$ input lines and $m + O_f + Z_f$ output lines then we can compute $f$ in a conservative way as follows: $$G_f({\underline{x}}, \underline{1}_{O_f}, \underline{0}_{Z_f}) =
(f({\underline{x}}), \underline{1}_{w({\underline{x}})}, \underline{0}_{z({\underline{x}})})$$ where $\underline 1_k$ (resp., $\underline 0_k$) is the $k$–tuple consisting of all $1$’s (resp., $0$’s), and the pair $(\underline{1}_{w({\underline{x}})},
\underline{0}_{z({\underline{x}})}) \in \{0,1\}^{O_f + Z_f}$ is such that $w({\underline{x}}) = {O_f +
E_n({\underline{x}}) - E_m(f({\underline{x}}))}$ and $z({\underline{x}}) = Z_f - E_n({\underline{x}}) + E_m(f({\underline{x}}))$.
As we can see, we use some additional input (resp., output) lines in order to provide (resp., remove) the required (resp., exceeding) energy that allows $G_f$ to compute $f$ in a conservative way. It is easy to see that the same trick can be applied to generic $d$–valued functions $f: L_d^n \to L_d^m$; instead of the number of missing or exceeding $1$’s, we just compute the missing or exceeding number of energy units, and we provide an appropriate number of additional input and output lines.
Conservative computations
=========================
Let us now introduce the notion of *conservative computation*. Let $G: L_d^n \to L_d^m$ be the function computed by an $(n, m, d)$–gate. Moreover, let $S_{in} = \langle {\underline{x}}_1, {\underline{x}}_2, \ldots$, ${\underline{x}}_k \rangle$ be a sequence of elements from $L_d^n$ to be used as input patterns for the gate, and let $S_{out} = \langle G({\underline{x}}_1), G({\underline{x}}_2), \ldots, G({\underline{x}}_k) \rangle$ be the corresponding sequence of output patterns from $L_d^m$. Let us consider the quantities $e_i = E_n({\underline{x}}_i) - E_m(G({\underline{x}}_i))$ for all $i \in \{1, 2, \ldots, k\}$; note that, without loss of generality, by an appropriate rescaling we may assume that all $e_i$’s are integer values. We say that the *computation* of $S_{out}$, obtained starting from $S_{in}$, is *conservative* if the following condition holds: $$\sum_{i=1}^k e_i
= \sum_{i=1}^k E_n({\underline{x}}_i) - \sum_{i=1}^k E_m(G({\underline{x}}_i)) = 0$$ This condition formalizes the requirement that the total energy provided by *all* input patterns of $S_{in}$ is used to build all output patterns of $S_{out}$. Of course it may happen that $e_i > 0$ or $e_i < 0$ for some $i \in \{1, 2,
\ldots, k\}$. In the former case the gate has an excess of energy that should be dissipated into the environment after the production of the value $G({\underline{x}}_i)$, whereas in the latter case the gate does not have enough energy to produce the desired output pattern. Since we want to avoid these situations, we assume to perform computations through gates which are equipped with an internal *accumulator* (also *storage unit*) which is able to store a maximum amount $C$ of energy units. We call $C$ the *capacity* of the gate. The amount of energy contained into the internal storage unit at a given time can thus be used during the next computation step if the energy of the output pattern that must be produced is greater than the energy of the corresponding input pattern.
If the output patterns $G({\underline{x}}_1)$, $G({\underline{x}}_2), \ldots, G({\underline{x}}_k)$ are computed exactly in this order then, assuming that the computation starts with no energy stored into the gate, it is not difficult to see that $\;st_1 := e_1, \;
st_2 := e_1 + e_2, \; \ldots, \; st_k := e_1 + e_2 + \ldots + e_k$ is the *sequence of the amounts of energy stored* into the gate during the computation of $S_{out}$. We say that a given conservative computation is *$C$–feasible* if $0 \le
st_i \le C$ for all $i \in \{1, 2, \ldots, k\}$. Notice that for conservative computations it always holds $st_k = 0$.
In some cases the order with which the output patterns of $S_{out}$ are computed does not matter. We can thus consider the following problem: Given an $(n, m, d)$–gate that computes the map $G: L_d^n \to L_d^m$, an input sequence ${\underline{x}}_1, \ldots, {\underline{x}}_k$ and the corresponding output sequence $G({\underline{x}}_1), \ldots$, $G({\underline{x}}_k)$, is there a permutation $\pi \in S_k$ (the symmetrical group of order $k$) such that the computation of $G({\underline{x}}_{\pi(1)}), G({\underline{x}}_{\pi(2)}), \ldots, G({\underline{x}}_{\pi(k)})$ is $C$–feasible? This is a decision problem, whose relevant information is entirely provided by the values $e_1, e_2, \ldots, e_k$, which can be formally stated as follows.
[Name]{}: [ConsComp]{}.
- [Instance]{}: a set ${\cal E} = \{e_1, e_2, \ldots, e_k\}$ of integer numbers such that $e_1 + e_2 + \ldots + e_k = 0$, and an integer number $C > 0$.
- [Question]{}: is there a permutation $\pi \in S_k$ such that $\forall \, i \in \{1, 2, \ldots, k\}$ $$0 \le \sum_{j=1}^i e_{\pi(j)} \le C \quad ?
\label{eq:constraints}$$
The [ConsComp]{} problem can be obviously solved by trying every possible permutation $\pi$ from $S_k$. However, this procedure requires an exponential time with respect to $k$, the length of the computation. A natural question is whether it is possible to give the correct answer in polynomial time. With the following theorem we show that the [ConsComp]{} problem is NP–complete. As it is well known [@garey-johnson], this means that if there would exist a polynomial time algorithm that solves the problem then we could immediately conclude that the two complexity classes P and NP coincide, a very unlikely situation.
[ConsComp]{} is [NP]{}–complete. \[teo:CONSCOMP\]
[ConsComp]{} is clearly in NP, since a permutation $\pi \in S_k$ has linear length and verifying whether $\pi$ is a solution can be done in polynomial time. In order to conclude that [ConsComp]{} is NP–complete, let us show a polynomial reduction from [Partition]{}, which is a well known NP–complete problem [@garey-johnson page 47].
Let $A = \{a_1, a_2, \ldots, a_k\}$ be a set of positive integer numbers, and let $m = \sum_{i=1}^k a_i$. The set $A$ is a positive instance of [Partition]{} if and only if there exists a set $A' \subseteq A$ such that $\sum_{a \in A'} a = \frac{m}{2}$. If $m$ is odd then $A$ is certainly a negative instance, and we can associate it to any negative instance of [ConsComp]{}. On the other hand, if $m$ is even we build the corresponding instance $({\cal E}, C)$ of [ConsComp]{} by putting $C = \frac{m}{2}$ and ${\cal E}
= \{e_1, e_2, \ldots, e_k, e_{k+1}, e_{k+2}\}$, where $e_i = -a_i$ for all $i \in \{1,2,\ldots,k\}$ and $e_{k+1} = e_{k+2} = \frac{m}{2}$. It is immediately seen that this construction can be performed in polynomial time.
We claim that $A$ is a positive instance of [Partition]{} if and only if $({\cal E}, C)$ is a positive instance of [ConsComp]{}. In fact, let us assume that $A$ is a positive instance of [Partition]{}. Then there exists a set $A' \subseteq A$ such that $\sum_{a \in A'} a =
\frac{m}{2}$, and the corresponding negative elements of ${\cal E}$ constitute a subset ${\cal E}'$ such that $\sum_{e \in {\cal E}'} e = -\frac{m}{2}$. We build a permutation $\pi \in S_k$ by selecting first the element $e_{k+1}$ followed by the elements of ${\cal E}'$ (chosen with any order), and then $e_{k+2}$ followed by the remaining elements of ${\cal E}$. It is immediately seen that $\pi$ satisfies the inequalities stated in , and hence $({\cal E}, C)$ is a positive instance of [ConsComp]{}. Conversely, let us assume that $({\cal E}, C)$ is a positive instance of [ConsComp]{}. Then there exists a permutation $\pi \in S_k$ that satisfies the inequalities stated in . Since the first chosen element cannot be negative, it must necessarily be $\frac{m}{2}$. Moreover, since $C = \frac{m}{2}$, the second $\frac{m}{2}$ can be chosen if and only if the energy stored into the gate is zero, that is, if and only if there exists a set ${\cal E}' \subseteq {\cal E}$ of negative elements whose sum is equal to $-\frac{m}{2}$. The opposites of these elements constitute a set $A' \subseteq A$ such that $\sum_{a \in A'} a = \frac{m}{2}$, and thus we can conclude that $A$ is a positive instance of [Partition]{}.
A framework for the study of energy–based properties of computations
====================================================================
In this section we introduce a framework which can be used to define and study energy–based properties of computations performed by $(n,m,d)$–gates. The crucial idea of our framework is that we look at computations as a sequence of *conditional movements* of energy. That is, the gate computes its output pattern as follows: for a given subset of input lines, a condition on their values is checked; if this condition is verified then a given action is performed, transforming such values, otherwise no transformation is applied. Successively, another condition is checked on another subset of lines (comprising the output lines from the first step of computation), which determines whether another action has to be performed, and so on until the required values are obtained on the output lines.
To realize the gate according to the above procedure, we need a (Boolean) *control equipment*, and two *primitives* to conditionally move energy from a given line to another one. We call these primitives *conditional up* ([CUp]{}) and *conditional down* ([CDown]{}). The realization of the gate can thus be viewed as a circuit composed by these simpler elements. Let us first describe [CUp]{} and [CDown]{} as $d$–valued gates. In the following, we will provide a quantum realization as formulas composed of creation and annihilation operators on ${{\mathbb C}}^d$, as we have done for the gates presented in [@cattaneo-leporati-leporini-qm].
The [CUp]{} gate is depicted in Figure \[fig:figures\] (a). It is a $(3,3,d)$–gate whose behavior is:
: $(c, a, b) \in L_d^3$
$c = 1$
$(c, \; a + \frac{1}{d-1}, \; b -
\frac{1}{d-1})$
$(c, a, b)$
As we can see, $c$ is a control line whose input value is returned unchanged. The condition $c=1$ enables the movement of a quantity $\delta$ of energy from the third to the second line. Of course, this action is performed only if possible, that is, only if $a \neq 1$ and $b \neq 0$ (equivalently, if the energy values associated to the second and third line are not $\varepsilon_1$ and $\varepsilon_0$, respectively). If these conditions are not satisfied, or if $c \neq 1$, then the gate behaves as the identity. Starting from this description, for any integer $d \ge 2$ we can easily write the truth table of the $d$–valued [CUp]{} gate.
Analogously, the behavior of the complementary $(3,3,d)$–gate [CDown]{} is:
: $(c, a, b) \in L_d^3$
$c = 1$
$(c, \; a - \frac{1}{d-1}, \; b +
\frac{1}{d-1})$
$(c, a, b)$
Let us note that $\text{\sc CDown}(c,a,b)$ can be obtained from $\text{\sc CUp}
(c,a,b)$ (and vice versa) by exchanging the second and the third line before and after the application of [CUp]{}.
![*(a) The Conditional Up [(CUp)]{} gate. (b) Realization of the Boolean Fredkin gate through two–valued [CUp]{}’s and [CDown]{}’s.*[]{data-label="fig:figures"}](figures.eps){width="12cm"}
Figure \[fig:figures\] (b) shows how, using the Boolean versions of [CUp]{} and [CDown]{} gates, we can implement the Boolean Fredkin (controlled switch) gate. Since the Fredkin gate is functionally complete for Boolean logic, using only two–valued [CUp]{} and [CDown]{} gates we can realize any Boolean circuit. In principle these Boolean circuits, together with $d$–valued [CUp]{}’s and [CDown]{}’s, can realize any conditional movement of energy, that is, any conceivable computation that can be performed by $(n,m,d)$–gates.
It is clear that implementing a gate, be it conservative or not, using only these primitives allows one to visualize the movement of energy between different parts of the gate during a computation. Such visualization may help us to optimize some aspects of the implementation of the gate, namely, the amount of energy moved and the extension of energy jumps. As shown in [@leporati-phd], such optimizations can be obtained by splitting (if possible) a given $(N, M, d)$–gate $H$ into $k$ blocks, so that its computation can be performed by an appropriate $(N/k, M/k, d)$–gate $G$ equipped with a storage unit of capacity $C$. However, the minimization of the amount of energy moved between different parts of $H$ during the computation is equivalent to the minimization of $C$, and hence it constitutes an NP–hard problem, whose decision version is the NP–complete problem [ConsComp]{}. This means that the reorganization of the internal machinery of $H$ to optimize the movements of energy is considered a difficult problem.
Now let us turn to the quantum realization of [CUp]{} and [CDown]{}. Generally, a quantum gate acts on memory cells that are $d$–level quantum systems called *qudits* (see [@cattaneo-leporati-leporini-qm] and [@gottesman]). A qudit is typically implemented using the energy levels of an atom or a nuclear spin. The mathematical description — independent of the practical realization — of a single qudit is based on the $d$–dimensional complex Hilbert space $\mathbb{C}^d$. In particular, the truth values of $L_d$ are represented by the unit vectors of the canonical orthonormal basis, called the *computational basis* of $\mathbb{C}^d$: $${\left\vert {0} \right\rangle} = \begin{bmatrix}
1\\ 0 \\ \vdots \\ 0 \\ 0
\end{bmatrix}, \hspace{0.5cm}
{\left\vert {\frac{1}{d-1}} \right\rangle} = \begin{bmatrix}
0 \\ 1 \\ \vdots \\0\\ 0
\end{bmatrix}, \hspace{0.5cm}
\cdots, \hspace{0.5cm}
{\left\vert {\frac{d-2}{d-1}} \right\rangle} = \begin{bmatrix}
0 \\ 0 \\ \vdots \\ 1 \\ 0
\end{bmatrix}, \hspace{0.5cm}
{\left\vert {1} \right\rangle} = \begin{bmatrix}
0 \\ 0 \\ \vdots \\ 0 \\ 1
\end{bmatrix}$$
A collection of $n$ qudits is called a *quantum register* of size $n$. It is mathematically described by the Hilbert space $\otimes^n{{\mathbb C}}^d=\underbrace{{{\mathbb C}}^d\otimes\ldots\otimes{{\mathbb C}}^d}_{\mbox{\small $n$
times}}$. An $n$–*configuration* is a vector ${\left\vert {x_1} \right\rangle}\otimes\ldots\otimes
{\left\vert {x_n} \right\rangle} \in \otimes^n{{\mathbb C}}^d$, simply written as ${\left\vert {x_1,\ldots,x_n} \right\rangle}$, for $x_i$ running on $L_d$. An $n$–configuration can be viewed as the quantum realization of the “classical” pattern $(x_1,\ldots, x_n ) \in L_d^n$. Let us recall that the dimension of $\otimes^n{{\mathbb C}}^d$ is $d^n$ and that the set $\{{\left\vert {x_1,\ldots,x_n} \right\rangle}:x_i\in L_d\}$ of all $n$–configurations is an orthonormal basis of this space, called the $n$–*register computational basis*.
Unlike the situation of the classical wired computer where voltages of a wire go over voltages of another, in quantum realizations of classical gates something different happens. First of all, in this setting every gate must have the same number of input and output lines (that is, they must be $(n, n, d)$–gates). Each qudit of a given register configuration ${\left\vert {x_1,\ldots,x_n} \right\rangle}$ (quantum realization of an input pattern) is in some particular quantum state ${\left\vert {x_i} \right\rangle}$ and an operation $G:\otimes^n{{\mathbb C}}^d\mapsto\otimes^n{{\mathbb C}}^d$ is performed which transforms this configuration into a new configuration $G({\left\vert {x_1,\ldots,x_n} \right\rangle}) = {\left\vert {y_1,\ldots,y_n} \right\rangle}$, which is the quantum realization of an output pattern. In other words, a quantum realization of an $(n, n, d)$–gate is a linear operator $G$ that transforms vectors of the $n$–register computational basis into vectors of the same basis. The action of $G$ on a non–factorized vector, expressed as a linear combination of the elements of the $n$–register basis, is obtained by linearity.
The collection of all linear operators on ${{\mathbb C}}^d$ is a $d^2$–dimensional linear space whose canonical basis is: $$\left\{ E_{x,y}={\left\vert {y} \right\rangle}{\left\langle {x} \right\vert}\;:\; x,y\in L_d \right\}$$ Since $E_{x,y} {\left\vert {x} \right\rangle} = {\left\vert {y} \right\rangle}$ and $E_{x,y} {\left\vert {z} \right\rangle} = \mathbf{0}$ for every $z \in L_d$ such that $z \neq x$, this operator transforms the unit vector ${\left\vert {x} \right\rangle}$ into the unit vector ${\left\vert {y} \right\rangle}$, collapsing all the other vectors of the canonical orthonormal basis of ${{\mathbb C}}^d$ into the null vector. For $i, j \in \{0, 1, \ldots, d-1\}$, the operator $E_{\frac{i}{d-1},
\frac{j}{d-1}}$ can be represented as an order $d$ square matrix having $1$ in position $(j+1,i+1)$ and $0$ in every other position: $$E_{\frac{i}{d-1},\frac{j}{d-1}} = \left(\delta_{r,j+1}\delta_{i+1,s}
\right)_{r,s=1,2,\ldots,d}$$
Each of the operators $E_{x,y}$ can be expressed, using the whole algebraic structure of the associative algebra of operators, as a suitable composition of creation and annihilation operators. An alternative approach, that uses spin–creation and spin–annihilation operators, is shown in [@cattaneo-leporati-leporini-qm]. We recall that the actions of the *creation* operator $a^\dag$ and of the *annihilation* operator $a$ on the vectors of the canonical orthonormal basis of ${{\mathbb C}}^d$ are $$\begin{aligned}
& a^\dagger {\left\vert {\frac{k}{d-1}} \right\rangle} = \sqrt{k+1} {\left\vert {\frac{k+1}{d-1}} \right\rangle}
& \text{for $k \in \{0, 1, \ldots, d-2\}$} \\
& a^\dagger {\left\vert {1} \right\rangle} = \mathbf{0}\end{aligned}$$ and $$\begin{aligned}
& a {\left\vert {\frac{k}{d-1}} \right\rangle} = \sqrt{k} {\left\vert {\frac{k-1}{d-1}} \right\rangle}
& \text{for $k \in \{1, 2, \ldots, d-1\}$} \\
& a {\left\vert {0} \right\rangle} = \mathbf{0}\end{aligned}$$ respectively. Hence, if denote by $A_{u,v}^{p,q,r}$ the expression $$\underbrace{v \cdots v}_r \underbrace{v^\ast \cdots v^\ast}_q
\underbrace{v \cdots v}_p u$$ where $u,v \in \{a^\dag,a\}$, $v^\ast$ is the adjoint of $v$, and $p,q,r$ are non negative integer values, then for any $i,j \in \{0,1,\ldots,d-1\}$ we can express the operator $E_{\frac{i}{d-1},\frac{j}{d-1}}$ in terms of creation and annihilation as follows: $$E_{\frac{i}{d-1},\frac{j}{d-1}} =
\begin{cases}
\frac{ \sqrt{j!}}{(d-1)!}
A_{a^\dag,a^\dag}^{d-2, d-1-j, 0} & \text{if $i = 0$} \\
\frac{ \sqrt{j!}}{(d-1)!}
A_{a,a^\dag}^{d-1, d-1-j, 0} & \text{if $i = 1$ and $j \ge 1$} \\
\frac{ \sqrt{i!}}{(d-1)! \sqrt{j!}}
A_{a^\dag,a^\dag}^{d-2-i, d-1, j} &
\begin{minipage}[t]{5.9cm}
if ($i = 1$, $j = 0$ and $d \ge 3$) or \\
\indent\hspace{0.25cm} ($1 < i < d-2$ and $j \le i$)
\end{minipage} \\
\frac{ \sqrt{j!}}{(d-1)! \sqrt{i!}}
A_{a,a}^{i-1, d-1, d-1-j} &
\begin{minipage}[t]{5.9cm}
if ($i = d-2$, $j = d-1$ and $d \ge 3$) \\
\indent\hspace{0.25cm} or ($1 < i < d-2$ and $j > i$)
\end{minipage} \\
\frac{1}{ \sqrt{(d-1)! j! (d-1)}}
A_{a^\dag,a}^{d-1, j, 0} & \text{if $i = d-2$ and $j \le d-2$} \\
\frac{1}{ \sqrt{(d-1)! j!}}
A_{a,a}^{d-2, j, 0} & \text{if $i = d-1$}
\end{cases}$$
Classical $(n,n,d)$–gates can be quantistically realized as sums of tensor products of the operators $E_{x,y}$ as follows. Let $x_1 x_2 \cdots x_n \mapsto y_1 y_2 \cdots y_n$ be a generic row of the truth table of an $(n,n,d)$–gate. For what we have said above, the operator $E_{x_1,y_1} \otimes E_{x_2,y_2}
\otimes \cdots \otimes E_{x_n,y_n}$ transforms the input configuration $x_1 x_2 \cdots x_n$ into the output configuration $y_1 y_2 \cdots y_n$, and collapses all the other input configurations of the $n$–register basis to the null vector. It is not difficult to see that if ${\cal O}_0, \ldots, {\cal O}_{d^n-1}$ are the “local” operators associated to the $d^n$ rows of the truth table, then the operator ${\cal O} = \sum_{i=0}^{d^n-1}{\cal O}_i$ is a quantum realization of the $(n,n,d)$–gate. Notice that the resulting operator ${\cal O}$ is not necessarily a unitary operator.
Starting from the truth tables of the $d$–valued gates [CUp]{} and [CDown]{} we can thus build the corresponding linear operators that realize them. For example, it is not difficult to see that the non–unitary linear operator — acting on the Hilbert space ${{\mathbb C}}^2 \otimes {{\mathbb C}}^2 \otimes {{\mathbb C}}^2$ — which realizes the Boolean [CUp]{} gate is: $${\rm Id} \otimes {\rm Id} \otimes {\rm Id} - c^\dagger c \otimes
a a^\dagger \otimes b^\dagger b +
({\rm Id} \otimes a^\dagger \otimes b) (c^\dagger c \otimes a a^\dagger
\otimes b^\dagger b)
\label{eq:CUp-formula}$$ where ${\rm Id}$ is the identity operator of ${{\mathbb C}}^2$ and, for the sake of clearness, we have written $c^\dagger$, $a^\dagger$, $b^\dagger$ (resp., $c$, $a$, $b$) to denote the creation (resp., annihilation) operator of ${{\mathbb C}}^2$ applied onto the subspaces of ${{\mathbb C}}^2 \otimes {{\mathbb C}}^2 \otimes {{\mathbb C}}^2$ corresponding to the first, second and third input, respectively. In fact, the gate behaves as the identity if the input pattern ${\left\vert {x_c,x_a,x_b} \right\rangle}$ is different from $(1,0,1)$, since in these cases $(c^\dagger c \otimes a a^\dagger \otimes b^\dagger b){\left\vert {x_c,x_a,x_b} \right\rangle} =
\mathbf{0}$, the null vector of ${{\mathbb C}}^2 \otimes {{\mathbb C}}^2 \otimes {{\mathbb C}}^2$. On the other hand $(c^\dagger c \otimes a a^\dagger \otimes b^\dagger b)
{\left\vert {1,0,1} \right\rangle} = {\left\vert {1,0,1} \right\rangle}$, hence the first two terms of dissapear and the operator $({\rm Id} \otimes a^\dagger \otimes b)$ is applied on ${\left\vert {1,0,1} \right\rangle}$, giving ${\left\vert {1,1,0} \right\rangle}$ as required.
In a completely analogous way we can see that the non–unitary linear operator which realizes the Boolean [CDown]{} gate is: $${\rm Id} \otimes {\rm Id} \otimes {\rm Id} - c^\dagger c \otimes
a^\dagger a \otimes b b^\dagger +
({\rm Id} \otimes a \otimes b^\dagger) (c^\dagger c \otimes a^\dagger a
\otimes b b^\dagger)$$
Let us note that the use of creation and annihilation operators allows for different physical implementations. For example, we can view computation not only as a conditional movement of energy but also as a conditional movement of particles between systems that may contain at most $d-1$ of particles. Alternatively, we can view computation as a sequence of conditional switches of the value of the $z$ component of the angular momentum of microscopical physical systems, using spin–creation and spin–annihilation instead of creation and annihilation operators [@cattaneo-leporati-leporini-qm].
Conclusions and directions for future work
==========================================
In this paper we have proposed the first steps towards a theory of conservative computing, where the amount of energy associated to the data which are manipulated during the computations is preserved.
The first obvious extension of our model is to take into account the energy used to perform computations, that is, to transform input values into output values. A first idea is to consider some additional *power source* input lines and *dissipation* output lines. Power source lines are fixed to a constant value from $L_d$ (usually $1$), and absorb energy from the environment. This energy is entirely consumed during the computation, whereas all the energy associated to the input pattern is returned by the output pattern. On the other hand, dissipation lines are used to model the release of energy into the environment; hence, their value is simply discarded. Conservative gates constitute a special case in our framework, where there are neither power source nor dissipation lines (under the hypothesis that we do not take into account the energy needed to perform the computation).
Since perfect conservation of energy can be obtained only in theory, a second possibility for future work could be to relax the conservativeness constraint , by assuming that the amount of energy dissipated during a computation step is not greater than a fixed value. Analogously, we can suppose that if we try to store an amount of energy that exceeds the capacity of the gate then the energy which cannot be stored is dissipated. In such a case it should be interesting to study trade-offs between the amount of energy dissipated and the hardness of the corresponding modified [ConsComp]{} problem.
Finally, it remains to study how to theoretically model and physically realize gates equipped with an internal storage unit. Here we just observe that, from a theoretical point of view, it seems appropriate to consider this kind of gates as finite state automata, by viewing the energy levels of the storage unit as their states.
[99]{} G. Cattaneo, A. Leporati and R. Leporini. Fredkin Gates for Finite–valued Reversible and Conservative Logics. *Journal of Physics A: Mathematical and General*, [**35**]{}, (2002) 9755–9785. G. Cattaneo, A. Leporati and R. Leporini. Quantum Conservative Gates for Finite–valued Logics. To appear on the International Journal of Theoretical Physics. E. Fredkin, T. Toffoli. Conservative Logic. *International Journal of Theoretical Physics*, **21**, (1982) 219–253. M. R. Garey, D. S. Johnson. *Computers and Intractability. A Guide to the Theory on NP–Completeness*. W. H. Freeman and Company, 1979. D. Gottesman. Fault–tolerant quantum computation with higher–dimensional systems. *Chaos, Solitons, and Fractals*, [**10**]{}, (1999) 1749–1758. A. Leporati. *Threshold Circuits and Quantum Gates*. Ph.D. Thesis, Computer Science Department, University of Milan, Italy, 2002. C. A. Petri. Gründsatzliches zur Beschreibung diskreter Prozesse. In Proceedings of the *3$^{\rm rd}$ Colloquium über Automatentheorie (Hannover, 1965)*, Birkhäuser Verlag, Basel, (1967) 121–140. English translation: Fundamentals of the Representation of Discrete Processes, ISF Report 82.04 (1982). N. Rescher. *Many–valued logics*. McGraw–Hill, 1969. J. B. Rosser, A. R. Turquette. *Many–valued logics*. North Holland, 1952.
[^1]: This work has been supported by MIUR$\backslash$ COFIN project “Formal Languages and Automata: Theory and Applications”.
|
---
abstract: 'We present Very Long Baseline Interferometry (VLBI) observations 217 days after the $\gamma$-ray burst of 2003 March 29. These observations provide further measurements of the size and position of GRB 030329 that are used to constrain the expansion rate and proper motion of this nearby GRB. The expansion rate appears to be slowing down with time, favoring expansion into a constant density interstellar medium, rather than a circumstellar wind with an r$^{-2}$ density profile. We also present late time Arecibo observations of the redshifted and OH absorption spectra towards GRB 030329. No absorption (or emission) is seen allowing us to place limits on the atomic neutral hydrogen of $N_H < 8.5 \times 10^{20}$ cm$^{-2}$, and molecular hydrogen of $N_{H_{2}} < 1.4 \times 10^{22}$ cm$^{-2}$. Finally, we present VLA limits on the radio polarization from the afterglow of $<$2% at late times.'
author:
- 'G. B. Taylor , E. Momjian , Y. Pihlström, T. Ghosh , & C. Salter'
nocite:
- '[@per98]'
- '[@gra05b]'
title: |
\
\
Late Time Observations of the Afterglow and Environment of GRB030329
---
Introduction
============
Our understanding of the origin of gamma-ray bursts (GRBs) has continued to advance rapidly in the years since the first X-ray [@cos97], optical [@van97] and radio [@fra97] afterglows were discovered. In particular the nearby afterglow from has solidified the GRB-supernova connection [@hjo03a; @mat03], and provided the first GRB with a well determined expansion rate [@tay04]. This event has presented a unique opportunity to test afterglow models [@ore04; @gra05], and to explore the environment around a GRB.
Afterglow models invoke gas-rich environments around at least some of the GRBs. Several basic properties of this circumburst medium are presently unknown, and need to be addressed in order to better understand the afterglow and its evolution. For example, in the simplest emission models, the density of the ambient medium is related to the afterglow flux density [@wax97]. Especially, if GRBs exist in both gas-poor and gas-rich environments, that could explain why afterglow is absent in some GRBs. Direct observations of the circumburst medium associated with GRBs would thus provide an important test of the fireball model.
Another issue is the density profile of the medium into which the GRB ejecta expands. Current models postulate a medium in which density is governed by the mass-loss of the progenitor star ($\rho\propto
r^{-2}$). At present however, some observations instead point to a uniform density in order to explain the evolution of the afterglow lightcurve [@bkp+03]. In a few cases, strong damped Ly$\alpha$ absorption has been found indicating column densities as high as 5 $\times$ 10$^{21}$ cm$^{-2}$ [@hjo03b]. During the expansion of the GRB ejecta, the circumburst medium is likely to go through different stages of ionization. Perna & Loeb (1998) suggest that as a consequence (optical) absorption lines will vary in their EW with time. The rate of this variation can constrain the size of the absorbing region. Similarly, in the radio the fraction of molecular and atomic gas should be governed by the effective ionization, and might therefore vary in a similar way.
One way to investigate the circumburst medium is via atomic or molecular absorption studies in the radio. Another way is to measure the deceleration of the expanding fireball. In this paper we present both late time Very Long Baseline Interferometry (VLBI) observations of the afterglow of and similar epoch and OH absorption observations taken with the Arecibo telescope.
Assuming a Lambda cosmology with $H_0 = 71$ km/s/Mpc, $\Omega_M =
0.27$ and $\Omega_\Lambda=0.73$, the angular-diameter distance of at $z=0.1685$ is $d_A=589\,$Mpc, and 1 milliarcsec corresponds to 2.85 pc.
Observations and Results {#sec:obs}
========================
Late Time VLBI Observations
---------------------------
[lrrrrrc]{} 20 Jun 2003 & 83 & 8.409 & 138 & 50 & 2 & Y27\
1 Nov 2003 & 217 & 8.409 & 138 & 50 & 2 & Y27\
1 Nov 2003 & 217 & 8.409 & 165 & 32 & 2 & VLBA+EB+Y27+WB+AR+MC+NT\
30 Nov 2003 & 247 & 1.216 & 30 & 12 & 2 & AR\
30 Nov 2003 & 247 & 1.428 & 50 & 12 & 2 & AR\
30 Nov 2003 & 247 & 1.473 & 30 & 12 & 2 & AR\
The VLBI observations were taken at 8.4 GHz on 2003 November 1, 217 days after the burst, with a global array including the Very Long Baseline Array (VLBA) of the NRAO[^1]. Other telescopes used were the Effelsberg 100-m telescope[^2], the phased VLA, the Green Bank Telescope (GBT), the 305 m Arecibo telescope[^3], the Westerbork (WSRT) tied array, and the Noto and Medecina telescopes of the Consiglio Nazionale delle Ricerche. In all some 56 individual and combined antennas each of 25m or more in diameter were employed with a combined collecting area of 0.12 km$^2$. Most antennas were on-source for a period of 5.5 hours. All stations recorded with 256 Mbps with 2 bit sampling in dual circular polarization with the exception of Noto which had only a right-circular polarization receiver available. The observations were correlated at the Joint Institute for VLBI in Europe (JIVE).
The nearby (1.5$^\circ$) source J1051+2119 was used for phase-referencing with a 2:1 minute cycle on source:calibrator. The weak calibrator J1048+2115 was observed hourly to check on the quality of the phase referencing. Self-calibration with a 1 hour solution interval was used to further refine the calibration and remove some slow-changing atmospheric phase errors. The final image has an rms noise of 24 $\mu$Jy/beam. This is substantially higher than the expected thermal noise of 8$\mu$Jy/beam, in part because of partial loss of signals from Arecibo and Westerbork for reasons not completely understood.
We fit a symmetric, two-dimensional Gaussian to the measured visibilities on and find a size of 0.176 $\pm$ 0.08 mas. As in [@tay04] the error of the size is estimated from signal-to-noise ratios and from Monte-Carlo simulations of the data using identical ($u$,$v$) coverage, similar noise properties, and a Gaussian component of known size added. The standard deviation of the recovered sizes, modelfitted in the same way as we treat the observations, was found to be 0.053 mas.
We also obtain a position for of R.A. 10$^h$44$^m$49.95955$^s$ and Dec. 21$^\circ$31’17.4377” with an uncertainty of 0.2 mas in each coordinate. Solving for proper motion using all the high frequency VLBI observations to date, we derive $\mu_{\rm r.a.}=-0.05\pm 0.41$ mas yr$^{-1}$ and $\mu_{\rm dec.}=-0.24\pm 0.41$ mas yr$^{-1}$, or an angular displacement over the first 217 days of 0.14$\pm$0.35 mas (Fig. 1). These observations are consistent with those reported by [@tay04], and impose an even stronger limit on the proper motion. This limit argues against the cannonball model for GRBs proposed by [@dado04].
Single Dish Arecibo Observations
--------------------------------
Single dish observations of GRB 030329 were carried out with the L-band Wide receiver of the 305 m Arecibo Radio Telescope on 2003 November 30, for a total of 2 hr. The simple position-switched observations utilized all four interim correlator boards to observe various L-band frequencies. The bandwidth of each board was 12.5 MHz. While the first two boards were set to observe the frequency of the redshifted $\lambda$21 cm H [I]{} line, recording each linear polarization with 2048 spectral channels, the other two boards were set to observe simultaneously the orthogonal polarizations at the frequencies of the redshifted $\lambda$18 cm mainline OH transitions (1665 & 1667 MHz) and the 1720 MHz satellite OH transition with 1024 spectral channels per polarization.
Following the editing out of data suffering from radio frequency interference, the total on-source integration time for the redshifted $\lambda$21 cm H [I]{} and the 1720 MHz satellite OH transitions was 30 min each. For the redshifted $\lambda$18 cm mainline OH transitions, the on-source integration time was 50 min.
Constraints on the Atomic and Molecular Gas
===========================================
Limits on Absorption
--------------------
Figures 2, 3, and 4 show pairs of total-intensity spectra of the GRB 030329 centered at the optical heliocentric velocity that corresponds to the frequency of the redshifted $\lambda$21 cm H [I]{} line, the redshifted $\lambda$18 cm OH mainlines, and the redshifted $\lambda$18 cm OH satellite line at 1720 MHz, respectively.
The top spectra in Figs. 2, 3, and 4, are Hanning-smoothed with spectral resolutions of 3.52 km s$^{-1}$ (12.2 kHz), 5.99 km s$^{-1}$ (24.4 kHz), and 5.80 km s$^{-1}$ (24.4 kHz), respectively. The rms noise level in each of these spectra is 1.02, 0.32, and 0.57 mJy beam$^{-1}$, respectively. The bottom plot in each figure is a five-channel smoothed version of the respective top spectra. The velocity resolution in these spectra is 8.80, 14.98, and 14.50 km s$^{-1}$, and the rms noise level of each spectrum is 0.482, 0.224, and 0.282 mJy beam$^{-1}$, respectively.
No emission or absorption is seen in any of the Arecibo spectra. We can derive a 3$\sigma$ limit on the opacity of $\tau < 0.53$ with a velocity resolution of 8.8 km s$^{-1}$. This corresponds to a column density limit of $N_H < 8.5 \times 10^{20}$ cm$^{-2}$ assuming a spin temperature of 100 K, and uniform coverage. If the spin temperature is higher or the line is shallow and wide, then this limit could be higher. Given the extreme energies involved in the GRB, enough to cause a sudden ionospheric disturbance in the Earth’s atmosphere [@sch03] some 740 Mpc away, it could well be that nearly all the along the line-of-sight in the host galaxy has been ionized and not yet recombined. Further research in this area is needed.
In a similar manner we can place a 3$\sigma$ upper limit on the 1667 MHz OH mainline opacity of $\tau < 0.11$ with a velocity resolution of 15.0 km s$^{-1}$. This corresponds to an OH column density limit of $N_{OH} < 1.4 \times 10^{15}$ cm$^{-2}$ assuming a uniform covering factor, and an excitation temperature of 10 K. We can further deduce a limit on molecular hydrogen using the relation $N_{H_{2}}$ $\sim$ 10$^7$ $N_{OH}$ [@kan02], of $N_{H_{2}} < 1.4 \times
10^{22}$ cm$^{-2}$.
The continuum flux density of the GRB 030329 at 1.4 GHz at the time of these observations was $\sim3.5$ mJy. The various spectra reported here show a higher continuum flux density, because of a 15 mJy continuum source located at an angular distance of about 3 arcmin from the GRB 030329, i.e., within the primary beam of the Arecibo radio telescope.
Angular Size Measurements {#sec:size}
-------------------------
Our measured size of 0.176 $\pm$ 0.08 mas is quite close to the size of 0.172 $\pm$ 0.043 found by [@tay04], indicating a possible slowing of the burst at late times. This trend was already apparent from the average expansion velocities derived from the angular size measurements on April 22 and June 20. The entire history of expansion for is shown in Fig. 5. The first measurement at 15 days comes from a model-dependent estimate of the quenching of the scintillation [@bkp+03]. The late time curvature may indicate that between 83 and 217 days, has transitioned into a non-relativistic expansion. Alternatively, it is possible (but somewhat contrived) that the intrinsic surface brightness profile has changed in a way to compensate for the expansion. For all epochs we fit a two-dimensional Gaussian model since our resolution does not permit us to discriminate between a Gaussian, ring, disk, or more complex profile.
A gamma-ray burst drives a relativistic blast wave into a circumburst medium of density $\rho$ whose radius $R$ is related to the energy of the explosion approximately by $E\sim R^3\rho c^2\gamma^2$, where $\gamma$ is the bulk Lorentz factor of the fireball. More complete treatments are given by [@bm76; @cl00; @gl03]. The density profile of the circumburst medium is generally taken to be either a $1/r^2$ wind, or a constant density ISM, such as one might find beyond the termination shock of the progenitor’s stellar wind [@che04]. The deceleration at late times currently favors the ISM model of [@gra05] which shows a break in the expansion rate (see their Figures 4 and 6). Unfortunately, the large uncertainty in our measurement of the size of at late times does not permit us to discriminate strongly between the various models.
[@che04] place the termination shock of the Wolf-Rayet wind at a distance of 0.4 pc from the progenitor. Beyond that they predict a fairly constant density out to the red supergiant shell at a radius of 1.7 pc. With a current diameter of 0.5 pc for (radius of 0.25 pc) at day 217 it is near the termination shock, especially if the progenitor had a shorter lifetime, or a relatively high density ISM has stalled the shock.
Implications of low Polarization
--------------------------------
From 8.4 GHz VLBA observations on April 6 [@tay04] derived a 3$\sigma$ limit on the linear polarization of 0.16 mJy/beam, corresponding to a limit on the fractional polarization of $<$1.0%. In a contemporaneous optical observation [@gre03] measure a polarization of 2.2+/-0.3%. The decrease in polarization at lower frequencies has been explained as the result of the source being optically thick at 8.4 GHz at these early times since the maximum degree of linear polarization of an optically thick synchrotron source is 12% while the maximum polarization of an optically thin synchrotron source is about 80% [@pac70]. To look for any change in polarization with time as the source transitions to optically thin at 8.4 GHz we have analyzed the very sensitive, late time phased VLA observations for polarimetry. We find no detection at either epoch and place 3$\sigma$ limits of $<$1.8% and $<$4.7% on 2003 Jun. 20 and 2003 Nov. 1 respectively. The total intensity measured with the VLA on these epochs is 3.11 $\pm$ 0.03 and 0.75 $\pm$ 0.02 mJy respectively.
The emission mechanism for the GRB afterglow is widely accepted to be synchrotron radiation, which is intrinsically linearly polarized if there is an ordered component to the magnetic field, or if the magnetic field is generated at the internal shocks [@med99]. The recent detection of 80 $\pm$ 20% polarization in the prompt $\gamma$-ray emission from GRB 021206 [@cob03], although controversial [@rut04; @wig04], has led to an increased interest in modeling the time and frequency dependent behavior of the polarization from the afterglow [@nak03; @gra03; @gra05b].
To have an observed polarization of $<$5% in the late time afterglow (see §2.2) could be explained as the result of highly disordered magnetic fields [@grak03].
Propagation effects can also reduce the intrinsic linear polarization below detectable levels. A Faraday screen produced by ionized gas and magnetic fields can cause gradients in the observed polarization angle across the source, leading to depolarization if the resolution element of the telescope, or the size of the source if unresolved, is large compared to the gradients. The rotation measures ($RM$) can be related to the line-of-sight magnetic field, $B_{\|}$, by
$$RM = 812\int\limits_0^L n_{\rm e} B_{\|} {\rm d}l ~{\rm
radians~m}^{-2}~,
\eqno(1)$$
where $B_{\|}$ is measured in mG, $n_{e}$ in cm$^{-3}$, d$l$ in pc, and the upper limit of integration, $L$, is the distance from the emitting source to the end of the path through the Faraday screen along the line of sight. We can estimate the minimum magnetic field needed to produce a gradient across the source. To produce a 90$^\circ$ rotation at 8.4 GHz requires a RM of 1200 rad m$^{-2}$. Assuming a density within the red supergiant shell of 0.2 cm$^{-3}$ [@che04] and a path length of 2 pc, then the magnetic field strength required is $B_{\|}=4$ mG. This field strength is similar to the equipartition field strengths [@dou03] have found from modeling the radio emission from colliding-wind Wolf-Rayet (WR) binary systems, although the field strengths in these systems have probably been enhanced by the collision. Faraday screens in the GRB environment are considered in more detail by @gra05b.
Conclusions
===========
While no detection of atomic () or molecular (OH) material is found towards , the limits of $N_H < 8.5 \times 10^{20}$ cm$^{-2}$ are not particularly constraining. Observations of more shielded molecules like NH$_3$ towards future bursts at early times would be of interest, and given the expected strength of a nearby afterglow of $\sim$50 mJy or more, could yield detections, or place interesting limits on the amount of material in the GRB environment.
Although GRB 030329 has faded considerably, it may still be detectable with VLBI techniques. Even crude estimates of the size could differentiate between the predictions of wind and constant-density environments at these late times. Radio re-brightening of has been predicted by [@gl03], and [@li04] estimate a level of 0.6 mJy 1.7 years after the burst. This re-brightening might occur as the counterjet becomes non-relativistic and therefore radiatively isotropic. This brightening will be accompanied by a temporary rapid growth in the size of the source as two, well-separated jets become visible, and by a change in the light centroid [@gl03]. If such a re-brightening occurs then a precise size estimate at late times becomes readily achievable with existing facilities.
Rather surprisingly, we find no detectable linear polarization from at cm wavelengths to limits as low as 1%. This could indicate less order than expected in the magnetic fields of the external shock that drives the afterglow, or a Faraday screen that depolarizes the radio emission. All of these limits are from 8 days or more after the burst. Given the RHESSI result [@cob03] of large polarization from the $\gamma$-rays, and predictions of some fireball models, it would be well worth searching for polarization from the prompt optical and radio emission ([*e.g.*]{}, Granot & Taylor 2005).
In the near future the [*Swift*]{} satellite[^4] should dramatically increase the number of GRBs with measured redshifts, revealing some that are nearby. In future VLBI studies of GRB afterglows at redshifts less than 0.1 it should be possible to image the structure of the afterglow. Fireball models of heating by a single relativistic shock front predict that at late times the fireball should look like a ring [@gps99b].
GBT thanks the Kavli Institute for Particle Astrophysics and Cosmology for hospitality and support. We thank Jonathon Granot and Avi Loeb for useful discussions. This research has made use of NASA’s Astrophysics Data System.
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[^1]: The National Radio Astronomy Observatory is operated by Associated Universities, Inc., under cooperative agreement with the National Science Foundation.
[^2]: The 100-m telescope at Effelsberg is operated by the Max-Planck-Institut f[ü]{}r Radioastronomie in Bonn.
[^3]: The Arecibo Observatory is part of the National Astronomy and Ionosphere Center, which is operated by Cornell University under a cooperative agreement with the National Science Foundation.
[^4]: see http://swift.gsfc.nasa.gov
|
---
abstract: 'The statement that Gibbs equilibrium ensembles are equivalent is a base line in many approaches in the context of equilibrium statistical mechanics. However, as a known fact, for some physical systems this equivalence may not be true. In this paper we illustrate from first principles the inequivalence between the canonical and microcanonical ensembles for a system with long range interactions. We make use of molecular dynamics simulations and Monte Carlo simulations to explore the thermodynamics properties of the self gravitating ring model and discuss on what conditions the Maxwell construction is applicable.'
author:
- 'T. M. Rocha Filho'
- 'C. H. Silvestre'
- 'M. A. Amato'
title: 'Ensemble Inequivalence and Maxwell Construction in the Self-Gravitating Ring Model'
---
Introduction
============
Equilibrium Statistical Mechanics is a hallmark of theoretical physics and an invaluable tool to study the properties of matter for more than a century. At its foundations lies Gibbs’ ensembles theory [@gibbs], which is an elegant formulation applicable to a very broad class of phenomena (for a brief history of ensemble theory see [@inaba]). As it is well known for any rounded practitioner, the microcanonical ensemble is harder to use than the canonical ensemble, and the grand-canonical ensemble is in a sense the more simple among them. Provided that the predictions of different ensembles coincide, one can then choose which one to use according to the needs in consideration. As a consequence, many authors devoted a considerable effort to the task of proving and establishing the limits of validity of the equivalence of the different ensembles (see [@lewis] and references therein). From the thermodynamic viewpoint ensemble equivalence is based on the fact that the Legendre transformation connecting different ensembles is invertible. This implies particularly that the entropy is a concave function of the energy in the whole physically accessible energy range. In statistical terms it means that all properties of the system are well described either in terms of energy or temperature, that is essentially Gibbs’ argument.
These respective ensembles are equivalent in the thermodynamic limit $N\rightarrow\infty$ if the interaction potential is tempered and stable, i. e.if the energy is additive and a stable equilibrium state exists [@ruelle]. The Helmholtz free energy and the grand-potential are then obtained from the microcanonical entropy by the usual Legendre transforms. Examples of physically relevant and non-stable potentials are the gravitational interaction, where for some specific cases the non-stability leads to the so-called gravothermal catastrophe [@binney], and multi-species plasmas [@ruelle]. As well known, equilibrium ensembles for self-gravitating systems are inequivalent, and an energy interval with negative heat capacity exists in the microcanonical ensemble [@padmanabhan]. The appearance of a convexity region in the entropy-energy curve breaks down the equivalence and any state of the system in the convex region cannot be realized in the canonical ensemble. The simplicity of Gibbs argument fails in this case [@ellis].
According to van Hove’s theorem [@hove], and under certain assumptions, the pressure $P$ in the canonical ensemble must be a decreasing function of the volume $V$ and becomes constant for the values of $V$ in the interval of phase coexistence. In the microcanonical ensemble this corresponds to the Maxwell construction prescription to replace the entropy by its concave envelope. One of the conditions required in van Hove’s theorem is that, for a three-dimensional system, the interparticle potential $V(r)$ satisfies $V(r)\geq r^{-3-\alpha}$ for $\alpha>0$ and large distances $r$, i. e. that the potential is short-ranged and the total energy is additive. Therefore, in equilibrium statistical mechanics calculations, a convex intruder in the entropy function can only exist as a result of approximations, e. g. using a mean-field approach for a system with short-range interactions, or as a consequence of finite size effects [@wales; @lyndenbell; @gross]. This point is very well illustrated for the two-dimensional Potts model with nearest neighbors interaction, where a convex dip is present for small lattice size, disappearing for increasing $N$ with the negative specific heat region being replaced by a flat curve, while for globally coupled spins, the convex dip remains even in the thermodynamic limit [@ispolatov].
Thus the additivity of energy and entropy, which follows from the temperedness of the potential, and its stability, ensure that equilibrium ensembles are equivalent. The important point is that there exist real systems for which these conditions are not met. However, this does not imply that ensembles are not equivalent as the conditions are sufficient but not necessary.
Besides self-gravitating systems, examples of real physical situations with the occurrence of a convex intruder in the entropy are two-dimensional quasi-geostrophic flows [@turkington], wave-particle interaction in a plasma in the presence of two harmonics [@elskens; @teles] and magnetically self-confined plasma torus [@kiessling]. They are also are examples of long-range interacting systems, with interacting potentials decaying at long-distances as $1/r^\alpha$ with $\alpha<D$, $D$ being the spatial dimension [@proc1; @proc2; @proc3; @booklri; @campa; @levin]. This definition implies a non-additive energy as the interaction energy between two subsystems is no longer negligible when compared to the bulk energy. It is worth noticing that this definition may be at variance with some works in the literature, as for instance in Ref. [@kac] where an interaction with an exponential dependence on distance, and therefore not long-ranged in the sense adopted here, is referred as long-ranged.
The main goal of the present paper is to illustrate with a specific model of a many particle system with dynamics, for the first time up to the author’s knowledge, the inequivalence of the microcanonical and canonical ensembles from first principles, i. e. by only solving the Hamilton equations of motion, and to show that the results so obtained are in agreement with previous theoretical studies and Monte-Carlos simulations. This allows us to discuss the physical origin of ensembles inequivalence as well as the meaning of the Maxwell construction if the interactions are long-ranged. The model system chosen is the one-dimensional self-gravitating ring model with Hamiltonian in Eq. (\[ringmodham\]), as its thermodynamic properties are well known, with a first order phase transition from a homogeneous to a non-homogeneous phase [@sota].
The structure of the paper is the following: the ring model is presented in Section \[ringmod\] and in Section \[mce\] we present our molecular dynamics results and compare them to Monte Carlo simulations and results from previous works. In Section \[disc\] we discuss our results and present some concluding remarks.
The self-gravitating ring model {#ringmod}
===============================
The Self-Gravitating Ring (SGR) model was introduced by Sota and collaborators [@sota] and describes a system of $N$ particles constrained to move on a circle and interacting by a gravitational potential regularized by a (usually small) softening parameter $\epsilon$ introduced in order to avoid the divergence of the potential at short distances. With a proper choice of units its Hamiltonian can be written as: $$H=\sum_{i=1}^N\frac{p_i^2}{2}-\frac{1}{2N}\sum_{i,j=1}^N\frac{2\sqrt{\epsilon}}{\sqrt{1-\cos(\theta_i-\theta_j)+\epsilon}},
\label{ringmodham}$$ with $\theta_i$ being the position angle on the circle of particle $i$ and $p_i$ its conjugate (angular) momentum. Here units have been chosen such that the minimum value of the energy per particle is $-1$ irrespective of the value of $\epsilon$. The factor $1/N$ in the potential energy term is known as the Kac factor, and can be introduced by a change of time units (as long as $N$ remains finite), in order for the total energy to be extensive, although remaining non-additive. It also facilitates the comparison of results with different numbers of particles. The analogous of a magnetization can be introduced here by its components: $$M_x=\frac{1}{N}\sum_{i=1}^N\cos\theta_i,\hspace{10mm} M_y=\frac{1}{N}\sum_{i=1}^N\sin\theta_i.
\label{magscompsdef}$$ Many properties of the model were studied in previous works [@eu1; @casetti; @monechi; @tatekawa; @nardini; @eu2]. It has a phase-transition from a low energy ferromagnetic phase to a high energy homogeneous (non-magnetic) phase. The order of the transition depends on the value of the softening parameter. For smaller values of $\epsilon$ the transition is first order and becomes continuous for higher values of the parameter. It is worth noticing that for systems with long-range interaction in the $N\rightarrow\infty$ limit particles are exactly uncorrelated [@chavanis].
Microcanonical and canonical ensembles for the self-gravitating ring model {#mce}
==========================================================================
We fix the value for the softening parameter as $\epsilon=10^{-2}$ such that the system has a continuous phase transition and a negative heat capacity for an energy interval, at the same time allowing for faster molecular dynamics simulations (smaller values of $\epsilon$ leads to higher values of numeric error at fixed time step in the integration algorithm). For reference purposes we first determine the caloric curve for the system for the chosen value of $\epsilon$, with very high accuracy, using an iterative numeric variational method by Tatekawa et al. that also applied it for studying the thermodynamics of the SGR model [@tatekawa]. It amount to maximizing the Gibbs entropy for uncorrelated particles: $$s\equiv S/N=-\int{\rm d}p\:{\rm d}\theta\:f_1(p,\theta)\ln f_1(p,\theta),
\label{gibbsent}$$ where the Boltzmann constant is set to unity and $f_1(p,\theta)$ is the one-particle distribution function. The results are shown in Fig. \[caloricmicrofig\], where a region of negative heat capacity is clearly visible. Note that the variational method is solely based on entropy maximization on the space of one-particle distribution functions, and the results obtained are thus a direct consequence of the second law of thermodynamics. Microcanonical Monte Carlo results are also shown in the figure and were obtained using the method described in Ref. [@ray], with a very good agreement with the variational method.
One may ask whether the system can relax to an equilibrium state if its energy belongs to the region with a negative heat capacity. In order to answer that question we numerically solve the Hamiltonian equations of motion for the $N$ particle system using a graphics processing unit parallel implementation [@eu3] of a fourth-order symplectic integrator [@yoshida]. The system is initially prepared in a [*waterbag*]{} non-equilibrium state with a uniform distribution in the intervals $-p_0<p<p_0$ and $-\theta_0<\theta<\theta_0$, with the constants $p_0$ and $\theta_0$ chosen for the system to have the required energy. Since the relaxation time to reach equilibrium is typically very long in long-range interacting systems, and scales with $N$ for non-homogeneous or $N^2$ for homogeneous states [@eu2; @eu4; @chris], very long computer runs are required. The left panel of Fig. \[exdynmicro\] shows the time evolution of the total kinetic $K$ and potential $V$ energies per particle for a homogeneous waterbag non-equilibrium initial state with $\theta_0=\pi$ and total energy per particle $e=-0.135$. The total simulation time for this case is roughly 12 hours on a NVIDIA GTX 690 graphic card. In order to measure the distance to equilibrium we use the kurtosis of the momentum distribution, i. e. the fourth reduced statistical moment of $p$ given by ${\cal K}=\langle p^4\rangle/\langle p^2\rangle^2$. Its value for any Gaussian distribution is given by ${\cal K}=3$. The right panel of Fig. \[exdynmicro\] shows the kurtosis of the momentum distribution, where we see that the system reaches equilibrium at $t\approx2\times10^5$. Data points obtained from such simulations and different energy values are also shown in Fig. \[caloricmicrofig\], with a very good agreement with both microcanonical Monte Carlo simulations and the variational method.
Thus, at least for the present model, equilibrium states with a negative heat capacity are accessible by relaxation from non-equilibrium states.
The next point to consider is to determine the caloric curve in the canonical ensemble. Canonical Monte Carlo results are show in the left panel of Fig. \[ensemblestufig\], alongside with points obtained from the minimization of the free energy. The latter was obtained using the usual relation $F=E-TS$ with the entropy computed from Eq. (\[gibbsent\]) by writing $$f_1(p,\theta)=\sqrt{\beta/2\pi}\:e^{-\beta p^2/2}\rho(\theta)
\label{onepartdf}$$ and the (normalized) spatial distribution $\rho(\theta)$ given a histogram of the particle positions from the microcanonical Monte Carlo simulation. Both results from canonical Monte Carlo and free energy minimization are in quite good agreement except for a few points corresponding to metastable states close to $e=-0.55$. Such states, being local minima of the free energy, trap the Monte Carlo evolution for a very large number of steps and are known to cause numerical difficulties. To circumvent the effects of such metastable states we used a relatively small number of particles $N=300$.
Now, if the system, in a microcanonical equilibrium state with a negative heat capacity, is put in contact with a thermal bath at the same temperature, it becomes unstable and evolves to the canonical equilibrium state that minimizes the free energy, for the same temperature of the initial state. We illustrate this, again from from first principles, by performing a molecular dynamics simulation of both system and thermal bath. Following the same approach as in Ref. [@nosprl], the bath is modeled by a Hamiltonian system with a short-range interaction formed by $M$ rotors with Hamiltonian: $$H_{bath}=\sum_{i=N+1}^{N+M}\frac{p_i^2}{2}+\sigma\sum_{i=N+1}^{N+M}\left[1-\cos(\theta_i-\theta_{i+1})\right],
\label{bathham}$$ where $\theta_i$ and $p_i$ the coordinates and the momenta of particle $i$, with $\theta_{N+M+1}\equiv\theta_{N+1}$ and $\sigma$ the interaction strength among first neighbors. The number of particles in the bath is chosen such that $M\gg N$ to guarantee that the main system only causes minor disturbances in the bath. The SGR model and the bath are coupled by the interaction potential: $$V_{int}=\lambda\sum_{i=1}^L\left[1-\cos(\theta_i-\theta_{i+N})\right],
\label{intpotsb}$$ with $\lambda$ the coupling parameter and $L$ chosen typically as a fraction of $N$. Since the initial conditions for both systems are randomly chosen from given initial distribution, any choice for which particles are coupled is equivalent, and we simply chose to couple the $L$ particles of the self-gravitating ring model with indices $i=1,\dots,L$ to the $L$ particles of the bath with indices $i=N+1,\dots,N+L$. The total Hamiltonian is then the sum of the Hamiltonian in Eq. (\[ringmodham\]), $H_{bath}$ and $V_{int}$. The choice of the parameters requires some experimentation, and the values used here are $N=512$, $M=204\:800$, $L=16$, $\sigma=5.0$ and $\lambda=1.0$. The left panel in Fig. \[energsfig\] shows the time evolution to equilibrium of the isolated SGR model with a waterbag initial state with energy $e=-0.2$, which lies inside the negative heat capacity energy interval. The resulting microcanonical equilibrium state is then coupled to the thermal bath. The time evolution of temperature as given by twice the kinetic energy, the kurtosis of the velocity distribution for the main system, and the interaction energy between the system and the bath are show in the right panel of the same figure. The ring model starts from the initial state become unstable and evolves to a canonical equilibrium while the temperature remains constant, up to small fluctuations. The kurtosis remains always close to the equilibrium value, indicating that the momentum distribution function remains a Gaussian during the whole time evolution. The spatial distribution function changes until the final state corresponds to the minimum of the free energy for the non-homogeneous phase.
The caloric curve in the canonical ensemble from MD simulations is obtained in the following way: the system is prepared in a low energy equilibrium state, corresponding to a positive heat capacity in contact with the heat bath. The temperature of the bath is then raised by a factor $\alpha$ by multiplying the velocities of the particles in the bath by $\sqrt{\alpha}$. The coupled system and bath are then left to evolve, after a very long time, into a new equilibrium state, and the total energy and temperature as twice the kinetic energy are obtained from a numeric average over a given time interval. The results are show in Fig. \[caloriccan\], where the jump in energy is clearly visible, in accordance with canonical Monte Carlo results shown in Fig. \[ensemblestufig\]. By comparing Figs. \[ensemblestufig\] and \[caloriccan\] we see that when rising the temperature from a stable non-homogeneous state, the jump occurs at an energy value dictated by the minimization of the free energy, as obviously expected. For systems with short-range interactions, states in the gap are accessible by considering different proportion of the particles in each one of the phases at each extremity of the gap. This leads to the Maxwell construction prescription and yield the dotted line in Fig. \[ensemblestufig\]. On the other hand, for long-range interacting systems and as extensively discussed in the pertaining literature (see for instance Ref. [@booklri]), there is no phase-coexistence and the line associated to the Maxwell construction is physically meaningless. Our numerical experiment for the SGR model shows this behavior clearly as a small increase in the temperature at the phase transition causes a discontinuous jump in the total energy. This is different to what is observed in the microcanonical ensemble. We stress here that this comes out directly and solely from the dynamics of the system, i. e. from the numeric solution of the Hamilton equations of motion.
Discussion and Conclusions {#disc}
==========================
Up to the authors’ knowledge, the verification for a specific model of ensemble inequivalence was never shown before from full $N$-body molecular dynamics molecular simulations and without any other assumption on the system. From the results above we see that for the ring model in the canonical ensemble, i. e. for a system coupled to a large energy reservoir, the whole energy interval corresponding to the Maxwell construction is not physically realizable in the canonical ensemble. If the temperature is raised starting from a non-homogeneous state with positive heat capacity, the energy jumps discontinuously, the energy interval of the jump corresponding to the usual Maxwell prescription, but without phase coexistence that would correspond to this inaccessible energies in the canonical ensemble. If the system is in a state with energy in this interval, it will absorb from or give energy to the bath until it reaches a stable canonical equilibrium that minimizes the free energy, as expected [@touchette], and as illustrated from our direct numerical simulations. This is at variance to systems with short range interactions where any energy in the interval where the Maxwell construction is required is realizable by a combination of the two different phases, as guaranteed by the additivity of energy and entropy valid for such systems but not for long-range interactions.
It is a common practice, when computing statistical mechanics properties of a short-range interacting system, to rely on some approximation that results in a van der Waals loop, e. g. using a mean-field approximation. In this case replacing the convex dip in the entropy function by a flat line according to the Maxwell construction prescription is fully justified, as the entropy of the original system is additive, although this property is usually not satisfied in a mean-field approximation. On the other hand, if the system has long-range interactions, or equivalently is not tempered and does not satisfies the condition of van Hove’s theorem, then the convex dip may not disappear in the thermodynamic limit. Using the Maxwell construction in this case is plainly wrong as it leads to physically non realizable states. In this case, the correct prescription is to take the whole region corresponding to a negative heat capacity as non physical. A purely thermodynamic description of long-range interacting systems remains possible along the lines discussed in Refs. [@latella1; @latella2].
Acknowledgments
===============
The authors would like to thank S. R. Salinas and V. B. Henriques for fruitful discussions. TMRF was partially financed by CNPq (Brazil) and CHS was financed by CAPES (Brazil). We also thank an anonymous referee for calling our attention to Ref. [@kiessling].
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abstract: 'We study locally gated silicene nanoribbons as spin active devices. We find that the gated segments of zigzag nanoribbons can be used for inversion of spins. The strong intrinsic spin-orbit coupling for low Fermi energy in presence of an external vertical electric field provides a fast spin precession around the axis perpendicular to the silicene plane. The spin inversion length can be as small as 10 nm. On the other hand in the armchair nanoribbons the spin inversion occurs via the Rashba effect which is weak and the spin inversion lengths are of the order of $\mu$m.'
author:
- Bartłomiej Rzeszotarski
- Bartłomiej Szafran
bibliography:
- 'bib\_silicene.bib'
title: Electron spin inversion in gated silicene nanoribbons
---
Introduction
============
Silicene [@chow] is buckled graphene-like structure with strong spin-orbit (SO) coupling [@Liu11; @Liu; @Ezawa] belonging to the 2D-Xenes group [@xene] that potentially can be used in spintronic devices [@Zutic04]. In silicene systems numerous quantum effects have been predicted, including the quantum spin Hall effect [@Liu11], anomalous Hall effect [@Ezawa] and its valley-polarized variant [@Pan14]. Also, an appearance of giant magnetoresistance is expected [@Xu12; @Rachel14]. Furthermore, in presence of the external perpendicular electric field topological phase transitions in the edge states are predicted [@Ezawa2012a; @romera; @Tabert13]. Spin-filtering applications are possible for silicene [@Tsai13; @wu15; @miso15; @shak15; @nunez16] with the local exchange field introduced by the ferromagnetic proximity effects [@an; @0yoko; @1wang; @2saxena; @3derak; @4wang] or magnetic impurities [@derak16; @rzang]. In recent research, the silicene field effect transistor that operates at room-temperature has been demonstrated [@Tao15] with Al$_2$O$_3$ dielectric substrate that only weakly modifies the free-standing silicene band structure near the Dirac points [@al2o3] in contrast to Ag substrates [@Aufray10; @Feng12; @Vogt12].
In the present paper we study electron spin inverter in silicene that exploits the SO interactions. The Rashba spin-orbit interaction due to the vertical electric field generates an in-plane effective magnetic field [@Meier2007] that in III-V two-dimensional electron gas induces precession of the spin that is injected perpendicular to the plane of confinement. However, we find that for zigzag silicene nanoribbon the precession in the Rashba effective field is blocked by the intrinsic SO interaction that generates strong internal magnetic field along the $z$ axis [@cum] that stabilizes the spin and stops its precession. The effects of the intrinsic SO coupling [@cum] are lifted for the armchair edge that introduces the intervalley scattering. The spin-precession in armchair ribbons is observed but since the Rashba SO interaction in slicene is weak the spin inversion lengths are of the order of 1 $\mu$m which may not be attractive from the point of the practical application.
We show that spin precession occurs very fast under intrinsic SO coupling in zigzag silicene nanoribbons when the spin is injected within the ribbon plane perpendicular to the electron momentum. The difference in wave vectors can be easily chosen by Fermi energy in presence of the external electric field. The spin precession length can be tuned by the electric fields to the values lower than 10 nm.
Theory
======
Hamiltonian
-----------
In calculations we use the $\pi$ band tight-binding Hamiltonian [@Liu] for the free standing silicene which takes the form $$\begin{aligned}
H_0=&-t\sum_{\langle k,l\rangle \alpha } c_{k \alpha}^\dagger c_{l \alpha}+e \sum_{k,\alpha}F_z \ell_k c^\dagger_{k,\alpha}c_{k,\alpha}, \nonumber \\
& -i\frac{2}{3}\lambda_{R}^{int.} \sum_{\langle \langle k,l \rangle \rangle \alpha,\beta } \mu_{kl} c^\dagger _{k\alpha}\left(\vec{\sigma}\times\vec{d}_{kl} \right)^z_{\alpha\beta} c_{l\beta} \nonumber \\
&+i\frac{\lambda_{SO}}{3\sqrt{3}} \sum_{\langle \langle k,l\rangle \rangle \alpha, \beta } \nu_{kl} c^\dagger_{k\alpha} \sigma^{z}_{\alpha,\beta}c_{l\beta} \nonumber \\
& +i\lambda_{R}^{ext.} (F_z) \sum_{\langle k,l \rangle\alpha,\beta} c_{k\alpha}^\dagger \left(\vec{\sigma}\times\vec{d}_{kl} \right)^z_{\alpha\beta} c_{l\beta} \nonumber \\
& + \sum_{k,\alpha,\beta}c^\dagger_{k,\alpha}(\mathbf{M}\cdot\pmb{\sigma})_{\alpha\beta}c_{k,\beta},
\label{eq:h0}\end{aligned}$$
where $c_{k \alpha}^\dagger$ ($c_{k \alpha}$) is the creation (annihilation) operator for an electron on site $k$ with spin $\alpha$. Summation over $\langle k,l\rangle$ and $\langle\langle k,l\rangle\rangle $ stands for the nearest and next nearest neighbor ions, respectively. (i) The first term of the Hamiltonian describes the hoppings between nearest atoms with $t=1.6$ eV [@Liu; @Ezawa]. (ii) The second term includes electrostatic potential due to electric field $F_z$ perpendicular to the system with $\ell_k=\pm \frac{0.46\text{\AA}}{2} $ with $+$ ($-$) sign for the ions of the A (B) sublattice. (iii) The third term describes the intrinsic Rashba interaction with parameter $\lambda_{R}^{int.}=0.7$ meV [@Liu; @Ezawa] due to the built-in electric field that emerges from the vertical shift of the $A$ and $B$ sublattices in silicene, where ${\bf d}_{kl}=\frac{{\bf r}_l-{\bf r_k}}{|{\bf r}_l-{\bf r_k}|}$ is the position of $k$-th ion and ${\bf r_k}=(x_k,y_k,z_k)$, with the lattice constant $a=3.86$ Å. The $\mu_{kl}=+1$ ($-1$) for $\langle \langle k,l\rangle \rangle$ ions within sublattice A (B). (iv) The fourth term represents the effective SO coupling with $\lambda_{SO}=3.9$ meV in the Kane-Mele form [@km; @km2] with $\nu_{kl}=+1$ ($-1$) for the counterclockwise (clockwise) next-nearest neighbor hopping. (v) The fifth term describes the extrinsic Rashba effect which results from the external electric field perpendicular to the silicene plane or broken mirror symmetry by e.g. the substrate. The parameter $\lambda_{R}^{ext.}$ varies linearly with the external field and for $F_z=17$ meV/Å the $\lambda_{R}^{ext.}(F_z)=10$ $\mu$eV [@Ezawa]. (vi) The last term introduces the local exchange field with magnetization described by exchange field $\mathbf{M}=(M_x,M_y,M_z)$ that may arise due to proximity of an insulating ferromagnetic substrate [@chow; @an; @0yoko; @1wang; @2saxena; @3derak; @4wang]. To solve the scattering problem for the atomistic system described by Hamiltonian (\[eq:h0\]) we use the wave function matching (WFM) technique as described in the appendix of Ref. [@bubel]. The transmission probability from the input lead to mode $m$ (output lead) cab be written as $$T^m = \sum_n \vert t^{mn} \vert ^2,$$
where $t^{mn}$ is the probability amplitude for the transmission from the mode $n$ in the input lead to mode $m$ in the output lead. We distinguish spin for each mode $p$ by quantum expectation values of the Pauli matrices $\langle S_\bullet \rangle = \langle \psi_i^p \vert \sigma_\bullet \vert \psi_i^p \rangle$ through each atom $i$ inside lead. The positive (negative) $\langle S_\bullet \rangle$ values are labeled by $u$,$\uparrow$ ($d$,$\downarrow$). With this notation the spin-dependent conductance can be put in form $$G_{wv} = G_0 \sum_{m,n} \vert t^{mn} \vert ^2 \delta_{w,\alpha(n)}\delta_{v,\beta(m)},$$ where $G_0 = e^2/h$ is the conductance quantum, $w$ ($v$) is the expected input (output) orientation of the spin, while $\alpha$ and $\beta$ correspond to determined sings of $\langle S_\bullet \rangle$ sign for a given mode. For example, for the incident spin polarized along the $z$ direction, the $\langle S_z \rangle = \langle \psi_i^p \vert \sigma_z \vert \psi_i^p \rangle$ is evaluated and the contribution to conductance that corresponds to the spin flip from $u$ to $d$ orientation is calculated as $G_{ud} = G_0 \sum_{m,n} \vert t^{mn} \vert ^2 \delta_{+,\alpha(n)}\delta_{-,\beta(m)}$. All other spin-dependent conductance components can be calculated in the same way.
![ Sketch of the device. The Fermi level electrons propagate from the source (S) to the drain (D). The voltage $V_G$ applied to the top gate produces a perpendicular electric field $F_z$ on length $L$ (see Eq. \[eq:h0\]). The role of the source (S) and drain (D) is played by homogeneous semi-infinite silicene ribbons outside the gated area. []{data-label="fig:0-sch"}](sch.pdf){width="45.00000%"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
(100,10)[${\bf B}=(0,0,b)$ ]{} (250,10)[${\bf B}=(b,0,0)$ ]{}
(-65,63)[(a)]{} (-65,80)[$F_z = 0$ mV/Å]{} (5,80)[$F_z = 100$ mV/Å]{} (-65,63)[(b)]{} (-65,63)[(c)]{} (-65,80)[$F_z = 0$ mV/Å]{} (5,80)[$F_z = 100$ mV/Å]{} (-95,63)[(d)]{}
(-65,74)[(e)]{} (-65,74)[(f)]{} (-50,-10)[k $\big[\frac{1}{2a}\big]$]{} (22,-10)[k $\big[\frac{1}{2a}\big]$]{} (-122,-10)[k $\big[\frac{1}{2a}\big]$]{} (95,-10)[k $\big[\frac{1}{2a}\big]$]{} (-65,74)[(g)]{} (-95,74)[(h)]{}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
We consider the vertical orientation of the system in $z=0$ plane where incident electron momentum is set in $y$ direction and the width of the ribbon is defined along the $x$ axis \[see Fig. \[fig:0-sch\]\]. External electric field is applied perpendicular to system (along $z$ direction).
Results and Discussion
======================
Perpendicular spin polarization
-------------------------------
### Zigzag lead
In Fig. \[fig:1-bands\] we show the dispersion relation and the average value of the spin for a zigzag silicene ribbon covered by an [*infinite*]{} top gate \[Fig. \[fig:0-sch\]\] that can produce a homogeneous vertical electric field. The ribbon is 28 atoms wide, which corresponds to a width of $\simeq 5$ nm. Note, that such narrow or even narrower – down to 0.8 nm – silicene nanoribbons have been grown experimentally [@hiraoka].
In order to probe the spin properties of the system we introduced a very small external magnetic field equal to 1 $\mu$T oriented in the $z$ \[Fig. 2(a,b,e,f)\] or $x$ \[Fig. 2(c,d,g,h)\] direction. In Fig. \[fig:1-bands\] all the bands are nearly two-fold degenerate with respect to the spin. The lifting of the spin degeneracy can only be observed on enlarged fragments which are discussed in detail below in the text.
For ${\bf B}=(0,0,b)$ with no external electric field the electron spin is polarized in the $\pm z$ direction for any Fermi energy \[Fig. \[fig:1-bands\](a,e)\]. When the external electric field of $F_z = 100$ mV/Å is switched on, the external Rashba interaction introduces an effective magnetic field $\mathbf{B}_{\lambda_R^{ext.}}= \xi ( \mathbf{p \times E} )= (\xi p_yF_z,0,0) $ [@Meier2007] ($\xi$ is a constant) that tends to set the spins parallel or antiparallel to the $x$ axis. On the other hand the intrinsic spin-orbit coupling of the Kane-Mele form in the absence of the intervalley scattering, tends to polarize the spins in the direction perpendicular to the silicene plane [@cum]. Figs. 2(b) and 2(f) illustrate the competition between the intrinsic spin-orbit coupling and the external Rashba interaction. The latter prevails for high Fermi energy for which the spins are polarized within silicene plane in the $\pm x$ direction. For lower Fermi energy the spin-diagonal intrinsic SO coupling keeps the spin polarized in $\pm z$ direction [@cum].
Now, let us consider a device, i.e. a system with a [*finite*]{} top gate \[see Fig. 1\] and the spin injected polarized in the $z$ direction. The scattering problem for the zigzag nanoribbon has been solved for finite length of the external electric field $F_z=1$ V/Å at $E_F=0.258$ eV in two cases: [*(i)*]{} For $\lambda_{SO}=0$ the splitting between the spin-polarized subbands is $\Delta k_1=0.00288 \frac{1}{2a}$. The Rashba SO interaction induces the spin precession with respect to the $x$ axis. The rotation of the electron spin along the $x$ axis upon transition along the length of $L_1$ can be evaluated as [@Datta90] $$\Delta \varphi = \Delta k L_1.
\label{eq:dat}$$ For $L_1(\pi)=842$ nm the spin rotates from $d$ to $u$ orientation \[Fig. \[fig:2-maps\](a)\]. [*(ii)* ]{} For $\lambda_{SO}=3.9$ meV the intrinsic spin-orbit coupling keeps the electron spin polarized along the $z$ direction with only weak oscillations due to $\Delta k_2=0.036 \frac{1}{2a}$ and the $L_2(\pi)=67.4$ nm \[Fig. \[fig:2-maps\](b)\].
![ Spin $S_z$ component maps for zigzag nanoribbon 28 atoms wide for the intrinsic SO coupling constant $\lambda_{SO}=0$ (a) and $\lambda_{SO}=3.9$ meV (b). In between dashed horizontal lines the external electric field $F_z=1$ V/Å is applied. The first propagating mode with initial spin up has been chosen at $E_F=258$ meV. []{data-label="fig:2-maps"}](2-szb.png "fig:") ![ Spin $S_z$ component maps for zigzag nanoribbon 28 atoms wide for the intrinsic SO coupling constant $\lambda_{SO}=0$ (a) and $\lambda_{SO}=3.9$ meV (b). In between dashed horizontal lines the external electric field $F_z=1$ V/Å is applied. The first propagating mode with initial spin up has been chosen at $E_F=258$ meV. []{data-label="fig:2-maps"}](2-sz.png "fig:") (-200,200)[(a)]{} (-100,200)[(b)]{}
We conclude that the realization of a perpendicular spin inverter in a zigzag nanoribbon at low Fermi energy is excluded by the strong intrinsic SO which keeps the spin polarized along the $z$ axis.
### Armchair lead
The effective magnetic field due to the intrinsic spin-orbit interaction that prevents the spin precession along the Rashba effective field is only present provided that the transport modes have a definite valley [@cum; @km]. The armchair edge of the ribbon introduces maximal valley mixing and removes the intrinsic spin-orbit effective magnetic field. In order to eliminate the effects of the intrinsic spin-orbit interaction we considered an armchair semiconducting ribbon 19 atoms wide (for metallic version precession occurs in the same manner). In the armchair nanoribbon with no electric field the initial spin $z$ is conserved \[Fig. \[fig:3-bands\_z\](a,c)\] but when a high electric field $F_z=1$ V/Å is applied, the available states correspond to spins polarized along $x$ axis \[Fig. \[fig:3-bands\_z\](b,d)\] due to the Rashba effective magnetic field. The conductance for $L=842$ nm and $F_z=1$ V/Å was calculated and presented in Fig. \[fig:3-trans\]. Total conductance (blue line) and spin-flipping conductance (red line) oscillates with peaks for integer number of wavelength halves within the gated length $L$ \[see Tab. \[tab:wl\]\]. For $F_z\neq 0$ a step potential appears in silicene sublattices that allows transmission for resonant modes only.
$E_F$ \[meV\] $k$ \[1/6$a_{Si}$\] $\lambda_m=\frac{2\pi}{k}$ \[nm\] $N=\frac{L}{\lambda_m}$ $\overline{N}$
--------------- ------- --------------------- ----------------------------------- ------------------------- ----------------
$m_1$ 0.1172 71.71 11.75
$m_2$ 0.1221 68.78 12.25
$m_1$ 0.1121 74.93 11.25
$m_2$ 0.1171 71.74 11.75
$m_1$ 0.1072 78.37 10.75
$m_2$ 0.1122 74.89 11.25
: Fermi wavelengths $\lambda_m$ in resonances \[see black dots in Fig. \[fig:3-trans\]\]. The results are obtained from the band structure for armchair ribbon (19 atoms width) with vertical electric field $F_z=1$ V/Å. $m_1$ and $m_2$ stand for the two modes in the first conductive subband at the Fermi level.
\[tab:wl\]
Figure \[fig:3-trans\] shows that for the armchair ribbon and the chosen length of the gated area the resonant electron transfer is accompanied by the spin-flip. In a wide range around $E_F=276.49$ meV the spin-subbands splitting remains almost the same $\Delta k \approx 0.005 \frac{1}{6a_{Si}}$ (where $a_{Si}=\frac{a}{\sqrt{3}}$ is the in-plane distance between nearest-neighbors Si atoms, thus $L_a(\pi) \approx 842$ nm) and provides a perfect spin inverter. We can see that the spin inversion length is very large even for an extreme value 1V/Å of the electric field applied here [@drumm; @Ni2011], which is not promising in the context of practical applications.
(-65,78)[$F_z = 0$ V/Å]{} (10,78)[$F_z = 1$ V/Å]{} (-95,75)[(a)]{} (90,75)[(b)]{} (-50,-10)[k $\big[\frac{1}{6a_{Si}}\big]$]{} (22,-10)[k $\big[\frac{1}{6a_{Si}}\big]$]{} (-95,75)[(c)]{} (90,75)[(d)]{}
![ Total conductance $G$ (blue) and the spin-flipping $G_{ud}$ contribution (red) for an armchair silicene nanoribbon with the length of the gated area $L=842$ nm and the external electric field $F_z=1$ V/Å. The basis of spin states perpendicular to the ribbon is considered here. The black dots marked the peaks described in Tab. \[tab:wl\]. []{data-label="fig:3-trans"}](3-trans.pdf){width="50.00000%"}
In-plane spin polarization
--------------------------
Figure \[fig:1-bands\](c,g) shows that for the electrons fed from the silicene lead in which the external electric field is absent one can polarize the spins in the $x$ direction with an infinitesimal magnetic field. For $F_z=0$, above energy $E_F=10$ meV, the electron spin is oriented parallel or antiparallel to the $x$ axis by the field of 1 $\mu$T. Moreover, Fig. 2(d) indicates that in presence of nonzero $F_z$ for low Fermi energy the spins of the transport modes are polarized along the $z$ axis. One can use this fact in order to arrange for a device which inverts the in-plane polarized incident spins for the electrons that enter a gated region. Furthermore if we apply external electric field $F_z=100$ mV/Å, the first subbands \[see Fig. \[fig:1-bands\](i)\] with specified spin $z$ states splits in a very wide $\Delta k$ spectrum \[zoom in Fig. \[fig:4-agressive\]\] between the two propagating modes.
![ The lowest conduction subbands for zigzag nanoribbon and $F_z=100$ mV/Å. The color scale denotes the spin $S_z$ component in $\hbar/2$ units. The difference between two opposite-spin right-going subbands at Fermi energy $E_F = 24.2$ meV is marked as $\Delta k = 1.02\; \frac{1}{2a}$.[]{data-label="fig:4-agressive"}](4-agressive_band_x_100_sz.pdf "fig:"){width="50.00000%"} (-145,-10)[k $\big[\frac{1}{2a}\big]$]{}
![ Total conductance $G$ (blue), the spin flipping $G_{ud}$ (red) and spin conserving $G_{uu}$ (orange) contributions. Here, $G_{ud}$ denotes the flip from $S_x=1$ to $S_x=-1$ in $\hbar/2$ units. The width of the zigzag ribbon was set to 28 atoms, the length is $L=5.5$ nm and $F_z=100$ mV/Å. The two peaks marked by black dots correspond to dashed lines in band structure in Fig. \[fig:4-agressive\].[]{data-label="fig:4-trans"}](4-trans.pdf){width="40.00000%"}
Figure \[fig:4-trans\] shows the conductance for the zigzag nanoribbon (28 atoms width) with external electric field $F_z=100$ mV/Å and $L=5.5$ nm. We show the total conductance $G$ and its the spin-resolved contributions to conductance $G=G_{ud}+G_{uu}+G_{du}+G_{dd}$. $G_{ud}$ stands for the flip from $1$ to $-1$ in $S_x [\hbar/2]$ component, and $G_{uu}$ for the transport with the spin kept parallel to the $x$ axis. Since the time-reversal symmetry is conserved one has $G_{du}=G_{ud}$ and $G_{dd}=G_{uu}$.
We find two peaks of $G_{ud}\simeq G_0 $ for $E_F=22$ meV and 24.2 meV. For the first peak the difference of the Fermi wave vectors is roughly twice larger than in the other \[Fig. 6\]. For higher $E_F$ the $\Delta k$ values drastically drop and the precession is too slow to invert spin on that short $L$ path and $G_{ud}$ in Fig. 7 drops to zero.
For the $E_F=24.2$ meV the spacing of the right-going wave vectors for opposite spin states is $\Delta k = 1.02\; \frac{1}{2a}$ which according to Eq. (\[eq:dat\]) provides the spin precession length $L_x(\pi) = 6a \approx 2.32$ nm. In Fig. \[fig:5-offset\] we plotted the spin-flipping conductance as a function of the length of the gated region for varied values of the $F_z$ and $\lambda_{SO}$. For each subplot in Fig. \[fig:5-offset\] the Fermi energy was tuned to maintain the same spin precession rate: $\Delta k = 1.02\; \frac{1}{2a}$, $L_x(\pi)\approx 2.32$ nm. We find that the subsequent peaks of the spin-flipping conductance $G_{ud}$ are separated by $2 L_x(\pi) = 4.64$ nm, corresponding to an additional full spin rotation from one peak to the other. The offset between the nominal $L_x(\pi)$ and the actual $L$ value for which the first peak of $G_{ud}$ occurs is due to the finite size of the top gate. Note, that the band structure of Fig. 6 is calculated for an infinite $L$. We find that this offset is dependent of the Fermi energy (the higher $E_F$, the lower wave sensitivity to the step of $F_z$ potential) and on intrinsic SO interaction strength. Increasing ten times the intrinsic SO factor $\lambda_{SO}$ we significantly shorten the offset lengths. At the start and at the end of $F_z$ area the spin precession is unsettled which extends the actual spin inversion length \[Fig. \[fig:5-maps\](a,b)\]. In the scattering spin density the local extrema of $S_x$ are indeed spaced by $2\cdot L_x(\pi) = 4.64$ nm \[Fig. \[fig:5-maps\](c)\]. The fact that the scattering density mostly occupies the left edge of the nanoribbon \[Fig. \[fig:5-maps\](d)\] is consistent with the results obtained for infinite $L$ \[Fig. \[fig:5-maps\](e,f)\].
, respectively. The applied Fermi energies: $E_{F}=24.2$ meV (a), $E_{F}=38.386$ meV (b), $E_{F}=231.653$ meV (c), and $E_{F}=252.989$ meV (d). []{data-label="fig:5-offset"}](5-trans-KM1-100Fz.pdf "fig:")(-260,50)[(a)]{}
, respectively. The applied Fermi energies: $E_{F}=24.2$ meV (a), $E_{F}=38.386$ meV (b), $E_{F}=231.653$ meV (c), and $E_{F}=252.989$ meV (d). []{data-label="fig:5-offset"}](5-trans-KM10-100Fz.pdf "fig:")(-260,50)[(b)]{}
, respectively. The applied Fermi energies: $E_{F}=24.2$ meV (a), $E_{F}=38.386$ meV (b), $E_{F}=231.653$ meV (c), and $E_{F}=252.989$ meV (d). []{data-label="fig:5-offset"}](5-trans-KM1-1000Fz.pdf "fig:")(-260,50)[(c)]{}
, respectively. The applied Fermi energies: $E_{F}=24.2$ meV (a), $E_{F}=38.386$ meV (b), $E_{F}=231.653$ meV (c), and $E_{F}=252.989$ meV (d). []{data-label="fig:5-offset"}](5-trans-KM10-1000Fz.pdf "fig:")(-260,80)[(d)]{}
(-33,160)[(a)]{} (-33,160)[(b)]{} (-80,160)[(c)]{} (-43,160)[(d)]{} (-42,160)[(e)]{}(-40,60) (-42,160)[(f)]{}(-40,60)
\
\
(-45,90)[(a)]{}(-50,-10)[k $\big[\frac{1}{2a}\big]$]{} (-45,90)[(b)]{}(-50,-10)[k $\big[\frac{1}{2a}\big]$]{}(-45,90)[(c)]{}(-50,-10)[k $\big[\frac{1}{2a}\big]$]{}(-45,90)[(d)]{}(-50,-10)[k $\big[\frac{1}{2a}\big]$]{}(-75,90)[(e)]{}(-80,-10)[k $\big[\frac{1}{2a}\big]$]{}
) compared to the $G_{ud}$ (dashed line) in nanoribbon without FM leads (see Fig. \[fig:4-trans\]). []{data-label="fig:FMtrans"}](FM-trans.pdf "fig:"){width="40.00000%"}\
![ Conductance $G$ with spin conserving $G_{uu}$ and spin-flipping $G_{ud}$ parts in disordered system for a random fluctuation of the vertical field $U_z$ including the left edge (a), and with no fluctuation on the left edge (c). Map of on-site energies that correspond to $\vert V_z \vert \equiv e(F_z+U_z) \vert \ell_k \vert$ has been presented for each case (b) and (d), respectively. Magnitude of the fluctuating field $U_z$ in both cases is comparable to $F_z$ and corresponds to energy $\approx$ 20meV. []{data-label="fig:fluct"}](edg_trans.pdf "fig:"){width="33.00000%"} ![ Conductance $G$ with spin conserving $G_{uu}$ and spin-flipping $G_{ud}$ parts in disordered system for a random fluctuation of the vertical field $U_z$ including the left edge (a), and with no fluctuation on the left edge (c). Map of on-site energies that correspond to $\vert V_z \vert \equiv e(F_z+U_z) \vert \ell_k \vert$ has been presented for each case (b) and (d), respectively. Magnitude of the fluctuating field $U_z$ in both cases is comparable to $F_z$ and corresponds to energy $\approx$ 20meV. []{data-label="fig:fluct"}](edg_flu.pdf "fig:"){width="14.00000%"}(-230,80)[(a)]{}(-60,90)[(b)]{}(-5,40)\
![ Conductance $G$ with spin conserving $G_{uu}$ and spin-flipping $G_{ud}$ parts in disordered system for a random fluctuation of the vertical field $U_z$ including the left edge (a), and with no fluctuation on the left edge (c). Map of on-site energies that correspond to $\vert V_z \vert \equiv e(F_z+U_z) \vert \ell_k \vert$ has been presented for each case (b) and (d), respectively. Magnitude of the fluctuating field $U_z$ in both cases is comparable to $F_z$ and corresponds to energy $\approx$ 20meV. []{data-label="fig:fluct"}](noedg_trans.pdf "fig:"){width="33.00000%"} ![ Conductance $G$ with spin conserving $G_{uu}$ and spin-flipping $G_{ud}$ parts in disordered system for a random fluctuation of the vertical field $U_z$ including the left edge (a), and with no fluctuation on the left edge (c). Map of on-site energies that correspond to $\vert V_z \vert \equiv e(F_z+U_z) \vert \ell_k \vert$ has been presented for each case (b) and (d), respectively. Magnitude of the fluctuating field $U_z$ in both cases is comparable to $F_z$ and corresponds to energy $\approx$ 20meV. []{data-label="fig:fluct"}](noedg_flu.pdf "fig:"){width="14.00000%"}(-230,80)[(c)]{}(-60,90)[(d)]{}(-5,40)
Spin-flip detection
-------------------
Above, we indicated the setup for the in-plane spin inversion. Let us now discuss a possible experimental setup that could detect the inversion. For that purpose we consider a system with ferromagnetic insulators placed above the silicene. Due to the proximity effect the exchange energy $\mathbf{M}$ appears in Hamiltonian (\[eq:h0\]) and modifies the band structure. In Fig. \[fig:FM-sch\](a,e) we plotted the dispersion relation for $M_x=\pm 7$ meV and $F_z=100$ mV/Å that can be achieved in a proximity of a ferromagnet magnetized along the $x$ axis \[see the sketch on top of Fig. \[fig:FM-sch\]\].
The configuration proposed in Fig. \[fig:FM-sch\] filters the spins polarized along the $x$ axis by choosing the Fermi energy that corresponds to only one conductive subband. Spin polarization along the $+x$ or $-x$ direction for that propagation is allowed on each lead and can be set by an adequate FM polarization. The length of the spacer $L_0$ was set to 11 nm and the length of the top gate \[Fig. \[fig:FM-sch\](c)\] in the middle remains the same as in the system without FM, $L=5.5$ nm, for the band structure of Fig. \[fig:4-agressive\].
The setup of Fig. \[fig:FM-sch\] works in the following way: the total conductance of the system is equal to 1 flux quantum only if the full spin-flip occurs \[Fig. \[fig:FMtrans\]\] and the maxima of $G$ coincide with the maxima of $G_{ud}$ that were obtained in Fig. 7 without the ferromagnets. Figure \[fig:FMtrans\] presents the results for varied length of the spacer $L_0$. Although the conductance varies with $L_0$, the total conductance stays at 1 $G_0$ for the Fermi energies that correspond to the spin flip under the gate of the fixed length $L$. The proposed device should allow to tune the Fermi energy to obtain a perfect spin inverter.
Note, that for the spin injection by the input lead alternatively to the magnetic proximity effect one can use the procedure for all-electrical generation of spin-polarized currents by an energy-dependent phase difference for the electron spin proposed recently [@tao17].
Electrostatic disorder
----------------------
In order to check the robustness of a spin-flipping conductance to a disorder we returned to the system without the FM proximity effect and we introduced inhomogeneities to the vertical electric field that produces Gaussian impurities in the potential [@fluct] $$U_z = \sum_{i}^{N} U_i \exp{(-\vert{} \mathbf{r_k}-\mathbf{R_i} \vert{}^2/2\eta^2)},$$ where $\mathbf{R_i}$ is the center of the $i$-th impurity and $\eta$ is set equal to the lattice constant. $U_i \in (-\mathfrak{m}F_z,\mathfrak{m}F_z) $ is a scale factor chosen randomly and multiplied by magnitude factor $\mathfrak{m}$ equal 1 or 0.1. This fluctuations is introduced in our calculations by modification of the vertical electric field $F^d_z(x,y)=F_z+U_z(x,y))$ in Hamiltonian $H_0$ (\[eq:h0\]).
We considered $N=5$ impurities in the gated area. When the magnitude of $U_i$ was set to $\mathfrak{m}=0.1$ we observe only a slight difference in the spin-flipping conductance. The results is still very similar to the case without the fluctuations \[Fig. \[fig:4-trans\]\]. For magnitude $\mathfrak{m}=1$ we can distinguish two cases: (i) when one of the impurity center is localized at the left edge of the nanoribbon then spin-flip conductance drastically falls down \[Fig. \[fig:fluct\](a,b)\] because most of wave function goes along left edge and any fluctuation on this path just blocks the electron propagation (ii) when all impurities centers are localized inside top gate excluding left edge the one-inversion conductance is still available but spin-flipping conductance with more than one inversion along path is hampered \[Fig. \[fig:fluct\](c,d)\].
Summary and conclusions
=======================
We considered a gated segment of a silicene nanoribbon as a spin inverter via precession of the incident spin in the effective magnetic field due to the SO interactions. The Rashba interaction due to the external electric field fails to induce the spin inversion in the zigzag ribbon for which the intrinsic SO interaction keeps the incident electron spin polarized along the $z$ direction. The perpendicular polarization of the electron spin is not present for the armchair ribbon that allows the Rashba interaction to drive the spin-precession. However, the resulting spin precession length is large, of the order of $\mu$m. We demonstrated that the gated zigzag nanoribbon can be used as an inverter of in-plane polarized incident spins and that spin precession length can be very short for low Fermi energy, e.g. less than 10 nm for a reasonable value of the external electric field $F_z=100$ meV/Å. That spin inversion length strongly depends of the $\Delta k$ chosen through the $E_F$ level that can be tuned in a large range due to the spin splitting in the band structure when the external electric field is applied. With the local exchange field it is possible to prepare spin-flip detection device that is transparent for the spin-polarized transport only for a perfect spin inversion in the gates area. Electric field fluctuations can suppress the precession if the center of impurity appears on the left edge of the nanoribbon and its magnitude is comparable to applied external electric field. In other cases precession is hampered but spin-flip is still observed.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Alina Mreńca-Kolasińska for helpful discussions. B.R. is supported by Polish government budget for science in 2017-2021 as a research project under the program “Diamentowy Grant” (Grant No. 0045/DIA/2017/46) and by the EU Project POWER.03.02.00-00-I004/16. B.S. acknowledges the support of NCN grant DEC-2016/23/B/ST3/00821. The calculations were performed on PL-Grid Infrastructure.
|
---
abstract: 'In this paper, which constitutes the first part of the series, we consider calculation of two-centre Coulomb and hybrid integrals over Slater-type orbitals (STOs). General formulae for these integrals are derived with no restrictions on the values of the quantum numbers and nonlinear parameters. Direct integration over the coordinates of one of the electrons leaves us with the set of overlap-like integrals which are evaluated by using two distinct methods. The first one is based on the transformation to the ellipsoidal coordinates system and the second utilises a recursive scheme for consecutive increase of the angular momenta in the integrand. In both methods simple one-dimensional numerical integrations are used in order to avoid severe digital erosion connected with the straightforward use of the alternative analytical formulae. It is discussed that the numerical integration does not introduce a large computational overhead since the integrands are well-behaved functions, calculated recursively with decent speed. Special attention is paid to the numerical stability of the algorithms. Applicability of the resulting scheme over a large range of the nonlinear parameters is tested on examples of the most difficult integrals appearing in the actual calculations including at most 7$i$-type functions ($l$=$6$).'
author:
- Michał Lesiuk
- Robert Moszynski
title: |
Calculation of two-centre two-electron integrals over Slater-type orbitals revisited.\
I. Coulomb and hybrid integrals
---
Introduction {#sec:intro}
============
Slater-type orbitals [@slater30; @slater32], or more general exponential-type orbitals, are the natural choice of basis set for applications in quantum chemistry and molecular or atomic physics. Their common origin is the analytical solution of the Schrödinger equation for the hydrogen atom. It can be shown that Slater-type orbitals behave correctly at the electron-nucleus coalescence points *i.e.* they satisfy Kato’s conditions [@kato57]. Additionally, the Slater-type orbitals decay exponentially when an electron is far from the nucleus. This is in line with the theoretical findings of the asymptotic form of the electron density [@agmon82]. It is obvious that Gaussian orbitals [@boys50], which have gained an enormous popularity in the last 50 years, are able to satisfy neither of the above conditions. Virtually the only issue which prohibited the widespread use of the Slater-type orbitals is calculation of the two-electron molecular integrals.
The main purpose of the present series of papers is to provide a complete set of methods for the evaluation of the two-electron two-centre integrals. The reliability of these methods needs to be sufficient to allow the use of Slater-type orbitals including high angular momentum functions for the diatomic systems. Our integral program based on the presented algorithms serves as a vehicle for the upcoming new *ab-initio* quantum chemistry program package <span style="font-variant:small-caps;">Kołos</span>. This program combines basis set of Slater-type orbitals with state-of-the-art quantum chemical *ab-initio* methods and is aimed at spectroscopically accurate (few $\mbox{cm}^{-1}$) results for the diatomic systems.
When considering our approach to the present problem, one issue needs to be clarified. To reach the spectroscopic accuracy it is not only necessary to use huge basis sets but also very accurate quantum chemistry methods. Let us now observe that calculations of the two-electron integral file scale as the fourth power of the size of the system ($N^4$) in the worst case scenario. This can be compared with the scaling of the accurate coupled cluster methods, $N^6$ for CCSD, $N^8$ for CCSDT *etc.* [@bartlett89; @raghavachari89; @noga87; @bartlett90]. As a result, one can expect that calculations of the integral file should not be a bottleneck in high-level calculations of the correlation energy. On the other hand, since we require the aforementioned accuracy in the molecular energy, we need the integrals to be calculated with higher precision than typical. We believe that the requirement for accuracy of 12 decimal places is reasonable.
The situation described above suggests that we should favour accuracy of the algorithms over their speed. In other words, if we had two algorithms - the first one being fast but less accurate, and the second one being somehow slower but significantly more accurate - we would pick up the second one. Of course, we still have limitations on the computational time and we cannot use arbitrary precision arithmetic, for instance. This philosophy of choosing and developing algorithms shall be perceptible throughout the whole series of papers.
This series of papers is organised as follows. In Paper I we deal with calculation of the Coulomb and hybrid integrals *i.e.* $(aa|bb)$ and $(aa|ab)$, respectively, where $a$ and $b$ denote the nuclei at which orbitals are located. We use direct integration over the second electron in the same spirit as several previous investigators but we differ in methods of computation of almost all nontrivial basic quantities. Final forms of the working expressions are also completely reformulated. Moreover, we present the results of demanding tests of the numerical performance. In Paper II we apply the Neumann expansion to calculation of the exchange integrals, $(ab|ab)$. We report new methods of calculation of the most difficult auxiliary quantities appearing in the theory. Additionally, we discuss how new algorithms can be sewed together to form a sufficiently general method. Finally, in Paper III we provide the first application of the presented theory - *ab-initio* calculations for the beryllium dimer which is an interesting system from both spectroscopic and theoretical point of view. In these calculations we use STO basis sets ranging from double to sextuple zeta quality combined with high level *ab-initio* methods in order to provide spectroscopically accurate results.
The literature dealing with evaluation of the molecular integrals over Slater-type orbitals is extensive and a full bibliography would count hundreds of positions. Its detailed review is undoubtedly beyond the scope of the present report. Therefore, our introduction is, by necessity, limited and subjective. Nonetheless, let us recall several prominent and the most widely used general techniques for computation of the aforementioned integrals.
Single-centre expansions allow to expand STOs located at some point of space around a different centre. These methods were pioneered by Barnett and Coulson as the widely known $\zeta$-function method [@coulson37; @barnett51; @barnett63] and later independently by Löwdin [@lowdin56] ($\alpha$-function method). In cases when the single-centre expansion terminates under the integral sign due to spherical symmetry of the integrand, it typically results in closed-form, compact and plausible expressions. However, in many cases, such as calculation of the exchange integrals, the single-centre expansions results in an infinite series which have a pathologically slow (*i.e.* logarithmic) convergence rate [@flygare66]. The problem does not have a satisfactory solution although several approaches [@barnett90] were adopted to overcome it. The second problem of the single-centre expansions is the catastrophic digital erosion during calculations of the auxiliary quantities [@jones92; @kennedy99] which seems to be extremely difficult to overcome. A promising work-around is use of the symbolic computational environments such as <span style="font-variant:small-caps;">Mathematica</span> [@jones94; @barnett00; @barnett002] but at present the symbolic methods are typically orders of magnitude slower than the numerical ones. Since the time the single-centre methods were first proposed, several new (or more general) expansion techniques has been developed. Examples are the works of Guseinov [@guseinov03], Harris and Michels [@harris65] and Rico *et al.* [@rico86; @rico90] and references therein.
The second class of methods which gained a significant interest is the Gaussian expansion methods and the Gaussian transform methods. The former is simply based on a least squares fit of a linear combination of Gaussian orbitals in order to mimic the shape of STOs. This idea, proposed first by Boys and Shavitt, [@boys56] was the dominant method used in the early versions of the SMILES program [@smiles]. The Gauss transform methods are more involved and use some integral representations in order to transform STO into a more computationally convenient form. The initial proposition of Shavitt and Karplus [@shavitt62; @shavitt65; @kern65] was to use the Laplace transform of the exponential function but now a handful of different schemes is in use, along with suitable discretisation techniques [@rico04].
The next prominent technique is the family of Fourier-transform methods which are usually used in conjunction with the so-called $B$-functions. These methods were primarily developed by the group of Steinborn [@steinborn78; @steinborn83a; @steinborn83b; @steinborn83c; @steinborn86a; @steinborn86b; @steinborn88a; @steinborn88b; @steinborn90; @steinborn91; @delhalle80] and applied to many difficult cases of the many-centre integrals. The fact that $B$-functions, being essentially a linear combination of STOs, possess an exceptionally simple Fourier transform can be used to evaluate the integrals in the momentum space and reduce many important integrals to the combination of some one-dimensional integrals. However, these integrals contain highly oscillatory integrands (including the Bessel functions) which make numerical integration extremely difficult with standard Gaussian quadrature techniques. Some approaches were adopted to accelerate the convergence towards the exact value with increasing number of quadrature nodes. The prominent method is the SD-transform, put forward by Sidi [@sidi81; @sidi82], and later applied by Safouhi [@safouhi04; @safouhi06]. Despite that, it seems that there is no general method reliable enough to evaluate the integrals in question in a black-box fashion.
There is also a number of less extensively studied techniques for evaluation of the molecular integrals over STOs. These include the Coulomb Sturmians introduced by Shull and Löwdin [@shull59] and used by some other authors [@smeyers66; @guseinov02; @avery04; @red04]. The shift operator technique [@rico00a; @rico00b; @rico01] is a very elegant method which generates integrals with arbitrary STOs starting with the simplest integrals with 1s functions by application of the so-called shift operator. Gill *et al.* [@gill08; @gill09] introduced the Coulomb resolution techniques where the interaction potential is expanded in terms of the so-called potential functions resulting from the the Poisson equation. This method has been recently pursued by Hoggan and coworkers [@hoggan09; @hoggan10] and included in their STOP program package [@stop].
Remarkably, it has not been a well-known fact yet that all two-centre integrals over STOs were integrated analytically in a closed-form. In a recent work Pachucki [@pachucki09; @pachucki12a] has shown that the so-called master integral with inverse powers of all interparticle distances can be obtained from the second order differential equation in the distance between the nuclei. The present authors also contributed to the development of this theory by extending it to the case of Slater geminals [@lesiuk12]. Pachucki used these expressions for calculations of the Born-Oppenheimer potential for the hydrogen molecule [@pachucki10; @pachucki13] and helium hydride ion [@pachucki12b]. However, an extreme level of complication of this theory along with drastic numerical instabilities occurring in the calculations have made its use limited to certain special forms of the basis set, applicable only to two-electron systems. We believe that some ingenious reformulation of this theory is necessary to circumvent the aforementioned difficulties.
We postpone the discussion of the methods based on the Neumann expansion of the interaction potential in the ellipsoidal coordinates. In the second paper of the series it is used to evaluate the exchange integrals and a proper separate introduction is given therein.
Let us now concentrate on methods designed specifically for treatment of the Coulomb and hybrid integrals. For the former ones there exists a plethora of independent methods which differ in both accuracy and speed. Probably the first attack on this problem was attempted by Barnett and Coulson [@coulson37] by using the single-centre expansion technique. Roothaan [@roothaan51] pioneered the direct integration method in the ellipsoidal coordinates which was later pursued by several authors [@ruedenberg56; @roothaan56; @ruedenberg66; @silver68a; @silver68b; @ruedenberg69; @eschrig79]. Later, it has become apparent that integration in the momentum space utilising the Fourier representation of STOs is very advantageous [@geller64; @szondy65; @silverstone67; @silverstone70; @harris69; @steinborn83c; @steinborn86a]. Gaussian transform techniques [@shavitt62; @shavitt65; @rico88], refined translation/expansion methods [@sharma76; @silverstone77; @pendls91] and several special approaches [@rico88; @tai92; @jones93; @hierse93; @hierse94; @magnasco89; @magnasco90] were also successfully applied. For hybrid integrals the number of available methods is modest. Several prominent techniques, such as the Fourier transform, cannot be applied straightforwardly. The biggest effort was aimed at the direct integration [@ruedenberg56; @ruedenberg67; @ruedenberg68; @miller68] or its combinations with the translation techniques [@guseinov77; @guseinov83; @guseinov89]. Our unified approach to the Coulomb and hybrid integrals is based on the earlier experiences with the direct integration. By using the Laplace expansion of the interaction potential and analytic integration over the coordinates of the second electron the problem is reduced to the calculation of the standard overlap integrals and a set of *overlap-like* integrals. To calculate these integrals two distinct approaches are used. The first one is integration in the ellipsoidal coordinates and the second method is based on recursive techniques. In both cases a simple, one-dimensional numerical integration is used to avoid drastic digital erosion. This indicates some connections with the method of Miller [@miller68]. Finally, we verify that when both methods are used together, in their respective regions of applicability, a loss of digits observed in the calculations by using some other methods can be avoided within a reasonable range of the nonlinear parameters.
Let us also note in passing that to perform actual calculations on the diatomic systems one also requires one-electron two-centre and two-electron one-centre (atomic) integrals. The former can be computed using various techniques among which the Fourier transform methods [@steinborn78; @steinborn83a; @steinborn83b; @steinborn83c; @steinborn86a; @steinborn86b; @steinborn88a; @steinborn88b; @steinborn90; @steinborn91; @delhalle80], recursive techniques for increasing the angular momenta in the integral [@guseinov05; @guseinov07; @rico88; @rico89b; @rico93; @rico91; @rico89c], and finally direct integration using the ellipsoidal coordinate system [@mulliken49; @corbato56; @roothaan51; @harris60] were intensively studied. The latter seems to be the method of choice for these integrals. Two-electron atomic integrals have been solved at least since the papers of Clementi and co-workers (see Refs. [@clementi74] and references therein). For the sake of completeness, a refined, simple, and numerically stable procedure for the computation of these integrals was included in the Supplementary Material [@supplement].
Preliminaries {#sec:pre}
=============
Let us consider a diatomic system with the nuclei A and B centred at the positions $\textbf{R}_A=(0,0,-R/2)$ and $\textbf{R}_B=(0,0,R/2)$, respectively, in the ordinary Cartesian coordinate system. Slater-type orbitals (STOs) have the following general form: $$\begin{aligned}
\label{sto1}
\chi_{nlm}(\textbf{r};\zeta)=S_n(\zeta) r^{n-1} e^{-\zeta r} Y_{lm}(\theta,\phi),\end{aligned}$$ Therefore, any STO is uniquely described by the quartet of parameters $(n,l,m,\zeta)$. We assume throughout that $n$, $l$ are restricted to the positive integers ($n>l$). The variables $r_a$, $\theta_a$, $\phi$ denote the spherical coordinates located on the atom $A$ with analogous notation for the centre $B$. In Eq. (\[sto1\]), $S_n(\zeta)$ is the radial normalisation constant: $$\begin{aligned}
S_n(\zeta)=\frac{(2\zeta)^{n+1/2}}{\sqrt{(2n)!}},\end{aligned}$$ and $Y_{lm}(\theta,\phi)$ are the spherical harmonics defined according to the Condon-Shortley phase convention [@messiah99]: $$\begin{aligned}
Y_{lm}(\hat{\bf r}) = \Omega_{lm} P_l^{|m|}(\cos \theta)\frac{e^{\dot{\imath}m\phi}}{\sqrt{2\pi}},\end{aligned}$$ where $P_l^m$ are the (unnormalised) associated Legendre polynomials [@stegun72] and $\Omega_{lm}$ is the angular normalisation constant: $$\begin{aligned}
\Omega_{lm} = \dot{\imath}^{m-|m|} \sqrt{\frac{2l+1}{2}\frac{(l-|m|)!}{(l+|m|)!}}\end{aligned}$$ In actual calculations it is typical to use real versions of the spherical harmonics. However, the complex spherical harmonics are more convenient in the derivations and thus we use them throughout the paper. Transfer to the real spherical harmonics can be performed on the top of the presented algorithms by using standard relations.
Let us now introduce the prolate ellipsoidal coordinates $(\xi,\eta,\phi)$ by means of the following relations: $$\begin{aligned}
\xi = \frac{r_a+r_b}{R},\;\; \eta = \frac{r_a-r_b}{R},\end{aligned}$$ so that $1\leq \xi \leq \infty$, $-1 \leq \eta \leq 1$ and $0 \leq \phi \leq 2\pi$. The spherical coordinates are expressed through the ellipsoidal coordinates by means of the well-known expressions $$\begin{aligned}
&r = \frac{R}{2}\left(\xi+\kappa\eta\right),\;\;
\cos \theta = \frac{1+\kappa\xi\eta}{\xi+\kappa\eta},\end{aligned}$$ where the value of $\kappa$ is equal to $+1$ if STO is located on the centre A or $-1$ if it is located on the centre B. The volume element becomes $d\textbf{r} = \left(\frac{R}{2}\right)^3(\xi^2-\eta^2)\,d\xi\,d\eta\,d\phi$. The simplest way to express the product of two STOs (*i.e.* the charge distribution) in the ellipsoidal coordinates is to proceed in two steps. First, we transfer the following scaled product of the Legendre polynomials by means of the expression $$\begin{aligned}
\label{trans1}
\small
\begin{split}
&P_{l_a}^{|m_a|}(\cos \theta_a)P_{l_b}^{|m_b|}(\cos
\theta_b)r_a^{l_a}r_b^{l_b}=\\
&\left(\frac{R}{2}\right)^{l_a+l_b}
\left[(\xi^2-1)(1-\eta^2)\right]^{|M|/2}
\sum_{p=0}^{\Gamma}\sum_{q=0}^{\Gamma} \left( {\bf \large \Xi}_{l_al_b}^M
\right)_{pq} \xi^p \eta^q
\end{split}\end{aligned}$$ where $M=m_a-m_b$, $\Gamma=l_a+l_b$ and ${\bf \large \Xi}_{l_al_b}^M$ are square matrices of dimension $\Gamma$. The values of the latter depend on the locations of the orbitals and their quantum numbers. Explicit forms of these matrices can easily be deduced from the general expressions available in the literature [@maslen90; @rico89; @yasui82; @guseinov70; @guseinov97]. We tabulated the values of ${\bf \large \Xi}_{l_al_b}^M$ up to the maximum value of $l_a+l_b$ equal to 24. These tables, along with <span style="font-variant:small-caps;">Mathematica</span> code [@math7] used for their generation, can be obtained from the authors on demand.
The remainder can be transferred to the ellipsoidal coordinates by using the following formula: $$\begin{aligned}
\label{trans2}
r_a^{n_a}r_b^{n_b}=\sum_{k=0}^{k_{max}} B_k^{n_an_b}\xi^k \eta^{k_{max}-k},\end{aligned}$$ with $k_{max} = n_a+n_b$. The above expression has been extensively used by many authors [@guseinov70; @rosen31; @ozdogan04] who presented explicit expressions for the coefficients $B_k^{n_an_b}$ (the so-called generalised binomial coefficients). We found it simpler to tabulate these coefficients as series of one-dimensional look-up tables.
Making use of the transfer relations (\[trans1\]) and (\[trans2\]) one can write down the explicit expression for the STOs charge distribution in terms of the ellipsoidal coordinates. The result reads (for convenience, we additionally included the Jacobian): $$\begin{aligned}
\label{trans12}
\begin{split}
&\left(\frac{R}{2}\right)^3(\xi^2-\eta^2)\chi_{n_al_am_a}^*(\textbf{r}_a;\zeta_a)\chi_{n_bl_bm_b}(\textbf{r}
_b;\zeta_b)=\\
&\frac{K_{ab}}{2\pi}e^{-\alpha\xi-\beta\eta} \left[(\xi^2-1)(1-\eta^2)\right]^{|M|/2}e^{\dot{\imath}M\phi}\times\\
&\sum_{k=0}^{k_{max}} B_k^{n_a-l_a,n_b-l_b}\sum_{p=0}^{\Gamma}\sum_{q=0}^{\Gamma} \left( {\bf \large \Xi}_{l_al_b}^M
\right)_{pq} \xi^{p+k}\; \eta^{q+k_{max}-k},
\end{split}\end{aligned}$$ with $M=m_a-m_b$, $k_{max}=n_a-l_a+n_b-l_b$ and $\Gamma=l_a+l_b$. Additionally, in Eq. (\[trans12\]) we introduced several new quantities: $\alpha=\frac{R}{2}(\zeta_a+\zeta_b)$, $\beta=\frac{R}{2}(\kappa_a\zeta_a+\kappa_b\zeta_b)$, $K_{ab}=S_{n_a}(\zeta_a) S_{n_b}(\zeta_b)\, \Omega_{l_am_a} \Omega_{l_bm_b} \left(\frac{R}{2}\right)^{n_a+n_b+1}$. The above formulation is quite explicit and rather transparent at the same time. Apart from that, it remains valid for “singular” orbitals such as $0s$ which is advantageous from the point of view of some developments.
Before passing further let us introduce three useful auxiliary functions: $$\begin{aligned}
\label{Ap}
A_p(\alpha) = \int_1^\infty d\xi\; \xi^p e^{-\alpha \xi},\end{aligned}$$ $$\begin{aligned}
\label{Bq}
B_q(\beta ) = \int_{-1}^{+1} d\eta\; \eta^q e^{-\beta \eta},\end{aligned}$$ $$\begin{aligned}
\label{ap}
a_p(\alpha) = \int_0^1 d\xi\; \xi^p e^{-\alpha \xi}.\end{aligned}$$ The first two of the above functions are the so-called Mulliken integrals [@mulliken49]. Accurate and stable calculation of these integrals was considered by many authors, the works of Corbató [@corbato56] and a recent paper of Harris [@harris04] need to be mentioned in this respect. The third integral, Eq. (\[ap\]), can be considered complementary to the first integral, Eq. (\[Ap\]), and has strong connections with the lower incomplete gamma functions. Integrals (\[ap\]) have to be computed by using the Miller algorithm [@gautschi67], as discussed by Harris [@harris02].
Coulomb and hybrid integrals {#sec:coulomb}
============================
In this section we attack the main objectives of this paper - calculation of the coulomb ($I_C$) and hybrid ($I_H$) integrals. With the notation developed in the previous section they take the following form: $$\begin{aligned}
\begin{split}
I_C = \int d\textbf{r}_1 \int d\textbf{r}_2\,
\chi_{n_1l_1m_1}^*(\textbf{r}_{1a};\zeta_1) \chi_{n_2l_2m_2}(\textbf{r}_{1a};\zeta_2)\\
\times\frac{1}{r_{12}}\,\chi_{n_3l_3m_3}^*(\textbf{r}_{2b};\zeta_3) \chi_{n_4l_4m_4}(\textbf{r}_{2b};\zeta_4),\\
\end{split}\end{aligned}$$ $$\begin{aligned}
\begin{split}
I_H = \int d\textbf{r}_1 \int d\textbf{r}_2\,
\chi_{n_1l_1m_1}^*(\textbf{r}_{1a};\zeta_1) \chi_{n_2l_2m_2}(\textbf{r}_{1b};\zeta_2)\\
\times\frac{1}{r_{12}}\,\chi_{n_3l_3m_3}^*(\textbf{r}_{2b};\zeta_3) \chi_{n_4l_4m_4}(\textbf{r}_{2b};\zeta_4).
\end{split}\end{aligned}$$ Let us note that in the above expressions we have adapted a particular, fixed location of the STOs. This convention is very useful from the point of view of the upcoming derivation. Other possible options for the orbitals location within the class of the Coulomb and hybrid integrals can be obtained by using the usual eightfold permutational symmetry of the integrals.
Initial reduction to the overlap-like integrals {#subsec:chinit}
-----------------------------------------------
Before proceeding with the integration of $I_C$ and $I_H$ let us simplify the formulae by using the Clebsh-Gordan expansion of the products of the spherical harmonics. In the case of the Coulomb integrals one expands pairs of the spherical harmonics on both centres; in case of the hybrid integrals, only the pair dependent on the coordinates of the second electron can be expanded. Once the Clebsh-Gordan expansion is used and the resulting integrals are written explicitly, it becomes obvious that the problem reduces now to the calculation of the following families of the integrals $$\begin{aligned}
\begin{split}
\widetilde{I}_C = &\int d\textbf{r}_1 \int d\textbf{r}_2\,
r_{1a}^{n_{12}-2}\, Y_{L_1M}^*(\cos \theta_{1a},\phi)
\,\frac{1}{r_{12}}\\
&\times r_{2b}^{n_{34}-2}\, Y_{L_2M}(\cos \theta_{2b},\phi)\, e^{-\zeta_{12}r_{1a}-\zeta_{34}r_{2b}},
\end{split}\end{aligned}$$ $$\begin{aligned}
\begin{split}
\widetilde{I}_H = &\int d\textbf{r}_1 \int d\textbf{r}_2\,
r_{1a}^{n_1-1}\, Y_{l_1m_1}^*(\cos \theta_{1a},\phi)e^{-\zeta_1r_{1a}}\\
&\times
r_{1b}^{n_2-1}\, Y_{l_2m_2} (\cos \theta_{1b},\phi)e^{-\zeta_2r_{1b}}
\,\frac{1}{r_{12}}\\
&\times r_{2b}^{n_{34}-2}\, Y_{L_2M}(\cos \theta_{2b},\phi)\, e^{-\zeta_{34}r_{2b}},
\end{split}\end{aligned}$$ where $n_{12}=n_1+n_2$, $\zeta_{12}=\zeta_1+\zeta_2$ *etc*. and $r_{ij}$ denote the interparticle distances. It is evident that any Coulomb integral ($I_C$) can be written as a linear combination of the pertinent integrals $\widetilde{I}_C$ and the correspondence between $I_H$ and $\widetilde{I}_H$ is analogous. For convenience, we have also skipped the normalisation constants $S_n$ since their multiplicative presence is obvious and does not change throughout the derivation. When considering the coefficients that relate $I_{C/H}$ and $\widetilde{I}_{C/H}$ there is an additional effort connected with calculation of the Wigner $3J$ symbols (or equivalently the Clebsh-Gordan coefficients). Computation of these quantities is not a trivial problem and has been considered many times in the literature, see Refs. [@venkatesh78; @tuzun98; @schulten76; @fang92; @lai90; @rasch03] as representative examples.
The first step of the integration proceeds in the usual manner - one integrates over the coordinates of the second electron. This is a quite natural approach since both orbitals of the second electron lie on the same centre (B, in our convention). The formula for the necessary integral exists in the literature and appears independently in many works. The simplest way to arrive at the final expression is to use the Laplace expansion of $1/r_{12}$ in spherical coordinates relative to the centre B. Independently of the derivation route, one arrives at
$$\begin{aligned}
\label{int2}
\begin{split}
\int d\textbf{r}_2\,\frac{1}{r_{12}} r_{2b}^{n_{34}-2}\, Y_{L_2M}(\cos \theta_{2b},\phi)\, e^{-\zeta_{34}r_{2b}}&=
\frac{4\pi}{2L_2+1}\frac{Y_{L_2M}(\cos \theta_{1b},\phi)}{\zeta_{34}^{n_{34}}}
\bigg[ (\zeta_{34}r_{1b})^{n_{34}}\,a_{n_{34}+L_2}(\zeta_{34}r_{1b})\\
&+(n_{34}-L_2-1)!\;e^{-\zeta_{34}r_{1b}}\sum_{j=L_2}^{n_{34}-1}\frac{(\zeta_{34}\,r_{1b})^j}{(j-L_2)!}\bigg],
\end{split}\end{aligned}$$
where $a_n$ is given by Eq. (\[ap\]). To bring the above expression into a more familiar and simplified form we could use the following obvious relationships $$\begin{aligned}
\label{illegal1}
a_n(\alpha) &= \frac{n!}{\alpha^{n+1}} - A_n(\alpha),\end{aligned}$$ $$\begin{aligned}
\label{illegal2}
A_n(\alpha) &= \frac{e^{-\alpha}\, n!}{\alpha^{n+1}} \sum_{k=0}^n \frac{\alpha^k}{k!}.\end{aligned}$$ By doing so, one expresses the integral (\[int2\]) explicitly through the elementary functions only. It seems to be advantageous but there are two main problems connected with use of Eqs. (\[illegal1\]) and (\[illegal2\]). Firstly, these expressions introduce spurious singularities (high inverse powers of $r_{1b}$) and generate integrals which have to be treated with special methods. Secondly, and more importantly, Eq. (\[illegal1\]) by itself is numerically badly conditioned and these problems propagate to the final expressions for the Coulomb and hybrid integrals. Precisely speaking, unless the relationship $n\gg \alpha$ holds, Eq. (\[illegal1\]) consists of subtraction of two large numbers to a relatively small result. Therefore, a huge digital erosion occurs, especially when large values of the quantum numbers are necessary.
This leads to the conclusion that in order to preserve a good numerical stability of the method, we have to abandon the use of Eqs. (\[illegal1\]) and (\[illegal2\]) and exploit Eq. (\[int2\]) as it stands. By inserting Eq. (\[int2\]) into the initial expressions for $\widetilde{I}_{C}$ one obtains the formula
$$\begin{aligned}
\label{ic}
\small
\begin{split}
\widetilde{I}_{C} &= \frac{4\pi}{2L_2+1} \frac{1}{\zeta_{34}^{n_{34}}}
\bigg[\zeta_{34}^{n_{34}} \int d\textbf{r}_1 \;r_{1a}^{n_{12}-2} \;Y_{L_1M}(\cos
\theta_{1a},\phi)\;e^{-\zeta_{12}r_{1a}}\;
r_{1b}^{n_{34}}\;Y_{L_2M}(\cos \theta_{1b},\phi)\;a_{n_{34}+L_2}(\zeta_{34}r_{1b})\\
&+ (n_{34}-L_2-1)! \sum_{j=L_2}^{n_{34}-1}
\frac{\zeta_{34}^j}{(j-L_2)!}
\int d\textbf{r}_1 \;r_{1a}^{n_{12}-2} \;Y_{L_1M}(\cos
\theta_{1a},\phi)\;e^{-\zeta_{12}r_{1a}}\;
r_{1b}^j\;Y_{L_2M}(\cos \theta_{1b},\phi)\;e^{-\zeta_{34}r_{1b}} \bigg].
\end{split}\end{aligned}$$
For the hybrid integrals, the manipulations are slightly more involved. After inserting Eq. (\[int2\]) into the formula for $\widetilde{I}_{H}$ one is left with three spherical harmonics under the integral sign. Two of these spherical harmonics are centred at the nucleus B and therefore can be expanded in the Clebsh-Gordan series. The result of this manipulations is as follows (the usual notation for the Wigner $3J$ symbols is used):
$$\begin{aligned}
\label{ih}
\small
\begin{split}
\widetilde{I}_{H} &= \frac{(-1)^{m_2}}{\zeta_{34}^{n_{34}}}\sqrt{(2l_2+1)(2L_2+1)}
\sum_{L_1=|l_2-L_2}^{l_2+L_2} \sqrt{\frac{4\pi}{2L_1+1}}
\left(\begin{array}{cccc} l_2 & L_2 & L_1\\ -m_2 & -M & m_1 \end{array}\right)
\left(\begin{array}{cccc} l_2 & L_2 & L_1\\ 0 & 0 & 0 \end{array}\right)\\
&\times \bigg[\zeta_{34}^{n_{34}} \int d\textbf{r}_1 \;r_{1a}^{n_1-1} \;Y_{l_1m_1}(\cos
\theta_{1a},\phi)\;e^{-\zeta_{1}r_{1a}}\;
r_{1b}^{n_{2}+n_{34}-1}\;Y_{L_1m_1}(\cos \theta_{1b},\phi)\;e^{-\zeta_2r_{1b}}\;a_{n_{34}+L_2}(\zeta_{34}r_{1b})\\
&+ (n_{34}-L_2-1)! \sum_{j=L_2}^{n_{34}-1}
\frac{\zeta_{34}^j}{(j-L_2)!}
\int d\textbf{r}_1 \;r_{1a}^{n_{1}-1} \;Y_{l_1m_1}(\cos
\theta_{1a},\phi)\;e^{-\zeta_{1}r_{1a}}\;
r_{1b}^{n_2+j-1}\;Y_{L_1m_1}(\cos \theta_{1b},\phi)\;e^{-(\zeta_2+\zeta_{34})r_{1b}} \bigg].
\end{split}\end{aligned}$$
Let us now investigate the above formulae in a greater detail. It is obvious that Eqs. (\[ic\]) and (\[ih\]) include two basic types of integrals which take the following general forms
$$\begin{aligned}
\label{ovrlp1}
S_{n_1l_1m}^{n_2l_2m}(\zeta_1,\zeta_2) =
\int d\textbf{r}_1 \;r_{1a}^{n_1-1} \;Y_{l_1m}(\cos
\theta_{1a},\phi)\;e^{-\zeta_1r_{1a}}\;
r_{1b}^{n_2-1}\;Y_{l_2m}(\cos \theta_{1b},\phi)\;e^{-\zeta_2r_{1b}},\end{aligned}$$
$$\begin{aligned}
\label{ovrlp2}
\widetilde{S}_{n_1l_1m}^{n_2l_2m}({n_3};\zeta_1,\zeta_2,\zeta_3) =
\int d\textbf{r}_1 \;r_{1a}^{n_1-1} \;Y_{l_1m}(\cos
\theta_{1a},\phi)\;e^{-\zeta_1r_{1a}}\;
r_{1b}^{n_2-1}\;Y_{l_2m}(\cos \theta_{1b},\phi)\;e^{-\zeta_2r_{1b}}\;a_{n_3}(\zeta_3r_{1b}),\end{aligned}$$
The first integral is simply an overlap integral between two-centre STO charge distributions and for the second one let us introduce the name *overlap-like integral*. The latter differs from the former only by the presence of $a_n$ function under the integral sign. Further, we concentrate solely on the overlap-like integrals and present two separate approaches. We shall verify that these two methods combined provide sufficient accuracy and reasonable speed to allow calculation of the desired Coulomb and hybrid integrals. We see no need to consider overlap integrals (\[ovrlp1\]) separately. As one can see shortly, they can be computed by using exactly the same algorithms as integrals (\[ovrlp2\]). The only differences lie in the fact that for the overlap-like integrals we use numerical integration to compute some of the basic quantities and for the overlap integrals, Eq. (\[ovrlp1\]), this numerical integration can simply be skipped due to absence of the $a_n$ factor.
Calculation of the overlap-like integrals by the ellipsoidal coordinates method {#subsec:chinit}
-------------------------------------------------------------------------------
For the calculation of the overlap-like integrals the use of ellipsoidal coordinates seems to be a natural approach because the standard one-electron integrals separate into a product of simple one-dimensional integrals. It is obvious, however, that due to the presence of the factor $a_{n}$ in Eq. (\[ovrlp2\]) this separation can no longer be performed straightforwardly. Not discouraged by this fact, we proceed in a conventional manner and utilise Eq. (\[trans12\]) to express the integrand in Eq. (\[ovrlp2\]) in elliptic coordinates. Noting that the axial symmetry of the integrand requires $M=0$ in the transfer formula (\[trans12\]) we arrive at the expression
$$\begin{aligned}
\label{transs}
\widetilde{S}_{n_1l_1m}^{n_2l_2m}({n_3};\zeta_1,\zeta_2,\zeta_3) =
K_{12}
\sum_{k=0}^{k_{max}} B_k^{n_1-l_1,n_2-l_2}\sum_{p=0}^{\Gamma}\sum_{q=0}^{\Gamma} \left( {\bf \large \Xi}_{l_1l_2}^0
\right)_{pq} \int_{+1}^\infty d\xi \int_{-1}^{+1} d\eta\; \xi^{p+k}\; \eta^{q+k_{max}-k}
e^{-\alpha\xi-\beta\eta}\;a_{n_3}\big[\gamma\big(\xi+\eta\big)\big],\end{aligned}$$
after an elementary integration over the angle $\phi$. In the above expression $K_{12}=(R/2)^{n_1+n_2+3}\,\Omega_{l1m1}\,\Omega_{l2m2}$, $k_{max}=n_1-l_1+n_2-l_2$, $\alpha$ and $\beta$ are defined analogously as in Eq. (\[trans12\]) and $\gamma=R\cdot\zeta_3/2$. Let us now consider the inner integrals in the above expression and define the auxiliary integrals class: $$\begin{aligned}
\label{jl1}
\begin{split}
J_\lambda(p,q;\alpha,\beta,\gamma)&= \int_{+1}^\infty d\xi \int_{-1}^{+1} d\eta\; \xi^p\; \eta^q \\
&\times e^{-\alpha\xi-\beta\eta}\;a_{\lambda}\big[\gamma\big(\xi+\eta\big)\big].
\end{split}\end{aligned}$$ The above integrals do not separate to a product of one-dimensional integrals and are also very resistant to the numerical integration. However, let us insert the integral representation (\[ap\]) and change the order of integration so that integrations over $\xi$ and $\eta$ are performed first. One easily recognises that the inner integrals are the Mulliken integrals defined in Eqs. (\[Ap\]), (\[Bq\]) and the integrals (\[jl1\]) can be written as $$\begin{aligned}
\label{jl2}
J_\lambda(p,q;\alpha,\beta,\gamma)=
\int_0^1 dt\; t^\lambda A_p(\alpha+\gamma t)\;B_q(\beta-\gamma t).\end{aligned}$$ Note that, apart from reducing the dimensionality of the integral, we have obtained a form which is very convenient for the numerical integration. The Mulliken integrals are smooth, continuous functions of the real variable with no singularities on the integration line or unwanted oscillatory behaviour. Therefore, there is no need to use numerical quadratures with overwhelmingly large number of points. Additionally, the Mulliken integrals can be calculated extremely efficiently in a recursive fashion for arbitrary values of the parameters.
Despite the obvious advantages of the numerical integration of Eq. (\[jl2\]) this approach still has to be justified to some extent. One may ask what is the point of using numerical integration since integrals (\[jl2\]) can be worked out analytically. One can do that, for instance, by inserting in Eq. (\[jl2\]) the explicit expressions for the Mulliken integrals, which are available in the literature [@mulliken49]. Next, the integral over $t$ can be expressed as a hypergeometric function of two integer parameters and with help of the so-called contiguous relations one can reduce the initial integrals to combinations of the well-known basic functions. This approach seems to be particularly attractive for the Coulomb integrals (when $\zeta_2=0$) since, as pointed out by Tai [@tai92], the final explicit expressions contain only elementary functions of the real variables. Therefore, the numerical approach to the integrals (\[jl2\]) seems to be an unwise decision at first glance.
However, the actual situation is more complicated. Taking Eq. (\[jl2\]) as a starting point, we note that the explicit expressions for $B_q$ functions are badly conditioned due to cancellation of two large terms to a relatively small result. That is why computation of $B_q$ from the analytic expressions is unstable and alternate methods need to be utilised [@corbato56; @harris04]. This instability propagates further to the integrals (\[jl2\]) and becomes more pronounced as the value of $q$ increases. Nonetheless, with help of the symbolic algebra package, such as <span style="font-variant:small-caps;">Mathematica</span>, one can derive explicit expressions for $J_\lambda$ in order to verify their usefulness. We found that for $\beta\approx\gamma$ the loss of digits is enormous, even when the values of $q$ are not large. Therefore, a prohibitively high arithmetic precision is required to obtain any useful information about the values of $J_\lambda$. Taking into consideration the philosophy presented in the introduction (favouring accuracy over speed within reasonable limits), the above observation seems to state a deadly argument against the analytic approach. In other words, the numerical integration can be understood as a simple way to avoid a severe digital erosion.
[c|ccccccccccccc]{} \[table1\] $\zeta_1$/$\zeta_3$ & 0.1250 & 0.2500 & 0.5000 & 1.0000 & 2.0000 & 4.0000 & 8.0000 & 16.000 & 32.000 & 64.000 & 128.00 & 256.00\
\
0.1250 & 9$-$16 & 9$-$16 & 8$-$16 & 9$-$16 & 8$-$16 & 7$-$16 & 8$-$16 & 5$-$16 & 1$-$16 & 0$-$13 & 0$-$5 & 0$-$0\
0.2500 & 8$-$16 & 8$-$16 & 8$-$16 & 8$-$16 & 8$-$16 & 9$-$16 & 7$-$16 & 5$-$16 & 1$-$16 & 0$-$13 & 0$-$5 & 0$-$0\
0.5000 & 9$-$16 & 8$-$16 & 8$-$16 & 8$-$16 & 8$-$16 & 7$-$16 & 6$-$16 & 5$-$16 & 1$-$16 & 0$-$12 & 0$-$4 & 0$-$0\
1.0000 & 9$-$16 & 8$-$16 & 9$-$16 & 8$-$16 & 8$-$16 & 7$-$16 & 7$-$16 & 4$-$16 & 0$-$16 & 0$-$12 & 0$-$5 & 0$-$0\
2.0000 & 8$-$16 & 8$-$16 & 8$-$16 & 9$-$16 & 8$-$16 & 7$-$16 & 8$-$16 & 5$-$16 & 0$-$16 & 0$-$11 & 0$-$4 & 0$-$0\
4.0000 & 8$-$16 & 8$-$16 & 8$-$16 & 9$-$16 & 7$-$16 & 8$-$16 & 7$-$16 & 5$-$16 & 0$-$16 & 0$-$12 & 0$-$5 & 0$-$0\
8.0000 & 8$-$16 & 8$-$16 & 8$-$16 & 6$-$16 & 7$-$16 & 8$-$16 & 7$-$16 & 5$-$16 & 1$-$16 & 0$-$14 & 0$-$7 & 0$-$1\
16.000 & 7$-$16 & 7$-$16 & 7$-$16 & 5$-$16 & 5$-$16 & 5$-$16 & 6$-$16 & 7$-$16 & 4$-$16 & 0$-$16 & 0$-$9 & 0$-$3\
32.000 & 4$-$16 & 4$-$16 & 4$-$16 & 3$-$16 & 2$-$16 & 2$-$16 & 1$-$16 & 4$-$16 & 6$-$16 & 0$-$16 & 0$-$11 & 0$-$4\
64.000 & 0$-$16 & 0$-$16 & 0$-$15 & 0$-$14 & 0$-$15 & 0$-$14 & 0$-$14 & 0$-$16 & 2$-$16 & 1$-$16 & 0$-$12 & 0$-$4\
128.00 & 0$-$10 & 0$-$10 & 0$-$8 & 0$-$8 & 0$-$7 & 0$-$7 & 0$-$8 & 0$-$11 & 0$-$14 & 1$-$15 & 0$-$12 & 0$-$4\
256.00 & 0$-$4 & 0$-$4 & 0$-$2 & 0$-$1 & 0$-$1 & 0$-$0 & 0$-$1 & 0$-$4 & 0$-$7 & 0$-$14 & 0$-$12 & 0$-$4\
For the benchmarking purposes, we show results of the calculation of two integrals, $\widetilde{S}_{1312m}^{1512m}({26};\zeta_1,0,\zeta_3)$ and $\widetilde{S}_{7,6,m}^{21,18,m}({26};\zeta_1,\zeta_1,\zeta_3)$, within the reasonable range of values of the nonlinear parameters $\zeta_1$, $\zeta_3$. We are free to set $R=1$ since an increase of $R$ results only in scaling of the nonlinear parameters by $R$ (up to a trivial multiplicative constant). All necessary $J_\lambda$ integrals were calculated numerically using 100 or 200 grid points of the Tanh-Sinh quadrature [@tanhsinh1; @tanhsinh2] for double and quadruple arithmetic precision, respectively. Under these conditions, $J_\lambda$ integrals are typically calculated with full precision allowed by the arithmetic.
The integrals, $\widetilde{S}_{1312m}^{1512m}({26};\zeta_1,0,\zeta_3)$ and $\widetilde{S}_{7,6,m}^{21,18,m}({26};\zeta_1,\zeta_1,\zeta_3)$, are the most difficult quantities (in terms of the angular momentum) encountered in the calculation of the Coulomb and hybrid integrals, respectively, including at most 7$i$ functions. We set $\zeta_2 = \zeta_1$ in the second integral for illustrative purposes - the overall picture changes very slightly when the value of $\zeta_2$ is distorted. The results are presented in Table 1 for the first integral and in Table 2 for the second integral. One observes a progressive loss of digits when one of the nonlinear parameters is large and the second is small. This digital erosion is due to the cancellation of large numbers during summations in Eq. (\[transs\]) and it cannot be avoided in the ellipsoidal coordinates method. The use of quadruple precision improves the situation a lot but it is not sufficient to cope with the most difficult cases. Of course, for lower angular momentum functions the changes are less sharp but the overall trend remains the same. Concluding, our observations signal that the ellipsoidal coordinates method alone is not sufficient to calculate the desired integrals with the prescribed accuracy and need to be supplemented by a different algorithm.
In the present series of papers we do not go into technical details of the implementation *etc.*, but let us give a short remark on the timings in the present algorithm. The numerical integration of the integrals $J_\lambda$ typically consumes about a half of the total time necessary to calculate a given shell of integrals. Only for the smallest values of the quantum numbers this ratio is higher, but these integrals are very cheap anyway. The remaining time is spent on the lengthy summations in Eq. (\[transs\]), formation of $\widetilde{I}_{C/H}$, Eqs. (\[ic\]) and (\[ih\]), and summation of the initial Clebsh-Gordan expansion to finally arrive at the value of $I_{C/H}$. Therefore, the numerical integration is not connected with a drastic overhead as might have been initially expected. A faster scheme for the calculation of $J_\lambda$ shall not result in a significant overall speed-up. Typically, the Coulomb and hybrid integrals are obtained in 1-100 $\mu$s per integral, depending on the values of quantum numbers, with hybrid integrals being slightly more expensive.
[c|ccccccccccccc]{} \[table2\] $\zeta_1$/$\zeta_3$ & 0.1250 & 0.2500 & 0.5000 & 1.0000 & 2.0000 & 4.0000 & 8.0000 & 16.000 & 32.000 & 64.000 & 128.00 & 256.00\
\
0.1250 & 0$-$9 & 0$-$10 & 0$-$12 & 0$-$16 & 1$-$16 & 6$-$16 & 7$-$16 & 7$-$16 & 6$-$16 & 6$-$16 & 4$-$16 & 4$-$16\
0.2500 & 0$-$10 & 0$-$10 & 0$-$13 & 0$-$16 & 2$-$16 & 5$-$16 & 8$-$16 & 7$-$16 & 6$-$16 & 7$-$16 & 4$-$16 & 4$-$16\
0.5000 & 0$-$10 & 0$-$10 & 0$-$13 & 0$-$16 & 3$-$16 & 6$-$16 & 7$-$16 & 7$-$16 & 6$-$16 & 7$-$16 & 4$-$16 & 4$-$16\
1.0000 & 0$-$11 & 0$-$11 & 0$-$14 & 0$-$16 & 2$-$16 & 5$-$16 & 7$-$16 & 7$-$16 & 7$-$16 & 7$-$16 & 5$-$16 & 5$-$16\
2.0000 & 0$-$12 & 0$-$13 & 0$-$15 & 0$-$16 & 2$-$16 & 5$-$16 & 7$-$16 & 6$-$16 & 6$-$16 & 6$-$16 & 5$-$16 & 5$-$16\
4.0000 & 0$-$14 & 0$-$15 & 0$-$16 & 0$-$16 & 2$-$16 & 4$-$16 & 6$-$16 & 5$-$16 & 6$-$16 & 6$-$16 & 5$-$16 & 5$-$16\
8.0000 & 0$-$16 & 0$-$16 & 0$-$16 & 0$-$16 & 2$-$16 & 3$-$16 & 4$-$16 & 4$-$16 & 4$-$16 & 4$-$16 & 7$-$16 & 5$-$16\
16.000 & 0$-$16 & 0$-$16 & 0$-$16 & 0$-$16 & 1$-$16 & 1$-$16 & 1$-$16 & 2$-$16 & 2$-$16 & 2$-$16 & 4$-$16 & 3$-$16\
32.000 & 0$-$13 & 0$-$14 & 0$-$13 & 0$-$14 & 0$-$15 & 0$-$15 & 0$-$15 & 0$-$16 & 0$-$16 & 0$-$16 & 0$-$15 & 0$-$14\
64.000 & 0$-$6 & 0$-$6 & 0$-$6 & 0$-$7 & 0$-$7 & 0$-$8 & 0$-$9 & 0$-$9 & 0$-$9 & 0$-$6 & 0$-$5 & 0$-$4\
128.00 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$1 & 0$-$2 & 0$-$2 & 0$-$4 & 0$-$3 & 0$-$2\
Calculation of the overlap-like integrals by the recursive method
-----------------------------------------------------------------
For the calculation of the overlap-like integrals by using the recursive method it is more convenient to introduce different basic integrals, so that the final expressions take a simpler form. Let us note that Eq. (\[ovrlp2\]) can be rewritten as $$\begin{aligned}
\widetilde{S}_{n_1l_1m}^{n_2l_2m}({n_3};\zeta_1,\zeta_2,\zeta_3) =
\frac{1}{R}\,\Omega_{l_1m}\Omega_{l_2m} \langle \varphi_{n_1}^{l_1m}|\varphi_{n_2}^{l_2m}\rangle,\end{aligned}$$ where $$\begin{aligned}
\begin{split}
\label{pertinent1}
\langle \varphi_{n_1}^{l_1m}|\varphi_{n_2}^{l_2m}\rangle &=
\int_0^\infty dr_a \int_{|r_a-R|}^{r_a+R} dr_b \;r_a^{n_1} r_b^{n_2}e^{-\zeta_1r_a-\zeta_2r_b}\\
&\times P_{l_1}^m(\cos \theta_a)\;P_{l_2}^m(\cos \theta_{b})
\;a_{n_3}(\zeta_3r_{1b}).
\end{split}\end{aligned}$$ In the second expression we changed the variables from the Cartesian coordinates to the internal coordinate system $(r_a,r_b,\phi)$ and integrated over the angle. Note, that the notation for the nonlinear parameters and for the variable $n_3$ was suppressed since these quantities do not change during the recursive process. We have to stress that all formulae presented here are valid only for $m>0$. There is no need to consider the negative values of $m$ because of the axial symmetry of the integrands.
Generally speaking, to establish a recursive process which is able to increase the values of $l_1$, $l_2$ and $m$, starting with provided values of $\langle \varphi_{n_1}^{00}|\varphi_{n_2}^{00}\rangle$ we need to use the well-known recursion relations for the Legendre polynomials $P_l^m$. A similar idea was applied by several authors to the calculation of various important matrix elements [@guseinov05; @guseinov07; @rico88; @rico89b; @rico93; @rico91; @rico89c]. Let us first derive a recursion relation connecting $\langle
\varphi_{n_1}^{mm}|\varphi_{n_2}^{mm}\rangle$ with different $m$ by recalling the following expression for the Legendre polynomials with $l=m$: $$\begin{aligned}
P_m^m(\cos \theta) = \frac{(2m)!}{2^m m!} \sin^m \theta,\end{aligned}$$ so that $$\begin{aligned}
P_{m+1}^{m+1}(\cos \theta) = P_{m}^{m}(\cos \theta) (2m+1)\sin \theta.\end{aligned}$$ By combining two expressions like the above for $\cos \theta_a$ and $\cos \theta_b$ and using the obvious relationship $r_a \sin \theta_a = r_b \sin \theta_b $ one finds $$\begin{aligned}
\begin{split}
&P_{m+1}^{m+1}(\cos \theta_a) P_{m+1}^{m+1}(\cos \theta_b)=\\
&P_{m}^{m}(\cos \theta_a) P_{m}^{m}(\cos \theta_b)
(2m+1)^2 \frac{r_a}{r_b} \sin^2 \theta_a,
\end{split}\end{aligned}$$ and the expression for $\sin^2 \theta_a$ in terms of $r_a$, $r_b$ is elementary. Finally, this leads to the recursion relation for the desired set of integrals $$\begin{aligned}
\begin{split}
&\langle \varphi_{n_1}^{m+1,m+1}|\varphi_{n_2}^{m+1,m+1}\rangle = \frac{(2m+1)^2}{2R^2} \times \\
&\bigg[ R^2 \langle \varphi_{n_1+1}^{mm}|\varphi_{n_2-1}^{mm}\rangle
+R^2\langle \varphi_{n_1-1}^{mm}|\varphi_{n_2+1}^{mm}\rangle\\
&+\langle \varphi_{n_1+1}^{mm}|\varphi_{n_2+1}^{mm}\rangle-
\frac{1}{2}R^4\langle \varphi_{n_1-1}^{mm}|\varphi_{n_2-1}^{mm}\rangle\\
&-\frac{1}{2}\langle \varphi_{n_1+3}^{mm}|\varphi_{n_2-1}^{mm}\rangle-
\frac{1}{2}\langle \varphi_{n_1-1}^{mm}|\varphi_{n_2+3}^{mm}\rangle \bigg].
\end{split}\end{aligned}$$
The second ingredient of the recursive process is a relation that allows to increase the values of $l_1$ and $l_2$ independently, starting with the just considered $\langle \varphi_{n_1}^{mm}|\varphi_{n_2}^{mm}\rangle$ integrals. The following recursion relation for the Legendre polynomials is useful $$\begin{aligned}
(l-m+1)P_{l+1}^m(x)+(l+m)P_{l-1}^m(x)=(2l+1)xP_l^m(x).\end{aligned}$$ If one uses the above relation for $P_{l_1}^m(\cos \theta_a)$ in Eq. (\[pertinent1\]) and subsequently expresses $\cos
\theta_a$ through $r_a$ and $r_b$ from the cosine theorem, the following recursion is obtained $$\begin{aligned}
\label{rec1}
\begin{split}
&\frac{1}{2R}\bigg[
\langle \varphi_{n_1+1}^{l_1m}|\varphi_{n_2}^{l_2m}\rangle-
\langle \varphi_{n_1-1}^{l_1m}|\varphi_{n_2+2}^{l_2m}\rangle
+R^2\langle \varphi_{n_1-1}^{l_1m}|\varphi_{n_2}^{l_2m}\rangle \bigg]\\
&=(l_1-m+1) \langle \varphi_{n_1}^{l_1+1,m}|\varphi_{n_2}^{l_2m}\rangle
+(l_1+m) \langle \varphi_{n_1}^{l_1-1,m}|\varphi_{n_2}^{l_2m}\rangle,
\end{split}\end{aligned}$$ which can be used to increase $l_1$ at cost of $n_1$ and $n_2$. A corresponding expression for increasing $l_2$ can be obtained by repeating the derivation for $P_{l_2}^m(\cos \theta_b)$. Therefore, by using Eq. (\[rec1\]) and its counterpart for the centre $b$, we can build all $\langle \varphi_{n_1}^{l_1m}|\varphi_{n_2}^{l_2m}\rangle$ starting with integrals with $l_1=l_2=m$ and higher $n_1$, $n_2$.
Having said this, the only thing that remains in question is the calculation of the pertinent integrals $\langle
n_100|n_200\rangle$. Let us return to Eq. (\[pertinent1\]) $$\begin{aligned}
\begin{split}
\label{pertinent2}
\langle \varphi_{n_1}^{00}|\varphi_{n_2}^{00}\rangle &=
\int_0^\infty dr_a \int_{|r_a-R|}^{r_a+R} dr_b \;r_a^{n_1} r_b^{n_2}\\
&\times e^{-\zeta_1r_a-\zeta_2r_b}\;a_{n_3}(\zeta_3r_{1b}),
\end{split}\end{aligned}$$ use the integral representation of $a_n$, Eq. (\[ap\]), and reverse the order of integration. By doing so we obtain an equivalent representation of the basic integrals $$\begin{aligned}
\label{pertinent3}
\langle \varphi_{n_1}^{00}|\varphi_{n_2}^{00}\rangle =
\int_0^1 dt\; t^{n_3}\; \Gamma_{n_1n_2}(R;\zeta_1,\zeta_2+t\zeta_3),\end{aligned}$$ where $\Gamma_{mn}$ are the usual overlap integrals between $ns$-type orbitals $$\begin{aligned}
\label{tmn}
\Gamma_{mn}(R;\zeta_1,\zeta_2) = \int_0^\infty dr_a \int_{|r_a-R|}^{r_a+R} dr_b \;r_a^m r_b^n\;
e^{-\zeta_1r_a-\zeta_2r_b}.\end{aligned}$$ In our approach, the outer integral in (\[pertinent3\]) is carried out numerically. The arguments for this approach are virtually the same as in the ellipsoidal coordinates method. Roughly speaking, numerical integration serves as a way to avoid numerical instabilities which inevitably appear when the analytic approaches are used. However, now we require a robust scheme for the calculation of $\Gamma_{mn}$, so that these integrals can be computed at each point of the grid without a great overhead. In fact, the main advantage of the numerical integration in the ellipsoidal coordinates method was that the integrand in Eq. (\[jl2\]) could be evaluated extremely efficiently and with a strictly controlled precision. On the other hand, the desired algorithm has to preserve a decent accuracy up to large values of $m$ and $n$ (several tens, say). Determination of such an algorithm still presents a challenge from the practical point of view.
[c|ccccccccccccc]{} \[table3\] $\zeta_1$/$\zeta_3$ & 0.1250 & 0.2500 & 0.5000 & 1.0000 & 2.0000 & 4.0000 & 8.0000 & 16.000 & 32.000 & 64.000 & 128.00 & 256.00\
\
0.1250 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$5 & 0$-$11 & 1$-$16 & 4$-$16 & 5$-$16 & 1$-$16 & 0$-$11\
0.2500 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$5 & 0$-$11 & 1$-$16 & 4$-$16 & 5$-$16 & 1$-$16 & 0$-$11\
0.5000 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$5 & 0$-$11 & 1$-$16 & 4$-$16 & 5$-$16 & 1$-$16 & 0$-$11\
1.0000 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$5 & 0$-$11 & 1$-$16 & 4$-$16 & 5$-$16 & 1$-$16 & 0$-$11\
2.0000 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$6 & 0$-$10 & 0$-$14 & 4$-$16 & 4$-$16 & 0$-$14 & 0$-$11\
4.0000 & 0$-$6 & 0$-$6 & 0$-$6 & 0$-$6 & 0$-$6 & 0$-$5 & 0$-$11 & 1$-$16 & 3$-$16 & 4$-$16 & 1$-$16 & 0$-$12\
8.0000 & 0$-$12 & 0$-$12 & 0$-$12 & 0$-$12 & 0$-$10 & 0$-$12 & 1$-$16 & 3$-$16 & 3$-$16 & 2$-$16 & 1$-$16 & 0$-$12\
16.000 & 1$-$16 & 1$-$16 & 1$-$16 & 1$-$16 & 0$-$14 & 1$-$16 & 1$-$16 & 2$-$16 & 2$-$16 & 1$-$16 & 0$-$15 & 0$-$11\
32.000 & 4$-$16 & 4$-$16 & 4$-$16 & 4$-$16 & 4$-$16 & 3$-$16 & 3$-$16 & 5$-$16 & 6$-$16 & 2$-$16 & 1$-$16 & 0$-$14\
64.000 & 5$-$16 & 5$-$16 & 5$-$16 & 5$-$16 & 4$-$16 & 3$-$16 & 4$-$16 & 7$-$16 & 9$-$16 & 6$-$16 & 1$-$16 & 0$-$15\
128.00 & 1$-$16 & 1$-$16 & 1$-$16 & 1$-$16 & 0$-$14 & 1$-$16 & 1$-$16 & 1$-$16 & 4$-$16 & 3$-$16 & 0$-$16 & 0$-$13\
256.00 & 0$-$12 & 0$-$12 & 0$-$12 & 0$-$12 & 0$-$10 & 0$-$12 & 0$-$12 & 0$-$12 & 0$-$13 & 0$-$13 & 0$-$14 & 0$-$12\
The basic integrals $\Gamma_{mn}$ are well-known in the literature. Many authors considered their computation by using several different algorithms which varied in accuracy and speed. Let us note, however, that in the calculation of the integrals (\[tmn\]) the main issue is the numerical stability. The actual expressions for these integrals are not difficult to derive and include only simple elementary functions. Unfortunately, these expressions consist of finite series with terms of alternating signs. When $m$, $n$ are increased these terms grow exponentially while the sum remains by orders of magnitude smaller. As a result, a gross digital erosion is inevitable. In a large fraction of works which considered calculation of the integrals (\[tmn\]), or used them as a part of different algorithms, the issue of numerical stability was completely disregarded or treated very lightly. The common justification for this fact is that authors were mainly interested in low quantum numbers or devised their algorithms to verify the correctness of the approach more than to perform general calculations.
Let us begin by noting that all integrals (\[tmn\]) can be generated by a consecutive differentiation of $\Gamma_{00}$ with respect to the nonlinear parameters $\zeta_1$, $\zeta_2$ *i.e.* $$\begin{aligned}
\label{Gmn}
\Gamma_{mn}(R;\zeta_1,\zeta_2) =
\left(-\frac{\partial}{\partial \zeta_1}\right)^m
\left(-\frac{\partial}{\partial \zeta_2}\right)^n \Gamma_{00}(\zeta_1,\zeta_2),\end{aligned}$$ which is, in substance, a trivial case of the so-called shift method of Fernández Rico *et al.* [@rico00a; @rico00b; @rico01]. The simplest integrals $\Gamma_{00}$ are elementary $$\begin{aligned}
\label{G00}
\Gamma_{00}(R;\zeta_1,\zeta_2) = \frac{2}{\zeta_1+\zeta_2} \frac{e^{-\zeta_2R}-e^{-\zeta_1R}}{\zeta_1-\zeta_2}.\end{aligned}$$ It is now convenient to define $g_{00}$ by $$\begin{aligned}
\label{g00}
g_{00}(R;\zeta_1,\zeta_2) = 2\,\frac{e^{-\zeta_2R}-e^{-\zeta_1R}}{\zeta_1-\zeta_2},\end{aligned}$$ so that $\Gamma_{00}=g_{00}/(\zeta_1+\zeta_2)$, and the definition of $g_{mn}$ is analogous $$\begin{aligned}
\label{gmn}
g_{mn}(R;\zeta_1,\zeta_2) =
\left(-\frac{\partial}{\partial \zeta_1}\right)^m
\left(-\frac{\partial}{\partial \zeta_2}\right)^n g_{00}(\zeta_1,\zeta_2).\end{aligned}$$ Let us now multiply both sides of Eq. (\[G00\]) by $\zeta_1+\zeta_2$, rewrite the result in terms of $g_{00}$ and differentiate both sides $m$ with respect to $-\zeta_1$ and $n$ times with respect to $-\zeta_2$. After some rearrangements, the final result can be written as $$\begin{aligned}
\label{Grec}
\Gamma_{mn} = \frac{1}{\zeta_1+\zeta_2}\bigg[g_{mn}+m\Gamma_{m-1,n}+n\Gamma_{m,n-1}\bigg],\end{aligned}$$ where the notation for the nonlinear parameters is suppressed for brevity. The above expression is an inhomogeneous linear recursion relation for $\Gamma_{mn}$. Note, that all integrals $\Gamma_{mn}$ are positive and so are the values of $g_{mn}$. Therefore, the above recursion relation is completely stable. This approach is reminiscent of the treatment of the one-centre integrals by Sack *et al.* [@sack67].
[c|ccccccccccccc]{} \[table4\] $\zeta_1$/$\zeta_3$ & 0.1250 & 0.2500 & 0.5000 & 1.0000 & 2.0000 & 4.0000 & 8.0000 & 16.000 & 32.000 & 64.000 & 128.00 & 256.00\
\
0.1250 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0\
0.2500 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0\
0.5000 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0\
1.0000 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0\
2.0000 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0\
4.0000 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0 & 0$-$0\
8.0000 & 0$-$5 & 0$-$5 & 0$-$5 & 0$-$5 & 0$-$5 & 0$-$5 & 0$-$5 & 0$-$6 & 0$-$6 & 0$-$6 & 0$-$5 & 0$-$4\
16.000 & 0$-$12 & 0$-$12 & 0$-$12 & 0$-$12 & 0$-$12 & 0$-$12 & 0$-$12 & 0$-$13 & 0$-$13 & 0$-$13 & 0$-$12 & 0$-$11\
32.000 & 6$-$16 & 6$-$16 & 6$-$16 & 6$-$16 & 6$-$16 & 6$-$16 & 6$-$16 & 6$-$16 & 7$-$16 & 7$-$16 & 4$-$16 & 0$-$14\
64.000 & 8$-$16 & 8$-$16 & 8$-$16 & 8$-$16 & 8$-$16 & 8$-$16 & 8$-$16 & 8$-$16 & 9$-$16 & 10$-$16 & 6$-$16 & 0$-$15\
128.00 & 4$-$16 & 4$-$16 & 4$-$16 & 4$-$16 & 4$-$16 & 4$-$16 & 4$-$16 & 4$-$16 & 6$-$16 & 7$-$16 & 2$-$16 & 0$-$14\
The problem is now reduced to an efficient calculation of $g_{mn}$. Explicit differentiation is not an option because of similar cancellations as for the initial $\Gamma_{mn}$ integrals. However, let us observe that $g_{00}$ can also be rewritten as $$\begin{aligned}
g_{00}(\zeta_1,\zeta_2) = \frac{R}{2} e^{-\zeta_1R} M\big[1,2,(\zeta_1-\zeta_2)R\big],\end{aligned}$$ where $M(a,b,z)$ is the confluent hypergeometric function [@stegun72] (denoted as $_1F_1$ by some authors). By using two differentiation formulae for $M(a,b,z)$ $$\begin{aligned}
\frac{\partial^n}{\partial z^n} M(a,b,z) &= \frac{(a)_n}{(b)_n} M(a+n,b+n,z),\\
\frac{\partial^n}{\partial z^n} \bigg[ e^{-z} M(a,b,z) \bigg] &= (-1)^n \frac{(b-a)_n}{(b)_n} e^{-z} M(a,b+n,z),\end{aligned}$$ one easily arrives at the new formula for $g_{mn}$ $$\begin{aligned}
\begin{split}
g_{mn}(\zeta_1,\zeta_2) &= \frac{1}{2}\;e^{-\zeta_1R}\;R^{m+n+1}\times\\
&\times M\big[1+n,2+m+n,(\zeta_1-\zeta_2)R\big].
\end{split}\end{aligned}$$ At this point the problem can be considered to be solved because methods of calculation of $M(a,b,z)$ for arbitrary real (or even complex) values of the parameters $a$, $b$, and $z$ exist. Let us note that here we deal with an exceptionally special case of $M(a,b,z)$ with both $a$ and $b$ being strictly positive integers, and additionally $b>a$ always holds. Moreover, we can use the symmetry of the initial integrals, $\langle
\varphi_{n_1}^{00}|\varphi_{n_2}^{00}\rangle = \langle \varphi_{n_2}^{00}|\varphi_{n_1}^{00} \rangle$, in order to impose the restriction $\zeta_1\geq\zeta_2$, which gives $z\geq 0$. All these conditions signal that we should design a dedicated procedure for the calculation of $M(a,b,z)$ in this special case and avoid using general algorithms which are drastically more complicated and involve a large computational overhead. In Appendix we present a recursive method which is able to calculate $M(a,b,z)$ in our special case with a decent speed, at the same time preserving full accuracy allowed by the arithmetic.
In Tables 3 and 4 we present results of the benchmark calculations for the same representative integrals, $\widetilde{S}_{1312m}^{1512m}({26};\zeta_1,0,\zeta_3)$ and $\widetilde{S}_{7,6,m}^{21,18,m}({26};\zeta_1,\zeta_1,\zeta_3)$, as in the previous subsection. We use the same numerical quadrature as before and typically a machine precision is obtained in Eq. (\[pertinent3\]). One sees that the recursive algorithm fails completely, even in the quadruple arithmetic precision, when nonlinear parameters are both small. On the other hand, as they get large the accuracy gradually improves which is exactly the opposite behaviour to the one found in the ellipsoidal method. Therefore, two methods presented in this paper can be considered fully complementary and together are able to cover a sufficiently large range of the nonlinear parameters. Outside this range, hybrid integrals are usually very small and are typically neglected in advance by the Schwarz screening technique or a similar scheme. Coulomb integrals with bigger values of the nonlinear parameters may still be non-negligible. However, they can be computed with different standard techniques such as the multipole expansion. It is mandatory for a general program to include such a method as an option.
Conclusions
===========
Concluding, we derived new expressions for the Coulomb and hybrid integrals over the Slater-type orbitals, with no restrictions on the values of the quantum numbers, starting by a direct integration over coordinates of the second electron. In this way the desired integrals reduce to combinations of ordinary overlap integrals and a set of the so-called overlap-like integrals. These basic integrals are evaluated by using two distinct methods - direct integration in the ellipsoidal coordinate system or with a recursive scheme for increasing angular momenta in the integrand. One of the biggest problems in actual computations is numerical stability of the resulting algorithms. Many formulations available in the literature contain numerically badly conditioned expressions which introduce a significant loss of digits when evaluated in a finite arithmetic precision. We show how these instabilities can be avoided if a simple, one-dimensional numerical integration is used instead. We discuss that this numerical approach introduces an acceptable computational overhead due to well-behaved and simple form of the integrands. We also show that the remaining numerical instabilities can be easily controlled. Extensive numerical tests are presented, verifying the usefulness and applicability of the method.
This work was supported by the Polish Ministry of Science and Higher Education, grant NN204 182840. ML acknowledges the Polish Ministry of Science and Higher Education for the support through the project *“Diamentowy Grant”*, number DI2011 012041. RM was supported by the Foundation for Polish Science through the *“Mistrz”* program. We would like to thank Bogumił Jeziorski for fruitful discussions, reading and commenting on the manuscript.
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Calculation of $M(a,b,z)$ for $a,b \in \mathbb{Z}_+$, $b>a$, $z\geq 0$ {#app:appa}
======================================================================
Let us start by recalling some of the useful formulae obeyed by $M(a,b,z)$. The first one is the Gautschi representation of the continued fraction (GCF) [@gautschi77] which states that $$\begin{aligned}
\label{gcf}
\begin{split}
&\frac{M(a+1,b,z)}{M(a,b,z)} = 1 + \frac{z}{a}\sum_{k=0}^\infty p_k,\\
&p_0 = 1,\;\;\; p_k = \prod_{i=1}^k r_i,\\
&r_0 = 0,\;\;\; r_k = -\frac{a_k(1+r_{k-1})}{1+a_k(1+r_{k-1})},\\
&a_k = \frac{(a+k)z}{(b-z+k-1)(b-z+k)}.
\end{split}\end{aligned}$$ The second useful expression is the recursion relation which allows to increase the value of $a$ at constant $b$: $$\begin{aligned}
\label{recm}
\begin{split}
&(b-a)M(a-1,b,z)+(2a-b+z)M(a,b,z)\\
&-aM(a+1,b,z)=0.
\end{split}\end{aligned}$$ The region $a,b \in \mathbb{Z}_+$, $b>a$, $z\geq 0$ needs to be divided into three subregions and different algorithms have to be used in each of them. They are as follows:
- $b\geq2a+z$,\
one first uses GCF, Eq. (\[gcf\]), in order to obtain the ratio $M(a+1,b,z)/M(a,b,z)$ for the maximal desired $b$ and $a=\lceil(b-z)/2\rceil$ ($\lceil * \rceil$ is the ceiling function). The recursion (\[recm\]) can be rewritten as $$\begin{aligned}
\label{miller}
r_{a-1} = \frac{b-a}{ar_a+b-2a-z},\end{aligned}$$ where $r_a = M(a+1,b,z)/M(a,b,z)$. This recursion is then carried out downward, starting with the value of the ratio obtained from GCF, until $r_0$ is reached. Since $M(0,b,z)=1$, it turns out that $r_0 = M(1,b,z)$ and other values can be obtained by using the definition of $r_a$ *e.g.* $M(2,b,z) = r_1 M(1,b,z)$.
- $b<2a+z$, $b\geq z$,\
again, the relation (\[recm\]) is transformed into a Miller-like two-step recursion $$\begin{aligned}
r_a=\frac{b-a}{a}\frac{1}{r_{a-1}}+2+\frac{z-b}{a},\end{aligned}$$ with $r_a$ being defined in the same way as previously. Starting with an arbitrary value of $r_0$, this recursion is carried out upward up to the line $a=b$ (corresponding to $r_{b-1}$). Using the exact relationship $M(b,b,z)=e^z$ one finds that actual values of $M(a,b,z)$ can be reconstructed as $M(b-1,b,z)=M(b,b,z)/r_{b-1}=e^z/r_{b-1}$, $M(b-2,b,z)=M(b-1,b,z)/r_{b-2}$ *etc.* until the value of $M(1,b,z)$ is reached.
- $b<2a+z$, $b<z$,\
this is the so-called anomalous convergence region of GCF *i.e.* the expression (\[gcf\]) converges to the wrong result [@gautschi77] and therefore cannot be used. However, in this region the initial upward recursion (\[recm\]) is totally stable since all terms in (\[recm\]) are positive. The starting (exact) values are $$\begin{aligned}
M(0,b,z) &= 1,\\
M(1,b,z) &= (b-1)e^z a_{b-2}(z),\end{aligned}$$ where $a_n$ are given by Eq. (\[ap\]). The second relationship breaks down when $b=1$ but in this case we obtain independently $M(1,1,z)=e^z$, as noted beforehand.
Let us also add in passing that the power series expansion of $M(a,b,z)$ around $z=0$ can additionally be used for small $z$ $$\begin{aligned}
M(a,b,z) = \sum_{s=0}^\infty \frac{(a)_s}{(b)_s s!}z^s,\end{aligned}$$ since it typically converges very fast in the vicinity of the origin, $z\approx 0$. Similar conclusion holds for the asymptotic expansion of $M(a,b,z)$ as $z$ is large. Remarkably, when the values of $M(a,b,z)$ are calculated as described in this Appendix, no loss of digits is observed, and thus $\langle \varphi_{n_1}^{00}|\varphi_{n_2}^{00}\rangle$ can be obtained with full precision up to very large values of $n_1$ and $n_2$.
|
---
abstract: 'Topological states in non-Hermitian systems are known to exhibit some anomalous features. Here, we find two new anomalous features of non-Hermitian topological states. We consider a one dimensional nonreciprocal Hamiltonian and show that topological robustness can be practically lost for a linear combination of topological eigenstates in non-Hermitian systems due to the non-Hermitian skin effect. We consider a two dimensional non-Hermitian Chern insulator and show that chirality of topological states can be broken at some parameters of the Hamiltonan. This implies that the topological states are no longer immune to backscattering in 2D.'
address: 'Department of Physics, Eskisehir Technical University, Eskisehir, Turkey '
author:
- 'C. Yuce'
title: ' Anomalous features of non-Hermitian topological states '
---
Introduction
============
Topological band theory and its non-Hermitian extension have attracted great attention in the past decades [@sondeney1; @aah1; @ghatakdas; @1d5; @ann01; @eklon]. The periodic table of Hermitian topological insulators is well-known, but its non-Hermitian counterpart has not yet been fully constructed. Fortunately, some recent attempts pave the way for constructing it [@ptTI; @cyek2]. Non-Hermitian topological edge states were initially found in simple models such as complex extensions of the Su-Schrieffer-Heeger (SSH) model [@1d1; @1d2; @1d6; @1d7; @1d8; @1d9; @1d10; @1d11; @1d3ekl; @1d12; @1d13; @1d14; @1d3; @1d15; @1d16; @floquet1; @floquet2; @bhjkl; @feng; @ann2] and Kitaev model [@kita1; @kita2; @kita3; @kita4; @kita5]. Then more complex models has been introduced and studied in the literature [@cyek1; @cyek3; @cyek4; @cyek5; @cyek6; @genel000; @genel001; @genel002; @genel003; @genel004; @genel005; @genel006; @genel007; @genel008; @genel009; @genel0010; @genel0011; @genel0012; @genel0013; @genel0014; @genel0015; @genel0016; @genel0017; @genel0019; @genel0020; @genel0021; @takata; @ann3; @ann4; @ann5; @ann6; @ann7]. In a recent paper, it was shown that topological phase can also arise in a non-Hermitian quasicrystal with parity-time symmetry [@nhquasi]. Note that topological edge states in a non-Hermitian system can have real or complex energy eigenvalues. The latter one can be used as a topological laser [@laser] or spontaneous topological pump at large times [@yucepump].\
One of the main problem in the theory of non-Hermitian topological systems is to understand the bulk-boundary correspondence [@bulkboun01; @bulkboun13; @ueda1a; @bulkboun02; @bulkboun02b; @bulkboun04b; @bulkboun06; @bulkboun07; @bulkboun08; @bulkboun09; @bulkboun10; @bulkboun11; @bulkboun12; @bulkboun14; @bulkboun15; @bulkboun16]. Unfortunately, the standard bulk-boundary correspondence fails in many non-Hermitian systems. As opposed to Hermitian systems, topological phase transition points can not be generally determined using periodical form of the insulating non-Hermitian Hamiltonian. This is because of the fact that energy eigenvalues of topological and bulk states can depend sensitively on boundary conditions. Therefore, one must study non-Hermitian systems with open edges to precisely explore topological states. Furthermore, the non-Hermitian topological invariants explored so far in the literature are generally model dependent in sharp contrast to Hermitian systems [@winding1; @winding3; @yenice]. In other words, one can find at least two different topological numbers in the literature that can predict different topological phase diagram for a given non-Hermitian gapped Hamiltonian. In [@pseudo], the existence of topological edge states for the SSH model with gain and loss was explained using the idea of pseudo topological insulator. The so called non-Hermitian skin effect was introduced in a nonreciprocal tight binding lattice with asymmetrical couplings [@bulkboun03; @bulkboun04]. It states that not only topological states but also bulk states are localized around either edge, which causes the density of states at the edge to be increased. The accumulation of the eigenstates around one edge can be understood as an amplification of them in one way and a corresponding decaying in the opposite way due to an imaginary gauge field [@bulkboun07]. Recently, the existence of hybrid skin-topological modes in a 2-dimensional system [@skin01] and non-Hermitian anomalous skin effect have been predicted [@anaskef; @robustbulks]. In the latter case, topological states becomes extended all over the system. This has no analog in Hermitian systems as Hermitian topological states always occur around the edges where topological phase transition occurs.\
It is well known that topological states in non-Hermitian systems show some anomalous features [@bulkboun01; @bulkboun13; @ueda1a]. In this paper, we explore some other anomalous features of non-Hermitian topological states. These are the superposition-induced loss of topological protection and chirality breakdown in a two dimensional non-Hermitian Chern insulator. Any linear combination of topological states is certainly topological in Hermitian systems. Here we find a one dimensional nonreciprocal Hamiltonian and show that a symmetric combination of its topological edge states is not topologically protected and grows unboundedly in time. As another anomalous feature, we consider a two dimensional non-Hermitian system. In a two dimensional Hermitian Chern insulator, topological edge states are protected and hence backscattering of topological edge states from symmetry protecting perturbative disorders are forbidden. We show that this is not always the case in non-Hermitian systems since the chirality can be broken due to the non-Hermitian skin effect.
Fragile topological zero energy state
=====================================
Consider the 1D SSH tight-binding chain with nonreciprocal hopping amplitudes for the first sub-lattice. The non-Hermitian Hamiltonian under the periodic boundary condition can be written as $$\begin{aligned}
\label{kiRYazm}
\mathcal{H} (k)=\left(\begin{array}{cc} 0& t_1+{t_2}~e^{-ik} \\ t_1^{\prime}+t_2~e^{ik} & 0 \end{array}\right)\end{aligned}$$ where the real-valued parameters $t_1$, $t_2$ and $t_1^{\prime}$ are hopping amplitudes. The Hamiltonian is Hermitian if $t_1={t_1}^{\prime}$. One can easily see that this Hamiltonian has chiral symmetry: $\ds{\sigma_z\mathcal{H} (k) \sigma_z=-\mathcal{H} (k) }$ where $\sigma_i$ refers to Pauli matrices. This implies that eigenvalues come in pairs at a given $k$. They are given by $E_{\mp}=\mp\sqrt{( t_1+{t_2}~e^{-ik}) ( t_1^{\prime}+t_2~e^{ik} ) }$. The corresponding non-Hermitian winding number can be found in [@genel0019; @winding1]. We refer the reader to [@genel0019] for the topological phase diagram of the system under both periodical and open boundary conditions.
![The wave packets of the topological zero energy edge states for the non-reciprocal lattice with the parameters $t_1=0.5$, $t_1^{\prime}=0.1$, $t_2=1$ and $N=30$. $\ds{\psi_{1}}$ (blue) and $\ds{\psi_{2}}$ (red) are topological zero energy states localized around the left edge due to the non-Hermitian skin effect. In (b), we plot the superposition states $ \ds{ \psi^{\mp}=P_0( \psi_{1}\mp \psi_{2} ) } $, where $P_0$ is a constant satisfying $\sum_n | \psi^{\mp}|^2=1$ ($P_0=1/2$ for $\ds{\psi^-}$ while $P_0=7.6 \times10^4$ for $\ds{\psi^+}$). []{data-label="thflks0938"}](SP1ek.jpeg "fig:"){width="4.5cm"} ![The wave packets of the topological zero energy edge states for the non-reciprocal lattice with the parameters $t_1=0.5$, $t_1^{\prime}=0.1$, $t_2=1$ and $N=30$. $\ds{\psi_{1}}$ (blue) and $\ds{\psi_{2}}$ (red) are topological zero energy states localized around the left edge due to the non-Hermitian skin effect. In (b), we plot the superposition states $ \ds{ \psi^{\mp}=P_0( \psi_{1}\mp \psi_{2} ) } $, where $P_0$ is a constant satisfying $\sum_n | \psi^{\mp}|^2=1$ ($P_0=1/2$ for $\ds{\psi^-}$ while $P_0=7.6 \times10^4$ for $\ds{\psi^+}$). []{data-label="thflks0938"}](SP2ek.jpeg "fig:"){width="4.5cm"}
![The total density $\ds{P_{\mp}(t)=\sum_{n} |\Psi^{\mp}(t) |^2}$ as a function of time for $\Psi^+$ (red) and $\Psi-$ (blue) in the presence of the disorder in (a). The density for the symmetric state grows in time at small times. But it does not grow unboundedly. Instead, it makes non-periodical oscillation with large amplitude as can be seen in the inset, which shows $\ds{P_{+}(t)}$ up to quite large times. The density profiles of $\Psi_n^+$ at $t=0$ (solid red) and $t=25$ (dashed red) are given in (b).[]{data-label="thflks09382d"}](SP3ek.jpeg "fig:"){width="4.2cm"} ![The total density $\ds{P_{\mp}(t)=\sum_{n} |\Psi^{\mp}(t) |^2}$ as a function of time for $\Psi^+$ (red) and $\Psi-$ (blue) in the presence of the disorder in (a). The density for the symmetric state grows in time at small times. But it does not grow unboundedly. Instead, it makes non-periodical oscillation with large amplitude as can be seen in the inset, which shows $\ds{P_{+}(t)}$ up to quite large times. The density profiles of $\Psi_n^+$ at $t=0$ (solid red) and $t=25$ (dashed red) are given in (b).[]{data-label="thflks09382d"}](SP4ek.jpeg "fig:"){width="4.25cm"}
\
To study topological edge states, consider that our system with an even number of lattice sites $\ds{N}$ is subjected to the open boundary conditions. It is well known that in the topologically nontrivial Hermitian case, $\ds{ t_1^{\prime}=t_1<t_2}$, there exists two topological zero energy eigenstates symmetrically localized around both edges. In the topologically nontrivial non-Hermitian case, the zero energy eigenstates can still appear. However, the non-reciprocity of the hopping amplitudes breaks the spatial symmetry, which implies that they are no longer symmetrically localized around the edges. In fact, not only topological eigenstates but also all of the bulk eigenstates move towards the same edge due to the non-Hermitian skin effect. Let $\ds{\psi_{1}}$ and $\ds{\psi_{2}}$ be the topological zero energy eigenstates. In Fig.\[thflks0938\](a), we plot them for the parameters $t_1=0.5$, $t_1^{\prime}=0.1$, $t_2=1$ and $N=30$. As can be seen from the figure, they are localized around the same edge. To check their topological robustness, we introduce disorder in hopping amplitudes for which the chiral symmetry of the system remains intact. The new hopping amplitudes become $\ds{t_1{\rightarrow}~t_1+{\delta}_{1,n}}$, $\ds{t_2{\rightarrow}~t_2+{\delta}_{2,n}}$ and $\ds{t_1^{\prime}{\rightarrow}~t_1^{\prime}+{\delta}_{3,n}}$, where $\delta_{1,n}$, $\delta_{2,n}$ and $\delta_{3,n}$ are site-dependent and real-valued random set of constants in the interval $(-0.1,0.1)$. Therefore, the hopping amplitudes between neighbouring sites become completely independent. In this case, the eigenstates are deformed but their eigenvalues are still equal to zero, which is a direct result of the topological protection.\
It is commonly believed that no topological state exists at either edge as a result of the non-Hermitian skin effect. This statement is true for topological eigenstates but one can still construct a topological state localized around the edge where none of the eigenstates is localized. To see this in our system, consider the following symmetric and antisymmetric superpositions of the topological eigenstates $$\label{mkxz294d}
\psi^{\mp}=P_0( \psi_{1}\mp \psi_{2} )$$ where $P_0$ is a constant and $\ds{\psi_{1}}$ and $\ds{\psi_{2}}$ are the topological zero energy eigenstates. Note that any linear combination of topological eigenstates is also a topological state since the system is linear.\
We assume that the total density is equal to one: $\ds{\sum_{n=1}^N | \psi^{\mp}|^2=1}$. We numerically find that $\ds{P_0=0.5}$ for $\ds{\psi^-}$ while $\ds{P_0=7.6 \times 10^4}$ for $\ds{\psi^+}$ for the parameters used in Fig.\[thflks0938\](a). Note this very large numerical difference between them. Such a huge difference is unique to non-Hermitian systems and has an interesting consequence as we will see below. Let us first plot the symmetric and antisymmetric states. In Fig.\[thflks0938\](b), we plot them as a function of the site number $n$. One can see that the symmetric state $\psi^+$ (in red) is localized not around the left but the right edge. This clearly shows the possibility of constructing a topological zero energy state localized around the edge where no eigenstate is localized due to the non-Hermitian skin effect. However, $P_0$ becomes very large for such a construction.\
Let us now discuss time evolution of the topological states (\[mkxz294d\]). Suppose first that there is no disorder in the system. In this case, no transition among the zero energy eigenstates occur in time and hence the density profiles of the symmetric and antisymmetric states do not change in time. It is well known that the topological zero energy eigenstates $\psi_{1,2}$ conserve their zero energy eigenvalue even in the presence of the symmetry preserving weak disorder (they are topologically protected). A question arises. Can topological protection be still observed for $\ds{ \psi^{\mp}}$? At first sight, this question seems odd because they are just linear combinations of the topological zero energy eigenstates. In Hermitian systems, such symmetric and antisymmetric states remain localized around the edges in the presence of the weak disorder and their zero energy values resist to the disorder. One may naively think that this is also true in non-Hermitian systems. But this is not the case and the picture changes drastically in the presence of even very weak disorder, which is inevitable in a real experiment. To see this clearly, let us start with the initial wave packets $\ds{\psi^{\mp}}$ and find their time evolutions $\Psi^{\mp}(t)$ ($\Psi^{\mp}(0)=\psi^{\mp}$) in the presence of the hopping amplitude disorder, which preserves the chiral symmetry. Let us define the time dependent total densities $\ds{P_{\mp}(t)=\sum_{n=1}^N |\Psi^{\mp} (t)|^2}$, where $P_{\mp}(t=0)=1$. In Fig.\[thflks09382d\](a), we plot the total densities $P_{\mp}(t)$ as a function of time. As can be seen, the total density oscillates slightly in time for the antisymmetric state (in blue) while it grows in time for the symmetric state (in red). In a non-Hermitian system with real spectrum, a linear combination of two eigenstates with different eigenvalues shows power oscillation (the total power oscillates in time) [@powosc]. This is attributed to the nonorthogonality of eigenstates. In our system, $\ds{\psi_{1}}$ and $\ds{\psi_{2}}$ have the same zero energy but some bulk states contribute perturbatively to the total wave packet in the presence of the disorder and consequently we expect slight power oscillation. Therefore, the density oscillation observed for the antisymmetric state is already expected. However, power growth of the symmetric state is quite unexpected. The Fig.\[thflks09382d\](b) shows the density profiles of the symmetric state at two different times. Surprisingly, it is no longer localized around the right edge at large times and becomes extended in time as opposed to the antisymmetric state, which remains localized around the left edge. The same disorder has little effect on one of the superposition state while it has a dramatic change on the other superposition state. This is interesting and has no analog in Hermitian systems. This leads to instability of the topological state $\psi^+$ from the dynamical perspective. This unexpected behaviour can be explained as follows. The constant $P_0$ is of the order of 4 for $\psi^+$ while it is small for $\psi^-$. The presence of the weak disorder makes perturbative changes on the eigenstates $\ds{\psi_{1}}$ and $\ds{\psi_{2}}$. This perturbative change is amplified by 4 orders of magnitude for $\psi^+$. This implies that weak disorder effectively becomes very large disorder only for the symmetric state. In other words, the symmetric topological state is extraordinarily sensitive to the noise in the system. It is well known in Hermitian topological systems that the topological robustness can be observed in the presence of symmetry protecting disorder weak enough not to close the band gap. In a similar way, we think that topological robustness for $\psi^+$ is hard to see in a real experiment because weak disorder effectively becomes very strong. In other words, $\psi^+$ is a topological state with practically missing topological robustness. We stress that the main physics is the same for the symmetric and antisymmetric states. For example, the power oscillation occurs for both of them. However, the large value of the constant $P_0$ makes the oscillation amplitude very large. In the inset of the Fig.2(a), we plot the total density up to a large time. One can see power oscillation with a large amplitude. In a typical experiment, such a large time is not practically realizable and hence one can observe power growth at small experimental times.\
The first main finding of this paper is that topological robustness can be practically lost for a superpositional topological state in non-Hermitian systems. Below, we study another interesting anomalous feature in a 2D non-Hermitian topological systems.
Broken Chirality
================
{width="5.6cm"} {width="5.6cm"} {width="5.6cm"}
{width="6cm"} {width="6cm"} \[s6saowa7c\]
{width="5.6cm"} {width="4cm"} {width="4cm"} {width="4cm"} {width="4cm"}
Consider the following non-Hermitian Chern insulator in two dimensions $$\mathcal{H} (\textbf{k})= \left( \begin{array}{cc} \Delta +\cos k_x +F_2 & A (\sin k_x -F_1 ) \\
A^{\prime} (\sin k_x +F_1 ) & - ( \Delta +\cos k_x+F_2 )
\end{array}\right) \label{yudj2}$$ where $\ds{F_1=\frac{e^{i k_y}-\beta~ e^{-i k_y}}{2}}$, $\ds{F_2=\frac{e^{i k_y}+\alpha ~e^{-i k_y}}{2}}$ and $A$, $A^{\prime}$, $\alpha$, $\beta$ and $\Delta $ are all real-valued parameters. Noe that this Hamiltonian reduces to the well-known Hermitian Chern insulator Hamiltonian when $A=A^{\prime}$ and $\alpha=\beta=1$ [@qwz]. The corresponding energy eigenvalues come in pair and are given by $\ds{E=\mp\sqrt { ( \Delta +\cos k_x+F_2 )^2-A A^{\prime} (\sin^2 k_x -F_1^2 ) }}$.\
Let us consider a finite system with open edges and study topological states. Suppose that the system is periodical along $x$-direction and has open edges along $y$-direction, which consists of $N$ sites. Since the system is translationally invariant along $x$, we can partially do Fourier transformation and the resulting Hamiltonian has discrete form indexed by the continuous parameter $k_x$. For our numerical computation, we take $A=A^{\prime}=1$, $\Delta=-1.2$, $\alpha=\beta=0.2$ and $N=24$. We find that the corresponding system has fully real spectrum. The Fig.3 (a) plots the energy spectrum as a function of $\ds{k_x}$. One can see the states in the band gap connecting the lower and upper bands across the bulk gap. They are topological states departed from nearly $k_x=\mp\pi/6$. They cross each other at $k_x = 0$ and switch places and enter the upper bulk band at the symmetrical point from where they departed. These states are edge states propagating in opposite directions (at a given $E$). So far, there is nothing interesting since a similar structure can also be seen in the Hermitian counterpart. Let us now plot the density profiles of the edge states at a given $k_x$. In the Fig. 3 (b,c), one can see that the edge states are localized around the same edge as opposed to the Hermitian system, where edge states are localized symmetrically at the opposite edges. In other words, forward and backward moving edge states (with positive and negative $k_x$ values) are localized on the same edge. This implies that the chirality is lost in our system (Fig. 4). What is special in our system is that the edge states are no longer robust against backscattering. In the Hermitian counterpart, the forward and backward moving edge states at the same energy are localized around the opposite edges. Therefore there is no available state moving in the opposite direction on the same edge. The topological states moving oppositely at the same energy are well separated so no transition occurs. This chirality of the edge states is mainly responsible for the robustness of them against backscattering in the presence of the weak disorder in Hermitian systems. But this is not the case in our system as the topological states are localized on the same edge and weak symmetry protecting disorder induces transition between forward and backward moving edge states. Consequently, the topological states have no immunity to backscattering. This is the second main finding of this paper. This poses a question of advantages of topological states in such a system.\
To make further exploration, we plot the energy spectrum when $A=A^{\prime}=1$, $\Delta=0.5$ and $\alpha=\beta=4$ in the Fig. 5.(a). We find that the corresponding spectrum is purely real. Apparently, topological states appear in the band gap for all values of $\ds{k_x}$. In this case, we can scan $k_x$ to study chirality of topological states. In Fig. 5.(b-e), we plot the density profiles for topological states for various values of $\ds{{\mp}k_x}$ with $E>0$. The topological edge state with negative $k_x$ (thick curve) is always localized around the right edge. However, the localization character of the topological state with positive $k_x$ (dashed curve) depends on $\ds{k_x}$. For example, the topological state is localized around the left (right) edge at $\ds{k_x=2\pi/3}$ ($\ds{k_x=-2\pi/3}$) as can be seen from (b). Therefore the topological edge states are chiral at this particular value of $k_x$. As $\ds{k_x}$ decreases, the topological states with positive $k_x$ are shifted towards the right edge as can be seen from (c-e). At around $\ds{k_x=\pi/3}$ (d), the topological state becomes extended all over the system (recall that it is also delocalized in $x$ direction due to the translational invariance along $x$ ) [@anaskef]. For small values of $\ds{|k_x|}$, both states are localized around the same edge as can be seen from (e) and chirality of them are broken. These show that chirality breaking can be energy dependent through the parameter $k_x$.
Conclusion
==========
In this paper, we have found two anomalous features of topological states in a non-Hermitian nonreciprocal system. Firstly, we have discussed that a superposition of topological states can be practically fragile against the disorder in a non-Hermitian system even if the topological eigenstates are robust. This superposition-induced loss of topological protection is unique to non-Hermitian systems. We further find power growth effect at small times for a superposed eigenstates in a non-Hermitian system with real spectrum. Secondly, we have shown that chirality of topological edge states can be lost in two dimensional non-Hermitian systems as a result of the non-Hermitian skin effect. This is also unique to non-Hermitian systems and can lead to the loss of robustness of topological edge states. We have discussed that chirality of topological edge states in a non-Hermitian two dimensional system can be energy dependent. We think that our findings will pave the way for the understanding of topological phase in non-Hermitian systems.
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|
---
abstract: 'We examine the mode functions of the electromagnetic field on spherically symmetric backgrounds with special attention to the subclass which allows for a foliation as open Friedmann-Lemaître (FL) spacetime. It is well-known that in certain scalar field theories on open FL background there can exist so-called supercurvature modes, their existence depending on parameters of the theory. Looking at specific open universe models, such as open inflation and the Milne Universe, we find that no supercurvature modes are present in the spectrum of the electromagnetic field. This excludes the possibility for superadiabatic evolution of cosmological magnetic fields within these models without relying on new physics or breaking the conformal invariance of electromagnetism.'
author:
- |
Julian Adamek,$^1$[^1] Claudia de Rham$^2$[^2] and Ruth Durrer$^2$[^3]\
$^1$Institut für Theoretische Physik und Astrophysik, Julius-Maximilians-Universität Würzburg,\
Emil-Fischer-Str. 31, 97074 Würzburg, Germany\
$^2$Département de Physique Théorique & Center for Astroparticle Physics, Université de Genève,\
24 Quai Ernest Ansermet, 1211 Genève 4, Switzerland
bibliography:
- 'supercurv.bib'
title: Mode Spectrum of the Electromagnetic Field in Open Universe Models
---
cosmology: theory – magnetic fields – early Universe.
Introduction
============
The generation of large scale coherent magnetic fields in the Universe which are observed in low and high redshift galaxies [@Kronberg:1993vk; @Pentericci:2000mp], clusters [@Clarke:2000bz], filaments [@Battaglia:2008ex] and even in voids [@Neronov:2010; @Taylor:2011bn], is still an unsolved problem in cosmology. Fields generated in the early Universe are generally small and, on large scales, they usually simply evolve via flux conservation, $B\propto a^{-2}$, where $a$ denotes the scale factor of cosmic expansion. One exception to this rule are helical magnetic fields which develop an inverse cascade moving power from small to larger scales [@Banerjee:2004df; @Campanelli:2007tc]. This can alleviate the problem of magnetic field generation somewhat but is still not sufficient [@Caprini:2009pr; @Durrer:2010mq].
Another idea has been put forward recently in @Barrow:2011ic: in an open universe, supercurvature modes decay slower that $1/a^2$ and can therefore remain relevant at late times. The question remains of how such supercurvature modes are generated. In this paper, we explore this proposal within explicit open universe models. Whilst open inflation is no longer the most favored model of inflation, it is the most explicit model that leads to an open universe, and we therefore start by studying the generation of supercurvature modes within that model. We show that within the Coleman-de Luccia bubble universe [@Coleman:1980aw; @Bucher:1994gb], supercurvature modes of the magnetic field are actually not part of the physical spectrum and can therefore not be generated. We show that the same results also hold for the Milne Universe. These are two explicit cases without Big Bang singularity. For them we can unambiguously specify the initial Cauchy surface needed to define the quantum vacuum and the physical spectrum. Of course we could arbitrarily pronounce any open Friedmann slicing $\{t= {\rm const.}\}$ as our initial Cauchy surface. But besides the fact that this surface does not allow supercurvature modes, such a definition is arbitrary, incomplete and not unique.
Let us first, in a brief paragraph, present the issue of supercurvature modes. The Laplacian on the spatial slices $\{ t=\mbox{const.}\}$ in an open Friedmann universe with curvature $K$ has eigenfunctions with eigenvalues $-k^2$, $$\Delta Y_k = -k^2 Y_k \,.$$ The functions with $k^2>|K|$ or, for symmetric, traceless tensors of rank $m$, with $k^2>(1+m)|K|$, form a complete set of functions on these slices which reduces to the usual Fourier modes in the limit $K\rightarrow 0$. There are, however, also so-called supercurvature modes, eigenfunctions of the Laplacian with eigenvalues in the range $0<k^2<|K|$. At first glance one might argue that, since every square integrable function can be expanded in terms of the subcurvature basis, supercurvature modes play no role. If we only consider the post-inflationary Universe, this view seems justified. However, it has been shown in @Sasaki:1994yt, that in open inflation under certain circumstances supercurvature modes can be present, see also @Lyth:1995cw and @GarciaBellido:1995wz for a discussion in a different context. The basic reason for this is that the $\{ t=\mbox{const.}\}$ slices of the post-inflationary Universe do not represent Cauchy surfaces of the entire spacetime containing the Coleman-de Luccia bubble. However, in order to discuss quantum fields and particle generation during inflation, we have to expand the fields in a complete basis on a Cauchy surface of the inflationary Universe and, as has been shown in @Sasaki:1994yt, in certain cases this can lead to the generation of modes which correspond to supercurvature modes after inflation.
In @Sasaki:1994yt the analysis is presented for scalar fields. In this paper we reduce the case of the electromagnetic field to the scalar field problem so that we can apply the results of @Sasaki:1994yt. We show that when expressing the electromagnetic field in terms of the Debye potentials [@TseChin:1973mp], these can be viewed as two conformally coupled fields for which no supercurvature modes exist. A definite statement can be made in any setup which allows for the Cauchy problem to be well-posed. On physical grounds, although there may be no unique mechanism to procure an open FL universe, we want to focus on the open inflation scenario. We think that a case study within this scenario is most useful because it is by far the most explicit and physically well-motivated setup which naturally leads to an open FL universe and, at the same time, comes with a complete description in the Cauchy sense. Other scenarios which, e.g., impose an open geometry to be realized ad hoc have to be supplemented by some arbitrary assumptions, and the question of supercurvature modes can therefore not seriously be addressed.
The open inflation scenario [@Bucher:1994gb; @GarciaBellido:1997te] was originally introduced at a time when observational data seemed to favor an open universe and it was therefore imperative to look for appropriate models. With the advent of precision measurements of the anisotropies in the cosmic microwave background, the evidence for considerable curvature to be present in our Universe has virtually evaporated [@Jaffe:2000tx; @Spergel:2003cb]. However, the scenario has recently attracted new interest in the context of eternal inflation [@Linde:1986fd; @Guth:2007ng] and the Landscape idea [@CarrWeinberg; @CarrSusskind]. From this new point of view, open inflation in the Coleman-de Luccia bubble Universe remains conceptually well-motivated, although the focus has shifted away from procuring non-vanishing curvature. In fact, the scenario allows that the curvature we observe today can be rather minuscule, see e.g. @DeSimone:2009dq for a discussion. As already pointed out, in this paper we exploit the fact that the setting contains enough information about the background spacetime such that the questions we want to study can be addressed in a meaningful way. Here we are not so much interested in the question of whether the spatial curvature of the observed Universe is negative, but we want to analyze the conceptual question of whether an open universe can allow for supercurvature modes of the electromagnetic field.
The remainder of this paper is organized as follows: in the next section we introduce the Debye potentials, write the electromagnetic Lagrangian in an open, closed or flat FL universe in these variables and derive a complete set of solutions to the Euler-Lagrange equations. We then analyze whether supercurvature modes are normalizable on a Cauchy surface, and by comparison with the pure scalar case, we conclude that no supercurvature modes are normalizable on an open de Sitter geometry. By conformal invariance we find that the same result holds for a Coleman-de Luccia bubble. Finally, for completeness, we provide the full quantization prescription of the Debye potentials, before summarizing our results. In an appendix, we present the explicit computation of the mode normalization also for the Milne model, which is one of the simplest open universe models.
General Formalism
=================
We consider background geometries of the Friedmann-Lemaître (FL) type. The line element reads $$\label{eq:metric}
ds^2~=~-dt^2 + a^2(t) \left[dr^2 + s^2(r) d\Omega^2\right]~,$$ where $s(r) = \sin r, r, \sinh r$ corresponds to closed, flat and open spatial geometry, respectively. In this work, we will focus on the latter two cases. In particular, the case $a(t) \equiv \mathrm{const.}, s(r) \equiv r$ gives the Minkowski metric, while $a(t) \equiv \sinh(H t) / H, s(r) \equiv \sinh r$ represents an open foliation of de Sitter space with $\Lambda = 3 H^2$. Note that with this convention, $r$ and $s$ have no units and spatial curvature is $K=\pm1$ or $0$, but $a$ and $t$ have units of length. As usual, we set $c = \hbar = 1$.
Because these types of backgrounds are spherically symmetric, they are appropriate for studying the electromagnetic field in terms of the Debye potentials [@TseChin:1973mp]. In this formalism, instead of making use of the usual $A^\mu$ vector potential, the electromagnetic field is decomposed into two potentials $U$ and $V$. The advantage of these Debye potentials is the fact that the equations completely decouple in any spherically symmetric background, while the components of $A^\mu$ are usually badly mixed if the spacetime is not flat. Therefore, the Debye potentials allow for even more general metrics than (\[eq:metric\]).
In equations ($4$) and ($5$) of @TseChin:1973mp, expressions for the physical electric and magnetic fields are given in terms of the Debye potentials. It will be useful for us to look at these fields in the helicity basis. Given an orthonormal basis $(\mathbf{e}_\theta, \mathbf{e}_\phi)$ on the sphere, the helicity basis reads $$\mathbf{e}_+ = \frac{1}{\sqrt{2}} \left(\mathbf{e}_\theta - i \mathbf{e}_\phi\right)~,\qquad
\mathbf{e}_- = \frac{1}{\sqrt{2}} \left(\mathbf{e}_\theta + i \mathbf{e}_\phi\right)~.$$ We find the following components of the physical electric and magnetic field in this new basis: $$\begin{aligned}
\label{eq:EB}
E_r &=& -\frac{1}{2 a s} \left(\eth\eth^\ast + \eth^\ast\eth\right) V~,\nonumber\\ B_r &=& -\frac{1}{2 a s} \left(\eth\eth^\ast + \eth^\ast\eth\right) U~,\nonumber\\
E_+ &=& -\frac{1}{\sqrt{2} a s} \left[\partial_r \left(s \eth V\right) + i \partial_t \left(a s \eth U\right)\right]~,\nonumber\\
B_+ &=& -\frac{1}{\sqrt{2} a s} \left[\partial_r \left(s \eth U\right) - i \partial_t \left(a s \eth V\right)\right]~,\nonumber\\
E_- &=& E_+^\ast~,\qquad B_- = B_+^\ast~.\end{aligned}$$ In these expressions, we make use of the *spin-raising* and *spin-lowering* operators $\eth$ and $\eth^\ast$. These are defined as $$\begin{aligned}
\eth \chi &=& -\sin^\sigma\!\theta \partial_\theta \left(\sin^{-\sigma}\!\theta \chi\right) - \frac{i}{\sin \theta} \partial_\phi \chi\nonumber\\
\eth^\ast \chi &=& -\sin^{-\sigma}\!\theta \partial_\theta \left(\sin^{\sigma}\!\theta \chi\right) + \frac{i}{\sin \theta} \partial_\phi \chi~,\end{aligned}$$ where $\sigma$ is the spin-weight of the field $\chi$. As the names imply, the spin-raising and lowering operators increase or decrease the spin-weight of a field by one unit. See, e.g., @Goldberg:1966uu for some details concerning these operators and the spherical harmonics used to expand a field of arbitrary spin-weight.
Using $F_{\mu\nu} F^{\mu\nu} = 2 \mathbf{B}^2 - 2 \mathbf{E}^2$, the Maxwell action in terms of the Debye potentials is $$\begin{gathered}
\label{eq:action}
\mathcal{S}_\mathrm{em}~=~\frac{1}{2} \int\! a^3 dt s^2 dr d\Omega \Biggl[- \frac{1}{a^2}
\partial_t \left(a \eth U\right) \partial_t \left(a \eth^\ast U\right) \Biggr.\\\Biggl.+ \frac{1}{a^2 s^2}
\left(\eth^\ast \eth U\right) \left(\eth \eth^\ast U\right) + \frac{1}{a^2 s^2}
\partial_r \left(s \eth U\right) \partial_r \left(s \eth^\ast U\right)\Biggr]\\ - \left\lbrace U \rightarrow V\right\rbrace~.\end{gathered}$$ It is evident from the action that the physical degrees of freedom are carried by the helicity-one representations $\eth U$ and $\eth V$. This is in agreement with the well-known properties of a photon. Furthermore, it is not surprising that the two helicity degrees of freedom decouple in a spherically symmetric background.
The equations of motion are[^4] $$\begin{gathered}
\label{eq:eom}
\frac{1}{a^3} \partial_t \left(a^3 \partial_t U\right) - \frac{1}{a^2} \Delta U \\+
\left(\frac{\left(\partial_t a\right)^2}{a^2} + \frac{\partial_t^2 a}{a} - \frac{\partial_r^2 s}{a^2 s}\right) U =\Box U + m^2_{\mathrm{eff}} U = 0~,\end{gathered}$$ and the same for $V$. Since $U$ and $V$ have identical properties, we will from now on only focus on $U$. All results apply to $V$ in exactly the same way. In the above expression, we have introduced the spatial Laplace operator $\Delta$ for a scalar field defined as $$\Delta~=~\frac{1}{s^2} \partial_r \left(s^2 \partial_r\right) + \frac{1}{2s^2} \left( \eth \eth^\ast +\eth^\ast \eth\right)~.$$ The operator $\Box$ in (\[eq:eom\]) is precisely the d’Alembertian for a Lorentz scalar. However, since $U$ is not itself a Lorentz scalar, it also acquires an effective mass $m_{\mathrm{eff}}$ which is precisely the one that corresponds to a conformal coupling to curvature.
In the case of spatial flatness, a complete set of eigenfunctions to $\Delta$ is given by $$\label{eq:flatmodes}
X_{p \ell m}(r, \theta, \phi)~=~p \sqrt{2 / \pi} j_\ell (p r) Y_{\ell m}(\theta, \phi)~,$$ where we have chosen the spherical Bessel functions $j_\ell$ which are regular at the origin. The eigenvalue equation on the flat three-dimensional surface is $$-\Delta X_{p \ell m}~=~p^2 X_{p \ell m}~,$$ and the eigenfunctions are normalized as $$\int\! dr r^2 d\Omega X_{p \ell m} X^\ast_{p'\ell'm'}~=~\delta(p - p') \delta_{\ell\ell'} \delta_{mm'}~.$$
In the case of an open geometry where $s = \sinh r$, the eigenfunctions are the harmonics on the three-hyperboloid. The eigenvalue equation reads $$-\Delta Y_{p \ell m}~=~\left(p^2 + 1\right) Y_{p \ell m}~,$$ where the eigenfunctions $Y_{p \ell m}$ which are regular at $r = 0$ are given by $$\begin{gathered}
\label{eq:openmodes}
Y_{p \ell m}(r, \theta, \phi)~=~f_{p\ell}(r) Y_{\ell m} (\theta, \phi)~,\\
f_{p\ell}(r) ~\equiv~\frac{\Gamma(i p + \ell + 1)}{\Gamma(i p + 1)} \frac{p}{\sqrt{\sinh r}} P_{i p - 1/2}^{-\ell - 1/2}(\cosh r)~,\end{gathered}$$ see, e.g., @Sasaki:1994yt. The normalization is again such that $$\int\! dr \sinh^2\!r d\Omega Y_{p\ell m} Y^\ast_{p'\ell'm'}~=~\delta(p - p') \delta_{\ell\ell'} \delta_{mm'}~,$$ this time on the three-hyperboloid.
The Debye potentials are conformally coupled to gravity, meaning that any conformal factor *which preserves the spherical symmetry of the geometry* can be absorbed into a redefinition of the fields. In particular this means that the equation of motion (\[eq:eom\]) is invariant under a time-dependent conformal rescaling $g_{\mu\nu} \rightarrow \omega^2(t) g_{\mu\nu}$ (with the corresponding redefinition of time) and a simple rescaling of the field as $U \rightarrow \omega^{-1}(t) U$.
No Supercurvature Modes {#sec:nosupercurv}
=======================
![\[fig:pcd\] Spacetime diagram of a one-bubble geometry which may be the result of a Coleman-de Luccia process. If one can neglect the geometric effect of the bubble (indicated as the white region), the spacetime is approximately de Sitter. There always exists a conformal map, given as a finite conformal factor, between the $O(3,1)$-symmetric one-bubble spacetime and the $O(4,1)$-symmetric de Sitter space. Region I resembles an open FL universe but does not contain any global Cauchy surface. Such a surface is indicated as $\Sigma_0$, which is entirely contained in region II. Region III is another open FL universe, similar to, but causally disconnected from region I. Some surfaces of constant radial or constant time coordinate are indicated as dashed lines.](pcd.eps){width="65mm"}
A special situation is given in the open universe models because the three-hyperboloids used in the foliation do not usually represent global Cauchy surfaces. Therefore, the failure of modes to be normalizable on the hyperboloids does not necessarily imply that they should be excluded from the physical spectrum. What really matters is the question of whether or not a mode is normalizable on a Cauchy surface. It is well-known that in certain scalar field models this leads to the occurrence of modes with discrete imaginary values of $p$ in the spectrum which are usually referred to as *supercurvature modes*.
In the case of the magnetic field, it was found in @Barrow:2011ic – see also references therein to earlier work, e.g. @Barrow:2008jp [@Tsagas:2005nn] – that supercurvature modes, if they exist, give rise to superadiabatic evolution and can therefore help to solve the problem of magnetogenesis. It is therefore of relevance whether or not the electromagnetic field can support supercurvature modes. With the formalism of the Debye potentials, we can now easily address this question.
In order to study whether such supercurvature modes are relevant, we have to check if there are some modes with imaginary $p$ which are normalizable on a Cauchy surface. As mentioned before, such a surface can usually not be found within the patch covered by the open coordinate chart. One therefore has to complete this chart, which means that one usually has to continue the coordinates across the initial singularity of the open chart (which is a coordinate singularity). As a specific example which is general enough, one can consider the creation of an open universe by the Coleman-de Luccia process [@Coleman:1980aw], as in the open inflation scenario [@Bucher:1994gb; @GarciaBellido:1997te]. A spacetime diagram is shown in figure \[fig:pcd\]. In this type of model, the open region (indicated as region I in figure \[fig:pcd\]) is contained fully within the lightcone of the nucleation event of a bubble which was created by a vacuum metastability transition. However, the entire one-bubble spacetime can easily be constructed from the instanton which is responsible for the transition. A Cauchy surface is then given, e.g., by the maximal three-section of the instanton, which represents the so-called turning-point geometry [@Coleman:1980aw]. It is located along the horizontal line indicated as $\Sigma_0$ in the figure. Any time-evolution of this surface is, of course, equally suitable.
In order to make the problem tractable analytically, let us ignore for a moment the geometric effects of the bubble altogether. That is, we consider an exact de Sitter geometry with $a(t) \equiv \sinh(H t) / H, s(r) \equiv \sinh r$. For de Sitter space, the question of supercurvature modes (in a scalar field setting) has been thoroughly studied in @Sasaki:1994yt. It turns out that their analysis can be easily applied to the present setup. Comparing our eq. (\[eq:eom\]) with the equation (2.7) of @Sasaki:1994yt, it is evident that the $\nu$-parameter is the one of the conformally coupled field. This should, of course, not come as a surprise since the electromagnetic field is conformally coupled. The analysis then proceeds with the calculation of the normalization of the modes on a Cauchy surface. While @Sasaki:1994yt work in de Sitter space, we will present this calculation for an even simpler toy model of an open FL universe, the Milne model, as pedagogical example in the appendix.
It is shown in @Sasaki:1994yt that *no supercurvature modes exist* in the conformally coupled case. By re-applying exactly the same arguments we can therefore conclude that there are no supercurvature modes for the Debye potentials as well. The result, so far, holds for the case of an exact open de Sitter background. However, one can show that it rigorously holds also for any one-bubble geometry like the ones produced in an arbitrary Coleman-de Luccia process. This can be seen by noting two facts. Firstly, the radial coordinate on the Cauchy surface corresponds to the analytic continuation of the time coordinate of the open chart. The continuation of the scale factor $a$ into the Euclidean domain therefore characterizes the geometric effects of the bubble. It is the behavior in Euclidean time which determines the normalizability of a mode. Secondly, we note that any bubble geometry can be mapped onto an exact de Sitter geometry by a finite conformal factor, which may depend on the radial coordinate on the Cauchy surface. However, we already pointed out that the mode equation is invariant under such a conformal transformation. In particular, the normalizability of a mode is not affected by any finite conformal rescaling. In fact this means that any $O(3, 1)$-symmetric geometry has the same spectrum of modes for the electromagnetic field. The non-existence of electromagnetic supercurvature modes then follows as a corollary from @Sasaki:1994yt.
Since the formalism we apply here is very different, it is worthwhile to explain the connection to @Barrow:2011ic in some more detail. Given our expression (\[eq:EB\]) for the magnetic field and using the mode expansion of eq. (\[eq:openmodes\]) for $U$ and $V$, one can show that the covariant three-dimensional (spatial) vector Laplacian acting on a magnetic mode with wavenumber $p$ yields $$-\Delta \mathbf{B}_{\left(p\right)}
= \frac{p^2 + 2}{a^2} \mathbf{B}_{\left(p\right)}~.$$ A comparison with eq. ($7$) of @Barrow:2011ic (see also their footnote $6$) then clarifies the relation between our wavenumber $p$ and their eigenvalue parameterization $n$. The superadiabatic modes with eigenvalues $n^2 < 2$ correspond to imaginary wavenumbers $p$. We have just shown that these modes are not included in the spectrum in any one-bubble open universe scenario.
Quantum Theory
==============
A canonical quantization prescription for the Debye potentials works as follows. First, we note that the true physical degrees of freedom which should be quantized are given by $\eth U$ and $\eth V$. Then, it is advised to rescale the fields such that the Hubble damping term in the mode equation disappears. To this end, we write $\eth U \equiv \upsilon / a$ and choose a conformal time coordinate defined by $dt = a~d\tau$. The action for the rescaled field $\upsilon$ reads $$\begin{gathered}
\mathcal{S}_\upsilon~=~\frac{1}{2} \int\!d\tau s^2 dr d\Omega \Bigl[-\partial_\tau \upsilon \partial_\tau \upsilon^\ast + \frac{1}{s^2}
\eth^\ast \upsilon \eth \upsilon^\ast\Bigr.\\ \Bigl. +\frac{1}{s^2} \partial_r \left(s \upsilon\right) \partial_r \left(s
\upsilon^\ast\right)\Bigr]\,.\end{gathered}$$ Following the usual rules of canonical quantization, the field $\upsilon$ is promoted to an operator $\hat{\upsilon}$ and can be expanded in terms of creation and annihilation operators of modes by writing $$\begin{gathered}
\hat{\upsilon} (\tau, r, \theta, \phi)~=~\int\!dp \sum_{\ell m} \frac{1}{\sqrt{2}}\Bigl[\hat{\mathrm{a}}_{p \ell m} \upsilon_p (\tau)
{}_{1\!}Y_{p \ell m}(r, \theta, \phi)\Bigr. \\ \Bigl. + \hat{\mathrm{a}}^\dagger_{p \ell m} \upsilon^\ast_p (\tau)
{}^{\phantom\ast\!}_{1\!}Y^\ast_{p \ell m}(r, \theta, \phi)\Bigr]~.\end{gathered}$$ Note that the field $\upsilon$ is of spin-weight one and should therefore be expanded in terms of the appropriate spherical harmonics. The eigenfunctions ${}_{1\!}Y_{p \ell m}$ of the spatial Laplace operator are the ones of eq. (\[eq:openmodes\]) with $Y_{\ell m}$ replaced by the corresponding spherical harmonic of spin-weight one, ${}_{1\!}Y_{\ell m}$. In case of spatial flatness, the eigenfunctions ${}_{1\!}Y_{p \ell m}$ have to be replaced by ${}_{1\!}X_{p \ell m}$, which are related to eq. (\[eq:flatmodes\]) in a similar way. In both cases, the mode functions $\upsilon_p(\tau)$ are governed by the mode equation $$\partial_\tau^2 \upsilon_p +p^2 \upsilon_p~=~0~,$$ and it is allowed to choose them as independent of $\ell$ and $m$. If one uses normalized mode functions $$\mathrm{Im} (\upsilon_p \partial_\tau \upsilon^\ast_p)~=~1~,$$ then the equal time commutator of field and canonical momentum, $$\begin{gathered}
\Bigl[\hat{\upsilon}(\tau, r, \theta, \phi), \partial_\tau \hat{\upsilon}^\ast(\tau, r', \theta', \phi')\Bigr]~=\\ i \frac{1}{s^2}
\delta(r - r') \delta(\cos\theta - \cos\theta') \delta(\phi - \phi')~,\end{gathered}$$ is equivalent to the standard commutation rules for the creation and annihilation operators, $$\left[\hat{\mathrm{a}}_{p \ell m}, \hat{\mathrm{a}}^\dagger_{p'\ell'm'}\right]~=~\delta(p - p') \delta_{\ell\ell'} \delta_{mm'}~.$$
The reader may wonder what would have been the difference if one had quantized the scalar Debye potentials directly instead of the helicity-one degrees of freedom which we obtained by applying a spin-raising operator. Firstly, by choosing to quantize the latter, we have avoided a quantization of the unphysical gauge modes with $\ell = 0$ which are present in the expansion of a scalar but not in the one of the helicity-one fields. Secondly, by noting that $\eth Y_{\ell m} = \sqrt{\ell \left(\ell + 1\right)} {}_{1\!}Y_{\ell m}$, one can see that some $\ell$-dependent factors may appear in the commutation rules for the scalar Debye potentials. A careful look at the action reveals that the scalar modes are not canonically normalized and that these factors are therefore expected. These differences, however, are completely inessential for the question of whether or not any supercurvature modes are part of the spectrum. In particular, the argument of section \[sec:nosupercurv\] works equally well for the helicity-one degrees of freedom, with rather obvious modifications when going through the detailed proof.
The standard choice of positive frequency mode functions is like in Minkowski space, $$\upsilon_p(\tau)~=~\frac{1}{\sqrt{p}} e^{-i p \tau}~,\qquad\qquad\text{(Minkowski)} \,.$$ This is not surprising as the rescaled electromagnetic fields, $B/a^2$ and $E/a^2$ are independent of the scale factor in conformal time. One can verify that some standard results of quantized electromagnetism are reproduced. We checked this for the vacuum two-point correlators $\langle E_a(t, r, \theta, \phi) E_b(t', r', \theta', \phi')\rangle$, which turn out to be the same as if obtained from a standard quantization of $A^\mu$ in Minkowski space.
However, the above quantization prescription is general enough to be applicable also in arbitrary flat or open FL backgrounds. For instance, one could obtain the primordial power spectrum of the electromagnetic field in the open inflation scenario. Since we have proven that no supercurvature modes are present, the result of this computation is, however, of pure academic interest, because one does not expect that significant perturbation amplitudes can be obtained after the inflationary era.
Conclusions
===========
In this letter we have studied the modes of the quantum vacuum of the electromagnetic field in an open, inflating Friedmann universe. Whilst subcurvature modes decay extremely fast after their generation, supercurvature modes, on the other hand, could remain relevant at the end of inflation and could then have a significant impact on the origin of large scale magnetic fields in the present Universe. Great care should therefore be taken in understanding under which circumstances supercurvature modes are expected to belong to the magnetic field spectrum. Here we have explored the eigenmodes of the electromagnetic field in an open universe and we have shown that supercurvature modes are not expected to be produced via any causal process, such as if the open universe is generated by bubble nucleation. This is a consequence of the conformal coupling of electromagnetism.
If one switches on some perturbative interaction later during inflation, conformal invariance may be broken and electromagnetic modes may be generated. But only modes which are present in the quantum vacuum can be excited by such a perturbative coupling, namely subcurvature modes.
We have focused on the open inflation scenario with the remark that it presently is the only physically motivated scenario which procures an open Friedmann universe and offers a complete enough framework to address the question of supercurvature modes. It also represents the scenario preferred by recent considerations of eternal inflation and the Landscape of string theory. Our argument shows already that no supercurvature modes can exist in all cases where the global spacetime carries the same $O(3, 1)$-symmetry as is displayed by the open patch. In less specific settings which are deprived of a global description, the question may be elusive, but the experience with the open inflation scenario teaches us that the physical viability of supercurvature modes in the context of standard electromagnetism has yet to be demonstrated. In a singular open Friedmann universe this question cannot be seriously addressed, and when addressed naively, choosing $\{t=\rm{const.}\}$ hypersurfaces, again no supercurvature modes are contained in the physical spectrum.
In view of this result, it appears more and more unlikely that standard electromagnetism can support any superadiabatic evolution of cosmological magnetic fields under physical conditions. Such an evolution can be obtained only by breaking the conformal invariance of electromagnetism (already at the time of bubble nucleation) or by introducing some other kind of new physics.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Misao Sasaki for interesting comments. JA wants to thank the University of Geneva for hospitality and the German Research Foundation (DFG) for financial support through the Research Training Group 1147 “Theoretical Astrophysics and Particle Physics.” CdR and RD are supported by the Swiss National Science Foundation.
Mode Spectrum in the Milne Universe
===================================
We present here the explicit computation of the mode spectrum in the Milne model, which is one of the simplest open FL geometries. It is obtained by rewriting the line element of Minkowski space as $$ds^2 = -dT^2 + dR^2 + R^2 d\Omega^2 = -dt^2 + t^2 \left[dr^2 + \sinh^2 r d\Omega^2\right]~.$$ While $-\infty < T < \infty$ and $0 \leq R < \infty$ are the coordinates of a standard (spatially flat) spherical coordinate system which covers the full Minkowski spacetime, using $0 < t < \infty$ and $0 \leq r < \infty$ one obtains a metric of the open FL type with $a = t$, cf. eq. (\[eq:metric\]). This new coordinate system, with $T=t\cosh r$ and $R=t\sinh r$, covers the interior of the future lightcone of $T = R = 0$. Within this very simple setting which yet has all the desired features, we want now to exemplify the reasoning of section \[sec:nosupercurv\].
As a first step, for the mode expansion of eq. (\[eq:openmodes\]) we can immediately solve the mode equation. The solutions to eq. (\[eq:eom\]) take the form $$U_{p \ell m \pm}(t, r, \theta, \phi) = N_{p\pm} t^{-1 \pm i p} Y_{p \ell m}(r, \theta, \phi)~,$$ where $N_{p\pm}$ is a normalization to be determined. To this end, we want to evaluate the Klein-Gordon inner product on a Cauchy surface. The whole point is that the open spatial hypersurfaces $\lbrace t = \mathrm{const.}\rbrace$ do not represent proper Cauchy surfaces and one should therefore make a better choice. We choose the surface $\lbrace T = 0\rbrace$ which is a proper global section of Minkowski space. This hypersurface lies entirely outside the coordinate patch covered by $t, r$, however, by making appropriate analytic continuations, we can complete the chart to include $\lbrace T = 0\rbrace$. More precisely, by taking $t \rightarrow i \rho$, $r \rightarrow \tau - i \pi / 2$, the region outside the lightcone is covered by $-\infty < \tau < \infty$ and $0 < \rho < \infty$. Furthermore, the hypersurface $\lbrace T = 0\rbrace$ coincides with the one defined by $\lbrace \tau = 0\rbrace$. The line element is given as $$ds^2 = d\rho^2 - \rho^2 d\tau^2 + \rho^2 \cosh^2\tau d\Omega^2~.$$
It is noteworthy that the role of time and radial distance have been interchanged by the analytic continuation, just as it was done by @Sasaki:1994yt for the case of de Sitter. The Klein-Gordon inner product is finally given by $$\begin{gathered}
\label{eq:KGproduct}
\langle U_{p \ell m \pm}, U_{p' \ell' m' \pm} \rangle_{\mathrm{K-G}} = i\! \int\limits_{T = 0}\!dR R^2 d\Omega U_{p \ell m \pm}^\ast \overset{\leftrightarrow}{\partial_T} U_{p' \ell' m' \pm}\\
= i \delta_{\ell\ell'} \delta_{mm'} N^\ast_{p\pm} N_{p'\pm} e^{\mp\left(p + p'\right) \pi / 2} \int\limits_0^\infty \frac{d\rho}{\rho} \rho^{\mp i \left(p - p'\right)} \\\times \cosh^2 \tau \left. f_{p\ell}^\ast \overset{\leftrightarrow}{\partial_\tau} f_{p'\ell'} \right|_{\tau = 0}\end{gathered}$$ By making a change of variables to $\ln \rho$, one can see that the $\rho-$integral is a representation of the delta function $\delta(p - p')$ for $p, p'$ real. For any imaginary $p$ or $p'$, the integral is badly divergent, which implies that the modes with imaginary $p$ have zero norm. In other words, there are no supercurvature modes of the Debye potentials in the Milne model.
For the regular modes with real values of $p$, the term printed in the last line of eq. (\[eq:KGproduct\]) can be easily evaluated once setting $p = p'$ and $\ell = \ell'$. One obtains $$\cosh^2 \tau \left. f_{p\ell}^\ast \overset{\leftrightarrow}{\partial_\tau} f_{p\ell} \right|_{\tau = 0} = -\frac{2 i p}{\pi} \sinh \pi p~.$$
[^1]: jadamek@physik.uni-wuerzburg.de
[^2]: claudia.derham@unige.ch
[^3]: ruth.durrer@unige.ch
[^4]: As in eqs. ($6$) and ($7$) of @TseChin:1973mp, we have omitted an overall spherical Laplacian $(\eth \eth^\ast + \eth^\ast \eth) / 2$ acting on the equation. The solutions are identical up to modes which are annihilated by this operator. These are exactly the modes of $U$ with zero angular momentum ($\ell = 0$). Noting that these modes are already annihilated by $\eth$ and $\eth^\ast$ individually, it is evident from inspecting eq. (\[eq:EB\]) that they are pure gauge modes which do not contribute to the physical electromagnetic field. Note also that with this operator, the equations of motion would appear to be fourth order in the angular coordinates. The equations for the true physical degrees of freedom $\eth U$ and $\eth V$, however, would remain second order in all coordinates.
|
---
abstract: 'We prove the second law of thermodynamics and the nonequilibirum fluctuation theorem for pure quantum states. The entire system obeys reversible unitary dynamics, where the initial state of the heat bath is not the canonical distribution but is a single energy-eigenstate that satisfies the eigenstate-thermalization hypothesis (ETH). Our result is mathematically rigorous and based on the Lieb-Robinson bound, which gives the upper bound of the velocity of information propagation in many-body quantum systems. The entanglement entropy of a subsystem is shown connected to thermodynamic heat, highlighting the foundation of the information-thermodynamics link. We confirmed our theory by numerical simulation of hard-core bosons, and observed dynamical crossover from thermal fluctuations to bare quantum fluctuations. Our result reveals a universal scenario that the second law emerges from quantum mechanics, and can experimentally be tested by artificial isolated quantum systems such as ultracold atoms.'
author:
- Eiki Iyoda
- Kazuya Kaneko
- Takahiro Sagawa
title: 'Fluctuation Theorem for Many-Body Pure Quantum States'
---
*Introduction.* Although the microscopic laws of physics do not prefer a particular direction of time, the macroscopic world exhibits inevitable irreversibility represented by the second law of thermodynamics. Modern researches has revealed that even a pure quantum state, described by a single wave function without any genuine thermal fluctuation, can relax to macroscopic thermal equilibrium by a reversible unitary evolution [@Neumann1929; @Jensen1985; @Tasaki1998; @Reimann2008; @Rigol2008; @Linden2009; @Polkovnikov2011; @Linden2012; @Goldstein2013; @Gogolin2016; @Tasaki2016Typ]. Thermalization of isolated quantum systems, which is relevant to the zeroth law of thermodynamics, is now a very active area of researches in theory [@Neumann1929; @Tasaki1998; @Reimann2008; @Linden2009; @Linden2012; @Goldstein2013], numerics [@Jensen1985; @Rigol2007; @Rigol2008; @Biroli2010; @Cassidy2011; @Calabrese2011; @Mallayya2017], and experiments [@Kinoshita2006; @Hofferberth2007; @Gring2012; @Trotzky2012; @Langen2015; @Clos2016; @Kaufman2016]. Especially, the concepts of typicality [@Popescu2006; @Goldstein2006; @Sugita2007; @Tasaki2016Typ] and the eigenstate thermalization hypothesis (ETH) [@Berry1977; @Peres1984; @Jensen1985; @Srednicki1994; @Rigol2008; @Rigol2009; @Biroli2010; @Kim2014; @Alba2015; @Beugeling2014; @Garrison2015; @DAlessio2016; @Mori2016] have played significant roles.
However, the second law of thermodynamics, which states that the thermodynamic entropy increases in isolated systems, has not been fully addressed in this context. We would emphasize that the informational entropy (i.e., the von Neumann entropy) of such a genuine quantum system never increases, but is always zero [@Nielsen2000]. In this sense, a fundamental gap between the microscopic and macroscopic worlds has not yet been bridged: How does the second law emerge from pure quantum states?
In a rather different context, a general theory of the second law and its connection to information has recently been developed even out of equilibrium [@Parrondo2015; @Sagawa2008; @Sagawa2010; @Funo2013], which has also been experimentally verified in laboratories [@Toyabe2010; @Berut2012; @Koski2014; @Vidrighin2016]. This has revealed that information contents and thermodynamic quantities can be treated on an equal footing, as originally illustrated by Szilard and Landauer in the context of Maxwell’s demon [@Landauer1961; @Leff2003]. This research direction invokes a crucial assumption that the heat bath is, at least in the initial time, in the canonical distribution [@Sagawa2012]; this special initial condition effectively breaks the time-reversal symmetry and leads to the second law of thermodynamics. The same assumption has been employed in various modern researches on thermodynamics, such as the nonequilibrium fluctuation theorem [@Jarzynski1997; @Crooks1999; @Jarzynski2000; @Kurchan2000; @Tasaki2000FT; @Esposito2009; @Campisi2011; @Sagawa2012; @Collin2005; @Batalhao2015; @An2015] and the thermodynamic resource theory [@Horodecki2013; @Brandao2015].
Based on these streams of researches, in this Letter we rigorously derive the second law of thermodynamics for isolated quantum systems in pure states. We consider a small system and a large heat bath, where the bath is initially in a single energy-eigenstate. Such an eigenstate is a pure quantum state, and does not include any statistical mixture as is the case for the canonical distribution. The second law that we show here is formulated with the von Neumann entropy of the system, ensuring the information-thermodynamics link, which is a characteristic of our study in contrast to previous approaches [@Tasaki2000; @Ikeda2015; @DAlessio2016]. Furthermore, we prove the integral fluctuation theorem [@Jarzynski1997; @Tasaki2000FT; @Esposito2009; @Jin2016], a universal relation in nonequilibrium statistical mechanics, which expresses the second law as an equality rather than an inequality.
The key of our theory is combining the following two fundamental concepts. One is the Lieb-Robinson bound [@Lieb1972; @Hastings2006], which characterizes the finite group velocity of information propagation in quantum many-body systems with local interaction. The other is the ETH, which states that even a single energy-eigenstate can behave as thermal [@Berry1977; @Peres1984; @Jensen1985; @Srednicki1994; @Rigol2008; @Rigol2009; @Biroli2010; @Kim2014; @Alba2015; @Beugeling2014; @Garrison2015; @DAlessio2016; @Mori2016]. In this Letter, we newly prove a variant of the ETH [@Biroli2010; @Mori2016], which is referred to as the [*w*eak]{} ETH and states that most of the energy eigenstate satisfies the ETH, if an eigenstate is randomly sampled from the microcanonical energy shell.
Our theory provides a rigorous scenario of the emergence of the second law from quantum mechanics, which is relevant to experiments of artificial isolated quantum systems. Furthermore, our approach to the second law would be applicable to quite a broad class of modern researches of thermodynamics, from thermalization in ultracold atoms [@Trotzky2012] to scrambling in black holes [@Hayden2007; @Sekino2008; @Maldacena2015; @Maldacena2016].
*Setup.* We first formulate our setup with a heat bath in a pure state. Suppose that the entire system consists of system S and bath B. We assume that bath B is a quantum many-body system on a $d$-dimensional hypercubic lattice with $N$ sites. The Hamiltonian is given by $$\hat{H}=\hat{H}_{{\mathrm{S}}}+\hat{H}_{{\mathrm{I}}}+\hat{H}_{{\mathrm{B}}},
\label{Main_total_Hamiltonian}$$ where $\hat{H}_{{\mathrm{S}}}$ and $\hat{H}_{{\mathrm{B}}}$ are respectively the Hamiltonians of system S and bath B, and $\hat{H}_{{\mathrm{I}}}$ represents their interaction. We assume that $\hat{H}_{{\mathrm{B}}}$ is translation invariant with local interaction, and system S is locally in contact with bath B (see Fig. 1(a)). We also assume that the correlation functions in the canonical distribution with respect to $\hat{H}_{{\mathrm{B}}}$ is exponentially decaying for any local observables, which implies that bath B is not on a critical point.
The initial state of the total system is given by $$\hat{\rho}(0) =\hat \rho_{\rm S} (0) \otimes | E_i \rangle \langle E_i |,
\label{Main_initial}$$ where $\hat \rho_{\rm S} (0)$ is the initial density operator of system S, and $| E_i \rangle$ is the initial energy eigenstate of bath B. We sample $| E_i \rangle$ from the set of the energy eigenstates in the microcanonical energy shell in a uniformly random way, as will be described in detail later. We can then define temperature $T$ of $| E_i \rangle$ as the temperature of the corresponding energy shell. We note that the initial correlation between S and B is assumed to be zero.
The total system then obeys a unitary time evolution by the Hamiltonian: $\hat{\rho}(t)=\hat{U}\hat{\rho}(0)\hat{U}^\dag$ with $\hat U:= \exp (- {\rm i} \hat H t / \hbar)$. Such a situation can experimentally be realized with ultracold atoms by quenching an external potential at time $0$. Let $\hat{\rho}_{{\mathrm{S}}}(t):={{\mathrm{tr}}}_{{\mathrm{B}}}{\left[\hat{\rho}(t)\right]}$ and $\hat{\rho}_{{\mathrm{B}}}(t):={{\mathrm{tr}}}_{{\mathrm{S}}}{\left[\hat{\rho}(t)\right]}$ be the density operators of system S and bath B at time $t$, respectively. The change in the von Neumann entropy of S is given by $\Delta S_{{\mathrm{S}}}:=S_{{\mathrm{S}}}(t)-S_{{\mathrm{S}}}(0)$ with $S_{{\mathrm{S}}}(t) :=-{{\mathrm{tr}}}_{{\mathrm{S}}}{\left[\hat{\rho}_{{\mathrm{S}}}(t) \ln \hat{\rho}_{{\mathrm{S}}}(t) \right]}$. We also define the heat emitted from bath B by $Q:=
-\mathrm{tr}_{{\mathrm{B}}}[
\hat{H}_\mathrm{B}(
\hat{\rho}_\mathrm{B}(t)
-
\hat{\rho}_\mathrm{B}(0)
)
]$.
If the initial state of system S is pure (i.e., $\hat \rho_{\rm S} (0) = | \psi \rangle \langle \psi |$), the total system is also pure, whose von Neumann entropy vanishes. In such a case, the final state $\hat \rho (t)$ remains in a pure state because of the unitarity, but is entangled. Correspondingly, the final state of S is mixed and has non-zero von Neumann entropy, which is regarded as the entanglement entropy.
*Second law.* We now discuss our first main result. If $| E_i \rangle$ is a typical energy eigenstate, that satisfies the ETH, the second law of thermodynamics is shown to hold within a small error: $$\begin{aligned}
\Delta S_{{\mathrm{S}}}- \beta Q \geq - \varepsilon_{\mathrm{2nd}},
\label{Main_Clausius}\end{aligned}$$ where $\varepsilon_\mathrm{2nd}$ is a positive error term. We can rigorously prove that $\varepsilon_\mathrm{2nd}$ can be arbitrarily small if bath B is sufficiently large. The left-hand side of inequality (\[Main\_Clausius\]) is regarded as the average entropy production $\langle \sigma \rangle :=\Delta S_{{\mathrm{S}}}- \beta Q$, where $\langle \cdots \rangle$ describes the ensemble average, and $\sigma$ is the stochastic entropy production that will be introduced later. We note that, if the initial state of bath B is not pure but in the canonical distribution $\hat{\rho}_{{\mathrm{B}}}^{\mathrm{can}}:=e^{-\beta\hat{H}_{{\mathrm{B}}}}/\mathrm{tr}[e^{-\beta\hat{H}_{{\mathrm{B}}}}]$, inequality (\[Main\_Clausius\]) exactly holds without any error [@Sagawa2012].
The second law (\[Main\_Clausius\]) implies that the information-thermodynamics link emerges in genuine quantum systems, if we look at the informational entropy of subsystem S, though that of the total system remains unchanged. A significant consequence of inequality (\[Main\_Clausius\]) is the Landauer erasure principle [@Landauer1961]. Suppose that the initial state of S stores one bit of information such that $S_{{\mathrm{S}}}(0) = \ln 2$, and it is erased in the final state: $S_{\rm S} (t) = 0$. We then have $\Delta S_{\rm S} = - \ln 2$, and the heat emission from S, represented by $-Q$, is bounded by $k_{\rm B}T \ln 2$ within a small error. While the Landauer principle and its generalizations have been derived in various ways [@Parrondo2015; @Sagawa2014; @Shizume1995; @Piechocinska2000; @Esposito2011; @Sagawa2011; @Reeb2014], we here showed that it emerges in the presence of a pure quantum bath.
We will prove inequality (\[Main\_Clausius\]) in Supplemental Material in a mathematically rigorous way. Here We only discuss the essentials of the proof, where the key ingredients are the Lieb-Robinson bound [@Lieb1972; @Hastings2006] and the weak ETH [@Biroli2010; @Mori2016].
*Lieb-Robinson bound.* The Lieb-Robinson bound gives an upper bound of the velocity of information propagation, and is applicable to any system on a generic lattice with local interaction. To apply the Lieb-Robinson bound, we divide bath B into ${{\mathrm{B}}}_1$ and ${{\mathrm{B}}}_2$, such that ${{\mathrm{B}}}_1$ is near system S and ${{\mathrm{B}}}_2$ is far from S (see Fig. 1(b)). Then, the Lieb-Robinson bound [@Lieb1972; @Hastings2006] sets the shortest time $\tau$, at which information about ${{\mathrm{B}}}_2$ reaches S across ${{\mathrm{B}}}_1$. We refer to $\tau$ as the Lieb-Robinson time.
The detailed formulation of the Lieb-Robinson bound is the following. Let $\tilde{{{\mathrm{S}}}}$ be the union of S and the support of $\hat{H}_{{\mathrm{I}}}$. Let $\hat{A}_\mathrm{\tilde{S}}$ and $\hat{B}_{\partial {{\mathrm{B}}}_1}$ be arbitrary operators with the supports $\tilde{{{\mathrm{S}}}}$ and $\partial {{\mathrm{B}}}_1$, respectively, where $\partial {{\mathrm{B}}}_1$ is the boundary between ${{\mathrm{B}}}_1$ and ${{\mathrm{B}}}_2$. Let $\mathrm{dist}(\mathrm{\tilde{S}},\partial\mathrm{B}_1)$ be the spatial distance between $\tilde{{{\mathrm{S}}}}$ and $\partial {{\mathrm{B}}}_1$ on the lattice, and let $| \rm \tilde{S} |$ and $| \partial {{\mathrm{B}}}_1 |$ be the numbers of the sites in $\tilde{{{\mathrm{S}}}}$ and $\partial {{\mathrm{B}}}_1$ respectively. The Lieb-Robinson bound is formulated in terms of the operator norm $\| \cdot \| $ as $$\begin{aligned}
\label{Main_eq:LRB}
\frac{
\|
[
\hat{A}_\mathrm{\tilde{S}}(t),\hat{B}_{\partial {{\mathrm{B}}}_1}
]
\|
}
{
\|\hat{A}_\mathrm{\tilde{S}}\|
\|\hat{B}_{\partial {{\mathrm{B}}}_1}\|
}
\leq
C
|\tilde{{{\mathrm{S}}}}||\partial {{\mathrm{B}}}_1|
e^{-\mu\mathrm{dist}(\tilde{{{\mathrm{S}}}}, \partial {{\mathrm{B}}}_1)}
(e^{v|t|}-1),\end{aligned}$$ where $\hat{A}_\mathrm{\tilde{S}}(t):=\hat{U}^\dag\hat{A}_\mathrm{\tilde{S}}\hat{U}$ represents the time evolution in the Heisenberg picture, and $C$, $v$, $\mu$ are positive constants. In particular, $v/\mu$ represents the velocity of information propagation. The Lieb-Robinson time is then given by $\tau : = \mu {{\mathrm{dist}}}(\tilde{S},\partial{{\mathrm{B}}}_1) / v$.
*Weak ETH.* We next consider the concept of the weak ETH. Let $D$ be the dimension of the Hilbert space of the microcanonical energy shell of bath B, which is exponentially large with respect to $N$, and let $\{ | E_i \rangle \}_{i=1}^D$ be an orthonormal set of the energy eigenstates of $\hat H_{{\mathrm{B}}}$ in the energy shell. Suppose that we choose $| E_i \rangle$ from $\{ | E_i \rangle \}_{i=1}^D$ in a uniformly random way. As proved in Supplemental Material, if ${{\mathrm{B}}}_2$ is sufficiently larger than ${{\mathrm{B}}}_1$, we typically have that $${{\mathrm{tr}}}_{{{\mathrm{B}}}}{\left[\hat{O}_{{{\mathrm{B}}}_1}| E_i \rangle \langle E_i | \right]}
\simeq
{{\mathrm{tr}}}_{{{\mathrm{B}}}}{\left[\hat{O}_{{{\mathrm{B}}}_1}\hat{\rho}_{{\mathrm{B}}}^{\mathrm{can}}\right]},
\label{Main_wETH}$$ which implies that $| E_i \rangle$ is indistinguishable from $\hat{\rho}_{{\mathrm{B}}}^{\mathrm{can}}$ if we only look at any operator $\hat{O}_{{{\mathrm{B}}}_1}$ on ${{\mathrm{B}}}_1$ with $\|\hat{O}_{{{\mathrm{B}}}_1}\|=1$. We refer to this theorem as the weak ETH, which is a variant of a theorem shown in Ref. [@Biroli2010; @Mori2016]. We note that the equivalence of the canonical and the microcanonical ensembles for reduced density operators [@Tasaki2016; @Brandao2015EOE] plays an important role here.
We note that the weak ETH is true even if bath B is an integrable system [@Kim2014; @Alba2015]. However, it has been shown that atypical states that do not satisfy Eq. (\[Main\_wETH\]) have large weights after quantum quench in the case of integrable systems [@Biroli2010], and the steady values of macroscopic observables are consistent with the generalized Gibbs ensemble (GGE) [@Rigol2006; @Rigol2007; @Vidmar2016] rather than the microcanonical ensemble. In this sense, the weak ETH is physically significant only for nonintegrable systems, though it is mathematically true even for integrable systems. The reason why the weak ETH is called “weak” is that there is another concept called the “strong” ETH (or just ETH) that is believed to be true only for nonintegrable systems, where every energy eigenstate satisfies Eq. (\[Main\_wETH\]) without exception [@Kim2014].
*Outline of the proof of (\[Main\_Clausius\]).* We are now in the position to discuss the outline of the proof of the second law (\[Main\_Clausius\]). In the short time regime $t\ll\tau$, system S cannot feel the existence of ${{\mathrm{B}}}_2$, and the heat bath effectively reduces to ${{\mathrm{B}}}_1$. From the weak ETH, if ${{\mathrm{B}}}_2$ is sufficiently larger, the initial state $|E_i\rangle$ of bath B is typically indistinguishable from the canonical distribution, if we only look at any operator in ${{\mathrm{B}}}_1$. Thus, the reduced density operators of system S at time $t\ll\tau$ are almost the same for the initial energy eigenstate and the initial canonical distribution. Therefore, the conventional proof of the second law with the canonical bath approximately applies to the present situation, leading to inequality (\[Main\_Clausius\]).
*Integral fluctuation theorem.* We next discuss the IFT for the case that bath B is initially in an energy eigenstate, which is our second main result. Let $\sigma$ be the stochastic entropy production defined as follows. Suppose that one performs projection measurements of $\hat{\sigma} (t) :=-\ln\hat{\rho}_{{\mathrm{S}}}(t)+\beta\hat{H}_{{\mathrm{B}}}$ at the initial and the final times, where the first and the second terms on the right-hand side respectively represent the informational and the energetic terms, corresponding to the first and the second terms on the left-hand side of inequality (\[Main\_Clausius\]). Then, $\sigma$ is given by the difference between the obtained outcomes of these measurements. The average of the stochastic entropy production is equivalent to the aforementioned average entropy production: ${\left\langle\sigma\right\rangle} =\Delta S_{{\mathrm{S}}}-\beta Q$.
The conventional IFT states that, if the initial state of bath B is the canonical distribution, $$\langle e^{-\sigma} \rangle = 1.$$ We note that the IFT holds even when system S is far from equilibrium. A crucial feature of the IFT is that it reproduces the second law $\langle \sigma \rangle \geq 0$ from the Jensen inequality $\langle e^{-\sigma} \rangle \geq e^{- \langle \sigma \rangle}$. Furthermore, the IFT can reproduce the fluctuation-dissipation theorem [@Esposito2009].
Our result is the IFT in the case that bath B is initially in a typical energy eigenstate (i.e., with initial condition (\[Main\_initial\])): $$\begin{aligned}
| \langle e^{-\sigma} \rangle -1 |\leq \varepsilon_{\mathrm{FT}},
\label{Main_FT_pure}\end{aligned}$$ where $\varepsilon_{\mathrm{FT}}$ is a positive error term. We can rigorously prove that $\varepsilon_{\mathrm{FT}}$ can be arbitrarily small if bath B is sufficiently large. We note that the detailed fluctuation theorem [@Esposito2009] also holds with an initial typical eigenstate, from which we can prove the IFT as a corollary (see Supplemental Material for details).
The central idea of the proof of the IFT (\[Main\_FT\_pure\]) is almost the same as that of the second law, which is outlined above. On the other hand, we need to make an additional assumption to prove inequality (\[Main\_FT\_pure\]) that $$\begin{aligned}
\label{Main_eq:assump_FT}
[\hat{H}_{{\mathrm{S}}}+\hat{H}_{{\mathrm{B}}},\hat{H}_{{\mathrm{I}}}]\simeq 0,\end{aligned}$$ which means that the sum of the energies of S and B is conserved at the level of fluctuations, and does not necessarily mean that $\hat H_{{\mathrm{I}}}$ itself is small. We note that assumption (\[Main\_eq:assump\_FT\]) is consistent with the concept of “thermal operation" in the thermodynamic resource theory [@Horodecki2013; @Brandao2015], where the left-hand side of Eq. (\[Main\_eq:assump\_FT\]) is assumed to be exactly zero. The rigorous meaning of “$\simeq$" is discussed in Supplemental Material. If the left-hand side of Eq. (\[Main\_eq:assump\_FT\]) is nonzero but small, a small positive error term $\varepsilon_{\rm I}$ should be added to the right-hand side of inequality (\[Main\_FT\_pure\]), which cannot be arbitrarily small even in the large-bath limit. However, we will numerically show later that this additional term is negligible in practice.
*Estimation of the error terms.* We evaluate the error terms in inequalities (\[Main\_Clausius\]) and (\[Main\_FT\_pure\]) with respect to the size $N$ of bath B. We set $| {{\mathrm{B}}}_1 | = \mathcal O (N^\alpha)$ with $0<\alpha < 1/2$. The error from the weak ETH is bounded by $\mathcal{O}(N^{-(1-2\alpha)/4+\delta})+\mathcal{O}(N^{-(1-2\alpha)/8+\delta/2}/\sqrt{\tilde{\varepsilon}})$, where $\delta>0$ is an unimportant constant that can be arbitrarily small, and $\tilde{\varepsilon}$ is the fraction of atypical eigenstates in the weak ETH. The error from the Lieb-Robinson bound is bounded by $e^{-\mu{{\mathrm{dist}}}(\tilde{{{\mathrm{S}}}},\partial{{\mathrm{B}}}_1)}(e^{vt}-vt-1)$, which is negligible compared with the error term from the weak ETH for sufficiently large $N$, but increases in time with $\mathcal{O}(t^2)$ up to $t\simeq 1/v$.
*Numerical simulation.* We performed numerical simulation of hard-core bosons with nearest-neighbor repulsion by exact diagonalization. System S is on a single site and bath B is on a two-dimensional lattice (see the inset of Fig. 2). The annihilation (creation) operator of a boson at site $i$ is written as $\hat{c}_i$ ($\hat{c}_i^\dag$), which satisfies the commutation relations $[\hat{c}_i,\hat{c}_j]=[\hat{c}^\dag_i,\hat{c}^\dag_j]=[\hat{c}_i,\hat{c}^\dag_j]=0$ for $i\neq j$, $\{\hat{c}_i,\hat{c}_i\}=\{\hat{c}^\dag_i,\hat{c}^\dag_i\}=0
$, and $
\{\hat{c}_i,\hat{c}^\dag_i\}=1$. The occupation number is defined as $\hat{n}_i:=\hat{c}^\dag_i\hat{c}_i$. Let $i=0$ be the index of the site of system S. The Hamiltonians in Eq. (\[Main\_total\_Hamiltonian\]) are then given by $$\begin{aligned}
\label{Main_eq_NS_Ham1}
\hat{H}_{{\mathrm{S}}}&=
\omega\hat n_0
,
\quad
\hat{H}_{{\mathrm{I}}}=
-\gamma^\prime
\sum_{<0,j>}
(\hat{c}^\dag_0 \hat{c}_j+\hat{c}^\dag_j \hat{c}_0),\\
\label{Main_eq_NS_Ham2}
\hat{H}_{{\mathrm{B}}}&=
\omega
\sum_i \hat n_i
-
\gamma
\sum_{<i,j>}
(\hat{c}^\dag_i \hat{c}_j+\hat{c}^\dag_j \hat{c}_i)
+
g
\sum_{<i,j>}
\hat{n}_i \hat{n}_j,\end{aligned}$$ where $\omega>0$ is the on-site potential, $-\gamma$ is the hopping rate in bath B, $-\gamma^\prime$ is the hopping rate between system S and bath B, and $g>0$ is the repulsion energy. The initial state of system S is given as $\ket{\psi}:=\hat{c}_0^\dag\ket{0}$. We set the size of bath B by $16=4\times 4$, and the initial number of bosons in bath B by $4$. To evaluate the Lieb-Robinson time, we set ${{\mathrm{dist}}}(\tilde{S},\partial {{\mathrm{B}}}_1)=1$. We can then evaluate that $v \simeq \gamma $ and $\mu=1$ if $g\ll \gamma$, and therefore the Lieb-Robinson time is given by $\tau\simeq\gamma^{-1}$. We set the inverse temperature of the initial eigenstate as $\beta=0.1$.
Figure 2 shows the time dependence of $\langle \sigma \rangle$, which implies that the second law indeed holds. The average entropy production gradually increases up to $t \simeq \tau$, and then saturates with some oscillations around the time average. We note that the oscillation for $\gamma^\prime=4\omega$ is the Rabi oscillation between system S and a part of B. Remarkably, we observed that the second law holds even in a much longer time regime $t \gg \tau$, though our theorem ensures the second law only up to $t \simeq \tau$. This implies that the second law is so robust against bare quantum fluctuations of pure quantum states.
We next confirmed the IFT (\[Main\_FT\_pure\]). As shown in Fig. 3, $\langle e^{-\sigma}\rangle$ is very close to unity up to $t \simeq \tau$, as predicted by our theorem. After $t \simeq \tau$, however, the deviation of $\langle e^{-\sigma} \rangle$ from unity becomes significant, which is consistent with our theorem. This deviation comes from the effect of bare quantum fluctuations of the initial state, because if the initial state was the canonical distribution, such deviation would never be observed. As time increases, system S more and more feels the effect of bare quantum fluctuations, and the deviation becomes larger. This is regarded as a dynamical crossover from emergent thermal fluctuations to bare quantum fluctuations across the Lieb-Robinson time $\tau$; The IFT holds only for the former. Such a crossover is not clearly observed in the second law (Fig. 2), because the second law only concerns the average of the stochastic entropy production, while its fluctuations play a significant role in the IFT. Our numerical result also clarifies that our theory indeed accounts for the validity of IFT in the short time regime, because the numerically observed time scale of the breakdown of the IFT is consistent with our theory.
As shown in the inset of Fig. 3, the error term of the IFT is proportional to $t^2$ up to $t \simeq 1/v \simeq \tau$ in our numerical simulation. In fact, our error evaluation based on the Lieb-Robinson bound predicts that an error term increases in time with $t^2$-dependence as discussed before, if the additional error term $\varepsilon_{\rm I}$, which could also increase in time, is zero (or equivalently, the left-hand side of (\[Main\_eq:assump\_FT\]) is zero). Therefore, our numerical result clarifies that the contribution from the left-hand side of (\[Main\_eq:assump\_FT\]) is negligible in our setup, though it is not exactly zero in our Hamiltonians (\[Main\_eq\_NS\_Ham1\]) and (\[Main\_eq\_NS\_Ham2\]).
*Concluding remarks.* In this Letter, we have established the second law (\[Main\_Clausius\]) and the IFT (\[Main\_FT\_pure\]) for unitary dynamics in the presence of a heat bath that is initially in a typical energy eigenstate. Our result implies that thermal fluctuations can emerge from quantum fluctuations in a short time regime, and the former crosses over to the latter in time. Our rigorous mathematical proof is based on the Lieb-Robinson bound (4) and the weak ETH (5). We also performed numerical simulation of two-dimensional hard-core bosons, and confirmed our theoretical results.
We remark that inequality (\[Main\_Clausius\]) only ensures the entropy increase between the initial and final times, and does not imply the monotonic entropy increase in continuous time. The extension of our result to the monotonic second law in continuous time could be possible with controlled approximations such as the Born-Markov approximation [@Breuer2002], as is the case for the standard master equation approach [@Horowitz2013; @Hekking2013]. This is a future direction of our work, though it would be technically nontrivial.
Our results can experimentally be tested with artificial isolated quantum systems such as ultracold atoms on an optical lattice, by employing the technique of single-site addressing [@Cheneau2012]. Another candidate of experimentally relevant systems is superconducting qubits, where fully-controlled dynamics of thermalization can be observed [@Pekola2015]. To examine the relevance of our theory to non-artificial complex materials in noisy open environment is an future issue.
The authors are most grateful to Hal Tasaki for valuable discussions, especially on the equivalence of ensembles. The authors also thank Takashi Mori and Jae Dong Noh for valuable discussions. E.I. and T.S. are supported by JSPS KAKENHI Grant Number JP16H02211. E.I. is also supported by JSPS KAKENHI Grant Number 15K20944. T.S. is also supported by JSPS KAKENHI Grant Number JP25103003.
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{width="0.8\linewidth"}
\[Main\_fig1\]
{width="0.7\linewidth"}
\[Main\_fig2\]
{width="0.7\linewidth"}
\[Main\_fig3\]
[Supplemental Material : Fluctuation Theorem for Many-Body Pure Quantum States]{}
\[1\] Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
\[2\] Department of Basic Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan
In this Supplemental Material, we prove the second law of thermodynamics and the fluctuation theorem for a system in contact with a heat bath in a pure state \[inequalities (3) and (7) in the main text\] in a mathematically rigorous way. For this purpose, we need several technical assumptions that will be discussed in detail. However, the bare essentials of the proof are the same as illustrated in the main text. In addition, we show supplementary results of numerical simulation.
The organization of Supplemental Material is as follows. In Sec. \[sec:review\], we briefly review the established proof of the conventional second law and the fluctuation theorem with a heat bath that is initially in the canonical distribution. In Sec. \[sec:inequality\], we remark on the operator norm and the trace norm. In Sec. \[sec:LRB\], we formulate the Lieb-Robinson bound that is a key of our proof. In Sec. \[sec:MicroCanonical\], we define the canonical ensemble and the microcanonical ensemble. In Sec. \[sec:weakETH\], we show our theorems on the weak eigenstate-thermalization hypothesis (ETH). In Sec. \[sec:setup\], we describe our basic setup of the proof of the second law and the fluctuation theorem. In Sec. \[sec:locality\], we prove two important lemmas. In particular, Lemma \[lemma:Reference\] plays a key role in the proof of our main results. In Secs. \[sec:2nd\] and \[sec:FT\], we prove our main results: the second law (Theorem 1) and the fluctuation theorem (Theorem 2), respectively. In Sec. \[sec:typicality\], we discuss the typicality in the Hilbert space. In Sec. \[sec:Discussion\], we remark on some assumptions in our setup. In Sec. \[sec:Numerical\], we show the details of our numerical calculation and supplementary numerical results.
Throughout Supplemental Material, we set $\hbar=1$ and $k_{\rm B} = 1$ for simplicity. We will only consider finite-dimensional Hilbert spaces.
The conventional second law and the fluctuation theorem {#sec:review}
=======================================================
In this section, we review the conventional proof of the second law and the fluctuation theorem for a system in contact with a heat bath that is initially in the canonical distribution.
Setup
-----
Suppose that the total system consists of system S and bath B as shown in Fig. \[fig:fig\_s\_1\]. The Hamiltonian of the composite system is given by $$\hat{H}
:=
\hat{H}_\mathrm{S}
+
\hat{H}_{\mathrm{I}}
+
\hat{H}_{\mathrm{B}},$$ where $\hat{H}_\mathrm{S}$ and $\hat{H}_\mathrm{B}$ are the Hamiltonian of system S and bath B, respectively. The interaction Hamiltonian between system S and bath B is denoted by $\hat{H}_\mathrm{I}$. We note that there is no additional assumption on $\hat{H}$ (such as locality) in this section. The coupling between system S and bath B can either be weak or strong.
The initial state is represented by a product state without any correlation between S and B: $$\hat{\rho}(0)=\hat{\rho}_{{\mathrm{S}}}(0)\otimes\hat{\rho}_{{\mathrm{B}}}^{{\mathrm{can}}}.
\label{canonical_initial}$$ The initial state of system S is arbitrary, and the initial state of bath B is described by the canonical distribution $\hat{\rho}_{{\mathrm{B}}}^{\mathrm{can}}:=\exp(-\beta\hat{H}_{{\mathrm{B}}})/Z_{{\mathrm{B}}}$, where $Z_{{\mathrm{B}}}:={{\mathrm{tr}}}_{{\mathrm{B}}}[\exp(-\beta\hat{H}_{{\mathrm{B}}})]$ is the partition function and $\beta$ is the inverse temperature.
The composite system obeys the unitary evolution described by $\hat{U}:=\exp(-i\hat{H}t)$, and the final state of the system is given by $\hat{\rho}(t)=\hat{U}\hat{\rho}(0)\hat{U}^\dag$. The reduced density operators for the final state of system S and bath B are defined as $\hat{\rho}_{{\mathrm{S}}}(t):={{\mathrm{tr}}}_{{{\mathrm{B}}}}[\hat{\rho}(t)]$ and $\hat{\rho}_{{\mathrm{B}}}(t):={{\mathrm{tr}}}_{{{\mathrm{S}}}}[\hat{\rho}(t)]$, respectively.
![ Schematic of the setup for the conventional proof of the second law and the fluctuation theorem. The composite system consists of system S and bath B, where the initial state of bath B is the canonical distribution. []{data-label="fig:fig_s_1"}](fig_s_1.pdf){width="0.3\linewidth"}
The second law of thermodynamics
--------------------------------
We first note that the von Neumann entropy of a density operator $\hat \rho$ is given by $$S(\hat{\rho}):=-\mathrm{tr}{\left[\hat{\rho} \ln\hat{\rho}\right]}.$$ The change in the von Neumann entropy of system S is represented as $\Delta S_{{\mathrm{S}}}:= S_{{\mathrm{S}}}(t)-S_{{\mathrm{S}}}(0):=S(\hat{\rho}_{{\mathrm{S}}}(t))-S(\hat{\rho}_{{\mathrm{S}}}(0))$. We also define the heat as $Q:=
-\mathrm{tr}_{{\mathrm{B}}}[
\hat{H}_\mathrm{B}(
\hat{\rho}_\mathrm{B}(t)
-
\hat{\rho}_\mathrm{B}(0)
)
]$. The second law of thermodynamics in this setup is then stated as follows:
[[\[prop:2nd\] [**Proposition** ]{}]{} (Second law of thermodynamics with the canonical bath [@S_Sagawa2012])]{} $$\label{eq:conv_2nd}
\Delta S_{{\mathrm{S}}}- \beta Q\geq 0.$$
[**Proof.**]{} The von Neumann entropy of the composite system in the initial and the final states are given by $$\begin{split}
S(\hat{\rho}(0))
&=
-\mathrm{tr}_{{\mathrm{S}}}{\left[\hat{\rho}_{{\mathrm{S}}}(0)\ln \hat{\rho}_{{\mathrm{S}}}(0)\right]}
-\mathrm{tr}_{{\mathrm{B}}}{\left[\hat{\rho}_{{\mathrm{B}}}^{\mathrm{can}}\ln \hat{\rho}_{\mathrm{B}}^{\mathrm{can}}\right]}
=
S_{\mathrm{S}}(0)
+\beta\mathrm{tr}_{{\mathrm{B}}}[\hat{\rho}_{{\mathrm{B}}}^{\mathrm{can}}
\hat{H}_{{\mathrm{B}}}]
+\ln Z_{{\mathrm{B}}},
\\
S(\hat{\rho}(t))
&=
-
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}
[
\hat{\rho}(t)\ln\hat{\rho}(t)
]
\leq
-
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}
{\left[
\hat{\rho}(t)\ln(\hat{\rho}_{\mathrm{S}}(t)\otimes \hat{\rho}_\mathrm{B}^{\mathrm{can}})
\right]}
=
S_{\mathrm{S}}(t)+\beta\mathrm{tr}_{{\mathrm{B}}}[\hat{\rho}_{{\mathrm{B}}}(t)\hat{H}_{{\mathrm{B}}}]
+\ln Z_{{\mathrm{B}}},
\end{split}$$ where we used the Klein inequality: $\mathrm{tr}{\left[\hat{\rho}\ln\hat{\rho}\right]}
\geq
\mathrm{tr}{\left[\hat{\rho}\ln\hat{\sigma}\right]}$. Since the von Neumann entropy is invariant under unitary dynamics $S(\hat{\rho}(0)) = S(\hat{\rho}(t))$, we obtain inequality (\[eq:conv\_2nd\]). $\Box$
The fluctuation theorem {#subsec:FT}
-----------------------
We consider the concept of stochastic entropy production [@S_Esposito2009]. We first define $$\hat{\sigma}(t){:=}\beta \hat{H}_\mathrm{B}-\ln\hat{\rho}_S(t)$$ in the Schrödinger picture, and consider the projection measurements of $\hat \sigma (t)$ at time $0$ and $t$. Let $\sigma_{\rm i}$ and $\sigma_{\rm f}$ be the measurement outcomes at time $0$ and $t$, respectively. The joint probability distribution of $\sigma_{\rm i}$ and $\sigma_{\rm f}$ is given by $$P_{{\mathrm{F}}}[\sigma_{\rm f}; \sigma_{\rm i}]
:=
{{\mathrm{tr}}}[
\hat{P}_{\sigma_{\rm f}}
\hat{U}
\hat{P}_{\sigma_{\rm i}}
\hat{\rho}(0)
\hat{P}_{\sigma_{\rm i}}
\hat{U}^\dag
\hat{P}_{\sigma_{\rm f}}
],$$ where $\hat{P}_{\sigma_{\rm i}}$ and $\hat{P}_{\sigma_{\rm f}}$ are the projection operators corresponding to eigenvalues $\sigma_{\rm i}$ and $\sigma_{\rm f}$ of $\hat \sigma (0)$ and $\hat \sigma (t)$, respectively. We note that $\hat \rho (0)$ and $\hat \sigma (0)$ are, and thus $\hat \rho (0)$ and $\hat{P}_{\sigma_{\rm i}}$ are, commutable. The index ${{\mathrm{F}}}$ in $P_{{\mathrm{F}}}[\sigma_{\rm f}; \sigma_{\rm i}]$ means the forward time evolution. The stochastic entropy production is then defined as the difference between the two measurement outcomes: $$\sigma := \sigma_{\rm f} - \sigma_{\rm i}.$$ The probability distribution of the stochastic entropy production is written as $P_{{\mathrm{F}}}(\sigma)$, which is given by $$P_{{\mathrm{F}}}(\sigma)
:=
\sum_{\sigma_{\rm f}, \sigma_{\rm i}} \delta(\sigma-(\sigma_{\rm f}-\sigma_{\rm i}))
P_{{\mathrm{F}}}[\sigma_{\rm f}; \sigma_{\rm i}],$$ where $\delta (\cdot )$ is the delta function. We note that in the proof of our main results in the subsequent sections, we will not use the concept of the delta function in order to avoid introducing advanced mathematical tools for the rigorous argument.
The characteristic function of the stochastic entropy production is given by the Fourier transformation of $P_{{\mathrm{F}}}(\sigma)$ [@S_Esposito2009]: $$\label{eq:def_Gu}
G_\mathrm{F}(u)
:=
\int_{- \infty}^{+\infty} d\sigma e^{iu\sigma}
P_{{\mathrm{F}}}(\sigma),$$ where $u$ is referred to as the counting field. The $n$th differential of the characteristic function gives the $n$th order moment of the stochastic entropy production: $${\left\langle\hat{\sigma}^n\right\rangle}
=
\left.
\frac{\partial^n G_\mathrm{F}(u)}{\partial(iu)^n}
\right|_{u=0}.$$
We note that the ensemble average of the entropy production is given by $$\langle \sigma \rangle =\Delta S_{{\mathrm{S}}}-\beta Q,$$ which is the left-hand side of the second law (\[eq:conv\_2nd\]). Therefore, we can rewrite the second law (\[eq:conv\_2nd\]) in terms of the entropy production by $$\langle \sigma \rangle \geq 0.
\label{conv_2nd_sigma}$$
By noting that $[\hat{\sigma}(0),\hat{\rho}(0)]=0$, the characteristic function is rewritten as $$\label{eq:CF_Gu}
G_\mathrm{F}(u)
=
{{\mathrm{tr}}}[
\hat{U}
e^{-iu\hat{\sigma}(0)}
\hat{\sigma}(0)
\hat{U}^\dag
e^{iu\hat{\sigma}(t)}
],$$ which can be regarded as the definition of $G_\mathrm{F}(u)$ in the case that we do not use the delta function.
We next consider the reversed time evolution that obeys unitary evolution $\hat U^\dagger$. The initial state of the reversed process is given by the product state with the final state of system S and the canonical distribution of bath ${{\mathrm{B}}}$: $
\hat{\rho}_{{\mathrm{R}}}(0)
:=
\hat{\rho}_{{\mathrm{S}}}(t)
\otimes
\hat{\rho}_{{\mathrm{B}}}^{\mathrm{can}}$, where index ${{\mathrm{R}}}$ represents the reversed processes. The density operator at time $t$ of the reversed process is given by $\hat \rho_{{\mathrm{R}}}(t) = \hat U^\dagger \hat{\rho}_{{\mathrm{R}}}(0) \hat U$.
We then define the stochastic entropy production in the reversed processes by the difference between the measurement outcomes of $\hat \sigma (t)$ and $\hat \sigma (0)$ at the initial and the final times of the reversed processes, respectively. Let $P_{{\mathrm{R}}}(\sigma)$ be the probability distribution of the stochastic entropy production in the reversed processes, and $G_{{\mathrm{R}}}(u)$ be the corresponding characteristic function. By noting that $[\hat{\sigma}(t),\hat{\rho}_{{\mathrm{R}}}(0)]=0$, we have $$G_\mathrm{R}(u)
=
{{\mathrm{tr}}}[
\hat{U}^\dag
e^{-iu\hat{\sigma}(t)}
\hat{\rho}_{{\mathrm{R}}}(0)
\hat{U}
e^{iu\hat{\sigma}(0)}
].$$
We note that ${\rm tr}_{{\mathrm{B}}}[\hat \rho_{{\mathrm{R}}}(t)] \neq \hat \rho_{{\mathrm{S}}}(0)$ in general. Therefore, the entropy production defined above is not necessarily related to the change in the von Neumann entropy of system S in the reversed process. In this sense, the physical interpretation of the entropy production is not very clear in the reversed process. If ${\rm tr}_{{\mathrm{B}}}[\hat \rho_{{\mathrm{R}}}(t)] = \hat \rho_{{\mathrm{S}}}(0)$ happens to be true, the entropy production in the reversed process has a clear interpretation related to the von Neumann entropy of the reversed process. Such a situation is physically realizable if the initial and the final states of system S are in thermal equilibrium in both of the forward and the reversed processes.
Apart from its physical interpretation, the entropy production in the reversed process is always mathematically well-defined, and the following argument including the reversed process is true for any case. In addition, the entropy production in the reversed process is a useful concept to prove some theorems that only include quantities in the forward process, such as the integral fluctuation theorem. Therefore, we will not go into details of the physical interpretation of the reversed processes in the following argument.
We now discuss the fluctuation theorem. The detailed fluctuation theorem characterizes a universal relationship between the probabilities of the entropy changes in the forward and the reversed processes. In the following argument, it is convenient to formulate the detailed fluctuation theorem in terms of the characteristic functions:
[[\[prop:FT\] [**Proposition** ]{}]{} (Detailed fluctuation theorem with the canonical bath [@S_Esposito2009])]{} $$\label{eq:FT_Gu}
G_\mathrm{F}(u)=G_\mathrm{R}(-u+i).$$
[**Proof.**]{} Letting $v:=-u+i$, we have $$\begin{split}
G_\mathrm{F}(u)
&=
{{\mathrm{tr}}}[
\hat{U}
e^{-i(-v+i)\hat{\sigma}(0)}
\hat{\rho}(0)
\hat{U}^\dag
e^{i(-v+i)\hat{\sigma}(t)}
]
\\
&=
{{\mathrm{tr}}}[
\hat{U}
e^{iv\hat{\sigma}(0)}
e^{\hat{\sigma}(0)}
\hat{\rho}(0)
\hat{U}^\dag
e^{-iv\hat{\sigma}(t)}
e^{-\hat{\sigma}(t)}
]
\\
&=
{{\mathrm{tr}}}[
\hat{U}
e^{iv\hat{\sigma}(0)}
\hat{U}^\dag
e^{-iv\hat{\sigma}(t)}
\hat{\rho}_{\mathrm{S}}(t)
\otimes
\hat{\rho}_\mathrm{B}^\mathrm{can}
]
\\
&=
{{\mathrm{tr}}}[
\hat{U}^\dag
e^{-iv\hat{\sigma}(t)}
\hat{\rho}_{\mathrm{S}}(t)
\otimes
\hat{\rho}_\mathrm{B}^\mathrm{can}
\hat{U}
e^{iv\hat{\sigma}(0)}
]
\\
&=
G_\mathrm{R}(-u+i),
\end{split}$$ which implies Eq. (\[eq:FT\_Gu\]). $\Box$
We next discuss the integral fluctuation theorem (IFT) [@S_Kurchan2000; @S_Tasaki2000]. We consider the following quantity: $${\left\langlee^{-\sigma}\right\rangle}
:=
\int_{- \infty}^{+\infty} d\sigma
P_{{\mathrm{F}}}(\sigma)e^{-\sigma},$$ or equivalently $${\left\langlee^{-\sigma}\right\rangle}
:=
\sum_{\sigma_{\rm f}, \sigma_{\rm i}}
P_{{\mathrm{F}}}[\sigma_{\rm f}; \sigma_{\rm i}] e^{-(\sigma_{\rm f} -\sigma_{\rm i} )}.$$ We then have the following corollary.
[[\[cor:IFT\] [**Corollary** ]{}]{} (Integral fluctuation theorem with the canonical bath)]{} $$\label{integralFT}
{\left\langlee^{-\sigma}\right\rangle} = 1.$$
[**Proof.**]{} We note that ${\left\langlee^{-\sigma}\right\rangle} = G_{\rm F}(i)$ and $G_{\rm R} (0) =1$. By substituting $u=i$ to the detailed fluctuation theorem (\[eq:FT\_Gu\]), we obtain the IFT (\[integralFT\]). $\Box$
We make some remarks on the fluctuation theorem. Applying the inverse Fourier transform to Eq. (\[eq:FT\_Gu\]), we obtain the detailed fluctuation theorem in terms of the probability distributions: $$\label{eq:FT_conv_P}
\frac{P_{{\mathrm{F}}}(\sigma)}{P_{{\mathrm{R}}}(-\sigma)}=e^{\sigma}.$$ By integrating the both-hand sides of $P_{{\mathrm{F}}}(\sigma)e^{-\sigma} = P_{{\mathrm{R}}}(-\sigma)$, which is equivalent to Eq. (\[eq:FT\_conv\_P\]), we again obtain the IFT: $${\left\langlee^{-\sigma}\right\rangle}
=
\int_{- \infty}^{+\infty} d\sigma
P_{{\mathrm{F}}}(\sigma)e^{-\sigma}
=
\int_{- \infty}^{+\infty} d\sigma
P_{{\mathrm{R}}}(-\sigma)
=
1.$$ By using the Jensen inequality to Eq. (\[integralFT\]), we obtain ${\left\langlee^{-\sigma}\right\rangle}\geq e^{-{\left\langle\sigma\right\rangle}}$, which reproduces the second law (\[conv\_2nd\_sigma\]).
Norms of operators {#sec:inequality}
==================
As a preliminary, we briefly review the operator norm and the trace norm for finite-dimensional Hilbert spaces. Let $| \psi \rangle$ be a vector of a Hilbert space, and $\|\ket{\psi}\|$ be its norm.
We consider operators on the Hilbert space, which is not necessarily Hermitian. First, the operator norm of an operator $\hat{X}$ is defined as $$\|\hat{X}\|
:=
\sup_{\|\ket{\Psi}\|=1}
\sqrt{\langle \Psi |
\hat X^\dagger
\hat{X}
| \Psi \rangle
},$$ which is equal to the largest singular value of $\hat X$. Second, the trace norm of an operator $\hat{X}$ is defined as $$\|\hat{X}\|_1:= {\rm tr}\left[ \sqrt{ \hat{X}^\dagger \hat X } \right],$$ which is the sum of the singular values of $\hat X$. If $\hat X$ is Hermitian, the above definition reduces to $$\|\hat{X}\|_1{:=}{\rm tr}[ | \hat{X} | ].$$ We note some useful (and well-known) properties of the norms:
[[\[prop:IneqNorm\]]{}]{} For any operators, $$| {\rm tr} [ \hat X ] | \leq \| \hat X \|_1,$$ $$\| \hat X \hat Y \|_1 \leq \| \hat X \| \| \hat Y \|_1.$$
The Lieb-Robinson bound {#sec:LRB}
=======================
In this section, we review the Lieb-Robinson bound [@S_Lieb1972; @S_Hastings2006] that plays a key role in our study. The Lieb-Robinson bound gives an upper bound of the velocity of information propagation in a quantum many-body system on a general graph, which is formulated in terms of the operator norm of the commutator of two operators in spatially distinct regions on the graph.
We consider a general setup for the Lieb-Robinson bound as follows. The system is defined on a finite graph, written as $\Lambda:=(\Lambda_\mathrm{v},\Lambda_\mathrm{e})$, where the sets of all vertices (sites) and all edges (bonds) are denoted by $\Lambda_\mathrm{v}$ and $\Lambda_\mathrm{e}$, respectively. A region X in the graph is defined as a set of sites: ${{\mathrm{X}}}\subset \Lambda_\mathrm{v}$. We write the number of elements of a set as $|\cdot|$. For example, $| \Lambda_\mathrm{v} |$ describes the number of all the sites in the graph.
We next define the spatial distance on the graph. The distance between two sites ${\rm x}, {\rm y} \in \Lambda_{\rm v}$ is defined as the number of bonds in the shortest path that connects x and y, which we denote as ${{\mathrm{dist}}}(\mathrm{x},\mathrm{y})$. The distance between two regions ${{\mathrm{X}}}$ and ${{\mathrm{Y}}}$ is correspondingly defined as $${{\mathrm{dist}}}({{\mathrm{X}}},{{\mathrm{Y}}}):=\min_{{{\mathrm{x}}}\in{{\mathrm{X}}},{{\mathrm{y}}}\in{{\mathrm{Y}}}}{{\mathrm{dist}}}({{\mathrm{x}}},{{\mathrm{y}}}).$$
The Hamiltonian in our setup is expressed as the sum of local Hamiltonians: $$\hat{H}=\sum_{{{\mathrm{Z}}}\subset\Lambda_\mathrm{v}} \hat{h}_{{\mathrm{Z}}},
\label{local_Hamiltonian}$$ where $\hat{h}_{{\mathrm{Z}}}$ is a local Hamiltonian on a bounded support ${{\mathrm{Z}}}$. The sum on the right-hand side of Eq. (\[local\_Hamiltonian\]) is over a particular set of bounded supports. We then make the following assumption.
[[\[assump:LRB\]]{} (Conditions on the local Hamiltonians and the graph [@S_Lieb1972; @S_Hastings2006])]{} There exist $\lambda_0 >0$, $\mu > 0$, and $p_0 > 0$, such that for any ${{\mathrm{x}}},{{\mathrm{y}}}\in\Lambda_\mathrm{v}$, $$\sum_{\mathrm{Z}\ni {{\mathrm{x}}},{{\mathrm{y}}}}
\|\hat{h}_\mathrm{Z}\|
\leq
\lambda_0\exp[-\mu\mathrm{dist}({{\mathrm{x}}},{{\mathrm{y}}})],$$ $$\label{eq:LRB_assump2}
\sum_{z\in\Lambda_\mathrm{v}}
\exp[-\mu(\mathrm{dist}({{\mathrm{x}}},{{\mathrm{z}}})+\mathrm{dist}({{\mathrm{z}}},{{\mathrm{y}}}))]
\leq
p_0
\exp[-\mu\mathrm{dist}({{\mathrm{x}}},{{\mathrm{y}}})].$$
The constants $\lambda_0$ and $\mu$ are determined by the Hamiltonian and the graph, which represent the interaction strength and the interaction range, respectively. For example, if we consider the case of the nearest-neighbor interaction on a hypercubic lattice, $\lambda_0$ and $\mu$ are determined to satisfy $\max_{{{\mathrm{Z}}}}\|\hat{h}_{{\mathrm{Z}}}\|\leq \lambda_0 e^{-\mu}$. If the interaction is bounded as $\|\hat{h}_{{\mathrm{Z}}}\|\leq J$ with $J>0$ being a constant, $\lambda_0$ is given by $\lambda_0=J e^{\mu}$ with arbitrary $\mu$. The constant $p_0$ is determined by the graph structure and $\mu$. For the case of nearest-neighbor interaction on the two-dimensional square lattice, $p_0$ is given by $2$.
The Lieb-Robinson bound states that there exists an upper bound of the velocity of information propagation for a general Hamiltonian which satisfies Assumption \[assump:LRB\]. In Sec. \[sec:setup\], we will formulate a more specific setup for our study, where Assumption \[assump:LRB\] is also assumed and the Lieb-Robinson bound is applicable.
We now state the Lieb-Robinson bound as follows.
[[\[prop:LRB\]]{} (Lieb-Robinson bound [@S_Lieb1972; @S_Hastings2006])]{} Let $\hat{A}$ and $\hat{A}^\prime$ be operators with supports A and ${{\mathrm{A}}}^\prime$, respectively. We consider the Heisenberg picture of $\hat{A}$, defined as $\hat{A}(t):=\hat{U}^\dag\hat{A}\hat{U}$ with $\hat{U}:=\exp(-i\hat{H}t)$. Under Assumption \[assump:LRB\], defining $C:=2/p_0$ and $v:=\lambda_0p_0$, the Lieb-Robinson bound states that $$\label{eq:LRB}
\|
[
\hat{A}(t),\hat{A}^\prime
]
\|
\leq
C
\|\hat{A}\|
\|\hat{A}^\prime\|
|{{\mathrm{A}}}||{{\mathrm{A}}}^\prime|
\exp[-\mu\mathrm{dist}({{\mathrm{A}}},{{\mathrm{A}}}^\prime)]
(e^{v|t|}-1).$$
The constants $C$ and $v$ depend on the interaction and the graph structure through $\lambda_0$ and $p_0$. Here, we refer to $v/\mu$ as the Lieb-Robinson velocity. We then define a characteristic time as $$\tau:= \frac{\mu{{\mathrm{dist}}}({{\mathrm{A}}},{{\mathrm{A}}}^\prime)}{v},$$ to which we refer as the Lieb-Robinson time. If $t\ll\tau$, the right-hand side of inequality (\[eq:LRB\]) becomes exponentially small with respect to the distance between ${{\mathrm{A}}}$ and ${{\mathrm{A}}}^\prime$.
In the case of nearest-neighbor interaction on the two-dimensional square lattice, we have $C=1/2$, $v=2Je^\mu$, and $\tau=\mu{{\mathrm{dist}}}({{\mathrm{A}}},{{\mathrm{A}}}^\prime) / (2e^\mu J)$. Since $\mu$ is arbitrary, we can choose it to maximize the Lieb-Robinson time, and $\tau$ is maximized when $\mu=1$, where $v=2eJ$ and $\tau={{\mathrm{dist}}}({{\mathrm{X}}},{{\mathrm{Y}}}) / (2eJ)$.
The canonical and the microcanonical ensembles {#sec:MicroCanonical}
==============================================
In this section, we rigorously define the canonical and the microcanonical ensembles, and discuss their equivalence in line with Ref. [@S_Tasaki2016]. We consider a quantum many-body system on a lattice, to which we refer as “bath B", as it will play a role of a heat bath in our main theorems in Secs. \[sec:setup\]-\[sec:typicality\]. We assume that bath B is on a $d$-dimensional hypercubic lattice with the periodic boundary condition. We denote the set of sites of bath B by the same notation, B. We denote the number of sites in bath B by $N:=|{{\mathrm{B}}}|$, and the side length of ${{\mathrm{B}}}$ by $L$ such that $L^d=N$. Let $\mathcal{H}_\mathrm{i}$ be the local Hilbert space on site ${{\mathrm{i}}}\in {{\mathrm{B}}}$. The total Hilbert space of B is denoted by $\mathcal H_{{\mathrm{B}}}:=\otimes_{{{\mathrm{i}}}\in {{\mathrm{B}}}}\mathcal{H}_\mathrm{i}$. The dimension of $\mathcal{H}_{{\mathrm{B}}}$ is denoted by $D_{{\mathrm{B}}}$.
We assume that the Hamiltonian of bath B is represented as the sum of local Hamiltonians: $$\hat{H}_\mathrm{B}
=
\sum_{\mathrm{Z}\subset \mathrm{B}} \hat{h}_{\mathrm{Z}},
\label{l_H}$$ where the sum is taken over a particular set of bounded region Z. We write the spectrum decomposition of $\hat H_{{\mathrm{B}}}$ as $$\hat H_{{\mathrm{B}}}:= \sum_{i=1}^{D_{{\mathrm{B}}}} E_i | E_i \rangle \langle E_i |,$$ where $E_i$ is an energy eigenvalue, and $| E_i \rangle$ is the corresponding eigenstate. We assume the locality of the interaction on the lattice, which implies that the interaction range of the Hamiltonian is independent of the size of bath B:
[[\[assump:LocalInt\]]{} (Locality of the interaction)]{} There exists an integer $k>0$ that is independent of $N$, such that for any $\hat h_{\rm Z}$ with support Z, ${{\mathrm{dist}}}({{\mathrm{i}}},{{\mathrm{j}}})\leq k$ holds for ${{\mathrm{i}}}, {{\mathrm{j}}}\in {{\mathrm{Z}}}$. We refer to $k$ as the interaction range inside bath B. In addition, we assume that $\| \hat h_{{{\mathrm{Z}}}} \|$ is independent of $N$ for any $\hat{h}_{{\mathrm{Z}}}$.
We also assume that bath ${{\mathrm{B}}}$ is translation invariant:
[[\[assump:TransInv\]]{} (Translation invariance)]{} The local Hilbert spaces $\mathcal{H}_\mathrm{i}$ with ${{\mathrm{i}}}\in {{\mathrm{B}}}$ are identical, and the local Hamiltonians $\hat h_{\rm Z}$ are the same for any Z in Eq. (\[l\_H\]), where ${{\mathrm{Z}}}$’s are defined over the lattice in a translation invariant way. Let $D_{\mathrm{loc}}$ be the dimension of $\mathcal{H}_{{{\mathrm{i}}}}$.
We now define the canonical ensemble.
[[\[def:Canonical\] [**Definition** ]{}]{} (Canonical ensemble)]{} For a given inverse temperature $\beta$, the density operator of the canonical ensemble is defined as $$\hat{\rho}^{\mathrm{can}}_{{{\mathrm{B}}}}
:=
\frac{1}{Z(\beta)}e^{-\beta \hat{H}_{{\mathrm{B}}}},$$ where $Z(\beta):={{\mathrm{tr}}}[e^{-\beta\hat{H}_{{\mathrm{B}}}}]$ is the partition function.
We also define the average energy density of the canonical ensemble at inverse temperature $\beta$ by $$u^\mathrm{can}(\beta ) :=\frac{1}{N} {{\mathrm{tr}}}[\hat{H}_{{\mathrm{B}}}\hat{\rho}_{{\mathrm{B}}}^{{{\mathrm{can}}}}].
\label{u_beta}$$
To define the microcanonical ensemble, we employ the framework of Ref. [@S_Tasaki2016], where the equivalence of the canonical and the microcanonical ensembles has rigorously been proved. Let $\Delta>0$ be the width of the microcanonical energy shell, which can depend on $N$ but is bounded from below by $N$-independent constant $\delta^\prime>0$. This implies that $\Delta$ can be in the order of $\mathcal{O}(1) \leq \Delta \leq \mathcal{O}(N)$.
[[\[def:EnergyShell\] [**Definition** ]{}]{} (Microcanonical energy shell)]{} The energy shell with energy $U$ and width $\Delta$ is defined as the following set of indices of energy eigenvalues of $\hat H_{{\mathrm{B}}}$: $$M_{U,\Delta}:=
\{
i:E_i\in (U-\Delta,U]
\}.$$ We also define the Hilbert space $\mathcal H_{U,\Delta}$ that is spanned by the energy eigenstates $\{ | E_i \rangle \}_{i\in M_{U,\Delta}}$ in the energy shell. We denote the dimension of the Hilbert space of the energy shell as $D(U,\Delta):=|M_{U,\Delta}|$.
To relate the microcanonical and the canonical ensembles, we define energy $U(\beta)$ that is determined by a given inverse temperature $\beta$. In line with Ref. [@S_Tasaki2016], we employ the following definition.
[[\[def:NDepEnergy\] [**Definition** ]{}]{} ]{} For a given inverse temperature $\beta$, $U(\beta)$ is defined as $$U(\beta)
:=
\delta^\prime
\operatorname*{arg\,max}_{\nu\in\mathbb{Z}}
D(\nu\delta^\prime,\delta^\prime)
e^{-\beta \nu\delta^\prime}.$$
The above definition is motivated by the Legendre transformation from $\beta$ to $U$, and is roughly rewritten as $$U(\beta) := \operatorname*{arg\,min}_U \left( \beta U - \ln D(U,\delta^\prime) \right),$$where $\ln D(U,\delta^\prime) $ is the thermodynamic entropy.
We now define the microcanonical ensemble for a given inverse temperature $\beta$.
[[\[def:Microcanonical\] [**Definition** ]{}]{} (Microcanonical ensemble)]{} The density operator of the microcanonical ensemble corresponding to the energy shell $M_{U(\beta),\Delta}$ is defined as $$\hat{\rho}^{\mathrm{MC}}_{{{\mathrm{B}}}}
:=
\frac{1}{D(U(\beta),\Delta)}\sum_{i\in M_{U(\beta),\Delta}} \ket{E_i}\bra{E_i}.$$
In the following, we write $D:=D(U(\beta),\Delta)$ for simplicity.
We next discuss two fundamental assumptions that are required to prove the equivalence of the ensembles. We first consider the correlation function.
[[\[def:Correlation\] [**Definition** ]{}]{} (Correlation function)]{} Let $\hat{\rho}$ be an arbitrary density operator. The correlation function of arbitrary two operators $\hat{A}$ and $\hat{A}^\prime$ is defined as $$\mathrm{cor}_{\hat{\rho}}(\hat{A},\hat{A}^\prime)
:=
{{\mathrm{tr}}}[\hat{\rho} \hat{A}\hat{A}^\prime]
-
{{\mathrm{tr}}}[\hat{\rho} \hat{A}]{{\mathrm{tr}}}[\hat{\rho} \hat{A}^\prime].$$
We then assume the following property of the correlation functions for the canonical ensemble.
[[\[assump:ExpDecayT\] [**Assumption** ]{}]{} (Exponential decay of correlation functions: Assumption I of [@S_Tasaki2016])]{} For a given inverse temperature $\beta$, there exist positive constants $\xi$ and $\gamma_1$ that satisfy the following. Let $\hat{A}$ and $\hat{A}^\prime$ be arbitrary operators with supports ${{\mathrm{A}}},{{\mathrm{A}}}^\prime\subset {{\mathrm{B}}}$. For any $N$, we have $$|\mathrm{cor}_{\hat{\rho}^\mathrm{can}_{{{\mathrm{B}}}}}(\hat{A},\hat{A}^\prime)|
\leq
C(\hat{A},\hat{A}^\prime)
\exp\left[
-{{\mathrm{dist}}}({{\mathrm{A}}},{{\mathrm{A}}}^\prime)/\xi
\right],$$ where $$C(\hat{A},\hat{A}^\prime)
:=
\gamma_1
\|\hat{A}\|
\|\hat{A}^\prime\|
|{{\mathrm{A}}}|
|{{\mathrm{A}}}^\prime|.$$
Assumption \[assump:ExpDecayT\] can be rigorously proved for any $d$ and sufficiently small $\beta$, by using the cluster expansion method [@S_Park1982; @S_Frolich2014]. We note that a slightly different version of the exponential decaying of correlation functions have been proved in Ref. [@S_Kliesch2014] for sufficiently small $\beta$.
We next consider the Massieu function $\varphi(\beta)$, which is related to the free energy by $f(\beta)=-\varphi(\beta)/\beta$.
[[\[def:Massieu\] [**Definition** ]{}]{} (Massieu function)]{} We define the Massieu function of $N$ and $\beta$ as $$\varphi_N(\beta)
:=
\frac{1}{N}
\log Z(\beta).$$ It is known rigorously in general [@S_Ruelle1999] that the following limit exists: $$\varphi(\beta)
:=
\lim_{N\rightarrow\infty}
\varphi_N(\beta).$$
We then assume the following properties of the Massieu function.
[[\[assump:Massieu\] [**Assumption** ]{}]{} (Properties of the Massieu function: Assumption II of [@S_Tasaki2016])]{} For a given inverse temperature $\beta$, there exist $\beta_1$ and $\beta_2$ such that $\beta_1<\beta<\beta_2$, and the following two properties are valid. First, there exists $\gamma_0$ such that $$|
\varphi_N(\beta^\prime)
-
\varphi(\beta^\prime)
|
\leq
\frac{\gamma_0}{N}$$ for any $\beta^\prime\in[\beta_1,\beta_2]$ and $N$. Second, the Massieu function $\varphi(\beta)$ is twice continuously differentiable, and satisfies $\varphi^{\prime\prime}(\beta)\geq c_0$ with a constant $c_0>0$ in interval $[\beta_1,\beta_2]$.
Assumption \[assump:Massieu\] can also be proved for any $d$ and sufficiently small $\beta$, by using the cluster expansion method [@S_Park1982; @S_Frolich2014].
We now define the thermodynamic limit with the inverse temperature being fixed:
[[\[def:TDlimit\] [**Definition** ]{}]{} (Thermodynamic limit)]{} The thermodynamic limit is given by $N \to \infty$ with $\beta$ being fixed. In the thermodynamic limit, the interaction range $k$ in Assumption \[assump:LocalInt\] is kept constant.
In the following argument, the phrase of “sufficiently large $N$” will be used in the sense of the above thermodynamic limit.
We are now in the position to state the equivalence of ensembles in terms of the reduced density operators of a subsystem.
[[\[prop:EquiEns\] [**Proposition** ]{}]{} (Equivalence of ensembles: Main theorem in [@S_Tasaki2016])]{} Let ${{\mathrm{B}}}_1$ be a hypercube or a pair of identical hypercubes in B, whose side length is $l=L^\alpha$ with $0\leq\alpha<1/2$. Under Assumptions \[assump:LocalInt\], \[assump:TransInv\], \[assump:ExpDecayT\], and \[assump:Massieu\], for a given inverse temperature $\beta>0$ and for any $\varepsilon_2>0$, $$\label{ineq:equi_ens}
\|
\mathrm{tr}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\hat{\rho}_{{{\mathrm{B}}}}^\mathrm{can}]
-
\mathrm{tr}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\hat{\rho}_{{{\mathrm{B}}}}^{\mathrm{MC}}]
\|_1
\leq
\varepsilon_2$$ holds for sufficiently large $N$. More precisely, the left-hand side above is bounded as $$\label{ineq:equi_ens_error}
\|
\mathrm{tr}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\hat{\rho}_{{\mathrm{B}}}^\mathrm{MC}]
-
\mathrm{tr}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\hat{\rho}^{\mathrm{can}}_\mathrm{B}]
\|_1
\leq
\mathcal{O} {\left(N^{-(1-2\alpha)/4+\delta}\right)},$$ where $\delta>0$ is an arbitrarily small constant.
The above theorem implies that the canonical and the microcanonical ensembles are locally indistinguishable, when we only look at ${{\mathrm{B}}}_1$ that is not too large.
Strictly speaking, $l$ should be an integer given by $\lfloor L^\alpha \rfloor $ (i.e., the maximum integer that is not larger than $L^\alpha$). To avoid too much complications of notations, however, we just omit $\lfloor \cdots \rfloor $ throughout this paper, and do not go into a technically strict argument that distinguishes whether $L^\alpha$ is an integer or not.
We note that Proposition \[prop:EquiEns\] in Ref. [@S_Tasaki2016] is a variant of Theorem 1 in Ref. [@S_Brandao2015]. We here employ the equivalence of the ensembles in the form of Ref. [@S_Tasaki2016] that is applicable up to $\alpha < 1/2$, while $\alpha < 1/(d+1)$ in Ref. [@S_Brandao2015].
Weak eigenstate-thermalization hypothesis (ETH) {#sec:weakETH}
===============================================
In this section, we formulate and prove the weak ETH in the setup introduced in the previous section. Let $\mathcal{M}_{U(\beta),\Delta} := \{ | E_i \rangle \}_{i=1}^D$ be the set of the eigenstates of $\hat{H}_{{\mathrm{B}}}$ in the energy shell $M_{U(\beta),\Delta}$. The goal of this section is to prove Eq. (5) in the main text, which states that $$\label{weak_ETH_main}
{{\mathrm{tr}}}[\hat{O}_{{{\mathrm{B}}}_1}
| E_i \rangle \langle E_i |]
\simeq
{{\mathrm{tr}}}[
\hat{O}_{{{\mathrm{B}}}_1}
\hat{\rho}_{{\mathrm{B}}}^{\mathrm{can}}]$$ holds for a typical choice of $| E_i \rangle$ from $\mathcal{M}_{U(\beta),\Delta}$, where ${{\mathrm{B}}}_1$ is a hypercube in B with side length $l=L^\alpha$ as is the case for Proposition \[prop:EquiEns\], and $\hat{O}_{{{\mathrm{B}}}_1}$ is any bounded operator on ${{\mathrm{B}}}_1$. We first note that $$\label{eq:wETH_split}
|
{{\mathrm{tr}}}[\hat{O}_{{{\mathrm{B}}}_1}
(
| E_i \rangle \langle E_i |
-
\hat{\rho}_{{\mathrm{B}}}^{\mathrm{can}}
)
]
|
\leq
|
{{\mathrm{tr}}}[\hat{O}_{{{\mathrm{B}}}_1}
(
| E_i \rangle \langle E_i |
-
\hat{\rho}_{{\mathrm{B}}}^{\mathrm{MC}}
)
]
|
+
|
{{\mathrm{tr}}}[\hat{O}_{{{\mathrm{B}}}_1}
(
\hat{\rho}_{{\mathrm{B}}}^{\mathrm{MC}}
-
\hat{\rho}_{{\mathrm{B}}}^{\mathrm{can}}
)
]
|
.$$ The second term on the right-hand side can be evaluated from the equivalence of the ensembles (i.e., Proposition \[prop:EquiEns\]). Therefore, we will focus on the first term on the right-hand side of inequality (\[eq:wETH\_split\]).
Weak ETH for operators
----------------------
We consider any operator $\hat O$ that is defined on the Hilbert space $\mathcal H_{{\mathrm{B}}}$. We define the expectation value of $\hat O$ for $| E_i \rangle \in \mathcal{M}_{U(\beta),\Delta}$ as $$O_i:=\bra{E_i}\hat{O}\ket{E_i}.$$ The microcanonical average of $\hat O$ is then rewritten as $$\overline{O}:= {\rm tr}[\hat O \hat{\rho}_{{{\mathrm{B}}}}^\mathrm{MC}]
= \frac{1}{D}\sum_{i\in M_{U(\beta),\Delta}} O_i.$$ We consider the fluctuation of $O_i$’s around $\overline{O}$, which can be quantified as $$(\Delta O)^2
:=
\frac{1}{D}
\sum_{i\in M_{U(\beta),\Delta}} (O_i)^2
-
\left( \overline{O} \right)^2.
\label{O_fluctuation}$$ If this fluctuation is very small, almost all the energy eigenstates (i.e., typical energy eigenstates) satisfy $O_i \simeq \overline{O}$, which is the main idea of the weak ETH:
[[\[def:WeakETH\] [**Definition** ]{}]{} (Weak ETH for operators [@S_Biroli2010])]{} We say that an observable $\hat{O}$ satisfies the weak ETH, if for any $\varepsilon>0$, $$(\Delta O)^2<\varepsilon
\label{eq:weakETH_def}$$ holds for sufficiently large $N$.
Biroli *et al.* [@S_Biroli2010] discussed the weak ETH for local operators, while their proof was not rigorous. We now make their argument rigorous, and extend it to quasi-local operators, where their argument for local operators is the case of $\alpha=0$ in the following lemma.
[[\[lemma:WeakETH\] [**Lemma** ]{}]{} (Weak ETH for quasi-local operators)]{} Let ${{\mathrm{B}}}_1$ be a hypercube in ${{\mathrm{B}}}$, whose side length is $l=L^\alpha$ with $0\leq \alpha< 1/2$. Under Assumptions \[assump:LocalInt\], \[assump:TransInv\], \[assump:ExpDecayT\], and \[assump:Massieu\], the weak ETH holds for any operator $\hat{O}_{{{\mathrm{B}}}_1}$ with support ${{\mathrm{B}}}_1$ and normalized as $\|\hat{O}_{{{\mathrm{B}}}_1}\|=1$.
![ Schematics for the proof of Lemma \[lemma:WeakETH\]. (a) Bath B is divided into hypercubes ${{\mathrm{C}}}_i$, which are identical copies of ${{\mathrm{C}}}_1$. (b) The set of blocks $\mathcal{C}$ is divided into $\mathcal{C}_{{\mathrm{a}}}$ and $\mathcal{C}_{{\mathrm{b}}}$ such that $\mathcal{C}_{{\mathrm{a}}}$ consists of ${{\mathrm{C}}}_1$ and its neighbors. []{data-label="fig:fig_s_blocks"}](fig_s_2.pdf){width="0.7\linewidth"}
[**Proof.**]{} Let $n:=|{{\mathrm{B}}}_1|$. We divide bath B into $N/n$ blocks, which are identical copies of ${{\mathrm{B}}}_1$. We label them as ${{\mathrm{C}}}_j$ with $j=1,\cdots,N/n$, where ${{\mathrm{C}}}_1$ is identified to ${{\mathrm{B}}}_1$ itself (see Fig. \[fig:fig\_s\_blocks\](a)). Correspondingly, $\hat{O}_{{{\mathrm{B}}}_1}$ is rewritten as $\hat{O}_{{{\mathrm{C}}}_1}$. Let $\mathcal{C}$ be the set of the blocks $\mathcal{C}:=\{{{\mathrm{C}}}_1,{{\mathrm{C}}}_2,\cdots,{{\mathrm{C}}}_{N/n}\}$. We also define $\hat{O}_{{{\mathrm{C}}}_j}~(j=2,3,\cdots, | \mathcal C|)$, which are translational copies of $\hat{O}_{{{\mathrm{C}}}_1}$ defined on ${{\mathrm{C}}}_j$. From Assumption \[assump:TransInv\] (translation invariance), the expectation values of $\hat{O}_{{{\mathrm{C}}}_j}$’s are the same: For any $| E_i \rangle$ ($i=1,2,\cdots, D$) and any $j$ ($=1,2,\cdots, | \mathcal C|$), $$\bra{E_i}
\hat{O}_{{{\mathrm{C}}}_j}
\ket{E_i}
=
\bra{E_i}
\hat{O}_{{{\mathrm{C}}}_1}
\ket{E_i}.$$ Therefore, for any $| E_i \rangle$, $$\bra{E_i}
\hat{O}_{{{\mathrm{B}}}_1}
\ket{E_i}
=
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}}
\bra{E_i}
\hat{O}_{{{\mathrm{C}}}_j}
\ket{E_i}.$$ By defining $$\hat O := \frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}}
\hat{O}_{{{\mathrm{C}}}_j},$$ we obtain $$\bra{E_i}
\hat{O}_{{{\mathrm{C}}}_1}
\ket{E_i}
=
\bra{E_i}
\hat{O}
\ket{E_i}.$$ We will then consider $\langle E_i | \hat O | E_i \rangle$ instead of $\langle E_i | \hat{O}_{{{\mathrm{C}}}_1} | E_i \rangle$ in the following.
We evaluate the fluctuation of $\langle E_i | \hat O | E_i \rangle$, quantified by $(\Delta O)^2$ in Eq. (\[O\_fluctuation\]), in line with Ref. [@S_Biroli2010]. From the Cauchy-Schwarz inequality, $(\Delta O)^2$ is bounded from above as $$(\Delta O)^2
\leq
\frac{1}{D}
\sum_{i\in M_{U(\beta),\Delta}}(O^2)_i
-
(\overline{O})^2
=
{\left\langle
\delta O^2
\right\rangle}_\mathrm{MC},$$ where $(O^2)_i := \langle E_i | \hat O^2 | E_i \rangle$ and $${\left\langle
\delta O^2
\right\rangle}_\mathrm{MC}
:=
{\rm tr} [
\hat O^2
\hat{\rho}^\mathrm{MC}_{{{\mathrm{B}}}}
] - \left( {\rm tr} [\hat O \hat \rho_{{\mathrm{B}}}^{\rm MC}] \right)^2.$$ Since $\hat{O}$ is the sum of $\hat{O}_{{{\mathrm{C}}}_j}$’s, we can expand ${\left\langle\delta O^2\right\rangle}_\mathrm{MC}$ as $$\begin{split}
{\left\langle
\delta O^2
\right\rangle}_\mathrm{MC}
&=
\frac{1}{|\mathcal{C}|^2}
\sum_{{{\mathrm{C}}}_j,{{\mathrm{C}}}_k\in\mathcal{C}}
{\left(
\langle
\hat{O}_{{{\mathrm{C}}}_j}\hat{O}_{{{\mathrm{C}}}_k}
\rangle_\mathrm{MC}
-
\langle
\hat{O}_{{{\mathrm{C}}}_j}
\rangle_\mathrm{MC}
\langle
\hat{O}_{{{\mathrm{C}}}_k}
\rangle_\mathrm{MC}
\right)}
\\
&=
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}}
{\left(
\langle
\hat{O}_{{{\mathrm{C}}}_j}\hat{O}_{{{\mathrm{C}}}_1}
\rangle_\mathrm{MC}
-
\langle
\hat{O}_{{{\mathrm{C}}}_j}
\rangle_\mathrm{MC}
\langle
\hat{O}_{{{\mathrm{C}}}_1}
\rangle_\mathrm{MC}
\right)}
\\
&=
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}}
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{MC}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
),
\end{split}
\label{eq:deltaOmc}$$ where $\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{MC}}(\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}) := \langle
\hat{O}_{{{\mathrm{C}}}_j}\hat{O}_{{{\mathrm{C}}}_1}
\rangle_\mathrm{MC}
-
\langle
\hat{O}_{{{\mathrm{C}}}_j}
\rangle_\mathrm{MC}
\langle
\hat{O}_{{{\mathrm{C}}}_1}
\rangle_\mathrm{MC}$, and we used translation invariance to obtain the second line.
The correlation functions on the right-hand side of Eq. (\[eq:deltaOmc\]) are bounded as $$\begin{split}
&
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}}
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{MC}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)
\\
\leq&
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}}
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)
+
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}}
{\left|
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{MC}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)
-
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)
\right|},
\end{split}
\label{eq:deltaOmc2}$$ where $\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}$ is the correlation function for the canonical ensemble.
To evaluate the first term on the right-hand side of inequality (\[eq:deltaOmc2\]), we devide $\mathcal{C}$ into $\mathcal{C}_{{\mathrm{a}}}$ and $\mathcal{C}_{{\mathrm{b}}}$ such that ${{\mathrm{C}}}_1$ and its neighbors are in $\mathcal{C}_{{\mathrm{a}}}$ (see Fig. \[fig:fig\_s\_blocks\](b)). We note that $|\mathcal{C}_{{\mathrm{a}}}|=3^d$. The first term on the right-hand side of inequality (\[eq:deltaOmc2\]) is written as $$\begin{split}
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}}
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)
&=
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}_{{\mathrm{a}}}}
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}
(
\hat{O}_{{{\mathrm{C}}}_1},\hat{O}_{{{\mathrm{C}}}_1}
)
+
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}_{{\mathrm{b}}}}
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
).
\end{split}
\label{eq:CorCanSplit}$$ From $|\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)|
\leq
2\|\hat{O}_{{{\mathrm{C}}}_j}\|\|\hat{O}_{{{\mathrm{C}}}_1}\|=2$, the first term on the right-hand side of Eq. (\[eq:CorCanSplit\]) is evaluated as $$\begin{split}
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}_{{\mathrm{a}}}}
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)
&\leq
2\frac{|\mathcal{C}_{{\mathrm{a}}}|}{|\mathcal{C}|}.
\end{split}$$ To evaluate the second term on the right-hand side of Eq. (\[eq:CorCanSplit\]), we use Assumption \[assump:ExpDecayT\] (exponential decay of correlation functions) and obtain $$|\mathrm{cor}_{\hat{\rho}^\mathrm{can}_{{{\mathrm{B}}}}}(\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1})|
\leq
\gamma_1
N^{2\alpha}
\exp\left[
-{{\mathrm{dist}}}({{\mathrm{C}}}_j,{{\mathrm{C}}}_1)/\xi
\right],$$ where $\gamma_1$ is a positive constant. If $\alpha=0$, the sum in the second term on the right-hand side of Eq. (\[eq:CorCanSplit\]) converges from Assumption \[assump:ExpDecayT\], and therefore, there exists a positive constant $\gamma_2$ such that $$\sum_{{{\mathrm{C}}}_j\in\mathcal{C}_{{\mathrm{b}}}}
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)
\leq
\gamma_2.$$ If $0<\alpha<1/2$, the sum in the second term on the right-hand side of Eq. (\[eq:CorCanSplit\]) converges to $0$ in the limit of $N\rightarrow\infty$: $$\begin{split}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}_{{\mathrm{b}}}}
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)
&\leq
\gamma_1
|\mathcal{C}_{{\mathrm{b}}}|
N^{2\alpha}
\exp
{\left[-\min_{k\in\mathcal{C}_{{\mathrm{b}}}}({{\mathrm{dist}}}({{\mathrm{C}}}_k,{{\mathrm{C}}}_1))/\xi\right]}
\\
&\leq
\gamma_1
N^{1-\alpha}
N^{2\alpha}
\exp
{\left[-\min_{k\in\mathcal{C}_{{\mathrm{b}}}}({{\mathrm{dist}}}({{\mathrm{C}}}_k,{{\mathrm{C}}}_1))/\xi\right]}
\rightarrow 0.
\end{split}$$ Thus, the sum in the first term on the right-hand side of inequality (\[eq:deltaOmc2\]) is bounded with a positive constant $\gamma_2$ as $$\begin{split}
\label{eq:wETH_Op_1}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}}
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)
&\leq
\gamma_2
\end{split}$$ for $0\leq\alpha<1/2$.
The second term on the right-hand side of inequality (\[eq:deltaOmc2\]) is bounded from Proposition \[prop:EquiEns\] (the equivalence of the ensembles): There exists a positive constant $\gamma_3$ such that $$\begin{split}
\label{eq:wETH_Op_2}
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}}
{\left|
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{MC}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)
-
\mathrm{cor}_{\hat{\rho}_{{\mathrm{B}}}^\mathrm{can}}
(
\hat{O}_{{{\mathrm{C}}}_j},\hat{O}_{{{\mathrm{C}}}_1}
)
\right|}
\leq&
\frac{1}{|\mathcal{C}|}
\sum_{{{\mathrm{C}}}_j\in\mathcal{C}}
\gamma_3 N^{-(1-2\alpha)/4+\delta}
\\
=&
\gamma_3 N^{-(1-2\alpha)/4+\delta}
\end{split}$$ holds for sufficiently large $N$.
By combining inequalities (\[eq:deltaOmc2\]), (\[eq:wETH\_Op\_1\]), and (\[eq:wETH\_Op\_2\]), we finally obtain $${\left\langle
\delta O^2
\right\rangle}_\mathrm{MC}
\leq
\gamma_2
N^{-(1-\alpha)}
+
\gamma_3 N^{-(1-2\alpha)/4+\delta}$$ for sufficiently large $N$. The right-hand side above can be arbitrarily small if $N$ is sufficiently large, which proves the lemma. $\Box$
From the proof of Lemma \[lemma:WeakETH\], we obtain $$(\Delta O)^2\leq
\mathcal{O}{\left(N^{-(1-\alpha)}\right)}
+
\mathcal{O}{\left(N^{-(1-2\alpha)/4+\delta}\right)},$$ where the dominant term on the right-hand side is $\mathcal{O}{\left(N^{-(1-2\alpha)/4+\delta}\right)}$.
Weak ETH with eigenstate-typicality
-----------------------------------
From Lemma \[lemma:WeakETH\], almost all the energy eigenstates give approximately the same expectation values of a quasi-local operator as that of the microcanonical ensemble. In this subsection, we formulate this in terms of typicality in the set of energy eigenstates.
Let ${\bf X}$ be a statement about the energy eigenstates in $\mathcal{M}_{U(\beta),\Delta}$ corresponding to the energy shell $M_{U(\beta),\Delta}$. We define the probability that $\bf X$ is true with respect to the uniform distribution on $\mathcal{M}_{U(\beta),\Delta}$: $$\begin{aligned}
P_{\mathcal{M}_{U(\beta),\Delta}}[{\bf X}]:=\frac{n_{\mathrm{true}}}{D},\end{aligned}$$ where $n_{\mathrm{true}}$ is the number of the energy eigenstates in $\mathcal{M}_{U(\beta),\Delta}$ for which statement ${\bf X}$ is true. We then define the concept of the eigenstate typicality as follows.
[[\[def:wETHTypical\] [**Definition** ]{}]{} ($\tilde{\varepsilon}$-eigenstate-typical statement)]{} Let $\tilde{\varepsilon}>0$. We say that statement $\bf X$ about the energy eigenstates in $\mathcal{M}_{U(\beta),\Delta}$ holds $\tilde{\varepsilon}$-eigenstate-typically, if $P_{\mathcal{M}_{U(\beta),\Delta}}[{\bf X}]>1-\tilde{\varepsilon}$ holds.
We now state the weak ETH for quasi-local observables in terms of the eigenstate typicality.
[[\[lemma:EigenMC\] [**Lemma** ]{}]{} (Weak ETH with eigenstate typicality)]{} Let ${{\mathrm{B}}}_1$ be a hypercube in ${{\mathrm{B}}}$, whose side length is $l=L^\alpha$ with $0\leq\alpha<1/2$. Let $\hat{O}$ be any operator on ${{\mathrm{B}}}_1$ with $\|\hat{O}\|=1$. Under Assumptions \[assump:LocalInt\], \[assump:TransInv\], \[assump:ExpDecayT\], and \[assump:Massieu\], for any $\varepsilon_1>0$ and any $\tilde{\varepsilon}>0$, $$\label{eq:LemmaWeakETH}
|O_i-\overline{O}|
\leq
\varepsilon_1$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$.
[**Proof.**]{} From the Chebyshev inequality, we have for any $\varepsilon_1 > 0$, $$\begin{aligned}
P_{\mathcal{M}_{U(\beta),\Delta}}[|O_i-\overline{O}|>\varepsilon_1]<
\frac{(\Delta O)^2}{\varepsilon_1^2}.\end{aligned}$$ From Lemma \[lemma:WeakETH\], for any $\varepsilon_1 > 0$ and any $\tilde \varepsilon > 0$, $$\begin{aligned}
\label{evaluate_N}
\frac{(\Delta O)^2}{\varepsilon_1^2} < \tilde \varepsilon\end{aligned}$$ holds for sufficiently large $N$. Therefore, $$\label{eq:weakETH_prob}
P_{\mathcal{M}_{U(\beta),\Delta}}[|O_i-\overline{O}|>\varepsilon_1]<\tilde{\varepsilon}$$ holds for sufficiently large $N$, which proves the lemma. $\Box$
From inequality (\[evaluate\_N\]) and the argument in Lemma \[lemma:WeakETH\], inequality (\[eq:LemmaWeakETH\]) holds $\tilde{\varepsilon}$-eigenstate-typically if $N$ satisfies $\mathcal{O}{\left(N^{-(1-2\alpha)/4+\delta}\right)} \leq \varepsilon_1^2 \tilde \varepsilon$. In other words, $$\label{ineq:LemmaWeakETH_error}
|O_i-\overline{O}|
\leq
\mathcal{O}{\left(\sqrt{N^{-(1-2\alpha)/4+\delta}/\tilde{\varepsilon}}\right)}$$ holds $\tilde{\varepsilon}$-eigenstate-typically.
By combining the weak ETH and the equivalence of the ensembles discussed in Sec. \[sec:MicroCanonical\], we obtain the weak ETH of the form (\[weak\_ETH\_main\]) (i.e., Eq. (5) in the main text). In fact, from Lemma \[lemma:EigenMC\] and Propositions \[prop:EquiEns\], we have the following corollary.
[[\[cor:EigenCan\] [**Corollary** ]{}]{}]{} Let ${{\mathrm{B}}}_1$ be a hypercube in ${{\mathrm{B}}}$, whose side length is $l=L^\alpha$ with $0\leq\alpha<1/2$. Let $\hat{O}$ be any operator on ${{\mathrm{B}}}_1$ with $\|\hat{O}\|=1$. Under Assumptions \[assump:LocalInt\], \[assump:TransInv\], \[assump:ExpDecayT\], and \[assump:Massieu\], for any $\varepsilon_{12} >0$ and any $\tilde \varepsilon > 0$, $$|
O_i
-
{{\mathrm{tr}}}_{{\mathrm{B}}}[
\hat{O}
\hat{\rho}^{\mathrm{can}}_\mathrm{B}
]
|
\leq
\varepsilon_{12}
\label{new_coro0}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$.
[**Proof.**]{} The left-hand side of inequality (\[new\_coro0\]) is bounded as $$\begin{aligned}
|
O_i-
{{\mathrm{tr}}}_{{\mathrm{B}}}[
\hat{O}
\hat{\rho}^{\mathrm{can}}_\mathrm{B}
]
|
\leq&
|
O_i
-\overline{O}
|
+
|
\overline{O}
-
{{\mathrm{tr}}}_{{\mathrm{B}}}[
\hat{O}
\hat{\rho}^{\mathrm{can}}_\mathrm{B}
]
|.
\label{ineq:WeakETHCan_proof1}\end{aligned}$$ From Lemma \[lemma:EigenMC\], for any $\varepsilon_{12}>0$ and any $\tilde{\varepsilon}>0$, $$\begin{aligned}
|
O_i
-\overline{O}
|
\leq
\varepsilon_{12}/2\end{aligned}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$. From Propositions \[prop:EquiEns\], for any $\varepsilon_{12}>0$, $$\begin{aligned}
|
\overline{O}
-
{{\mathrm{tr}}}_{{\mathrm{B}}}[
\hat{O}
\hat{\rho}^{\mathrm{can}}_\mathrm{B}
]
|
\leq&
\|\hat{O}\|
\|
{{\mathrm{tr}}}_{{{\mathrm{B}}}_2}[\hat{\rho}^{\mathrm{MC}}_\mathrm{B}]
-
{{\mathrm{tr}}}_{{{\mathrm{B}}}_2}[\hat{\rho}^{\mathrm{can}}_\mathrm{B}]
]
\|_1
\leq
\varepsilon_{12}/2\end{aligned}$$ holds for sufficiently large $N$. By summing up the above inequalities, we prove the corollary. $\Box$
From inequality (\[ineq:equi\_ens\_error\]) in Proposition \[prop:EquiEns\] and inequality (\[ineq:LemmaWeakETH\_error\]) in Lemma \[lemma:EigenMC\], inequality (\[new\_coro0\]) in Corollary \[cor:EigenCan\] holds if $N$ satisfies $\mathcal{O}{\left(N^{-(1-2\alpha)/4+\delta}\right)}\leq \varepsilon_{12} /2$ and $\mathcal{O}{\left(N^{-(1-2\alpha)/4+\delta}\right)} \leq \varepsilon_{12}^2 \tilde \varepsilon / 4$. In other words, $$\begin{aligned}
|
O_i
-
{{\mathrm{tr}}}_{{\mathrm{B}}}[
\hat{O}
\hat{\rho}^{\mathrm{can}}_\mathrm{B}
]
|
\leq
\mathcal{O}{\left(N^{-(1-2\alpha)/4+\delta}\right)}
+
\mathcal{O}{\left(\sqrt{N^{-(1-2\alpha)/4+\delta}/\tilde{\varepsilon}}\right)}\end{aligned}$$ holds $\tilde{\varepsilon}$-eigenstate-typically.
If $\alpha=0$, Corollary \[cor:EigenCan\] can be described in terms of the trace norm of the density operators of a subsystem.
[[\[cor:EigenCan\_TraceNorm\] [**Corollary** ]{}]{}]{} Let ${{\mathrm{B}}}_1$ be a hypercube in ${{\mathrm{B}}}$ with side length $l$ that is independent of the size $N$ of bath B. Under Assumptions \[assump:LocalInt\], \[assump:TransInv\], \[assump:ExpDecayT\], and \[assump:Massieu\], for any $\varepsilon_{14} >0$ and any $\tilde \varepsilon > 0$, $$\|
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\ket{E_i}\bra{E_i}]
-
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\hat{\rho}^{\mathrm{can}}_\mathrm{B}]
\|_1
\leq
\varepsilon_{14}
\label{new_coro0_tracenorm}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$.
[**Proof.**]{} Let $D_{{{\mathrm{B}}}_1}$ be the dimension of the Hilbert space of ${{\mathrm{B}}}_1$. Let $\{\hat{O}_{k}\}$ ($k=1,\dots, 2D_{{{\mathrm{B}}}_1}^2$ and $\|\hat{O}_{k}\|=1$) be a linear basis of the space of linear operators on ${{\mathrm{B}}}_1$. From Corollary \[cor:EigenCan\], for each $k$, for any $\varepsilon_{14,1}>0$ and any $\tilde{\varepsilon}>0$, we find that $$\begin{aligned}
\left|\mathrm{tr}_{{{\mathrm{B}}}_1}
{\left[
\hat{O}_{k}
{\left(
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\ket{E_i}\bra{E_i}]
-
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\hat{\rho}^{\mathrm{can}}_\mathrm{B}]
\right)}
\right]}
\right|
&\leq
\varepsilon_{14,1}\end{aligned}$$ holds $\tilde{\varepsilon}/(2D_{{{\mathrm{B}}}_1}^2)$-eigenstate-typically for sufficiently large $N$. Because $D_{{{\mathrm{B}}}_1}$ is independent of $N$, for any $\varepsilon_{14,1}>0$ and any $\tilde{\varepsilon}>0$, we find that $$\begin{aligned}
\left|\mathrm{tr}_{{{\mathrm{B}}}_1}
{\left[
\hat{O}_{k}
{\left(
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\ket{E_i}\bra{E_i}]
-
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\hat{\rho}^{\mathrm{can}}_\mathrm{B}]
\right)}
\right]}
\right|
&\leq
\varepsilon_{14,1}
\label{eq:cor:EigenCan_TN}\end{aligned}$$ holds for all $k=1,\cdots,2D_{{{\mathrm{B}}}_1}^2$, $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$.
Let $\hat{O}$ be any operator with support ${{\mathrm{B}}}_1$ and with $\|\hat{O}\|=1$. We can expand $\hat{O}$ as $\hat{O}=\sum_{k=1}^{2D_{{{\mathrm{B}}}_1}^2} c_k \hat{O}_{k}$ with $c_k\in\mathbb{C}$. We have $$\begin{aligned}
\begin{split}
{\left|
\mathrm{tr}_{{{\mathrm{B}}}_1}
{\left[
\hat{O}
{\left(
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\ket{E_i}\bra{E_i}]
-
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\hat{\rho}^{\mathrm{can}}_\mathrm{B}]
\right)}
\right]}
\right|}
&=
{\left|\mathrm{tr}_{{{\mathrm{B}}}_1}
{\left[
\sum_{k=1}^{2D_{{{\mathrm{B}}}_1}^2}
c_k
\hat{O}_{k}
{\left(
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\ket{E_i}\bra{E_i}]
-
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\hat{\rho}^{\mathrm{can}}_\mathrm{B}]
\right)}
\right]}
\right|}
\\
&\leq
\sum_{k=1}^{2D_{{{\mathrm{B}}}_1}^2}
|c_k|
{\left|
\mathrm{tr}_{{{\mathrm{B}}}_1}
{\left[
\hat{O}_{k}
{\left(
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\ket{E_i}\bra{E_i}]
-
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\hat{\rho}^{\mathrm{can}}_\mathrm{B}]
\right)}
\right]}
\right|}.
\end{split}\end{aligned}$$ From inequality (\[eq:cor:EigenCan\_TN\]), for any $\varepsilon_{14,1}>0$ and any $\tilde{\varepsilon}>0$, $$\begin{aligned}
\begin{split}
{\left|
\mathrm{tr}_{{{\mathrm{B}}}_1}
{\left[
\hat{O}
{\left(
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\ket{E_i}\bra{E_i}]
-
{{\mathrm{tr}}}_{{{\mathrm{B}}}\backslash{{\mathrm{B}}}_1}[\hat{\rho}^{\mathrm{can}}_\mathrm{B}]
\right)}
\right]}
\right|}
\leq
\sum_{k=1}^{2D_{{{\mathrm{B}}}_1}^2}
|c_k|
\varepsilon_{14,1}
\end{split}
\label{eq:cor:EigenCan_TN2}\end{aligned}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$. Because the set of $\hat{O}_k$ with $\|\hat{O}_k\|=1$ is compact, $$\begin{aligned}
c:=\max_{\|\hat{O}_{{{\mathrm{B}}}_1}\|=1}{\left(\sum_{k=1}^{2D_{{{\mathrm{B}}}_1}^2}|c_k|\right)}\end{aligned}$$ is finite, which is independent of $N$. Therefore, the right-hand side of inequality (\[eq:cor:EigenCan\_TN2\]) is bounded as $\sum_{k=1}^{2D_{{{\mathrm{B}}}_1}^2}
|c_k|
\varepsilon_{14,1}
\leq
\varepsilon_{14,1}c.
$ By letting $\varepsilon_{14} := \varepsilon_{14,1}c$, we prove the corollary. $\Box$
Our setup {#sec:setup}
=========
We now formulate the setup of our main theorems on the second law and the fluctuation theorem. We also introduce some definitions that are crucial in the proof of our main results.
![ Schematic of our setup. The total system consists of system S and bath B. System S is locally attached to a part of bath B. Bath B is a quantum many-body system on the $d$-dimensional hypercubic lattice with the periodic boundary condition and translation invariance. []{data-label="fig:fig_s_2a"}](fig_s_3.pdf){width="0.3\linewidth"}
We consider a composite system that consists of system S and bath B (see Fig. \[fig:fig\_s\_2a\] for a schematic). Let $\Lambda=(\Lambda_\mathrm{v},\Lambda_\mathrm{e})$ be the graph of the composite system, where $\Lambda_{\rm v}$ is the set of vertices (sites) and $\Lambda_{\rm e}$ is the set of edges (bonds), as is the case for Sec. \[sec:LRB\]. The sets of sites of system S and bath B are denoted by the same notations: ${{\mathrm{S}}}\subset \Lambda_{{{\mathrm{v}}}}$ and ${{\mathrm{B}}}\subset \Lambda_{{{\mathrm{v}}}}$, respectively. We note that ${{\mathrm{S}}}\cup {{\mathrm{B}}}= \Lambda_{{{\mathrm{v}}}}$ and ${{\mathrm{S}}}\cap {{\mathrm{B}}}= \phi$ with $\phi$ being the empty set. In the same manner as in Sec. \[sec:MicroCanonical\], we assume that B is the $d$-dimensional hypercubic lattice with the periodic boundary condition, where $N:=|{{\mathrm{B}}}|$. On the other hand, S can be an arbitrary graph.
The Hamiltonian of the total system is written as $$\hat{H}
:=
\hat{H}_\mathrm{S}
+
\hat{H}_{\mathrm{I}}
+
\hat{H}_{\mathrm{B}},
\label{eq:Hamiltonian_total}$$ where $\hat{H}_{\mathrm{S}}$ and $\hat{H}_{\mathrm{B}}$ are the Hamiltonians of system S and bath B, whose supports are ${{\mathrm{S}}}\subset \Lambda_{{{\mathrm{v}}}}$ and ${{\mathrm{B}}}\subset \Lambda_{{{\mathrm{v}}}}$, respectively. The interaction between system S and bath ${{\mathrm{B}}}$ is represented by $\hat{H}_\mathrm{I}$. We write the support of $\hat{H}_\mathrm{I}$ in bath B as ${{\mathrm{I}}}\subset \Lambda_{\rm v}$: ${{\mathrm{I}}}:={{\mathrm{B}}}\cap {{\mathrm{supp}}}(\hat{H}_{{\mathrm{I}}})$. We define $\tilde{{{\mathrm{S}}}}:={{\mathrm{S}}}\cup{{\mathrm{I}}}$. The Hilbert space of $\tilde{{{\mathrm{S}}}}$ is denoted by $\mathcal H_{\tilde{{{\mathrm{S}}}}}:=\otimes_{{{\mathrm{i}}}\in {\tilde {{\mathrm{S}}}}}\mathcal{H}_\mathrm{i}$ and the dimension of $\mathcal H_{\tilde{{{\mathrm{S}}}}}$ is denoted by $D_{\tilde{{{\mathrm{S}}}}}$. We also define $\tilde{{{\mathrm{B}}}}:= {{\mathrm{B}}}\backslash {{\mathrm{I}}}=({{\mathrm{S}}}\cup{{\mathrm{B}}})\backslash \tilde{{{\mathrm{S}}}}$.
We assume that the Hamiltonian of bath B is represented as the sum of local Hamiltonians, as in Eq. (\[l\_H\]). We further assume all of the foregoing Assumptions \[assump:LRB\], \[assump:LocalInt\], \[assump:TransInv\], \[assump:ExpDecayT\], and \[assump:Massieu\], such that all of the foregoing Propositions and Lemmas are applicable to bath B. We also assume that the interaction between S and B is local:
[[\[assump:LocalIntS\]]{} (Locality of the interaction between S and B)]{} We assume that $\rm I$ and $\hat{H}_{{\mathrm{I}}}$ are independent of $N$. We refer to $k' :={{\mathrm{dist}}}({{\mathrm{S}}}, \tilde{{{\mathrm{B}}}})$ as the interaction range between system S and bath B.
In the present setup with system S and bath B, the thermodynamic limit is defined as follows.
[[**Definition \[def:TDlimit\]’**]{} (Thermodynamic limit)]{} The thermodynamic limit is given by $N \to \infty$ with $\beta$ being fixed. In the thermodynamic limit, interaction ranges $k$ (in Assumption \[assump:LocalInt\]) and $k'$ (in Assumption \[assump:LocalIntS\]) are kept constant, and the graph structure and the Hamiltonian of system S do not change. We note that $D_{{\mathrm{S}}}$ and $D_{\tilde{{{\mathrm{S}}}}}$ do not change in the thermodynamic limit.
![ Schematic of regions on the graph. Bath B is divided into two regions: ${{\mathrm{B}}}_1$ and ${{\mathrm{B}}}_2$. We also define $\tilde{{{\mathrm{S}}}}:= {{\mathrm{S}}}\cup {{\mathrm{I}}}$, where $\tilde {{\mathrm{S}}}$ is localized around S. We assume that ${{\mathrm{I}}}\subset {{\mathrm{B}}}_1$. The information propagation from ${{\mathrm{B}}}_2$ is illustrated by the blue arrows, which does not reach $\tilde {{\mathrm{S}}}$ in a sufficiently short time regime. Before the information about ${{\mathrm{B}}}_2$ reaches $\tilde {{\mathrm{S}}}$, $\tilde {{\mathrm{S}}}$ does not feel the existence of ${{\mathrm{B}}}_2$. System S cannot distinguish whether the state of bath B is a typical energy eigenstate or the canonical distribution in the sufficiently short time regime. []{data-label="fig:fig_s_3"}](fig_s_4.pdf){width="0.5\linewidth"}
As discussed in the main text, the initial state of the composite system is given by $$\hat \rho (0) := \hat \rho_{{\mathrm{S}}}(0) \otimes \hat \rho_{{\mathrm{B}}}(0),
\label{setup_initial}$$ where $\hat \rho_{{\mathrm{S}}}(0) $ is an arbitrary state of system S. On the other hand, $$\hat{\rho}_{{\mathrm{B}}}(0) := | E_i \rangle \langle E_i |$$ is a typical energy eigenstate in the energy shell $M_{U(\beta),\Delta}$ of bath B, in the sense of the weak ETH as discussed in Sec. \[sec:weakETH\].
For the proof of our main results, we divide bath B into two regions: $\mathrm{B}_1$ and $\mathrm{B}_2$ with ${{\mathrm{B}}}={{\mathrm{B}}}_1\cup {{\mathrm{B}}}_2$ and ${{\mathrm{B}}}_1\cap{{\mathrm{B}}}_2=\phi$. We assume that ${{\mathrm{B}}}_1$ is a hypercube in ${{\mathrm{B}}}$, whose side length is $l=L^\alpha$ with $0<\alpha<1/2$. As shown in Fig. \[fig:fig\_s\_3\], $\mathrm{B}_1$ is near S, and $\mathrm{B}_2$ is far from S. By noting that ${{\mathrm{I}}}$ is localized around S because of the local interaction between S and B, we assume that ${{\mathrm{I}}}\subset {{\mathrm{B}}}_1$. The Hamiltonians of these regions are defined as $$\hat{H}_\mathrm{X}
=
\sum_{\mathrm{Z}\subset \mathrm{X}} \hat{h}_{\mathrm{Z}}
\quad(\mathrm{X}={{\mathrm{B}}}_1,{{\mathrm{B}}}_2).$$ We then define the interaction Hamiltonian between ${{\mathrm{B}}}_1$ and ${{\mathrm{B}}}_2$ as $$\hat{H}_{\partial\mathrm{B}_{1}}
:=
\hat{H}_{{\mathrm{B}}}-
\hat{H}_{\mathrm{B}_1}
-
\hat{H}_{\mathrm{B}_2},$$ and define the boundary of the regions as the support of the interaction Hamiltonian: $$\partial\mathrm{B}_{1}
:=
{{\mathrm{supp}}}(\hat{H}_{\partial\mathrm{B}_{1}}).$$
Truncated dynamics around system S {#sec:locality}
==================================
We now go into the main part of our proof. In this section, we introduce a [*reference dynamics*]{} with the initial canonical ensemble, to which the conventional proof of the second law and the fluctuation theorem is applicable. We then show that the dynamics around system S is almost the same for both the actual dynamics and the reference dynamics. In the following, we assume all of the foregoing Assumptions \[assump:LRB\], \[assump:LocalInt\], \[assump:TransInv\], \[assump:ExpDecayT\], \[assump:Massieu\], and \[assump:LocalIntS\].
Truncation of the dynamics
--------------------------
First of all, we compare the dynamics with the same initial condition $\hat{\rho}(0)$ as in Eq. (\[setup\_initial\]), but with different Hamiltonians: the actual Hamiltonian $\hat H$ and a truncated Hamiltonian $$\hat{H}_{{\mathrm{T}}}:=\hat{H}_\mathrm{S}+\hat{H}_{\mathrm{I}}+\hat{H}_{\mathrm{B}_1}.$$ The subscript ${{\mathrm{T}}}$ represents that the dynamics is “truncated”. The density operators at time $t$ under unitary evolutions with these Hamiltonians are given by $$\begin{split}
\hat{\rho}(t)
&{:=}e^{-i\hat{H}t}
\hat{\rho}(0)
e^{i\hat{H}t},
\\
\hat{\rho}_{{\mathrm{T}}}(t)
&{:=}e^{-i\hat{H}_{{\mathrm{T}}}t}
\hat{\rho}(0)
e^{i\hat{H}_{{\mathrm{T}}}t}.
\end{split}$$
We focus on the dynamics around system S (i.e., $\tilde{{{\mathrm{S}}}}$ defined in Sec. \[sec:setup\]). The corresponding reduced density operators are defined as $\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t):=\mathrm{tr}_{\tilde{{{\mathrm{B}}}}}[\hat{\rho}(t)]$ and $\hat{\rho}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t):=\mathrm{tr}_{\tilde{{{\mathrm{B}}}}}[\hat{\rho}_{{\mathrm{T}}}(t)]$. We then have the following lemma.
[[\[lemma:TruncDynamics\] [**Lemma** ]{}]{}]{} For any $\varepsilon_4>0$ and $t> 0$, $$\label{eq:LRB_DM1}
\|
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)
-
\hat{\rho}_{{\tilde{{{\mathrm{S}}}}},{{\mathrm{T}}}}(t)
\|_1
\leq
\varepsilon_4$$ holds for sufficiently large $N$.
[**Proof.**]{} Let $\hat{O}_{\tilde{{{\mathrm{S}}}}}$ be an arbitrary operator with support ${\tilde{{{\mathrm{S}}}}}$ and with operator norm $\|\hat{O}_\mathrm{\tilde{{{\mathrm{S}}}}}\|=1$. We have $$\label{eq:LRB_Lemma1_eq1}
\begin{split}
&
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}}}
(\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)-\hat{\rho}_{{\tilde{{{\mathrm{S}}}}},{{\mathrm{T}}}}(t)
)
]
|
\\
=&
|\mathrm{tr}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}
[
(
e^{i\hat{H}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}t}
-
e^{i\hat{H}_{{\mathrm{T}}}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}t}
)
\hat{\rho}_{{\tilde{{{\mathrm{S}}}}}}(0)
]
|
\\
\leq&
\|
(
e^{i\hat{H}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}t}
-
e^{i\hat{H}_{{\mathrm{T}}}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}t}
)
\|
\cdot
\|
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(0)
\|_1
\\
=&
\|
(
e^{i\hat{H}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}t}
-
e^{i\hat{H}_{{\mathrm{T}}}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}t}
)
\|,
\end{split}$$ where we used Proposition \[prop:IneqNorm\] from the second to the third line. To evaluate the final line of (\[eq:LRB\_Lemma1\_eq1\]), we calculate as $$\begin{split}
&
e^{i\hat{H}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}t}
-
e^{i\hat{H}_{{\mathrm{T}}}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}t}
\\
=&
\int_0^t ds\frac{d}{ds}
{\left(
e^{i\hat{H}s}
e^{i\hat{H}_{{\mathrm{T}}}(t-s)}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}(t-s)}
e^{-i\hat{H}s}
\right)}
\\
=&
i
\int_0^t
ds
\{
e^{i\hat{H}s}
(\hat{H}-\hat{H}_{{\mathrm{T}}})
e^{i\hat{H}_{{\mathrm{T}}}(t-s)}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}(t-s)}
e^{-i\hat{H}s}
\\
&\qquad
+
e^{i\hat{H}s}
e^{i\hat{H}_{{\mathrm{T}}}(t-s)}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
(\hat{H}_{{\mathrm{T}}}-\hat{H})
e^{-i\hat{H}_{{\mathrm{T}}}(t-s)}
e^{-i\hat{H}s}\}
\\
=&
i
\int_0^t
ds
\{
e^{i\hat{H}s}
(\hat{H}_{\partial\mathrm{B}_{1}}+\hat{H}_{\mathrm{B}_2})
e^{i\hat{H}_{{\mathrm{T}}}(t-s)}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}(t-s)}
e^{-i\hat{H}s}
\\
&\qquad
-
e^{i\hat{H}s}
e^{i\hat{H}_{{\mathrm{T}}}(t-s)}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}(t-s)}
(\hat{H}_{\partial\mathrm{B}_{1}}+\hat{H}_{\mathrm{B}_2})
e^{-i\hat{H}s}
\}
\\
=&
-i
\int_0^t
ds
e^{i\hat{H}s}
[
e^{i\hat{H}_{{\mathrm{T}}}(t-s)}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}(t-s)}
,
\hat{H}_{\partial\mathrm{B}_{1}}
]
e^{-i\hat{H}s},
\end{split}$$ where we used $[\hat{H}_{\mathrm{B}_2},\hat{O}_{\tilde{{{\mathrm{S}}}}}]=0$ and $[\hat{H}_{\mathrm{B}_2},\hat{H}_{{\mathrm{T}}}]=0$. We thus obtain $$\label{eq:norm_diff_short_S2}
\|
e^{i\hat{H}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}t}
-
e^{i\hat{H}_{{\mathrm{T}}}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}t}
\|
\leq
\int_0^t
ds
\|
[
e^{i\hat{H}_{{\mathrm{T}}}(t-s)}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}(t-s)}
,
\hat{H}_{\partial\mathrm{B}_{1}}
]
\|.$$ The Lieb-Robinson bound (Proposition \[prop:LRB\]) is now applicable to the right-hand side of inequality (\[eq:norm\_diff\_short\_S2\]). Let us set in Proposition \[prop:LRB\] $\hat{A}(t) := \hat{O}_{\tilde{{{\mathrm{S}}}}}(t-s)$ and $\hat{A}^\prime := \hat{H}_{\partial{{\mathrm{B}}}_{1}}$, and identify $t$ to $t-s$, A to $\tilde {{\mathrm{S}}}$, and ${{\mathrm{A}}}^\prime$ to $\partial {{\mathrm{B}}}_1$. By noting that the Hamiltonian is given by $\hat{H}_{{\mathrm{T}}}$, we obtain $$\label{eq:LRB_S_dynamics}
\|
[
e^{i\hat{H}_{{\mathrm{T}}}(t-s)}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}(t-s)}
,
\hat{H}_{\partial\mathrm{B}_{1}}
]
\|
\leq
C
\|
\hat{O}_{\tilde{{{\mathrm{S}}}}}
\|
\cdot
\|
\hat{H}_{\partial\mathrm{B}_{1}}
\|
\cdot
|{\tilde{{{\mathrm{S}}}}}|
\cdot
|\partial\mathrm{B}_{1}|
\cdot
e^{-\mu \mathrm{dist}({\tilde{{{\mathrm{S}}}}},\partial\mathrm{B}_{1})}
(e^{v|t-s|}-1),$$ where $C$ is the constant defined in Proposition \[prop:LRB\]. By substituting this to inequality (\[eq:norm\_diff\_short\_S2\]), we obtain $$\label{LR_S1}
\|
e^{i\hat{H}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}t}
-
e^{i\hat{H}_{{\mathrm{T}}}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}t}
\|
\leq
C
\|
\hat{O}_{\tilde{{{\mathrm{S}}}}}
\|
\cdot
\|
\hat{H}_{\partial\mathrm{B}_{1}}
\|
\cdot
|{\tilde{{{\mathrm{S}}}}}|
\cdot
|\partial\mathrm{B}_{1}|
\cdot
e^{-\mu\mathrm{dist}({\tilde{{{\mathrm{S}}}}},\partial\mathrm{B}_{1})}
(e^{vt}-vt-1).$$ By noting that $\| \hat{O}_{\tilde{{{\mathrm{S}}}}}
\| = 1$, we finally obtain, from inequalities (\[eq:LRB\_Lemma1\_eq1\]) and (\[LR\_S1\]) $$\label{eq:LRB_DM2}
\|
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)
-
\hat{\rho}_{{\tilde{{{\mathrm{S}}}}},{{\mathrm{T}}}}(t)
\|_1
\leq
C
\frac{
\|
\hat{H}_{\partial\mathrm{B}_{1}}
\|
}{v}
\cdot
|{\tilde{{{\mathrm{S}}}}}|
\cdot
|\partial\mathrm{B}_{1}|
\cdot
e^{-\mu\mathrm{dist}({\tilde{{{\mathrm{S}}}}},\partial\mathrm{B}_{1})}
(e^{vt}-vt-1).$$ The right-hand side of inequality (\[eq:LRB\_DM2\]) exponentially decreases with respect to the distance between $\tilde{{{\mathrm{S}}}}$ and $\partial{{\mathrm{B}}}_{1}$. Therefore, the right-hand side of inequality (\[eq:LRB\_DM2\]) can be arbitrarily small for sufficiently large $N$, and the lemma is proved. $\Box$
In our setup, the Lieb-Robinson time $\tau$ is defined as $$\tau:=\mu{{\mathrm{dist}}}(\tilde{{{\mathrm{S}}}},\partial{{\mathrm{B}}}_{1})/v
\simeq (\mu/v)N^{\alpha/d}
.$$ From the proof of Lemma \[lemma:TruncDynamics\], the left-hand side of inequality (\[eq:LRB\_DM1\]) is bounded by a term proportional to $
e^{-\mu N^{\alpha/d}}
(e^{vt}-vt-1).$ This term increases in time with $\mathcal{O}(t^2)$ up to $t\simeq 1/v$.
The reference dynamics {#sec:ref_dynamics}
----------------------
We now introduce the *reference dynamics*, written as $\hat \rho^{{\mathrm{R}}}(t)$, where the initial state is given by $$\hat \rho^{{\mathrm{R}}}(0) := \hat{\rho}_{{\mathrm{S}}}(0)\otimes\hat{\rho}_{{{\mathrm{B}}}}^{{\mathrm{can}}},
\label{ref_initial}$$ and the Hamiltonian is given by the actual one: $\hat{H}$. We note that the inverse temperature of $\hat{\rho}^\mathrm{can}_{{\mathrm{B}}}$ in $\hat{\rho}^{{\mathrm{R}}}(0)$ is determined by that of the corresponding energy shell defined in Definition \[def:Microcanonical\].
In the reference dynamics, the argument in Sec. \[sec:review\] (the conventional proof of the second law and the fluctuation theorem) is applicable. In this subsection, we will prove Lemma \[lemma:Reference\], which states that the difference between the actual and the reference dynamics can be arbitrarily small in the thermodynamic limit. Based on Lemma \[lemma:Reference\], we will prove the second law and the fluctuation theorem in Secs. \[sec:2nd\] and \[sec:FT\], respectively.
Before going to Lemma \[lemma:Reference\], we show a straightforward extension of Corollary \[cor:EigenCan\] to any operator in the composite system ${{\mathrm{S}}}\cup {{\mathrm{B}}}_1$.
[[\[cor:EigenCan2\] [**Corollary** ]{}]{}]{} Let $\hat{\rho}_{{\mathrm{S}}}$ be any density operator of system S. Let $\hat{O}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}_1}$ be any operator defined on ${{\mathrm{S}}}\cup{{\mathrm{B}}}_1$ with $\|\hat{O}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}_1}\|=1$. For any $\varepsilon_{13} >0$ and any $\tilde \varepsilon > 0$, $${\left|
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}{\left[
\hat{O}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}_1}
{\left(
\hat{\rho}_{{\mathrm{S}}}\otimes \ket{E_i}\bra{E_i}
-
\hat{\rho}_{{\mathrm{S}}}\otimes \hat{\rho}^{\mathrm{can}}_\mathrm{B}
\right)}\right]}\right|}
\leq
\varepsilon_{13}
\label{new_coro3}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$.
[**Proof.**]{} From Corollary \[cor:EigenCan\], for any $\varepsilon_{13} >0$ and any $\tilde \varepsilon > 0$, $${\left|
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}{\left[
\hat{O}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}_1}
{\left(
\hat{\rho}_{{\mathrm{S}}}(0) \otimes \ket{E_i}\bra{E_i}
-
\hat{\rho}_{{\mathrm{S}}}(0) \otimes \hat{\rho}^{\mathrm{can}}_\mathrm{B}
\right)}\right]}\right|}
\leq
\|{{\mathrm{tr}}}_{{\mathrm{S}}}[
\hat{O}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}_1}\hat{\rho}_{{\mathrm{S}}}(0)
]\|
\varepsilon_{13}
\label{eq:new_coro3_1}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$. Let $\hat{\rho}_{{{\mathrm{S}}}}=\sum_{k=1}^{D_{{{{\mathrm{S}}}}}}\rho_{kk}
\ket{\varphi_k}
\bra{\varphi_k}$ be the spectrum decomposition of $\hat{\rho}_{{{\mathrm{S}}}}$, where $\{\ket{\varphi_k}\}$ is an orthonormal basis of $\mathcal{H}_{{\mathrm{S}}}$. By defining $\hat{O}_{kj}:=\bra{\varphi_k}\hat{O}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}_1}\ket{\varphi_j}$, we have $$\begin{split}
\|{{\mathrm{tr}}}_{{\mathrm{S}}}[
\hat{O}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}_1}
\rho_{{\mathrm{S}}}]
\|
=&
{\left\|
\sum_{k=1}^{D_{{{{\mathrm{S}}}}}}
\rho_{kk}
\hat{O}_{kk}
\right\|}
\leq
\sum_{k=1}^{D_{{{{\mathrm{S}}}}}}
\rho_{kk}
\|
\hat{O}_{kk}
\|
\leq
\sum_{k=1}^{D_{{{{\mathrm{S}}}}}}
\rho_{kk}
\|
\hat{O}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}_1}
\|
=
1,
\end{split}
\label{eq:new_coro3_2}$$ where we used $\|\hat{O}_{kk}\|
\leq
\|\hat{O}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}_1}\|
=1
$ and $\sum_{k=1}^{D_{{\mathrm{S}}}}\rho_{kk}=1$. By combining inequalities (\[eq:new\_coro3\_1\]) and (\[eq:new\_coro3\_2\]), we prove the corollary. $\Box$
The reduced density operator of $\tilde{{{\mathrm{S}}}}$ in the reference dynamics is defined as $\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t):={{\mathrm{tr}}}_{\tilde{{{\mathrm{B}}}}}[\hat{\rho}^{{\mathrm{R}}}(t)]$. The difference between $\hat{\rho}_{\tilde{\mathrm{S}}}(t)$ and $\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t)$ is then evaluated as follows.
[[\[lemma:Reference\] [**Lemma** ]{}]{}]{} For any $\varepsilon_5>0$, any $\tilde{\varepsilon}>0$, and $t>0$, $$\label{eq:LRB_DM3}
\|
\hat{\rho}_{\tilde{\mathrm{S}}}(t)
-
\hat{\rho}^{{\mathrm{R}}}_{\tilde{\mathrm{S}}}(t)
\|_1
\leq
\varepsilon_5$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$.
[**Proof.**]{} Let $\{\hat{O}_{\tilde{{{\mathrm{S}}}},k}\}$ ($k=1,\dots, 2D_{\tilde{{{\mathrm{S}}}}}^2$ and $\|\hat{O}_{\tilde{{{\mathrm{S}}}},k}\|=1$) be a linear basis of the space of linear operators of $\mathcal{H}_{\tilde{{{\mathrm{S}}}}}$. We first note that $$\begin{aligned}
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t)
)
]
|
&\leq
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)-\hat{\rho}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t)
)
]
|
+
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
\hat{\rho}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t)
)
]
|
\nonumber
\\
&+
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t)
)
]
|.
\label{eq:LRB2_S_tilde}\end{aligned}$$ The first and the third terms on the right-hand side of inequality (\[eq:LRB2\_S\_tilde\]) are in the form of the first line of inequality (\[eq:LRB\_Lemma1\_eq1\]), to which we can apply Lemma \[lemma:TruncDynamics\]. Therefore, for any $\varepsilon_{5,1}>0$ and any $\varepsilon_{5,2}>0$, $$\begin{aligned}
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)-\hat{\rho}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t)
)
]
|
\leq \varepsilon_{5,1},
\label{eq:Lem4_51}
\\
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t)
)
]
|
\leq \varepsilon_{5,2}
\label{eq:Lem4_52}\end{aligned}$$ holds for sufficiently large $N$.
The second term on the right-hand side of inequality (\[eq:LRB2\_S\_tilde\]) is evaluated as $$\begin{split}
&|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
\hat{\rho}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t)
)
]
|
\\
=&
|\mathrm{tr}_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
{\rm tr}_{{{\mathrm{B}}}_2} [
e^{-i\hat{H}_{{\mathrm{T}}}t}
\hat{\rho}(0)
e^{i\hat{H}_{{\mathrm{T}}}t}
]
-
{\rm tr}_{{{\mathrm{B}}}_2} [
e^{-i\hat{H}_{{\mathrm{T}}}t}
\hat{\rho}^{{\mathrm{R}}}(0)
e^{i\hat{H}_{{\mathrm{T}}}t}
]
)
]
|
\\
=&
|\mathrm{tr}_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(e^{-i\hat{H}_{{\mathrm{T}}}t}\hat \rho_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}(0)e^{i\hat{H}_{{\mathrm{T}}}t}
-
e^{-i\hat{H}_{{\mathrm{T}}}t}
\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}(0)
e^{i\hat{H}_{{\mathrm{T}}}t}
)
]
|
\\
=&
|\mathrm{tr}_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}
[
e^{i\hat{H}_{{\mathrm{T}}}t}
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
e^{-i\hat{H}_{{\mathrm{T}}}t}
(\hat{\rho}_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}(0)
-
\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}(0)
)
]
|,
\end{split}
\label{eq:LRB2_S_tilde2}$$ where we used $\mathrm{tr}_{{{\mathrm{B}}}_2}[ e^{-i\hat{H}_{{\mathrm{T}}}t} \hat{\rho}(0) e^{i\hat{H}_{{\mathrm{T}}}t} ] = e^{-i\hat{H}_{{\mathrm{T}}}t}\hat \rho_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}(0)e^{i\hat{H}_{{\mathrm{T}}}t} $ with $\hat \rho_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}(0) := {\rm tr}_{{{\mathrm{B}}}_2} [\hat \rho (0)]$ from the second to the third line. By applying Corollary \[cor:EigenCan2\], for any $\varepsilon_{5,3}>0$ and any $\tilde{\varepsilon}>0$, $$|\mathrm{tr}_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}
[
e^{i\hat{H}_{{\mathrm{T}}}t}
\hat{O}_{\tilde{{{\mathrm{S}}}}}
e^{-i\hat{H}_{{\mathrm{T}}}t}
(\hat{\rho}_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}(0)
-
\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}\cup {{\mathrm{B}}}_1}(0)
)
]
|
\leq
\varepsilon_{5,3}$$ holds $\tilde{\varepsilon}/(2D_{\tilde{{{\mathrm{S}}}}}^2)$-eigenstate-typically for sufficiently large $N$.
By summarizing the foregoing argument, and by letting $\varepsilon_{5,4} := \varepsilon_{5,1}+\varepsilon_{5,2}+\varepsilon_{5,3}$, we find that for each $k$, for any $\varepsilon_{5,4}>0$ and any $\tilde{\varepsilon}>0$, $$\begin{aligned}
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t)
)
]
|
&\leq
\varepsilon_{5,4}
\label{eq:Lemma:Reference1}\end{aligned}$$ holds $\tilde{\varepsilon}/(2D_{\tilde{{{\mathrm{S}}}}}^2)$-eigenstate-typically for sufficiently large $N$. Because $D_{\tilde{{{\mathrm{S}}}}}$ is independent of $N$, for any $\varepsilon_{5,4}>0$ and any $\tilde{\varepsilon}>0$, $$\begin{aligned}
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t)
)
]
|
&\leq
\varepsilon_{5,4}
\label{eq:Lemma:Reference2}\end{aligned}$$ holds for all $k=1,\cdots,2D_{\tilde{{{\mathrm{S}}}}}^2$, $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$.
Let $\hat{O}_\mathrm{\tilde{{{\mathrm{S}}}}}$ be an arbitrary operator with support $\tilde{{{\mathrm{S}}}}$ and with operator norm $\|\hat{O}_\mathrm{\tilde{{{\mathrm{S}}}}}\|=1$. We can expand $\hat{O}_\mathrm{\tilde{{{\mathrm{S}}}}}$ as $\hat{O}_\mathrm{\tilde{{{\mathrm{S}}}}}=\sum_{k=1}^{2D_{\tilde{{{\mathrm{S}}}}}^2} c_k \hat{O}_{\tilde{{{\mathrm{S}}}},k}$ with $c_k\in\mathbb{C}$. We note that $$\begin{aligned}
\begin{split}
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}}}
(
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t)
)
]
|
&=
{\left|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
{\left[
\sum_{k=1}^{2D_{\tilde{{{\mathrm{S}}}}}^2}
c_k
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
{\left(
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t)
\right)}
\right]}
\right|}
\\
&\leq
\sum_{k=1}^{2D_{\tilde{{{\mathrm{S}}}}}^2}
|c_k|
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t)
)
]
|
\end{split}\end{aligned}$$ From inequality (\[eq:Lemma:Reference2\]), for any $\varepsilon_{5,4}>0$ and any $\tilde{\varepsilon}>0$, $$\begin{aligned}
\begin{split}
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}}}
(
\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t)
)
]
|
\leq
\sum_{k=1}^{2D_{\tilde{{{\mathrm{S}}}}}^2}
|c_k|
\varepsilon_{5,4}
\end{split}
\label{eq:Lemma:Reference3}\end{aligned}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$. Because the set of $\hat{O}_{\tilde{{{\mathrm{S}}}}}$ with $\|\hat{O}_{\tilde{{{\mathrm{S}}}}}\|=1$ is compact, $$\begin{aligned}
c:=\max_{\|\hat{O}_{\tilde{{{\mathrm{S}}}}}\|=1}{\left(\sum_{k=1}^{2D_{\tilde{{{\mathrm{S}}}}}^2}|c_k|\right)}\end{aligned}$$ is finite, which is independent of $N$ because $\tilde{{{\mathrm{S}}}}$ does not change in the thermodynamic limit. Therefore, the right-hand side of inequality (\[eq:Lemma:Reference3\]) is bounded as $\sum_{k=1}^{2D_{\tilde{{{\mathrm{S}}}}}^2}
|c_k|
\varepsilon_{5,4}
\leq
\varepsilon_{5,4}c.
$ By letting $\varepsilon_{5} := \varepsilon_{5,4}c$, the lemma is proved. $\Box$
In the above proof, we used the Lieb-Robinson bound (Lemma \[lemma:TruncDynamics\]) for evaluating the first and the third terms on the right-hand side of inequality (\[eq:LRB2\_S\_tilde\]). They are proportional to $
e^{-\mu\mathrm{dist}({\tilde{{{\mathrm{S}}}}},\partial\mathrm{B}_{1})}
(e^{vt}-vt-1)
$, which is exponentially small with respect to ${{\mathrm{dist}}}(\tilde{{{\mathrm{S}}}},\partial {{\mathrm{B}}}_1)\simeq N^{\alpha/d}$. We can evaluate the second term on the right-hand side of inequality (\[eq:LRB2\_S\_tilde\]) by Corollary \[cor:EigenCan2\] as $$\begin{aligned}
|\mathrm{tr}_{\tilde{{{\mathrm{S}}}}}
[
\hat{O}_{\tilde{{{\mathrm{S}}}},k}
(
\hat{\rho}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}},{{\mathrm{T}}}}(t)
)
]
|
\leq
\mathcal{O}{\left(N^{-(1-2\alpha)/4+\delta}\right)}
+
\mathcal{O}{\left(\sqrt{N^{-(1-2\alpha)/4+\delta}/\tilde{\varepsilon}}\right)}\end{aligned}$$ for sufficiently large $N$.
We note that $0<\alpha<1/2$ can be arbitrarily chosen. If $\alpha$ is smaller, $e^{-\mu N^{\alpha/d}}$ is larger but $N^{-(1-2\alpha)/4+\delta}$ is smaller. For sufficiently large $N$, $e^{-\mu N^{\alpha/d}}$ is smaller than $N^{-(1-2\alpha)/4+\delta}$ for any $0<\alpha<1/2$. Therefore, a good error evaluation for sufficiently large $N$ is obtained by setting $0<\alpha\ll 1/2$.
By summing up, we have the following observation:
[[\[obs:SizeDep\] [**Observation** ]{}]{} (Size and time dependence of Lemma \[lemma:Reference\])]{} Let $\tilde{\varepsilon}>0$ be the probability of atypical eigenstates. The left-hand side of inequality (\[eq:LRB\_DM3\]) in Lemma \[lemma:Reference\] is then bounded as $$\begin{aligned}
\label{eq:obs:SizeDep}
\|
\hat{\rho}_{\tilde{\mathrm{S}}}(t)
-
\hat{\rho}^{{\mathrm{R}}}_{\tilde{\mathrm{S}}}(t)
\|_1
\leq
\mathcal{O}{\left(N^{-(1-2\alpha)/4+\delta}\right)}
+
\mathcal{O}{\left(\sqrt{N^{-(1-2\alpha)/4+\delta}/\tilde{\varepsilon}}\right)}\end{aligned}$$ in the short time regime $t\ll\tau$ and for sufficiently large $N$, where we neglected the error term proportionate to $e^{-\mu\mathrm{dist}({\tilde{{{\mathrm{S}}}}},\partial\mathrm{B}_{1})}(e^{vt}-vt-1)$. We note that the neglected term increases in time with $\mathcal{O}(t^2)$ up to $t\simeq 1/v$.
Proof of the second law of thermodynamics {#sec:2nd}
=========================================
In this section, we prove the second law of thermodynamics \[inequality (3) in the main text\] in the foregoing setup.
Before going to the proof, we note the following general inequality:
[[\[prop:vNEntIneq\] [**Proposition** ]{}]{} (Inequality for the von Neumann entropy [@S_Andenaert2007])]{} Let $\hat{\rho}$, $\hat{\rho}^\prime$ be density operators on a finite-dimensional Hilbert space with dimension $D$. They satisfy $$\label{eq:ineq1}
|S(\hat{\rho})-S(\hat{\rho}^\prime)|
\leq
\delta_\rho\ln (D-1)+H(\delta_\rho),$$ where $\delta_\rho{:=}{\|\hat{\rho}-\hat{\rho}^\prime\|_1}/2$ and $H$ is the Shannon entropy: $$H(x)
:=
-x\ln x-(1-x)\ln(1-x).$$
We now go into the main argument. We define the following quantities as in the main text:
[[\[def:vNEntHeat\] [**Definition** ]{}]{} (The change in the von Neumann entropy of system S and the heat)]{} The change in the von Neumann entropy of system S and the heat absorbed by system S are respectively defined as $$\begin{split}
\Delta S_{{\mathrm{S}}}&:=
S(\hat{\rho}_{{\mathrm{S}}}(t))-S(\hat{\rho}_{{\mathrm{S}}}(0)),
\\
Q
&:=
-\mathrm{tr}_{{\mathrm{B}}}[
\hat{H}_\mathrm{B}(
\hat{\rho}_\mathrm{B}(t)
-
\hat{\rho}_\mathrm{B}(0)
)
].
\end{split}$$
The second law of thermodynamics is now stated as follows.
[[\[th:2nd\] [**Theorem** ]{}]{} (Second law of thermodynamics: inequality (3) in the main text)]{} For any $\varepsilon_{\mathrm{2nd}}>0$, any $\tilde{\varepsilon}>0$, and $t>0$, $$\label{eq:2nd}
\Delta S_{{\mathrm{S}}}-\beta {Q}\geq -\varepsilon_{\mathrm{2nd}}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$.
[**Proof.**]{} As in Lemma \[lemma:Reference\], we consider the reference dynamics introduced in Sec. \[sec:locality\]. The proof is divided into five parts.
*1. Second law for the reference dynamics.* From Proposition \[prop:2nd\], the second law holds without any error for the reference dynamics. We define the heat in the reference dynamics as $
Q^{{\mathrm{R}}}{:=}-{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[
\hat{H}_{{{\mathrm{B}}}}(\hat{\rho}^{{\mathrm{R}}}(t)-\hat{\rho}^{{\mathrm{R}}}(0))
],
$ and define the change in the von Neumann entropy in the reference dynamics as $\Delta S^{{\mathrm{R}}}_{{{\mathrm{S}}}}:= S(\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(t))-S(\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(0))$. The second law for the reference dynamics is then given by $$\label{eq:2nd_law_R}
\Delta S^{{\mathrm{R}}}_{{{\mathrm{S}}}}-\beta Q^{{\mathrm{R}}}\geq 0.$$
*2. Division of the von Neumann entropy.* The change in the von Neumann entropy can be written as $$\begin{split}
\Delta S_{{\mathrm{S}}}&=
S(\hat{\rho}_{{\mathrm{S}}}(t))-S(\hat{\rho}_{{\mathrm{S}}}(0))
\\
&=
S(\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(t))-S(\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(0))
+S(\hat{\rho}_{{\mathrm{S}}}(t))-S(\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(t))
-S(\hat{\rho}_{{\mathrm{S}}}(0))+S(\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(0))
\\
&=
\Delta S^{{\mathrm{R}}}_{{{\mathrm{S}}}}
+{\left(S(\hat{\rho}_{{\mathrm{S}}}(t))-S(\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(t)\right)},
\end{split}
\label{eq:diff_vonNeumann_S}$$ where we used $\hat{\rho}_{{\mathrm{S}}}(0) = \hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(0)$ from the second to the third line. The first term in the last line of Eq. (\[eq:diff\_vonNeumann\_S\]) satisfies the second law (\[eq:2nd\_law\_R\]) of the reference dynamics. We will then evaluate $S(\hat{\rho}_{{\mathrm{S}}}(t))-S(\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(t))$ as follows.
*3. Approximation of the von Neumann entropy.* By using Proposition \[prop:vNEntIneq\] \[Eq. (\[eq:ineq1\])\], we have $$\label{eq:2nd_2}
|
S(\hat{\rho}_{{\mathrm{S}}}(t))-S(\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(t))
|
\leq
\delta_{{\mathrm{S}}}\ln (D_{{\mathrm{S}}}-1)
+
H(\delta_{{\mathrm{S}}})
,$$ where $\delta_{{\mathrm{S}}}:=\|\hat{\rho}_{{\mathrm{S}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(t)\|_1 /2$. We have $\|\hat{\rho}_{{\mathrm{S}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(t)\|_1 \leq \|\hat{\rho}_{\tilde {{\mathrm{S}}}}(t)-\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t)\|_1$ from the monotonicity of the trace norm. Therefore, Lemma \[lemma:Reference\] is applicable to evaluate $\delta_{{\mathrm{S}}}$. In addition, the right-hand side of inequality (\[eq:2nd\_2\]) is continuous with respect to $\delta_{{\mathrm{S}}}$, and is equal to zero if $\delta_{{\mathrm{S}}}=0$. Therefore, for any $\varepsilon_{\mathrm{2nd}}>0$ and any $\tilde{\varepsilon}>0$, $$|
S(\hat{\rho}_{{\mathrm{S}}}(t))-S(\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(t))
|\leq\varepsilon_{\mathrm{2nd}}/3$$ holds $\tilde{\varepsilon}/3$-eigenstate-typically for sufficiently large $N$.
*4. Approximation of the heat.* We next compare ${Q^{{\mathrm{R}}}}$ with ${Q}$. The total average energy ${\mathrm{tr}_{\mathrm{S}\cup\mathrm{B}}}[ \hat{H} \hat \rho (t) ]$ is conserved with Hamiltonian $\hat{H}$. Therefore, we have $$\label{eq:Q1}
\begin{split}
{Q^{{\mathrm{R}}}}
&:=
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[
(\hat{H}_{{\mathrm{S}}}+\hat{H}_{{\mathrm{I}}})(\hat{\rho}^{{\mathrm{R}}}(t)-\hat{\rho}^{{\mathrm{R}}}(0))
],
\\
{Q}
&:=
{\mathrm{tr}_{\mathrm{S}\cup\mathrm{B}}}[
(\hat{H}_{{\mathrm{S}}}+\hat{H}_{{\mathrm{I}}})(\hat{\rho}(t)-\hat{\rho}(0))
].
\end{split}$$ By noting that the support of $\hat{H}_{{\mathrm{S}}}+\hat{H}_\mathrm{I}$ is $\tilde{\mathrm{S}}$, the deviation between ${Q}$ and ${Q^{{\mathrm{R}}}}$ is evaluated as $$\label{eq:diff_Q}
\begin{split}
|{Q}-{Q^{{\mathrm{R}}}}| \leq
|{{\mathrm{tr}}}_{\tilde{{{\mathrm{S}}}}}[
(\hat{H}_{{\mathrm{S}}}+\hat{H}_\mathrm{I})
(\hat{\rho}_{\tilde {{\mathrm{S}}}}(t)-\hat{\rho}_{\tilde {{\mathrm{S}}}}^{{\mathrm{R}}}(t))
]|
+
|{{\mathrm{tr}}}_{\tilde{{{\mathrm{S}}}}}[
(\hat{H}_{{\mathrm{S}}}+\hat{H}_\mathrm{I})
(\hat{\rho}_{\tilde {{\mathrm{S}}}}(0)-\hat{\rho}_{\tilde {{\mathrm{S}}}}^{{\mathrm{R}}}(0))
]|.
\end{split}$$
We evaluate the first term on the right-hand side of inequality (\[eq:diff\_Q\]) as $$\begin{split}
|{{\mathrm{tr}}}_{\tilde{{{\mathrm{S}}}}}[
(\hat{H}_{{\mathrm{S}}}+\hat{H}_\mathrm{I})
(\hat{\rho}_{\tilde {{\mathrm{S}}}}(t)-\hat{\rho}_{\tilde {{\mathrm{S}}}}^{{\mathrm{R}}}(t))
]|
\leq
\| \hat{H}_{{\mathrm{S}}}+\hat{H}_\mathrm{I} \| \| \hat{\rho}_{\tilde {{\mathrm{S}}}}(t)-\hat{\rho}_{\tilde {{\mathrm{S}}}}^{{\mathrm{R}}}(t) \|_1,
\end{split}$$ to which Lemma \[lemma:Reference\] is straightforwardly applicable. We note that $\| \hat{H}_{{\mathrm{S}}}+\hat{H}_\mathrm{I} \|$ is assumed to be independent of $N$. Therefore, for any $\varepsilon_{\mathrm{2nd}}>0$ and any $\tilde{\varepsilon}>0$, $$|{{\mathrm{tr}}}_{\tilde{{{\mathrm{S}}}}}[
(\hat{H}_{{\mathrm{S}}}+\hat{H}_\mathrm{I})
(\hat{\rho}_{\tilde {{\mathrm{S}}}}(t)-\hat{\rho}_{\tilde {{\mathrm{S}}}}^{{\mathrm{R}}}(t))
]|
\leq\varepsilon_{\mathrm{2nd}}/3$$ holds $\tilde{\varepsilon}/3$-eigenstate-typically for sufficiently large $N$.
The second term on the right-hand side of (\[eq:diff\_Q\]) can also be evaluated from Lemma \[cor:EigenCan2\]. Therefore, for any $\varepsilon_{\mathrm{2nd}}>0$ and any $\tilde{\varepsilon}>0$, $$|{{\mathrm{tr}}}_{\tilde{{{\mathrm{S}}}}}[
(\hat{H}_{{\mathrm{S}}}+\hat{H}_\mathrm{I})
(\hat{\rho}_{\tilde {{\mathrm{S}}}}(0)-\hat{\rho}_{\tilde {{\mathrm{S}}}}^{{\mathrm{R}}}(0))
]|
\leq \varepsilon_{\mathrm{2nd}}/3$$ holds $\tilde{\varepsilon}/3$-eigenstate-typically for sufficiently large $N$.
*5. Finishing the proof.* By summarizing the foregoing argument, for any $\varepsilon_{\mathrm{2nd}}>0$ and any $\tilde{\varepsilon}>0$, $$| (\Delta S_{{\mathrm{S}}}-\beta {Q}) - ( \Delta S^{{\mathrm{R}}}_{{{\mathrm{S}}}}-\beta Q^{{\mathrm{R}}}) | \leq \varepsilon_{\mathrm{2nd}}
\label{second_f}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$. By combining inequality (\[second\_f\]) with inequality (\[eq:2nd\_law\_R\]), we prove the theorem. $\Box$
The error term $\varepsilon_{\mathrm{2nd}}$ comes from the difference between the von Neumann entropies in system S, $S(\hat{\rho}_{{\mathrm{S}}}(t))-S(\hat{\rho}^{{\mathrm{R}}}_{{{\mathrm{S}}}}(t))$, and the heat $Q-Q^{{\mathrm{R}}}$ between the actual dynamics and the reference dynamics. From Observation \[obs:SizeDep\], we evaluate the error as $$| (\Delta S_{{\mathrm{S}}}-\beta {Q}) - ( \Delta S^{{\mathrm{R}}}_{{{\mathrm{S}}}}-\beta Q^{{\mathrm{R}}}) | \leq
\mathcal{O}{\left(N^{-(1-2\alpha)/4+\delta}\right)}
+
\mathcal{O}{\left(\sqrt{N^{-(1-2\alpha)/4+\delta}/\tilde{\varepsilon}}\right)}$$ in the short time regime $t\ll \tau$ and for sufficienly large $N$.
Proof of the fluctuation theorem {#sec:FT}
================================
In this section, we prove the integral fluctuation theorem with the heat bath in an energy eigenstate \[inequality (7) in the main text\] in the foregoing setup. We first prove the detailed fluctuation theorem, and then obtain the integral fluctuation theorem as a corollary. By noting that the initial state of bath B is an energy eigenstate, we can define the characteristic functions as follows:
[[\[def:CharaFunc\] [**Definition** ]{}]{} (Characteristic functions of the entropy production)]{} The characteristic functions of the entropy production in the forward and the reversed processes are respectively defined as $$\begin{split}
&
G_{{\mathrm{F}}}(u)
:=
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[
\hat{U}
e^{-iu\beta\hat{H}_{{{\mathrm{B}}}}}e^{iu\ln\hat{\rho}_{{\mathrm{S}}}(0)}
\hat{\rho}_{{\mathrm{S}}}(0)\otimes\hat{\rho}_{{\mathrm{B}}}(0)
\hat{U}^\dag
e^{-iu\ln\hat{\rho_{{\mathrm{S}}}}(t)}
e^{iu\beta\hat{H}_{{{\mathrm{B}}}}}
],
\\
&
G_{{\mathrm{R}}}(u)
:=
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[
\hat{U}^\dag
e^{-iu\beta\hat{H}_{{{\mathrm{B}}}}}e^{iu\ln\hat{\rho}_{{\mathrm{S}}}(t)}
\hat{\rho}_{{\mathrm{S}}}(t)\otimes\hat{\rho}_{{\mathrm{B}}}(0)
\hat{U}
e^{-iu\ln\hat{\rho_{{\mathrm{S}}}}(0)}
e^{iu\beta\hat{H}_{{{\mathrm{B}}}}}
],
\end{split}$$ where $u \in \mathbb C$.
We next give an important, but rather technical definition.
[[\[def:EpsilonI\] [**Definition** ]{}]{}]{} For $t>0$ and $u \in \mathbb C$, we define $\varepsilon_{{\mathrm{I}}}\geq 0$ by $$\label{eq:G_UV_assump}
\frac{\varepsilon_{{\mathrm{I}}}}{6}
:=
\limsup_{N \to \infty}
\|
e^{iu\hat{H}_{{\mathrm{B}}}}\hat{U} e^{-iu\hat{H}_{{\mathrm{B}}}}
-
e^{-iu\hat{H}_{{\mathrm{S}}}}
\hat{U}
e^{iu\hat{H}_{{\mathrm{S}}}}
\|,$$ where $N \to \infty$ means the thermodynamic limit.
We discuss the meaning of $\varepsilon_{{\mathrm{I}}}$. In the main text (Eq. (8)), we discussed an assumption on the interaction Hamiltonian: $$[\hat{H}_{{\mathrm{S}}}+\hat{H}_{{\mathrm{B}}},\hat{H}_{{\mathrm{I}}}]\simeq 0.$$ The precise meaning of the above assumption is now formulated as $$\varepsilon_{\rm I} \ll 1.$$ In fact, if $[\hat{H}_{{\mathrm{S}}}+\hat{H}_{{\mathrm{B}}},\hat{H}_{{\mathrm{I}}}]$ is exactly zero, $\|
e^{iu\hat{H}_{{\mathrm{B}}}}\hat{U} e^{-iu\hat{H}_{{\mathrm{B}}}}
-
e^{-iu\hat{H}_{{\mathrm{S}}}}
\hat{U}
e^{iu\hat{H}_{{\mathrm{S}}}}
\|$ is also exactly zero.
Logically speaking, we do not need to assume $\varepsilon_{\rm I} \ll 1$ in the following mathematical argument. However, if $\varepsilon_{\rm I}$ is not small, the error term can become large in the fluctuation theorem; our main theorem is physically meaningless in such a situation. Fortunately, our numerical simulation shown in the main text demonstrates that the contribution from $\varepsilon_{\rm I}$ is indeed negligible in our model of numerical simulation. In particular, the inset of Fig. 3 in the main text shows that the main contribution to the error term of the fluctuation theorem is not $\varepsilon_{\rm I}$.
The fluctuation theorem is now stated as follows.
[[\[th:FT\] [**Theorem** ]{}]{} (Detailed fluctuation theorem)]{} For any $\varepsilon_{\mathrm{FT}}>0$, any $\tilde{\varepsilon}>0$, $t>0$ and $u \in \mathbb C$, $$\label{eq:FT}
|G_{{\mathrm{F}}}(u)-G_{{\mathrm{R}}}(-u+i)|
\leq
\varepsilon_{\mathrm{FT}}+\varepsilon_{{\mathrm{I}}}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$.
[**Proof.**]{} We again consider the reference dynamics introduced in Sec. \[sec:locality\]. The proof is similar to that of the second law, which is divided into four parts.
*1. Fluctuation theorem for the reference dynamics.* We define the characteristic functions for the reference dynamics: $$\begin{split}
G^{{\mathrm{R}}}_{{{\mathrm{F}}}}(u)
&:=
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[
\hat{U}
e^{-iu\beta\hat{H}_{{{{\mathrm{B}}}}}}e^{iu\ln\hat{\rho}_{{{\mathrm{S}}}}^{{\mathrm{R}}}(0)}
\hat{\rho}_{{{\mathrm{S}}}}^{{\mathrm{R}}}(0)\otimes
\hat{\rho}^{{\mathrm{can}}}_{{{\mathrm{B}}}}
\hat{U}^\dag
e^{-iu\ln\hat{\rho}_{{{\mathrm{S}}}}^{{\mathrm{R}}}(t)}
e^{iu\beta\hat{H}_{{{{\mathrm{B}}}}}}
],
\\
G^{{\mathrm{R}}}_{{{\mathrm{R}}}}(u)
&:=
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[
\hat{U}^\dag
e^{-iu\beta\hat{H}_{{{{\mathrm{B}}}}}}e^{iu\ln\hat{\rho}_{{{\mathrm{S}}}}^{{\mathrm{R}}}(t)}
\hat{\rho}_{{{\mathrm{S}}}}^{{\mathrm{R}}}(t)\otimes\hat{\rho}_{{{\mathrm{B}}}}^{\mathrm{can}}
\hat{U}
e^{-iu\ln\hat{\rho}_{{{\mathrm{S}}}}^{{\mathrm{R}}}(0)}
e^{iu\beta\hat{H}_{{{{\mathrm{B}}}}}}
].
\end{split}$$ From Proposition \[prop:FT\], the fluctuation theorem holds without any error for the reference dynamics: $$\label{eq:FT_Reference}
G_{{{\mathrm{F}}}}^{{\mathrm{R}}}(u)-G_{{{\mathrm{R}}}}^{{\mathrm{R}}}(-u+i)=0.$$
*2. Rewriting the characteristic functions.* We define $$\begin{split}
\delta_{{\mathrm{I}}}&:=
\|
e^{iu\hat{H}_{{\mathrm{B}}}}\hat{U} e^{-iu\hat{H}_{{\mathrm{B}}}}
-
e^{-iu\hat{H}_{{\mathrm{S}}}}
\hat{U}
e^{iu\hat{H}_{{\mathrm{S}}}}
\|.
\end{split}$$ From Definition \[def:EpsilonI\] and the property of the limit superior, we obtain $$\delta_{{\mathrm{I}}}\leq \varepsilon_{{\mathrm{I}}}/4
\label{e_I}$$ for sufficiently large $N$. The characteristic functions can then be evaluated as $$\begin{split}
|
G_{{\mathrm{F}}}(u)
&-
{\mathrm{tr}_{\mathrm{S}\cup\mathrm{B}}}[
\hat{U}
\hat{V}_0
\hat{\rho}(0)
\hat{U}^\dag
\hat{V}_t^\dag
]
|
\leq \delta_{{\mathrm{I}}},
\\
|
G_{{{\mathrm{F}}}}^{{\mathrm{R}}}(u)
&-
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[
\hat{U}
\hat{V}_0
\hat{\rho}^{{\mathrm{R}}}(0)
\hat{U}^\dag
\hat{V}_{t}^{{{\mathrm{R}}}\dag}
]
|
\leq \delta_{{{\mathrm{I}}}},
\end{split}$$ where we defined $\hat{V}_t:=e^{iu\beta\hat{H}_{{\mathrm{S}}}}e^{iu\ln\hat{\rho}_{{\mathrm{S}}}(t)}$ and $\hat{V}_{t}^{{\mathrm{R}}}:=e^{iu\beta\hat{H}_{{\mathrm{S}}}}e^{iu\ln\hat{\rho}_{{{\mathrm{S}}}}^{{\mathrm{R}}}(t)}$. We can also evaluate $G_{{\mathrm{R}}}(u)$ and $G_{{{\mathrm{R}}}}^{{\mathrm{R}}}(u)$ in the same manner. The key point here is that we have rewritten the characteristic functions within a small error in such a way that Lemma \[lemma:Reference\] is applicable.
*3. The Lieb-Robinson bound.* We evaluate the difference between $G_{{{\mathrm{F}}}}(u)$ and $G^{{\mathrm{R}}}_{{{\mathrm{F}}}}(u)$ as follows: $$\begin{split}
|G_{{{\mathrm{F}}}}(u)-G_{{{\mathrm{F}}}}^{{\mathrm{R}}}(u)|
\leq&
2\delta_{{\mathrm{I}}}\\
&+
|
{\mathrm{tr}_{\mathrm{S}\cup\mathrm{B}}}[\hat{U}\hat{V}_0\hat{\rho}(0)\hat{U}^\dag\hat{V}_t^\dag]
-
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[\hat{U}\hat{V}_0\hat{\rho}^{{\mathrm{R}}}(0)\hat{U}^\dag\hat{V}_t^\dag]
|
\\
&+
|
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[\hat{U}\hat{V}_0\hat{\rho}^{{\mathrm{R}}}(0)\hat{U}^\dag\hat{V}_t^\dag]
-
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[\hat{U}\hat{V}_0\hat{\rho}^{{\mathrm{R}}}(0)\hat{U}^\dag\hat{V}_{t}^{{{\mathrm{R}}}\dag}]
|.
\end{split}
\label{eq:FT_diff_proof}$$ Since the support of $\hat{V}_t^\dag$ is S, Lemma \[lemma:Reference\] is applicable to the second term on the right-hand side of inequality (\[eq:FT\_diff\_proof\]), by replacing $\tilde {{\mathrm{S}}}$ with S, $\hat{\rho}(0)$ with $\hat{V}_0\hat{\rho}(0)$, and $\hat{\rho}^{{\mathrm{R}}}(0)$ with $\hat{V}_0\hat{\rho}^{{\mathrm{R}}}(0)$. Therefore, for any $\varepsilon_{\rm FT} > 0$ and any $\tilde \varepsilon >0$, $$|
{\mathrm{tr}_{\mathrm{S}\cup\mathrm{B}}}[\hat{U}\hat{V}_0\hat{\rho}(0)\hat{U}^\dag\hat{V}_t^\dag]
-
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[\hat{U}\hat{V}_0\hat{\rho}^{{\mathrm{R}}}(0)\hat{U}^\dag\hat{V}_t^\dag]
|
\leq
\varepsilon_{\rm FT} / 4
\label{F_proof1}$$ holds $\tilde \varepsilon /4$-eigenstate-typically for sufficiently large $N$.
The third term on the right-hand side of inequality (\[eq:FT\_diff\_proof\]) is evaluated as $$\begin{split}
&
|
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[\hat{U}\hat{V}_0\hat{\rho}^{{\mathrm{R}}}(0)\hat{U}^\dag\hat{V}_t^\dag]
-
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[\hat{U}\hat{V}_0\hat{\rho}^{{\mathrm{R}}}(0)\hat{U}^\dag\hat{V}_{t}^{{{\mathrm{R}}}\dag}]
|
\\
\leq&
\|
e^{-iu\beta\hat{H}_{{\mathrm{S}}}}\hat{U}\hat{V}_0\hat{\rho}^{{\mathrm{R}}}(0)\hat{U}^\dag
\|
\|
e^{iu\ln\hat{\rho}_{{\mathrm{S}}}(t)}
-
e^{iu\ln\hat{\rho}_{{{\mathrm{S}}}}^{{\mathrm{R}}}(t)}
\|_1.
\end{split}
\label{eq:FT_diff_proof2}$$ Since $e^{iu\ln x}$ is a continuous function of $x$, if $\|
\hat{\rho}_{{\mathrm{S}}}(t)
-
\hat{\rho}_{{{\mathrm{S}}}}^{{\mathrm{R}}}(t)
\|_1$ is small, $\|
e^{iu\ln\hat{\rho}_{{\mathrm{S}}}(t)}
-
e^{iu\ln\hat{\rho}_{{{\mathrm{S}}}}^{{\mathrm{R}}}(t)}
\|_1$ is also small. By applying Lemma \[lemma:Reference\], $\|
e^{iu\ln\hat{\rho}_{{\mathrm{S}}}(t)}
-
e^{iu\ln\hat{\rho}_{{{\mathrm{S}}}}^{{\mathrm{R}}}(t)}
\|_1$ is bounded from above. Therefore, for any $\varepsilon_{\rm FT} > 0$ and any $\tilde \varepsilon >0$, $$|
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[\hat{U}\hat{V}_0\hat{\rho}^{{\mathrm{R}}}(0)\hat{U}^\dag\hat{V}_t^\dag]
-
{{\mathrm{tr}}}_{{{\mathrm{S}}}\cup{{\mathrm{B}}}}[\hat{U}\hat{V}_0\hat{\rho}^{{\mathrm{R}}}(0)\hat{U}^\dag\hat{V}_{t}^{{{\mathrm{R}}}\dag}]
|
\leq
\varepsilon_{\rm FT} / 4
\label{F_proof2}$$ holds $\tilde \varepsilon /4$-eigenstate-typically for sufficiently large $N$.
By summing up inequalities (\[e\_I\]), (\[F\_proof1\]), and (\[F\_proof2\]), for any $\varepsilon_{\rm FT} > 0$ and any $\tilde \varepsilon >0$, $$| G_{{{\mathrm{F}}}}(u)-G_{{{\mathrm{F}}}}^{{\mathrm{R}}}(u)|
\leq
\varepsilon_{\rm FT}/2 + \varepsilon_{{\mathrm{I}}}/2$$ holds $\tilde \varepsilon /2$-eigenstate-typically for sufficiently large $N$. In the same manner, we can show that and for any $\varepsilon_{\rm FT} > 0$ and any $\tilde \varepsilon >0$, $$|G_{{{\mathrm{R}}}}(-u+i)-G_{{{\mathrm{R}}}}^{{\mathrm{R}}}(-u+i)|
\leq
\varepsilon_{\rm FT}/2 + \varepsilon_{{\mathrm{I}}}/2$$ holds $\tilde \varepsilon /2$-eigenstate-typically for sufficiently large $N$.
*4. Finishing the proof.* By combining the above results in the forward and the reversed processes, we finally prove that for any $\varepsilon_{\mathrm{FT}}>0$ and any $\tilde{\varepsilon}>0$, $$|G_{{\mathrm{F}}}(u)-G_{{\mathrm{R}}}(-u+i)|
\leq
\varepsilon_{\mathrm{FT}}+\varepsilon_{{\mathrm{I}}}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$. $\Box$
We note that the convergence of the left-hand side of inequality (\[eq:FT\]) is not uniform with respect to $u$ and $t$. As a corollary of Theorem \[th:FT\], the integral fluctuation theorem is obtained.
[[\[cor:IFT\_pure\] [**Corollary** ]{}]{} (Integral fluctuation theorem: inequality (7) in the main text)]{} For any $\varepsilon_{\mathrm{FT}}>0$, any $\tilde{\varepsilon}>0$, and $t>0$, $$\label{eq:IFT}
|\langle e^{-\sigma}\rangle-1|
\leq
\varepsilon_{\mathrm{FT}}+\varepsilon_{{\mathrm{I}}}$$ holds $\tilde{\varepsilon}$-eigenstate-typically for sufficiently large $N$.
[**Proof.**]{} As discussed in Corollary \[cor:IFT\], ${\left\langlee^{-\sigma}\right\rangle} = G_{\rm F}(i)$ and $G_{\rm R} (0) =1$. By substituting $u=i$ to the detailed fluctuation theorem (\[eq:FT\]), we obtain inequality (\[eq:IFT\]). $\Box$
The size dependence and the time dependence of the left-hand side of inequalities (\[eq:FT\]) or (\[eq:IFT\]) can be evaluated by the same manner as Observation \[obs:SizeDep\] in Sec. \[sec:ref\_dynamics\], but $\varepsilon_{{\mathrm{I}}}$ should be added to the right-hand side of inequality (\[eq:obs:SizeDep\]).
Typicality in the Hilbert space {#sec:typicality}
===============================
In the foregoing sections, we have discussed the second law and the fluctuation theorem in the case that the initial state of bath B is an energy eigenstate. In this section, we will discuss a slightly different setup in terms of the typicality in the whole Hilbert space of the energy shell, and prove a similar lemma to Lemma \[lemma:Reference\] in Sec. \[sec:ref\_dynamics\]. We note that we still assume that bath B is on a hypercubic lattice with the periodic boundary condition.
We suppose that the initial state of bath B is given by a pure state $$\begin{aligned}
\ket{\Psi}=\sum_{i\in M_{U(\beta),\Delta}}c_i \ket{E_i},
\label{eq:initial_Haar}\end{aligned}$$ where $c_i\in\mathbb{C}$ and $\sum_{i\in M_{U(\beta),\Delta}}|c_i|^2=1$. We note that $\ket{\Psi}$ is not necessarily a single energy eigenstate. The initial state of the total system is given by $$\begin{aligned}
\hat{\rho}(0)
=
\hat{\rho}_{{\mathrm{S}}}(0)\otimes\hat{\rho}_{{\mathrm{B}}}(0),\end{aligned}$$ where $\hat{\rho}_{{\mathrm{B}}}(0):=|\Psi\rangle \langle \Psi |$. The total system then obeys the unitary evolution with Hamiltonian $\hat{H}$ defined in Eq. (\[eq:Hamiltonian\_total\]), where all of the foregoing Assumptions \[assump:LRB\], \[assump:LocalInt\], \[assump:TransInv\], \[assump:ExpDecayT\], \[assump:Massieu\], and \[assump:LocalIntS\] are satisfied.
Let $\mathcal H_{{U(\beta),\Delta}}^\prime$ be the set of unit vectors in the Hilbert space $\mathcal H_{{U(\beta),\Delta}}$ of the energy shell. We consider the Haar measure (i.e., the uniform measure) of $\mathcal H_{{U(\beta),\Delta}}^\prime$. We denote by $| \mathcal V |$ the volume of a subset $\mathcal V \subset \mathcal H_{{U(\beta),\Delta}}^\prime$ with the Haar measure. We introduce the concept of $\tilde \varepsilon$-typical statement:
[[\[def:Typical\] [**Definition** ]{}]{} ($\tilde{\varepsilon}$-typical statement)]{} Let $\tilde{\varepsilon}>0$. We say that a statement $\bf X$ about a unit vector in $\mathcal H_{{U(\beta),\Delta}}^\prime$ holds $\tilde{\varepsilon}$-typically, if there exists a subset $\mathcal V\subset \mathcal H_{{U(\beta),\Delta}}^\prime$ such that ${|\mathcal V|}/{|\mathcal H_{{U(\beta),\Delta}}^\prime |}>1-\tilde{\varepsilon}$ and $\bf X$ holds for any $|\Psi\rangle\in\mathcal{V}$.
As discussed in Sec. \[subsec:FT\], it is necessary to employ the projection measurements on the initial and the final states in order to define the stochastic entropy production $\sigma$. If the initial state (\[eq:initial\_Haar\]) is not an energy eigenstate, it is projected to the diagonal ensemble $$\hat{\rho}_{{\mathrm{B}}}^\mathrm{diag}:=\sum_{i\in M_{U(\beta),\Delta}} |c_i|^2 \ket{E_i}\bra{E_i}.
\label{eq:initial_diag}$$ From Lemma \[lemma:EigenMC\] in Sec. \[sec:weakETH\], we obtain the following corollary.
[[\[cor:DiagMC\] [**Corollary** ]{}]{}]{} Let $\hat{O}$ be any operator on ${{\mathrm{B}}}_1$ with $\|\hat{O}\|=1$, where ${{\mathrm{B}}}_1$ is a hypercube in B with side length $l=L^\alpha$ with $0\leq \alpha<1/2$. For any $\varepsilon_6>0$ and any $\tilde{\varepsilon}>0$, $$|
{{\mathrm{tr}}}_{{\mathrm{B}}}[
\hat{O}
\hat{\rho}_{{\mathrm{B}}}^\mathrm{diag}
]
-
\overline{O}
|
\leq
\varepsilon_6
\label{eq:cor:DiagMC}$$ holds $\tilde{\varepsilon}$-typically for sufficiently large $N$.
[**Proof.**]{} By defining $O_i:=\langle E_i| \hat{O}|E_i\rangle$ with $|E_i\rangle\in\mathcal{M}_{U(\beta),\Delta}$, we have $$\label{eq:rhodist_diagonal1}
\begin{split}
|
{{\mathrm{tr}}}_{{\mathrm{B}}}[
\hat{O}
\hat{\rho}_{{\mathrm{B}}}^\mathrm{diag}
]
-
\overline{O}
|
&=
{\left|
\sum_{i\in M_{U(\beta),\Delta}}
|c_i|^2
(
O_i-\overline{O}
)
\right|}
\\
&\leq
\sum_{i\in M_{U(\beta),\Delta}}
|c_i|^2
|O_i-\overline{O}|.
\end{split}$$ From Lemma \[lemma:EigenMC\], for any $\varepsilon_{6,1}>0$ and any $\tilde{\varepsilon}_6>0$, we have $$\label{eq:diagonal_1}
|O_i-\overline{O}|
\leq
\varepsilon_{6,1}$$ $\tilde{\varepsilon}_6$-eigenstate-typically for sufficiently large $N$. We then split $M_{U(\beta),\Delta}$ into a typical subset $M_{{\mathrm{t}}}$ and an atypical subset $M_{{{\mathrm{a}}}}:=M\setminus M_{{\mathrm{t}}}$, where inequality (\[eq:diagonal\_1\]) is satisfied if and only if $i\in M_{{\mathrm{t}}}$. We note that $|M_{{{\mathrm{a}}}}|\leq\tilde{\varepsilon}_6 D$. The right-hand side of inequality (\[eq:rhodist\_diagonal1\]) is then evaluated as $$\label{eq:rhodist_diagonal2}
\begin{split}
\sum_{i\in M_{U(\beta),\Delta}}
|c_i|^2
|O_i-\overline{O}|
=&
\sum_{i\in M_{{\mathrm{t}}}}
|c_i|^2
|O_i-\overline{O}|
+
\sum_{i\in M_{{\mathrm{a}}}}
|c_i|^2
|O_i-\overline{O}|
\\
\leq &
\varepsilon_{6,1}
+
\sum_{i\in M_{{\mathrm{a}}}}
|c_i|^2
|O_i-\overline{O}|.
\end{split}$$
From the property of the Haar measure of $\mathcal H_{{U(\beta),\Delta}}^\prime$ [@S_Sugita2007], we have $$\begin{split}
\overline{
|c_i|^2
}
&=\frac{1}{D},
\\
\overline{
|c_i|^4
}
&=\frac{2}{D(D+1)},
\\
\overline{
|c_i|^2|c_j|^2
}
&=\frac{1}{D(D+1)}\quad (i\neq j),
\end{split}$$ where $\overline{\cdots}$ describes the average with respect to the Haar measure. Therefore, we obtain $$\begin{aligned}
\overline{
\sum_{i\in M_{{\mathrm{a}}}}|c_i|^2
}
&=
\frac{|M_{{\mathrm{a}}}|}{D},
\\
\overline{
{\left(\sum_{i\in M_{{\mathrm{a}}}}|c_i|^2\right)}^2
}
-
{\left(\overline{
\sum_{i\in M_{{\mathrm{a}}}}|c_i|^2
}\right)}^2
&=
\frac{1}{D}\frac{|M_{{\mathrm{a}}}|(D-|M_{{\mathrm{a}}}|)}{D(D+1)}
<
\frac{1}{D}.\end{aligned}$$
From the Chebyshev inequality, we have that for any $\varepsilon_{6,2} > 0$, $$\begin{aligned}
P{\left[
{\left|
\sum_{i\in M_{{\mathrm{a}}}}|c_i|^2
-
\frac{|M_{{\mathrm{a}}}|}{D}
\right|}
\leq\varepsilon_{6,2}
\right]}
\geq
1- \frac{1}{D\varepsilon_{6,2}^2}\end{aligned}$$ holds for sufficiently large $N$. Since $D$ exponentially increases in $N$, for any $\varepsilon_{6,2} > 0$ and any $\tilde \varepsilon > 0$, we have $$\begin{aligned}
\frac{1}{D\varepsilon_{6,2}^2}
< \tilde \varepsilon\end{aligned}$$ for sufficiently large $N$. By noting that $\|\hat{O}\|=1$, for any $\varepsilon_{6,2}$ and for any $\tilde{\varepsilon}$, we obtain $$\begin{split}
\sum_{i\in M_{{\mathrm{a}}}}
|c_i|^2
|O_i-\overline{O}|
&\leq
2
\frac{|M_{{\mathrm{a}}}|}{D}
+
2
{\left|
\sum_{i\in M_{{\mathrm{a}}}}
|c_i|^2-\frac{|M_{{\mathrm{a}}}|}{D}
\right|}
<
\varepsilon_{6,2}
\end{split}
\label{eq:rhodist_diagonal3}$$ $\tilde{\varepsilon}$-typically for sufficiently large $N$. Letting $\varepsilon_6:=\varepsilon_{6,1}+\varepsilon_{6,2}$, we prove the corollary from inequalities (\[eq:rhodist\_diagonal2\]) and (\[eq:rhodist\_diagonal3\]). $\Box$
We now discuss a variant of Lemma \[lemma:Reference\] in Sec. \[sec:ref\_dynamics\]. Let $\hat{\rho}(t):=e^{-i\hat{H}t}\hat{\rho}(0)e^{i\hat{H}t}$ with $\hat{\rho}(0):=\hat{\rho}_{{\mathrm{S}}}(0)\otimes \hat{\rho}_{{\mathrm{B}}}^\mathrm{diag}$. We again introduce the reference dynamics $\hat{\rho}^{{\mathrm{R}}}(t):=e^{-i\hat{H}t}\hat{\rho}^{{\mathrm{R}}}(0)e^{i\hat{H}t}$ with $\hat{\rho}^{{\mathrm{R}}}(0)$ defined in Eq. (\[ref\_initial\]). We define the reduced density operators on $\tilde{{{\mathrm{S}}}}$ as $\hat{\rho}_{\tilde{{{\mathrm{S}}}}}(t):={{\mathrm{tr}}}_{\tilde{{{\mathrm{B}}}}}[\hat{\rho}(t)]$ and $\hat{\rho}^{{\mathrm{R}}}_{\tilde{{{\mathrm{S}}}}}(t):={{\mathrm{tr}}}_{\tilde{{{\mathrm{B}}}}}[\hat{\rho}^{{\mathrm{R}}}(t)]$. From Proposition \[prop:EquiEns\], Corollary \[cor:DiagMC\], and Lemma \[lemma:TruncDynamics\], we can prove a similar lemma to Lemma \[lemma:Reference\] for the initial diagonal ensemble (\[eq:initial\_diag\]), by replacing $\tilde{\varepsilon}$-eigenstate-typically by $\tilde{\varepsilon}$-typically.
[[**Lemma \[lemma:Reference\]$^\prime$**]{}]{} For any $\varepsilon_5>0$, any $\tilde{\varepsilon}>0$, and $t>0$, $$\|
\hat{\rho}_{\tilde{\mathrm{S}}}(t)
-
\hat{\rho}^{{\mathrm{R}}}_{\tilde{\mathrm{S}}}(t)
\|_1
\leq
\varepsilon_5$$ holds $\tilde{\varepsilon}$-typically for sufficiently large $N$.
From Lemma \[lemma:Reference\]$^{\prime}$, we can prove our main results (Theorems \[th:2nd\] and \[th:FT\], and Corollary \[cor:IFT\_pure\]) also for typical pure states in the whole Hilbert space, by again replacing $\tilde{\varepsilon}$-eigenstate-typically to $\tilde{\varepsilon}$-typically. We note that the initial state is projected to the diagonal ensemble, and therefore we should replace $\hat{\rho}_{{\mathrm{B}}}$ by $\hat{\rho}_{{\mathrm{B}}}^\mathrm{diag}$ in Definitions \[def:vNEntHeat\] and \[def:CharaFunc\].
Discussion {#sec:Discussion}
==========
We have proved the second law of thermodynamics and the fluctuation theorem with a pure-state bath. To make the proof mathematically rigorous, we made several technical assumptions. For example, we assumed that the heat bath is on a hypercubic lattice with the periodic boundary condition and translation invariance. Although these assumptions are technically necessary for our proof, we can physically expect that the lattice structure is neither restricted to hypercubes nor to the periodic boundary condition, in order to obtain essentially the same results.
On the other hand, there is a subtle problem on the assumption of translation invariance. We can again physically expect that local breaking of translation invariance (by for example a local defect) does not affect the essentials of our results. However, if there is strong global disorder that leads to the Anderson localization or the many-body localization (MBL), it has been shown that the weak ETH is no longer valid [@S_Pal2010; @S_Imbrie2016]. In this sense, the absence of localization is crucial to obtain our results.
We also emphasize that the spatial locality of interaction, represented by Assumptions \[assump:LRB\], \[assump:LocalInt\], and \[assump:LocalIntS\], is crucially important. The spatial nonlocality leads to anomalous Lieb-Robinson time that is proportional to $\ln N$ with system size $N$ [@S_Lashkari2013], as is the case for the Sachdev-Ye-Kitaev model [@S_Maldacena2016].
We next remark on the concept of typicality. In the proof of our main results, we used the concept of typicality in Corollary \[cor:EigenCan\] for a typical energy eigenstates or in Corollary \[cor:DiagMC\] for a typical pure state in the whole Hilbert space. However, if we assume that inequality (\[new\_coro0\]) in Corollary \[cor:EigenCan\] holds for a given state, we can prove our main results without invoking the concept of typicality. More explicitly, we can prove the following corollary.
[[\[cor:WithoutTypicality\] [**Corollary** ]{}]{} (Main results without typicality)]{} Suppose that there exists a sequence of state vectors of bath B, written as $\{\ket{\Psi_N}\}_{N\in\mathbb{N}}$, which satisfies the following. Let ${{\mathrm{B}}}_1$ be a hypercube in ${{\mathrm{B}}}$, whose side length is $l=L^\alpha$ with $0\leq\alpha<1/2$. Let $\hat{O}$ be any operator on ${{\mathrm{B}}}_1$ with $\|\hat{O}\|=1$. For any $\varepsilon >0$, $${\left|
{{\mathrm{tr}}}_{{\mathrm{B}}}{\left[
\hat{O}{\left(
\hat{\rho}^\mathrm{diag}[\ket{\Psi_N}]
-
\hat{\rho}_{{\mathrm{B}}}^{{\mathrm{can}}}\right)}
\right]}
\right|}
\leq
\varepsilon$$ holds for sufficiently large $N$, where $\hat{\rho}^\mathrm{diag}[\ket{\Psi_N}]:=\sum_{i\in M_{U(\beta),\Delta}} \ket{E_i} \bra{E_i} |\langle\Psi_N|E_i\rangle|^2$. Then, under Assumptions \[assump:LRB\], \[assump:LocalInt\], and \[assump:LocalIntS\], our main results (Theorems \[th:2nd\] and \[th:FT\], and Corollary \[cor:IFT\_pure\]) hold true for $\{| \Psi_N \rangle\}$, by removing the phrases that “for any $\tilde \varepsilon> 0$” and that “$\tilde \varepsilon$-eigenstate-typically” or “$\tilde \varepsilon$-typically”.
Details of the numerical simulation {#sec:Numerical}
===================================
In this section, we show the details of our numerical simulation and the supplementary numerical results.
The Hamiltonian
---------------
We considered hard-core bosons with nearest-neighbor repulsions, whose Hamiltonian is given by: $$\begin{aligned}
\hat{H}_{{\mathrm{S}}}&=
\omega\hat n_0
,
\quad
\hat{H}_{{\mathrm{I}}}=
-\gamma^\prime
\sum_{<0,j>}
(\hat{c}^\dag_0 \hat{c}_j+\hat{c}^\dag_j \hat{c}_0),\\
\hat{H}_{{\mathrm{B}}}&=
\omega
\sum_i \hat n_i
-
\gamma
\sum_{<i,j>}
(\hat{c}^\dag_i \hat{c}_j+\hat{c}^\dag_j \hat{c}_i)
+
g
\sum_{<i,j>}
\hat{n}_i \hat{n}_j,\end{aligned}$$ where $\omega>0$ is the on-site potential, $-\gamma$ is the hopping rate in bath B, $-\gamma^\prime$ is the hopping rate between system S and bath B, and $g>0$ is the repulsion energy. The annihilation (creation) operator of bosons at site $i$ is written as $\hat{c}_i$ ($\hat{c}_i^\dag$), which satisfies the commutation relations $[\hat{c}_i,\hat{c}_j]=[\hat{c}^\dag_i,\hat{c}^\dag_j]=[\hat{c}_i,\hat{c}^\dag_j]=0$ for $i\neq j$, $\{\hat{c}_i,\hat{c}_i\}=\{\hat{c}^\dag_i,\hat{c}^\dag_i\}=0$, $\{\hat{c}_i,\hat{c}^\dag_i\}=1$.
Bath B is on the two-dimensional lattice with the open boundary condition. We here employed the open boundary condition for our numerical simulation in order to make the Lieb-Robinson time larger, while the periodic boundary condition is assumed in our rigorous proof. We expect that this difference does not matter as discussed in Sec. \[sec:Discussion\]. System S is on a single site that is attached to one of the sites in bath B.
The initial state is a product state of system S and bath B : $$\label{eq:initial_numerical}
\hat{\rho}(0)=\ket{\psi}\bra{\psi}\otimes\ket{\Psi}\bra{\Psi},$$ where $\ket{\psi} :=\hat{c}_0^\dag\ket{0}$ and $| \Psi \rangle$ is a pure state of bath B. The time evolution operator $\hat{U}=\exp(-i\hat{H}t/\hbar)$ is calculated by the full exact diagonalization of the total Hamiltonian $\hat H$.
Temperature of bath B
---------------------
![ The inverse temperatures against the energy eigenvalues. The bath is on $16=4\times4$ sites with 4 bosons. The hopping rate within bath B is taken as $\gamma/\omega=1$, and the repulsion energy within bath B is taken as $g/\omega=0.1$. []{data-label="fig:fig4"}](fig_s_5.pdf){width="0.5\linewidth"}
In our rigorous theory, temperature of bath B is defined by the corresponding energy shell in Definition \[def:EnergyShell\]. In practice, we numerically calculate the temperature of pure state $\ket{\Psi}$ as follows. The average energy density of $| \Psi \rangle$ is given by $e (\Psi):= \langle \Psi | \hat{H}_{{\mathrm{B}}}| \Psi \rangle / N$. Then, there exists $\beta$ ($0<\beta < \infty$) such that $$\begin{aligned}
\label{eq:def_temp_supppl}
e(\Psi )=u^\mathrm{can}(\beta),\end{aligned}$$ where $u^\mathrm{can}(\beta)$ is given by Eq. (\[u\_beta\]). We refer to $\beta$ in Eq. (\[eq:def\_temp\_supppl\]) as the inverse temperature of $| \Psi \rangle$. Figure \[fig:fig4\] shows the inverse temperatures of energy eigenstates, which are plotted against the energy eigenvalues.
The Lieb-Robinson time
----------------------
We consider parameters $\lambda_0$, $p_0$, $C$, $v$, and $\tau$ discussed in Sec. \[sec:LRB\] in terms of the Lieb-Robinson bound. Because of the nearest neighbor interaction and the two-dimensional square lattice in our model, the above parameters are given as follows: $\lambda_0=\sqrt{\gamma^2+g^2}$, $p_0=2$, $C=1/2$, $v=2\sqrt{\gamma^2+g^2}$, and $\tau={{\mathrm{dist}}}(\tilde{S},\partial{{\mathrm{B}}}_1) /( 2\sqrt{\gamma^2+g^2})$. We choose $\mu=1$ to maximize $\tau$. In addition, we set ${{\mathrm{B}}}_1$ such that ${{\mathrm{dist}}}(\tilde{S},\partial{{\mathrm{B}}}_1) =1$, and we have the Lieb-Robinson time $\tau=( 2\sqrt{\gamma^2+g^2})^{-1}$.
Numerical verification of the detailed fluctuation theorem
----------------------------------------------------------
![ The eigenstate dependence and the time dependence of $\delta G(t)$. The bath is on $12=4\times3$ sites with 4 bosons, and the hopping rate within bath B is taken as $\gamma/\omega=1$, where the Lieb-Robinson time is $\tau \simeq 1/\omega$. The repulsion energy within bath B is given by $g/\omega=0.1$, and the system-bath coupling is given by $\gamma^\prime/\omega=0.1$. (a) The eigenstate dependence of $\delta G(t)$ with $\omega t=0.01$. The eigenstates are chosen within $123\leq i\leq 222$. The inverse temperatures of the initial eigenstates with $i=123$ and $i=222$ are $0.292$ and $0.052$, respectively. The red arrows indicate $i=175, 168$ and $160$. (b) The time dependence of $\delta G(t)$. The eigenstates are taken with $i=175, 168$ and 160. In the short time regime, the time dependence of the error is consistent with Observation \[obs:SizeDep\]: $\delta G(t)\propto t^2$. []{data-label="fig:dft2"}](fig_s_6.pdf){width="\linewidth"}
In the main text, we showed the numerical results on the second law and the integral fluctuation theorem. In this subsection, we numerically study the detailed fluctuation theorem (Theorem \[th:FT\]), where the initial state is an energy eigenstate (i.e., $|\Psi \rangle = |E_i\rangle$ in Eq. (\[eq:initial\_numerical\])).
We first discuss the eigenstate dependence of the difference between $G_{{\mathrm{F}}}(u)$ and $G_{{\mathrm{R}}}(-u+i)$. We define an integrated error $\delta G(t)$ as $$\delta G(t):=
\int_0^{2\pi}
du|G_{{\mathrm{F}}}(u)-G_{{\mathrm{R}}}(-u+i)|.$$ Figure \[fig:dft2\] (a) shows $\delta G(t)$ with $\varepsilon t=0.01$, which is plotted against index $i$ of the initial eigenstate $|E_i\rangle$ of bath B. While $\delta G(t)$ is smaller than $3\times10^{-6}$ for most of the energy eigenstates, there are atypical eigenstates with larger $\delta G(t)$, which is consistent to Theorem \[th:FT\]. Figure \[fig:dft2\] (b) shows time dependence of $\delta G(t)$, which is proportional to $t^2$ in the short time regime. This is consistent with the theoretical prediction based on the Lieb-Robinson bound (Observation \[obs:SizeDep\]).
We focus on the case that the initial state of bath B is taken as $\ket{E_i}$ with $i=175$, whose $\delta G(t)$ is smallest in Fig. \[fig:dft2\] (b). Figures \[fig:dft3\] (a)-(c) show the case of weak coupling between system S and bath B, and Figs. \[fig:dft3\] (d)-(f) show the case of strong coupling. In the short time regime, the detailed fluctuation theorem holds within a small error for the both cases of the weak and the strong couplings. As time increases, the fluctuation theorem tends to be violated as in Fig. \[fig:dft3\] (c) and (f).
![ The real parts of $G_{{\mathrm{F}}}(u)$ and $G_{{\mathrm{R}}}(-u+i)$ for energy eigenstate $i=175$. The bath is on $12=4\times3$ sites with 4 bosons, and the hopping rate within bath B is taken as $\gamma/\omega=1$, where the Lieb-Robinson time is $\tau \simeq 1/\omega$. The repulsion energy within bath B is given by $g/\omega=0.1$. The upper panels (a)-(c) and the lower panels (d)-(f) show the weak and strong coupling cases, respectively. In the short time regime (i.e., $\omega t\leq1$), the fluctuation theorem holds within small error. As time increases, the fluctuation theorem tends to be violated as in (c) and (f). []{data-label="fig:dft3"}](fig_s_7.pdf){width="\linewidth"}
Numerical results on typical pure states in the Hilbert space
-------------------------------------------------------------
In this subsection, we show numerical results for the case that the initial state of bath B is the diagonal ensemble (\[eq:initial\_diag\]), which is obtained by projecting $|\Psi\rangle$ in Eq. (\[eq:initial\_numerical\]) that is sampled from the whole Hilbert space of the energy shell with respect to the Haar measure. As discussed in Sec. \[sec:typicality\], we can prove the second law and the fluctuation theorem for this setup. Because the computational cost for mixed states is higher than that for pure states, we consider a smaller bath ($9=3\times3$ sites with 3 bosons) in this subsection.
Figures \[fig:dft4\] (a) and (b) show the verifications of the second law and the integral fluctuation theorem, respectively. Since bath B is smaller, the temporal fluctuations of $\langle \sigma\rangle$ and $\langle e^{-\sigma}\rangle$ are larger.
![ Numerical verification of the second law of thermodynamics and the fluctuation theorem for the diagonal ensemble, which is obtained from typical pure state $|\Psi\rangle$ that is sampled with respect to the Haar measure. The bath is on $9=3\times3$ sites with 3 bosons, and the hopping rate within bath B is taken as $\gamma/\omega=1$, where the Lieb-Robinson time is $\tau \simeq 1/\omega$. The repulsion energy within bath B is given by $g/\omega=0.1$. The initial inverse temperature of bath B is $\beta=0.1$. (a) Entropy production plotted against $\omega t$. (b) The deviation of $\langle e^{-\sigma}\rangle$ from unity plotted against $\omega t$. The deviation is proportional to $t^2$ in the short time regime. []{data-label="fig:dft4"}](fig_s_8.pdf){width="\linewidth"}
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abstract: 'We study nonequilibrium phase transitions in a mass-aggregation model which allows for diffusion, aggregation on contact, dissociation, adsorption and desorption of unit masses. We analyse two limits explicitly. In the first case mass is locally conserved whereas in the second case local conservation is violated. In both cases the system undergoes a dynamical phase transition in all dimensions. In the first case, the steady state mass distribution decays exponentially for large mass in one phase, and develops an infinite aggregate in addition to a power-law mass decay in the other phase. In the second case, the transition is similar except that the infinite aggregate is missing.'
author:
- |
Satya N. Majumdar$^{1}$, Supriya Krishnamurthy$^{2}$ and Mustansir Barma$^{1}$\
[1. *Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India*]{}\
[2. *PMMH, ESPCI, 10 Rue Vauquelin, 75231 Paris Cedex 05, France*]{}
title: 'Nonequilibrium Phase Transitions in Models of Aggregation, Adsorption and Dissociation'
---
[2]{} The steady state of a system in thermal equilibrium is described by the Gibbs distribution. Phase transitions which occur in such equilibrium systems as one changes the external fields such as temperature or magnetic field are by now well understood. On the other hand there is a wide variety of inherently [ *nonequilibrium*]{} systems in nature whose steady states are not described by the Gibbs distribution, but are determined by the underlying microscopic dynamical processes and are often hard to determine. Examples include systems exhibiting self-organized criticality[@BTW], several reaction-diffusion systems[@ZGB], fluctuating interfaces[@edw] and many others. As one changes the rates of the underlying dynamical processes, the steady states of such systems may undergo nonequilibrium phase transitions. As compared to their equilibrium counterparts, these nonequilibrium steady states and the transitions between them are much less understood due to the lack of a general framework. It is therefore important and necessary to study simple models amenable to analysis in order to understand the mechansims of such phase transitions.
Here we study the nonequilibrium phase transitions in an important class of systems which involve microscopic processes of diffusion and aggregation, dissociation, adsorption and desorption of masses. These processes are ubiquitous in nature, and arise in a variety of physical settings, for example, in the formation of colloidal suspensions[@White] and polymer gels[@Ziff] on the one hand, and aerosols and clouds[@Fried] on the other. They also enter in an important way in surface growth phenomena involving island formation[@Lewis]. In this Letter, we introduce a simple lattice model incorporating these microscopic processes and study the nonequilibrium steady states and the transitions between them both analytically within mean field theory and numerically in one dimension.
Our lattice model, which evolves in continuous time, is defined as follows. For simplicity we define the model on a one-dimensional lattice with periodic boundary conditions although generalizations to higher dimensions are quite straightforward. Beginning with a state in which the masses are placed randomly, a site is chosen at random. Then one of the following events can occur:
1. Adsorption: With rate $q$, a unit mass is adsorbed at site $i$; thus $m_i\to m_i+1$.
2. Desorption: With rate $p$, a unit mass desorbs from site $i$; thus $m_i\to m_i-1$ provided $m_i\geq 1$.
3. Chipping (single-particle dissociation): With rate $w$, a bit of the mass at the site “chips” off, [*i.e.*]{} provided $m_i\geq 1$, a single particle leaves site $i$ and moves with equal probability to one of the neighbouring sites $i-1$ or $i+1$; thus $m_i\to m_i-1$ and $m_{i\pm 1}+1$.
4. Diffusion and Aggregation: With rate $1$, the mass $m_i$ at site $i$ moves either to site $i-1$ or to site $i+1$. If it moves to a site which already has some particles, then the total mass just adds up; thus $m_i\to 0$ and $
m_{i\pm 1}\to m_{i\pm 1}+m_i$.
Note that we have assumed that both desorption and diffusion rates are independent of the mass. In a more realistic situation these rates would depend upon the mass. However, our aim here is not to study this model in full generality, but rather to identify the mechanism of a dynamical phase transition in the simplest possible scenario involving these microscopic processes. Indeed we show below that even within this simplest scenario, novel dynamical phase transitions occur which are nontrivial yet amenable to analysis.
Though the model can be studied in the full parameter space of all four basic processes, for simplicity we restrict ourselves here to two limiting cases: (i) $p=0$, $q=0$, i.e. only chipping, diffusion and aggregation moves are allowed. In this limit, mass is locally conserved by the moves and we call this model the conserved-mass aggregation model (CMAM) (ii) $w=0$, i.e. all moves except for chipping are allowed. In this case, adsorption and desorption lead to violation of local mass conservation. We call this the In-out model. In this Letter, we analyse the CMAM model in some detail and only outline the main results for the In-out model.
Let us summarize our main results: (i) In the CMAM, single particles are allowed to chip off from massive conglomerates. This move corresponds to the physical process of single functional units breaking off from larger clusters in the polymerization problem. It leads to a replenishment of the lower end of the mass spectrum, and competes with the tendency of the coalescence process to produce more massive aggregates. The result of this competition is that two types of steady states are possible, and there is a dynamical phase transition between the two. In one state, the steady state mass distribution $P(m)$ decays exponentially, while the other is more striking and interesting: $P(m)$ decays as a power law for large $m$ but in addition develops a delta function peak at $m=\infty$. Physically this means that an infinite aggregate forms that subsumes a finite fraction of the total mass, and coexists with smaller finite clusters whose mass distribution has a power law tail. In the language of sol-gel transitions, the infinite aggregate is like the gel while the smaller clusters form the sol. However, as opposed to the models of irreversible gelation where the sol disappears in the steady state, in our model the sol coexists with the gel even in the steady state. Interestingly, the mechanism of formation of the infinite aggregate in the steady state resembles Bose-Einstein condensation (BEC), though the condensate (the infinite aggregate here) forms in real space rather than momentum space as in conventional BEC. (ii) In the In-out model too we find a phase transition in the steady state as the adsorption ($q$) and desorption ($p$) rates are varied. In one phase (low values of $q$) $P(m)$ decays exponentially whereas in the other phase (high $q$) it has a power law tail. This power law phase is similar to that of the Takayasu model[@Takayasu] of particle injection and aggregation.
We first analyse the CMAM within the mean field approximation, ignoring correlations in the occupancy of adjacent sites. Then we can directly write down equations for $ P(m,t)$, the probability that any site has a mass $m$ at time $t$.\
$$\begin{aligned}
\frac{dP(m,t)} {dt} &=& -(1+w)[1+s(t)] P(m,t)
+ w P(m+1,t) \nonumber \\
&+&w s(t) P(m-1,t)+ P*P ;\;\;\; m \geq 1~~ \label{eq:mft1}\\
\frac{dP (0,t)} {dt} &=& - (1+w)s(t) P(0,t) + wP(1,t) +
s(t) \label{eq:mft2}. \end{aligned}$$ Here $ s(t) \equiv 1-P(0,t)$ is the probability that a site is occupied by a mass and $P*P=\sum_{m^{\prime}=1}^{m}P(m^{\prime},t)P(m-m^{\prime},t)$ is a convolution term that describes the coalescence of two masses.
The above equations enumerate all possible ways in which the mass at a site might change. The first term in Eq. (\[eq:mft1\]) is the “loss” term that accounts for the probability that a mass $m$ might move as a whole or chip off to either of the neighbouring sites, or a mass from the neighbouring site might move or chip off to the site in consideration. The probability of occupation of the neighbouring site, $s(t) = \sum_{m=1} P(m,t)$, multiplies $P(m,t)$ within the mean-field approximation where one neglects the spatial correlations in the occupation probabilities of neighbouring sites. The remaining three terms in Eq. (\[eq:mft1\]) are the “gain” terms enumerating the number of ways that a site with mass $m^{\prime} \neq m$ can gain the deficit mass $m -m^{\prime}$. The second equation Eq. (\[eq:mft2\]) is a similar enumeration of the possibilities for loss and gain of empty sites. Evidently, the mean field equations conserve the total mass.
To solve the equations, we compute the generating function, $Q(z,t) = \sum_{m=1}^{\infty} P(m,t)z^{m}$ from Eq. (\[eq:mft1\]) and set $ \partial Q / \partial t =0$ in the steady state. We also need to use Eq. (\[eq:mft2\]) to write $P(1,t)$ in terms of $s(t)$. This gives us a quadratic equation for $Q$ in the steady state. Choosing the root that corresponds to $Q(z=0) = 0$, we find\
$$\begin{aligned}
Q(z) &=&{{w+2s+ws}\over {2}}-{w\over {2z}}-{wsz\over {2}}
\nonumber \\
&+& ws{(1-z)\over {2z}}\sqrt {(z-z_1)(z-z_2)}. \label{eq:qsol}\end{aligned}$$ where $z_{1,2}=(w+2\mp 2\sqrt {w+1})/ws$. The value of the occupation probability $s$ is fixed by mass conservation which implies that $\sum mP(m)=M/L\equiv \rho$. Putting ${\partial}_zQ (z=1)=\rho$, the resulting relation between $\rho$ and $s$ is $$2\rho = w(1-s) - ws\sqrt {(z_1-1)(z_2-1)}~.
\label{eq:defq}$$
The steady state probability distribution $P(m)$ is the coefficient of $z^m$ in $Q(z)$ and can be obtained from $Q(z)$ in Eq. (\[eq:qsol\]) by evaluating the integral $$P(m) = {1\over {2\pi i}}\int_{C_o} \frac {Q(z)} {z^{ m+1}} dz
\label{eq:contour}$$ over the contour $C_o$ encircling the origin. The singularities of the integrand govern the asymptotic behaviour of $P(m)$ for large $m$. Clearly the integrand has branch cuts at $z=z_{1,2}$. For fixed $w$, if one increases the density $\rho$, the occupation probability $s$ also increases as evident from Eq. (\[eq:defq\]). As a result, both the roots $z_{1,2}$ start decreasing. As long as the lower root $z_1$ is greater than $1$, Eq. (\[eq:defq\]) is well defined and the analysis of the contour integration around the branch cut $z=z_1$, yields for large $m$, $$P(m) \sim e^{-m/{m^{*}}}/m^{3/2} ~,$$ where the characteristic mass, $m^{*}=1/{\log (z_1)}$ and diverges as $\sim (s_c-s)^{-1}$ as $s$ approaches $s_c =(w+2-2\sqrt {w+1})/w$. $s_c$ is the critical value of $s$ at which $z_1=1$. This exponentially decaying mass distribution is the signature of “disordered" phase which occurs for $s<s_c$ or equivalently from Eq. (\[eq:defq\]) for $\rho < {\rho}_c={\sqrt {w+1}} -1$.
When $\rho={\rho}_c$, we have $z_1=1$, and analysis of the contour around $z=z_1=1$ yields a power law decay of $P(m)$, $$P(m)\sim m^{-5/2}.$$ As $\rho$ is increased further beyond ${\rho}_c$, $s$ cannot increase any more because if it does so, the root $z_1$ would be less than $1$ (while the other root $z_2$ is still bigger than $1$) and Eq. (\[eq:defq\]) would be undefined. The only possibility is that $s$ sticks to its critical value $s_c$ or equivalently the lower root $z_1$ sticks to $1$. Physically this implies that adding more particles does not change the occupation probability of sites. This can happen only if all the additional particles (as $\rho$ is increased) aggregate on a vanishing fraction of sites, thus not contributing to the occupation of the others. Hence in this “infinite-aggregate" phase $P(m)$ has an infinite-mass aggregate, in addition to the power law decay $m^{-5/2}$. Concomitantly Eq. (\[eq:defq\]) ceases to hold, and the relation now becomes $$\rho = {w\over {2}}(1-s_c) + \rho_{\infty}$$ where $\rho_{\infty}$ is the fraction of the mass in the infinite aggregate. The mechanism of formation of the aggregate is reminiscent of Bose Einstein condensation. In that case, for temperatures in which a macroscopic condensate exists, particles added to the system do not contribute to the occupation of the excited states; they only add to the condensate, as they do to the infinite aggregate here.
Thus the mean field phase diagram (see inset of Fig. 1) of the system consists of two phases, “Exponential" and “Aggregate", which are separated by the phase boundary, $\rho_c={\sqrt {w+1}}-1$. While this phase diagram remains qualitatively the same even in $1$-d, the exponents characterizing the power laws are different from their mean field values (see Fig. 1).
We have studied this model using Monte Carlo simulations on a one-dimensional lattice. Although we present results here for a relatively small size lattice, $L=1024$, we have checked our results for larger sizes as well. We confirmed that all the qualitative predictions of the mean-field theory remain true, by calculating $P(m)$ numerically in the steady state. Figure 1 displays two numerically obtained plots of $P(m)$. The existence of both the “Exponential" (denoted by $\times$) for $\rho<{\rho}_c$ and the “Aggregate" phase (denoted by $+ $) for $\rho>{\rho}_c$ is confirmed. In particular, the second curve shows clear evidence of a power-law behaviour of the distribution, which is cut off by finite-size effects, and for an ‘infinite’ aggregate beyond. We confirmed that the mass $M_{agg}$ in this aggregate grows linearly with the size, and that the spread $\delta M_{agg}$ grows sublinearly, implying that the ratio $\delta M_{agg}/M_{agg}$ approaches zero in the thermodynamic limit. The exponent $ \tau_{CMAM} $ which characterizes the finite-mass fragment power law decay for $\rho>{\rho}_c$ is numerically found to be $ 2.33\pm
.02$ and remains the same at the critical point $\rho=\rho_c$.
We note that in conserved-aggregation models studied earlier within mean field theory[@Vigil; @Krapiv], the steady state mass distribution also changed from an exponential distribution to a power law as the density was increased beyond a critical value. However, the existence of the striking infinite aggregate in the steady state for $\rho>{\rho}_c$ was not identified earlier.
We next study the steady state phase diagram of the In-out model in the $q$-$p$ plane. In this model, mass is not locally conserved. The mass per site $M(t)$ evidently obeys the exact equation $$\frac{dM}{dt}=q-ps(t)
\label{eq:mass}$$ where $ s(t)$ is the fraction of sites occupied by a mass $m_i \geq 1$. In the steady state, let the mean value of $s(t)$ be $s$. If $q/p$ is low, $s$ adjusts to make $q-ps$ vanish, and the mean mass reaches a time-independent value $M$. This defines the finite-mass phase. As we will see below, as $q/p$ increases beyond a critical value, $s$ never catches up with $q/p$ and reaches a steady state value which is less than $q/p$; in this phase, $M$ increases linearly in time while $P(m,t)\sim m^{-\tau_{T}}f(mt^{-x})$ which in the long time limit converges to a time-independent form, decaying as a power law with exponent $\tau_{T}$, even though the moments of this distribution diverge as time increases to infinity. We call this the growing-mass or the Takayasu phase. In fact, for $p=0$ the In-out model reduces exactly to that of the Takayasu model (TM) of injection and aggregation of masses[@Takayasu] which has found widespread applications ranging from river models[@river] to granular systems[@SNC]. Indeed what we find here is that the growing mass phase of the TM at $p=0$ persists up to a nonzero critical value $p_c(q)$ for a given $q$, while for $p>p_c(q)$ the mass stops growing and $P(m)$ decays exponentially for large $m$ in the steady state.
The mean field analysis of the In-out model is similar to that of the CMAM model though a little bit trickier. We defer the details for a future publication[@KMB] and only outline the results here. We find that the critical line $p_c(q)=q+2{\sqrt {q}}$ separates two phases in the $q$-$p$ plane. For $p>p_c$, $P(m)\sim m^{-3/2}\exp (-m/m^{*})$ for large $m$. For $p=p_c$, $P(m)\sim m^{-5/2}$ and for $p<p_c$, $P(m)\sim m^{-3/2}$ for large $m$. For a fixed $q$, the steady state occupation density $s(p,q)$ develops an interesting cusp as $p$ crosses $p_c(q)$. For example, at $q=1$, where $p_c=3$, $s(p)=1/p$ for $p>3$ as follows simply from Eq. (9); but for $p<3$, the determination of $s(p)$ is nontrivial[@KMB] and is given by the positive root of the cubic equation, $16ps^3+(8p^2+4p-25)s^2+(p^3-11p^2-43p-25)s-p^3+2p^2+17p+25=0$.
The qualitative predictions of mean field theory – the existence of a power-law (Takayasu) phase ($P(m) \sim m^{-\tau_T}$) and a phase with exponential mass distribution, with a different critical behaviour at the transition ($P(m) \sim m^{-\tau_c}$) – are found to hold in 1-d as well. The Takayasu exponent $\tau_T$ is known exactly to be $4/3$ in $1$-d and $3/2$ within mean field theory[@Takayasu]. Figure 2 shows the results of numerical simulations in $1$-d for the phase diagram and the decay of the mass distribution in the two phases and at the transition point. The values obtained, $\tau_T=4/3$ and $\tau_c \simeq 1.833$, are quite different from their mean-field values, $\tau_T=3/2$ and $\tau_c =5/2$, reflecting the effects of correlations between masses at different sites.
We may reinterpret the configuration of masses in the In-out model as an interface profile on regarding $m_i$ as a local height variable. While the model may have some unphysical features in the context of an interface due to the columns of masses moving as a whole, the analogy helps however to understand physically the nature of the transition in the In-out model. In the interface language this corresponds to a wetting transition; the key factor responsible for the occurrence of the smooth phase is a substrate, implicit in the constraint $m_i\geq 0$ in the In-out model. The wet phase is identified with a growing mass phase, which has a rough profile, with exact roughness exponent $\chi_{T}=5/2$[@KMB] in $1$-d. Since $\chi_T>1$, the interface in the wet phase is not self-affine. Recently a nonequilibrium wetting transition was also observed in an interface model[@Hinrichsen] where the interface in the wet phase is self-affine due to surface tension effects which are absent in our model. Interestingly, however, in our model the substrate is able to induce a self-affine interface at the critical point with roughness exponent $\chi =1/3$ within mean field theory and $\chi\approx 0.7$ in $1$-d[@KMB], despite the anomalously large roughness of the wet phase.
We thank Deepak Dhar for useful discussions and S. Cueille and S. Redner for pointing out references [@Vigil] and [@Krapiv] to us.
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abstract: 'The forecast method introduced by [@korsos2014] is generalised from the horizontal magnetic gradient ($G_{M}$), defined between two opposite polarity spots, to all spots within an appropriately defined region close to the magnetic neutral line of an active region. This novel approach is not limited to searching for the largest $G_{M}$ of two single spots as in previous methods. Instead, the pre-flare conditions of the evolution of spot groups is captured by the introduction of the [*weighted*]{} horizontal magnetic gradient, or $WG_{M}$. This new proxy enables the potential of forecasting flares stronger than M5. The improved capability includes (i) the prediction of flare onset time and (ii) an assessment whether a flare is followed by another event within about 18 hours. The prediction of onset time is found to be more accurate here. A linear relationship is established between the duration of converging motion and the time elapsed from the moment of closest position to that of the flare onset of opposite polarity spot groups. The other promising relationship is between the maximum of the $WG_{M}$ prior to flaring and the value of $WG_{M}$ at the moment of the initial flare onset in the case of multiple flaring. We found that when the $WG_{M}$ decreases by about 54%, then there is no second flare. If, however, when the $WG_{M}$ decreases less than 42%, then there will be likely a follow-up flare stronger than M5. This new capability may be useful for an automated flare prediction tool.'
author:
- 'M. B. Korsós, A. Ludmány, R. Erdélyi and T. Baranyi'
title: On flare predictability based on sunspot group evolution
---
Introduction
============
The endeavour to establish reliable flare forecast methods has resulted in numerous attempts to identify promising physical quantities, proxies, features and behavioural patterns to predict an imminent flare [see e.g. @sawyer1986; @hochedez2005; @benz2008]. The ultimate task is to construct a diagnostic tool that determines the unique conditions leading to flaring events, triggered by magnetic reconnection [see e.g. @yamada2010]. The empirical study presented here focuses on features at the solar surface, namely investigating the pre-flare dynamics of sunspot groups.
Most of the attempts that developed flare forecast tools employ suitably defined (and derived) quantities from magnetograms. The majority of previous efforts studied the behaviour of the horizontal gradient of the line-of-sight component of the magnetic field, usually a well-observed property of an active region (AR). By investigating the Solar and Heliospheric Observatory’s Michelson Doppler Imager (SOHO/MDI) magnetograms, @schrijver2007 found that energetic flares are connected to the separation line of opposite polarity regions with high magnetic gradient. Further works followed suit, including e.g. @mason2010 who studied the gradient-weighted inversion line length which exhibits a significant increase prior to flares. The maximum horizontal gradient and the length of the neutral line were considered by @cui2006, @jing2006, @huang2010 and @yu2010a. The fractal structure was addressed by @crisc2009, however, @georg2012 was sceptical about this approach. In the literature, there are a number of other indicators proposed to measure non-potentiality. @leka07 suggested 8 categories of different distributions with an impressive list of 29 types of variables defined on them. All these methods above (and others) have advantages and caveats, however, none are yet accepted as a universal and reliable prediction tool.
Surprisingly, sunspots themselves are rarely considered as possible holders of information for flare forecast. Arguably, the simplest way to apply sunspot data is to approach non-potentiality by using their McIntosh classification [@bloom12]. However, the McIntosh classes are only based on morphology, and they do not contain perhaps the most important information relevant to the present context, i.e. the distribution of opposite magnetic polarities. Our preceding study (@korsos2014, hereafter Paper I) is the first attempt to track the details of the evolution of sunspots (umbrae) in order to identify signatures of flare imminence. Paper I implemented a proxy horizontal magnetic gradient, $G_M$, defined between two [*single*]{} spots of opposite polarities close to the magnetic neutral line. The pre-flare behaviour of this proxy quantity exhibited characteristic and unique patterns: steep rise, high maximum and a gradual decrease prior to flaring. These properties may yield a tool for the assessment of flare probability and intensity within a 2-10 hour window.
The method developed here allows us to elaborate on some of the most important properties of an imminent flare; its intensity and the onset time. The prediction of intensity is found to be more reliable, as a linear relationship has been found between the pre-flare maximum of $G_M$ and the peak intensity emitted in the 1-8 Å range, according to Geostationary Operational Environmental Satellite (GOES) x-ray measurements[^1]. This result may be considered to be an indicator of the existing relationship between the proxies of free and released energies. Next, the onset time prediction is found to be somewhat less precise, as the most probable time of flare onset is between 2 and 10 hours after the $G_{M}$ maximum. The aim of the present work is to find more reliable forecasting methods for the onset time as well as the likelihood of consecutive flaring.
Method of Examination
=====================
The empirical basis of this study, similar to Paper I, is the SDD[^2] (SOHO/MDI-Debrecen Data) sunspot catalogue, the most detailed of its kind covering the years of MDI operations (1996-2010). In addition, the intensity peaks of the examined flares the GOES solar flare database[^3] is employed. The SDD is efficient for tracking the internal dynamics of spot groups. It contains data on position, area and magnetic field for all spots with a cadence of 1.5 hrs [@gyori2011].
Let us now consider the involvement of [*all*]{} magnetic spots in an appropriately selected area at the region of a Polarity Inversion Line (PIL). When a new spot (min. 3 MSH) emerges close (within 40 $\pm 5$ Mm, which is always enough to capture the emergence) to existing spots of opposite polarity, we determine the maximum $G_M$ between the emerging and existing spots. Once a pair of spots of opposite polarities with a maximum $G_M$ is found, we compute the area-weighted centre between them. Next, there is a defined circular area, around this weighted location where $G_M$ is highest, whose diameter is 3$^{\circ}$$\pm 0.5$$^{\circ}$ in Carrington heliographic coordinates. The center of the circle is fixed and spot groups within this area are now monitored.
We assume that the underlying process driving a flare is a collective one between nearby spots. Therefore, a new proxy parameter, a generalisation of the $G_M$, called the weighted horizontal magnetic gradient ($WG_{M}$) is defined to account for this collective behaviour:
$$WG_{M} = \left | \frac {\sum_{i} B_{p,i}\cdot A_{p,i} - \sum_{j} B_{n,j}\cdot A_{n,j}}{d_{pn}} \right |.
\label{proxy}$$
Here, $B$ (determined by $f(A)$ in Paper I) and $A$ denote the mean magnetic field and area of umbra. The indices $p$ and $n$ denote positive and negative polarities, $i$ and $j$ are their running indices in the selected spot cluster and $d_{pn}$ is the distance between the area-weighted centers of two subgroups of opposite polarities in this cluster.
We have analysed 45 single and 16 multiple flare cases which are stronger than M5 between 1996 and 2010. This limitation is based on the findings of Paper I, i.e. the present method seems to be suitable for energetic flares above M5. Besides some similarities between the behaviour of $G_{M}$ and $WG_{M}$, the current approach results in significant, previously unseen pre-flare behavioural patterns. Let us demonstrate the key features with two randomly selected, but typical, examples.
Figure \[8771\] shows a typical active region, AR 8771, with a single flare. The right-hand panels of Fig. \[8771\] are: the white light image (top), magnetogram (bottom) and a cartoon reconstructing the AR from the SDD catalogue (middle). The $WG_{M}$ (top left) shows a steep rise and high maximum (called $WG_M^{max}$) followed by its decrease until the flare. However, and most importantly, we found characteristic and appealing differences from the result of the single spot-pair method, as demonstrated by e.g. the distance diagnostics panel (middle left) of the spot groups of opposite fluxes. This plot contains a conspicuous dip, indicating a duration of converging-diverging motion of the area-weighted centres that seem to be indicative of the next flare for [*all*]{} cases we investigated. The total unsigned magnetic flux (bottom left) in this part of the AR shows some increase before the flare but it has no special, identifiable characteristic feature. Additionally, we tested a large sample of non-flaring spot groups with opposite polarities to determine whether this behaviour is found in these cases as well. We found no such behaviour. Therefore, the evolution of the distance between the area-weighted centers of spot groups of opposite polarities shows potentials for flare forecast.
Figure \[9393\] is another typical example, i.e. AR 9393 also examined in Paper I, but now with multiple flare activities linked only to Area 2 because the highest variability of $WG_M$ is in Area 2, while Area 1 and 3 do not show such a property (see in Paper I). A series of flares is considered multiple, if after the occurring flare there is another flare, within an 18-hr window, belonging to the same preceding rising phase of $WG_M$ and at the same time belonging to the [*same*]{} decreasing phase after $WG_M^{max}$ (see e.g. the X1.4 flare followed by the X20 flare in the upper left panel of Fig. \[9393\]). The behaviour of $WG_{M}$ is analogous to that of the single spot-pair (compare to $G_M$ of Fig. 2 in Paper I), showing a steep rise, high maximum and a slower decrease prior to the X1.7 single flare (29 March), and, in spite of the limited temporal resolution, arguably a similar pattern is found leading to the follow-up ndependent multiple flares (i.e., X1.4 and X20, 2 April). Next, the distance diagnostics (middle left) panel of the spot groups of opposite fluxes do show similar conspicuous dips as in Fig. \[8771\]. Note, these multiple flares are likely unrelated to the single flare as they do not fall within the 18-hour window we propose as a requirement for flares to be connected. Again, this pre-flare behaviour pattern was not identifiable in the distance diagram of the single spot-pair method in Paper I (see its Fig. 2, left column, middle panel). Finally, we could not conclude any special and immediate pre-flare behaviour on a daily time-scale by investigating the variation of unsigned flux within Area 2 (bottom left) because it was decreasing before the X1.7 flare while it was increasing before the X1.4 flare. A statistical study also shows that the temporal variation of the unsigned flux does not exhibit characteristic pre-flare signatures. This, however, does not contradict the findings of @schrijver2007 about a statistical relationship between the likelihood of X- or M-flares and the unsigned flux at SPIL (Strong-gradient Polarity Inversion Line) because selected actual states of ARs were investigated and not the time profile of the examined quantities.
By the above examples of case studies, we are now encouraged to conduct a statistical study to confirm (or refute) the above findings for all cases of flaring spots with a strength of over M5 available in the SDD catalogue.
Diagnostic potentials with spot dynamics
========================================
We test the proposed diagnostics on a statistical sample by applying the following requirements: Firstly, the examined pre-flare variation is within $\pm 70^{\circ}$ from the central meridian to avoid geometrical foreshortening close to the limb. Next, the flare onset is no further eastward from the central meridian than $-40^{\circ}$ to have sufficient time to follow the development of $WG_{M}$.
In Paper I, a linear relationship was found between the maximum value of $G_{M}$ preceding a flare and the peak intensity of flares. This behaviour is confirmed for $WG_{M}$ by Fig. \[invmax\]. 45 single flares (crosses, left panel) and 45 single with additional 16 largest of multiple flares (circles, right panel) show a linear relationship between $WG_{M}^{max}$ and the corresponding GOES flare intensity. Here, we restrict the empirical analysis for flares between M5 and X4 classes only.
Next, the new method revealed further important connections, which is conspicuous, in Figs. \[8771\] and \[9393\]. This connection is between the durations of converging-diverging motion of the centers of opposite polarities. This intriguing pattern was found in all 61 cases investigated here. The question rises, whether there is a relationship between the duration of the converging motion (the duration from the moment of the first point when the distance began decreasing to the moment of the minimum point of the parabolic curve) and the time elapsed from the moment of minimum distance until the flare onset (duration of the diverging motion and the follow-up time until the flare onset). To determine these two time intervals for each flare, parabolic curves were fitted to their distance data. For a sample see the top left panel of Fig. \[pushpull\] which is a parabolic fit to the distance data from the left middle panel of Fig. \[8771\], showing the converging-diverging behaviour of this relative motion.
Figure \[pushpull\] gives further insight into the relation between these intervals by plotting the time from the moment of minimum distance to the flare onset as a function of the duration of converging motion. First, the upper right diagram depicts the duration of diverging motion as function of the duration of converging motion for the 45 single (crosses) and 16 multiple (circles) flare cases. Note that the duration of diverging motion is shorter than the time period from the moment of minimum distance to that of the flare onset (see bottom panels of Fig. \[pushpull\]). However, the converging-motion phase and the diverging-motion phase have the same duration. [@yamada2010] found similar properties in laboratory reconnection experiments, and called it the “push and pull-mode”. The present observations are a confirmation of the laboratory experiment.
The lower diagrams plot the time from the moment of closest position to the flare onset as function of the duration of the converging motion phase. In the cases of multiple flares, we investigated the time from the moment of closest position to the [*first*]{} flare onset as function of the duration of the converging motion phase. The left/right panel contains those cases when the spot groups are younger/older than three days at the time of flare onset. The regression lines of the two cases are, surprisingly, different. By estimating the time the magnetic fields younger than three days should be distinguished from the older ones, the relevant formulae are given in the lower panels of Figure \[pushpull\]. One may be able to estimate a rough onset time of the flare. Note the considerable dispersion. If the study area is younger than three days then about a mere hour is needed to be added to the duration of the corresponding scaled duration of converging motion, where the scale-factor is 1.3 for younger ones, to obtain the flare onset time. However, if the area is older than three days then the scale-factor is 0.85, and one needs to add 12 $\pm 3$ hours to the scaled duration of converging motion.
Next, a relationship is found between the values of $WG_{M}$ at its maximum prior to flaring ($WG_{M}^{max}$) and at the time of flare onset ($WG_{M}^{Flare}$), as visualised in Figure \[maxflaregm\]. We investigate separately the 45 cases when only a single flare took place after the maximum of $WG_{M}$, and the 16 when more flares erupted after the maximum within an 18-hour window after the occurring flare on the decreasing phase of the $WG_{M}$. The left panel depicts the cases of single energetic flares (crosses). The right panel depicts the first flares (circles) of those ARs where multiple flares are produced. The plots are interpreted as follows: a single flare erupts when the $WG_{M}^{max}$ was less than $5\cdot10^{6}$ Wb/m and the $WG_{M}$ decreases by more than half of the $WG_{M}^{max}$, in the pre-flare phase (for example the X1.4 flare in the NOAA 8771 on Fig. \[8771\] and the X1.7 flare in the AR NOAA 9393 on Fig. \[9393\]). This is likely due to the fact that the magnetic energy in the region decreases so significantly during this first flare that there is simply not enough energy left to release another flare. In that case, if the decrease is smaller than half (about $42$%) by the onset of the first flare, some further flaring may be expected, meaning that the disturbance of this first flare could be forcing opposite polarity fields together in the solar atmosphere leading, for example to the ‘homologous’ flaring that is often observed. On the other hand, the $WG_{M}^{max}$ larger than $5\cdot10^{6}$ Wb/m seems to be enough in itself to predict a multiple flaring event. We cannot comment yet on further flares (i.e. third, etc.) in the case of multiple flares as the temporal resolution of the SDD catalogue is too coarse. For an example, see the case of AR NOAA 9393 where an X 1.4 flare was followed by an X 20 one within 12 hours (Fig. \[9393\]).
Discussion
==========
In this paper, we present advancements in the classification of pre-flare conditions with an application to flare prediction. 61 cases were investigated in the vicinity of PILs of ARs. We assumed the flare onset to be a collective response to their dynamics. This assumption needs further investigations both observationally (with higher resolution) and theoretically (e.g. numerical simulations).
First, we found that the pre-flare behaviour of the weighted horizontal magnetic gradient ($WG_{M}$) exhibits similar patterns to those found with the single spot-pair method: steep rise, high maximum and gradual decrease until the flare onset (Figs. \[8771\]-\[9393\]). Next, Figure \[invmax\] corroborates the relationship, reported in Paper I, between the maximum of $G_{M}$ and the intensity of single flares. Here, this relationship is modelled as linear one, however, the dispersion is considerable and theoretical (e.g. numerical) modelling may be necessary to confirm or refute this relation. There may be a yet unknown physical parameter, therefore not accounted for, that would reduce the dispersion. This is the reason of restriction on the currently considered GOES classes. However, the relationship found can be still regarded to be a link between the proxy measures of the free energy and the released energy. A shortfall of the single spot-pair method was that one could not deduce the flare intensity in the case of multiple flares. This is now rectified by the introduction of the weighted horizontal magnetic gradient. The spot-group method is now capable of providing a rough estimate of the expected largest flare intensity from $WG_{M}^{max}$ (Fig. \[invmax\]). In addition, this method also gives a better estimate of the expected time of flare onset (Fig. \[pushpull\]), and, may be able to predict whether an energetic flare after $WG_{M}^{max}$ is the only flare or further flare events can be expected (Fig. \[maxflaregm\]).
Let us assume that $WG_{M}$ is a proxy of the available non-potential (i.e. free) energy to be released in a spot group. In this case, we may conclude from Fig. \[maxflaregm\]: (i) if the maximum of the released energy may be over half of the maximum of the accumulated (free) energy, no further energetic flare(s) can be expected; (ii) If the maximum of the released flare energy is less than about $\sim$42%, further flares are more probable. In short, Fig. \[maxflaregm\] allows us to track the variation of the energy balance of ARs and to assess the probabilities of consecutive flares and their intensities.
Last but not least we provide some notes on the estimate of the onset time of an imminent flare. Here, its determination is refined. Paper I only presented a statistics that 60% of observed energetic flares are between 2-10 hrs after the maximum of $G_{M}$. Figure \[pushpull\], however, allows a much stronger statement on the expected time of onset due to $WG_{M}$. The figure uncovers the relationship between the duration of the converging motion of opposite polarities (their compression) and the time elapsed between the closest position and flare onset following the diverging motion [see earlier motions @yamada2010]. By determining the duration of the converging motion the flare onset can now be assessed for [*all*]{} cases. We also found that the data points of the motions of younger spot groups have smaller dispersion (left of Fig. \[pushpull\]).
Acknowledgment
==============
The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2012-2015) under grant agreement No. 284461 (eHEROES project). MBK and RE are grateful to Science and Technology Facilities Council (STFC) UK for the financial support received. RE would like to thank for the invitation, support and hospitality received from the Hungarian Academy of Sciences under their Distinguished Guest Scientists Fellowship Programme (ref. nr. 1751/44/2014/KIF) that has allowed him to stay three months at the Debrecen Heliophysical Observatory (DHO) of the Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences. RE is also grateful to NSF, Hungary (OTKA, Ref. No. K83133).
Abramenko, V. I., Yurchyshyn, V. B., Wang, H., et al. 2003, , 597, 1135 Benz, A.O. 2008, Liv. Rev. Sol. Phys, 5, 1 Bloomfield, D. S., Higgins, P. A., McAteer, R. T. J., et al. 2012, , 747, L41 Criscuoli, S., Romano, P., Giorgi, F., et al. 2009, , 506, 1429 Cui, Y., Li, R., Zhang, L., et al. 2006, , 237, 45 Georgoulis, M. K. 2012, , 276, 161 Győri, L., Baranyi, T., & Ludmány, A. 2011, IAUS 273, 403 Hochedez, J.-F., Zhukov, A., Robbrecht, E., et al. 2005, Ann. Geophys., 23, 3149 Huang, X., Yu, D., Hu, Q., et al. 2010, , 263, 175 Jing, J., Song, H., Abramenko, V., et al. 2006, , 644, 1273 Korsós, M. B., Baranyi, T., & Ludmány, A. 2014, , 789, 107 Leka, K. D. & Barnes, G. 2007, , 656, 1173 Mason, J. P. & Hoeksema, J. T. 2010, ApJ, 723, 634 Messerotti, M., Zuccarello, F., Guglielmino, S.L., et al. 2009, , 147, 121
Sawyer, C., [Solar Flare Prediction]{}, Univ Press Colorado, 1986 Schrijver, C. 2007, , 655, L117 Yamada, M., Kulsrud, R., Ji, H. 2010, Rev. Mod. Phy., 82, 603 Yu, D., Huang, X., Hu, Q., et al. 2010, , 709, 321
[^1]: http://www.ngdc.noaa.gov/stp/satellite/goes/dataaccess.html
[^2]: http://fenyi.solarobs.unideb.hu/SDD/SDD.html
[^3]: http://www.ngdc.noaa.gov/stp/satellite/goes/dataaccess.htm
|
---
abstract: 'We prove ultradifferentiable Chevelley restriction theorems for a wide range of ultradifferentiable classes. As a special case we find that isotropic functions, i.e., functions defined on the vector space of real symmetric matrices invariant under the action of the special orthogonal group by conjugation, possess some ultradifferentiable regularity if and only if their restriction to diagonal matrices has the same regularity.'
address: 'Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria & University of Education Lower Austria, Campus Baden Mühlgasse 67, A-2500 Baden'
author:
- Armin Rainer
title: |
Ultradifferentiable Chevalley theorems\
and isotropic functions
---
[^1]
Introduction
============
Let the special orthogonal group ${\operatorname}{SO}(n)$ act by conjugation on the vector space ${\operatorname}{Sym}(n)$ of real symmetric $n \times n$ matrices. Functions $f : {\operatorname}{Sym}(n) \to {\mathbb{R}}$ that are invariant under this action are called *isotropic*, i.e.$$f(S A S^t) = f(A) \quad \text{ for all } A \in {\operatorname}{Sym}(n), ~ S \in {\operatorname}{SO}(n).$$ By the spectral theorem, every ${\operatorname}{SO}(n)$-orbit ${\operatorname}{SO}(n) \cdot A$ intersects the subspace ${\operatorname}{Diag}(n) \cong {\mathbb{R}}^n$ of diagonal matrices orthogonally with respect to the invariant inner product $\langle A,B \rangle = {\operatorname}{Tr}(AB^t) = {\operatorname}{Tr}(AB)$. The intersection is the orbit of the symmetric group ${\operatorname}{S}_n$ which acts by permuting the eigenvalues of $A$. Then $f : {\operatorname}{Sym}(n) \to {\mathbb{R}}$ is isotropic if and only if $$f(A) = F(a_1,\ldots,a_n), \quad A \in {\operatorname}{Sym}(n),$$ for some unique symmetric function $F : {\mathbb{R}}^n \to {\mathbb{R}}$, where $a_1,\ldots,a_n$ are the eigenvalues of $A$ repeated according to their multiplicity. Isotropic functions are important in continuum mechanics, in particular, elasticity.
It is well-known that the map that assigns to a real symmetric matrix $A$ its $n$-tuple of eigenvalues (e.g. in decreasing order) is Lipschitz (even difference-convex) but not $C^1$. Thus it is surprising that $f$ is smooth if and only if $F$ is smooth. This follows from a result of Glaeser [@Glaeser63F] (see also Schwarz [@Schwarz75] and Mather [@Mather77] for generalizations): every symmetric $F \in C^\infty({\mathbb{R}}^n)$ can be written in the form $$F(x) = G({\sigma}_1(x),\ldots,{\sigma}_n(x)),$$ where $G \in C^\infty({\mathbb{R}}^n)$ and ${\sigma}_i$ are the elementary symmetric functions in $n$ variables. Consequently, $$f(A) = G((-1)^1{\operatorname}{Tr} (\wedge^1 A) , \ldots, (-1)^n{\operatorname}{Tr}( \wedge^n A))$$ is the composite of two $C^\infty$-maps. Glaeser’s theorem for finite differentiability involves an unavoidable loss of regularity and can hence not be used to determine the regularity of $f$. Indeed, for a symmetric $C^{n r}$-function $F$ the function $G$ is in general only of class $C^r$; see Barban[ç]{}on [@Barbancon72] and Rumberger [@Rumberger98]. Nevertheless it is true that, for any $r \in {\mathbb{N}}$, an isotropic function $f(A) = F(a_1,\ldots,a_n)$ is $C^r$ if and only if $F$ is $C^r$. This was proved by Ball [@Ball84] for $r=0,1,2,\infty$ and by Sylvester [@Sylvester:1985aa] for all $r = 0,1,2,\cdots,\infty$. Later Šilhavý [@Silhavy2000] gave a simple elementary proof of Sylvester’s result and an inductive formula for the derivatives of $f$; see also Scheuer [@Scheuer:2018aa]. The simpler Hölder case $C^{r,{\alpha}}$ for $0 < {\alpha}<1$ is already contained in [@Ball84]. For applications in elasticity see the discussion of the stored energy function in [@Ball84 Section 6].
In this paper we will show that the same phenomenon “permanence of regularity” between an isotropic function $f$ and its symmetric companion $F$ holds for *ultradifferentiable functions*. These are $C^\infty$-function with certain growth restrictions for their iterated derivatives which define the ultradifferentiable class. For instance, the real analytic class is defined by the Cauchy estimates. Modification of the Cauchy estimates in terms of a weight sequence gives rise to the classical Denjoy–Carleman classes (among them the Gevrey classes which are important in PDEs). Braun–Meise–Taylor classes arose from measuring the regularity in terms of prescribing the decay of the Fourier transform.
We will work in a very general framework for ultradifferentiable analysis which comprises all classically studied classes (notably, the aforementioned ones). Similarly as for finite differentiability the corresponding Glaeser theorem involves a strict loss of regularity and is hence not applicable; see Bronshtein[@Bronshtein86; @Bronshtein87] and [@RainerDC]. It is worth mentioning that for Denjoy–Carleman classes we require that the weight sequence has moderate growth whereas in the case of Braun–Meise–Taylor classes our results apply for all standard weight functions. (Precise definitions are given in .) It does not matter for the problem whether the class is quasianalytic or not. In particular, we get as a corollary that an isotropic function $f$ is real analytic if and only if its symmetric companion $F$ is real analytic. Furthermore we shall also deduce a version of the result for *Gelfand–Shilov classes*, i.e., ultradifferentiable rapidly decreasing functions for which the defining bounds are global in contrast to the aforementioned regularity classes.
The results for isotropic functions () will be special cases of the ultradifferentiable Chevalley restriction theorems ( and ) that we shall prove in . The latter are formulated for Cartan decompositions of real semisimple Lie algebras of noncompact type; see the setting in . The proof follows closely the one given by Dadok [@Dadok:1982aa] for the $C^\infty$-case which is based on the analysis of the Laplace operator on invariant functions and a weak elliptic regularity result. We shall combine this analysis with lacunary regularity results for ultradifferentiable classes. These lacunary regularity results are reviewed and adapted to our general ultradifferentiable setting in . For the Chevalley theorem in Gelfand–Shilov classes we will use their invariance under the Fourier transform.
Notation {#notation .unnumbered}
--------
We will use multiindex notation with $D_j = - i {\partial}_j$, where $i = \sqrt{-1}$, and $D^{\alpha}= D_1^{{\alpha}_1} D_2^{{\alpha}_2} \cdots D_n^{{\alpha}_n}$ such that the Fourier transform $$\widehat f(\xi) = \int f(x) e^{- i \langle x,\xi\rangle } \, dx$$ on the Schwartz class ${\mathcal{S}}({\mathbb{R}}^n)$ satisfies $\widehat {D_jf}(\xi) = \xi_j \widehat f(\xi)$ and $\widehat{x_j f(x)} = - D_j \widehat f$.
Ultradifferentiable functions {#sec:ultra}
=============================
Denjoy–Carleman classes {#sec:DC}
-----------------------
Let $U \subseteq {\mathbb{R}}^n$ be open. Let $M=(M_k)$ be a positive sequence. For ${\rho}>0$ and compact $K\subseteq U$ consider the seminorm $$\|f\|^M_{K,{\rho}} := \sup_{\substack{x \in K\\{\alpha}\in {\mathbb{N}}^n}} \frac{|D^{\alpha}f(x)|}{{\rho}^{|{\alpha}|} M_{|{\alpha}|}}, \quad f \in {\mathcal{C}}^\infty(U).$$ The *Denjoy–Carleman class of Roumieu type* ${\mathcal{E}}^{\{M\}}$ is defined by $${\mathcal{E}}^{\{M\}}(U) := \big\{f \in {\mathcal{C}}^\infty(U) : {\;\forall}K \Subset U {\;\exists}{\rho}>0 : \|f\|^M_{K,{\rho}} <\infty\big\}$$ and the *Denjoy–Carleman class of Beurling type* ${\mathcal{E}}^{(M)}$ by $${\mathcal{E}}^{(M)}(U) := \big\{f \in {\mathcal{C}}^\infty(U) : {\;\forall}K \Subset U {\;\forall}{\rho}>0 : \|f\|^M_{K,{\rho}} <\infty\big\},$$ We endow these spaces with their natural locally convex topologies. The study of Denjoy–Carleman classes started around 1900 with the work of E. Borel.
We shall assume that the sequence $M=(M_k)$ is
1. logarithmically convex, i.e. $M_k^2 \le M_{k-1} M_{k+1}$ for all $k$, and satisfies
2. $M_0 = 1 \le M_1$ and
3. $M_k^{1/k} \to \infty$.
In that case we say that $M$ is a *weight sequence*. It is easy to see that for a weight sequence $M$ the sequence $M_k^{1/k}$ (and thus also $M$) is increasing and $M_{k}M_{\ell} \le M_{k+\ell}$ for all $k,\ell$.
Let $M=(M_k)$ and $N=(N_k)$ be positive sequences. Then boundedness of the sequence $(M_k/N_k)^{1/k}$ is a sufficient condition for the inclusions ${\mathcal{E}}^{\{M\}} \subseteq {\mathcal{E}}^{\{N\}}$ and ${\mathcal{E}}^{(M)} \subseteq {\mathcal{E}}^{(N)}$ (this means that the inclusions hold on all open sets). The condition is also necessary provided that $M$ satisfies (1), see [@Thilliez08] and [@Bruna80/81]. For instance, stability of the classes ${\mathcal{E}}^{\{M\}}$ and ${\mathcal{E}}^{(M)}$ by derivation is equivalent to boundedness of the sequence $(M_{k+1}/M_k)^{1/k}$ (for the necessity we assume that $M$ satisfies (1)). If $(M_k/N_k)^{1/k} \to 0$ then ${\mathcal{E}}^{\{M\}} \subseteq {\mathcal{E}}^{(N)}$, and conversely provided that $M$ satisfies (1). Hence sequences $M$ and $N$ satisfying (1) are called *equivalent* if there is a constant $C>0$ such that $C^{-1} \le (M_k/N_k)^{1/k}\le C$; this is precisely the case if they defined the same Denjoy–Carleman classes.
Of particular importance in the theory of differential equations are the *Gevrey classes* ${\mathcal{G}}^s$, for $s \ge 1$. These are by definition the Roumieu type classes associated with the weight sequence $M_k= k!^s$, i.e., ${\mathcal{G}}^s = {\mathcal{E}}^{\{(k!^s)_k\}}$. For $s=1$ we get the class of real analytic functions ${\mathcal{G}}^1 = {\mathcal{E}}^{\{(k!)_k\}} = {\mathcal{C}}^{\omega}$ in the Roumieu case and the restrictions of entire functions ${\mathcal{E}}^{((k!)_k)}$ in the Beurling case.
The inclusion $C^{{\omega}} \subseteq {\mathcal{E}}^{\{M\}}$ (as well as the inclusion ${\mathcal{E}}^{((k!)_k)} \subseteq {\mathcal{E}}^{(M)}$) is equivalent to the condition
1. $(k!/M_k)^{1/k}$ is bounded.
The inclusion $C^{\omega}\subseteq {\mathcal{E}}^{(M)}$ is equivalent to the stronger condition
1. $(k!/M_k)^{1/k} \to 0$ as $k \to \infty$.
A positive sequence $M=(M_k)$ is said to have *moderate growth* if
1. $\exists C>0 {\;\forall}k,\ell \in {\mathbb{N}}: M_{k+\ell} \le C^{k+\ell} M_k M_\ell$.
This condition entails that the corresponding classes ${\mathcal{E}}^{\{M\}}$ and ${\mathcal{E}}^{(M)}$ are stable by derivation.
Note that the Gevrey classes ${\mathcal{G}}^s$, for $s\ge 1$, satisfy all of the conditions (1)–(5), if $s>0$ also $(4')$ is satisfied.
General ultradifferentiable classes
-----------------------------------
By a *weight matrix* we mean a family ${\mathfrak{M}}$ of weight sequences $M$ which is totally ordered with respect to the pointwise order relation on sequences. For a weight matrix ${\mathfrak{M}}$ and an open subset $U \subseteq {\mathbb{R}}^n$ we consider the *Roumieu class* $$\begin{aligned}
{\mathcal{E}}^{\{{\mathfrak{M}}\}}(U) &:= \big\{f \in {\mathcal{C}}^\infty(U) : {\;\forall}K \Subset U {\;\exists}M \in {\mathfrak{M}}{\;\exists}{\rho}>0 : \|f\|^M_{K,{\rho}} <\infty\big\}\end{aligned}$$ and the *Beurling class* $$\begin{aligned}
{\mathcal{E}}^{({\mathfrak{M}})}(U) &:= \big\{f \in {\mathcal{C}}^\infty(U) : {\;\forall}K \Subset U {\;\forall}M \in {\mathfrak{M}}{\;\forall}{\rho}>0 : \|f\|^M_{K,{\rho}} <\infty\big\}\end{aligned}$$ with their natural locally convex topologies. Clearly every Denjoy–Carleman class ${\mathcal{E}}^{\{M\}}$ and ${\mathcal{E}}^{(M)}$ is a ultradifferentiable class of this kind (then ${\mathfrak{M}}$ consists just of the weight sequence $M$).
In the following we will consider weight matrices ${\mathfrak{M}}$ with additional properties which will depend on the type, i.e. Beurling or Roumieu, of the class:
1. (Beurling case) For all $M \in {\mathfrak{M}}$ we have $$\begin{gathered}
\tag{$B_1$}
(k!/ M_k)^{1/k} \to 0 \quad \text{ as } k \to \infty,
\\
\tag{$B_2$}
{\;\forall}M \in {\mathfrak{M}}{\;\exists}L \in {\mathfrak{M}}{\;\exists}C>0 {\;\forall}k,\ell \in {\mathbb{N}}: L_{k+\ell} \le C^{k + \ell} M_k M_\ell.
\end{gathered}$$
2. (Roumieu case) For all $M \in {\mathfrak{M}}$ we have that $(k!/ M_k)^{1/k}$ is bounded and $$\begin{gathered}
\tag{$R_1$}
(k!/ M_k)^{1/k} \quad \text{ is bounded, }
\\ \tag{$R_2$}
{\;\forall}M \in {\mathfrak{M}}{\;\exists}N \in {\mathfrak{M}}{\;\exists}C>0 {\;\forall}k,\ell \in {\mathbb{N}}: M_{k+\ell} \le C^{k + \ell} N_k N_\ell.
\end{gathered}$$
We will call a weight matrix ${\mathfrak{M}}$ \[*regular*\] if it satisfies ($B$) in the Beurling case and ($R$) in the Roumieu case. Under these conditions the classes ${\mathcal{E}}^{[{\mathfrak{M}}]}$ contain the class of real analytic functions and are stable under differentiation. We use square brackets $[~]$ in statements that hold in the Beurling case $(~)$ as well as in the Roumieu case $\{~\}$ under the respective assumptions.
For Denjoy–Carleman classes either of the conditions ($B_2$) and ($R_2$) reduces to the moderate growth condition \[sec:DC\](5).
All our results will apply for *Braun–Meise–Taylor classes* ${\mathcal{E}}^{[{\omega}]}$. Here ${\omega}$ is a *weight function*, i.e., a continuous increasing functions ${\omega}: [0,\infty) \to [0,\infty)$ with ${\omega}(0) =0$ and $\lim_{t \to \infty} {\omega}(t) = \infty$ such that
1. ${\omega}(2t) = O({\omega}(t))$ as $t \to \infty$, \[om1\]
2. $\log t = o({\omega}(t))$ as $t \to \infty$, and \[om3\]
3. ${\varphi}(t) := {\omega}(e^t)$ is convex. \[om4\]
The classes ${\mathcal{E}}^{[{\omega}]}$ are defined by $${\mathcal{E}}^{\{{\omega}\}}(U) := \{f \in {\mathcal{C}}^\infty(U) : {\;\forall}K \Subset U {\;\exists}{\rho}>0 : \|f\|^{\omega}_{K,{\rho}} <\infty\}$$ and $${\mathcal{E}}^{({\omega})}(U) := \{f \in {\mathcal{C}}^\infty(U) : {\;\forall}K \Subset U {\;\forall}{\rho}>0 : \|f\|^{\omega}_{K,{\rho}} <\infty\}$$ by means of the seminorms $$\|f\|^{\omega}_{K,{\rho}} := \sup_{x \in K,\,{\alpha}\in {\mathbb{N}}^n} |D^{\alpha}f(x)| \exp(-\tfrac{1}{{\rho}} {\varphi}^*({\rho}|{\alpha}|)),
\quad f \in {\mathcal{C}}^\infty(U),$$ where ${\varphi}^*(t) := \sup_{s\ge 0} \big(st-{\varphi}(s)\big)$, for $t>0$, is the *Young conjugate* of ${\varphi}$.
These classes were originally introduced by Beurling [@Beurling61] and Björck [@Bjoerck66] in terms of decay properties of the Fourier transform. The description that we used above is due to Braun, Meise, and Taylor [@BMT90].
Braun–Meise–Taylor classes ${\mathcal{E}}^{[{\omega}]}$ can be identified as classes ${\mathcal{E}}^{[{\mathfrak{M}}]}$ for suitable weight matrices ${\mathfrak{M}}$. In fact, by [@RainerSchindl12], setting $$W^x_k := \exp(\tfrac{1}{x} {\varphi}^*(x k)), \quad \text{ for } k \in {\mathbb{N}}\text{ and } x >0,$$ defines a weight matrix ${\mathfrak{W}}= \{W^x : x >0\}$ such that $${\mathcal{E}}^{[{\omega}]} = {\mathcal{E}}^{[{\mathfrak{W}}]} \quad \text{ algebraically and topologically.}$$ The weight matrix ${\mathfrak{W}}$ always satisfies ($B_2$) as well as ($R_2$); cf. [@RainerSchindl12 (5.6)]. It fulfills ($B_1$) (resp. ($R_1$)) if and only if ${\omega}(t) = o(t)$ (resp. ${\omega}(t) = O(t)$) as $t \to \infty$; cf. [@RainerSchindl12 Corollary 5.17].
Due to [@BMM07], ${\mathcal{E}}^{[{\omega}]} = {\mathcal{E}}^{[M]}$ for some weight sequence $M$ if and only if $$\begin{aligned}
{\;\exists}H\ge 1 {\;\forall}t\ge 0 :
2{\omega}(t) \le {\omega}(Ht) + H.
\end{aligned}$$ This is the case if and only if some (equivalently each) $W^x$ has moderate growth, and then ${\mathcal{E}}^{[{\omega}]} = {\mathcal{E}}^{[W^x]}$ for all $x>0$.
Lacunary regularity {#sec:lacunary}
-------------------
In this section we review some results of Liess [@Liess:1990aa] which are based on earlier work by Baouendi and Metivier [@Baouendi:1982aa] and Bolley, Camus, and Metivier [@Bolley:1991ab]. We shall need only one direction of the characterization of Liess (in a special case) and try to get by with the minimal assumptions on the weight sequences, but we will otherwise not strive for utmost generality. Moreover we show the result in the framework of weight matrices.
Let ${\mathfrak{M}}$ be a weight matrix and let $P$ be a linear partial differential operator with analytic coefficients of order $m$ on an open subset $U$ of ${\mathbb{R}}^n$. Moreover, let $k=(k_j)$ be a strictly increasing sequence of positive integers. Then ${\mathcal{E}}^{[{\mathfrak{M}}]}_{P,k}(U)$ denotes the space of all $f \in C^\infty(U)$ such that for all $K \Subset U$ there exist $M \in {\mathfrak{M}}$ and $C,{\rho}>0$ such that (resp. for all $M \in {\mathfrak{M}}$ and all ${\rho}>0$ there is $C>0$ such that) $$\|P^{k_j} f\|_{L^2(K)} \le C {\rho}^{m k_j} M_{m k_j} \quad \text{ for all } j \in {\mathbb{N}}.$$
\[lem:A1\] Assume that ${\mathfrak{M}}$ is a \[regular\] weight matrix and that $k=(k_j)$ satisfies $$\label{eq:reccurence}
a k_j \le k_{j+1} \le a k_j + b \quad \text{ for all } j,$$ for some $a\in {\mathbb{N}}_{\ge 1}$ and $b \in {\mathbb{N}}$. Then for $f \in C^\infty(U)$ the following conditions are equivalent:
1. $f \in {\mathcal{E}}^{[{\mathfrak{M}}]}(U)$.
2. for all $K \Subset U$ there exist $M \in {\mathfrak{M}}$ and $C,{\rho}>0$ such that (resp. for all $M \in {\mathfrak{M}}$ and all ${\rho}>0$ there is $C>0$ such that) $$\|d_v^{k_j} f\|_{L^2(K)} \le C {\rho}^{k_j} M_{k_j} \quad \text{ for all } j \in {\mathbb{N}}\text{ and all } v \in S^{n-1},$$
where $d_v f(x) = {\partial}_t|_{t=0} f(x+tv)$ is the directional derivative.
Only the implication (2) $\Rightarrow$ (1) requires an argument. It follows from the following interpolation formula (see [@Liess:1990aa (5)]): if $U' \Subset U'' \Subset U$ are open subsets of ${\mathbb{R}}^n$ and $\ell \le k$, then for $i = 1,\ldots,n$, $$\label{eq:interpolation}
\|D_i^\ell u\|_{L^2(U')}
\le C(U',U'')^k \|u\|_{L^2(U'')}^{1-\ell/k} \Big(k^\ell \|u\|_{L^2(U'')}^{\ell/k} + \|D_i^k u\|_{L^2(U'')}^{\ell/k} \Big).$$ For each fixed $v \in S^{n-1}$ we may choose a basis in which $v$ is the first basis vector. Now it suffices to choose $j$ such that $k_j \le \ell \le k_{j+1}$ and to use for $k = k_{j+1}$. In the Roumieu case the conditions and ($R_2$) guarantee that there exist $N,N' \in {\mathfrak{M}}$ such that $$M_{k_{j+1}} \le M_{a k_j +b} \le C^{a k_j +b} N_b N_{a k_j} \le C^{a k_j +b + a(a+1)k_j/2} N'_b (N'_{k_j})^a,$$ and hence using again we conclude $$M_{k_{j+1}}^{1/k_{j+1}} \lesssim (N'_{k_{j}})^{1/k_{j}} \le (N'_{\ell})^{1/\ell}.$$ Moreover $k_{j+1}^\ell \le A^\ell k_j^\ell \le A^\ell \ell^\ell \le (A')^\ell M_\ell$ for all $M \in {\mathfrak{M}}$ and suitable constants $A,A'$, since $(k!/M_k)^{1/k}$ is bounded by ($R_1$). In the Beurling case we use ($B$) instead of ($R$) in a similar way. Since $(k!/M_k)^{1/k} \to 0$ in this case, we find that $k_{j+1}^\ell \le {\epsilon}^\ell M_\ell$ for all $M \in {\mathfrak{M}}$ and all ${\epsilon}>0$.
In any case we may conclude that (2) actually holds for the sequence $k_j =j$. By the polarization formula [@KM97 Lemma 7.3(1)], we have $$\|d^j f(x)\|_{L_j} \le (2e)^j \sup_{v \in S^{n-1}} |d_v^j f(x)|,$$ where $\|d^j f(x)\|_{L_j}$ denotes the operator norm of the Fréchet derivative of order $j$ of $f$ at $x$. Together with the Sobolev inequality and the fact that ${\mathfrak{M}}$ is stable by derivation (in order to switch from $L^2$- to $L^\infty$-estimates) we find that $f \in {\mathcal{E}}^{[{\mathfrak{M}}]}(U)$.
\[lem:A2\] Let $P$ be some elliptic linear partial differential operator of order $m$ with real analytic coefficients on an open set $U \subseteq {\mathbb{R}}^n$. Let $k=(k_j)$ be a strictly increasing sequence of positive integers satisfying . Assume that ${\mathfrak{M}}$ is a \[regular\] weight matrix. Then $${\mathcal{E}}^{[{\mathfrak{M}}]}(U) = {\mathcal{E}}^{[{\mathfrak{M}}]}_{P,k}(U).$$
The inclusion ${\mathcal{E}}^{[{\mathfrak{M}}]}(U) \subseteq {\mathcal{E}}^{[{\mathfrak{M}}]}_{P,k}(U)$ is clear.
For the converse inclusion we follow the arguments of [@Bolley:1991ab Proposition 3.3]. Let $U' \Subset U'' \Subset U$ be open subsets of ${\mathbb{R}}^n$. By [@Baouendi:1982aa Theorem 1.2], there is a constant $C>0$ such that for all $k \in {\mathbb{N}}$ there is ${\chi}_k \in C^\infty_c(U'')$ with ${\chi}_k = 1$ on $U'$ and with values in $[0,1]$ such that for all $u \in C^\infty({\overline}{U''})$ $$|\xi|^{m k} |\widehat{({\chi}_k u)}(\xi)| \le C^{k+1} \Big( \|P^k u\|_{L^2(U'')} + k!^m \|u\|_{L^2(U'')}\Big), \quad \xi \in {\mathbb{R}}^n, k \in {\mathbb{N}}.$$ Thus if $u \in {\mathcal{E}}^{[{\mathfrak{M}}]}_{P,k}(U)$ then there exist $M \in {\mathfrak{M}}$ and $C,{\rho}>0$ (resp. for all $M \in {\mathfrak{M}}$ and all ${\rho}>0$ there is $C$) such that $$|\widehat{({\chi}_{k_j} u)}(\xi)| \le C {\rho}^{m k_j}|\xi|^{-m k_j} M_{m k_j}.$$ Fix $v \in S^{n-1}$. In view of $$\begin{aligned}
(-i d_v)^{mk_j - n-1} u(x) =
\frac{1}{(2\pi)^n} \int e^{ i \langle x ,\xi \rangle} \langle v,\xi\rangle^{mk_j - n-1} \widehat{{\chi}_{k_j} u}(\xi)\, d\xi
\end{aligned}$$ we conclude that, for all $j$, $$\begin{aligned}
\|d_v^{mk_j - n-1} u\|_{L^\infty(U')} \le C {\rho}^{mk_j} M_{m k_j} \le \tilde C {\tilde {\rho}}^{mk_j} N_{m k_j-n-1},
\end{aligned}$$ where we used ($R_2$) in the Roumieu case. In the Beurling case we use ($B_2$) instead. The constants $\tilde C, \tilde {\rho}$ are independent of $v \in S^{n-1}$. It remains to apply for the sequence $j \mapsto mk_j - n-1$.
It is possible to just work with ${\partial}_1,\ldots,{\partial}_n$ instead of all directional derivatives $d_v$. Then the equivalence of (1) and (2) in would read: $${\mathcal{E}}^{[{\mathfrak{M}}]}(U) = \bigcap_{i=1}^n {\mathcal{E}}^{[{\mathfrak{M}}]}_{{\partial}_i,k}(U).$$ The proof is similar, but in the end one has to show the stronger statement $$\bigcap_{i=1}^n {\mathcal{E}}^{[{\mathfrak{M}}]}_{{\partial}_i,k}(U) = \bigcap_{i=1}^n {\mathcal{E}}^{[{\mathfrak{M}}]}_{{\partial}_i,(j)_j}(U).$$
Gelfand–Shilov classes
----------------------
Let ${\mathfrak{M}}$ be a weight matrix. We consider the *Gelfand–Shilov classes* ${\mathcal{S}}^{[{\mathfrak{M}}]}({\mathbb{R}}^n)$ consisting of all $f \in {\mathcal{S}}({\mathbb{R}}^n)$ such that there exist $M \in {\mathfrak{M}}$ and ${\rho}>0$ with (resp. for all $M \in {\mathfrak{M}}$ and all ${\rho}>0$) $$\sup_{k,\ell \in {\mathbb{N}}}
\sup_{|{\alpha}| = \ell} \frac{\| |x|^k D^{\alpha}f \|_{L^\infty({\mathbb{R}}^n)}}{{\rho}^{k + \ell} M_k M_\ell} < \infty.$$ The next lemma is a slight generalization of [@Chung:1996aa Theorem 2.3].
\[lem:A3\] Assume that ${\mathfrak{M}}$ is a \[regular\] weight matrix. Then the following are equivalent.
1. $f \in {\mathcal{S}}^{[{\mathfrak{M}}]}({\mathbb{R}}^n)$.
2. There exist $M\in {\mathfrak{M}}$ and constants $C,{\rho},{\sigma}>0$ (for all $M \in {\mathfrak{M}}$ and for all ${\rho},{\sigma}>$ there is $C>0$) such that $$\sup_x |x^{\alpha}f(x)| \le C {\rho}^{|{\alpha}|} M_{|{\alpha}|} \quad \text{ and } \quad \sup_x |D^{\beta}f(x)| \le C {\sigma}^{|{\beta}|} M_{|{\beta}|}$$ for all ${\alpha},{\beta}\in {\mathbb{N}}^n$.
3. There exist $M\in {\mathfrak{M}}$ and constants $C,{\rho},{\sigma}>0$ (for all $M \in {\mathfrak{M}}$ and for all ${\rho},{\sigma}>$ there is $C>0$) such that $$\sup_x |x^{\alpha}f(x)| \le C {\rho}^{|{\alpha}|} M_{|{\alpha}|} \quad \text{ and } \quad \sup_{\xi} |\xi^{\beta}\widehat f(\xi)| \le C {\sigma}^{|{\beta}|} M_{|{\beta}|}$$ for all ${\alpha},{\beta}\in {\mathbb{N}}^n$.
It is straightforward to adapt the proof of [@Chung:1996aa Theorem 2.3].
An obvious consequence of the lemma is that ${\mathcal{S}}^{[{\mathfrak{M}}]}({\mathbb{R}}^n)$ is invariant under the Fourier transform: $f \in {\mathcal{S}}^{[{\mathfrak{M}}]}({\mathbb{R}}^n)$ if and only if $\widehat f \in {\mathcal{S}}^{[{\mathfrak{M}}]}({\mathbb{R}}^n)$.
Ultradifferentiable Chevalley theorems {#sec:Chevalley}
======================================
The setting
-----------
The following facts can be found in [@Helgason:1962aa]; see also [@Dadok:1982aa] whose arguments we follow closely.
Let ${\mathfrak{g}}$ be a real semisimple Lie algebra of noncompact type and let ${\mathfrak{g}}= {\mathfrak{k}}\oplus {\mathfrak{p}}$ be a Cartan decomposition. Then ${\mathfrak{k}}$ is a subalgebra of ${\mathfrak{g}}$ which is the Lie algebra of the maximal compact subgroup $K$ of the adjoint group ${\operatorname}{Int}({\mathfrak{g}})$. The decomposition is direct with respect to the Killing form. Let ${\mathfrak{a}}\subseteq {\mathfrak{p}}$ be a maximal abelian subspace and $${\mathfrak{g}}= {\mathfrak{g}}^0 \oplus \bigoplus_{{\alpha}\in {\Sigma}} {\mathfrak{g}}^{\alpha}$$ the root space decomposition with respect to ${\mathfrak{a}}$, where $${\mathfrak{g}}^{\alpha}= \{ X \in {\mathfrak{g}}: ({\operatorname}{ad} H) X = {\alpha}(H) X \text{ for all } H \in {\mathfrak{a}}\}$$ and ${\Sigma}$ is the set of roots $0 \ne {\alpha}\in {\mathfrak{a}}^*$ with ${\mathfrak{g}}^{\alpha}\ne 0$. We set $m_{\alpha}:= \dim {\mathfrak{g}}^{\alpha}$.
Choose a Weyl chamber ${\mathfrak{a}}^+ \subseteq {\mathfrak{a}}$, i.e., a connected component of the complement of the union of hyperplanes in ${\mathfrak{a}}$ defined by the roots ${\alpha}\in {\Sigma}$. Let ${\Sigma}^+$ denote the collection of positive roots w.r.t. ${\mathfrak{a}}^+$. The adjoint action of $K$ on ${\mathfrak{p}}$ preserves the inner product induced by the Killing form. Every $K$-orbit intersects ${\mathfrak{a}}$ orthogonally in an orbit of the Weyl group. The Weyl group $$W = N_K({\mathfrak{a}})/Z_K({\mathfrak{a}}),$$ where $$N_K({\mathfrak{a}}) = \{ k \in K : {\operatorname}{Ad}(k) {\mathfrak{a}}= {\mathfrak{a}}\}$$ and $$Z_K({\mathfrak{a}}) = \{ k \in K : {\operatorname}{Ad}(k) H = H \text{ for all } H \in {\mathfrak{a}}\},$$ is a finite group of linear automorphisms of ${\mathfrak{a}}$ which is generated by the reflections in the hyperplanes $\{H \in {\mathfrak{a}}: {\alpha}(H) = 0\}$, for ${\alpha}\in {\Sigma}^+$.
If $M$ denotes the centralizer of ${\mathfrak{a}}$ in $K$, then $$K/M \times {\mathfrak{a}}^+ \ni (kM,H) \mapsto {\operatorname}{Ad}(k)H$$ is a diffeomorphism onto an open and dense subset of ${\mathfrak{p}}$. There is the following integral formula for $f \in C^\infty_c({\mathfrak{p}})$, $$\label{change}
\int_{\mathfrak{p}}f(x) \, dx = \int_{{\mathfrak{a}}^+} \int_{K/M} f({\operatorname}{Ad}(k)H) \prod_{{\alpha}\in {\Sigma}^+} {\alpha}(H)^{m_{\alpha}} \, dk \, dH,$$ where $dx$ and $dH$ are the Lebesgue measures on ${\mathfrak{p}}$ and ${\mathfrak{a}}$, respectively, and $dk$ is an invariant measure on $K/M$, all of them with suitable normalizations; see [@Helgason:1962aa p.380].
We denote by $C^\infty({\mathfrak{p}})^K$ the space of $C^\infty$-functions on ${\mathfrak{p}}$ which are invariant under the adjoint action of $K$ on ${\mathfrak{p}}$. Similarly $C^\infty({\mathfrak{a}})^W$ is the space of $W$-invariant $C^\infty$-functions on ${\mathfrak{a}}$. By ${\Delta}_{{\mathfrak{p}}}$ we mean the flat Euclidean Laplace operator on ${\mathfrak{p}}$. For $f \in C^\infty({\mathfrak{p}})^K$ we have $$({\Delta}_{{\mathfrak{p}}} f)|_{{\mathfrak{a}}^+} = {\operatorname}{rad}({\Delta}_{{\mathfrak{p}}}) (f|_{{\mathfrak{a}}^+}),$$ where ${\operatorname}{rad}({\Delta}_{{\mathfrak{p}}})$ is a differential operator on ${\mathfrak{a}}^+$ called the *radial part* of ${\Delta}_{{\mathfrak{p}}}$. Then (see [@Dadok:1982aa Proposition 1.1]) $$\label{rad}
{\operatorname}{rad}({\Delta}_{{\mathfrak{p}}}) = {\Delta}_{{\mathfrak{a}}} + \sum_{{\alpha}\in {\Sigma}^+} m_{\alpha}\frac{{\operatorname}{grad} {\alpha}}{{\alpha}},$$ where ${\Delta}_{{\mathfrak{a}}}$ is the Laplace operator on ${\mathfrak{a}}$ and $({\operatorname}{grad} {\alpha})(f) := \langle {\operatorname}{grad} {\alpha}, {\operatorname}{grad} f\rangle$.
Chevalley’s theorem in local ultradifferentiable classes
--------------------------------------------------------
\[Chevalley\] Let ${\mathfrak{M}}$ be a \[regular\] weight matrix. Then the restriction mapping ${\mathcal{E}}^{[{\mathfrak{M}}]}({\mathfrak{p}})^K \to {\mathcal{E}}^{[{\mathfrak{M}}]}({\mathfrak{a}})^W$ is an isomorphism.
Every $f \in C^\infty({\mathfrak{a}})^W$ can be extended to a continuous function $\widetilde f$ on ${\mathfrak{p}}$, by making it constant on the $K$-orbits. The continuity of $\widetilde f$ follows from [@Helgason:1980aa Proposition 2.4]; in fact, for $X,Y \in {\mathfrak{p}}$ one has $${\operatorname}{dist}({\operatorname}{Ad}(K) X,{\operatorname}{Ad}(K) Y) = {\operatorname}{dist}({\operatorname}{Ad}(K) X \cap {\mathfrak{a}}^+,{\operatorname}{Ad}(K) Y \cap {\mathfrak{a}}^+).$$ Using and , it is not hard to see that $$\label{key}
{\Delta}_{{\mathfrak{p}}} \widetilde f = ({\operatorname}{rad}({\Delta}_{{\mathfrak{p}}}) f){\,\widetilde{~~~}}$$ in the sense of distributions; for details see [@Dadok:1982aa pp.124-125].
Since $f$ is invariant with respect to the reflection through the hyperplane $\{{\alpha}= 0\}$, the function $({\operatorname}{grad} {\alpha})(f)$ vanishes on $\{{\alpha}= 0\}$ whence $({\operatorname}{grad} {\alpha})(f)/{\alpha}$ is smooth. Then, by , we may conclude that ${\operatorname}{rad}({\Delta}_{{\mathfrak{p}}}) f \in C^\infty({\mathfrak{a}})^W$, where now ${\operatorname}{rad}({\Delta}_{{\mathfrak{p}}})$ is the obvious extension to ${\mathfrak{a}}$ as a $W$-invariant differential operator (with singularities along the hyperplanes $\{{\alpha}= 0\}$, for ${\alpha}\in {\Sigma}^+$).
Iterating yields $$\label{key2}
{\Delta}_{{\mathfrak{p}}}^m \widetilde f = ({\operatorname}{rad}({\Delta}_{{\mathfrak{p}}})^m f){\,\widetilde{~~~}}, \quad m \ge 1.$$ This implies that $\widetilde f \in C^\infty({\mathfrak{p}})^K$ by means of elliptic regularity; cf. [@Dadok:1982aa p.124].
Now suppose that $f \in {\mathcal{E}}^{[{\mathfrak{M}}]}({\mathfrak{a}})^W$. By the above, the $K$-invariant extension $\widetilde f$ is smooth. Let $U$ be a relatively compact open subset of ${\mathfrak{p}}$ and let $V$ be its saturation with respect to the $K$-action, i.e., the union of all $K$-orbits that meet $U$. By , $$\label{relation}
\|{\Delta}_{{\mathfrak{p}}}^m \widetilde f\|_{L^\infty(V)} = \| ({\operatorname}{rad}({\Delta}_{{\mathfrak{p}}})^m f){\,\widetilde{~~~}}\, \|_{L^\infty(V)} =
\| {\operatorname}{rad}({\Delta}_{{\mathfrak{p}}})^m f \|_{L^\infty(V \cap {\mathfrak{a}})}.$$ We claim that there exist $M \in {\mathfrak{M}}$ and constants $C,{\rho}>0$ (resp. for all $M \in {\mathfrak{M}}$ and all ${\rho}>0$ there is $C>0$) such that $$\label{estimate}
\| {\operatorname}{rad}({\Delta}_{{\mathfrak{p}}})^m f \|_{L^\infty(V \cap {\mathfrak{a}})} \le C {\rho}^{2m} M_{2m}, \quad \text{ for all } m \ge 1.$$ Then, by , we may conclude $$\|{\Delta}_{{\mathfrak{p}}}^m \widetilde f\|_{L^\infty(V)} \le C {\rho}^{2m} M_{2m}, \quad \text{ for all } m \ge 1.$$ That implies that $\widetilde f \in {\mathcal{E}}^{[{\mathfrak{M}}]}_{{\Delta}_{{\mathfrak{p}}},(j)_j}({\mathfrak{p}})$. By , we find that $\widetilde f \in {\mathcal{E}}^{[{\mathfrak{M}}]}({\mathfrak{p}})^K$.
It remains to show the claim . We use that, for any $f \in C^\infty({\mathfrak{a}})^W$, the function $$F:= \frac{({\operatorname}{grad} {\alpha})(f)}{|{\operatorname}{grad} {\alpha}|}$$ vanishes on the hyperplane $\{{\alpha}=0\}$. Although ${\operatorname}{rad}({\Delta}_{{\mathfrak{p}}})$ is a differential operator with singularities along the hyperplanes $\{{\alpha}= 0\}$, for ${\alpha}\in {\Sigma}^+$, its action on $W$-invariant functions $f$ is well-behaved. Indeed, if we set $$v_{\alpha}= \frac{{\operatorname}{grad} {\alpha}}{|{\operatorname}{grad} {\alpha}|}$$ and denote by $$y_{\alpha}(x) := x - \frac{{\alpha}(x)}{|{\operatorname}{grad} {\alpha}|} v_{\alpha},$$ the orthogonal projection on the hyperplane $\{{\alpha}= 0\}$, then since $F$ vanishes on $\{{\alpha}= 0\}$ we have for all $x \in {\mathfrak{a}}$, $$\begin{aligned}
F(x) &= \int_0^1 \frac{d}{dt} F\big(t \tfrac{{\alpha}(x)}{|{\operatorname}{grad} {\alpha}|} v_{\alpha}+ y_{\alpha}(x)\big) \,dt
\\
&= \int_0^1 dF\big(t \tfrac{{\alpha}(x)}{|{\operatorname}{grad} {\alpha}|} v_{\alpha}+ y_{\alpha}(x)\big) (\tfrac{{\alpha}(x)}{|{\operatorname}{grad} {\alpha}|} v_{\alpha}) \,dt
\\
&=\frac{{\alpha}(x)}{|{\operatorname}{grad} {\alpha}|} \int_0^1 d_{v_{\alpha}} F\big(t \tfrac{{\alpha}(x)}{|{\operatorname}{grad} {\alpha}|} v_{\alpha}+ y_{\alpha}(x)\big) \,dt.
\end{aligned}$$ Since $F =\frac{({\operatorname}{grad} {\alpha})(f)}{|{\operatorname}{grad} {\alpha}|} = d_{v_{\alpha}} f$, we obtain $$\begin{aligned}
\Big(\frac{{{\operatorname}{grad}}{\alpha}}{{\alpha}}\Big) (f)(x)
&= \int_0^1 d_{v_{\alpha}} \big(d_{v_{\alpha}} f\big)\big(t \tfrac{{\alpha}(x)}{|{\operatorname}{grad} {\alpha}|} v_{\alpha}+ y_{\alpha}(x)\big) \,dt
\\
&= \int_0^1 d_{ v_{\alpha}}^2 f \big(t \tfrac{{\alpha}(x)}{|{\operatorname}{grad} {\alpha}|} v_{\alpha}+ y_{\alpha}(x)\big) \,dt.
\end{aligned}$$ By , we see that for $f \in C^\infty({\mathfrak{a}})^W$ we have $${\operatorname}{rad}({\Delta}_{{\mathfrak{p}}})f(x) = ({\Delta}_{{\mathfrak{a}}} f)(x) + \sum_{{\alpha}\in {\Sigma}^+} m_{\alpha}\int_0^1 d_{ v_{\alpha}}^2 f \big(t \tfrac{{\alpha}(x)}{|{\operatorname}{grad} {\alpha}|} v_{\alpha}+ y_{\alpha}(x)\big) \,dt.$$ Since ${\operatorname}{rad}({\Delta}_{{\mathfrak{p}}})f \in C^\infty({\mathfrak{a}})^W$ we can replace $f$ in the above formula by ${\operatorname}{rad}({\Delta}_{{\mathfrak{p}}})f$ and iterate this procedure in order to express ${\operatorname}{rad}({\Delta}_{{\mathfrak{p}}})^m(f)$ for $m \ge 1$ in terms of combinations of powers of differential operators ${\Delta}_{\mathfrak{a}}$ and $d_{v_{\alpha}}$ for ${\alpha}\in {\Sigma}^+$ applied to $f$. Using the linearity of the operators and the linearity of ${\alpha}$ and $y_{\alpha}$ and computing $L^\infty$-norms on balls centered at the origin in ${\mathfrak{a}}$, it is then straightforward to conclude .
\[cor1\] Let $M$ be a weight sequence with moderate growth. Then the restriction mapping ${\mathcal{E}}^{[M]}({\mathfrak{p}})^K \to {\mathcal{E}}^{[M]}({\mathfrak{a}})^W$ is an isomorphism, if we assume $(k!/M_k)^{1/k}$ is bounded in the Roumieu case and $(k!/M_k)^{1/k} \to 0$ as $k \to \infty$ in the Beurling case.
\[cor2\] Let ${\omega}$ be a weight function. Then the restriction mapping ${\mathcal{E}}^{[{\omega}]}({\mathfrak{p}})^K \to {\mathcal{E}}^{[{\omega}]}({\mathfrak{a}})^W$ is an isomorphism, if we assume ${\omega}(t) = O(t)$ as $t \to \infty$ in the Roumieu case and ${\omega}(t) = o(t)$ as $t \to \infty$ in the Beurling case.
\[cor3\] The restriction mapping $C^{\omega}({\mathfrak{p}})^K \to C^{\omega}({\mathfrak{a}})^W$ is an isomorphism.
Chevalley’s theorem in Gelfand–Shilov classes
---------------------------------------------
As a consequence of the $C^\infty$ Chevalley theorem Dadok [@Dadok:1982aa Corollary 1.5] showed that every $W$-invariant Schwartz function $f \in {\mathcal{S}}({\mathfrak{a}})^W$ extends to a Schwartz function $\widetilde f \in {\mathcal{S}}({\mathfrak{p}})^K$; see also Helgason [@Helgason:1980aa Proposition 2.3] for a different proof.
\[GelfandShilov\] Let ${\mathfrak{M}}$ be a \[regular\] weight matrix. Then the restriction mapping ${\mathcal{S}}^{[{\mathfrak{M}}]}({\mathfrak{p}})^K \to {\mathcal{S}}^{[{\mathfrak{M}}]}({\mathfrak{a}})^W$ is an isomorphism.
Let $f \in {\mathcal{S}}^{[{\mathfrak{M}}]}({\mathfrak{a}})^W$. We want to show that the $K$-invariant extension $\widetilde f$ of $f$ to ${\mathfrak{p}}$ is of class ${\mathcal{S}}^{[{\mathfrak{M}}]}$. We already know that $\widetilde f$ is smooth. Choose linear coordinates in ${\mathfrak{p}}$ such that the $K$-invariant inner product induced by the Killing form is given by $\langle x,y \rangle = x_1 y_1 + \cdots + x_n y_n$. Then $|x|^2 = \langle x, x \rangle$ is a $K$-invariant polynomial. By , it suffices to check that there exist $M \in {\mathfrak{M}}$ and constants $C,{\rho},{\sigma}>0$ (resp. for all $M \in {\mathfrak{M}}$ and all ${\rho},{\sigma}>0$ there is $C>0$) such that $$\label{GSestimate}
\sup_x |x|^{2k} |\widetilde f(x)| \le C {\rho}^{2k} M_{2k} \quad \text{ and }
\quad \sup_\xi |\xi|^{2k} |(\widetilde f)^\wedge(\xi)| \le C {\sigma}^{2k} M_{2k}$$ for all $k$. Here $(\widetilde f)^\wedge$ denotes the Fourier transform of $\widetilde f$. (We use that for any ${\alpha}\in {\mathbb{N}}^n$ we have $|x^{\alpha}| \le |x|^{|{\alpha}|} \le |x|^{2k} \max\{1,|x|^2\}$, where $k = \lfloor |{\alpha}|/2 \rfloor$, and that ${\mathfrak{M}}$ is \[regular\].)
The first estimate in simply follows from the assumption $f \in {\mathcal{S}}^{[{\mathfrak{M}}]}({\mathfrak{a}})^W$ and the fact that $$\big\| | \cdot |^{2k} \widetilde f(\cdot) \big\|_{L^\infty({\mathfrak{p}})}
= \big\| | \cdot |^{2k} f(\cdot)\big\|_{L^\infty({\mathfrak{a}})}.$$ For the second estimate in we observe that for all $\xi \in {\mathfrak{p}}$ (where $n = \dim {\mathfrak{p}}$) $$\begin{aligned}
|\xi|^{2k} | (\widetilde f)^\wedge(\xi)| \le |({\Delta}_{\mathfrak{p}}^k \widetilde f)^\wedge(\xi)|
&\le \int_{\mathfrak{p}}(1+ |x|^{2n})|{\Delta}_{\mathfrak{p}}^k \widetilde f(x)| \, \frac{dx}{(1+ |x|^{2n})}
\\
&\le \big\|(1+ |\cdot|^{2n}) {\Delta}_{\mathfrak{p}}^k \widetilde f(\cdot)\big\|_{L^\infty({\mathfrak{p}})} \int_{\mathfrak{p}}\, \frac{dx}{(1+ |x|^{2n})}
\\
&\le C(n)\, \big\|(1+ |\cdot |^{2n}) {\operatorname}{rad}({\Delta}_{\mathfrak{p}})^k f(\cdot)\big\|_{L^\infty({\mathfrak{a}})},\end{aligned}$$ where in the last inequality we used . The assumption $f \in {\mathcal{S}}^{[{\mathfrak{M}}]}({\mathfrak{a}})^W$ together with the fact that ${\mathfrak{M}}$ is \[regular\] and the justification for yields the required estimate.
Isotropic functions
===================
Let ${\mathfrak{g}}=\mathfrak{sl}(n,{\mathbb{R}})$ be the Lie algebra of $n \times n$ real matrices with trace zero and consider the Cartan decomposition ${\mathfrak{g}}= {\mathfrak{k}}\oplus {\mathfrak{p}}$, where ${\mathfrak{k}}= \mathfrak{so}(n,{\mathbb{R}})$ is the Lie algebra of skew-symmetric matrices and ${\mathfrak{p}}= {\operatorname}{Sym}(n)_0$ are the symmetric matrices with trace zero. Then $K = {\operatorname}{SO}(n)$ acts by conjugation on ${\mathfrak{p}}$ and the maximal subalgebra ${\mathfrak{a}}= {\operatorname}{Diag}(n)_0$ consists of the diagonal matrices with trace zero. The Weyl group is isomorphic to the symmetric group ${\operatorname}{S}_n$ and acts on ${\mathfrak{a}}$ be permuting the diagonal entries.
Let $F : {\operatorname}{Diag}(n) \cong {\mathbb{R}}^n \to {\mathbb{R}}$ be a symmetric function. We extend $F$ to an isotropic function $f : {\operatorname}{Sym}(n) \to {\mathbb{R}}$ by setting $f(A) = F(a_1,\ldots,a_n)$, where $a_1 \ge a_2 \ge \cdots \ge a_n$ are the eigenvalues of $A$. Setting $B= A - \tfrac{1}{n}({{\operatorname}{Tr}}A) \mathbb I$ and $b_j := a_j - \tfrac{1}{n}\sum_{i=1}^n a_i$, for $j=1,\ldots,n$, we have $$\begin{aligned}
g(B) &:= f(B + \tfrac{1}{n}({{\operatorname}{Tr}}A) \mathbb I)
\\
&= f(A)
\\
&= F(a_1,\ldots,a_n)
\\
&= F\Big(b_1 + \tfrac{1}{n}\sum_{i=1}^n a_i,\ldots,b_n +\tfrac{1}{n}\sum_{i=1}^n a_i\Big)
\\
&=: G(b_1,\ldots,b_n),
\end{aligned}$$ where $g : {\operatorname}{Sym}(n)_0 \to {\mathbb{R}}$ is isotropic and $G : {\operatorname}{Diag}(n)_0 \to {\mathbb{R}}$ is symmetric. Moreover, if $F$ is $C^\infty$, of class ${\mathcal{E}}^{[{\mathfrak{M}}]}$, or of class ${\mathcal{S}}^{[{\mathfrak{M}}]}$ (for a \[regular\] weight matrix), then $G$ is of the same class, since $F$ is the composite of $G$ with the orthogonal projection onto ${\operatorname}{Diag}(n)_0$. Then and imply that $g$, and thus $f$, is of the corresponding class. So we obtain
\[isotropic\] Let ${\mathfrak{M}}$ be a \[regular\] weight matrix. An isotropic function $f : {\operatorname}{Sym}(n) \to {\mathbb{R}}$ is of class ${\mathcal{E}}^{[{\mathfrak{M}}]}$ (resp. ${\mathcal{S}}^{[{\mathfrak{M}}]}$) if and only if its symmetric companion $F : {\mathbb{R}}^n \to {\mathbb{R}}$ is of class ${\mathcal{E}}^{[{\mathfrak{M}}]}$ (resp. ${\mathcal{S}}^{[{\mathfrak{M}}]}$).
Clearly, we immediately get isotropic versions of .
Alternatively, it is possible to use the inductive formula for the derivatives of $f$ derived in [@Silhavy2000] in order to give a direct proof of .
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[10]{}
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[^1]: The author was supported by the Austrian Science Fund (FWF), START Programme Y963.
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---
abstract: 'We study properties of collective radiations of coherently driven two three-level ladder-type atoms trapped in a single-mode cavity. Using the electromagnetically induced transparency technique, we show that the three-photon blockade effect can be observed and the properties of collective radiations are strongly dependent on the phase between two atoms. In the case of in-phase radiations, the frequency range to realize the three-photon blockade can be broadened as the control field increases. However, in the regime of the three-photon blockade, the property of collective radiations changes from hyperradiance to subradiance. In the case of out-of-phase radiations, hyperradiance accompanied with the three-photon blockade can be observed. The results presented in this paper show that our scheme is an attractive candidate to generate antibunched photon pairs and control the properties of collective radiations.'
author:
- 'Y. F. Han'
- 'C. J. Zhu'
- 'J. P. Xu'
- 'Y. P. Yang'
bibliography:
- 'supRadiance\_hyf.bib'
title: 'Electromagnetic control of the collective radiations with three-photon blockade'
---
Introduction
============
As one of the most fascinating topics in the field of quantum optics, superradiance, and subradiance have arisen extensive attention in both theory and experiments since its discovery by Dicke [@dicke1954coherence; @rehler1971superradiance; @scully2006directed; @akkermans2008photon; @rohlsberger2010collective; @monz201114; @svidzinsky2013quantum; @feng2014effect; @scully2015single; @longo2016tailoring]. In earlier years, Agarwal et al. ascribed the physical mechanism of the superradiant collective emission to strong quantum correlations among atoms lying in symmetric Dicke states [@agarwal1974quantum; @gross1982superradiance]. Later, it is found that the superradiance or subradiance results from the multiparticle entanglement of the Dicke state via the interference of quantum pathways [@wiegner2011quantum; @nienhuis1987spontaneous]. The experimental works by Blatt et al. also show that the entanglement plays an indispensable role in the collective emission of radiation [@monz201114; @devoe1996observation].
In recent years, Scully et al. [@scully2015single] theoretically studied the dynamics of single photon superradiance and subradiance, and Röhlsberger et al. [@rohlsberger2010collective] carried out an experimental work by embedding an ensemble of resonant atoms in the center of a planar cavity. Due to the back reaction in cavity QED systems, many novel features of the collective radiations have been studied theoretically and demonstrated experimentally [@friedberg1973frequency; @agarwal1974quantum; @gross1979maser; @rempe1990observation; @reimann2015cavity], which help us to get a further understanding of superradiant or subradiant collective emission. In particular, Pleinert et al. predict the possibility of hyperradiance arising from the collective radiations in a strongly coupled two atoms cavity QED system [@pleinert2017hyperradiance]. As the hyperradiance occurs, the mean photon number can be up to $2$ orders of magnitude larger than the case of a single atom. Xu et al. [@Xu2017Hyperradiance] show that the hyperradiance can also be observed when the pump field drives two atoms under the non-resonant condition [@agarwal1984vacuum; @raimond2001manipulating].
Compared with the cavity driving schemes, which is generally considered in cavity QED systems, the atom driving enhances the optical nonlinearity so that many interesting quantum and nonlinear features of cavity photons can be observed, including the improvement of two-photon blockade [@zhu2017collective], observation of three-photon blockade [@hamsen2017two], and realization of hyper-radiance phenomenon [@pleinert2017hyperradiance; @Xu2017Hyperradiance; @pleinert2018phase]. In particular, we show that the asymmetry in the atom-cavity coupling strengths open new pathways for multiphoton blockade which result in the three-photon blockade phenomenon with reasonable mean photon number over a broad frequency regime.
Combining the electromagnetically induced transparency technique with the two atoms cavity QED system, In this paper, we show that the collective radiation properties are strongly dependent on the control field Rabi frequency and the phase shift between two atoms. It is shown that the three-photon blockade can be observed not only under the condition of out-of-phase radiations but also in-phase radiations. The frequency range for achieving the three-photon blockade can be broadened by increasing the control field Rabi frequency. In particular, the property of collective radiations changes from subradiance to hyperradiance accompanying with the three-photon blockade effect. Therefore, it is possible to generate photon pairs efficiently in this system.
Model
=====
![(a) The sketch of the two atoms cavity QED system, where two three-level ladder-type atoms are trapped in a single mode cavity. The energy levels are labeled as $|\alpha\rangle\ (\alpha=g,m,e)$ with energy $\hbar\omega_\alpha$. A pump (control) field $\eta$ ($\Omega_L$) with angular frequency $\omega_p$ ($\omega_L$) couples the $|g\rangle\leftrightarrow|m\rangle$ ($|m\rangle\leftrightarrow|e\rangle$) transition. Here, $\Gamma_{me}$ ($\Gamma_{gm}$) is the spontaneous decay rate of state $|e\rangle$ ($|m\rangle$), and the cavity decay rate is $\kappa$. $\Delta_\alpha$ is the detuning of state $|\alpha\rangle$. (b) and (c) show the dressed state pictures of $\phi_z=0$ and $\pi$, respectively.[]{data-label="fig:fig1"}](fig1.eps){width="50.00000%"}
The configuration of this two atoms cavity QED system is shown in Fig. 1(a), where two identical three-level ladder-type atoms are trapped in a single mode cavity at different positions $z_i$ ($i=1,2$). For each atom, the energy levels are labeled as $|g\rangle$, $|m\rangle$ and $|e\rangle$, respectively. As shown in Fig. 1(a), a weak pump field with angular frequency $\omega_p$ drives the $|g\rangle\leftrightarrow|m\rangle$ transition, but a strong control field with angular frequency $\omega_L$ couples $|m\rangle\leftrightarrow|e\rangle$ transition. In our system, we assume that the cavity mode with resonant frequency $\omega_{\rm cav}=2\pi c/\lambda_{\rm cav}$ only couples the $|g\rangle\leftrightarrow|m\rangle$ transition. The position-dependent coupling strengths between the atom and cavity are then given by $g_i=g\cos{(2\pi z_i/\lambda_{\rm cav})}$ where $z_i$ is the position of the $i$-th atom ($i=1,2$), $g$ is the maximum atom-cavity coupling strength and $\lambda_{\rm cav}$ is the wavelength of the cavity mode. For mathematic simplicity, we also assume that one atom is fixed at the antinode of the cavity (i.e., $z_1=0$), while the position of the other atom can be varied along the axis of the cavity. Therefore, the atom-cavity coupling strengths are given by $g_1=g$ and $g_2=g\cos{(\phi_z)}$ with $\phi_z=2\pi\Delta z/\lambda_{\rm cav}=2\pi(z_2-z_1)/\lambda_{\rm cav}$.
Under the electric dipole and rotating wave approximations, the dynamical behavior of the whole system shown in Fig. 1(a) can be treated by a master equation given by $$\frac{d}{dt}\rho=-\frac{i}{\hbar}[H,\rho]+{\cal L}_{\kappa}\rho+{\cal L}_{\Gamma}\rho,
\label{eq:master}$$ where $\rho$ is the density operator of the atom-cavity system. The Hamiltonian of the whole system $H=H_{0}+H_{I}+H_{D}$, where $H_0$, $H_I$ and $H_D$ denotes the energy of the atoms and cavity, the interaction between the atom and cavity, and the driving terms, respectively. In our system, we have $H_{0}=\hbar\sum_{j=1}^2(\Delta_{m}S^{j}_{mg}S^{j}_{gm}+\Delta_{e}S^{j}_{em}S^{j}_{me})+\hbar\Delta_{\rm cav}a^{\dag} a$, $H_{I}=\hbar\sum_{j=1}^2[g_{j}(aS^{j}_{mg}+a^{\dag} S^{j}_{gm})+\Omega_{L}(S^{j}_{em}+S^{j}_{me})]$ and $H_{D}=\hbar \eta\sum_{j=1}^2(S^{j}_{mg}+S^{j}_{gm})$. Here, $\Omega_L$ and $\eta$ are the Rabi frequency of the control field and pump field, respectively. $S^j_{\alpha\beta}=|\alpha\rangle_j\langle\beta|\ (\alpha,\beta={g,m,e})$ is the atomic raising and lowering operator of the $j$-th atom ($j=1,2$). $a$ and $a^{\dag}$ are the annihilation and creation operator of the cavity mode. The detunings are defined as $\Delta_{m}=\omega_{m}-\omega_g-\omega_{p}$, $\Delta_{e}=\omega_{e}-\omega_g-(\omega_p+\omega_c)=\Delta_m+\Delta_{L}$ with $\Delta_L=\omega_e-\omega_m-\omega_L$ and $\Delta_{\rm cav}=\omega_{\rm cav}-\omega_{p}$, respectively. It is worth to point out that, to avoid the dipole-dipole interactions [@goldstein1997dipole], the separation of two atoms must be larger than the cavity wavelength $\lambda_{\rm cav}$.
The spontaneous decays of the atomic states are introduced by the Liouvillian operators, i.e., ${\cal L}_{\Gamma}\rho=\sum_{j=1}^2[\Gamma_{gm}(2S_{gm}^{j}\rho S_{mg}^{j}-S_{mg}^{j}S_{gm}^{j}\rho-\rho S_{mg}^{j}S_{gm}^{j})+\Gamma_{me}(2S_{me}^{j}\rho S_{em}^{j}-S_{em}^{j}S_{me}^{j}\rho-\rho S_{em}^{j}S_{me}^{j})]$, where $\Gamma_{\alpha\beta}$ denotes the spontaneous decay rate from state $|\beta\rangle$ to state $|\alpha\rangle$. The Liouvillian term describing the cavity decay at rate $\kappa$ is given by ${\cal L}_{\kappa}\rho=\kappa(2a\rho a^{\dag}-a^{\dag} a \rho-\rho a^{\dag} a)$. Solving Eq. (1) numerically, one can examine the features of the collective radiations and the quantum properties of the cavity field simultaneously.
In general, the quantum properties of an optical field can be characterized by the field correlation function, which can also be used to describe the statistical properties of the fields, such as bunching and anti-bunching behaviors. In quantum field theory, the steady-state second-order and third-order field correlation functions are defined as $$g_{\rm ss}^{(2)}(0)=\frac{\langle a^{\dag} a^{\dag} a a \rangle}{\langle a^{\dag} a \rangle^{2}},$$ and $$g_{\rm ss}^{(3)}(0)=\frac{\langle a^{\dag} a^{\dag} a^{\dag} a a a \rangle}{\langle a^{\dag} a \rangle^{3}},$$ respectively. Here, $g^{(2)}_{\rm ss}(0)>1$ denotes the bunched photons, whereas $g^{(2)}_{\rm ss}(0)<1$ denotes the anti-bunched photons. Indeed, $g^{(2)}_{\rm ss}(0)<1$ is an important witness for the two-photon blockade effect and is used to characterize the quality of single-photon sources in applications. $g^{(3)}_{\rm ss}(0)$ is the third-order field correlation function, which is used to characterize the probability of simultaneous arrival of three photons. If $g^{(3)}_{\rm ss}(0)<1$ but $g^{(2)}_{\rm ss}(0)>1$, one can observe that two photons arrive together, but the third photon arrives at a different time. This condition implies the three-photon blockade effect and can be used to characterize the quality of two-photon sources [@zhu2017collective].
{width="\textwidth"}
On the other hand, the behavior of collective radiations can be characterized by a witness parameter $R$, which is given by [@pleinert2017hyperradiance], $$\label{eq:R}
% \nonumber to remove numbering (before each equation)
R=\frac{\langle a^{\dag} a \rangle _{2}-2\langle a^{\dag} a \rangle _{1}}{2\langle a^{\dag} a \rangle _{1}}$$ Here, $\langle a^{\dag} a \rangle _{2}$ is the mean photon number in two atoms cavity QED system, and $\langle a^{\dag} a \rangle _{1}$ is the mean photon number in a single atom-cavity QED system. Obviously, $R<0$ means that the collective radiations are suppressed (i.e., subradiance), and $R=0$ indicates that the radiation rate is the same as that of the single atom case. However, $R>0$ shows that the collective radiations are enhanced compared with the single atom case. Particularly, $R=1$ (i.e., $\langle a^{\dag} a \rangle _{2}=4\langle a^{\dag} a \rangle _{1}$) indicates that the radiation rate scales with the square of the number of atoms $\propto N^{2}$, corresponding to the superradiance phenomenon [@auffeves2011few]. Furthermore, $R>1$ denotes that the collective radiations are significantly enhanced, which is defined as the hyperradiance [@pleinert2017hyperradiance].
In-phase radiations
===================
We first consider the case that two atoms radiate in phase (i.e., $\phi_z=0$), where $g_{1}=g_{2}=g$. In the absence of the control field, both theoretical and experimental works show that the three-photon blockade is hard to be observed [@hamsen2017two; @zhu2017collective]. As shown in Fig. 2(a), there exist four peaks in the cavity excitation spectrum, corresponding to the one-photon and two-photon excitations, respectively. In a very narrow frequency regime near the two-photon excitation, the three-photon blockade can be achieved with the superradiance and even hyper-radiance behavior \[$R\geq1$, see the gray area in Fig. 2(a) and (b)\]. Here, the system parameters are given by $g=20\kappa,\Gamma_{gm}=\kappa,\Gamma_{me}=\kappa/100,\eta=2\kappa$[@pritchard2010cooperative; @petrosyan2011electromagnetically]. In the presence of the control field, the left side peak in the cavity excitation spectrum is split into two peaks due to the coupling of the control field \[see Fig. 2(c)\]. As a result, the frequency range to realize the three-photon blockade is broadened and the collective radiation is still enhanced ($0<R<1$) as shown in Fig. 2(c) and (d) (gray areas). Further increasing the control field Rabi frequency, although the behavior of the collective radiations changes from superradiance to subradiance ($R<0$), the three-photon blockade can still be observed over a wide frequency range \[see Fig. 2(e) and 2(f)\].
The physical mechanism can be explained by exploring the eigenstates of this multilevel system. Assuming that the pump and control fields are not very strong and can be treated as perturbations of the system, we first consider the interaction between the cavity field and states $|g\rangle$ and $|m\rangle$. In this case, the eigenvalues and eigenstates can be obtained easily by using the collective states $|gg\rangle$, $|\pm\rangle=(|mg\rangle\pm|gm\rangle)/\sqrt{2}$ and $|mm\rangle$ to rewrite the Hamiltonian [@pleinert2017hyperradiance; @zhu2017collective; @han2018electromagnetic]. As shown in Fig. 1(b), we have the eigenstates $\Psi_{\pm}^{(1)}=(\pm|gg,1\rangle+|+,0\rangle)/\sqrt{2}$ with eigenvalues $E_{\pm}^{(1)}=\hbar\omega_{\rm cav}\pm\sqrt{2}g\hbar$ in one-photon space. In two-photon space, the eigenstates are given by $\Psi_{\pm}^{(2)}=|gg,2\rangle/\sqrt{3}\pm|+,1\rangle/\sqrt{2}+|mm,0\rangle/\sqrt{6}$ with eigenvalues $E_{\pm}^{(2)}=2\hbar\omega_{\rm cav}\pm\sqrt{6}g\hbar$ and $\Psi_{0}^{(2)}=(-\sqrt{3}|gg,2\rangle+\sqrt{6}|mm,0\rangle)/3$ with eigenvalues $E_{0}^{(2)}=2\hbar\omega_{\rm cav}$. Taking $\Delta_{L}=\sqrt{6}g/2$, we find that the control field couples the $\Psi_{+}^{(2)}\rightarrow|ee,0\rangle$ transition resonantly via two-photon process. Therefore, the state $\Psi_{+}^{(2)}$ is split into a doublet, corresponding to two left side peaks shown in Fig. 2(c) and 2(e).
{width="\textwidth"}
Out-of-phase radiations
=======================
Now, we consider that two atoms radiate out-of-phase, i.e., $\phi_{z}=\pi$ yielding $g_{1}=-g_{2}=g$. Neglecting the pump and control field, one can obtain three eigenstates in one-photon space, labeled as $\Psi_{\pm}^{(1)}=(\pm|gg,1\rangle+|-,0\rangle)/\sqrt{2}$ with eigenvalues $E_{\pm}^{(1)}=\hbar\omega_{\rm cav}\pm \sqrt{2}g\hbar$ and $\Psi_{0}^{(1)}=|+,0\rangle$ with eigenvalues $E_{0}^{(1)}=\hbar\omega_{\rm cav}$, respectively. In two-photon space, there exist four eigenstates, i.e., $\Psi_{\pm}^{(2)}=-|gg,2\rangle/\sqrt{3}\mp|-,1\rangle/\sqrt{2}+|mm,0\rangle/\sqrt{6}$ with eigenvalues $E_{\pm}^{(2)}=2\hbar\omega_{\rm cav}\pm \sqrt{6}g\hbar$, two degenerate states $\Psi_{0}^{(2)}=|gg,2\rangle/\sqrt{3}+\sqrt{6}|mm,0\rangle/3$ and $\Phi_{0}^{(2)}=|+,1\rangle$ with the same eigenvalue $E_{0}^{(2)}=2\hbar\omega_{\rm cav}$, as depicted in Fig. 1(c). Different from the case of $\phi_z=0$, the asymmetry coupling strengths cause that the one-photon excitation pathways are forbidden [@zhu2017collective]. Therefore, two-photon (two side peaks) and multiphoton (central peak) excitations are dominant and three-photon blockade effect can be observed over a wide frequency range as shown in Fig. 3(a) and (b). When the control field is tuned to be resonant with the $|ee,0\rangle\rightarrow \Psi^{(2)}_{+}$ transition via the two-photon process as shown in Fig. 1(c), the two-photon excitation state $\Psi^{(2)}_{+}$ will be split into a doublet so that one can observe four peaks in the cavity excitation spectrum \[see Fig. 3(b)\]. As shown in panels (d) and (f), the three-photon blockade can always be observed over a wide frequency range when the control field is turned on. However, the behavior of the collective radiation changes from superradiance to hyperraidance as the control field Rabi frequency increases \[see panels (c) and (e)\]. It must be pointed out that these features of the collective radiations and the quantum properties of the cavity field exhibited in out-of-phase radiation case can not be observed in single atom-cavity QED systems.
Conclusion
==========
We have studied the behavior of collective radiations of cavity field under the condition of the three-photon blockade in a two atoms cavity QED system. By using the EIT technique, we show that the behavior of the collective radiation and quantum properties of the cavity field can be greatly changed by tuning the control field Rabi frequency. Even in the case of in phase radiations, the three photon blockade effect can be observed over a wide frequency range. Under the condition of three-photon blockade, more interestingly, the behavior of the collective radiations can be changed from superradiance to hyperraidance by just increasing the control field Rabi frequency in the case of out phase radiations. The results presented here shows that our scheme is a good candidate to generate photon pairs via the three-photon blockade effect.
We acknowledge the National Key Basic Research Special Foundation (Grant No. 2016YFA0302800); the Shanghai Science and Technology Committee (Grant No. 18JC1410900); the National Nature Science Foundation (Grants No. 11774262, 11474003, 11504003, 61675006).
|
---
abstract: 'We have performed a set of 11 three-dimensional magnetohydrodynamical core-collapse supernova simulations in order to investigate the dependencies of the gravitational wave signal on the progenitor’s initial conditions. We study the effects of the initial central angular velocity and different variants of neutrino transport. Our models are started up from a 15$M_{\odot}$ progenitor and incorporate an effective general relativistic gravitational potential and a finite temperature nuclear equation of state. Furthermore, the electron flavour neutrino transport is tracked by efficient algorithms for the radiative transfer of massless fermions. We find that non- and slowly rotating models show gravitational wave emission due to prompt- and lepton driven convection that reveals details about the hydrodynamical state of the fluid inside the protoneutron stars. Furthermore we show that protoneutron stars can become dynamically unstable to rotational instabilities at $T/|W|$ values as low as $\sim2\%$ at core bounce. We point out that the inclusion of deleptonization during the postbounce phase is very important for the quantitative GW prediction, as it enhances the absolute values of the gravitational wave trains up to a factor of ten with respect to a lepton-conserving treatment.'
address: 'Institute of Physics, Basel University Klingelbergstrasse 82 CH-4056 Basel , Switzerland'
author:
- 'S. Scheidegger, S.C. Whitehouse, R. Käppeli, and M. Liebendörfer'
title: Gravitational waves from supernova matter
---
Introduction {#section:intro}
============
Stars in the mass range $8M_{\odot}\lesssim M \lesssim 40M_{\odot}$ end their lives in a core-collapse supernova (CCSN). However, at present the fundamental explosion mechanism, which causes a star to lose its envelope by a yet uncertain combination of factors including neutrino heating, rotation, hydrodynamical instabilities, core g-mode oscillations and magnetic fields, is still under debate (for a review, see e.g. [@2007PhR...442...38J]). As strong indications both from theory and observations exist that CCSNe show aspherical, multidimensional features [@1994ApJ...435..339H; @2006Natur.440..505L], there is a reasonable hope that a small amount of the released binding energy will also be emitted as gravitational waves (GWs), thus delivering us first-hand information about the dynamics and the state of matter at the centre of the star. GW emission from CCSNe were suggested to arise from i) axisymmetric rotational core collapse and bounce ii) prompt-, neutrino-driven postbounce convection and anisotropic neutrino emission , iii) protoneutron star (PNS) g-mode oscillations [@2006PhRvL..96t1102O] and iv) nonaxisymmetric rotational instabilities For recent reviews with a more complete list of references, see [@2006RPPh...69..971K; @2009CQGra..26f3001O]. However, only i) can be considered as being well understood as far as the physics of the collapse is concerned, since only theses models incorporate all relevant input physics known at present [@2008PhRvD..78f4056D] (there are, though, still large uncertainties with respect to the progenitor star, e.g. rotation profiles, magnetic fields, and inhomogenities from convection). The prediction of all other suggested emission scenarios (ii-iv) still neglect, to a certain extent, dominant physics features due to the diversity and complexity of the CCSN problem on the one hand side and restrictions of available computer power on the other side. Hence, the computational resources were so far either spent on highly accurate neutrino transport (e.g. ) while neglecting other physical degrees of freedom such as magnetic fields, or focus on a general relativistic treatment and/or 3D fluid effects such as accretion funnels, rotation rate and convection, but approximate or even neglect the important micro physics. Only recently have detailed 3D computer models of CCSN become feasible with the emerging power of tens of thousands CPUs unified in a single supercomputer. Such detailed simulations are absolutely indispensable for the following reasons: a) GW astronomy requires not only very sensitive detectors, but also depends on extensive data processing of the detector output on the basis of reliable GW estimates [@abbott]. b) The temperatures and densities inside a supernova core exceed the range that is easily accessible by terrestrial experiments. Thus, it will be impossible for the foreseeable future to construct a unique finite temperature equation of state (EoS) for hot and dense matter based on experimentally verified data. Therefore, models with different parameter settings must be run and their computed wave form output then can be compared with actual detector data. Hence, modelling will bridge the gap between theory and measurement and allowing the use of use of CCSNe as laboratory for exotic nuclear and particle physics [@2009NuPhA.827..573L]. In this paper, we will present the GW analysis of a set of 11 three-dimensional ideal magnetohydrodynamical (MHD) core-collapse simulations. We will focus our study on the imprint of 3D nonaxisymmetric features onto the GW signature. Our calculations include presupernova models from stellar evolution calculations, a finite-temperature nuclear EoS and a computationally efficient treatment of deleptonization during the collapse phase. General relativistic corrections to the spherically symmetric Newtonian gravitational potential are taken into account. Moreover, while several models incorporate long-term neutrino physics by means of a leakage scheme, we also present the first results of a model which includes a neutrino transport approximation in the postbounce phase that takes into account both neutrino heating and cooling. As for the progenitor, we systematically investigate the effects of the spatial grid resolution, the neutrino transport physics and the precollapse rotation rate with respect to its influence on the nonaxisymmetric matter dynamics.
This paper is organised as follows. In section 2 we briefly describe the initial model configurations and the numerical techniques employed for their temporal evolution. Furthermore, we review the tools used for the GW and data analysis. Section 3 collects the results of our simulations. Finally, section 4 contains our conclusions and an outlook of our future research.
Description of the magnetohydrodynamical models {#section:MHD}
-----------------------------------------------
For the 3D Newtonian ideal MHD CCSN simulations presented in this paper, we use the `FISH` code [@2009arXiv0910.2854K]. The gravitational potential is calculated via a spherically symmetric mass integration that includes radial general relativistic corrections [@Marek2006]. The 3D computational domain consists of a central cube of either $600^3$ or $1000^3$ cells, treated in equidistant Cartesian coordinates with a grid spacing of 1km or 0.6km. It is, as explained in , embedded in a larger spherically symmetric computational domain that is treated by the time-implicit hydrodynamics code ‘Agile’ [@Liebend2002]. Closure for the MHD equations is obtained by the softest version of the finite-temperature nuclear EoS of [@Lattimer1991]. The inclusion of neutrino physics is an essential ingredient of CCSNe simulations, as $\sim$99% of the released binding energy is converted into neutrinos of all flavours. Their complex interactions with matter (e.g. [@2007PhR...442...38J]) are believed to drive the supernova explosion dynamics in the outer layers as well as deleponizing the PNS to its compact final stage as a neutron star. As the Boltzmann neutrino transport equation can only be numerically solved in a complete form in spherical symmetry on today’s supercomputers [@Mezzacappa2005], our 3D simulations must rely on several feasible approximations which capture the dominant features of the neutrino physics. As for the treatment of the deleptonization during the collapse phase, we apply a simple and computationally efficient $Y_{e}$ vs. $\rho$ parametrization scheme which is based on data from detailed 1D radiation-hydrodynamics calculations [@Liebendorfer2005]. For this we use the results obtained with the `Agile-Boltztran` code [@Liebend2005], including the above-mentioned EoS and the electron capture rates from [@1985ApJS...58..771B]. Around core bounce, this scheme breaks down as it cannot model the neutronization burst. After core bounce, the neutrino transport thus is tracked for several models via a partial (i.e. a leakage scheme) or full implementation of the isotropic diffusion source approximation scheme (IDSA, [@2009ApJ...698.1174L]). The IDSA decomposes the distribution function $f$ of neutrinos into two components, a trapped component $f^{t}$ and a streaming component $f^{s}$, representing neutrinos of a given species and energy which find the local zone opaque or transparent, respectively. The total distribution function is the sum of the two components, $f=f^{t} + f^{s}$. The two components are evolved using separate numerical techniques, coupled by a diffusion source term $\Sigma$. The source term $\Sigma$ converts trapped into streaming particles and vice versa. We determine it from the requirement that the temporal change of $f^{t}$ has to reproduce the diffusion limit in the limit of small mean free path. Note that our leakage scheme significantly overestimates the deleptonization in and around the neutrinosphere region, as it neglects any absorption of transported neutrinos by discarding the streaming component ($f^{s}=0$).
\[tab1\]
[@\*[8]{}[l]{}]{} $\0\0Model$ & $\Omega_{c,i}$ & $\beta_{i}$ & $\beta_{b}$ &$\rho_{c,b}$ &$\0f_{TW}$&$E_{GW}$ & $t_{f}$ R0 & 0 & 0 & 0 & 4.39 & - & 0.02 &130 R0$_{IDSA}$ & 0 & 0 & 0 & 4.34 & - & 0.01 & 81
R1$_{HR}$ & 0.3 & 0.6$\cdot10^{-5}$ & 1.7$\cdot10^{-4}$ & 4.36 & - & 0.24 & 25 R1$_{L}$ & 0.3 & 0.6$\cdot10^{-5}$ & 1.8$\cdot10^{-4}$ & 4.38 & - & 0.10 & 93
R2 & 3.14 & 0.6$\cdot10^{-3}$ & $1.6\cdot10^{-2}$ & 4.27 & - & 5.5 &127
R3 & 3.93 & 1.0$\cdot10^{-3}$ & $2.3\cdot10^{-2}$ & 4.16 & 670 & 14&106
R4 & 4.71 & 1.4$\cdot10^{-3}$ & $3.2\cdot10^{-2}$ & 4.04 & 615 & 35&64
R5 & 6.28 & 2.6$\cdot10^{-3}$ & $5.2\cdot10^{-2}$ & 3.80 & 725 & 59&63 R5$_{L}$ & 6.28 & 2.6$\cdot10^{-3}$ & $5.1\cdot10^{-2}$ & 3.65 & 909 &214&197
R6 & 9.42 & 5.7$\cdot10^{-3}$ & $8.6\cdot10^{-2}$ & 3.22 & 662 & 77&99 R7 & 12.57 & 1.0$\cdot10^{-2}$ & $10.2\cdot10^{-2}$ & 2.47 & 727& 12&93
The presupernova stellar models stem from Newtonian 1D stellar evolution calculations and hence may not cover all possible states prior to the collapse of a multidimensional star. Therefore we construct the initial conditions of our simulations by a parametric approach. We employ a solar-metallicity 15$M_{\odot}$ progenitor of [@Woosley1995], and set it into rotation according to a shell-type rotation law of with a shellular quadratic cutoff at 500km radius. The initial magnitude of the magnetic field strength for all models is fixed at values suggested by [@2005ApJ...626..350H].
Gravitational Wave extraction {#section:GW}
-----------------------------
We employ the Newtonian quadrupole formula in the *first-moment of momentum density formulation* [@1990ApJ...351..588F] to extract the GWs from our simulation data. Note that the quadrupole formula is not gauge invariant and only valid in the Newtonian slow-motion limit [@1973grav.book.....M]. Nevertheless, it was shown by [@2003PhRvD..68j4020S] in comparative tests to work sufficiently well compared to more sophisticated methods, as it preserves phase while being off in amplitude by $\sim$10%.
Results {#section:results}
=======
Non- or slowly rotating core collapse
-------------------------------------
In order to study the influence of neutrino transport on the stochastic matter dynamics in the early supernova stages ($t\lesssim100$ms after bounce), without having other different physical parameters interfering, we carried out three simulations: R0 (purely hydrodynamical postbounce evolution), R1$_{L}$ (includes a leakage scheme) and R0$_{IDSA}$ (incorporates both neutrino cooling and heating). Non- and slowly rotating progenitors ($\Omega_{c,i} \leq 0.3$rads$^{-1}$ in our model set) all undergo quasi-spherically symmetric core collapse. As the emission of GWs intrinsically depends on dynamical processes that deviate from spherical symmetry, the collapse phase therefore does not provide any kind of signal, as shown in Fig.\[fig:1a\] for $t-t_{b}<0$. However, subsequent pressure-dominated core bounce, where the collapse is halted due to the stiffening of the EoS at nuclear density $\rho_{nuc}\approx 2\times 10^{14}$gcm$^{-3}$, launches a shock wave that plows through the infalling layers, leaving behind a negative entropy gradient. Moreover, as soon as the shock breaks through the neutrino sphere $\sim5$ms after bounce, the immediate burst of electron neutrinos causes a negative lepton gradient at the edge of the PNS. The combination of these two gradients form a convectively unstable region according to the Schwarzschild-Ledoux criterion [@1959flme.book.....L; @1988PhR...163...63W], which in turn induces a GW burst due to this so-called ‘prompt’ convection. A detailed comparison of the models R0, R1$_{L}$, R0$_{IDSA}$ shows that all of them follow a similar dynamical behaviour until about 20ms after bounce. At this stage, aspherities leading to GW emission are predominantly driven by the negative entropy gradient and not by the lepton gradient. Hence, the wave trains of all three models, which are based on stochastic processes, fit each other relatively well both in amplitude (several cm) and spectra ($\sim150 - 500$Hz). However, ‘prompt’ convection depends, as it was pointed out by [@2009CQGra..26f3001O], not on the negative entropy gradient alone, but also on numerical seed perturbations which are introduced by the choice of the computational grid. Hence, in order to test the dependence of our findings on the spatial resolution, we carried out model R1$_{HR}$. This better resolved simulation shows considerably smaller seed perturbations around $t-t_{b}\sim$ 0, as grid alignment effects are better suppressed at core bounce; hence prompt convection then is much weaker and a smaller GW amplitude ($\sim50\%$) is emitted, as shown in Fig.\[fig:1a\]. However, better numerical resolution also leads to less numerical dissipation in the system, which eases the dynamical effects that follow. Thus, we find for $\sim10$ms $\lesssim t \lesssim20$ms considerably stronger GW emission in R1$_{HR}$ compared to the 1km resolved models, as indicated in Fig.\[fig:1a\]. The three representative simulation results diverge strongly in the later postbounce evolution ($t\gtrsim$ 20ms). Convective overturn causes a smoothing of the entropy gradient. As a result, the GW amplitude in the hydrodynamical model R0 quickly decays ($t\lesssim30$ms after bounce) and is not revived during the later evolution. On the other hand, the negative radial lepton gradient (see Figs.\[fig:2a\] and \[fig:2b\]) which is caused by the neutronization burst and the subsequent deleptonization, which we model only in R1$_{L}$ and R0$_{IDSA}$, now starts to drive convection inside the PNS. For the latter models, the so-called PNS convection [@2006ApJ...645..534D] exhibits similar maximum amplitudes of $\sim$1-2cm (Fig.\[fig:1a\]), while differing from each other strongly in the corresponding spectra, as displayed in Fig.\[fig:1b\]. R1$_{L}$’s spectrum peaks between $\sim$600 -1000Hz, while R0$_{IDSA}$’s frequency band peaks at values as low as $\sim$100Hz. This affects the total energy $E_{GW}$ emitted ($\mathcal{O}(10^{-10})M_{\odot}c^2$ vs. $\mathcal{O}(10^{-11})M_{\odot}c^2$, see Tab.\[tab1\]), being one order of magnitude higher for R1$_{L}$ due to $dE_{GW}/df\propto f^2$. We found the key controlling factor of this behaviour to be the radial location of the convectively unstable zones and the related dynamical characteristical timescales $t_{dyn}$ involved. If we use as rough estimate t$_{dyn} \sim \Delta_{r}/\overline{c_{s}}$, [^1], and apply typical values for the models R0$_{IDSA}$ and R1$_{L}$, we confirm the obtained values. Furthermore, our leakage scheme significantly overestimates neutrino cooling processes, as one can see in Fig.\[fig:2b\]. There, the convectivly unstable layer is extended to radii above nuclear densities, where matter still is opaque for neutrinos and where the local speed of sound assumes values far larger than in the case of model R0$_{IDSA}$. Hence, the dynamical timescale of R1$_{L}$ is considerably shorter and the spectral distribution is peaked at higher values. When comparing the results found for model R0$_{IDSA}$ with a very recent 2D study of , where they carried out one simulation (cf. their model M15LS-2D) with comparable input physics (same $15M_{\odot}$ progenitor; same underlying finite-temperature EoS) and a very sophisticated neutrino transport scheme, we find very good agreement both in the amplitudes and frequencies. Hence we conclude that the primary ingredient for supernova simulations which attempt a quantitative prediction of GWs from ‘prompt’ and early PNS convection ($t\lesssim100$ms after bounce) is the accurate radial location and size of convectively unstable layers. It defines the dynamical behaviour and timescale of overturning matter in this early supernova stage.
Rotational core collapse & nonaxisymmetric instability at low $T/|W|$
---------------------------------------------------------------------
Recently it has been argued based on numerical simulations of equilibrium neutron star models or full core-collapse simulations that differentially rotating PNS can be subject to non-axiymmetric rotational instabilities (see Fig.\[fig:3a\]) at $\beta$ values ($\hat{=}T/|W|$, the ratio of rotational to gravitational energy) far below the ones known from the classical dynamical bar mode instability with a threshold of $\beta_{dyn} = 27\%$, or the secular instability, which is triggered at $\beta_{sec}\sim$ 14% [@1978trs..book.....T], leading to strong, narrow-band GW emission, as displayed in Fig.\[fig:3b\] . At present little is known about the true nature of the so-called low $T/|W|$ instability. Previous work has so far failed to establish (for example) an analytical instability criterion, as was pointed out by [@2009CQGra..26f3001O]. We addressed two relevant questions regarding the so-called ‘low $T/|W|$’ instability in the context of stellar core collapse: i) Which is the minimum $\beta$ value required in self-consistent core-collapse simulation to trigger the onset of the instability? This is important to know, since most stars which undergo a core collapse rotate only slowly [@2005ApJ...626..350H]; furthermore, it was pointed out by [@2008PhRvD..78f4056D] that even fast rotating PNS can never accrete enough angular momentum to reach the $\beta_{dyn}$ value required for the onset of the classical bar mode instability. ii) How does the inclusion of deleptonization in the postbounce phase quantitatively alter the GW signal? So far, 3D models have not included spectral neutrino physics in the postbounce phase. To study i), we systematically change the rotation rate while keeping the other model parameters fixed. The minimum $T/|W|$ value we found in our parameter range to trigger the instability was $\beta_{b}\sim2.3\%$ at core bounce (model R3), which is considerably lower than seen in previous studies ([@2007PhRvL..98z1101O] found $\beta_{b}\sim9\%$, while found $\beta_{b}\sim5\%$). Furthermore, we find that centrifugal forces set a limit to the maximum frequency of the GW signal around $\sim$ 900Hz. The faster the initial rotation rate, the stronger the influence of centrifugal forces, which slow down the postbounce advection of angular momentum onto the PNS. The result is a slower rotation rate, a lower pattern speed and thus GW emission at lower frequencies (see Tab.\[tabone\]). In order to address ii), we carried out ‘leakage’ model R5$_{L}$. This model shows 5-10 $\times$ larger maximum amplitudes due to the nonaxisymmetric dynamics compared to its hydrodynamical counterpart R5 that neglects neutrino cooling (see Fig. \[fig:3b\]). This suggests that the treatment of postbounce neutrino cooling plays an important role when it comes to the quantitative prediction of GW signals from a low $\beta$ instability. The neutrino cooling during the postbounce phase leads to a more compact PNS with a shorter dynamical timescale compared to the purely hydrodynamical treatment. This in turn is reflected in the dynamical evolution. The shock wave stalls at considerably smaller radii and becomes more quickly unstable to azimuthal fluid modes. Since there is much more matter in the unstable region of this model, the unstable modes grow faster, causing the emission of much more powerful GWs. However, we again point out that our leakage scheme overestimates the compactification of the PNS due to neutrino cooling. The ‘reality’ for the strength of GW emission therefore should lay in between the results from the pure hydrodynamical- and leakage treatment.
Summary and outlook
===================
We have presented the GW signature of eleven 3D core-collapse simulations with respect to variations in the spatial grid resolution, the underlying neutrino transport physics and the initial rotation rate. Our results show that in case of non- and slowly rotating models the GWs emitted during the first 20ms after bounce are predominantly due to entropy driven ‘prompt’ convection. It turns out that the crucial parameter to study this stochastic phenomenon is the choice of the spatial resolution and not the inclusion of a neutrino transport scheme. This parameter has a twofold effect: Firstly, it governs the influence of numerical noise, since a better resolution leads to lower numerical seed perturbations and thus smaller grid alignment effects. Therefore, the GW amplitude right at core bounce is smaller for higher spatial resolution. Secondly, it enhances the ability to follow dynamical features, as better numerical resolution causes less numerical dissipation in the system, which eases the dynamical effects which follow, leading to larger GW amplitudes after the core bounce compared to less resolved models. The lepton driven convection is the central engine for the later dynamical postbounce evolution of the PNS ($t\gtrsim 20$ms) and hence the GW emission. Our findings and comparisons with state of the art 2D simulations of suggest that the radial location and size of the convectively unstable layers are the key controlling factor for the outcome of the GW prediction, as they define the timescale and the dynamical behaviour of the overturning matter. Here we find a large sensitivity to the numerical approach of the neutrino transport scheme. In our rotational core-collapse simulations, nonaxisymmetric dynamics develops for models with a rotation rate of $\beta_{b} \gtrsim 2.3\%$ at core bounce. Beyond this value, which is considerably lower than found in previous studies (e.g. ), all models become subject to the ’low $T/|W|$’ instability of dominant m=1 or m=2 character within several ms after bounce (Fig.\[fig:3a\]). The fact that the effectively measured GW amplitude scales with the number of GW cycles $N$ as $h_{eff}\propto h\sqrt{N}$ [@1989thyg.book.....H] suggests that the detection of such a signal is tremendously enhanced. Moreover, we point out that the inclusion of deleptonization during the postbounce phase causes a compactification of the PNS which enhances the absolute values of the GW amplitudes up to a factor of ten with respect to a lepton-conserving treatment.
The major limitation of our code now is in the monopole treatment of gravity, since it cannot account for spiral structures, which could be reflected in GW. We are currently working on the improvement of this issue. Furthermore, the IDSA includes at present only the dominant reactions relevant to the neutrino transport problem (see [@2009ApJ...698.1174L] for details). Future upgrades will also include contributions from electron-neutrino scattering, which are indispensable during the collapse phase. The inclusion of this reaction will also make the cumbersome switch of the parametrization to the IDSA at bounce obsolete. Finally, we work on the inclusion of $\mu$ and $\tau$ neutrinos, which are very important for the cooling of the PNS to its final stage as neutron star.
The authors would like to thank C. D. Ott for stimulating and useful discussions and F.-K. Thielemann for his support. This work was supported by a grant from the Swiss National Supercomputing Centre-CSCS under project ID s168. We acknowledge support by the Swiss National Science Foundation under grant No. 200020-122287 and PP00P2-124879. Moreover, this work was supported by CompStar, a Research Networking Programme of the European Science Foundation. Further thanks go to John Biddiscombe and Sadaf Alam from the Swiss Supercomputing Centre CSCS for the smooth and enjoyable collaboration.
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[^1]: $\overline{c_{s}}=1/\Delta_{r}\int_{r}c_{s}(r)dr$ is the radially averaged sound speed of a convectively unstable layer with a radial extension of $\Delta_{r}$.
|
---
abstract: '[The interpretation of upcoming weak gravitational lensing surveys depends critically on our understanding of the matter power spectrum on scales $k < \SI{10}{{\si{\h.\mega\parsec^{-1}}}}$, where baryonic processes are important. We study the impact of galaxy formation processes on the matter power spectrum using a halo model that treats the stars and gas separately from the dark matter distribution. We use empirical constraints from X-ray observations (hot gas) and halo occupation distribution modelling (stars) for the baryons. Since X-ray observations cannot generally measure the hot gas content outside $r{\ensuremath{_{\mathrm{500c}}}}$, we vary the gas density profiles beyond this radius. Compared with dark matter only models, we find a total power suppression of $\SI{1}{\percent}$ ($\SI{5}{\percent}$) on scales $\SIrange{0.2}{1}{{\si{\h.\mega\parsec^{-1}}}}$ ($\SIrange{0.5}{2}{{\si{\h.\mega\parsec^{-1}}}}$), where lower baryon fractions result in stronger suppression. We show that groups of galaxies ($\num{e13} < m{\ensuremath{_{\mathrm{500c}}}} / (\si{{\si{\h^{-1}.{\si{\Msun}}}}}) < \num{e14}$) dominate the total power at all scales $k \lesssim \SI{10}{{\si{\h.\mega\parsec^{-1}}}}$. We find that a halo mass bias of $\SI{30}{\percent}$ (similar to what is expected from the hydrostatic equilibrium assumption) results in an underestimation of the power suppression of up to $\SI{4}{\percent}$ at $k=\SI{1}{{\si{\h.\mega\parsec^{-1}}}}$, illustrating the importance of measuring accurate halo masses. Contrary to work based on hydrodynamical simulations, our conclusion that baryonic effects can no longer be neglected is not subject to uncertainties associated with our poor understanding of feedback processes. Observationally, probing the outskirts of groups and clusters will provide the tightest constraints on the power suppression for $k \lesssim \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$.]{}'
author:
- |
Stijn N.B. Debackere[^1], Joop Schaye, Henk Hoekstra\
Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, The Netherlands\
bibliography:
- 'halo\_model.bib'
date: 'Last updated –; in original form –'
title: The impact of the observed baryon distribution in haloes on the total matter power spectrum
---
\[firstpage\]
cosmology: observations, cosmology: theory, large-scale structure of Universe, cosmological parameters, gravitational lensing: weak, surveys
Introduction {#sec:introduction}
============
Since the discovery of the Cosmic Microwave Background (CMB) [@Penzias1965; @Dicke1965], cosmologists have continuously refined the values of the cosmological parameters. This resulted in the discovery of the accelerated expansion of the Universe [@Riess1998; @Perlmutter1999] and the concordance Lambda cold dark matter ([$\Lambda$CDM]{}) model. Future surveys such as Euclid[^2], the Large Synoptic Survey Telescope (LSST)[^3], and the Wide Field Infra-Red Survey Telescope (WFIRST)[^4] aim to constrain the nature of this mysterious acceleration to establish whether it is caused by a cosmological constant or dark energy. This is one of the largest gaps in our current understanding of the Universe.
To probe the physical cause of the accelerated expansion, and to discern between different models for dark energy or even a modified theory of gravity, we require precise measurements of the growth of structure and the expansion history over a range of redshifts. This is exactly what future galaxy surveys aim to do, e.g. using a combination of weak gravitational lensing and galaxy clustering. Weak lensing measures the correlation in the distortion of galaxy shapes for different redshift bins, which depends on the matter distribution in the Universe, and thus on the matter power spectrum [for reviews, see e.g. @Hoekstra2008; @Kilbinger2015; @Mandelbaum2017]. The theoretical matter power spectrum is thus an essential ingredient for a correct interpretation of weak lensing observations.
The matter power spectrum can still not be predicted well at small scales ($k \gtrsim \SI{0.3}{{\si{\h.\mega\parsec^{-1}}}}$) because of the uncertainty introduced by astrophysical processes related to galaxy formation [@Rudd2008; @VanDaalen2011; @Semboloni2011]. In order to provide stringent cosmological constraints with future surveys, the prediction of the matter power spectrum needs to be accurate at the sub-percent level [@Hearin2012].
Collisionless N-body simulations, i.e. dark matter only (DMO) simulations, can provide accurate estimates of the non-linear effects of gravitational collapse on the matter power spectrum. They can be performed using a large number of particles, and in big cosmological boxes for many different cosmologies (e.g. [@Heitmann2009; @Heitmann2010]; [@Lawrence2010]; [@Angulo2012]). The distribution of baryons, however, does not perfectly trace that of the dark matter: baryons can cool and collapse to high densities, sparking the formation of galaxies. Galaxy formation results in violent feedback that can redistribute gas to large scales. Furthermore, these processes induce a back-reaction on the distribution of dark matter [e.g. @VanDaalen2011; @VanDaalen2019; @Velliscig2014]. Hence, the redistribution of baryons and dark matter modifies the power spectrum relative to that from DMO simulations.
Weak lensing measurements obtain their highest signal-to-noise ratio on scales $k \approx \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$ (see §1.8.5 in [@Amendola2018]). @VanDaalen2011 used the [OWLS]{} suite of cosmological simulations [@Schaye2010] to show that the inclusion of baryon physics, particularly feedback from Active Galactic Nuclei (AGN), influences the matter power spectrum at the level between $0.3 < k/(\si{{\si{\h.\mega\parsec^{-1}}}}) < 1$ in their most realistic simulation that reproduced the hot gas properties of clusters of galaxies. Further studies [e.g. @Vogelsberger2014a; @Hellwing2016; @Springel2017; @Chisari2018; @Peters2018; @VanDaalen2019] have found similar results. @Semboloni2011 have shown, also using the [OWLS]{} simulations, that ignoring baryon physics in the matter power spectrum results in biased weak lensing results, reaching a bias of up to in the dark energy equation of state parameter $w_{0}$ for a Euclid-like survey.
Current state-of-the-art hydrodynamical simulations allow us to study the influence of baryons on the matter power spectrum, but cannot predict it from first principles. Due to their computational cost, these simulations need to include baryon processes such as star formation and AGN feedback as “subgrid” recipes, as they cannot be directly resolved. The accuracy of the subgrid recipes can be tested by calibrating simulations to a fixed set of observed cosmological scaling relations, and subsequently checking whether other scaling relations are also reproduced [see e.g. @Schaye2015; @McCarthy2017; @Pillepich2017a]. However, this calibration strategy may not result in a unique solution, since other subgrid implementations or different parameter values can provide similar predictions for the calibrated relation but may differ in some other observable. Thus, the calibrated relations need to be chosen carefully depending on what we want to study.
A better option is to calibrate hydrodynamical simulations using the observations that are most relevant for the power spectrum, such as cluster gas fractions and the galaxy mass function [@McCarthy2017] and to include simulations that span the observational uncertainties [@McCarthy2018]. The calibration against cluster gas fractions is currently only implemented in the [<span style="font-variant:small-caps;">bahamas</span>]{} suite of simulations [@McCarthy2017]. Current high-resolution hydrodynamical simulations, such as e.g. EAGLE [@Schaye2015], Horizon-AGN [@Chisari2018] and IllustrisTNG [@Springel2017], do not calibrate against this observable. Moreover, the calibrated subgrid parameters required to reproduce their chosen observations result in gas fractions that are too high in their most massive haloes [@Schaye2015; @Barnes2017; @Chisari2018]. This is a problem, because both halo models [@Semboloni2013] and hydrodynamical simulations [@VanDaalen2019] have been used to demonstrate the existence of a strong link between the suppression of the total matter power spectrum on large scales and cluster gas fractions. As a result, these state-of-the-art simulations of galaxy formation are not ideal to study the baryonic effects on the matter power spectrum.
Focussing purely on simulation predictions risks underestimating the possible range of power suppression due to baryons, since the simulations generally do not cover the full range of possible physical models. Hence, given our limited understanding of the astrophysics of galaxy formation and the computational expense of hydrodynamical simulations, it is important to develop other ways to account for baryonic effects and observational constraints upon them.
One possibility is to make use of phenomenological models that take the matter distribution as input without making assumptions about the underlying physics. Splitting the matter into its dark matter and baryonic components allows observations to be used as the input for the baryonic component of the model. This bypasses the need for any model calibrations but may require extrapolating the baryonic component outside of the observed range. Such models can be implemented in different ways. For instance, @Schneider2015 and @Schneider2018 use a “baryon correction model” to shift the particles in DMO simulations under the influence of hydrodynamic processes which are subsumed in a combined density profile including dark matter, gas and stars with phenomenological parameters for the baryon distribution that are fit to observations. Consequently, the influence of a change in these parameters on the power spectrum can be investigated. Since this model relies only on DMO simulations, it is less computationally expensive while still providing important information on the matter distribution.
We will take a different phenomenological approach and use a modified version of the halo model to predict how baryons modify the matter power spectrum. We opt for this approach because it gives us freedom in varying the baryon distribution at little computational expense. We do not aim to make the most accurate predictor for baryonic effects on the power spectrum, but our goal is to systematically study the influence of changing the baryonic density profiles on the matter power spectrum and to quantify the uncertainty of the baryonic effects on the power spectrum allowed by current observational constraints.
The halo model describes the clustering of matter in the Universe starting from the matter distribution of individual haloes. We split the halo density profiles into a dark matter component and baryonic components for the gas and the stars. We assume that the abundance and clustering of haloes can be modelled using DMO simulations, but that their density profiles, and hence masses, change due to baryonic effects. This assumption is supported by the findings of @VanDaalen2014a, who used [OWLS]{} to show that matched sets of subhaloes cluster identically on scales larger than the virial radii in DMO and hydrodynamical simulations. We constrain the gas component with X-ray observations of groups and clusters of galaxies. These observations are particularly relevant since the matter power spectrum is dominated by groups and clusters on the scales affected by baryonic physics and probed by upcoming surveys $0.3 \lesssim k/({\si{\h.\mega\parsec^{-1}}}) \lesssim 10$, [e.g. @VanDaalen2015]. For the stellar component, we assume the distribution from Halo Occupation Distribution (HOD) modelling.
Earlier studies have used extensions to the halo model to include baryon effects, either by adding individual matter components from simulations [e.g. @White2004; @Zhan2004; @Rudd2008; @Guillet2010; @Semboloni2011; @Semboloni2013; @Fedeli2014; @Fedeli2014a], or by introducing empirical parameters inspired by the predicted physical effects of galaxy formation [see @Mead2015; @Mead2016]. However, these studies were based entirely on data from cosmological simulations, whereas we stay as close as possible to the observations and thus do not depend on the uncertain assumptions associated with subgrid models for feedback processes.
There is still freedom in our model because the gas content of low-mass haloes and the outskirts of clusters cannot currently be measured. We thus study the range of baryonic corrections to the dark matter only power spectrum by assuming different density profiles for the unobserved regions. Our model gives us a handle on the uncertainty in the matter power spectrum and allows us to quantify how different mass profiles of different mass haloes contribute to the total power for different wavenumbers, whilst simultaneously matching observations of the matter distribution. Moreover, we can study the impact of observational uncertainties and biases on the resulting power spectrum.
We start of by describing our modified halo model in § \[sec:hm\]. We describe the observations and the relevant halo model parameters in § \[sec:obs\_xray\]. We show our resulting model density components in § \[sec:components\] and report our results in § \[sec:results\]. We discuss our model and compare it to the literature in § \[sec:discussion\]. Finally, we conclude and provide some directions for future research in § \[sec:summary\_conclusions\]. This work assumes the *WMAP* 9 year [@Hinshaw2013] cosmological parameters $\{{\ensuremath{\Omega_\mathrm{m}}}, {\ensuremath{\Omega_\mathrm{b}}}, {\ensuremath{\Omega_\Lambda}}, \sigma_{8}, n{\ensuremath{_{\mathrm{s}}}}, h\} = \{0.2793, 0.0463,
0.7207, 0.821, 0.972, 0.7\}$ and all of our results are computed for $z=0$. All of the observations that we compare to assumed $h=0.7$, so we quote their results in units of $H_{0} = 70 \si{\hs.\km.\s^{-1}.{\si{\mega\parsec}}^{-1}}$ with $h_{70} = 1$. Whenever we quote units without any $\si{\h}$ or $\si{\hs}$ scaling, we assume $\si{\h}=0.7$ or, equivalently, $\si{\hs}=1$ [for a good reference and arguments on making definitions explicit, see @Croton2013]. When fitting our model to observations, we always use $\si{\h}=0.7$ to ensure a fair comparison between model and observations.
Halo Model {#sec:hm}
==========
Theory {#sec:hm_theory}
------
The halo model (e.g. [@Peacock2000; @Seljak2000a]; but the basis was already worked out in [@McClelland1977a] and [@Scherrer1991]; review in [@Cooray2002]) is an analytic prescription to model the clustering properties of matter for a given cosmology through the power spectrum [for a clear pedagogical exposition, see @Vandenbosch2013]. It gives insight into non-linear structure formation starting from the linear power spectrum and a few simplifying assumptions.
The spherical collapse model of non-linear structure formation tells us that any over-dense, spherical region will collapse into a virialized dark matter halo, with a final average density ${\left\langle \rho_{\mathrm{f}} \right\rangle} =
\Delta_{\mathrm{vir}}\rho_{\mathrm{c}}(z_{\mathrm{vir}})$, where $\Delta_{\mathrm{vir}}$ in general depends on cosmology, but is usually taken as $\Delta_{200}=200$, rounded from the Einstein-de Sitter value of $\Delta_{\mathrm{vir}} = 18 \pi^{2}$, with $\rho_{\mathrm{c}}(z_{\mathrm{vir}})$ the critical density of the Universe at the redshift of virialization. The fundamental assumption of the halo model is that all matter in the Universe has collapsed into virialized dark matter haloes that grow hierarchically in time through mergers. Throughout the paper we will adhere to the notation $m{\ensuremath{_{\mathrm{500c}}}}$ and $m{\ensuremath{_{\mathrm{200m}}}}$ to indicate regions enclosing an average density ${\left\langle \rho \right\rangle}{\ensuremath{_{\mathrm{500c}}}} = 500 \rho{\ensuremath{_{\mathrm{c}}}}(z)$ and ${\left\langle \rho \right\rangle}{\ensuremath{_{\mathrm{200m}}}} = 200 {\ensuremath{{\bar{\rho}}_{\mathrm{m}}}}(z)$, with ${\ensuremath{{\bar{\rho}}_{\mathrm{m}}}}(z) = {\ensuremath{\Omega_\mathrm{m}}}\rho{\ensuremath{_{\mathrm{c}}}}(z=0) (1+z)^3$, respectively.
At a given time, the halo mass function $n(m{\ensuremath{_{\mathrm{h}}}}, z)$ determines the co-moving number density of dark matter haloes in a given halo mass bin centered on $m{\ensuremath{_{\mathrm{h}}}}$. This function can be derived from analytic arguments, like for instance the Press-Schechter and Extended Press-Schechter (EPS) theories [e.g. @Press1974; @Bond1991; @Lacey1993], or by using DMO simulations [e.g. @Sheth1999b; @Jenkins2001; @Tinker2008]. Furthermore, assuming that the density profile of a halo is completely determined by its mass and redshift, i.e. $\rho(r) = \rho(r|m{\ensuremath{_{\mathrm{h}}}}, z)$, we can then calculate the statistics of the matter distribution in the Universe, captured by the power spectrum, by looking at the correlations between matter in different haloes (the two-halo or 2h term which probes large scales) and between matter within the same halo (the one-halo or 1h term which probes small scales).
Splitting the contributions to the power spectrum up into the 1h and 2h terms, we can rewrite $$\begin{aligned}
\label{eq:power}
P(k,z) & = V{\ensuremath{_{\mathrm{u}}}} {\left\langle |\hat{\delta}{\ensuremath{_{\mathrm{m}}}}(k,z)|^{2} \right\rangle} \\
\label{eq:power_12h}
& = P{\ensuremath{_{\mathrm{1h}}}}(k,z) + P{\ensuremath{_{\mathrm{2h}}}}(k,z) \, .\end{aligned}$$ Here $V{\ensuremath{_{\mathrm{u}}}}$ is the volume under consideration and $\hat{\delta}{\ensuremath{_{\mathrm{m}}}}(k,z)$ is the Fourier transform of the matter overdensity field $\delta{\ensuremath{_{\mathrm{m}}}}({\ensuremath{\mathbf{x}}},z) \equiv \rho({\ensuremath{\mathbf{x}}},z) / {\ensuremath{{\bar{\rho}}_{\mathrm{m}}}}(z) -
1$, with ${\ensuremath{{\bar{\rho}}_{\mathrm{m}}}}(z)$ the mean matter background density at redshift $z$. We define the Fourier transform of a halo as $$\begin{aligned}
\label{eq:rho_k}
\hat{\rho}(k|m{\ensuremath{_{\mathrm{h}}}}, z) & = 4\pi \int_0^{r{\ensuremath{_{\mathrm{h}}}}} {\ensuremath{\mathrm{d}}}r
\, \rho(r|m{\ensuremath{_{\mathrm{h}}}},
z) r^2 \frac{\sin(kr)}{kr} \, .\end{aligned}$$ The 1h and 2h terms are given by [for detailed derivations, see @Cooray2002; @Mo2010] $$\begin{aligned}
\label{eq:p_1h}
P{\ensuremath{_{\mathrm{1h}}}}(k,z) & = \int {\ensuremath{\mathrm{d}}}m{\ensuremath{_{\mathrm{200m,dmo}}}} \,
\begin{aligned}[t]
& n{\ensuremath{_{\mathrm{dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}(z),z) \\
\times & \frac{|\hat{\rho}(k|m{\ensuremath{_{\mathrm{h}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}),z)|^{2}}{{\ensuremath{{\bar{\rho}}_{\mathrm{m}}}}^{2}(z)} \\
\end{aligned} \\
\nonumber
P{\ensuremath{_{\mathrm{2h}}}}(k,z) & = P_{\mathrm{lin}}(k,z)
\begin{aligned}[t]
\Bigg[ \int {\ensuremath{\mathrm{d}}}m{\ensuremath{_{\mathrm{200m,dmo}}}} \, &
n{\ensuremath{_{\mathrm{dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}(z),z) \\
\times & b{\ensuremath{_{\mathrm{dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}(z),z) \\
\times & \frac{\hat{\rho}(k|m{\ensuremath{_{\mathrm{h}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}),z)}{{\ensuremath{{\bar{\rho}}_{\mathrm{m}}}}(z)}
\Bigg]^{2}
\end{aligned} \\
\label{eq:p_2h}
& \simeq P_{\mathrm{lin}}(k,z) \, .\end{aligned}$$ Our notation makes explicit that because our predictions rely on the halo mass function and the bias obtained from DMO simulations, we need to correct the true halo mass $m{\ensuremath{_{\mathrm{h}}}}$ to the DMO equivalent mass $m{\ensuremath{_{\mathrm{200m,dmo}}}}$, as we will explain further in § \[sec:hm\_modifications\]. The 2h term contains the bias $b{\ensuremath{_{\mathrm{dmo}}}}(m,z)$ between haloes and the underlying density field. For the 2h term, we simply use the linear power spectrum, which we get from CAMB[^5] for our cosmological parameters. For the halo mass function, we assume the functional form given by @Tinker2008, which is calibrated for the spherical overdensity halo mass $m{\ensuremath{_{\mathrm{200m,dmo}}}}$.
We assume $P{\ensuremath{_{\mathrm{2h}}}} \approx P{\ensuremath{_{\mathrm{lin}}}}$ since not all of our haloes will be baryonically closed. This would result in Eq. \[eq:p\_2h\] not returning to the linear power spectrum at large scales for models that have missing baryons within the halo radius. Assuming that the 2h term follows the linear power spectrum is equivalent to assuming that all of the missing baryons will be accounted for in the cosmic web, which we cannot accurately capture with our simple halo model.
We will use our model to predict the quantity $$\begin{aligned}
\label{eq:power_ratio}
R_{i}(k,z) \equiv \frac{P_{i}(k,z)}{P_{i,\mathrm{dmo}}(k,z)} \,
,\end{aligned}$$ the ratio between the power spectrum of baryonic model $i$ and the corresponding DMO power spectrum assuming the same cosmological parameters. This ratio has been given various names in the literature, e.g. the “response” [@Mead2016], the “reaction” [@Cataneo2018], or just the “suppression” [@Schneider2018]. We will refer to it as the power spectrum response to the presence of baryons. It quantifies the suppression or increase of the matter power spectrum due to baryons. If non-linear gravitational collapse and galaxy formation effects were separable, and baryonic effects were insensitive to the underlying cosmology, knowledge of this ratio would allow us to reconstruct a matter power spectrum from any DMO prediction. These last two assumptions can only be tested by comparing large suites of cosmological N-body and hydrodynamical simulations. We do not attempt to address them in this paper. However, @VanDaalen2011 [@VanDaalen2019], @Mummery2017, @McCarthy2018, and @Stafford2019 have investigated the cosmology dependence of the baryonic suppression. @Mummery2017 find that a separation of the cosmology and baryon effects on the power spectrum is accurate at the $\SI{3}{\percent}$ level between $\SI{1}{{\si{\h.\mega\parsec^{-1}}}} \lesssim k \lesssim \SI{10}{{\si{\h.\mega\parsec^{-1}}}}$ for cosmologies varying the neutrino masses between $0 < M_\nu / \si{\eV} < 0.48$. Similarly, @VanDaalen2019 find that varying the cosmology between *WMAP* 9 and Planck 2013 results in at most a $\SI{4}{\percent}$ difference for $k < \SI{10}{{\si{\h.\mega\parsec^{-1}}}}$.
Our model does not include any correction to the power spectrum due to halo exclusion. Halo exclusion accounts for the fact that haloes cannot overlap by canceling the 2h term at small scales [@Smith2011a]. It also cancels the shot-noise contribution from the 1h term at large scales. In our model, the important effect occurs at scales where the 1h and 2h terms are of similar magnitude, since the halo exclusion would suppress the 2h term. However, since we look at the power spectrum response to baryons $R_i(k)$, which is the ratio of the power spectrum including baryons to the power spectrum in the DMO case, our model should not be significantly affected, since the halo exclusion term modifies both of these terms in a similar way. We have checked that subtracting a halo exclusion term that interpolates between the 1h term at large scales and the 2h term at small scales only affects our predictions for $R_i(k)$ by at most $\SI{1}{\percent}$ at $k \approx \SI{3}{{\si{\h.\mega\parsec^{-1}}}}$.
Linking observed halo masses to abundances {#sec:hm_modifications}
------------------------------------------
Our model is similar to the traditional halo model as described by @Cooray2002. We make two important changes, however. Firstly, we split up the density profile into a dark matter, a hot gas, and a stellar component $$\begin{aligned}
\label{eq:rho_r}
\rho(r|m{\ensuremath{_{\mathrm{h}}}}, z) & = \rho{\ensuremath{_{\mathrm{dm}}}}(r|m{\ensuremath{_{\mathrm{h}}}}, z) +
\rho{\ensuremath{_{\mathrm{gas}}}}(r|m{\ensuremath{_{\mathrm{h}}}}, z) +
\rho{\ensuremath{_{\mathrm{\star}}}}(r|m{\ensuremath{_{\mathrm{h}}}}, z) \, .\end{aligned}$$ We will detail our specific profile assumptions in § \[sec:hm\_profiles\]. Secondly, we include a mapping from the observed halo mass $m{\ensuremath{_{\mathrm{h}}}}$ to the dark matter only equivalent halo mass $m{\ensuremath{_{\mathrm{200m,dmo}}}}$, as shown in Eqs. \[eq:p\_1h\] and \[eq:p\_2h\].
This second step is necessary for two reasons. First, the masses of haloes change in hydrodynamical simulations. In simulations with the same initial total density field, haloes can be linked between the collisionless and hydrodynamical simulations, thus enabling the study of the impact of baryon physics on individual haloes. @Sawala2013, @Velliscig2014 and @Cui2014 found that even though the abundance of individual haloes does not change, their mass does, especially for low-mass haloes (see Fig. 10 in [@Velliscig2014]). Feedback processes eject gas from haloes, lowering their mass at fixed radius. However, once this mass change is accounted for, the clustering of the matched haloes is nearly identical in the DMO and hydrodynamical simulations [@VanDaalen2014a]. Since the halo model relies on prescriptions for the halo mass function that are calibrated on dark matter only simulations, we need to correct our observed halo masses to predict their abundance.
Second, observed halo masses are not equivalent to the underlying true halo mass. Every observational determination of the halo mass carries its own intrinsic biases. Masses from X-ray measurements are generally obtained under the assumption of spherical symmetry and hydrostatic equilibrium, for example. However, due to the recent assembly of clusters of galaxies, sphericity and equilibrium assumptions break down in the halo outskirts [see @Pratt2019 and references therein]. In most weak lensing measurements, the halo is modeled assuming a Navarro-Frenk-White (NFW) profile [@Navarro1996] with a concentration-mass relation $c(m)$ from simulations. This profile does not necessarily accurately describe the density profile of individual haloes due to asphericity and the large scatter in the concentration-mass relation at fixed halo mass.
In our model, each halo will be labeled with four different halo masses. We indicate the cumulative mass profile of the observed and DMO equivalent halo with $m{\ensuremath{_{\mathrm{obs}}}}(\leq r)$ and $m{\ensuremath{_{\mathrm{dmo}}}}(\leq r)$, respectively. Firstly, we define the total mass inside $r{\ensuremath{_{\mathrm{500c,obs}}}}$ inferred from observations $$\begin{aligned}
\label{eq:m500c_xray}
m{\ensuremath{_{\mathrm{500c,obs}}}} & \equiv
m{\ensuremath{_{\mathrm{obs}}}}(\leq r{\ensuremath{_{\mathrm{500c,obs}}}}) \, .\end{aligned}$$ This mass will provide the link between our model and the observations. We work with $r{\ensuremath{_{\mathrm{500c,obs}}}}$ in this paper because it is similar to the radius up to which X-ray observations are able to measure the halo mass. However, any other radius can readily be used in all of the following definitions. Secondly, we have the true total mass inside the halo radius $r{\ensuremath{_{\mathrm{h}}}}$ for our extrapolated profiles $$\begin{aligned}
\label{eq:m_h}
m{\ensuremath{_{\mathrm{h}}}} & \equiv m{\ensuremath{_{\mathrm{obs}}}}(\leq r{\ensuremath{_{\mathrm{h}}}}) \, .\end{aligned}$$ Thirdly, we define the total mass in our extrapolated profiles such that the mean enclosed density is ${\left\langle \rho \right\rangle}{\ensuremath{_{\mathrm{200m}}}}$ $$\begin{aligned}
\label{eq:m200m_obs}
m{\ensuremath{_{\mathrm{200m,obs}}}} & \equiv m{\ensuremath{_{\mathrm{obs}}}}(\leq r{\ensuremath{_{\mathrm{200m,obs}}}}) \, .\end{aligned}$$ We differentiate between $r{\ensuremath{_{\mathrm{h}}}}$ and $r{\ensuremath{_{\mathrm{200m,obs}}}}$ because for some of our models we will extrapolate the density profile further than $r{\ensuremath{_{\mathrm{200m,obs}}}}$. Fourthly, we define the dark matter only equivalent mass for the halo $$\begin{aligned}
\label{eq:m200m_dmo}
m{\ensuremath{_{\mathrm{200m,dmo}}}} & \equiv m{\ensuremath{_{\mathrm{dmo}}}}(\leq
r{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}},
c{\ensuremath{_{\mathrm{dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}))) \, ,\end{aligned}$$ which depends on the observed halo mass $m{\ensuremath{_{\mathrm{500c,obs}}}}$ and the assumed DMO concentration-mass relation $c{\ensuremath{_{\mathrm{dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}})$, as we will discuss below. In each of our models for the baryonic matter distribution there is a unique monotonic mapping between all four of these halo masses. In the rest of the paper we will thus express all dependencies as a function of $m{\ensuremath{_{\mathrm{h}}}}$, unless our calculation explicitly depends on one of the three other masses (as we indicate in Eqs. \[eq:p\_1h\] and \[eq:p\_2h\] where the halo mass function requires the DMO equivalent mass from Eq. \[eq:m200m\_dmo\] as an input).
The DMO equivalent mass, Eq. \[eq:m200m\_dmo\], requires more explanation. We determine it from the following, simplifying but overall correct, assumption: the inclusion of baryon physics does not significantly affect the distribution of the dark matter. This assumption is corroborated by the findings of @Duffy2010, @Velliscig2014 and @Schaller2015, who all find that in hydrodynamical simulations that are able to reproduce many observables related to the baryon distribution, the baryons do not significantly impact the dark matter distribution. This assumption breaks down on galaxy scales where the dark matter becomes more concentrated due to the condensation of baryons at the center of the halo. However, these scales are smaller than the scales of interest for upcoming weak lensing surveys. Moreover, at these scales the stellar component typically dominates over the dark matter. Assuming that the dark matter component will have the same scale radius as its DMO equivalent halo, we can convert the observed halo mass into its DMO equivalent. The first step is to compute the dark matter mass in the observed halo, $$\begin{aligned}
\label{eq:m500c_dm}
m{\ensuremath{_{\mathrm{500c,dm}}}} & = m{\ensuremath{_{\mathrm{500c,obs}}}}
\begin{aligned}[t]
(1 & - f{\ensuremath{_{\mathrm{gas,500c,obs}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}}) \\
& -
f{\ensuremath{_{\mathrm{\star,500c,obs}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}}))
\, .
\end{aligned}\end{aligned}$$ The dark matter mass is obtained by subtracting the observed gas and stellar mass inside $r{\ensuremath{_{\mathrm{500c,obs}}}}$ from the observed total halo mass. The stellar fraction depends on the DMO equivalent halo mass since we take the stellar profiles from the `iHOD` model by @Zu2015 [hereafter ], which also uses a halo model that is based on the @Tinker2008 halo mass function. This requires us to iteratively solve for the DMO equivalent mass $m{\ensuremath{_{\mathrm{200m,dmo}}}}$. Next, we assume that the DMO equivalent halo mass at the radius $r{\ensuremath{_{\mathrm{500c,obs}}}}$ is given by $m{\ensuremath{_{\mathrm{500c,dmo}}}} = (1 - {\ensuremath{\Omega_\mathrm{b}}}/ {\ensuremath{\Omega_\mathrm{m}}})^{-1} m{\ensuremath{_{\mathrm{500c,dm}}}}$, which is consistent with our assumption that baryons do not change the distribution of dark matter. Subsequently, we can determine the halo mass $m{\ensuremath{_{\mathrm{200m,dmo}}}}$ by assuming a DMO concentration-mass relation, an NFW density profile, and solving $m{\ensuremath{_{\mathrm{dmo}}}}(\leq
r{\ensuremath{_{\mathrm{500c,obs}}}};c{\ensuremath{_{\mathrm{dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}})) =
m{\ensuremath{_{\mathrm{500c,dmo}}}}$ for $m{\ensuremath{_{\mathrm{200m,dmo}}}}$. Thus, we determine $m{\ensuremath{_{\mathrm{200m,dmo}}}}$ (Eq. \[eq:m200m\_dmo\]) by solving the following equation: $$\begin{aligned}
\label{eq:m200m_dmo_constraints}
& 4 \pi \int_0^{r{\ensuremath{_{\mathrm{500c,obs}}}}}
\rho{\ensuremath{_{\mathrm{NFW}}}}(r;c{\ensuremath{_{\mathrm{dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})))
r^{2} {\ensuremath{\mathrm{d}}}r \\
\nonumber
& = \frac{m{\ensuremath{_{\mathrm{500c,obs}}}}}{1 - {\ensuremath{\Omega_\mathrm{b}}}/ {\ensuremath{\Omega_\mathrm{m}}}}
\begin{aligned}[t]
(1 & - f{\ensuremath{_{\mathrm{gas,500c,obs}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}}) \\
& -
f{\ensuremath{_{\mathrm{\star,500c,obs}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})))
\, .
\end{aligned}\end{aligned}$$ We determine the stellar fraction at $r{\ensuremath{_{\mathrm{500c,obs}}}}$ by assuming the stellar profiles detailed in § \[sec:hm\_profiles\_stars\]. Finally, we obtain the relation $m{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})$ that assigns a DMO equivalent mass to each observed halo with mass $m{\ensuremath{_{\mathrm{500c,obs}}}}$.
We initiate our model on an equidistant log-grid of halo masses $\SI{e10}{{\si{\h^{-1}.{\si{\Msun}}}}} \leq m{\ensuremath{_{\mathrm{500c,obs}}}} \leq \SI{e15}{{\si{\h^{-1}.{\si{\Msun}}}}}$, which we sample with 101 bins. We show that our results are converged with respect to our chosen mass range and binning in App. \[app:results\_mass\_range\]. For each halo mass, we get the DMO equivalent mass $m{\ensuremath{_{\mathrm{200m,dmo}}}}$, the stellar fraction $f_{\star,i}(m{\ensuremath{_{\mathrm{200m,dmo}}}})$, with $i \in \{\mathrm{cen,sat}\}$, and the concentration of the DMO equivalent halo $c{\ensuremath{_{\mathrm{dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}})$. We will specify all of our different matter component profiles in § \[sec:hm\_profiles\].
Matter density profiles {#sec:hm_profiles}
-----------------------
In this section, we give the functional forms of the density profiles that we use in our halo model. We assume three different matter components: dark matter, gas and stars. The dark matter and stellar profiles are taken directly from the literature, whereas we obtain the gas profiles by fitting to observations from the literature. In our model, we only include the hot, X-ray emitting gas with $T > \SI{e7}{\K}$, thus neglecting the interstellar medium (ISM) component of the gas. The ISM component is confined to the scale of individual galaxies, where it can provide a similar contribution to the total baryonic mass as the stars. The only halo masses for which the total baryonic mass of the galaxy may be similar to that of the surrounding diffuse circum-galactic medium (CGM) are Milky Way-like galaxies, or even lower-mass haloes [@Catinella2010; @Saintonge2011]. However, these do not contribute significantly to the total power at our scales of interest, as we will show in § \[sec:results\_masses\].
### Hot gas {#sec:hm_profiles_gas}
For the density profiles of hot gas, we assume traditionally used beta profiles [@Cavaliere1978] inside $r{\ensuremath{_{\mathrm{500c,obs}}}}$ where we have observational constraints. We will extrapolate the beta profile as a power-law with slope $-\gamma$ outside $r{\ensuremath{_{\mathrm{500c,obs}}}}$. In our models with $r{\ensuremath{_{\mathrm{h}}}} > r{\ensuremath{_{\mathrm{200m,obs}}}}$, we will assume a constant density outside $r{\ensuremath{_{\mathrm{200m,obs}}}}$ until $r{\ensuremath{_{\mathrm{h}}}}$, which will then be the radius where the halo reaches the cosmic baryon fraction. This results in the following density profile for the hot gas: $$\label{eq:beta_gas}
\rho{\ensuremath{_{\mathrm{gas}}}}(r|m{\ensuremath{_{\mathrm{h}}}}) =
\begin{cases}
\rho_{0}\left(1 + (r/r{\ensuremath{_{\mathrm{c}}}})^{2}\right)^{-3\beta/2}, & r <
r{\ensuremath{_{\mathrm{500c,obs}}}} \\
\rho{\ensuremath{_{\mathrm{500c,obs}}}}
\left(\frac{r}{r{\ensuremath{_{\mathrm{500c,obs}}}}}\right)^{-\gamma},
& r{\ensuremath{_{\mathrm{500c,obs}}}} \leq r < r{\ensuremath{_{\mathrm{200m,obs}}}} \\
\rho{\ensuremath{_{\mathrm{500c,obs}}}}
\left(\frac{r{\ensuremath{_{\mathrm{200m,obs}}}}}{r{\ensuremath{_{\mathrm{500c,obs}}}}}\right)^{-\gamma},
& r{\ensuremath{_{\mathrm{200m,obs}}}} \leq r < r{\ensuremath{_{\mathrm{h}}}} \\
0, & r \geq r{\ensuremath{_{\mathrm{h}}}} \, .
\end{cases}$$ The normalisation $\rho_{0}$ is determined by the gas fractions inferred from X-ray observations and normalises the profile to $m{\ensuremath{_{\mathrm{gas,500c,obs}}}}$ at $r{\ensuremath{_{\mathrm{500c,obs}}}}$: $$\begin{aligned}
\nonumber
\rho_{0} & = \frac{m{\ensuremath{_{\mathrm{gas,500c,obs}}}}}{4/3 \pi r{\ensuremath{_{\mathrm{500c,obs}}}}^{3}
{}_{2}F_{1}(3/2,3\beta/2;5/2;-(r{\ensuremath{_{\mathrm{500c,obs}}}}/r{\ensuremath{_{\mathrm{c}}}})^{2})}
\\
\label{eq:rho_0_gas}
& = \frac{500\rho{\ensuremath{_{\mathrm{c}}}} f{\ensuremath{_{\mathrm{gas,500c,obs}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})}{
{}_{2}F_{1}(3/2,3\beta/2;5/2;-(r{\ensuremath{_{\mathrm{500c,obs}}}}/r{\ensuremath{_{\mathrm{c}}}})^{2})}
\, .\end{aligned}$$ Here ${}_{2}F_{1}(a,b;c;d)$ is the Gauss hypergeometric function. The values for the core radius $r{\ensuremath{_{\mathrm{c}}}}$, the slope $\beta$, and the hot gas fraction $f{\ensuremath{_{\mathrm{gas,500c,obs}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})$ are obtained by fitting observations, as we explain in § \[sec:obs\_xray\]. The outer power-law slope $\gamma$ is in principle a free parameter of our model, but as we explain below, it is constrained by the total baryon content of the halo. We choose a parameter range of $0 \leq \gamma \leq 3$.
For each halo, we determine $r{\ensuremath{_{\mathrm{200m,obs}}}}$ by determining the mean enclosed density for the total mass profile (i.e. dark matter, hot gas and stars). In the most massive haloes, a large part of the baryons is already accounted for by the observed hot gas profile. As a result, we need to assume a steep slope in these systems, since otherwise their baryon fraction would exceed the cosmic one before $r{\ensuremath{_{\mathrm{200m,obs}}}}$ is reached. Since the parameters of both the dark matter and the stellar components are fixed, the only way to prevent this is by setting a maximum value for the slope $-\gamma$ once the observational best-fit parameters for the hot gas profile have been determined. For each $\rho(r|m{\ensuremath{_{\mathrm{h}}}})$ we can calculate the value of $\gamma$ such that the cosmic baryon fraction is reached at $r{\ensuremath{_{\mathrm{200m,obs}}}}$. This will be the limiting value and only equal or steeper slopes will be allowed. We will show the resulting $\gamma(m{\ensuremath{_{\mathrm{500c,obs}}}})$-relation in § \[sec:components\], since it depends on the best-fit density profile parameters from the observations that we will describe in § \[sec:obs\_xray\]. Being the only free parameter in our model, $\gamma$ provides a clear connection to observations. Deeper observations that can probe further into the outskirts of haloes, can thus be straightforwardly implemented in our model.
We will look at two different cases for the size of the haloes, motivated by the observed hot gas fractions in § \[sec:obs\_xray\] and by the lack of observational constraints outside $r{\ensuremath{_{\mathrm{500c,obs}}}}$. We aim to include enough freedom in the halo outskirts such that the actual baryon distribution will be encompassed by the models. In both cases, we leave the power-law slope $\gamma$ free outside $r{\ensuremath{_{\mathrm{500c,obs}}}}$. The models differ outside $r{\ensuremath{_{\mathrm{200m,obs}}}}$ since there are no firm observational constraints on the extent of the baryonic distribution around haloes. In the first case, we will truncate the power-law as soon as $r{\ensuremath{_{\mathrm{200m,obs}}}}$ is reached, thus enforcing $r{\ensuremath{_{\mathrm{h}}}} = r{\ensuremath{_{\mathrm{200m,obs}}}}$. This corresponds to the halo definition that is used by @Tinker2008 in constructing their halo mass function. For the least massive haloes in our model, this will result in haloes that are missing a significant fraction of their baryons at $r{\ensuremath{_{\mathrm{200m,obs}}}}$, with lower baryon fractions $f{\ensuremath{_{\mathrm{bar,200m,obs}}}}$ for steeper slopes, i.e. higher values of $\gamma$. Since we assume the linear power spectrum for the 2h term, we will still get the clustering predictions on the large scales right. We will denote this case with the quantifier [$\mathtt{nocb}$]{}, since the cosmic baryon fraction $f{\ensuremath{_{\mathrm{b}}}}={\ensuremath{\Omega_\mathrm{b}}}/{\ensuremath{\Omega_\mathrm{m}}}$ is not reached for most haloes in this case. In the second case, we will set $r{\ensuremath{_{\mathrm{h}}}} = r{\ensuremath{_{\mathrm{f{\ensuremath{_{\mathrm{b}}}}}}}} > r{\ensuremath{_{\mathrm{200m,obs}}}}$ such that all haloes reach the cosmic baryon fraction at $r{\ensuremath{_{\mathrm{h}}}}$, we will denote this case with the quantifier [$\mathtt{cb}$]{}.
The [$\mathtt{nocb}$]{} and the [$\mathtt{cb}$]{} cases for each $\gamma$ result in the same halo mass $m{\ensuremath{_{\mathrm{200m,obs}}}}$, since they only differ for $r > r{\ensuremath{_{\mathrm{200m,obs}}}}$. Thus, they have the same DMO equivalent halo mass and the same abundance $n(m{\ensuremath{_{\mathrm{200m,dmo}}}}(m))$ in Eq. \[eq:p\_1h\]. The difference between the two models is the normalization and the shape of the Fourier density profile $\hat{\rho}(k|m)$ which depends on the total halo mass $m{\ensuremath{_{\mathrm{h}}}}$ and the distribution of the hot gas. The halo mass $m{\ensuremath{_{\mathrm{h}}}}$ will be higher in the [$\mathtt{cb}$]{} case due to the added baryons between $r{\ensuremath{_{\mathrm{200m,obs}}}} < r < r{\ensuremath{_{\mathrm{h}}}}$, resulting in more power from the 1h term. Since the baryons in the [$\mathtt{cb}$]{} case are added outside $r{\ensuremath{_{\mathrm{200m,obs}}}}$ there will also be an increase in power on larger scales.
For our parameter range $0 \leq \gamma \leq 3$, the [$\mathtt{nocb}$]{} and [$\mathtt{cb}$]{}cases encompass the possible power suppression in the Universe. For massive systems, we have observational constraints on the total baryon content inside $r{\ensuremath{_{\mathrm{500c,obs}}}}$ and our model variations capture the possible variation in the outer density profiles. The distribution of the baryons in the hot phase outside $r{\ensuremath{_{\mathrm{500c,obs}}}}$ is not known observationally. However, it most likely depends on the halo mass. For the most massive haloes, Sunyaev-Zel’dovich (SZ) measurements of the hot baryons indicate that most baryons are accounted for inside $5 \, r{\ensuremath{_{\mathrm{500c,obs}}}} \approx 2 \, r{\ensuremath{_{\mathrm{200m,obs}}}}$ [e.g. @Planck2013; @LeBrun2015]. This need not be the case for lower-mass systems where baryons can be more easily ejected out to even larger distances. Moreover, there are also baryons that never make it into haloes and that are distributed on large, linear scales. The main uncertainty in the power suppression at large scales stems from the baryonic content of the low-mass systems. The 1h term of low-mass haloes becomes constant for $k \lesssim \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$. Hence, on large scales we capture the extreme case where the low-mass systems retain no baryons ([$\mathtt{nocb}$]{} and $\gamma=3$) and all the missing halo baryons are distributed on large, linear scales in the cosmic web. We can also capture the other extreme where the low-mass systems retain all of their baryons in the halo outskirts ([$\mathtt{cb}$]{} and $\gamma=0$), since the details of the density profile do not matter on scales $k < \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$. Thus, the matter distribution in the Universe will lie somewhere in between these two extremes captured by our model.
### Dark matter {#sec:hm_profiles_dm}
We assume that the dark matter follows a Navarro-Frenk-White (NFW) profile [@Navarro1996] with the concentration determined by the $c{\ensuremath{_{\mathrm{200c,dmo}}}}(m{\ensuremath{_{\mathrm{200c,dmo}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}}))$ relation from @Correa2015c, which is calculated using `commah`[^6], assuming Eq. \[eq:m200m\_dmo\] to get the DMO equivalent mass. We assume a unique $c(m)$ relation with no scatter. We discuss the influence of shifting the concentration-mass relation within its scatter in App. \[app:results\_concentration\].
The concentration in `commah` is calculated with respect to $r{\ensuremath{_{\mathrm{200c,dmo}}}}$ (the radius where the average enclosed density of the halo is $200 \, \rho{\ensuremath{_{\mathrm{c}}}}$), so we convert the concentration to our halo definition by multiplying by the factor $r{\ensuremath{_{\mathrm{200m,dmo}}}}/r{\ensuremath{_{\mathrm{200c,dmo}}}}$ (for the DMO equivalent halo). This needs to be solved iteratively for haloes with different concentration $c{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}})$, since for each input mass $m{\ensuremath{_{\mathrm{200c,dmo}}}}$ and resulting concentration $c{\ensuremath{_{\mathrm{200c,dmo}}}}$, we need to find the corresponding $m{\ensuremath{_{\mathrm{200m,dmo}}}}$ to convert $c{\ensuremath{_{\mathrm{200c,dmo}}}}$ to $c{\ensuremath{_{\mathrm{200m,dmo}}}}$. We thus have for the dark matter component in Eq. \[eq:rho\_r\] $$\label{eq:nfw_dm}
\rho{\ensuremath{_{\mathrm{dm}}}}(r|m{\ensuremath{_{\mathrm{h}}}}) =
\begin{cases}
\frac{m{\ensuremath{_{\mathrm{x}}}}}{4 \pi r
{\ensuremath{_{\mathrm{x}}}}^{3}}\frac{c{\ensuremath{_{\mathrm{x}}}}^{3}}{Y(c{\ensuremath{_{\mathrm{x}}}})} \left(
\frac{c{\ensuremath{_{\mathrm{x}}}}r}{r{\ensuremath{_{\mathrm{x}}}}} \right)^{-1} \left(1 +
\frac{c{\ensuremath{_{\mathrm{x}}}}r}{r{\ensuremath{_{\mathrm{x}}}}}\right)^{-2}, & r \leq r{\ensuremath{_{\mathrm{h}}}} \\
0, & r > r{\ensuremath{_{\mathrm{h}}}} \, .
\end{cases}$$ The halo radius $r{\ensuremath{_{\mathrm{h}}}}$ depends on the hot gas density profile and is either $r{\ensuremath{_{\mathrm{h}}}}=r{\ensuremath{_{\mathrm{200m,obs}}}}$ in the case [$\mathtt{nocb}$]{}, or $r{\ensuremath{_{\mathrm{h}}}}=r{\ensuremath{_{\mathrm{f{\ensuremath{_{\mathrm{b}}}}}}}}$, the radius where the cosmic baryon fraction is reached, in the case [$\mathtt{cb}$]{}. We define $Y(c{\ensuremath{_{\mathrm{x}}}}) = \log(1 + c{\ensuremath{_{\mathrm{x}}}}) - c{\ensuremath{_{\mathrm{x}}}}/(1 +
c{\ensuremath{_{\mathrm{x}}}})$ and the concentration $c{\ensuremath{_{\mathrm{x}}}} = r{\ensuremath{_{\mathrm{x}}}}/r{\ensuremath{_{\mathrm{s}}}}$ with the scale radius $r{\ensuremath{_{\mathrm{s}}}}$ indicating the radius at which the NFW profile has logarithmic slope $-2$. The subscript ‘x’ indicates the radius at which the concentration is calculated, e.g. $\mathrm{x=200m}$. All of the subscripted variables are a function of the halo mass $m{\ensuremath{_{\mathrm{500c,obs}}}}$. The normalization factor in our definition ensures that the NFW profile has mass $m{\ensuremath{_{\mathrm{x}}}}$ at radius $r{\ensuremath{_{\mathrm{x}}}}$. For the dark matter component in our baryonic model, we require the mass at $r{\ensuremath{_{\mathrm{500c,obs}}}}$ to equal the dark matter fraction of the total observed mass $m{\ensuremath{_{\mathrm{500c,obs}}}}$ $$\begin{aligned}
\label{eq:mx_dm}
m{\ensuremath{_{\mathrm{x}}}} &= m{\ensuremath{_{\mathrm{500c,obs}}}}
\begin{aligned}[t]
(1 & - f{\ensuremath{_{\mathrm{gas,500c,obs}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}}) \\
& -
f{\ensuremath{_{\mathrm{\star,500c,obs}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})))
\, .
\end{aligned}\end{aligned}$$ We require the scale radius for the dark matter to be the same as the scale radius of the equivalent DMO halo, thus $$\begin{aligned}
\label{eq:c500c_dm}
c{\ensuremath{_{\mathrm{x}}}} & = c{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})) \cdot
\frac{r{\ensuremath{_{\mathrm{500c,obs}}}}}{r{\ensuremath{_{\mathrm{200m,dmo}}}}} \, .\end{aligned}$$ For the DMO power spectrum that we compare to in Eq. \[eq:power\_ratio\], we assume $\mathrm{x=200m,dmo}$ in Eq. \[eq:nfw\_dm\] and we use both the halo mass and the concentration derived for Eq. \[eq:m200m\_dmo\]. The halo radius for the dark matter only case is the same as in the corresponding baryonic model. This is the logical choice since this means that in the case where our model accounts for all of the baryons inside $r{\ensuremath{_{\mathrm{h}}}}$, the DMO halo and the halo including baryons will have the same total mass, only the matter distributions will be different. In the case where not all the baryons are accounted for, we can then see the influence on the power spectrum of baryons missing from the haloes.
### Stars {#sec:hm_profiles_stars}
For the stellar contribution we do not try to fit density profiles to observations. We opt for this approach since it allows for a clear separation between centrals and satellites. Moreover, it provides the possibility of a self-consistent framework that is also able to fit the galaxy stellar mass function and the galaxy clustering. Our model can be straightforwardly modified to take stellar fractions and profiles from observations, as we did for the hot gas. We implement stars similarly to HOD methods, specifically the `iHOD` model by . We will assume their stellar-to-halo mass relations for both centrals and satellites. The `iHOD` model can reproduce the clustering and lensing of a large sample of SDSS galaxies spanning 4 decades in stellar mass by self-consistently modelling the incompleteness of the observations. Moreover, the model independently predicts the observed stellar mass functions. In our case, since we have assumed a different cosmology, these results will not necessarily be reproduced. However, we have checked that shifting the halo masses at fixed abundance between the cosmology of and ours only results in relative shifts of the stellar mass fractions of $\approx \SI{10}{\percent}$ at fixed halo mass.
We split up the stellar component into centrals and satellites $$\label{eq:rho_star}
\rho{\ensuremath{_{\mathrm{\star}}}}(r|m{\ensuremath{_{\mathrm{h}}}}) =
\rho{\ensuremath{_{\mathrm{cen}}}}(r|m{\ensuremath{_{\mathrm{h}}}}) + \rho{\ensuremath{_{\mathrm{sat}}}}(r|m{\ensuremath{_{\mathrm{h}}}}) \, .$$ The size of typical central galaxies in groups and clusters is much smaller than our scales of interest, so we can safely assume them to follow delta profile density distributions, as is done in $$\label{eq:delta_cen}
\rho{\ensuremath{_{\mathrm{cen}}}}(r|m{\ensuremath{_{\mathrm{h}}}}) = f{\ensuremath{_{\mathrm{cen,200m,dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}) \,
m{\ensuremath{_{\mathrm{200m,dmo}}}} \, \delta{\ensuremath{^{\mathrm{D}}}}({\ensuremath{\mathbf{r}}})
\, ,$$ here $f{\ensuremath{_{\mathrm{cen}}}}(m)$ is taken directly from the `iHOD` fit and $\delta{\ensuremath{^{\mathrm{D}}}}({\ensuremath{\mathbf{r}}})$ is the Dirac delta function.
For the satellite galaxies, we assume the same profile as and put the stacked satellite distribution at fixed halo mass in an NFW profile $$\label{eq:nfw_sat}
\rho{\ensuremath{_{\mathrm{sat}}}}(r|m{\ensuremath{_{\mathrm{h}}}}) =
\begin{cases}
\frac{m{\ensuremath{_{\mathrm{x}}}}}{4 \pi r
{\ensuremath{_{\mathrm{x}}}}^{3}}\frac{c{\ensuremath{_{\mathrm{x}}}}^{3}}{Y(c{\ensuremath{_{\mathrm{x}}}})} \left(
\frac{c{\ensuremath{_{\mathrm{x}}}}r}{r{\ensuremath{_{\mathrm{x}}}}} \right)^{-1} \left(1 +
\frac{c{\ensuremath{_{\mathrm{x}}}}r}{r{\ensuremath{_{\mathrm{x}}}}}\right)^{-2}, & r \leq r{\ensuremath{_{\mathrm{h}}}} \\
0, & r > r{\ensuremath{_{\mathrm{h}}}} \, ,
\end{cases}$$ which is the same NFW definition as Eq. \[eq:nfw\_dm\]. The profile also becomes zero for $r>r{\ensuremath{_{\mathrm{h}}}}$. Clearly, there will still be galaxies outside of this radius in the Universe. However, in the halo-based picture, we need to truncate the halo somewhere. Since the stellar contribution is always subdominant to the gas and the dark matter at the largest scales, we can safely truncate the profiles without affecting our predictions at the percent level. We will take our reference values in Eq. \[eq:nfw\_sat\] at $\mathrm{x=200m,dmo}$. As in , the satellites are less concentrated than the parent dark matter halo by a factor $0.86$ $$\begin{aligned}
\label{eq:m200m_sat}
m{\ensuremath{_{\mathrm{x}}}} & = f{\ensuremath{_{\mathrm{sat}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}}) \, m{\ensuremath{_{\mathrm{200m,dmo}}}} \\
\nonumber
c{\ensuremath{_{\mathrm{x}}}} & = f{\ensuremath{_{\mathrm{c,sat}}}} \,
c{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}})\\
\label{eq:c200m_sat}
& = 0.86 \,
c{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{200m,dmo}}}})\, .\end{aligned}$$ We take the stellar fraction from the best fit model of .
This less concentrated distribution of satellites is also found in observations for massive systems in the local Universe [@Lin2004; @Budzynski2012; @VanderBurg2015a]. However, the observations generally find a concentration of $c{\ensuremath{_{\mathrm{sat}}}} \approx \numrange{2}{3}$ for group and cluster mass haloes, which is about a factor 2 lower than the dark matter concentration. Similar results are found in the [<span style="font-variant:small-caps;">bahamas</span>]{} simulations [@McCarthy2017]. In low-mass systems, on the other hand, the satellites tend to track the underlying dark matter profile quite closely [@Wang2014a] with $c{\ensuremath{_{\mathrm{sat}}}}(m) \approx c(m)$. The value of $f{\ensuremath{_{\mathrm{c,sat}}}} = 0.86$ is thus a good compromise between these two regimes. We have checked that assuming $f{\ensuremath{_{\mathrm{c,sat}}}} = 1$ results in differences $< \SI{0.03}{\percent}$ at all $k$, with the maximum difference reached at $k \approx \SI{30}{{\si{\h.\mega\parsec^{-1}}}}$.
X-ray observations {#sec:obs_xray}
==================
![Stacked histogram for the masses of the haloes in our sample. The XXL-100-GC [@Eckert2016] data probe lower masses than the REXCESS [@Croston2008] data set, but it is clear that most of the haloes are clusters of galaxies with $m{\ensuremath{_{\mathrm{500c,obs}}}} >
\SI{e14}{{\si{\Msun}}}$.[]{data-label="fig:obs_masses_hist"}](figures/obs_masses_hist_stacked){width="\columnwidth"}
![The X-ray hydrostatic gas fractions as a function of halo mass. The different data sets are explained in the text. The median $f_{\mathrm{gas,500c}}-m_{\mathrm{500c,obs}}$ relation (black, solid lines) and the best fit (red, dashed lines) using Eq. \[eq:fgas\_sigmoid\] are shown. We indicate the 15[$^{\mathrm{th}}$]{} and 85[$^{\mathrm{th}}$]{} percentile range by the red shaded region. We show the hydrostatic (thick lines) and bias corrected ($1-b=0.7$, thin lines) relations. Since, in the latter case, halo masses increase more than the gas masses, under the assumption of the best-fit beta profile to the hot gas density profiles, the gas fractions shift down. The fits deviate at low masses because we force $f_{\mathrm{gas,500c}} \to 0$ for $m_{\mathrm{500c,obs}} \to 0$.[]{data-label="fig:obs_fgas"}](figures/obs_gas_fractions+debiased_bias0p7){width="\columnwidth"}
{width="\textwidth"}
{width="\textwidth"}
We choose to constrain the halo model using observations of the hot, X-ray emitting gas in groups and clusters of galaxies, since these objects provide the dominant contribution to the power spectrum at our scales of interest and their baryon content is dominated by hot plasma.
We combine two data sets of X-ray observations with *XMM-Newton* of clusters for which the individually measured electron density profiles were available, namely REXCESS [@Croston2008] and the XXL survey [more specifically the XXL-100-GC subset, @Pacaud2016; @Eckert2016]. This gives a total of 131 ($31+100$) unique groups and clusters (there is no overlap between the two data sets) with masses ranging from $m{\ensuremath{_{\mathrm{500c,obs}}}} \approx \SI{1e13}{{\si{\Msun}}}$ to $m{\ensuremath{_{\mathrm{500c,obs}}}} \approx \SI{2e15}{{\si{\Msun}}}$, with the XXL sample probing lower masses, as can be seen in Fig. \[fig:obs\_masses\_hist\]. We extend our data with more sets of observations for the hydrostatic gas fraction of groups and clusters of galaxies, as shown in Fig. \[fig:obs\_fgas\]. We use a set of hydrostatic masses determined from *Chandra* archival data [@Vikhlinin2006b; @Maughan2008b; @Sun2009; @Lin2012a] and from the NORAS and REFLEX (of which REXCESS is a subset) surveys [@Pratt2009; @Lovisari2015].
REXCESS consists of a representative sample of clusters from the REFLEX survey [@Bohringer2007]. It includes clusters of all dynamical states and aims to provide a homogeneous sampling in X-ray luminosity of clusters in the local Universe ($z < 0.2$). Since all of the redshift bins are approximately volume limited [@Bohringer2007], we do not expect significant selection effects for the massive systems ($m{\ensuremath{_{\mathrm{500c}}}} > \SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}}$) as it has been shown by @Chon2017 that the lack of disturbed clusters in X-ray samples [@Eckert2011] is generally due to their flux-limited nature. The XXL-100-GC sample is flux-limited [@Pacaud2016] and covers a wider redshift range ($z \lesssim 1$). Since it is flux-limited, there is a bias to selecting more massive objects. At low redshifts, however, there is a lack of massive objects due to volume effects [@Pacaud2016]. From @Chon2017 we would also expect the sample to be biased to select relaxed systems.
Assuming an optically thin, collisionally-ionized plasma with a temperature $T$ and metallicity $Z$, the deprojected surface brightness profile can be converted into a 3-D electron density profile $n_{e}$, which is the source of the thermal bremsstrahlung emission [@Sarazin1986]. For the REXCESS sample, the spectroscopic temperature within $r_{\mathrm{500c,obs}}$ was chosen with the metallicity also deduced from a spectroscopic fit, whereas for the XXL sample the average temperature within $r<\SI{300}{kpc}$ was used with a metallicity of $Z= \SI{0.3}{\Zsun}$. We get the corresponding hydrogen and helium abundances by interpolating between the sets of primordial abundances, $(X_{0},Y_{0},Z_{0}) = (0.75, 0.25, 0)$, and of solar abundances, $(\mathrm{X_{\odot}},\mathrm{Y_{\odot}},\mathrm{Z_{\odot}})=(0.7133,0.2735,0.0132)$. We then find $(X,Y,Z)=(0.73899,0.25705,0.00396)$ for $Z=\SI{0.3}{\Zsun}$. To convert this electron density into the total density, we will assume these interpolated abundances, since in general for clusters the metallicity $Z \approx \SI{0.3}{\Zsun} = 0.00396$ [@Voit2005b; @Grevesse2007]. This is also approximately correct for the @Croston2008 data, since for their systems the median metallicity (bracketed by 15[$^{\mathrm{th}}$]{} and 85[$^{\mathrm{th}}$]{} percentiles) is $Z/\si{\Zsun}=0.27\substack{+0.09 \\ -0.05}$. Moreover, we assume the gas to be fully ionized. We know that the total gas density is given by $$\begin{aligned}
\nonumber
\rho_{\mathrm{gas}} & = \mu m_{{\text{H\,\textsc{\lowercase{}}}}} (n_{e} + n_{{\text{H\,\textsc{\lowercase{}}}}} + n_{{\text{He\,\textsc{\lowercase{}}}}})\\
\nonumber
& = \frac{1+Y/X}{2+3Y/(4X)} \frac{2 + 3Y/(4X)}{1
+ Y/(2X)} m_{{\text{H\,\textsc{\lowercase{}}}}} n_{e}\\
\label{eq:ne_to_rho}
& \approx 0.6 \cdot 1.93 \, m_{{\text{H\,\textsc{\lowercase{}}}}} n_{e}\end{aligned}$$ This results in the gas density profiles shown in Fig. \[fig:obs\_profiles\]. It is clear that at large radii the scatter is smaller for more massive systems. We bin the XXL data in 20 mass bins as the individual profiles have a large scatter at fixed radius. For each mass bin we only include the radial range where each profile in the bin is represented.
The two surveys derived the halo mass $m{\ensuremath{_{\mathrm{500c,obs}}}}$ differently. For REXCESS, the halo masses for the whole sample were determined from the $m_{\mathrm{500c,obs}}-Y{\ensuremath{_{\mathrm{X}}}}$ relation of @Arnaud2007, where $Y{\ensuremath{_{\mathrm{X}}}} = m{\ensuremath{_{\mathrm{gas,500c}}}} \, T{\ensuremath{_{\mathrm{X}}}}$ is the thermal energy content of the intracluster medium (ICM). @Arnaud2007 determined $m_{\mathrm{500c,obs}}$ under the assumption of spherical symmetry and hydrostatic equilibrium [see e.g. @Voit2005b]. @Eckert2016 take a different route. They determine halo masses using the $m_{\mathrm{500c,obs}}-T{\ensuremath{_{\mathrm{X}}}}$ relation calibrated to weak lensing mass measurements of 38 clusters that overlap with the CFHTLenS shear catalog, as described in @Lieu2016. As a result, the REXCESS halo mass estimates rely on the assumption of hydrostatic equilibrium, whereas @Eckert2016 actually find a hydrostatic bias $m{\ensuremath{_{\mathrm{X-ray}}}}/m{\ensuremath{_{\mathrm{WL}}}} = 1-b=0.72$, consistent with the analyses of [@VonderLinden2014] and [@Hoekstra2015]. Recently, @Umetsu2019 used the Hyper Suprime-Cam (HSC) survey shear catalog, which overlaps the XXL-North field almost completely, to measure weak lensing halo masses with a higher limiting magnitude and, hence, number density of source galaxies than CHFTLenS. They do not rederive the gas fractions of @Eckert2016, but they note that their masses are systematically lower by a factor $\approx 0.75$ than those derived in @Lieu2016, a finding which is consistent with @Lieu2017, who find a factor $\approx 0.72$. These lower weak lensing halo masses result in a hydrostatic bias of $b < 0.1$.
To obtain a consistent analysis, we scale the halo masses from @Eckert2016 back onto the hydrostatic $f{\ensuremath{_{\mathrm{gas,500c}}}}-m{\ensuremath{_{\mathrm{500c,obs}}}}$ relation, which we show in Fig. \[fig:obs\_fgas\]. We thus assume halo masses derived from the assumption of hydrostatic equilibrium. It might seem strange to take the biased result as the starting point of our analysis. However, we argue that this is an appropriate starting point. First, current estimates for the hydrostatic bias range from $0.58 \pm 0.04 \lesssim 1-b \lesssim 0.71 \pm 0.10$ corresponding to the results from Planck SZ cluster counts [@PlanckXXIV2016; @Zubeldia2019], or $0.688\pm 0.072 \lesssim 1-b \lesssim 0.80 \pm 0.14$ from weak lensing mass measurements of Planck clusters [@VonderLinden2014; @Hoekstra2015; @Medezinski2018]. Second, we are not able to determine the mass dependence of the relation for groups of galaxies from current observations. We will check how our results change when assuming a constant hydrostatic bias of $1-b=0.7$ in § \[sec:results\_bias\]. The thin, black line in Fig. \[fig:obs\_fgas\] shows the shift in the $f{\ensuremath{_{\mathrm{gas,500c}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})$ relation when assuming this constant hydrostatic bias.
We fit the cluster gas density profiles with beta profiles, following Eq. \[eq:beta\_gas\], within $[0.15,1]\,r_{\mathrm{500c,obs}}$, excising the core as usual in the literature, since it can deviate from the flat slope in the beta profile. In observations, it is common to assume a sum of different beta profiles to capture the slope in the inner $0.15 r{\ensuremath{_{\mathrm{500c,obs}}}}$. However, we correct for the mass that we miss in the core by fixing the normalization to reproduce the total gas mass of the profile, which is captured by the gas fraction $f{\ensuremath{_{\mathrm{gas,500c}}}}$. (This is equivalent to redistributing the small amount of mass that we would miss in the core to larger scales.) The slope at large radii, $\beta$, and the core radius, $r{\ensuremath{_{\mathrm{c}}}}$, are the final two parameters determining the profile. We show the residuals of the profile fits in Fig. \[fig:obs\_fits\] where we also include the residuals of the cumulative mass profile. It is clear from the residuals in the top panels of Fig. \[fig:obs\_fits\] that the beta profile cannot accurately capture the inner density profile of the hot gas. @Arnaud2010 show that the inner slope can vary from shallow to steep in going from disturbed to relaxed or cool-core clusters. This need not concern us because the deviations from the fit occur at such small radii that they will not be able to significantly affect the power at our scales of interest where the normalization of $\hat{\rho}{\ensuremath{_{\mathrm{gas}}}}(k)$ and, thus, the total mass of the hot gas component is the important parameter. In the bottom panel of Fig. \[fig:obs\_fits\] we show the residuals for the cumulative mass. The left-hand panel of the figure clearly shows that we force $m{\ensuremath{_{\mathrm{gas,500c}}}}$ in the individual profiles to equal the observed mass.
![The mass dependence of the core radius $r{\ensuremath{_{\mathrm{c}}}}$ of the beta hot gas density profile fits, Eq. \[eq:beta\_gas\]. We indicate the 15[$^{\mathrm{th}}$]{} and 85[$^{\mathrm{th}}$]{} percentiles with the gray shaded region and the median by the solid line. The error bars indicate the standard deviation in the best-fit parameter. We have binned the @Eckert2016 sample into 20 mass bins. There is no clear mass dependence.[]{data-label="fig:obs_rc_fit"}](figures/obs_rc_fit){width="\columnwidth"}
![As Fig. \[fig:obs\_rc\_fit\] but for the slope $\beta$ of the beta hot gas density profile fits, Eq. \[eq:beta\_gas\].[]{data-label="fig:obs_beta_fit"}](figures/obs_beta_fit){width="\columnwidth"}
We show the core radii, $r{\ensuremath{_{\mathrm{c}}}}$, and slopes, $\beta$, that we fit to our data set in Figs. \[fig:obs\_rc\_fit\] and \[fig:obs\_beta\_fit\], respectively. There is no clear mass dependence in the both of the fit parameters. Thus, we decided to use the median value for both parameters for all halo masses. This significantly simplifies the model, keeping the total number of parameters low.
We show the hydrostatic gas fractions from our observational data in Fig. \[fig:obs\_fgas\]. We fit the median $f_{\mathrm{gas,500c}}-m_{\mathrm{500c,obs}}$ relation with a sigmoid-like function given by $$\label{eq:fgas_sigmoid}
f_{\mathrm{gas,500c}}(m{\ensuremath{_{\mathrm{500c,obs}}}}) = \frac{{\ensuremath{\Omega_\mathrm{b}}}/{\ensuremath{\Omega_\mathrm{m}}}}{2} \left(1 +
\tanh\left(\frac{\log_{10}(m{\ensuremath{_{\mathrm{500c,obs}}}}/m_\mathrm{t})}{\alpha}
\right) \right) \, ,$$ under the added constraint $$\label{eq:fgas_constraint}
f_{\mathrm{gas,500c}}(m{\ensuremath{_{\mathrm{500c,obs}}}}) \leq
f{\ensuremath{_{\mathrm{b}}}} - f{\ensuremath{_{\mathrm{\star,500c}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}}) \,.$$ The function has as free parameters the turnover mass, $m{\ensuremath{_{\mathrm{t}}}}$, and the sharpness of the turnover, $\alpha$. We fix the gas fraction for $m{\ensuremath{_{\mathrm{500c,obs}}}}\to \infty$ to the cosmic baryon fraction $f{\ensuremath{_{\mathrm{b}}}} = {\ensuremath{\Omega_\mathrm{b}}}/{\ensuremath{\Omega_\mathrm{m}}}\approx \num{0.166}$, which is what we expect for deep potential wells and what we also see for the highest-mass clusters. However, we shift down the final $f{\ensuremath{_{\mathrm{gas,500c}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})$ relation at halo masses where the cosmic baryon fraction would be exceeded after including the stellar contribution. We also fix the gas fraction for $m \to 0$ to 0 since we know that low-mass dwarfs eject their baryons easily and are mainly dark matter dominated [e.g. @Silk2012a; @Sawala2015]. Moreover, their virial temperatures are too low for them to contain X-ray emitting gas. Fixing $f{\ensuremath{_{\mathrm{gas,500c}}}}(m\to0) = 0$ is probably not optimal, especially since we know that the lower mass haloes will contain a significant warm gas ($\SI{e4}{\K} \lesssim T \lesssim \SI{e6}{\K}$) component which should increase their baryonic mass. However, since we will use our freedom in correcting the gas fraction at $r{\ensuremath{_{\mathrm{h}}}}$ by assuming profiles outside $r{\ensuremath{_{\mathrm{500c,obs}}}}$, this choice should not significantly impact our results as we already discussed at the end of § \[sec:hm\_profiles\_gas\]. For our scales of interest, the shape of the profiles of low-mass systems will not matter as much as their total mass. Forcing the gas fraction to go to 0 for low halo masses causes a deviation from the observations at low halo masses. However, at low halo mass the X-ray observations will always be biased to systems with high gas masses, since these will have the highest X-ray luminosities.
In Fig. \[fig:obs\_fgas\] we also show fits to the data $f_{\mathrm{gas,500c}}-m_{\mathrm{500c,obs}}$ relation assuming a constant hydrostatic mass bias of $\frac{m{\ensuremath{_{\mathrm{hydro}}}}}{m{\ensuremath{_{\mathrm{true}}}}} = 1-b = 0.7$. In § \[sec:results\_bias\] we discuss how we compute this relation and the influence of this assumption on our results.
Model density components {#sec:components}
========================
![The allowed values for the extrapolated slope $\gamma$ of the beta density profile, Eq. \[eq:beta\_gas\], as a function of halo mass $m{\ensuremath{_{\mathrm{500c,obs}}}}$. We colour each line by the value $\gamma_0 = \gamma(m{\ensuremath{_{\mathrm{500c,obs}}}} \to 0)$. Since we extrapolate haloes to $r{\ensuremath{_{\mathrm{200m,obs}}}}$, the most massive haloes would contain too many baryons if $\gamma$ would be too small. Hence, for each halo mass, we compute the limiting $\gamma$ for which the halo is baryonically closed at $r{\ensuremath{_{\mathrm{200m,obs}}}}$. This limit is indicated by the dashed line. For each halo mass only slopes steeper than this limit are allowed.[]{data-label="fig:gamma_0_vs_m500c"}](figures/plaw_var_gamma_gamma_vs_m500c){width="\columnwidth"}
{width="\textwidth"}
![The halo baryon fraction at $r{\ensuremath{_{\mathrm{200m,obs}}}}$ as a function of halo mass $m{\ensuremath{_{\mathrm{200m,obs}}}}$. The baryon fraction $f{\ensuremath{_{\mathrm{bar,200m,obs}}}}$ is the same for both model [$\mathtt{nocb}$]{}, which effectively assumes that the missing halo baryons are redistributed far beyond $r{\ensuremath{_{\mathrm{200m,obs}}}}$ on linear scales, and model [$\mathtt{cb}$]{}, which adds the missing halo baryons in a uniform profile outside but near $r{\ensuremath{_{\mathrm{200m,obs}}}}$. The lines are colour-coded by $\gamma_0 \equiv \gamma(m{\ensuremath{_{\mathrm{500c}}}} \to 0)$, the extrapolated power-law slope of the hot gas density profiles between $r{\ensuremath{_{\mathrm{500c,obs}}}}$ and $r{\ensuremath{_{\mathrm{200m,obs}}}}$, with lower values of $\gamma_0$ corresponding to flatter slopes. The shape is set by the observed constraints on the baryon fractions at $r{\ensuremath{_{\mathrm{500c,obs}}}}$. As $\gamma_0$ decreases to 0, the halo baryon fractions increase. The knee at $m{\ensuremath{_{\mathrm{200m,obs}}}} \approx \SI{e12}{{\si{\h^{-1}.{\si{\Msun}}}}}$ is caused by the peak of the stellar mass fractions. The decreased range of possible baryon fractions for low-mass haloes is the consequence of their low gas fractions and the fixed prescription for the stellar component.[]{data-label="fig:fb_vs_m200m_obs"}](figures/plaw_var_gamma_fbar200m_obs_vs_m200m_obs){width="\columnwidth"}
![The radius where the cosmic baryon fraction is reached in units of $r{\ensuremath{_{\mathrm{200m,obs}}}}$ as a function of halo mass $m{\ensuremath{_{\mathrm{200m,obs}}}}$ for model [$\mathtt{cb}$]{}, which adds the missing halo baryons in a uniform profile outside $r{\ensuremath{_{\mathrm{200m,obs}}}}$. The lines are colour-coded by $\gamma_0 \equiv \gamma(m{\ensuremath{_{\mathrm{500c}}}} \to 0)$, the extrapolated power-law slope of the hot gas density profiles between $r{\ensuremath{_{\mathrm{500c,obs}}}}$ and $r{\ensuremath{_{\mathrm{200m,obs}}}}$, with lower values of $\gamma_0$ corresponding to flatter slopes. As $\gamma_0$ decreases to 0, the cosmic baryon fraction is reached closer to the halo radius $r{\ensuremath{_{\mathrm{200m,obs}}}}$.[]{data-label="fig:rfb_vs_m200m_obs"}](figures/plaw_var_gamma_rfb_vs_m200m_obs){width="\columnwidth"}
We determined the best-fit parameters for the beta profile, Eq. \[eq:beta\_gas\], in § \[sec:obs\_xray\]. The only remaining free parameter in our model is now the slope $\gamma$ of the extrapolated profile outside $r{\ensuremath{_{\mathrm{500c,obs}}}}$. As we explained in § \[sec:obs\_xray\], not all values of $\gamma$ are allowed for each halo mass $m{\ensuremath{_{\mathrm{500c,obs}}}}$, since the most massive haloes contain a significant fraction of their total baryon budget inside $r{\ensuremath{_{\mathrm{500c,obs}}}}$. Consequently, these haloes need steeper slopes $\gamma$, since otherwise they would exceed the cosmic baryon fraction before they reach the halo radius $r{\ensuremath{_{\mathrm{200m,obs}}}}$. We thus determine the relation $\gamma{\ensuremath{_{\mathrm{min}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})$ that limits the extrapolated slope such that, given the best-fit beta profile parameters, the halo reaches exactly the cosmic baryon fraction at $r{\ensuremath{_{\mathrm{200m,obs}}}}$. For each halo mass only slopes steeper than this limiting value are allowed. We show the resulting relation $\gamma{\ensuremath{_{\mathrm{min}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})$ in Fig. \[fig:gamma\_0\_vs\_m500c\]. We colour the curves by $\gamma_0=\gamma(m{\ensuremath{_{\mathrm{500c,obs}}}} \to 0)$. Since low-mass haloes have low baryon fractions at $r{\ensuremath{_{\mathrm{500c,obs}}}}$, we find that all values of $\gamma_0$ are allowed. For the most massive haloes, only the steepest slopes $\gamma \gtrsim 2.8$ are allowed. The handful of observations that are able to probe clusters out to $r{\ensuremath{_{\mathrm{200m,obs}}}}$ indeed find that the slope steepens in the outskirts [@Ghirardini2018].
Now we have all of the ingredients of our model at hand. We show the resulting profiles for our different matter components for 3 halo masses in Fig. \[fig:obs\_rho\_extrapolated\]. We show both the [$\mathtt{nocb}$]{}and [$\mathtt{cb}$]{} models, where the latter are just the former extended beyond $r{\ensuremath{_{\mathrm{200m,obs}}}}$ until the cosmic baryon fraction is reached. We colour the curves by $\gamma_0$. Given $\gamma_0$, the actual value of the slope $\gamma$ for each halo mass can be determined by following the tracks in Fig. \[fig:gamma\_0\_vs\_m500c\] from low to high halo masses, e.g. for the $m{\ensuremath{_{\mathrm{500c,obs}}}} = \SI{e15}{{\si{\h^{-1}.{\si{\Msun}}}}}$ halo all slopes $\gamma_0 \leq 2.8$ correspond to the actual slope $\gamma=2.8$. Besides the hot gas profiles, we also show the dark matter and stellar (satellite, since the central is modelled as a delta function) profiles. These profiles only depend on the value $\gamma_0$ through their maximum radius, since the halo radius $r{\ensuremath{_{\mathrm{h}}}}$ is determined by how fast the cosmic baryon fraction is reached and thus depends on $\gamma_0$.
It is clear that models with flatter slopes reach their baryon budget at smaller radii. These models will thus capture the influence of a compact baryon distribution on the matter power spectrum. We show the halo baryon fraction at $r{\ensuremath{_{\mathrm{200m,obs}}}}$ for different values of $\gamma_0$ in Fig. \[fig:fb\_vs\_m200m\_obs\]. The main shape of the gas fractions at $r{\ensuremath{_{\mathrm{200m,obs}}}}$ is set by the constraints on the gas fractions at $r{\ensuremath{_{\mathrm{500c,obs}}}}$. The group-size haloes have the largest spread in baryon fraction with changing slope $\gamma_0$. Our model is thus able to capture a large range of different baryon contents for haloes that all reproduce the observations at $r{\ensuremath{_{\mathrm{500c,obs}}}}$. The baryon fractions rise steeply between $\num{e11} < m{\ensuremath{_{\mathrm{200m,obs}}}} / (\si{{\si{\h^{-1}.{\si{\Msun}}}}}) < \num{e12}$ due to the peak in the stellar mass fraction in this halo mass range. For the low-mass haloes, the spread in baryon fraction is smaller at $r{\ensuremath{_{\mathrm{200m,obs}}}}$ because we hold the stellar component fixed in our model and their gas fractions are low. As a result, the low-mass systems do not differ much in the [$\mathtt{nocb}$]{} model. (In the [$\mathtt{cb}$]{} model they will differ due to the different halo radii $r{\ensuremath{_{\mathrm{h}}}}$ where the cosmic baryon fraction is reached.) For the slope $\gamma$ between $\numrange{0}{3}$ we will have $\approx \SIrange{20}{50}{\percent}$ of the total baryons in the Universe outside haloes in the [$\mathtt{nocb}$]{} model.
We have checked that the density profiles with varying $\gamma_0$ for for haloes with $\num{e14} < m{\ensuremath{_{\mathrm{500c,obs}}}} / \si{{\si{\h^{-1}.{\si{\Msun}}}}} < \num{e15}$ only cause a maximum deviation of $\approx \pm \SI{5}{\percent}$ in the surface brightness profiles for projected radii $R < r{\ensuremath{_{\mathrm{500c,obs}}}}$ compared to the fiducial model with $\gamma_0 = 3\beta$. This variation is within the error on the surface brightness counts and the density profiles with varying $\gamma_0$ are thus indistinguishable from the fiducial model in the investigated mass range. For haloes with $m{\ensuremath{_{\mathrm{500c,obs}}}} \leq \SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}}$, the deviations increase for lower values of $\gamma_0$, reaching $\SI{10}{\percent}$ for $\gamma_0=1.5$ and $m{\ensuremath{_{\mathrm{500c,obs}}}} = \SI{e13}{{\si{\h^{-1}.{\si{\Msun}}}}}$, but the observed hot gas density profiles at these halo masses also show a larger scatter.
We also have the [$\mathtt{cb}$]{} model where we force all haloes to include all of the missing baryons in their outskirts. In Fig. \[fig:rfb\_vs\_m200m\_obs\] we show how extended the baryon distribution needs to be in the [$\mathtt{cb}$]{} case as a function of the slope $\gamma_0$. The variations in the power-law slope paired with the [$\mathtt{cb}$]{}and [$\mathtt{nocb}$]{} models allow us to investigate the influence on the matter power spectrum of a wide range of possible baryon distributions that all reproduce the available X-ray observations for clusters with $m{\ensuremath{_{\mathrm{500c,obs}}}} \geq \SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}}$.
Results {#sec:results}
=======
In this section we show the results and predictions of our model for the matter power spectrum and we discuss their implications for future observational constraints. First, we show the influence of assuming different distributions for the unobserved hot gas in § \[sec:results\_outer\]. We show the influence of correcting observed halo masses to the dark matter only equivalent halo masses in order to obtain the correct halo abundances in § \[sec:results\_hmf\]. In § \[sec:results\_masses\], we show which halo masses dominate the power spectrum for which wavenumbers. Finally, we show the influence of varying the best-fit observed profile parameters in § \[sec:results\_prms\] and we investigate the effects of a hydrostatic bias in the halo mass determination in § \[sec:results\_bias\].
Influence of the unobserved baryon distribution {#sec:results_outer}
-----------------------------------------------
{width="\textwidth"}
{width="\textwidth"}
![The X-ray hydrostatic gas fractions as a function of halo mass, as in Fig. \[fig:obs\_fgas\]. The curves show the sigmoid-like fit from Eq. with the best-fit value for $\alpha=1.35$, coloured by the value $\log_{10}m{\ensuremath{_{\mathrm{t}}}}/(\si{{\si{\h^{-1}.{\si{\Msun}}}}}) \in \{13,13.5,14,14.5,15\}$ (the best-fit value is 13.94). The shaded green region indicates the area that is broadly in agreement with observations.[]{data-label="fig:fgas_vs_mt"}](figures/obs_gas_fractions+log10mt){width="\columnwidth"}
![The power suppression due to the inclusion of baryons at the fixed scale $k = \SI{0.5}{{\si{\h.\mega\parsec^{-1}}}}$ as a function of the baryon fraction of haloes with $m{\ensuremath{_{\mathrm{500c,obs}}}}=\SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}}$. The shaded green region indicates the gas fractions that broadly agree with observations. The [$\mathtt{cb}$]{} (dashed, connected triangles) and [$\mathtt{nocb}$]{} (connected circles) models are coloured by $\gamma_0$, i.e. the value of the extrapolated power-law slope of the hot gas density profiles between $r{\ensuremath{_{\mathrm{500c,obs}}}}$ and $r{\ensuremath{_{\mathrm{200m,obs}}}}$. We show the relation found by @VanDaalen2019 for hydrodynamical simulations and its $\pm \SI{1}{\percent}$ variation (black line with grey, shaded region). We indicate the value of $\log_{10} m{\ensuremath{_{\mathrm{t}}}}/(\si{{\si{\h^{-1}.{\si{\Msun}}}}})$ in Eq. along the top x-axis. Both our model and VD19 predict a positive correlation between the power suppression at fixed scale and the halo baryon fraction at fixed halo mass. However, it is clear that our model allows for a larger range in possible power suppression at fixed halo baryon fraction than is found in the simulations.[]{data-label="fig:power_vs_fbar"}](figures/plaw_var_log10mt+gamma_fbar_vs_dpower_fb+nofb){width="\columnwidth"}
In this section, we will investigate the influence of the distribution of the unobserved baryons inside and outside haloes on the matter power spectrum. Since we currently have only a very tenuous grasp of the whereabouts of the missing baryons, it is important to explore how their possible distribution impacts the matter power spectrum.
As stated in § \[sec:hm\_profiles\_gas\], our model is characterized by the extrapolated power-law slope $-\gamma$ for the hot gas density profile and by whether we assume the missing halo baryons to reside in the vicinity of the halo (model [$\mathtt{cb}$]{}) or not (model [$\mathtt{nocb}$]{}). As explained in § \[sec:hm\_profiles\_gas\], these two types of models only differ in $\hat{\rho}(k|m)$ due to the inclusion of more mass outside the traditional halo definition of $r{\ensuremath{_{\mathrm{200m,obs}}}}$ in the [$\mathtt{cb}$]{} case (see Figs. \[fig:obs\_rho\_extrapolated\], \[fig:rfb\_vs\_m200m\_obs\]). When discussing our model predictions for the power spectrum, we consider the range $\SI{0.1}{{\si{\h.\mega\parsec^{-1}}}} \leq k \leq \SI{5}{{\si{\h.\mega\parsec^{-1}}}}$ to be the vital regime since future surveys will gain their optimal signal-to-noise for $k \approx \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$ [@Amendola2018].
We show the response of the matter power spectrum to baryons for the [$\mathtt{nocb}$]{} and [$\mathtt{cb}$]{} models in, respectively, the top-left and top-right panels of Fig. \[fig:power\_ratio\]. The lines are coloured by the assumed value of $\gamma_0$. We indicate our fiducial model, which extrapolates the best-fit $\beta=0.71^{+0.20}_{-0.12}$, i.e. $\gamma_0 = 3\beta = 2.14$, from the X-ray observations, with the thick, black line. All models show a suppression of power on large scales with respect to the DMO prediction. All of our models have an upturn in the response for $k \gtrsim \SI{10}{{\si{\h.\mega\parsec^{-1}}}}$ and and enhancement of power for $k \gtrsim \SI{50}{{\si{\h.\mega\parsec^{-1}}}}$ due to the stellar component. This upturn is not present in other halo model approaches that only modify the dark matter profiles [e.g. @Smith2003; @Mead2015]. We shade the region $k > \SI{10}{{\si{\h.\mega\parsec^{-1}}}}$ in red because the range in responses of our model does not span the range allowed by observations there. On the contrary, on these small scales all of our models behave the same, since the hot gas is completely determined by the best-fit beta profile to the X-ray observations, and the stellar component is held fixed.
The total amount of power suppression at large scales depends sensitively on the halo baryon fractions, since models with the highest values of $\gamma_0$ also have the lowest baryon fractions $f{\ensuremath{_{\mathrm{bar,200m,obs}}}}$ at all halo masses (see Fig. \[fig:fb\_vs\_m200m\_obs\]). Our results confirm the predictions from hydrodynamical simulations, which have shown similar trends [@VanDaalen2011; @VanDaalen2019; @Hellwing2016; @McCarthy2017; @Springel2017; @Chisari2018]. However, our results do not rely on the uncertain assumptions associated with subgrid models for feedback processes. Our phenomenological model simply requires that we reproduce the density profiles of clusters without any assumptions about the underlying physics that resulted in the profiles.
The [$\mathtt{nocb}$]{} model, shown in the left-hand panel of Fig. \[fig:power\_ratio\], results in a larger spread of possible responses because the final total halo mass is not fixed to account for all the baryons as in [$\mathtt{cb}$]{}. The [$\mathtt{nocb}$]{} models with the steepest extrapolated density profiles, i.e. the highest values for $\gamma_0$, function as upper limits on the response, since the missing halo baryons are in reality likely to reside in the vicinity of the haloes and because low-mass haloes likely contain more gas than predicted by our extrapolated relation. However, this gas may not be well described by our beta profile assumption derived from the hot gas properties of clusters. On the other hand, the [$\mathtt{cb}$]{} models with flatter slopes (lower values for $\gamma_0$), shown in the right-hand panel of Fig. \[fig:power\_ratio\], function as lower limits on the response of the power spectrum to baryons, since it is likely that a significant fraction of the baryons does not reside inside haloes but rather in the diffuse, warm-hot, intergalactic medium [WHIM, as has been predicted by simulations and recently inferred from observations, see e.g. @Cen1999; @Dave2001; @Nicastro2018]. Hence, we find that the (minimum, fiducial, maximum) value of the minimum wavenumber for which the baryonic effect reaches $\SI{1}{\percent}$ is ($0.2$, $0.3$, $0.9$) $\si{{\si{\h.\mega\parsec^{-1}}}}$ in the [$\mathtt{nocb}$]{} models and ($0.5$, $0.8$, $1$) $\si{{\si{\h.\mega\parsec^{-1}}}}$ in the [$\mathtt{cb}$]{} models. The $\SI{5}{\percent}$ threshold is reached for ($0.5$, $0.8$, $2$) $\si{{\si{\h.\mega\parsec^{-1}}}}$ and ($1$, $1.4$, $2$) $\si{{\si{\h.\mega\parsec^{-1}}}}$, respectively, for the [$\mathtt{nocb}$]{} and [$\mathtt{cb}$]{}models.
We indicate the results from the [<span style="font-variant:small-caps;">bahamas</span>]{} simulation run $\mathtt{AGN\_TUNED\_nu0\_L400N1024\_WMAP9}$, which has been shown to reproduce a plethora of observations for massive systems [@McCarthy2017; @Jakobs2017], and the result for the [OWLS]{} AGN simulation [@Schaye2010; @VanDaalen2011] which has been widely used as a reference model in weak lensing analyses and is also consistent with the observed cluster gas fractions [@McCarthy2010]. We show the ratio between our models and the [<span style="font-variant:small-caps;">bahamas</span>]{} prediction of the power spectrum response to the presence of baryons in the bottom row of Fig. \[fig:power\_ratio\]. Our models encompass both the [<span style="font-variant:small-caps;">bahamas</span>]{} and [OWLS]{} predictions for $k \lesssim \SI{5}{{\si{\h.\mega\parsec^{-1}}}}$, which is the range of interest here. In the [$\mathtt{cb}$]{} case, our models all predict less power suppression than the simulations on large scales $k \lesssim \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$, which is most likely due to the fact that in the simulations there are actually baryons in the cosmic web that should not be accounted for by haloes, thus suggesting that models [$\mathtt{nocb}$]{} may be more realistic. However, since there are no observational constraints on the location of the missing halo baryons, we cannot exclude the models [$\mathtt{cb}$]{}. We stress that we did not fit our model to reproduce these simulations. The overall similarity is caused by the simulations reproducing the measured X-ray hot gas fractions that we fit our model to.
In Fig. \[fig:power\_ratio\_sims\], we compare predictions for the power spectrum response to baryons from a large set of higher-resolution, but smaller-volume, cosmological simulations to the prediction of our fiducial model. We compare the EAGLE [@Schaye2015; @Hellwing2016], IllustrisTNG [@Springel2017], Horizon-AGN [@Chisari2018], and Illustris [@Vogelsberger2014] simulations. We can see that in all of these simulations, except for Illustris, which is known to have AGN feedback that is too violent on group and cluster scales [@Weinberger2017], the baryonic suppression becomes significant only at much smaller scales than in [OWLS]{}, [<span style="font-variant:small-caps;">bahamas</span>]{} and our own model. From the halo model it is clear that the total baryon content of haloes, and thus the cluster gas fractions, are the dominant cause of baryonic power suppression on large scales $k \lesssim \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$, since $\hat{\rho}(k|m) \to m$ there. Indeed, @VanDaalen2019 explicitly demonstrated the link between cluster gas fractions and power suppression on large scales for a large set of hydrodynamical simulations including these. Since [<span style="font-variant:small-caps;">bahamas</span>]{}and [OWLS]{} AGN reproduce the cluster hot gas fractions, they predict the same large-scale behaviour for the power spectrum response to baryons. However, the other small-volume, high-resolution simulations overpredict the baryon content of groups and clusters as was shown for EAGLE, IllustrisTNG, and Horizon-AGN by, respectively, @Barnes2017, @Barnes2018, and @Chisari2018. We thus stress the importance of using simulations that are calibrated towards the relevant observations when training or comparing models aimed at predicting the matter power spectrum.
The small-scale behaviour of the power spectrum response to baryons is very sensitive to the stellar density profiles and as a result we see a large variation between the different simulation predictions in Fig. \[fig:power\_ratio\_sims\]. As is shown by @VanDaalen2019, the small-scale power turnover in the simulations depends strongly on the resolution and subgrid physics of the simulation. We mentioned earlier that our model is fixed at these scales by the best-fit beta profiles to the X-ray observations and the fixed stellar component.
Recently, @VanDaalen2019 analyzed 92 hydrodynamical simulations, including all the ones shown in Fig. \[fig:power\_ratio\_sims\], and showed that there is a strong correlation between the total power suppression at a fixed scale $k \lesssim \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$ and the baryon fraction at $r{\ensuremath{_{\mathrm{500c}}}}$ of haloes with $m{\ensuremath{_{\mathrm{500c}}}} = \SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}}$. We investigate the same relation with our model. We show the different relations that we assume for the gas fraction $f{\ensuremath{_{\mathrm{gas,500c}}}}(m{\ensuremath{_{\mathrm{500c,obs}}}})$ in Fig. \[fig:fgas\_vs\_mt\]. For these relations we assume the best-fit value $\alpha=1.35$ from our fit to the observed gas fractions in Eq. , but we vary the turnover mass from its best-fit value of $\log_{10}m{\ensuremath{_{\mathrm{t}}}}/(\si{{\si{\h^{-1}.{\si{\Msun}}}}})=13.94$. Thus, we can capture a large range of possible gas fractions at $r{\ensuremath{_{\mathrm{500c,obs}}}}$, allowing us to encompass both the observed and the simulated gas fractions of $m{\ensuremath{_{\mathrm{500c,obs}}}}=\SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}}$ haloes. For all these relations we then compute the power spectrum response due to the inclusion of baryons at the fixed scale $k=\SI{0.5}{{\si{\h.\mega\parsec^{-1}}}}$. We show the power suppression at this scale as a function of the halo baryon fraction in $m{\ensuremath{_{\mathrm{500c,obs}}}}=\SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}}$ haloes in Fig. \[fig:power\_vs\_fbar\]. Similarly to @VanDaalen2019, we find that higher baryon fractions at fixed halo mass result in smaller power suppression at fixed scale. In the [$\mathtt{nocb}$]{} ([$\mathtt{cb}$]{}) case, the model with $\gamma_0=1.125$ ($\gamma_0=3$) most closely tracks the prediction from the hydrodynamical simulations. However, since our model has complete freedom for the gas density profile in the halo outskirts, the range of possible power suppression is much larger than that found in the simulations analyzed by @VanDaalen2019. The matter distribution in simulations is constrained by the subgrid physics that is assumed. Hence, relying only on simulation predictions might result in an overly constrained and model-dependent parameter space, since other subgrid recipes might result in differences in the matter distribution at large scales.
We conclude that the total baryon fraction of massive haloes is of crucial importance to the baryonic suppression of the power spectrum. Our model and hydrodynamical simulations that reproduce the cluster gas fractions are in general agreement about the total amount of suppression at scales $k \lesssim \SI{5}{{\si{\h.\mega\parsec^{-1}}}}$, with the exact amplitude depending on the details of the missing baryon distribution and varying by $\approx \pm \SI{5}{\percent}$ around our fiducial model. Observations of the total baryonic mass for a large sample of groups and clusters would provide a powerful constraint on the effects of baryons on the matter power spectrum, provided we are able to reliably measure the cluster masses. Cluster gas masses can be determined with X-ray observations and their outskirts can be probed with SZ measurements. Groups are subject to a significant Malmquist bias in the X-ray regime and SZ measurements from large surveys like Planck [@PlanckXXVII2016], the Atacama Cosmology Telescope [ACTPol, @Hilton2017], and the South Pole Telescope [SPT, @Bleem2015] generally do not reach a high enough Signal-to-Noise ratio (SNR) to reliably measure the hot gas properties of group-mass haloes. Constraining the total baryon fraction of these haloes is thus challenging. However, progress could be made by adopting cross-correlation approaches between SZ maps and large redshift surveys as in @Lim2018. Finally, accurately determining the baryon fraction relies on accurate halo mass determinations for the observed systems. Halo masses can be determined from scaling relations between observed properties (e.g. the hot gas mass, the X-ray temperature, or the X-ray luminosity) and the total halo mass. However, these relations need to be calibrated to a direct measurement of the halo mass through e.g. a weak lensing total mass profile. We will investigate the influence of a hydrostatic bias in the halo mass determination in § \[sec:results\_bias\].
Influence of halo mass correction due to baryonic processes {#sec:results_hmf}
-----------------------------------------------------------
![The ratio of the enclosed observed halo mass to the dark matter only equivalent mass at the fixed radius $r{\ensuremath{_{\mathrm{200c,dmo}}}}$ as a function of $m{\ensuremath{_{\mathrm{200c,dmo}}}}$. The shaded red region shows the spread in our models for all values of $\gamma_0$, i.e. the extrapolated power-law slope of the hot gas density profiles between $r{\ensuremath{_{\mathrm{500c,obs}}}}$ and $r{\ensuremath{_{\mathrm{200m,obs}}}}$, with red lines indicating $\gamma_0=0$ (red, dashed line) and $\gamma_0=3$ (red, solid line). The thin, black, dotted line indicates the ratio $1 - f{\ensuremath{_{\mathrm{b}}}}$ that our model converges to when the halo baryon fraction reaches 0. The mass ratios at fixed radius $r{\ensuremath{_{\mathrm{200c,dmo}}}}$ converge towards high halo masses since not all values of $\gamma$ are allowed for massive haloes. For low masses, the ratios converge because the stellar component is held fixed and the gas fractions are low. The thick, blue, dash-dotted line shows the same relation at fixed radius $r{\ensuremath{_{\mathrm{200c,dmo}}}}$ in the (cosmo-)[OWLS]{} AGN simulation [@Velliscig2014]. The thin, blue, dash-dotted line shows the simulation relation corrected for changes in the dark matter mass profiles at $r{\ensuremath{_{\mathrm{200c,dmo}}}}$ with respect to the DMO equivalent haloes, since our model assumes that baryons do not affect the dark matter profile. The remaining difference in the mass ratio is due to differing baryon fractions between our model and the simulations.[]{data-label="fig:mobs_r200c_dmo_m200c_dmo_ratio"}](figures/plaw_var_gamma_m_obs_r200c_dmo_vs_m200c_dmo){width="\columnwidth"}
{width="\textwidth"}
Since halo abundances are generally obtained from N-body simulations, it is crucial that we are able to correctly link observed haloes to their dark matter only equivalents. However, astrophysical feedback processes result in the ejection of gas and, consequently, a modification of the halo profile and the halo mass $m{\ensuremath{_{\mathrm{200m}}}}$ [e.g. @Sawala2013; @Velliscig2014; @Schaller2015]. Thus, not accounting for the change in halo mass due to baryonic feedback would result in the wrong relation between halo density profiles and halo abundances in our model. Generally, feedback results in lower extrapolated halo masses $m{\ensuremath{_{\mathrm{200m,obs}}}}$ for the observed haloes than the DMO equivalent halo masses $m{\ensuremath{_{\mathrm{200m,dmo}}}}$. Thus, using the observed mass instead of the DMO equivalent mass in the halo mass function would result in an overprediction of the abundance of the observed halo since $n(m)$ decreases with increasing halo mass.
We described how we link $m{\ensuremath{_{\mathrm{200m,obs}}}}$ to $m{\ensuremath{_{\mathrm{200m,dmo}}}}$ in § \[sec:hm\_modifications\]. We remind the reader that we assume that baryons do not significantly alter the distribution of dark matter. Thus, the dark matter component of the observed halo has the same scale radius as its DMO equivalent and a mass that is a factor $1-{\ensuremath{\Omega_\mathrm{b}}}/{\ensuremath{\Omega_\mathrm{m}}}$ lower. The baryonic component of the observed halo is determined by the observations and our different extrapolations for $r>r{\ensuremath{_{\mathrm{500c,obs}}}}$. Then, from the total and rescaled DM density profiles of the observed halo, we can determine the masses $m{\ensuremath{_{\mathrm{200m,obs}}}}$ and $m{\ensuremath{_{\mathrm{200m,dmo}}}}$, respectively. These two masses will differ because the baryons do not follow the dark matter. The haloes have the abundance $n(m{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{200m,obs}}}}))$ for which we use the halo mass function determined by @Tinker2008. In this section, we test how this correction, i.e. using $n(m{\ensuremath{_{\mathrm{200m,dmo}}}}(m{\ensuremath{_{\mathrm{200m,obs}}}}))$ instead of $n(m{\ensuremath{_{\mathrm{200m,obs}}}})$, modifies our results.
We show the ratio of the observed halo mass to the DMO equivalent halo mass at fixed radius $r{\ensuremath{_{\mathrm{200c,dmo}}}}$ in Fig. \[fig:mobs\_r200c\_dmo\_m200c\_dmo\_ratio\]. This ratio does not depend on the model type, i.e. [$\mathtt{cb}$]{} or [$\mathtt{nocb}$]{}, since their density profiles are the same for $r < r{\ensuremath{_{\mathrm{200m,obs}}}}$. We indicate the range spanned by our models with $0 \leq \gamma_0 \leq 3$ by the red shaded region. They converge at the high-mass end because not all slopes $\gamma$ are allowed for high-mass haloes, as shown in § \[sec:components\]. At the low-mass end, our models converge because the stellar component is fixed and hence does not depend on $\gamma_0$, and the gas fractions approach 0. The thin, black, dotted line indicates the ratio $1 - f{\ensuremath{_{\mathrm{b}}}}$ that our model converges to when the halo baryon fraction reaches 0.
We also show the same relation found in the [OWLS]{} AGN [low-mass haloes, @Schaye2010] and [cosmo-OWLS]{} [high-mass haloes, @LeBrun2014a] simulations from @Velliscig2014. There are systematic differences between the predictions from the simulations and our model. These differences occur for two reasons. First, our assumption that the baryons do not alter the distribution of the dark matter with respect to the DMO equivalent halo, does not hold in detail. @Velliscig2014 show that at the fixed radius $r{\ensuremath{_{\mathrm{200c,dmo}}}}$ there is a difference of up to $\SI{4}{\percent}$ between the dark matter mass of the observed halo and the dark matter mass of the DMO equivalent halo, rescaled to account for the cosmic baryon fraction. The dark matter in low-mass haloes expands due to feedback expelling baryons outside $r{\ensuremath{_{\mathrm{200c,dmo}}}}$. In the highest-mass haloes, feedback is less efficient and the dark matter contracts in response to the cooling baryons. The thin, blue, dash-dotted line shows the relation in [OWLS]{} AGN when forcing the dark matter mass of the halo to equal the rescaled DMO equivalent halo mass. Hence, the contraction due to the presence of baryons of the dark matter component for high-mass haloes explains the difference between our model and the simulations. For the low-mass end, the expansion of the dark matter component is not sufficient to explain all of the difference. The remaining discrepancy results from the higher baryons fractions in our model compared to the simulations.
We will neglect the response of the DM to the redistribution of baryons throughout the rest of the paper. We have checked that scaling the halo density profiles of the DMO equivalent haloes to match the mass ratios from the [OWLS]{} AGN simulation only affects our predictions of the power suppression at the $\approx \SI{1}{\percent}$ level at our scales of interest (i.e. $k < \SI{10}{{\si{\h.\mega\parsec^{-1}}}}$). However, even this small correction is an upper limit because we have assumed a fixed ratio between the DM and rescaled DMO density profiles that exceeds the correction for the cosmic baryon fraction. Hence, even at large distances $r \gg r{\ensuremath{_{\mathrm{200c,dmo}}}}$ the mass ratio between the halo and its DMO equivalent does not converge, whereas the mass difference between hydrodynamical haloes and their DMO equivalents eventually decreases to 0 [see e.g. @Velliscig2014; @VanDaalen2014a]. We find such a small effect because the low-mass haloes, whose mass ratio differs the most between our model and the [OWLS]{} AGN simulation, only have a small effect on the total power at large scales, as we will show in § \[sec:results\_masses\].
We show how the correction of the halo abundance for the change in halo mass due to baryonic processes affects the predicted power spectrum response to baryons in Fig. \[fig:power\_ratio\_dndm\]. In the top row, we show the power spectrum response for both the [$\mathtt{nocb}$]{}(left panel) and [$\mathtt{cb}$]{} models (right panel) with (our fiducial models, solid lines) and without (dashed lines) the halo mass correction. When not correcting the halo abundance for the change in halo mass (i.e. when using $n(m{\ensuremath{_{\mathrm{200m,obs}}}})$), we actually find an increase in power with respect to the DMO model at scales $k \lesssim \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$, i.e. $R_i(k) > 1$, for both the [$\mathtt{nocb}$]{}and [$\mathtt{cb}$]{} models, since the inferred abundances for observed haloes with masses $m{\ensuremath{_{\mathrm{200m,obs}}}} \lesssim \SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}}$ are too high. At these scales, the Fourier profiles become constant in the 1h term, i.e. $\hat{\rho}(k|m{\ensuremath{_{\mathrm{h}}}}) \to m{\ensuremath{_{\mathrm{h}}}}$ in Eq. \[eq:p\_1h\], and the power spectrum behaviour is thus dictated entirely by the halo abundance. Hence, the power suppression that we find in our fiducial models at these scales is the consequence of correcting the DMO equivalent halo masses to account for the ejection of matter due to feedback. We stress that our implementation of this effect is purely empirical and does not rely on any assumptions about the physics involved in baryonic feedback processes.
In the bottom row of Fig. \[fig:power\_ratio\_dndm\], we show the ratio between the power spectrum response to the inclusion of baryons with and without the DMO equivalent halo mass correction. The correction is most significant for the steepest extrapolated density profile slopes, i.e. the highest values of $\gamma_0$, for which we see the smallest ratios. For $\gamma_0 = 3$, even the most massive haloes do not reach the cosmic baryon fraction inside $r{\ensuremath{_{\mathrm{200m,obs}}}}$ (see Fig. \[fig:fb\_vs\_m200m\_obs\]), and, hence, even their abundances would be calculated wrongly if the observed total mass was used, instead of the rescaled, observed DM mass, to compute the halo abundance. In the case of the [$\mathtt{cb}$]{} models, there is an extra increase in power for scales $k \lesssim \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$ due to the more extended baryon distribution.
It is striking how the halo mass correction modifies the suppression of power in the way required to encompass the simulation predictions at large scales. The correction to the DMO equivalent halo masses is necessary for this match.
Contribution of different halo masses {#sec:results_masses}
-------------------------------------
![The contribution of the 1-halo term for different halo mass ranges to the total power spectrum at all scales for our fiducial models. The [$\mathtt{nocb}$]{} (solid lines) and [$\mathtt{cb}$]{} (dashed lines) models are shown and the mass ranges are indicated by the colours. We also show the contribution of the 2-halo term, i.e. the linear power spectrum (black lines). The stellar contribution (1h term and all cross-correlations between matter and stars, connected stars) is also included, but only for the [$\mathtt{nocb}$]{} case, since the [$\mathtt{cb}$]{} case traces the [$\mathtt{nocb}$]{} lines. The red, lightly-shaded region for $k > \SI{10}{{\si{\h.\mega\parsec^{-1}}}}$ indicates the scales where our model is is not a good indicator of the uncertainty because the stellar component is not varied. The 1h term dominates the total power for $k \gtrsim \SI{0.5}{{\si{\h.\mega\parsec^{-1}}}}$. For the scales of interest here ($k \lesssim \SI{5}{{\si{\h.\mega\parsec^{-1}}}}$), most of the power is contributed by groups and clusters with $\SI{e13}{{\si{\h^{-1}.{\si{\Msun}}}}} \leq m{\ensuremath{_{\mathrm{500c,obs}}}} \leq \SI{e15}{{\si{\h^{-1}.{\si{\Msun}}}}}$. For the [$\mathtt{cb}$]{} models, low-mass ($\approx \SI{e13}{{\si{\h^{-1}.{\si{\Msun}}}}}$) haloes contribute more and clusters ($>\SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}}$) less compared with the [$\mathtt{nocb}$]{} models. The total stellar contribution to the power response is $\lesssim \SI{1}{\percent}$ for all scales $k \lesssim \SI{5}{{\si{\h.\mega\parsec^{-1}}}}$ and only exceeds $\SI{2}{\percent}$ for $k \gtrsim \SI{10}{{\si{\h.\mega\parsec^{-1}}}}$.[]{data-label="fig:power_mass_contrib"}](figures/plaw_var_m_bins_power_1h_vs_dmo_stars_fb+nofb){width="\columnwidth"}
To determine the observables that best constrain the matter power spectrum at different scales, it is important to know which haloes dominate the suppression of power at those scales. The dominant haloes will be determined by the interplay between the total mass of the halo and its abundance.
The halo model linearly adds the contributions from haloes of all masses to the power at each scale. We show the contributions for five decades in mass in Fig. \[fig:power\_mass\_contrib\] for our fiducial model in the [$\mathtt{nocb}$]{} and [$\mathtt{cb}$]{} cases. We integrate the 1-halo term, Eq. \[eq:p\_1h\], over 5 different decades in mass, spanning $\SI{e10}{{\si{\h^{-1}.{\si{\Msun}}}}} < m{\ensuremath{_{\mathrm{500c,obs}}}} < \SI{e15}{{\si{\h^{-1}.{\si{\Msun}}}}}$, and then divide each by the DMO power spectrum, showing the contribution of different halo masses to the power spectrum. We also show the contribution of the 2-halo term, i.e. the linear power spectrum. The mass dependence of our model comes entirely from the 1h term, which dominates the total power for $k \gtrsim \SI{0.5}{{\si{\h.\mega\parsec^{-1}}}}$.
We want to quantify the stellar contribution to the power spectrum to gauge whether we are allowed to neglect the ISM component of the gas. As explained in the beginning of § \[sec:hm\_profiles\], we can safely neglect the ISM if the stellar component contributes negligibly to the total power at our scales of interest ($k \lesssim \SI{5}{{\si{\h.\mega\parsec^{-1}}}}$). To this end, we also include the 1h term for the stellar component with all cross-correlations $|\rho_\star(k|m{\ensuremath{_{\mathrm{h}}}})\rho_i(k|m{\ensuremath{_{\mathrm{h}}}})|$ in Eq. with $i \in \{\mathrm{dm,gas}\}$. We only show this contribution for the [$\mathtt{nocb}$]{} case, since the [$\mathtt{cb}$]{} results are nearly identical. Fig. \[fig:power\_mass\_contrib\] clearly shows that the stellar component contributes negligibly to the power for all scales $k\lesssim \SI{5}{{\si{\h.\mega\parsec^{-1}}}}$ and, hence, we are justified in neglecting the contributions of the ISM to the gas component. However, for making predictions at the $\SI{1}{\percent}$ level on small scales ($k \gtrsim \SI{5}{{\si{\h.\mega\parsec^{-1}}}}$), the ISM and stellar components will become important and will need to be modelled more accurately.
At all scales $k \lesssim \SI{10}{{\si{\h.\mega\parsec^{-1}}}}$, the total power is dominated by groups ($\SI{e13}{{\si{\h^{-1}.{\si{\Msun}}}}} \leq m{\ensuremath{_{\mathrm{500c,obs}}}} \leq \SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}}$) and clusters ($\SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}} \leq m{\ensuremath{_{\mathrm{500c,obs}}}} \leq \SI{e15}{{\si{\h^{-1}.{\si{\Msun}}}}}$) of galaxies, with groups providing a similar or greater contribution than clusters. Similar results have been found in DMO simulations by @VanDaalen2015. Group-mass haloes have the largest range in possible baryon fractions in our model, depending on the slope $\gamma_0$ of the gas density profile for $r>r{\ensuremath{_{\mathrm{500c,obs}}}}$. We conclude that groups are crucial contributors to the power at large scales and thus measuring the baryon content of group-mass haloes will provide the main observational constraint on predictions of the baryonic suppression of the matter power spectrum.
Influence of density profile fitting parameters {#sec:results_prms}
-----------------------------------------------
{width="\textwidth"}
So far, we have shown the impact of the baryon distribution and haloes of different masses on the matter power spectrum for different wavenumbers when assuming our model that best fits the observations. However, since we assume the median values for the parameters $r{\ensuremath{_{\mathrm{c}}}}$ and $\beta$, and the median relation $f{\ensuremath{_{\mathrm{gas,500c}}}}-m{\ensuremath{_{\mathrm{500c,obs}}}}$ for the observed hot gas density profiles and there is a significant scatter around these medians, it is important to see how sensitive our predictions are to variations in the parameter values. In this section, we investigate the isolated effect of each observational parameter on the predicted matter power spectrum.
We remind the reader of the beta profile in Eq. \[eq:beta\_gas\] and the best fits for its parameters determined from the observations in Figs. \[fig:obs\_fgas\], \[fig:obs\_rc\_fit\], and \[fig:obs\_beta\_fit\]. In those figures, we indicated the median relations, which are used in our model, and the $15{\ensuremath{^{\mathrm{th}}}}$ and $85{\ensuremath{^{\mathrm{th}}}}$ percentiles of the observed values. We will test the model response to variations in the hot gas observations by varying each of the best-fit parameters between its $15{\ensuremath{^{\mathrm{th}}}}$ and $85{\ensuremath{^{\mathrm{th}}}}$ percentiles while keeping all other parameters fixed.
We show the result of these parameter variations for our fiducial model ($\gamma_0=2.14$) in the [$\mathtt{nocb}$]{} and [$\mathtt{cb}$]{} cases in Fig. \[fig:power\_var\_prms\]. We indicate the $15{\ensuremath{^{\mathrm{th}}}}$ ($85{\ensuremath{^{\mathrm{th}}}}$) percentile envelope with a dashed (solid), coloured line and shade the region enclosed by these percentiles. For both the [$\mathtt{cb}$]{} and [$\mathtt{nocb}$]{} cases, the parameters $\beta$ and $f{\ensuremath{_{\mathrm{gas,500c}}}}$ are the most important at large scales. Flatter outer slopes for the hot gas density profile, i.e. smaller values of $\beta$, will result in more baryons out to $r{\ensuremath{_{\mathrm{200m,obs}}}}$, yielding a smaller suppression of power on large scales. Higher gas fractions within $r{\ensuremath{_{\mathrm{500c,obs}}}}$ will result in haloes that are more massive and contain more of the baryons, again yielding a smaller suppression of power on large scales. The core radius $r{\ensuremath{_{\mathrm{c}}}}$ is the least important parameter. Increasing the size of the core requires a lower density in the core to reach the same gas fraction at $r{\ensuremath{_{\mathrm{500c,obs}}}}$ and yields more baryons in the halo outskirts. Hence, we see more power at large scales and less power on small scales when increasing the value of $r{\ensuremath{_{\mathrm{c}}}}$ similarly to decreasing the value $\beta$. However, the core is relatively close to the cluster center and thus has no impact on the matter distribution at large scales.
There is an important difference between the [$\mathtt{nocb}$]{} and [$\mathtt{cb}$]{} cases, however. The fit parameters only start having an effect on the power suppression at scales $k \gtrsim \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$ in the [$\mathtt{cb}$]{} case, whereas in the [$\mathtt{nocb}$]{} case they already start mattering around $k \approx \SI{0.3}{{\si{\h.\mega\parsec^{-1}}}}$. If all baryons are accounted for in the halo outskirts, as in the [$\mathtt{cb}$]{} case, the details of the baryon distribution do not matter for the power at the largest scales, since here the 1h term is fully determined by the mass inside $r{\ensuremath{_{\mathrm{h}}}}$, which does not change for different values of $\beta$ and $f{\ensuremath{_{\mathrm{gas,500c}}}}$. The [$\mathtt{nocb}$]{} model is a lot more sensitive to the baryon distribution within the halo, since depending on the value of $\beta$, or how many baryons can already be accounted for inside $r{\ensuremath{_{\mathrm{500c,obs}}}}$, the haloes can have large variations in mass $m{\ensuremath{_{\mathrm{200m,obs}}}}$.
In conclusion, the most important parameter to pin down is the gas fraction of the halo, as we already concluded in § \[sec:results\_outer\] and § \[sec:results\_masses\]. It has the largest effect on all scales in both the [$\mathtt{nocb}$]{} and [$\mathtt{cb}$]{} cases and varying its value within the observed scatter results in a $\approx \pm \SI{5}{\percent}$ variation around the power spectrum response predicted by our fiducial model. At the scales of interest to future surveys, the effect of $\beta$ is of similar amplitude. However, this parameter will be harder to constrain observationally than the gas content of the halo, especially for group-mass haloes, because X-ray observations cannot provide an unbiased sample and SZ observations cannot observe the density profile directly.
Influence of hydrostatic bias {#sec:results_bias}
-----------------------------
{width="\textwidth"}
All of our results so far assumed gas fractions based on halo masses derived from X-ray observations under the assumption of hydrostatic equilibrium (HE) and pure thermal pressure. Under the HE assumption non-thermal pressure and large-scale gas motions are neglected in the Euler equation [see e.g. the discussion in §2.3 of @Pratt2019]. However, in massive systems in the process of assembly, there is no a priori reason to assume that simplifying assumption to hold. We expect the most massive clusters to depart from HE, since we know from the hierarchical structure formation paradigm, that they have only recently formed. Moreover, the pressure can have a non-negligible contribution from non-thermal sources such as turbulence [@Eckert2018].
Investigating the relation between hydrostatically derived halo masses and the true halo mass requires hydrodynamical simulations [e.g. @Nagai2007; @Rasia2012; @Biffi2016; @Brun2017; @McCarthy2017; @Henson2017] or weak gravitational lensing observations [@Mahdavi2013; @VonderLinden2014; @Hoekstra2015; @Medezinski2018]. In both cases, the pressure profile of the halo is derived from observations of the hot gas. Under the assumption of spherical symmetry and HE, this pressure profile is then straightforwardly related to the total mass profile of the halo. Subsequently, this hydrostatic halo mass can be compared to an unbiased estimate of the halo mass, i.e. the true mass in hydrodynamical simulations, or the mass derived from weak lensing observations.
The picture arising from both simulations and observations is that hydrostatic masses, $m{\ensuremath{_{\mathrm{HE}}}}$, are generally biased low with respect to the weak lensing or true halo mass, $m{\ensuremath{_{\mathrm{WL}}}}$, with $m{\ensuremath{_{\mathrm{HE}}}}/m{\ensuremath{_{\mathrm{WL}}}} = 1-b \simeq \numrange{0.6}{0.9}$ (e.g. [@Mahdavi2013]; [@VonderLinden2014]; [@Hoekstra2015]; [@Brun2017]; [@Henson2017]; [@Medezinski2018]). The detailed behaviour of this bias depends on the deprojected temperature and density profiles, with more spherical systems being less biased.
Correcting for the observationally determined bias would result in higher halo masses and, consequently, a shift in the gas fractions away from the assumed best-fit $f{\ensuremath{_{\mathrm{gas,500c}}}}-m{\ensuremath{_{\mathrm{500c,obs}}}}$ relation. We argued previously that this is the most relevant observable to determine the suppression of power at scales $k \lesssim \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$. Thus, it is important to investigate how the HE assumption affects our predictions. Previously, @Schneider2018 have shown for three different levels of hydrostatic bias ($1 - b \in \{0.71, 0.83, 1\}$) that the predicted power suppression at large scales $k < \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$ can vary by up to $\SI{5}{\percent}$.
Staying in tune with § \[sec:results\_prms\], we adopt a single value for the bias to investigate its influence on our predictions. We will take $1-b=0.7$ which is consistent with both @VonderLinden2014 and @Hoekstra2015. Moreover, although the bias tends to be higher for higher-mass systems because of the presence of cooler gas in their outskirts [@Henson2017], we conservatively adopt this value for all halo masses. Correcting for the bias will influence our model in two ways. First, the inferred gas masses will increase slightly, since the true $r{\ensuremath{_{\mathrm{500c,obs}}}}$ will be larger than the value assumed from the hydrostatic estimate. We thus recompute the gas masses from our best-fit beta models to the observations. Second, the halo mass will increase by the bias factor which will result in new estimates for the gas fractions, which we show as the thin, solid, black line in Fig. \[fig:obs\_fgas\]. We then fit the median $f{\ensuremath{_{\mathrm{gas,500c}}}}-m{\ensuremath{_{\mathrm{500c,obs}}}}$ relation again, assuming Eq. \[eq:fgas\_sigmoid\], resulting in the thin, red, dashed line.
We show the resulting effect on the baryonic suppression of the power spectra in Fig. \[fig:power\_ratio\_debiased\]. The results are similar to varying $f{\ensuremath{_{\mathrm{gas,500c}}}}$ in Fig. \[fig:power\_var\_prms\], since the bias-corrected relation is similar to the 15[$^{\mathrm{th}}$]{} percentile $f{\ensuremath{_{\mathrm{gas,500c}}}}-m{\ensuremath{_{\mathrm{500c,obs}}}}$ relation, but with a more dramatic suppression of the baryonic mass for clusters and hence more suppression of the power at large scales. In the bottom panels of Fig. \[fig:power\_ratio\_debiased\], we find a maximum extra suppression of $\approx \SI{4}{\percent}$ due to the hydrostatic bias at $k = \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$ in both the [$\mathtt{nocb}$]{} and [$\mathtt{cb}$]{} cases, which is consistent with the findings of @Schneider2018. The magnitude of the suppression is lower for lower values of $\gamma_0$ since these models compensate for the lower baryon fraction within $r{\ensuremath{_{\mathrm{500c,obs}}}}$ by adding baryons between $r{\ensuremath{_{\mathrm{500c,obs}}}}$ and $r{\ensuremath{_{\mathrm{h}}}}$.
Accounting for the bias breaks the overall agreement with the simulations on large scales for the models with high values of $\gamma_0$. However, in [<span style="font-variant:small-caps;">bahamas</span>]{} and [OWLS]{} AGN, a hydrostatic bias of $1-b = 0.84$ and $1-b=0.8$ is found, respectively, for groups and clusters [@McCarthy2017; @LeBrun2014a]. When we assume $1-b=0.8$, we find a maximum extra suppression of $\approx \SI{2}{\percent}$ at $k = \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$ instead of $\approx \SI{4}{\percent}$. At other scales the effect of the hydrostatic bias is similarly reduced.
In conclusion, it is crucial to obtain robust constraints on the hydrostatic bias of groups and clusters of galaxies. Current measurements of this bias suggest that hydrostatic halo masses underestimate the true masses and that this bias results in a downward shift of the cluster gas fractions that is more severe than the observational scatter in the relation. Because the shift affects cluster-mass haloes, it results in an additional power suppression of up to $\approx \SI{4}{\percent}$ at $k = \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$, depending on how our model distributes the outer baryons. There are ways of measuring halo masses that do not rely on making the hydrostatic assumption, such as weak lensing observations, but these also carry their own intrinsic biases [@Henson2017]. Making mock observations in simulations allows us to characterize these separate biases [e.g. @Henson2017; @Brun2017], but the simulations still do not make a full like-for-like comparison with the observations. Finally, joint constraints on X-ray, SZ, and weak lensing halo mass scaling relations, including possible biases, as was done in @Bocquet2019, could provide more robust halo mass estimates.
Discussion {#sec:discussion}
==========
We have presented an observationally constrained halo model to estimate the power suppression due to baryons without any reliance on subgrid recipes for the unresolved physics of baryons in hydrodynamical simulations. We reiterate that our main goal is not to provide the most accurate predictions of the matter power spectrum, but to investigate the possibility of using observations to constrain it. The fact that the clustering of matched haloes does not change between DMO and hydrodynamical simulations [@VanDaalen2014a] implies that changes in the density profiles due to the baryons determine the change of the matter power spectrum. Hence, even though the halo model does not accurately predict the matter power spectrum, it can accurately predict the relative effect of baryonic processes on the power spectrum. The overall agreement between our model and hydrodynamical simulations that reproduce the observed distribution of baryons in groups and clusters, confirms that our model captures the first-order impact of baryons simply by reproducing the observed baryon content for groups and clusters.
In conclusion, the main strength of the model is that it allows us to quantify the impact of different halo masses, different halo baryon density distributions and observational biases and uncertainties on the baryonic suppression of the matter power spectrum without any necessity for uncertain subgrid recipes for feedback processes. This in turn allows us to provide a less model-dependent estimate of the range of possible baryonic suppression and to predict which observations would provide the strongest constraints on the matter power spectrum.
There are other models in the literature that aim to model the effect of baryon physics on the matter power spectrum. `HMcode` by @Mead2015 is widely used to include baryon effects in weak lensing analyses. Although `HMcode` is also based on the halo model, its aim is different from ours. @Mead2015 modify the dark matter halo profiles and subsequently fit the parameters of their halo model to hydrodynamical simulations to provide predictions for the baryonic response of the power spectrum that are accurate at the $\sim \SI{5}{\percent}$ level for $k \lesssim \SI{5}{{\si{\h.\mega\parsec^{-1}}}}$ with 2 free parameters related to the baryonic feedback (for a similar approach, see [@Semboloni2013]). These feedback parameters can then be jointly constrained with the cosmology using cosmic shear data. However, even though the modifications to the dark matter profile are phenomenologically inspired, there is no guarantee that the final best-fit parameters correspond to the actual physical state of the haloes. We obtain similar accuracy in the predicted power response when viewing $\gamma_0$ as a fitting parameter and comparing to hydrodynamical simulations. However, in our case, fitting $\gamma_0$ preserves the agreement with observations. Indeed, the most important difference between our approach and that of @Mead2015 is that we fit to observations instead of simulations.
The investigation of @Schneider2018 most closely matches our goal. @Schneider2015 and @Schneider2018 developed a *baryon correction model* to investigate the influence of baryon physics on the matter power spectrum. Their model shifts particles in DMO simulations according to the physical expectations from baryonic feedback processes. Since the model only relies on DMO simulations, it is not as computationally expensive as models that require hydrodynamical simulations to calibrate their predictions. Our simple analytic halo model is cheaper still to run, but it only results in a statistical description of the matter distribution, whereas the *baryon correction model* predicts the total matter density field for the particular realization that was simulated. Because our model combines the universal DMO halo mass function with observed density profiles, it can easily be applied to a wide variety of cosmologies without having to run an expensive grid of DMO simulations.
In the baryon correction model, the link to observations can also be made, making it similar to our approach. @Schneider2018 fit a mass-dependent slope of the gas profile, $\beta$ (note that their slope is not defined the same way as our slope $\beta$), and the maximum gas ejection radius, $\theta{\ensuremath{_{\mathrm{ej}}}}$, to the observed hot gas profiles of the XXL sample of @Eckert2016 and a compendium of X-ray gas fraction measurements. They also include a stellar component that is fit to abundance matching results, similar to our `iHOD` implementation. They show that their model can reproduce the observed relations as well as hydrodynamical simulations when fit to their gas fractions. @Schneider2018 use the observations to set a maximum range on their model parameters to then predict both the matter power spectrum and the shear correlation function. Our work, on the other hand, focusses on the impact of isolated properties of the baryon distribution on the power spectrum. Similarly to @Schneider2018, we find that the power suppression on large scales is very sensitive to the baryon distribution in the outskirts of the halo. However, our model allows us to clearly show that the halo baryon fractions are the crucial ingredient in setting the total power suppression at large scales, $k \lesssim \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$. Also similarly to @Schneider2018, we find that the hydrostatic mass bias significantly affects the total power suppression at large scales.
So far, we have not included redshift evolution. @Schneider2018 have found that the most important evolution of clusters and groups in cosmological simulations stems from the change in their abundance due to the evolution of the halo mass function in time, and not due to the change of the density profiles with time. This evolution can be readily implemented into our halo model.
Summary and conclusions {#sec:summary_conclusions}
=======================
Future weak lensing surveys will be limited in their accuracy by how well we can predict the matter power spectrum on small scales [e.g. @Semboloni2011; @Copeland2018; @Huang2019]. These scales contain a wealth of information about the underlying cosmology of our Universe, but the interpretation of the signal is complicated by baryon effects. Our current theoretical understanding of the impact of baryons on the matter power spectrum stems from hydrodynamical simulations that employ uncertain subgrid recipes to model astrophysical feedback processes. This uncertainty can be bypassed by adopting an observational approach to link the observed distribution of matter to the matter power spectrum.
We have provided a detailed study of the constraints that current observations of groups and clusters of galaxies impose on the possible influence of the baryon distribution on the matter power spectrum. We introduced a modified halo model that includes dark matter, hot gas, and stellar components. We fit the hot gas to X-ray observations of clusters of galaxies and we assumed different distributions for the missing baryons outside $r{\ensuremath{_{\mathrm{500c,obs}}}}$, the maximum radius probed by X-ray observations of the hot gas distribution. Subsequently, we quantified (i) how the outer, unobserved baryon distribution modifies the matter power spectrum (Fig. \[fig:power\_ratio\]). We also investigated (ii) how the change in halo mass due to baryonic effects can be incorporated into the halo model (Fig. \[fig:power\_ratio\_dndm\]). We showed (iii) the contributions to the matter power spectrum of haloes of different masses at different spatial scales (Fig. \[fig:power\_mass\_contrib\]), (iv) the influence of varying the individual best-fit parameters to the observed density profiles within their allowed range (Fig. \[fig:power\_var\_prms\]), and (v) the influence of a hydrostatic mass bias on the matter power spectrum (Fig. \[fig:power\_ratio\_debiased\]).
Our model has one free parameter, $\gamma_0$, related to the slope of the hot gas density profile for $r{\ensuremath{_{\mathrm{500c,obs}}}} \leq r \leq r{\ensuremath{_{\mathrm{200m,obs}}}}$, where observational constraints are very poor. We considered two extreme cases for the baryons. First, the [$\mathtt{nocb}$]{} models assume that haloes of size $r{\ensuremath{_{\mathrm{200m,obs}}}}$ do not necessarily reach the cosmic baryon fraction at this radius and that any missing baryons are located at such large distances that they only contribute to the 2-halo term. Second, in the [$\mathtt{cb}$]{} models the missing baryons inside $r{\ensuremath{_{\mathrm{200m,obs}}}}$ are distributed with an assumed uniform density profile outside this radius until the cosmic baryon fraction is reached. These cases provide, respectively, the maximum and minimum power suppression of large-scale power due to baryonic effects.
All of our observationally constrained models predict a significant amount of suppression on the scales of interest to future surveys ($\num{0.2} \lesssim k / (\si{{\si{\h.\mega\parsec^{-1}}}}) \lesssim \num{5}$). We find a total suppression of $\SI{1}{\percent}$ ($\SI{5}{\percent}$) on scales $\SIrange{0.2}{0.9}{{\si{\h.\mega\parsec^{-1}}}}$ ($\SIrange{0.5}{2}{{\si{\h.\mega\parsec^{-1}}}}$) in the [$\mathtt{nocb}$]{} case and on scales $\SIrange{0.5}{1}{{\si{\h.\mega\parsec^{-1}}}}$ ($\SIrange{1}{2}{{\si{\h.\mega\parsec^{-1}}}}$) in the [$\mathtt{cb}$]{} case for values $\gamma_0 = \numrange{3}{0}$ (Fig. \[fig:power\_ratio\]), where $\gamma_0$ is the low-mass limit of the power-law slope $\gamma$ between $r{\ensuremath{_{\mathrm{500c,obs}}}}$ and $r{\ensuremath{_{\mathrm{200m,obs}}}}$, i.e. $\gamma_0 = \gamma(m{\ensuremath{_{\mathrm{500c,obs}}}} \to 0)$. This large possible range of scales corresponding to a fixed suppression factor for each case illustrates the importance of the baryon distribution outside $r{\ensuremath{_{\mathrm{500c,obs}}}}$ (which is parameterised by $\gamma_0$) in setting the total power suppression.
We found that massive groups of galaxies ($\SI{e13}{{\si{\h^{-1}.{\si{\Msun}}}}} < m{\ensuremath{_{\mathrm{500c,obs}}}} < \SI{e14}{{\si{\h^{-1}.{\si{\Msun}}}}}$) provide a larger contribution than clusters to the total power at all scales (Fig. \[fig:power\_mass\_contrib\]). This is unfortunate, since we have shown that the baryonic content of group- and cluster-sized haloes, which is set by the observed gas fractions $f{\ensuremath{_{\mathrm{gas,500c}}}}$, determines the large-scale ($k \lesssim \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$) power suppression (Figs. \[fig:power\_vs\_fbar\] and \[fig:power\_var\_prms\]). However, observations of the hot gas content of groups are scarcer than those of clusters and are also subject to a considerable Malmquist bias. Current X-ray telescopes cannot solve this problem, but a combined approach with Sunyaev-Zel’dovich or gravitational lensing observations could provide a larger sample of lower mass objects.
We found that our observationally constrained models only encompass the predictions of hydrodynamical simulations that reproduce the hot gas content of groups and clusters of galaxies (Fig. \[fig:power\_ratio\_sims\]). Thus, we stress the importance of using simulations that reproduce the relevant observations when using such models to predict the baryonic effects on the matter distribution.
We found that accurately measuring the halo masses is of vital importance when trying to place observational constraints on the matter power spectrum. An unrecognized hydrostatic halo mass bias of $1-b=0.7$ would result in an underestimate of the total power suppression by as much as $\SI{4}{\percent}$ at $k = \SI{1}{{\si{\h.\mega\parsec^{-1}}}}$ (Fig. \[fig:power\_ratio\_debiased\]). In addition, it is critical to correct the observed halo masses for the redistribution of baryons when estimating their abundance using halo mass functions based on DM only simulations (Fig. \[fig:power\_ratio\_dndm\]).
All in all, it is encouraging that we are able to quantify the baryonic suppression of the matter power spectrum with a simple, flexible but physical approach such as our modified halo model. Our investigation allows us to predict the observations that will be most constraining for the impact of baryonic effects on the matter power spectrum.
Acknowledgements {#acknowledgements .unnumbered}
================
It is a pleasure to thank Ian McCarthy and Marcel van Daalen for useful discussions and comments. We would also like to thank Ian McCarthy and Dominique Eckert for providing the observational data and the REXCESS team for making their density profiles publically available. Finally, we would like to thank the referee for a constructive report that helped clarify the paper. The authors acknowledge support from: the Netherlands Organisation for Scientific Research (NWO) under grant numbers 639.043.512 (SD, HH) and 639.043.409 (SD, JS).
Influence of the halo mass range {#app:results_mass_range}
================================
In this section, we investigate how our choice of mass grid influences our predictions. We have chosen an equidistant log-grid of halo masses $\SI{e11}{{\si{\h^{-1}.{\si{\Msun}}}}} \leq m{\ensuremath{_{\mathrm{500c,obs}}}} \leq \SI{e15}{{\si{\h^{-1}.{\si{\Msun}}}}}$, sampled with 101 bins. Doubling or halving the number of bins only affects our predictions at the $< \SI{0.1}{\percent}$ level for all $k$. Similarly, increasing the maximum halo mass to $m{\ensuremath{_{\mathrm{500c,max}}}} = \SI{e16}{{\si{\h^{-1}.{\si{\Msun}}}}}$ only results in changes at the $< \SI{0.1}{\percent}$ level for all $k$. The only significant change occurs when decreasing the minimum halo mass to $m{\ensuremath{_{\mathrm{500c,min}}}} = \SI{e6}{{\si{\h^{-1}.{\si{\Msun}}}}}$, but this only affects scales smaller than of interest here. In this case, our baryonic models predict less power compared to the higher minimum mass case, since the low-mass haloes have no stars and gas. Hence, they will always contain less matter than their DMO equivalents and the DMO power will be boosted relative to the baryonic one. However, our predictions only change at the $\SI{1}{\percent}$ level for $k \gtrsim \SI{60}{{\si{\h.\mega\parsec^{-1}}}}$, thus our fiducial mass range is converged for our scales of interest, $k < \SI{10}{{\si{\h.\mega\parsec^{-1}}}}$.
Influence of concentration-mass relation {#app:results_concentration}
========================================
In this section, we investigate how changes in the concentration at fixed halo mass influences our predictions. While the concentration-mass relation does not show a strong mass dependence, the scatter about the median relation is significant [@Jing2000; @Bullock2001; @Duffy2008; @Dutton2014]. To investigate the potential influence of this scatter, we tested how our predictions for the power response due to baryons change when assuming the $c(m)$ relation shifted up and down by its log-normal scatter $\sigma_{\log_{10} c} = 0.15$ [@Duffy2008; @Dutton2014]. Increasing the concentration results in more (less) power at small (large) scales and thus a lower (higher) power suppression. Adopting this extreme shift in the concentration-mass relation results in a maximum variation of $\pm \SI{3}{\percent}$ in suppression at scales $k \lesssim \SI{20}{{\si{\h.\mega\parsec^{-1}}}}$. This variation is smaller than any of the hot gas density profile best-fit parameter variations in § \[sec:results\_prms\].
\[lastpage\]
[^1]: Contact e-mail: <debackere@strw.leidenuniv.nl>
[^2]: <http://www.euclid-ec.org>
[^3]: <http://www.lsst.org/>
[^4]: <http://wfirst.gsfc.nasa.gov>
[^5]: <http://camb.info/>
[^6]: <https://github.com/astroduff/commah>
|
---
abstract: 'We develop an arithmetic analogue of linear partial differential equations in two independent “space-time” variables. The spatial derivative is a Fermat quotient operator, while the time derivative is the usual derivation. This allows us to “flow” integers or, more generally, points on algebraic groups with coordinates in rings with arithmetic flavor. In particular, we show that elliptic curves have certain canonical “flows” on them that are the arithmetic analogues of the heat and wave equations. The same is true for the additive and the multiplicative group.'
address: |
University of New Mexico\
Albuquerque, NM 87131
author:
- 'Alexandru Buium and Santiago R. Simanca'
title: Arithmetic partial differential equations
---
Introduction
============
In this paper, we consider arithmetic partial differential equations in two “space-time” variables, a higher dimensional analogue of the theory of arithmetic ordinary differential equations developed in [@char; @difmod; @book]. In the ordinary case, the rôle of functions of one variable is played by integers, and that of the derivative operator is played by a “Fermat quotient operator” with respect to a fixed prime $p$. Instead, we now take power series in a variable $q$ with integer coefficients as the analogues of functions of two variables, and while maintaining the idea that a Fermat quotient type operator with respect to $p$ is the analogue of the derivative in the “arithmetic direction,” we now add the usual derivative with respect to $q$ to play the rôle of a derivative in the “geometric direction.” This leads to the study of some “arithmetic flows” of remarkable interest.
In the ordinary case [@char; @difmod; @book] the “arithmetic direction” was viewed as a “temporal direction.” In the present paper, the “arithmetic direction” is viewed as a “spatial direction,” and the “geometric direction” is the “time.” Under this interpretation, we will be able to think of points on algebraic varieties with coordinates in number theoretic rings as “functions of space,” and we will be able to flow these points using the geometric parameter $q$, parameter that morally speaking plays the rôle of (the exponential of) time.
We proceed to explain our idea in some detail, and begin by discussing some of the basic aspects of evolution partial differential equations as they appear in classical analysis, discussion that out of necessity will be carried out in a non-rigorous fashion. In particular, the word “function” will be used to refer to functions (or even distributions) belonging to unspecified classes, and we will ignore all questions on convergence, as well as those concerning the proper definition of certain products or convolutions. Instead, we will concentrate exclusively on the formal aspects of the story, and examine only those concepts whose arithmetic analogue will later play a rôle in our study. We will then describe qualitatively what these arithmetic analogues are, and will end the introduction by a presentation of the basic problems and results of our theory.
Evolution equations in analysis
-------------------------------
We denote by ${\mathbb R}_x$ the real line with “space” coordinate $x$ and ${\mathbb R}_t$ the real line with “time” coordinate $t$. We let ${\mathcal F}({\mathbb R}_x)$ be the ring of complex valued functions $f(x)$ on ${\mathbb R}_x$, and we let $\cF({\mathbb R}_x \times {\mathbb R}_t)$ be the ring of complex valued functions $v(x,t)$ on ${\mathbb R}_x \times
{\mathbb R}_t$. Both of these rings are equipped with pointwise addition and multiplication. We will sometimes evaluate functions at complex values of $t$ by “analytic continuation.”
### Linear partial differential operators
We consider first a general $r$-th order partial differential operator “in $1+1$ variables.” This is just an operator of the form $$\label{shapteshpe}
\begin{array}{rcl}
\cF({\mathbb R}_x \times {\mathbb R}_t) & \stackrel{P}{\ra} &
\cF({\mathbb
R}_x \times {\mathbb R}_t)\\
Pu & := & P\left(x,t,u,Du, D^2 u, \ldots, D^r u \right)\, ,
\end{array}$$ acting on functions $u=u(x,t)$. In this expression, $D^n u$ stands for the $n+1$ functions $\partial^{n-k}_x
\partial_t^k u$, $0\leq k \leq n$, where $\partial_x$ and $\partial_t$ are the corresponding partial derivative in the $x$ and $t$ directions, and $P(x,t,z)$ is a complex valued function of $\frac{(r+1)(r+2)}{2}+2$ complex variables. The operator is said to be [*linear*]{} if $P(x,t,z)$ is a linear function in the vector variable $z$. In that case, we define the [*full symbol*]{} $\sigma(P)(x,t,\xi,\tau)$ of $P$, a polynomial in $(\xi,\tau)$ with coefficients that are functions of $(x,t)$, by replacing $\partial^{n-k}_x \partial^k_t u$ in $P$ with the monomials $i^n
\xi^{n-k}\tau^k$. Modulo terms of degree $r-1$ or less, $\sigma(P)$ is an invariantly defined function on the cotangent bundle $T^{*}( {\mathbb R}_x \times {\mathbb R}_t)$. If $P(x,t,z)$ is independent of $(x,t)$, we say that the operator has constant coefficients. Standard examples of partial differential operators that are linear and have constant coefficients are $$\label{ceipat}
\begin{array}{llll}
Pu & = & \partial_t u -c \partial_x u\, , &
\textit{the convection operator}.\\
Pu & = & \partial_t u- c \partial_x^2 u \, , & \textit{the heat operator},\\
Pu & = & \partial^2_t u-c \partial^2_x u\, , & \textit{the wave
operator},
\end{array}$$ respectively. Here, $c$ is a constant that in the last two cases is assumed to be positive. These operators are of particular importance to us as their arithmetic analogues play a significant rôle in this article.
In regard to these examples, some remarks are in order:
1. If in the heat operator, we replace the real parameter $c$ by a purely imaginary constant, then we obtain the Schrödinger operator. As the results of this paper will suggest, our arithmetic analogue of the heat operator may also deserve attention as an analogue of the Schrödinger operator.
2. If in the heat operator we interchange $t$ and $x$, then we obtain the [*sideways heat operator*]{}. This operator will also have an analogue in our arithmetic theory.
3. If in the wave operator we replace the positive constant $c$ by a negative one, then we obtain the Laplace operator. Similarly, if in the convection operator $c$ is replaced by a purely imaginary constant, we obtain the Cauchy-Riemann operator. These are the typical examples of elliptic operators, and they do not have analogues among the arithmetic partial differential operators discussed here.
Given a linear partial differential operator $P$, we may consider the [*linear partial differential equation*]{} $Pu=0$, and its [*space of solutions*]{}, $$\cU=\cU_P:=\{u \in \cF(\bR_x \times \bR_t): \; Pu=0\}\, ,$$ that is to say, the kernel of $P$. One of the main problems of the theory is to describe the structure of this space, and to describe the image of the linear mapping on functions defined by $P$. Typically, a function $u$ in $\cU$ will depend on $\rho$ arbitrary functions $C_1(\xi),\ldots,
C_{\rho}(\xi)$, $\rho$ some integer not exceeding the order $r$ of $P$, while the description of the image of $P$ deals with the [*inhomogeneous equation*]{} $Pu=\varphi$ for a given $\varphi$, and the set of conditions on it under which the said equation has a solution. By duality, the latter problem is usually reduced to the consideration of the space of solutions for the dual linear operator $P^{*}$.
### Exponential solutions and characteristic roots
Assume $P$ has constant coefficients and that $u$ is in its space of solutions. The dependence of $u$ on the functions $C_j(\xi)$ above is easily obtained by using Fourier transform, as follows. Let $\hat{u}(\xi,t)$ denote the Fourier transform of $u$ in the $x$ variable. Then the equation $Pu=0$ yields $$\sigma(P)(-\xi,-i \partial_t)\hat{u}(\xi,t)=0\, , \label{q0}$$ an ordinary differential equation in the variable $t$ with parameter $\xi$. By solving this equation and applying the inverse Fourier transform in the parameter $\xi$, we obtain that $$\label{furh} u(x,t)=\sum_{j=1}^{\rho} \int C_j(\xi) e^{-i \xi
x-i\tau_j(\xi)t} d \xi\, ,$$ for some functions $C_j(\xi)$, where, for each $\xi$, the numbers $\tau_1(\xi),\ldots
,\tau_{\rho}(\xi)$ are the [*characteristic roots*]{} of $P$, that is to say, the (complex) roots of the [*characteristic polynomial*]{} $\sigma(P)(-\xi,-\tau)\in \bR[\tau]$, chosen to depend continuously on $\xi$. (We ignore here the problems arising from the possible presence of multiple roots.)
The “kernels” $e^{-i \xi x-i\tau_j(\xi)t}$ in (\[furh\]) are the [*exponential solutions*]{} in the space of solutions for $P$, and the formula exhibits the general element of this space as a sum of $\rho$ functions. There is one exponential solution per characteristic root, and the general solution $u$ depends $\bC$-linearly on one arbitrary function per root. (An important analytic aspect ignored here is that, in order to produce suitable distributional solutions through the formal manipulations above, we may need to choose some of the functions $C_j(\xi)$ to vanish identically. This is dictated by the behavior of the characteristic roots, and could make $u$ depend on fewer than $\rho$ arbitrary functions. For instance, think of the case on an elliptic operator, where some of the exponential solutions grow exponentially fast.)
We observe that for any $\xi$ and $\tau$, the exponentials $u_{\xi,\tau}(x,t):=e^{-i \xi x-i\tau t}$ “diagonalize” $P$. Indeed, we have $$\label{pirz}
Pu_{\xi,\tau}=\sigma(P)(-\xi,- \tau) \cdot u_{\xi,\tau}\, .$$ This fact leads naturally to the study of the inhomogeneous equation $Pu=\varphi$ by way of Fourier inversion.
### Boundary value problem
The classical approach to pinning down the functions $C_j(\xi)$ in (\[furh\]) is by imposing “boundary conditions” on the solution of $u$. For suppose we have a $\rho$-tuple $B=(B_1, \ldots ,B_{\rho})$ of linear partial differential operators $\cF(\bR_x \times \bR_t) \stackrel{B_j}{\ra}
\cF(\bR_x \times \bR_t)$. We consider the [*restriction operator*]{} $$\begin{array}{rcl}
\cF(\bR_x \times \bR_t) & \stackrel{\gamma}{\ra} & \cF(\bR_x)\\
\gamma v & := & v_{|t=0}\, ,
\end{array}$$ and if we let $B^0_j$ stand for the composition $\gamma \circ B_j$, we obtain the [*boundary value operator*]{} $$\begin{array}{rcl}
\cF(\bR_x \times \bR_t) & \stackrel{B^0}{\ra} & \cF(\bR_x)^{\rho}\\
B^0 u & := & (B^0_1 u,\ldots ,B^0_{\rho} u) \, .
\end{array}$$ We then say that the [*boundary value problem for $(P,B)$ is well posed*]{} if for any $g \in \cF(\bR_x)^{\rho}$ there exists a unique element $u$ in the space of solutions ${\mathcal U}_P$ whose boundary value $B^0 u$ is equal to $g$. In other words, the mapping $$B^0_P:\cU_P \ra \cF(\bR_x)^{\rho}$$ given by the restriction of $B^0$ to $\cU_P$ is a $\bC$-linear isomorphism. (Classically, the domain and range are endowed with some topology, and the continuity of both, the mapping and its inverse, are also required; we ignore that consideration here.) In the case where $P$ and $B_j$ have constant coefficients, a formal computation shows that the functions $C_j(\xi)$ and $g_j(x):=B^0 u$ are related by the equalities $$\sum_{k=1}^{\rho} \sigma(B_j)(-\xi,- \tau_k(\xi))
C_k(\xi)=\hat{g}_j(\xi),\ \ \ 1 \leq j \leq \rho\, .$$ The determinant of the matrix of this system is the [*Lopatinski determinant*]{}. Its non-vanishing is “morally” equivalent to the well posedness condition; cf. [@egorov], pp 321-322, or [@hormander].
A classical choice for the operators $B_j$ (corresponding to the [*Cauchy problem*]{}) is $$\label{inival}
B_j u =\partial^{j-1}_t u,\ \ \ 1 \leq j \leq \rho\, .$$
For the classical operators $P$ listed in (\[ceipat\]) and the operators $B_j$ in (\[inival\]), the corresponding boundary value problems are all well posed.
### Propagator and Huygens principle
Let us suppose that we have given operators $P,B_1,\ldots,B_{\rho}$ such that the boundary value problem for $(P,B)$ is well posed. Assume further that $P,B_1,\ldots
,B_{\rho}$ commute with the time translation operators $$\begin{array}{c}
\cF(\bR_x \times \bR_t) \stackrel{L_{t_0}}{\ra}
\cF(\bR_x \times \bR_t) \\ L_{t_0}(g(x,t))=g(x,t+t_0)
\end{array}$$ for all $t_0$. (This is the case, for instance, if the operators $P,B_1,\ldots ,B_{\rho}$ have constant coefficients.) Then $\cU_P$ is stable under all $L_t$, and we have the $\bC$-linear isomorphisms $$B^t_{P}:=B^0_P \circ L_{t}\, : \,
\cU_P \ra \cF(\bR_x)^{\rho} \, ,$$ explicitly given by $$B^{t_0}_{P}u=(B_1 u,\ldots ,B_{\rho} u)_{|t=t_0}\, .$$ For any pair $t_1$ and $t_2$, we obtain the [*evolution*]{} or [*propagator*]{} operator, defined as the $\bC$-linear isomorphism $$S_{t_1,t_2}:=B_{P}^{t_2} \circ (B_{P}^{t_1})^{-1}=
B_P^0 \circ L_{t_2-t_1} \circ (B_P^0)^{-1}: \cF(\bR_x)^{\rho} \ra
\cF(\bR_x)^{\rho}\, .$$ This family of operators satisfies the $1$-parameter group property $$S_{0, t_1+t_2}=S_{0,t_1} \circ S_{0,t_2} \, ,$$ a weak form of the “Huygens principle.”
### Fundamental solutions
The idea of evolution operator above is closely related to the concept of [*fundamental solution*]{}. In order to review this concept, let $\cF_{\star}(\bR_x)$ be the Abelian group $\cF(\bR_x)$, viewed as a ring with respect to convolution $$(f \star g)(x):=\int f(y)g(x-y)dy\, .$$ We also let $\cF_{\star}(\bR_x \times \bR_t)$ be the Abelian group $\cF(\bR_x \times \bR_t)$, viewed as a module over $\cF_{\star}(\bR_x)$ with respect to convolution in the variable $x$. Then, if $P$ commutes with the translation operators in the variable $x$ (for instance, if $P$ has constant coefficients), then the space of solutions $\cU_P$ is a $\cF_{\star}(\bR_x)$-submodule of $\cF_{\star}(\bR_x \times
\bR_t)$. In this situation, we will assume further that $B^0$ is a $\cF_{\star}(\bR_x)$-module homomorphism. This is the case for the operators $B_j$ in (\[inival\]).
Under these circumstances, if the boundary value problem for $(P,B)$ is well posed, $B_P^0$ is an $\cF_*(\bR_x)$-module isomorphism so the space of solutions $\cU_P$ is a free $\cF_{\star}(\bR_x)$-module of rank $\rho$ with a unique basis $$\label{fundulet} u_{1}(x,t), \ldots ,u_{\rho}(x,t)$$ (that generally speaking, will consist of distributions) such that the $\rho \times \rho$ matrix $(B_j^0 u_i)$ is diagonal, with diagonal entries the Dirac delta function $\delta_0=\delta_0(x)$ centered at $0$. This basis is the [*system of fundamental solutions*]{} of $(P,B)$. Clearly, for any $u \in \cU_P$, we have that $$\label{uaah} u(x,t)=\sum_{i=1}^{\rho} (B_i^0 u)(x) \star u_i(x,t)\, ,$$ and so the fundamental solutions $u_i$ appear as kernels in this integral representation for $u$. Conversely, let us assume that the operators $B_j$ are as in (\[inival\]), and that we can find $K_1,\ldots ,K_{\rho} \in \cU_P$ such that any $u \in \cU_P$ can be written uniquely as $$u(x,t)=\sum_{i=1}^{\rho} (\partial_t^{i-1} u)(x,0) \star K_i(x,t) \, .$$ (Cf. [@rauch], p. 138, for the case of the operators listed in (\[ceipat\]).) Applying $\partial_t^{j-1}$ to this identity, $1
\leq j \leq \rho$, and letting $t\ra 0$, we get that the matrix $K=K(x,t)$, whose entries are given by $K_{ij}=\partial^{j-1}_t
K_i$, $1 \leq i,j \leq \rho$, has the property that $K(x,0)$ is equal to ${\rm diag}(\delta_0,\ldots ,\delta_0)$. Thus, we conclude that $K_1,\ldots ,K_{\rho}$ is the system of fundamental solutions of $(P,B)$. The matrix $K$ is the [*fundamental solution matrix*]{}.
For simplicity, let us assume further that $Pu=\partial_t^{\rho} u-c
\partial_x^s u$. As $K_{\rho}$ is in the space of solutions of $P$, we see that $\partial_t^{\rho-1} K_{\rho},
\partial_t^{\rho-2}K_{\rho}, \ldots ,\partial_t^0K_{\rho}$ is also a system of fundamental solutions for $(P,B)$. The uniqueness implies that $$K_1=\partial_t^{\rho-1}K_{\rho},\; K_2 =\partial_t^{\rho-2}K_{\rho},
\; \ldots ,\; K_{\rho-1}=
\partial_t K_{\rho}\, .$$ Notice that $$S_{0,t}(g(x))=g(x) \star K(x,t)$$ for any $g(x) \in \cF(\bR_x)^{\rho}$, that is, the propagator operator is given by convolution with the fundamental solution matrix.
It is worth recalling that $K_{\rho}$ above may be used to solve the [*inhomogeneous equation*]{} $Pu=\varphi$, as in the following discussion where for simplicity we take once again $Pu=\partial_t^{\rho} u-c
\partial_x^s u$. Cf. [@hormander], pp. 80, 109, or [@egorov], pp. 142, 235. Let $H \in \cF(\bR_t)$ be the characteristic function of the interval $[0, \infty)$ (the [*Heaviside function*]{}). If we set $K_+(x,t):=K_{\rho}(x,t) \cdot H(t)$ where, of course, we are implicitly assuming that the product of the distributions in the right hand side is well defined, then we see that $$P K_+(x,t)=\delta_0(x) \delta_0(t)\, .$$ A function of $(x,t)$ satisfying this equation is said to be a [*fundamental solution of the inhomogeneous equation*]{}. A formal computation shows that for any $\varphi(x,t) \in \cF(\bR_x \times
\bR_t)$, the function $K_+ \star \varphi$ (where $\star$ denotes now the convolution with respect to both variables $x$ and $t$,) is a solution to the equation $Pu=\varphi$, that is to say, $$P(K_+ \star \varphi)=\varphi\, .$$
Evolution equations in arithmetic
---------------------------------
The main purpose of this paper is to propose an arithmetic analogue of the “$1+1$ evolution picture” above. In the remaining portion of this introduction, we informally present our main concepts, problems, and results.
### Main concepts
In our arithmetic theory, the ring $\cF({\mathbb R}_x)$ of functions in $x$ is replaced by a “ring of numbers” $R$. A natural choice [@char; @difmod; @book] for this $R$ is given by the completion of the maximum unramified extension of the ring $\bZ_p$ of $p$-adic integers. We will think of $p$ as a “space variable,” the analogue of $x$. And the analogue of the ring $\cF({\mathbb R}_{t}\times {\mathbb R}_x)$ of functions of space-time is the ring of formal power series $A=R[[q]]$, whose elements are viewed as “superpositions” of the “plane waves” $aq^n$, $a \in R$, analogues of the plane waves $a(x)e^{-2 \pi i n
t}$ of frequency $n$. (We will use other rings also, for instance, $R[[q^{-1}]]$. Series in $R[[q]]$ will be viewed as superpositions of plane waves “involving non-negative frequencies only,” whereas series in $R[[q^{-1}]]$ will be superpositions of plane waves “involving non-positive frequencies only.” It will be interesting to further enlarge these by considering the rings of Laurent power series and their $p$-adic completions, $R((q))\wh$ and $R((q^{-1}))\wh$, respectively.)
The rôle of the partial derivative $-(2 \pi i)^{-1}\partial_t$ is to be played by the derivation $$\begin{array}{rcl}
A & \stackrel{\dd}{\ra} & A \\
\dd u & := & {\displaystyle q \partial_q u}
\end{array}\, ,$$ where $\partial_q$ is the usual derivative with respect to $q$. On the other hand, the analogue of the partial derivative $(2 \pi
i)^{-1}\partial_x$, which should be interpreted as a derivative with respect to the prime $p$, is obtained by following the idea in [@char; @difmod; @book]. Indeed, we propose to define this derivative with respect to $p$ as the “Fermat quotient operator” given by $$\label{defde}
\begin{array}{rcl}
A & \stackrel{\d}{\ra} & A \\
\d u & := & {\displaystyle \frac{u^{(\phi)}(q^p)-u(q)^p}{p}}
\end{array}\, ,$$ where the upper index $(\phi)$ stands for the operation of twisting the coefficients of a series by the unique automorphism $\phi: R \ra R$ that lifts the $p$-th power Frobenius automorphism of $R/pR$. Note that the restriction of $\d$ to $R$ is the mapping $$\label{defcu}
\begin{array}{rcl}
R & \stackrel{\d}{\ra} & R \\
\d a & = & {\displaystyle \frac{\phi(a)-a^p}{p}}
\end{array}\, ,$$ which is the arithmetic analogue of a derivation, as discussed in [@char; @difmod; @book]. Note that the set of its [*constants*]{}, $R^{\d}:=\{a \in R;\d a=0\}$ consists of $0$ and the roots of unity in $R$. So $R^{\d}$ is a multiplicative monoid but not a subring of $R$.
In order to proceed, we need to describe the analogues of linear partial differential equations. We start, more generally, with maps of the form $$\label{baiie} \begin{array}{rcl} A & \stackrel{P}{\ra} & A\\
Pu & := &P(u,Du,\ldots ,D^r u),
\end{array}$$ where $P=P(z)$ is a $p$-adic limit of polynomials with coefficients in $A$ in $\frac{(r+1)(r+2)}{2}$ variables, and $D^n
u$ stands for the $n+1$ series $\d^i \dd^{n-i} u$, $0\leq i \leq
n$. These maps are the “partial differential operators” of order $r$ in this article. What remains to be done is to define the notion of linearity for them.
The naive requirement that $P(z)$ be a linear form in $z$ is not appropriate. Indeed, the property that linearity should really capture is that differences of solutions be again solutions, and this is not going to happen since $\d$ itself is not additive. We could insist upon asking that the map $u \mapsto Pu$ be additive, but as such, this condition would lead to a rather restricted class of examples. In order to find what we suggest is the right concept (which, in particular, will provide sufficiently many interesting examples), we proceed by generalizing our setting as follows (cf. [@char; @difmod; @book] for the ordinary differential case).
Firstly, we consider mappings $A^N \ra A$ as in (\[baiie\]), where now $u$ is an $N$-tuple. Secondly, we restrict such maps to subsets $X(A) \subset \bA^N(A)=A^N$, where $X\subset {\mathbb
A}^N$ is a closed subscheme of the affine $N$-space $\bA^N$ over $A$, and where $X(A)$ denotes the set of points of $X$ with coordinates in $A$. If $X$ has relative dimension $n$ over $A$, the induced maps $X(A) \ra A$ will be viewed as “partial differential operators” on $X$ in $1+1$ “independent variables” and $n$ “dependent variables.” Using a gluing procedure, we then derive the concept of a “partial differential operator,” $X(A)
\ra A$, on (the set of $A$-points of) an arbitrary scheme $X$ of finite type over $A$ (which need not be affine). Finally, when we take $X$ in this general set-up to be a commutative group scheme $G$ over $A$, we define a [*linear partial differential operator*]{} on $G$ to be a “partial differential operator” $G(A)
\ra A$ on $G$ that is also a group homomorphism, where $A$ is viewed with its additive group structure. By making this “set-theoretical” definition one that is “scheme-theoretical” (varying $A$), we arrive at the notion of [*linear partial differential operator*]{} that we propose in here. And once again, we are able to associate to any linear partial differential operator so defined a polynomial $\sigma(\xi_p,\xi_q)$ in two variables with $A$-coefficients, which we refer to as the ([*Picard-Fuchs*]{}) [*symbol*]{} of the operator.
The construction above is motivated by points of view adopted in analysis and mathematical physics. Indeed, we view ${\rm Spec}\, A$ as the analogue of $\bR_x \times \bR_t$, and we view schemes $X$ as the analogues of manifolds $M$, so the set $X(A)={\rm Hom}({\rm
Spec}\, A,X)$ is the analogue of the set $\cF(\bR_x \times
\bR_t,M)$ of maps $\bR_x \times \bR_t \ra M$ (which we require here to be at least continuous). The commutative group schemes $G$ of relative dimension $1$ (which will be the main concern of this paper) are the analogues of Lie groups of the form $\bC/\Gamma$ where $\Gamma$ is a discrete subgroup of $\bC$. (There are $3$ cases, those corresponding to a subgroup $\Gamma$ of rank $0$, $1$, or $2$ respectively. The corresponding groups $\bC/\Gamma$ are the additive group $\bC$, the multiplicative group $\bC^{\times}$, and the elliptic curves $E$ over $\bC$) On the other hand, any linear partial differential operator $ \cF(\bR_x \times
\bR_t) \ra \cF(\bR_x \times \bR_t)$ with symbol that vanishes at $(0,0)$ induces a homomorphism $$\cF(\bR_x \times \bR_t)/\Gamma \ra \cF(\bR_x \times \bR_t)\, .$$ We can consider then the composition $$\label{nuinca} \cF(\bR_x \times \bR_t,\bC/\Gamma) \simeq \cF(\bR_x
\times \bR_t)/\Gamma \ra \cF(\bR_x \times \bR_t).$$ Our groups $G(A)$ are the analogues of the groups $\cF(\bR_x \times
\bR_t,\bC/\Gamma)$, and our linear partial differential operators $G(A) \ra A$ are the analogues of the operators in (\[nuinca\]).
Given a linear partial differential operator $G(A) \stackrel{P}{
\ra} A$ in the arithmetic setting, we may consider the [*group of solutions*]{} consisting of those $u \in G(A)$ such that $Pu=0$. This is a subgroup of $G(A)$. Those elements in this space that “do not vary with time” will define the concept of [*stationary solutions*]{}. Notice that if $G$ descends to $R$, [*stationary*]{} will simply mean “belonging to $G(R)$.”
The linear partial differential operators in the sense above will be called $\D$-[*characters*]{}, and we will denote them by $\psi$ rather than $P$. This will put our terminology and notation in line with that used in [@char; @difmod; @book], where the case of arithmetic ordinary differential equations was treated.
As a matter of fact, some comments on the ordinary case are in order here. If all throughout the theory sketched above we were to insist that the operators $u \mapsto Pu$ in (\[baiie\]) be given by polynomials in $u, \dd u, \ldots , \dd^r u$ only, we would then be led to the Ritt-Kolchin theory of “ordinary differential equations” with respect to $\dd$, cf. [@ritt; @kolchin; @cassidy]. In particular, this would lead to the notion of $\dd$-[*character*]{} of an algebraic group, which in turn should be viewed as the analogue of a linear ordinary differential operator on an algebraic group (cf. to the Kolchin logarithmic derivative of algebraic groups defined over $R$, [@kolchin; @cassidy], and the Manin maps of Abelian varieties defined over $R[[q]]$, [@man; @annals]).
If on the other hand we were to insist throughout the theory that the operators $u \mapsto Pu$ in (\[baiie\]) be given by $p$-adic limits of polynomials in $u, \d u,\ldots ,\d^r u$ only, we would then be led to the arithmetic analogue of the ordinary differential equations in [@char; @difmod; @book]. In particular, we would then arrive at the notion of a $\d$-character of a group scheme, which is the arithmetic analogue of a linear ordinary differential equation on a group scheme.
### Main problems
We present, in what follows, a sample of the main problems to be treated in this paper:
1. Find all $\D$-characters on a given group scheme $G$.
2. For any $\D$-character $\psi$, describe the kernel of $\psi$, that is to say, the group of solutions of $\psi u=0$, and study the behavior of the solutions in terms of “convolution,” “boundary value problems,” “characteristic polynomial,” “propagators,” “Huygens’ principle,” etc.
3. For any $\D$-character $\psi$, describe the image of $\psi$, that is to say, the group of all the $\varphi$ such that the inhomogeneous equation $\psi u=\varphi$ has a solution.
These will be discussed in detail for the case of one dependent variable, that is to say, for groups of dimension one. Indeed, we shall thoroughly analyze the additive group $G={\mathbb G}_a$, the multiplicative group $G={\mathbb G}_m$, and elliptic curves $G=E$ over $A$, cases where $G(A)$ corresponds to the additive group $(A,+)$, the multiplicative group $(A^{\times},\cdot)$ of invertible elements of $A$, and the group $(E(A),+)$ of points with coordinates in $A$ of a projective non-singular cubic, with addition operation given by the chord-tangent construction, respectively.
### Main results
In regard to Problem 1, we will start by proving that the $\D$-characters of fixed order on a fixed group scheme form a finitely generated $A$-module. We will then provide a rather complete picture of the space of $\D$-characters in the cases where $G$ is either ${\mathbb G}_a$, or ${\mathbb G}_m$, or an elliptic curve $E$ over $A$. In particular, it will turn out that these groups possess certain remarkable $\D$-characters that are, roughly speaking, the analogues of the classical operators listed in (\[ceipat\]) above.
In the cases where $G$ is either ${\mathbb G}_a$, or ${\mathbb
G}_m$, or an elliptic curve $E$ defined over $R$, the $\D$-characters $\psi$ of $G$ are “essentially” built from $\dd$-characters $\psi_{\ddi}$, and $\d$-characters $\psi_{\di}$ of $G$. This situation is analogous to the one in classical analysis in $\bR_x \times \bR_t$, where linear differential operators are “built” from $\partial_x$ and $\partial_t$, respectively. Remarkably, however, if $G$ is an elliptic curve $E$ over $A$ that is sufficiently “general,” then $E$ possesses a $\D$-character $\psi^1_{\di \ddi}$ that cannot be built from $\dd$- and $\d$-characters alone. Thus, from a global point of view, this $\D$-character $\psi^1_{\di \ddi}$ is a “pure partial differential object,” in the sense that it cannot be built from “global ordinary linear differential objects.”
This will all unravel in the following manner. We will first prove that for any elliptic curve $E$ over $A$, there always exists a non-zero $\D$-character $\psi^1_{\di \ddi}$ of order one. We will then show that for a general elliptic curve $E$ over $A$, the $A$-module of $\D$-characters of order one has rank one, and so $\psi^1_{\di \ddi}$ is essentially unique. We view it as a [*canonical convection equation*]{} on the elliptic curve. Let us note that $\psi^1_{\di \ddi}$ cannot be decomposed as a linear combination of $\dd$- and $\d$-characters alone because, on these elliptic curves, all such characters of order one are trivial.
The $\D$-character $\psi^1_{\di \ddi}$ will turn out to be a factor of a canonical order two $\D$-character that can be expressed as $\psi^2_{\ddi}+\lambda \psi^2_{\di}$, the sum of a $\dd$-character $\psi^2_{\ddi}$ of order two, and of $\lambda$ times a $\d$-character $\psi^2_{\di}$ of order two also, $\lambda \in A$. We will view this $\D$-character of order two as a [*canonical wave equation*]{} on the elliptic curve.
Once again, for a general elliptic curve $E$ over $R$, we will encounter [*heat equations*]{} on $E$ also. These will be sums of the form $\psi^1_{\ddi}+\lambda \psi^2_{\di}$, where $\psi^1_{\ddi}$ is a $\dd$-character of order one, $\psi^2_{\di}$ is a $\d$-character of order two, and $\lambda \in
R$.
In regard to Problem 2, a first remark is that the “generic” $\D$-characters do not admit non-stationary solutions hence a natural question is to characterize those that admit such solutions. We succeed in giving such a characterization under a mild non-degeneracy condition on the symbol $\sigma(\xi_p,\xi_q)$, a condition that is satisfied “generically.” Roughly speaking, we will show that a non-degenerate $\D$-character $\psi$ of a group $G$ over $R$ has non-stationary solutions if, and only if, the polynomial $\sigma(0,\xi_q)$ has an integer root. The effect of this criterion is best explained if we consider families $\psi_{\lambda}$ of $\D$-characters of low order (usually $1$ or $2$) that depend linearly on a parameter $\lambda \in R$, and ask for the values of this parameter for which $\psi_{\lambda}$ possesses non-stationary solutions. We then discover a “quantization” phenomenon according to which, the only values of $\lambda$ for which this is so form a “discrete” set parameterized by integers $\kappa \in \bZ$. This singles out certain $\D$-characters as “canonical equations” on our groups, and produces, for instance, a [*canonical heat equation*]{} on a general elliptic curve over $R$.
A different kind of “quantization” will be encountered in our study of Tate curves with parameter $\beta q$, where $\beta \in
R$. (These curves are defined over $R((q))$ but not over $R$.) In that case, we obtain that the canonical convection equation has non-stationary solutions if and only if the values of $\beta$ are themselves “quantized,” that is to say, parameterized by integers $\kappa \in \bZ$.
The criterion above on the existence of non-stationary solutions will be a consequence of a more detailed analysis of spaces of solutions of $\psi u=0$. In order to explain this, we take a non-degenerate $\D$-character $\psi$ of a group $G$ over $R$, and consider first the groups $\cU_{\pm 1}$ of solutions of $\psi u=0$ in $G(A)$ ($A=R[[q^{\pm 1}]]$) vanishing at $q^{\pm 1}=0$, respectively. Intuitively, these are the analogues of those solutions in analysis that “decay to zero” as $t \ra \mp i
\infty$, respectively. We will prove that $\cU_{\pm 1}$ are finitely generated free $R$-modules under a natural convolution operation denoted by $\star$. The ranks of these $R$-modules are given by the cardinalities $\rho_{+}$ and $\rho_{-}$ of the sets ${\mathcal K}_{\pm}$ of positive and negative integer roots of the polynomial $\sigma(0,\xi_q)$, where $\sigma(\xi_p,\xi_q)$ is the symbol of $\psi$. The integers in $\cK_{\pm}$ are the [*characteristic integers*]{} of $\psi$.
For instance, if $\psi$ is the arithmetic analogue of the convection or heat equation then one of the spaces $\cU_{\pm 1}$ is zero and the other has rank one over $R$ under convolution. If $\psi$ is the arithmetic analogue of the wave equation then both spaces $\cU_{\pm 1}$ have rank one over $R$, an analogue of the picture in d’Alembert’s formula where one has $2$ waves traveling in opposite directions.
Going back to the general situation of a non-degenerate $\D$-character $\psi$, we will consider a [*boundary value problem at $q^{\pm 1}=0$*]{} as follows. Firstly, we will consider the operators $$\begin{array}{rcl}
R[[q^{\pm 1}]] & \stackrel{\Gamma_{\kappa}}{\ra} & R \vspace{1mm} \\
\Gamma_{\kappa}(\sum a_n q^n) & := & a_{\kappa}=\frac{1}{\kappa
!} (\partial_{q^{\pm 1}}^{\kappa} u)_{|q^{\pm 1}=0}
\end{array}\, ,$$ where, of course, $\partial_{q^{-1}}:=-q^2 \partial_q$. Secondly, we note that, up to an invertible element in $R$, there is a unique non-zero $\dd$-character $\psi_q$ of $G$ of minimal order. The $\D$-character $\psi_q$ has order $0$ if $G=\bG_a$, and order $1$ if $G$ is either $\bG_m$ or an elliptic curve over $R$; in the latter case $\psi_q$ is the Kolchin logarithmic derivative. For $\kappa \in \cK_{\pm}$ we denote by $B_{\kappa}^0$ the composition $\Gamma_{\kappa} \circ \psi_q$. We will then prove that the [*boundary value operator at $q^{\pm 1}=0$*]{}, $$\begin{array}{rcl}
\cU_{\pm 1} & \stackrel{B_{\pm}^0}{\ra} & R^{\rho_{\pm}}\\
B_{\pm}^0 u & = & (B_{\kappa}^0 u)_{\kappa \in \cK_{\pm}}
\end{array}\, ,$$ is an $R$-module isomorphism, and, furthermore, that there exist solutions $u_{\kappa} \in \cU_{\pm}$ such that for any $u \in \cU_{\pm 1}$ we have the formula $$\label{macbet} u=\sum_{\kappa \in {\mathcal
K}_{\pm}} (B_{\kappa}^0 u) \star u_{\kappa}\, .$$ This can be viewed as analogue of the expression (\[furh\]) because it exhibits $u$ as a sum of $\rho_{\pm}$ terms indexed by the characteristic integers, with each term depending $\bZ$-linearly on one arbitrary “function of space.” It can also be viewed as an analogue of (\[uaah\]) because of its formulation in terms of convolution. Then we interpret the bijectivity of $B_{\pm}^0$ by saying that the “boundary value problem at $q^{\pm
1}=0$” is [*well posed*]{}. (The language chosen here is a bit lax since no direct analogue of this boundary value problem at $q^{\pm 1}=0$ seems to be available in real analysis; indeed, such an analogue would prescribe boundary values at (complex) infinity, which does not appear as a natural condition to be imposed on solutions of linear partial differential equations in analysis.)
The solutions $u_{\kappa}$ ($\kappa \in \cK$) will be referred to as [*basic solutions*]{} of the $\D$-character $\psi $. In some sense, these elements of the kernel of $\psi$ are the analogues of both the exponential solutions and the fundamental solutions of the homogeneous equations in real analysis. Of course, these analogies have significant limitations.
An interesting feature of the solutions $u$ in the kernel of $\psi$ is the following “algebraicity modulo $p$” property. Let us denote by $k$ the residue field of $R$, hence $k=R/pR$. Then, for any such $u$, the reduction modulo $p$, $\overline{\psi_q u} \in k[[q^{\pm 1}]]$, of the series $\psi_q u$ is integral over the polynomial ring $k[q^{\pm 1}]$, and the field extension $$k(q) \subset k(q,\overline{\psi_q u})$$ is Abelian with Galois group killed by $p$. On the other hand, under certain general assumptions that are satisfied, in particular, by our “canonical” equations, we will prove that the solutions $u \neq 0$ of $\psi u=0$ are transcendental over $R[q]$. This transcendence result can be viewed as a (weak) incarnation of Manin’s Theorem of the Kernel [@man].
Some of the results above about groups $G$ over $R$ have analogues for groups not defined over $R$, such as the Tate curves. For the latter the $\dd$-character $\psi_q$ will now be the Manin map, which has order $2$.
If we instead consider solutions that do not necessarily decay to $0$ as $q^{\pm 1} \ra 0$, we are able to show that, in case $\rho_+=1$, the appropriate boundary value problem at $q \neq
0$ is well posed. (The condition $\rho_+=1$ is usually satisfied by our “canonical equations”.) Roughly speaking, for groups $G$ over $R$ and $\D$-characters $\psi$ with $\rho_+=1$, this boundary value problem has the following meaning. We consider $q_0 \in
pR^{\times}$ and $g \in G(R)$. If $A=R[[q]]$, we ask if there exists a (possibly unique) solution $u \in G(A)$ of the equation $\psi
u=0$ that satisfies the condition $B^{q_0}u =g$, where $$\begin{array}{rcl}
G(A) & \stackrel{B^{q_0}}{\ra} & G(R) \\
B^{q_0} u & := & u(q_0) \end{array}$$ is the group homomorphism induced by the ring homomorphism $A=R[[q]] \ra R$ that sends $q$ into $q_0$. We view $B^{q_0}$ as a [*boundary value operator at $q_0$*]{}, and we view the corresponding embedding ${\rm Spec}\, R \ra {\rm Spec}\, A$ as the “curve” $q=q_0$ in the $(p,q)$-plane, along which we are given our boundary values.
Our point of view here is reminiscent of that in real analysis, where we replace real time by a complex number whose real part is small in order to avoid singularities on the real axis. Indeed the condition “$q_0$ invertible” (that is to say, $v_p(q_0)=0$, where $v_p$ is the $p$-adic valuation) is an analogue of the condition “real time;” the condition “$q_0$ non-invertible with small $v_p(q_0)$” is an analogue of “complex time close to the real axis;” “$q_0$ close to $0$” (that is, $v_p(q_0)$ big) is an analogue of “time close to $-i \infty$.”
For equations with $\rho_+=1$, we will investigate also the “limit of solutions as $q \rightarrow 0$” (or intuitively, as $t
\rightarrow -i \infty$). This limit will sometimes exist, and if so, the limit will usually be a torsion point of $G(R)$. In this sense, torsion points tend to play the role of “equilibrium states at (complex) infinity.”
These principles do not hold uniformly in all examples. For instance, in the case of elliptic curves $E$ over $R$, we will need to replace $E(R)$ by a suitable subgroup of it, $E'(R)$, in order to avoid points whose orders are powers of $p$. And for elliptic curves over $A$, rather than those over $R$ (such as the Tate curves), the boundary value problem at $q \neq 0$ will take a slightly different form.
Once the boundary value problem at $q \neq 0$ has been solved, we can construct [*propagators*]{} as follows. Again, we fix $\psi$ with $\rho_+=1$, choose “complex times” $q_1, q_2 \in
pR^{\times}$, and consider the endomorphism $S_{q_1,q_2}$ of the group $G(R)$ that sends any $g_1 \in G(R)$ into $g_2:=u(q_2)$, where $u \in G(A)$ is the unique solution to the boundary value problem $$\begin{array}{ccl} \psi u & = & 0\\
u(q_1) & = & g_1
\end{array}.$$ This construction does not work uniformly in all examples the way it is described here. In order to make it so, we need an appropriate modification of the given recipe. Nevertheless, in all situations, we find that the propagator $S$ satisfies a $1$-parameter group property. For it turns out that given “complex times” $q_i=\zeta_i q_0$ with $\zeta_i \in R$ a root of unity, $i=1,2$, we have that $$S_{q_0,\zeta_1 \zeta_2 q_0}=S_{q_0,\zeta_2 q_0} \circ
S_{q_0,\zeta_1 q_0}\, ,$$ identity that can be viewed as a weak incarnation of Huygens’ principle.
In regard to Problem 3, we will consider non-degenerate $\D$-characters $\psi$ of $G$, and we will find sufficient conditions on a given series $\varphi \in
A=R[[q]]$ ensuring that the inhomogeneous equation $\psi
u=\varphi$ have a solution $u \in G(A)$. Specifically, let us define the support of the series $\varphi=\sum c_nq^n$ as the set $\{ n\, : \; c_n \neq 0\}$. We will then prove that if $\varphi$ has support contained in the set $\cK'$ of [*totally non-characteristic*]{} integers, then the inhomogeneous equation above, with $\varphi$ as right hand side, has a unique solution $u \in G(A)$ for which the support of $\psi_q u$ is disjoint from the set $\cK$ of characteristic integers. Here, $\cK'$ is defined by an easy congruence involving the symbol, and, as suggested by the terminology, $\cK'$ is disjoint from $\cK$. Furthermore, if $\bar{\varphi}$, the reduction mod $p$ of $\varphi$, is a polynomial, then we will prove a corresponding “algebraicity mod $p$” property for $u$ stating that the series $\overline{\psi_q u}$ is integral over $k[q^{\pm
1}]$, and that the field extension $k(q)\subset k(q,\overline{\psi_q
u})$ is Abelian with Galois group killed by $p$. On the other hand, for a “canonical” $\psi$ and for a $\varphi$ with [*short*]{} support, we will show that the solutions of $\psi u=\varphi$ are transcendental over $R[q]$.
The idea behind the results above is to construct, for all integers $\kappa$ coprime to $p$, a solution $u_{\kappa}$ of the equation $$\label{obbossesc} \psi u_{\kappa}=\frac{\sigma(0,\kappa)}{p} \cdot
q^{\kappa} \, ,$$ which we shall call a [*basic*]{} solution of the inhomogeneous equation. For $\kappa$ a characteristic integer, the right hand side of (\[obbossesc\]) vanishes and our basic solutions are the previously mentioned basic solutions of the homogeneous equation. For $\kappa$ a totally non-characteristic integer, the right hand side of (\[obbossesc\]) is a unit in $R$ times $q^{\kappa}$, and that leads to the desired result about the inhomogeneous equation. Notice that (\[obbossesc\]) should be viewed as an analogue of (\[pirz\]), and the $u_{\kappa}$’s (for $\kappa \in \bZ
\backslash p\bZ$) should be viewed as (partially) diagonalizing $\psi$.
Concluding remarks
------------------
It is natural to ask for an extension of the theory in the present paper to the case of $d+e$ independent variables and $n$ dependent variables, that is to say, to the case of $d$ time variables $q_1,
\ldots ,q_d$, $e$ space variables $p_1, \ldots ,p_e$, where $p_i$ are prime numbers, and groups $G$ of (relative) dimension $n$. It is not hard to perform such an extension to the case $e=1$, $d
\geq 1$, $n \geq 1$ (that is to say, one prime $p$ as space variable, $d \geq 1$ indeterminates $q_i$ as time variables, and groups of dimension $n \geq 1$). In fact, all the difficulties of this more general case are already present in ours here, where $d=e=n=1$. On the other hand, there seems to be no obvious way to extend the theory in a non-trivial way even to the case $d=1$, $e > 1$, $n=1$ (that is to say, two or more primes $p_i$ as space variables, one time variable $q$, and groups of dimension $1$). The main obstruction lies in the fact that when at least two primes are made to interact in the same equation, the solutions exhibit a rather fundamental divergent form.
We end our discussion here by summarizing (cf. the tables below) some of the similarities and differences between the set-ups of classical real analysis [@hormander; @folland; @rauch], classical $p$-adic analysis [@dwork; @koblitz], and arithmetic (in the spirit of [@char; @difmod; @book] for the ordinary differential case, and the present paper for the partial differential case). For the ordinary differential case we have:
$\ $ real analysis $p$-adic analysis arithmetic
-------------------------- ------------------------ ------------------- ------------
$1$-dimensional manifold ${\mathbb R}_{x}$ $R$ $R^{\d}$
ring of functions $\cF({\mathbb R}_{x})$ $R[[x]]$ $R$
operator on functions $\partial_x$ $\partial_x$ $\d$
For the partial differential case we have:
real analysis $p$-adic analysis arithmetic
-------------------------- ---------------------------------------------- ------------------------- -------------------
$2$-dimensional manifold ${\mathbb R}_{t}\times {\mathbb R}_{x}$ $R\times R$ $R^{\d} \times R$
ring of functions $\cF({\mathbb R}_{x}\times {\mathbb R}_{t})$ $R[[x,t]]$ $R[[q]]$
operators on functions $\partial_x,\partial_t$ $\partial_x,\partial_t$ $\d, \dd$
In the last column, the set $R^{\d}$ plays a role similar to that of the set of “geometric points of the spectrum of the field with one element” in the sense of Deninger, Kurokawa, Manin, Soulé, and others [@kurokawa; @manin; @soule]. (For comments on differences between our approach here and the ideology of the “field with one element,” we refer to the Introduction of [@book].) In particular $R^{\d}$ can be viewed as an object of dimension zero. Notice that the third column appears to be obtained from the second one by “decreasing dimensions by one;” this reflects the fact that, in contrast to the second column, the third one treats numbers as functions. Note also that the interpretation of $R^{\d}$ as a space of dimension zero is morally consistent with the (otherwise) curious fact that the groups of solutions $\cU_{1}$ and $\cU_{-1}$ are modules over $R$ with respect to a “convolution” operation $\star$. Indeed, this suggests that “pointwise” multiplication and “convolution” of functions on $R^{\d}$ coincide, which in turn, is compatible with viewing $R^{\d}$ as having dimension $0$.
Plan of the paper
-----------------
We begin in §2 by introducing our main concepts, and where, in particular, we define $\D$-characters, their various spaces of solutions, and the convolution module structure on these spaces. In §3 and §4, we construct and study partial differential jet spaces of schemes and formal groups, respectively. These are arithmetic-geometric analogues of the standard jet spaces of differential geometry, arithmetic in the space direction, and geometric in the time direction. The geometry of these jet spaces controls the structure of the spaces of $\D$-characters. Among several other constructions, we define in §4 the Picard-Fuchs symbol of a $\D$-character. In §5, we develop an analogue of the Fréchet derivative, and use it to define the Fréchet symbol of a $\D$-character. We then establish a link between the Fréchet symbol and the Picard-Fuchs symbol that will be useful in applications. We also develop in this section a brief analogue of the Euler-Lagrange formalism. In §6, §7 and §8, we present our main results about $\D$-characters and their solution spaces in the cases of $\bG_a$, $\bG_m$, and elliptic curves, respectively.
[**Acknowledgements.**]{} We would like to acknowledge useful conversations with F. J. Voloch and D. Thakur on the subject of transcendence.
Main concepts
=============
In this section, we begin by introducing the main algebraic concepts in our study, especially the $\D$-rings and $\D$-prolongation sequences. These are then used to define $\D$-morphisms of schemes, and, eventually, $\D$-characters of group schemes. We introduce various solution spaces of $\D$-characters, and discuss the convolution module structure on them.
Let $p$ be a prime integer that we fix throughout the entire paper. For technical purposes related to the use of logarithms of formal groups, we need to assume that $p \neq
2$. Later on, in our applications to elliptic curves, we will need to assume that $p\neq 3$ also. All throughout, $A$ shall be a ring, and $B$ an algebra over $A$. If $x$ is an element of $A$, we shall denote by $x$ its image in $B$ also.
We let $C_p(X,Y)$ stand for $$C_p(X,Y):=\frac{X^p+Y^p-(X+Y)^p}{p}\, ,$$ an element of the polynomial ring ${\mathbb Z}[X,Y]$.
After [@char], we say that a map $\delta: A \rightarrow B$ is a $p$-[*derivation*]{} if $\d(1)=0$, and $$\begin{array}{rcl}
\d(x+y) & = & \d x + \d y +C_p(x,y)\, , \\
\d(xy) & = & x^p \d y +y^p \d x +p \d x \d y\, ,
\end{array}$$ for all $x,y \in A$, respectively. We will always write $\d x$ instead of $\d(x)$.
Given a $\d$-derivation, we define $\phi:=\phip:=
\phi_{\d}: A \rightarrow B$ by $$\phip(x)= x^p +p\d x \, , \label{me0}$$ a map that turns out to be a homomorphism of rings. Sometimes, we will write $x^{\phi}$ instead of $\phip(x)$, and when omitted from the notation, the context will indicate the $p$-derivation $\d$ that is being used.
We recall that a map $\dd : A\rightarrow B$ is said to be a [*derivation*]{} if $$\begin{array}{rcl}
\dd(x+y) & = & \dd x + \dd y\, , \\
\dd(xy) & = & x\dd y+y\dd x\, ,
\end{array}$$ for all $x,y \in A$, respectively. For the time being, the index $q$ will not be given any interpretation. Later on, we will encounter the situation where $q$ is an element of $A$, and in that case, we will think of $\dd$ as a derivation in the “direction” $q$.
Let $\d:A \ra B$ be a $p$-derivation. A derivation $\dd : A
\rightarrow B$ is said to be a $\d$-[*derivation*]{} if $$\label{didi} \dd \d x=p\d \dd x + (\dd x)^p-x^{p-1}\dd x$$ for all $x \in A$.
In particular, for a $\d$-derivation $\dd : A \rightarrow
B$, we have that $$\label{cucu} \dd \circ \phip =p \cdot \phip \circ \dd\, .$$ Conversely, if $p$ is a non-zero divisor in $A$, then (\[cucu\]) implies (\[didi\]).
A $\{\d,\dd\}$-[*ring*]{} $A$ is one equipped with a $p$-derivation $\d:A \ra A$ and a $\d$-derivation $\dd : A \ra A$. A [*morphisms of $\{\d,\dd\}$-rings*]{} is a ring homomorphism that commutes with $\d$ and $\dd$. A $\{\d, \dd\}$-ring $B$ is said to be a $\{\d,\dd\}$-[*ring over the $\{\d,\dd\}$-ring*]{} $A$ if it comes equipped with a $\{\d,\dd\}$-ring homomorphism $A\rightarrow B$. We say that a $\{\d,\dd\}$-ring $A$ is a $\D$-[*subring*]{} of the $\{\d,\dd\}$-ring $B$ if $A$ is a subring of $B$ such that $\d A \subset A$ and $\dd A \subset A$, respectively.
We describe next the basic examples of $\{\d,\dd\}$-rings that will play crucial rôles in our paper. All throughout, we shall let $R$ stand for $R:=R_p:=\widehat{\mathbb Z}_p^{ur}$, the completion of the maximum unramified extension of ${\mathbb Z}_p$, $k$ for the residue field $k:=R/pR$, and $K$ for the fraction field $K:=R[1/p]$. Furthermore, we will let $\mu(R)$ be the multiplicative group of roots of unity in $R$, and recall that the reduction mod $p$ mapping $$\mu (R) \rightarrow k^{\times}$$ defines an isomorphism whose inverse is the [*Teichmüller lift*]{}. Any element of the ring $R$ can be represented uniquely as a series $\sum_{i=0}^{\infty}
\zeta_i p^i $, where $\zeta_i \in \mu(R)\cup \{ 0\}$. There is a unique ring isomorphism $$\phi : R \rightarrow R \label{fr}$$ that lifts the $p$-th power Frobenius isomorphism on $k$, and for $\zeta \in \mu(R)$, we have that $\phi(\zeta)=\zeta^p$.
The ring $R$ is isomorphic to the Witt ring on the algebraic closure ${\mathbb F}_p^a$ of ${\mathbb F}_p$, and for each $s\geq 1$, the ring $$R^{\phi^s}:=\{ x\in R \, \mid \; x^{\phi^s}=x\}$$ is isomorphic to the Witt ring on the field ${\mathbb F}_{p^s}$ with $p^s$ elements. Notice that the ring $R^{\phi}={\mathbb Z}_p$ is simply the ring of $p$-adic integers. As usual, we denote by ${\mathbb Z}_{(p)}$ the ring of all fractions $a/b\in {\mathbb Q}$, where $a\in {\mathbb Z}$ and $b\in {\mathbb Z}\setminus (p)$. We have the inclusions $${\mathbb Z}_{(p)} \subset {\mathbb Z}_{p} \subset
R^{\phi^s}\subset R \, .$$
\[unex\] The ring $R$ carries a unique $p$-derivation $\d: R \ra R$ given by (see (\[me0\])) $$\d x=\frac{\phip(x)-x^p}{p}\, .$$ The [*constants of*]{} $\d$ are defined to be $$R^{\d}:=\{x\in R\, : \; \d x =0\}\, ,$$ set that coincides with $\mu(R)\cup \{0\}$, the roots of unity in $R$ together with $0$. Notice that we have the trivial $\d$-derivation $\dd =0$ on $R$, and that the pair $(\d, 0)$ equips $R$ with a $\D$-ring structure.
We denote by $R[[q]]$ and $R[[q^{-1}]]$ the power series rings in $q$ and $q^{-1}$, respectively, and we embed them into the rings $$R((q))^{\wh } =R[[q]][q^{-1}]^{\wh }=
\left\{ \sum_{n=-\infty}^{\infty} a_nq^n
\, | \; \lim_{n\rightarrow -\infty} a_n =0 \right\}$$ and $$R((q^{-1}))^{\wh}=R[[q^{-1}]][q]^{\wh}=
\left\{ \sum_{n=-\infty}^{\infty} a_nq^n \, | \;
\lim_{n\rightarrow \infty} a_n =0 \right\}\, ,$$ respectively.
The rings $R((q))^{\wh}$ and $R((q^{-1}))^{\wh}$ have unique structures of $\D$-rings that extend that of $R$, such that $$\d q=0\, ,\; \dd q=q\, , \; \d(q^{-1})=0\, , \; \dd(q^{-1})=-q^{-1}\, .$$ The rings $R[[q]]$ and $R[[q^{-1}]]$ are $\D$-subrings of $R((q))^{\wh}$ and $R((q^{-1}))^{\wh}$, respectively. Also the ring $$R[q,q^{-1}]^{\wh}= \left\{ \sum_{n=-\infty}^{\infty} a_nq^n \, | \;
\lim_{n\rightarrow \pm \infty} a_n =0 \right\}\,$$ is a $\D$-subring of both $R((q))^{\wh}$ and $R((q^{-1}))^{\wh}$.
We describe these $\D$-structures in further detail.
The automorphism (\[fr\]) extends to a unique homomorphism $\phi: R[[q]]\ra R[[q]]$ such that $\phi(q)=q^p$. Similarly, it extends to a unique homomorphism $\phi: R[[q^{-1}]]\ra
R[[q^{-1}]]$ such that $\phi(q^{-1})=q^{-p}$, to which for obvious reasons we give the same name. Then the expression $$\d F:=\frac{F^{\phi}-F^p}{p}\, .$$ defines a $p$-derivation $\d$ on both, $R[[q]]$ and $R[[q^{-1}]]$, respectively. On the other hand, the expression $$\dd F:=q \frac{dF}{dq}=q\partial_q F$$ defines a $\d$-derivation on $R((q))^{\wh}$ and $R((q^{-1}))^{\wh}$. Here, $$\partial_q \left( \sum a_n q^n \right)=\frac{d}{dq}
\left( \sum a_n q^n \right):= \sum na_n q^n \, .$$ The operators $\d$ and $\dd$ so defined provide the various rings above with their respective $\D$-ring structures.
Observe that the $R$-algebra mapping $$\begin{array}{ccc}
R((q))^{\wh} & \rightarrow & R((q^{-1}))^{\wh} \\
q & \mapsto & q^{-1}
\end{array}$$ is a ring isomorphism that fails to be a $\D$-ring isomorphism.
In the sequel, we will repeatedly use the following “Dwork’s Lemma,” which we record here for convenience.
\[dwor\] Let $v \in 1+qK[[q]]$ be such that $v^{\phi}/v^p \in 1+pqR[[q]]$. Then $v\in 1+qR[[q]]$.
[*Proof*]{}. For $v \in 1+q{\mathbb Q}_p[[q]]$, this is proved in [@koblitz], p. 93. The general case follows by a similar argument.
We will also need the following case of Hazewinkel’s Functional Equation Lemma. This special version of this result implies Dwork’s Lemma.
\[haze\] [@haze] Let $\mu_0 \in R^{\times}$, and $\mu_1,\ldots ,\mu_s \in
R$. Consider two series $f,g \in K[[q]]$ such that $f \equiv
\lambda q$ mod $q^2$ in $K[[q]]$, for some $\lambda \in
R^{\times}$. Let $f^{-1} \in qK[[q]]$ be the compositional inverse of $f$. For $$\Lambda:=\mu_0+\frac{\mu_1}{p} \phip+\cdots
+\frac{\mu_s}{p}\phip^s \in K[\phip]\, ,$$ assume that both, $\Lambda f$ and $\Lambda g$, belong to $R[[q]]$. Then $f^{-1} \circ g \in R[[q]]$.
Closely related to these results is the following.
\[floricica\] Let $\mu_0 \in R^{\times}$, $\mu_1,\ldots ,\mu_s
\in R$, and $f \in K[[q]]$. If for $$\Lambda:=\mu_0+\mu_1\phip+\cdots +\mu_s \phip^s \in R[\phip]$$ we have that $\Lambda f \in qR[[q]]$, then $f \in qR[[q]]$. If on the other hand we have that $\Lambda f \in pqR[[q]]$, then $f \in pqR[[q]]$.
[*Proof*]{}. We may assume $\mu_0=1$. Set $\Lambda_1:=1-\Lambda \in \phip
R[\phip]$, and note that $\Lambda_1 q R[[q]] \subset q^pR[[q]]$. Set $g=\Lambda f$. Then $$f=g+\Lambda_1 g +\Lambda_1^2 g +\cdots \, ,$$ and the result follows.
Formally speaking, the $\D$-rings $R((q))\wh$ and $R((q^{-1}))\wh$, and their various $\D$-subrings introduced above, can be viewed as rings of “Fourier series.” These rings allow us to take limits of solutions to arithmetic partial differential equations in the “time” direction, as $t\rightarrow
\pm i\infty$. On the other hand, the examples of $\D$-rings of “Iwasawa series,” which we discuss next, control limits “as time goes to $0$.”
\[ninna\] Let $R[[q-1]]$ be the completion of the polynomial ring $R[q]$ with respect to the ideal $(q-1)$. Hence, $R[[q-1]]$ can be identified with the power series ring $R[[\tau]]$ in an indeterminate $\tau$, and $R[q]\wh$ embeds into $R[[q-1]]=R[[\tau]]$ via the map $q \mapsto 1+\tau$.
There is a unique $\D$-ring structure on $R[[q-1]]$ that extends that of $R[q]\wh$. Indeed, in terms of $\tau$ we have $$\begin{array}{rcl}
\d \tau & = & {\displaystyle \frac{(1 + \tau)^p -1 -\tau^p}{p}}\, ,
\vspace{1mm} \\
\dd \tau & = & 1+\tau\, .
\end{array}$$ Note that although $R[[q]]$ and $R[[q-1]]$ are isomorphic as rings, they are not isomorphic as $\D$-rings. In fact, they are quite “different” in this latter context.
Similarly, let $R[[q^{-1}-1]]$ be the completion of $R[q^{-1}]$ with respect to the ideal $(q^{-1}-1)$. Then there is a natural embedding $R[q^{-1}]\wh \subset R[[q^{-1}-1]]$, and a unique structure of $\D$-ring on $R[[q^{-1}-1]]$ that extends that of $R[q^{-1}]\wh$.
From here on, we consider the ring $A$ provided with a fixed structure of $\D$- ring. For simplicity, we always assume that $A$ is a $p$-adically complete Noetherian integral domain of characteristic zero, and we shall let $L$ be its fraction field. Example \[unex\] above illustrates the cases where $A$ is equal to $R$, $R[[q]]$, $R((q))^{\wh}$, $R[[q^{-1}]]$, $R((q^{-1}))^{\wh}$, and $R[q,q^{-1}]^{\wh}$, respectively, while Example \[ninna\] illustrates the cases where $A$ is equal to $R[[q-1]]$ and $R[[q^{-1}-1]]$, respectively. Whenever any of these rings is considered below, it will be given the $\D$-ring structure defined in these two examples.
In Examples \[unneg\] and \[ex2\] below we explain some basic general constructions that can be performed using a fixed $\D$-ring $A$.
\[unneg\] We will denote by $(A[\phip,\dd],+, \, \cdot \, )$ the ring generated by $A$ and the symbols $\phip$ and $\dd$ subject to the relations $$\begin{array}{rcl}
\dd \cdot \phip & = & p\cdot \phip \cdot \dd\, , \\
\mbox{} [\dd,a]=\dd \cdot a -a\cdot \dd & = & \dd a\, , \\
\phip \cdot a & = & a^{\phi}\cdot \phip \, ,
\end{array}$$ for all $a\in A$. Let $\xi_p$, $\xi_q$ be two variables, which we view as “duals to $p$ and $q$,” respectively. If $\mu=\mu(\xi_p,\xi_q)=\sum
\mu_{ij} \xi_p^i \xi_q^j \in A[\xi_p,\xi_q]$ is a polynomial, then we define $$\mu(\phip,\dd):=\sum \mu_{ij} \phip^i \dd^j \in
A[\phip,\dd ] \, .$$ The map $$\begin{array}{rcl}
A[\xi_p,\xi_q] & \ra & A[\phip,\dd] \\
\mu(\xi_p,\xi_q) & \mapsto & \mu(\phip,\dd)
\end{array}$$ is a left $A$-module isomorphism.
Given a $\mu(\xi_p,\xi_q)$ as above, we define the polynomial $$\mu^{(p)}(\xi_p,\xi_q):=\mu(p\xi_p,\xi_q)\, .$$ We have the following useful formula: $$\dd \mu(\phip,\dd)=\mu^{(p)}(\phip,\dd) \dd\, .$$
\[ex2\] Let $y$ be an $N$-tuple of indeterminates over $A$, and let $y^{(i,j)}$ be an $N$-tuple of indeterminates over $A$ parameterized by non-negative integers $i$, $j$, such that $y^{(0,0)}=y$. We set $$A\{ y\}:=A[y^{(i,j)}\mid_{i\geq 0, j\geq 0}]$$ for the polynomial ring in the indeterminates $y^{(i,j)}$. This ring has a natural $\D$-structure that we now discuss.
For let $\phip : A\{ y\} \rightarrow A\{ y\}$ be the unique ring homomorphism extending $\phip : A \rightarrow A$, and satisfying the relation $$\phip (y^{(i,j)})= (y^{(i,j)})^p + p y^{(i+1,j)} \, .$$ Then, we may define a $p$-derivation $\d: A\{ y\} \rightarrow A\{ y\}$ by the expression (see (\[me0\]) above) $$\d F: = \frac{\phip(F)-F^{p}}{p} \, .$$ In particular, we have that $\d y^{(i,j)}=y^{(i+1,j)}$.
We let $L\{y\}$ be $L\{y\}:=L\otimes_A A\{ y\}$, $L$ the field of fractions of $A$. Notice that $$L\{y\}=L[y^{(i,j)}\mid_{i\geq 0, j\geq 0}]=L[\phip^{i}y^{(0,j)}
\mid_{i\geq 0,
j\geq 0}]\, .$$ The endomorphism $\phip$ extends uniquely to an endomorphism $\phip$ of $L\{y\}$. Since the polynomials $\phi^{i}y^{(0,j)}$ are algebraically independent over $L$, there exists a unique derivation $\dd: L\{ y\} \rightarrow L\{ y\}$ that extends the derivation $\dd : A \rightarrow A$, and satisfies the relation $$\dd (\phip^{i}y^{(0,j)})=p^i \phip^{i}y^{(0,j+1)}\, .$$ We claim that $D:=\dd \circ \phip -p\cdot \phip \circ \dd$ vanishes on $L\{ y\}$.
Indeed, $D: L\{ y\} \rightarrow L\{ y\}$ is a derivation if we view $L\{ y\}$ as an algebra over itself via $\phip$. Thus, it suffices to observe that $D$ vanishes on $L$ and $\phip^{i}y^{(0,j)}$, and both of these are clear.
We now claim also that $\dd y^{(i,j)}\in A\{y\}$, and this will imply that $\dd$ induces a derivation of the ring $A\{ y\}$. For, proceeding by induction on the index $i$, with the case $i=0$ being clear, we assume it for $i$, and prove the desired statement for $i+1$. We have $$\begin{array}{rcl}
\dd y^{(i+1,j)} & = & {\displaystyle \dd \left( \frac{\phip(y^{(i,j)})-
(y^{(i,j)})^p}{p}\right)}\vspace{1mm} \\ & = &
\phip(\dd y^{(i,j)})-(y^{(i,j)})^{p-1}
\dd y^{(i,j)} \, ,
\end{array}$$ and this is clearly in $A\{ y\}$ by the induction hypothesis.
The morphisms $\d$, $\dd $ endow $A\{ y\}$ with a $\D$-ring structure. Clearly, $y^{(i,j)}=\d^i \dd^j y$, and so $$A\{ y\} = A[y, Dy, D^2y, \ldots ]\, ,$$ where for any whole number $n$, $D^ny$ stands for the $(n+1)$-tuple with components $\d^i \dd^{n-1}y$, $0\leq i\leq n$.
Let $S^{*}=\{S^n\}_{n\geq 0}$ be a sequence of rings. Suppose we have ring homomorphisms $\varphi: S^n \ra S^{n+1}$, $p$-derivations $\d:S^n \ra S^{n+1}$, and $\d$-derivations $\dd
:S^n \ra S^{n+1}$ such that $\d \circ \varphi=\varphi \circ \d$, and $\dd \circ \varphi=\varphi \circ \dd$. We then say that $(S^{*}, \varphi,\d,\dd)$, or simply $S^{*}$, is a $\{\d,\dd\}$-[*prolongation sequence*]{}. A [*morphism*]{} $u^*:S^* \ra \tilde{S}^*$ of $\D$-prolongation sequences is a sequence $u^n:S^n \ra \tilde{S}^n$ of ring homomorphisms such that $\d \circ u^n=u^{n+1} \circ \d$, $\dd \circ u^n=u^{n+1} \circ
\dd$, and $\varphi \circ u^n=u^{n+1} \circ \varphi$.
Let us consider the $\D$-ring structure on the ring $A$. We obtain a natural $\D$-prolongation sequence $A^{*}$ by setting $A^n=A$ for all $n$, and taking the ring homomorphisms $\varphi$ to be all equal to the identity. This leads to the natural concept of morphisms of $\D$-prolongation sequences over $A$.
We say that a $\D$-prolongation sequence $S^*$ is a [*$\D$-prolongation sequence over*]{} $A$ if we have a morphism of prolongation sequences $A^* \ra S^*$.
There is a natural notion of morphisms of $\D$-prolongation sequences over $A$ that we do not explicitly state. In the sequel, all $\D$-prolongation sequences, and morphisms of such, will be prolongation sequences over $A$.
Let $A\{ y\}$ the ring discussed in Example \[ex2\]. We consider the subrings $$S^n:=A[y, Dy, \ldots, D^n y ]\, .$$ We view $S^{n+1}$ as an $S^n$-algebra via the inclusion homomorphism, and observe that $\d S^n \subset S^{n+1}$, and $\dd S^n \subset S^{n+1}$, respectively. Therefore, $S^{*}=\{ S^n\}$ defines a $\D$-prolongation sequence. We then obtain the $p$-adic completion prolongation sequence $$A[y, Dy, \ldots, D^ny ]^{\widehat{\mbox{\phantom{p}}}}
\, ,$$ and the prolongation sequence $$A[[Dy, \ldots, D^n y ]]$$ of formal power series ring, with their corresponding $\D$-structures.
Let $X$ and $Y$ be smooth schemes over the fixed $\D$-ring $A$. By a [*$\D$-morphism of order $r$*]{} we mean a rule $f:X \ra Y$ that attaches to any $\D$-prolongation sequence $S^*$ of $p$-adically complete rings, a map of sets $X(S^0) \ra Y(S^r)$ that is “functorial” in $S^*$ in the obvious sense.
For any $\D$-prolongation sequence $S^*$, the shifted sequence $S^{*}[i]$, $S[i]^n:=S^{n+i}$, is a new $\D$-prolongation sequence. Thus, any morphism $f:X \ra Y$ of order $r$ induces maps of sets $X(S^i) \ra Y(S^{r+i})$ that are functorial in $S^*$. We can compose $\D$-morphisms $f:X \ra Y$, $g:Y \ra Z$ of orders $r$ and $s$, respectively, and get a $\D$-morphisms $g \circ f:X \rightarrow Z$ of order $r+s$. There is a natural map from the set of $\D$-morphisms $X \ra Y$ of order $r$ into the set of $\D$-morphisms $X \ra Y$ of order $r+1$, induced by the maps $Y(S^r) \ra Y(S^{r+1})$ arising from the $S^r$-algebra structure of $S^{r+1}$.
Recall that given a scheme $X$ over the ring $A$, if $B$ is an $A$-algebra we let $X(B)$ denote the set of all morphisms of $A$-schemes ${\rm Spec}\, B \rightarrow X$, and any such morphism is called a $B$-[*point*]{} of $X$. For instance, recall that if $X={\mathbb A}^1={\mathbb G}_a={\rm Spec}\, A[y]$, then $X(B)$ is simply the set $B$ itself because a morphism ${\rm Spec}\, B
\rightarrow {\rm Spec}\, A[y]$ is the same as a morphism $A[y]
\rightarrow
B$, and the latter is uniquely determined by the image of $y$ in $B$. If on the other hand, $X={\mathbb G}_m={\rm Spec}\, A[y,y^{-1}]={\rm Spec}\,
A[x,y]/(xy-1)$, then $X(B)=B^{\times}$ because ${\rm Hom}_{A}(A[y,y^{-1}],
B) =B^{\times}$ via the map $f \mapsto f(y)$. Finally, if $X={\rm Spec}\, A[x,y]/(f(x,y))$, then $X(B)=\{(a,b) \in B^2: \, f(a,b)=0\}$.
Let $G$ and $H$ be smooth group schemes over $A$. We say that $G
\ra H$ is a $\D$-[*homomorphism of order*]{} $r$ if it is a $\D$-morphism of order $r$ such that, for any prolongation sequence $S^{*}$, the maps $X(S^0) \ra Y(S^r)$ are group homomorphisms. A $\D$-[*character*]{} of order $r$ of $G$ is a $\D$-homomorphism $G \rightarrow {\mathbb G}_a$ of order $r$, where ${\mathbb G}_a={\rm Spec}\, A[y]$ is the additive group scheme over $A$. The group of $\D$-characters of order $r$ of $G$ will be denoted by $\bX^r_{\di \ddi}(G)$.
Note that the group $\bX^r_{\di \ddi}(G)$ has a natural structure of $A[\phip,\dd]$-module. We shall view the $\D$-characters of $G$ as “linear partial differential operators” on it. Of course, they are highly “non-linear” in the affine coordinates around various points of $G$.
Let $\psi:G \ra {\mathbb G}_a$ be a $\D$-character of a commutative smooth group scheme $G$ over $A$. For any $\D$-ring $B$ over $A$, $\psi$ induces a ${\mathbb Z}$-linear map $$\label{9991} \psi:G(B) \ra {\mathbb G}_a(B)=B\, .$$
The [*group of solutions*]{} of $\psi$ in $B$ is the kernel of the map [(\[9991\])]{}, that is to say, the group $$\{u \in G(B)\ | \ \psi u=0\}\, .$$ Given a subgroup $\Gamma$ of $G(B)$, the [*group of solutions*]{} of $\psi$ in $\Gamma$ is will be the group $$\{u \in \Gamma\ | \ \psi u=0\}\, .$$
\[cucurigu\] Let us assume that $A=R$, and that $G$ is a commutative smooth group scheme over $R$. We may consider the natural ring homomorphisms $$\begin{array}{rcl}
R[[q]] & \ra & R \\
U(q)=\sum_{n \geq 0} a_n q^n & \mapsto & U(0):=\left( \sum_{n \geq 0}
a_n q^n \right)_{|q=0}= a_0
\end{array}$$ and $$\begin{array}{rcl}
R[[q^{-1}]] & \ra & R \\
U(q)=\sum_{n \leq 0} a_n q^n & \mapsto &
U(\infty):=\left( \sum_{n \leq 0} a_n q^n \right)_{|q^{-1}=0}=
a_0 \,
\end{array}$$ respectively, and the corresponding induced group homomorphisms $$\label{mabuze}
\begin{array}{ccc}
G(R[[q]]) & \ra & G(R) \\
u & \mapsto &
u(0)
\end{array}$$ and $$\label{mabuze2}
\begin{array}{ccc}
G(R[[q^{-1}]]) & \ra & G(R) \\
u & \mapsto & u(\infty)
\end{array}\, .$$ If $e$ denotes the identity, we have the natural diagram of groups $$\label{dia}
\begin{array}{ccccc}
G(qR[[q]]) & \ra & G(R[[q]]) & \ra & G(R((q))\wh)\\
\ua & \ & \ua & \ & \ua\\
\{e\} & \ra & G(R) & \ra & G(R[q,q^{-1}]\wh)\\
\da & \ & \da & \ & \da\\
G(q^{-1}R[[q^{-1}]]) & \ra & G(R[[q^{-1}]]) & \ra &
G(R((q^{-1}))\wh)
\end{array}\, ,$$ where $G(qR[[q]])$ and $G(q^{-1}R[[q^{-1}]])$ are the kernels of the maps in (\[mabuze\]) and (\[mabuze2\]), respectively.
We will think of the spectrum of the ring $A=R((q))^{\widehat{\phantom{p}}}$ as the $pq$ “plane.” The $p$-axis is the “arithmetic” direction, while the $q$-axis is “geometric;” $q$ will be viewed as the “exponential of $-2 \pi it$” where $t$ is “time.” The operators $\d$ and $\dd$ are “vector fields” along these two directions.
Under these interpretations, the elements of $R((q))^{\widehat{\phantom{p}}}$ play the role of “functions in the variables” $p,q$. By considering (infinitely many) negative powers of $q$, we allow the corresponding “function” to have an “essential singularity as time approaches $-i \infty$.” The elements $u=\sum_{n \geq 0} a_nq^n$ of the power series ring $R[[q]]$ inside $R((q))^{\widehat{\phantom{p}}}$ are viewed as the analogues of functions in two variables that, as time goes to infinity, approach a well defined limit function of the spatial variable, the function $a_0=u(0)$. Here, the elements of the monoid $R^{\d}=\{ x\in R : \; \d x=0\}$ are to be thought of as the “constant functions.”
Let $\psi$ be a $\D$-character of $G$. Considering the groups of solutions of $\psi$ in the various groups of (\[dia\]) above, we get a diagram $$\label{dia2}
\begin{array}{ccccc}
\cU_1 & \ra & \cU_+ & \ra & \cU_{\ra}\\
\ua & \ & \ua & \ & \ua\\
\{e\} & \ra & \cU_0 & \ra & \cU_{\da}\\
\da & \ & \da & \ & \da\\
\cU_{-1} & \ra & \cU_- & \ra & \cU_{\la}
\end{array}\, .$$ We will usually denote by $\cU_*$ the groups in the above diagram. The elements of $\cU_0$ will be referred to as [*stationary solutions*]{} of $\psi$, and are interpreted as solutions that “do not depend on time.”
For $q_0 \in pR$, we may consider the ring homomorphism $$\begin{array}{ccc}
R[[q]] & \ra & R \\
q & \mapsto & q_0
\end{array}\, .$$ This is a $\D$-ring homomorphism for $q_0=0$, but not for $q_0 \neq 0$. The embedding ${\rm Spec}\, R \ra {\rm Spec}\, R[[q]]$ will be viewed as a “curve” in the “$pq$-plane.”
Corresponding to the homomorphism above, we may consider the specialization homomorphism $$\begin{array}{ccc}
G(R[[q]]) & \ra & G(R) \\
u & \mapsto & u(q_0)
\end{array}\, .$$ The “curve” associated to $q_0 \neq 0$ will be thought of as a curve along which we impose “boundary conditions.” For if $u_0
\in G(R)$, we will consider the “boundary value problem” with that initial data, that is to say, the problem of finding a solution (possibly unique) $u \in \cU_{+}$ such that $u(q_0)=u_0$. When $q_0=0$, $u(0) \in G(R)$ is thought of as the “limit of $u$ as time goes to infinity,” as mentioned earlier. This situation can be similarly stated for $R[[q^{-1}]]$ instead of $R[[q]]$.
We claim that the following hold: $$\label{moderate}
\begin{array}{rclll}
\psi(u(0)) & = & (\psi u)(0) & \text{for} & u \in G(R[[q]])\, ,\\
\psi(u(\infty)) & = & (\psi u)(\infty) & \text{for} & u \in
G(R[[q^{-1}]])\, .
\end{array}$$ Indeed, in order to prove the first of these identities, it is enough to observe that, by the functorial definition of $\psi$, we have a commutative diagram $$\begin{array}{ccc}
G(R[[q]]) & \stackrel{\psi}{\ra} & R[[q]]\\
\da & \ & \ \da\\
G(R) & \stackrel{\psi}{\ra} & R
\end{array}\, ,$$ where the vertical arrows are induced by the $\D$-ring homomorphism $$\begin{array}{ccc}
R[[q]] & \ra & R \\ q & \mapsto & 0
\end{array}\, .$$ The argument for the second of the identities is similar.
The identities in (\[moderate\]) imply that $$\label{sarrah} \cU_{\pm} = \cU_{0} \times \cU_{\pm 1},$$ where $\times$ denotes the “internal direct product.”
We may unify this picture by considering the various groups of solutions $\cU_*$ as subgroups of a single group. Indeed, let us consider the direct sum homomorphism $$\psi \oplus \psi\, : \; \frac{G(R((q))\h) \oplus
G(R((q^{-1}))\h)}{G(R[q,q^{-1}]\h)} \ra \frac{R((q))\h \oplus
R((q^{-1}))\h}{R[q,q^{-1}]\h}\, ,$$ where the denominators in the domain and range are diagonally embedded into the numerators. We then set $$\cU := {\rm ker}\, (\psi \oplus \psi) \, ,$$ and refer to this group as the [*group of generalized solutions*]{} of $\psi$. Its elements are the analogues of the generalized solutions of Sato.
The groups $\cU_{\la}$ and $\cU_{\ra}$ naturally embed into $\cU$ via the maps $x \mapsto
(-x,0)$ and $x \mapsto (0,x)$, respectively, and the restrictions of these two embeddings to $\cU_{\da}$ coincide. Thus, all the groups $\cU_*$ in the diagram (\[dia2\]) can be identified with subgroups of $\cU$, and we have $$\begin{array}{rcl}
\cU_{\ra} \cap \cU_{\la} & = & \cU_{\da}\, , \\
\cU_+ \cap \cU_- & = & \cU_0\, , \\
\cU_{1} \cap \cU_{-1} & = & \{e\}\, .
\end{array}$$ Note that, a priori, we do not have $\cU=\cU_{\ra} \cdot
\cU_{\la}$.
Next we introduce a concept of [*convolution*]{} in our setting. Let ${\mathbb
Z}\mu(R)$ be the group ring of the group $\mu(R)$. That is to say, ${\mathbb Z} \mu(R)$ is the set of all functions $f:\mu(R) \ra
{\mathbb Z}$ of finite support, equipped with pointwise addition, and multiplication $\star$ given by convolution $$(f_1 \star f_2)(\zeta):=\sum_{\zeta_1 \zeta_2=\zeta}
f_1(\zeta_1) f_2(\zeta_2)\, .$$ For any integer $\kappa \in \bZ$, we consider the ring homomorphism $$\label{cuofi}
\begin{array}{ccc}
\bZ \bmu(R) & \stackrel{[\kappa]}{\ra} & \bZ \bmu(R)\\
f & \mapsto & f^{[\kappa]}
\end{array}\, ,$$ where $f^{[\kappa]}(\zeta):= f(\zeta^{\kappa})$. If ${\mathbb
Z}[\mu(R)] \subset R$ is the subring generated by $\mu(R)$, then we have a natural surjective (but not injective) ring homomorphism $$\begin{array}{ccc}
{\mathbb Z} \mu(R) & \rightarrow & {\mathbb Z}[\mu(R)] \\
f & \mapsto & f^{\sharp}:=\sum_{\zeta} f(\zeta) \zeta
\end{array}\, .$$
Now let $R((q^{\pm 1}))\wh$ be either $R((q))\wh$ or $R((q^{-1}))\wh$, respectively. For $\zeta \in \mu(R)$, we may consider the $R$-algebra isomorphism $$\begin{array}{ccc}
\sigma_{\zeta}:R((q^{\pm 1}))\wh & \ra & R((q^{\pm 1}))\wh \\
u(q) & \mapsto & u(\zeta q)
\end{array}$$ that induces a group isomorphism $$\sigma_{\zeta}:G(R((q^{\pm 1}))\wh ) \ra G(R((q^{\pm 1}))\wh)\, .$$ Hence, the group $G(R((q^{\pm 1}))\wh)$ acquires a natural structure of ${\mathbb Z}
\mu(R)$-module via convolution. Indeed, for any $f \in {\mathbb Z} \mu(R)$ and $u \in G(R((q^{\pm 1}))\wh)$, we have $$f \star u:=\sum_{\zeta} f(\zeta) \sigma_{\zeta}(u)\, ,$$ essentially a “superposition" of translates of $u$.
Note now that the homomorphism $\sigma_{\zeta}:R((q^{\pm 1}))\wh \ra R
((q^{\pm 1}))\wh $ is actually a $\D$-ring homomorphism: the commutation of $\sigma_{\zeta}$ and $\dd$ is clear, while the commutation of $\sigma_{\zeta}$ and $\d$ follows because $$\phip(\sigma_{\zeta}(q^{\pm1}))=\phip(\zeta^{\pm1} q^{\pm1})=
\zeta^{\pm \phi}
q^{\pm p}=\zeta^{\pm p}q^{\pm p}=\sigma_{\zeta}(q^{\pm p})=\sigma_{\zeta}(
\phip(q^{\pm}))\, .$$
If $\psi$ is a $\D$-character of $G$, by the functorial definition of $\D$-characters we obtain a commutative diagram $$\begin{array}{ccc}
G(R((q^{\pm 1}))\wh ) & \stackrel{\psi}{\ra} & R((q^{\pm 1}))\wh\\
\sigma_{\zeta} \da & \ & \da \sigma_{\zeta}\\
G(R((q^{\pm 1}))\wh ) & \stackrel{\psi}{\ra} & R((q^{\pm 1}))\wh
\end{array}\, .$$ Hence, for any $f \in {\mathbb Z} \mu(R)$ and $u \in G(R((q^{\pm
1}))\wh)$, we have $$\psi(f \star u)=f \star \psi(u)\, .$$ That is to say, $\psi$ is $\bZ \bmu(R)$- module homomorphism. In particular, the groups $\cU_{\pm 1}$ of solutions of $\psi$ in $G(R((q^{\pm 1}))\wh)$ are ${\mathbb Z} \mu(R)$- submodules of $G(R(( q^{\pm 1}))\wh)$ respectively. Morally speaking, this says that a superposition of translates of a solution is again a solution.
Let us additionally assume that $G$ has relative dimension one over $R$. Let $T$ be an étale coordinate on a neighborhood of the zero section in $G$ such the zero section is given scheme theoretically by $T=0$. Then the ring of functions on the completion of $G$ along the zero section is isomorphic to a power series ring $R[[T]]$. We fix such an isomorphism. Then we have an induced group isomorphism $$\label{hacha}
\iota: q^{\pm 1}R[[q^{\pm 1}]] \ra G(q^{\pm 1}R[[q^{\pm 1}]])\, ,$$ where $q^{\pm 1}R[[q^{\pm 1}]]$ is a group relative to a formal group law $\cF(T_1,T_2) \in R[[T_1,T_2]]$ attached to $G$. We recall that the series $[p^n](T) \in R[[T]]$, defined by multiplication by $p^n$ in the formal group $\cF$, belongs to the ideal $(p,T)^n$. We equip $G(q^{\pm 1}R[[q^{\pm 1}]])$ with the topology induced via $\iota$ from the $(p,q^{\pm 1})$-adic topology on $R[[q^{\pm 1}]]$. Since $G(q^{\pm 1}R[[q^{\pm 1}]])$ is complete in this topology, the ${\mathbb Z} \mu(R)$-module structure of $G(q^{\pm 1}R[[q^{\pm 1}]])$ extends uniquely to a ${\mathbb Z} \mu(R)\wh$-module structure on $G(q^{\pm 1}R[[q^{\pm
1}]])$ in which multiplication by scalars is continuous. Here, ${\mathbb Z} \mu(R)\wh$ is, of course, the $p$-adic completion of ${\mathbb Z} \mu(R)$. We observe that the ring homomorphism $$\label{notinj} \begin{array}{rcl}
{\mathbb Z} \mu(R)\wh & \stackrel{\sharp}{\ra} & R \\
f & \mapsto & f^{\sh}:=\sum_{\zeta} f(\zeta)\zeta
\end{array}$$ is surjective (but not injective). Clearly, the groups $\cU_{\pm
1}$ of solutions of $\psi$ are ${\mathbb Z} \mu(R)\wh$- submodules of $\cU_{\pm}$, respectively.
Also, we note that a Mittag-Leffler argument shows that for any $\kappa$ coprime to $p$ the map $[\kappa]:\bZ \bmu(R)\h \ra \bZ
\bmu(R) \h$ is surjective.
We end our discussion here by introducing boundary value operators at $q^{\pm 1}=0$. We start by considering, for any $0 \neq \kappa
\in \bZ$, operators $$\begin{array}{rcl}
\Gamma_{\kappa}:R[[q^{\pm 1}]] & \ra & R \vspace{1mm} \\
u=\sum a_n q^n & \mapsto & \Gamma_{\kappa} u=a_{\kappa}
\end{array}\, .$$ We fix also a $\dd$-character $\psi_q$ of $G$. In the applications, for $G=\bG_a$, $\psi_q$ will be the identity; and for $G$ either $\bG_m$ or an elliptic curve $E$ over $R$, $\psi_q$ will be the “Kolchin logarithmic derivative.” If we fix a collection of non-zero integers ${\mathcal K}_{\pm} \subset
\bZ_{\pm}$, and set $\rho_{\pm}:=\sharp {\mathcal K}_{\pm}$, then we may consider the [*boundary value operator at $q^{\pm
1}=0$*]{}, $$\label{vocea}
\begin{array}{rcl}
G(R[[q^{\pm 1}]]) & \stackrel{B_{\pm}^0}{\ra} & R^{\rho_{\pm}}\\
B_{\pm}^0 u & = & (B_{\kappa}^0 u)_{\kappa \in \cK_{\pm}}
\end{array} \, ,$$ where $B_{\kappa}^0 u:= \Gamma_{\kappa} \psi_q u$. Here we view $R^{\rho_{\pm}}$ as a direct product $R^{\rho_{\pm}}=\prod_{\kappa \in \cK_{\pm}} R_{\kappa}$ of copies $R_{\kappa}$ of $R$ indexed by $\cK_{\pm}$.
It is easy to check that $B_{\pm}^0$ is a $\bZ \bmu(R)$-module homomorphism provided that $R^{\rho_{\pm}}$ be viewed as a $\bZ \bmu(R)$-module with the following structure: for each $\kappa \in {\mathcal K}_{\pm}$, the copy of $R$ indexed by $\kappa$ in $R^{\rho_{\pm}}$ is a $\bZ \bmu(R)$-module via the ring homomorphism $$\label{notinjj}
\begin{array}{c}
\bZ \bmu(R) \stackrel{[\kappa]}{\ra} \bZ \bmu(R)
\stackrel{\sh}{\ra} R\, .
\end{array}$$ In other words, for $f \in \bZ \bmu(R)$ and $(r_{\kappa})_{\kappa}
\in R^{\rho_{\pm}}$, we have that $$f \cdot
(r_{\kappa})_{\kappa}:=((f^{[\kappa]})^{\sharp}
r_{\kappa})_{\kappa}.$$ The restriction of $B_{\pm}^0$ to $G(q^{\pm 1}R[[q^{\pm 1}]])$, $$\begin{array}{rcl}
B_{\pm}^0:G(q^{\pm 1}R[[q^{\pm 1}]]) & \ra & R^{\rho_{\pm}},
\end{array}$$ is a $\bZ \bmu(R)\h$-module homomorphism where $R^{\rho_{\pm}}$ is viewed as a $\bZ \bmu(R)\h$-module with the following structure: for each $\kappa \in {\mathcal K}_{\pm}$, the copy of $R$ indexed by $\kappa$ in $R^{\rho_{\pm}}$ is a $\bZ \bmu(R)\h$-module via the ring homomorphism $$\label{notinjjj}
\begin{array}{c}
\bZ \bmu(R)\h \stackrel{[\kappa]}{\ra} \bZ \bmu(R)\h
\stackrel{\sh}{\ra} R\, .
\end{array}$$ It our applications, it will sometimes be the case that the $\bZ
\bmu(R)\h$-module structure of $\cU_{\pm 1}$ induces a certain structure of $R$-module, and the restriction of $B_{\pm}^0$ to $\cU_{\pm 1}$ becomes an $R$-module homomorphism.
The discussion in the previous Example applies to the case where $G$ is one of the following group schemes over $A=R$:
1. $G={\mathbb G}_a:={\rm Spec}\, R[y]$, the additive group.
2. $G={\mathbb G}_m:={\rm Spec}\, R[y,y^{-1}]$, the multiplicative group.
3. $G=E$, an elliptic curve over $R$.
The discussion, however, does not apply to the following case, which will also interest us later:
1. $G=E$, an elliptic curve over $A=R((q))\h$ that does not descend to $R$, for instance, the Tate curve.
In this latter case, we will need new definitions for some of the groups (\[dia2\]) (cf. the discussion in our subsection on the Tate curve).
Partial differential jet spaces of schemes
==========================================
In this section we introduce arithmetic-geometric partial differential jet spaces of schemes. We do this by analogy with the ordinary differential ones in geometry [@hermann] and arithmetic [@char; @difmod; @book], respectively. We also record some of their general properties (that can be proven in a manner similar to the corresponding proofs in the ordinary case [@book], proofs that, therefore, will be omitted here.) We recall that the $\D$-ring $A$ we have fixed is a $p$-adically complete Noetherian integral domain of characteristic zero.
For any scheme $X$ of finite type over $A$, we define a projective system of $p$-adic formal schemes $$\label{fite}
\cdots \ra J^r_{\di \ddi}(X) \ra J^{r-1}_{\di \ddi}(X) \ra
\cdots \ra J^1_{\di \ddi}(X) \ra J^0_{\di \ddi}(X)=\hat{X}\, ,$$ called the $\D$-[*jet spaces*]{} of $X$.
Let us assume first that $X$ is affine, that is to say, $X={\rm Spec}\, A[y]/I$, where $y$ is a tuple of indeterminates. We then set $$J^n_{\di \ddi}(X):=Spf\ A[y,Dy,\ldots ,D^n y]\h/(I,DI,\ldots ,D^n I)\, .$$ The sequence of rings $\{\cO(J^n_{\di \ddi}(X))\}_{n \geq 0}$ has a natural structure of $\D$-prolongat- ion sequence $\cO(J^*_{\di \ddi}(X))$, and the latter has the following universality property that can be easily checked.
\[universality\] For any $\D$-prolongation sequence $S^*$ that consists of $p$-adically complete rings, and for any homomorphism $u:\cO(X) \ra S^0$ over $A$, there is a unique morphism of $\D$-prolongation sequences $$u_*=(u_n):\cO(J^*_{\di \ddi}(X)) \ra S^*\, ,$$ with $u_0=u$.
[*Proof*]{}. Similar to [@book], Proposition 3.3.
As a consequence we get that, for affine $X$, the construction $X \mapsto J^r(X)$ is compatible with localization in the following sense:
\[ma duc acum\] If $X={\rm Spec}\, B$ and $U={\rm Spec}\, B_f$, $f \in B$, then $$\cO(J^r_{\di \ddi}(U))=(\cO(J^r_{\di \ddi}(X))_f)\wh\, .$$ Equivalently, $$J^r_{\di \ddi}(U)=J^r_{\di \ddi}(X) \times_{\hat{X}} \hat{U}\, .$$
[*Proof*]{}. Similar to [@book], Corollary 3.4.
Consequently, for $X$ that is not necessarily affine, we can define a formal scheme $J^r_{\di \ddi}(X)$ by gluing the various schemes $J^r_{\di
\ddi}(U_i)$ for $\{U_i\}$ an affine Zariski open covering of $X$. The resulting formal scheme will have a corresponding universality property, whose complete formulation and verification we leave to the reader.
If $\bG_a ={\rm Spec}\, A[y]$ is the additive group scheme over $A$, then $$J^n_{\di \ddi}(\bG_a)=Spf\ A[y,Dy,\ldots,D^n y]\wh \, .$$ If $\bG_m ={\rm Spec}\, A[y,y^{-1}]$ is the multiplicative group scheme over $A$, then $$J^n_{\di \ddi}(\bG_m)=Spf\ A[y,y^{-1},Dy,\ldots,D^n y]\wh \, .$$
By the universality property of jet spaces, we have that $$J^n_{\di \ddi}(X \times Y) \simeq J^n_{\di \ddi}(X)
\times J^n_{\di \ddi}(Y)$$ where the product on the left hand side is taken in the category of schemes of finite type over $A$, and the product on the right hand side is taken in the category of formal schemes over $A$. In the same vein, if $X$ is a group scheme of finite type over $A$, then (\[fite\]) is a projective system of groups in the category of $p$-adic formal schemes over $A$.
By the universality property of jet spaces, the set of order $r$ $\D$-morphisms $X \ra Y$ between two schemes of finite type over $A$ naturally identifies with the set of morphisms over $A$ of formal schemes $J^r_{\di \ddi}(X) \ra J^0_{\di \ddi}(Y)=\hat{Y}$. In particular, the set $\cO^r_{\di \ddi}(X)$ of all order $r$ $\D$-morphisms $X \ra \hat{\bA}^1$ identifies with the ring of global functions $\cO(J^r_{\di \ddi}(X))$. If $G$ is a group scheme of finite type over $A$, then the group $\bX_{\di \ddi}^r(G)$ of order $r$ $\D$-characters $G \ra \bG_a$ identifies with the group of homomorphisms $J^r_{\di \ddi}(G) \ra \hat{\bG}_a$, and thus, it identifies with an $A$-submodule of $\cO(J^r_{\di
\ddi}(G))$. Let $$\cO^{\infty}_{\di \ddi}(X):=\lim_{\ra} \cO^r_{\di \ddi}(X)$$ be the $\D$-ring of all $\D$-morphisms $X \ra \bA^1$, and let $$\bX^{\infty}_{\di \ddi}(X):=\lim_{\ra} \bX^r_{\di \ddi}(X)$$ be the group of $\D$-characters $G \ra \bG_a$. Then $\cO^{\infty}_{\di \ddi}(X)$ has a natural structure of $A[\phip,\dd]$-module, and $\bX^{\infty}_{\di \ddi}(X)$ is an $A[\phip,\dd]$-submodule.
\[local\] Let $X$ be a smooth affine scheme over $A$, and let $u:A[y] \ra
\cO(X)$ be an étale morphism, where $y$ is a $d$-tuple of indeterminates. Let $y^{(i,j)}$ be $d$-tuples of indeterminates, where $i,j \geq 0$. Then the natural morphism $$\cO(\hat{X})[y^{(i,j)}\mid _{1 \leq i+j \leq n} ]\wh \ra \cO(J^n_{\di
\ddi}(X))$$ that sends $y^{(i,j)}$ into $\d^i \dd^j (u(y))$ is an isomorphism. In particular, we have an isomorphism of formal schemes over $A$ $$J^n_{\di \ddi}(X) \simeq \hat{X} \times \hat{\bA}^{\frac{n(n+3)}{2}d}\, .$$
[*Proof*]{}. The argument is similar to that used in Proposition 3.13 of [@book].
If $Y \ra X$ is an étale morphism of smooth schemes over $A$ then $$J^n_{\di \ddi}(Y) \simeq J^n_{\di \ddi}(X) \times_{\hat{X}}
\hat{Y}\, .$$
The jet spaces with respect to a derivation [@annals] that arises in the Ritt-Kolchin theory [@kolchin] [@ritt] are “covered" by of our $\D$-jet spaces here. The same holds for the $p$-jet spaces (with respect to a $p$-derivation) constructed in [@char]. More precisely, if we for affine $X/A$ we set $$\begin{array}{rcl}
J^n_{\ddi}(X) & := & {\rm Spec}\, A[y,\dd y,\ldots,\dd^n y]/(I,\dd I,
\ldots,\dd^n I)\, , \\
J^n_{\di}(X) & := & Spf\ A[y,\d y,\ldots,\d^n y]\h/(I,\d I,\ldots,\d^n I)\, ,
\end{array}$$ then we have natural morphisms $$\begin{array}{rcl}
J^n_{\di \ddi}(X) & \ra & J^n_{\ddi}(X)\wh \, , \\
J^n_{\di \ddi}(X) & \ra & J^n_{\di}(X)\, .
\end{array}$$ The same holds then for any scheme $X$, not necessarily affine. The elements of $\cO^r_{\ddi}(X):= \cO(J^r_{\ddi}(X))$ identify with the $\dd$-morphisms $X \ra \bA^1$; the elements of $\cO^r_{\di}(X):=\cO(J^r_{\di}(X))$ identify with the $\d$-morphisms $X \ra \bA^1$. The homomorphisms $J^r_{\ddi}(G) \ra
\hat{\bG}_a$ identify with the $\dd$-characters $G \ra \bG_a$; the homomorphisms $J^r_{\di}(G) \ra \hat{\bG}_a$ identify with the $\d$-characters $G \ra \bG_a$.
\[ginge\] The functors $X \mapsto J^r_{\di \ddi}(X)$ from the category ${\mathcal C}$ of $A$-schemes of finite type to the category $\hat{\mathcal C}$ of $p$-adic formal schemes naturally extends to a functor from ${\mathcal B}$ to $\hat{\mathcal C}$, where ${\mathcal B}$ is the category whose objects as the same as those in ${\mathcal C}$, hence $Ob\, {\mathcal C}=Ob\, {\mathcal
B}$, and whose morphisms are defined by $${\rm Hom}_{\mathcal B}(X,Y):={\rm Hom}_{\hat{\mathcal C}}(\hat{X},\hat{Y})$$ for all $X,Y \in Ob \, {\mathcal B}$.
Partial differential jet spaces of formal groups
================================================
In this section we attach to any formal group law $\cF$ in one variable, certain formal groups in several variables that should be thought of as arithmetic-geometric partial differential jets of $\cF$. We use the interplay between these and the partial differential jet spaces of schemes introduced in the previous section to prove that the module of $\D$-characters is finitely generated. We also define the notion of Picard-Fuchs symbol of a $\D$-character, which will play a key rôle later.
Let $A$ be our fixed $\D$-ring and $y$ a $d$-tuple of variables. For any $G \in A[[y,Dy,\ldots,D^n y]]$, we let $G_{|y=0} \in
A[[Dy,\ldots,D^n y]]$ denote the series obtained from $G$ by setting $y=0$, while keeping $\d^i \dd^j y$ unchanged for $i+j \geq 1$. We recall that $L$ stands for the field of fractions of $A$.
\[carmen\] For $a,b \in \bZ_+$, we set $B:=A[Dy,\ldots,D^{a+b} y]\wh$. Then:
1. If $G \in A[[y]][\dd y,\ldots,\dd^b y]$, then $(\phi^a G)_{|y=0}
\in B$.
2. If $F \in A[[y]]$, then $(\d^a \dd^b F)_{|y=0} \in B$.
3. In the case where $y$ is a single variable, if $F =\sum_{n \geq 1}
c_n y^n \in yL[[y]]$ and $nc_n \in A$, then $(\phip^a \dd^b F)_{|y=0} \in B$ and $(\phip^a F)_{|y=0} \in pB$.
[*Proof*]{}. For the proof assertion 1), we may assume that $G=a(y) \cdot (\dd y)^{i_1}\ldots (\dd^b y)^{i_b}$, $a(y) \in A[[y]]$. We have that $$\label{dif}
y^{\phi^a}-y^{p^a}=p \Phi_a\, , \quad
\Phi_a \in \bZ[y,\d y,\ldots,\d^a y]\, ,$$ and so we get $(\phip^a G)_{|y=0} = (\phip^a a(y))_{|y=0} \cdot [(\phip^a \dd
y)^{i_1}\ldots (\phip^a \dd^b y)^{i_b}]_{|y=0}$.
It is clear that $[(\phip^a \dd y)^{i_1}\ldots(\phip^a \dd^b y)^{i_b}]_{|y=0}
\in \bZ[Dy,\ldots,D^{a+b} y]$. On the other hand, if $a(y)=\sum_{n \geq
0} a_n y^n$, by (\[dif\]) we then have that $$\begin{array}{rcl}
(\phip^a a(y))_{|y=0} & = & (\sum_{n \geq 0} a_n^{\phi^a}
(y^{p^a}+p\Phi_a)^n)_{|y=0}\\
& = & \sum_{n \geq 0} a_n^{\phi^a} p^n((\Phi_a)_{|y=0})^n \, ,
\end{array}$$ which is an element of $A[\d y,\ldots,\d^a y]\wh$.
For the proof of the second part, we first use induction to check that $\dd^b F \in A[[y]][\dd y,\ldots,\dd^b y]$. By the part of the Lemma already proven, we conclude that $(\phip^a \dd^b F)_{|y=0} \in A[Dy,\ldots,D^{a+b} y]\wh $. Assertion 2) then follows because $p^a \d^a \dd^b F$ belongs to the ring generated by $\phip^i \dd^b F$ with $i \leq a$.
For assertion 3), we first use induction to check that $$\label{ssst}
\dd^b F=\sum_{n \geq 1} (\dd^b c_n) y^n +G_b,\quad G_b \in
A[[y]][\dd y,\ldots,\dd^b y]\, .$$ Indeed, this is true for $b=0$, with $G_0=0$, and if true for some $b$, then $$\dd^{b+1}F=\sum_{n \geq 1} (\dd^{b+1} c_n) y^n +
[\sum_{n \geq 1} (\dd^b c_n) n y^{n-1}]\dd y +\dd G_b\, ,$$ by the induction hypothesis and the fact that $nc_n \in A$. By (\[ssst\]), we derive that $$(\phip^a \dd^b F)_{|y=0}=
[\sum_{n \geq 1} (\phip^a \dd^b c_n)(y^{\phi^a})^n]_{|y=0}+
(\phip^a G_b)_{|y=0}\, .$$ By the second part of the Lemma proven above, $(\phip^a G_b)_{|y=0} \in A[\d^i \dd^j
y\mid_{1 \leq i+j \leq a+b} ]\wh$. On the other hand $$\begin{array}{rcl}
[\sum_{n \geq 1} (\phip^a \dd^b c_n)(y^{\phip^a})^n]_{|y=0} & = &
[\sum_{n\geq 1} (\phip^a \dd^b c_n)(y^{p^a}+p \Phi_a)^n]_{|y=0}\\
\ & = & \sum_{n\geq 1} (\phip^a \dd^b c_n)p^n
((\Phi_a)_{|y=0})^n\, ,
\end{array}$$ which belongs to $pA[Dy,\ldots,D^{a+b} y]\wh $ because $c_n p^n=n c_n (p^n/n) \in pA$, and it defines a sequence that converge to $0$ as $n \ra \infty$.
Let now $T$ be one variable, $(T_1,T_2)$ a pair of “copies” of $T$, and $\cF:=\cF(T_1,T_2) \in A[[T_1,T_2]]$ be a formal group law in the variable $T$. Then, as in [@book], p. 124, the tuple $$\label{popo} (\cF,D \cF,\ldots,D^n \cF) \in
A[[T_1,T_2,DT_1,DT_2,\ldots,D^n T_1,D^n T_2]]^{\frac{(n+1)(n+2)}{2}}$$ is a formal group law over $A$ in the variables $T,DT,\ldots,D^nT$. Consequently, the tuple $$\label{popopo} (D \cF_{|T_1=T_2=0},\ldots,D^n \cF_{|T_1=T_2=0})
\in A[[DT_1,DT_2,\ldots,D^nT_1,D^nT_2]]^{\frac{n(n+3)}{2}}$$ is a formal group law over $A$ in the variables $DT,\ldots,D^nT$. By the second part of Lemma \[carmen\], this series (\[popopo\]) belong to $A[DT_1,DT_2,\ldots,D^nT_1, D^nT_2]\wh$, so they define a structure of group objects in the category of formal schemes over $A$, $$\label{papa}
(\hat{\bA}^{\frac{n(n+3)}{2}},[+]).$$ Let $l=l(T) \in L[[T]]$ be the logarithm of $\cF$ (cf. [@sil]), and let $a+b \leq n$. By assertion 3) of Lemma \[carmen\], we have that $$L^{[a,b]}:=p^{\epsilon(b)}(\phip^a \dd^b l)_{|T=0} \in
A[DT,\ldots,D^nT]\wh \, ,$$ where $\epsilon(b)$ is either $0$ or $-1$ if $b>0$ or $b=0$, respectively. So $L^{[a,b]}$ defines a morphism of formal schemes $$L^{[a,b]}:\hat{\bA}^{\frac{n(n+3)}{2}} \ra \hat{\bA}^1\, .$$ As in [@book], p. 125, the morphism $L^{[a,b]}$ above is actually a homomorphism $$\label{romanu}
L^{[a,b]}:(\hat{\bA}^{\frac{n(n+3)}{2}},[+]) \ra
\hat{\bG}_a=(\hat{\bA}^1,+)$$ of group objects in the category of formal schemes.
We let $G$ be $\bG_a$, $\bG_m$, or an elliptic curve (defined by a Weierstrass equation) over $A$, and let $\cF$ be the formal group law naturally attached to $G$. In the case $G=\bG_a={\rm Spec}\, A[y]$, we let $T=y$. In the case $G=\bG_m={\rm Spec}\, A[y,y^{-1}]$, we let $T=y-1$. And if $G=E$, we let $T$ be an étale coordinate in a neighborhood $U$ of the zero section $0$ such that $0$ is the zero scheme of $T$ in $U$. The group $${\rm ker} (J^n_{\di \ddi}(G) \ra J^0_{\di \ddi}(G)=\hat{G})$$ is isomorphic to the group (\[papa\]).
Let $e(T) \in L[[T]]$ be the exponential of the formal group law $\cF$ (that is to say, the compositional inverse of $l(T) \in L[[T]]$). We have $e(pT) \in pTA[T]\h$. We may consider the map $$\label{prost}
\begin{array}{rcl}
A[[T]][DT,\ldots,D^r T]\wh & \stackrel{\circ
e(pT)}{\longrightarrow} & A[T,DT,\ldots,D^rT]\wh \\
G \mapsto G \circ e(pT) & := & G(\d^i\dd^j (e(pT))\mid_{0 \leq i+j \leq r})
\end{array}\, .$$ On the other hand, by Proposition \[local\] we may consider the natural map $$\cO(J^r_{\di \ddi}(G)) \ra A[[T]][DT,\ldots,D^r T]\wh \, ,$$ which by the said Proposition is injective with torsion free cokernel. We shall view this map as an inclusion.(Note that this is the case, more generally, when $G$ is replaced by a smooth scheme over $A$ and $T$ is an étale coordinate.)
\[xxzz\] For any $\D$-character $\psi \in \cO(J^r_{\di
\ddi}(G))$, the series $\psi \circ e(pT)$ is in the $A$-linear span of $\{\phip^i \dd^j T\mid_{0 \leq i+j \leq
r}\}$.
[*Proof*]{}. Clearly $F:=\psi \circ e(pT)$ is [*additive*]{}, that is to say, $$F(T_1+T_2, \ldots,D^r(T_1+T_2)) = F(T_1,\ldots,D^rT_1)
+ F(T_2,\ldots,D^rT_2)\, .$$ But the only additive elements in $$A[1/p][[T,DT,\ldots,D^rT]]=
A[1/p][[\phip^i \dd^jT\mid_{0 \leq i+j \leq r}]]$$ are those in the $A[1/p]$-linear span of $\{\phip^i \dd^j T\mid_{0 \leq i+j \leq r}\}$. We are thus left to show that if $$\sum_{ij} a_{ij} \phip^i \dd^j T \in pA[T,DT,\ldots,D^rT]\wh$$ for $a_{ij} \in A$, then $a_{ij} \in pA$ for all $i,j$. Let $\bar{a}_{ij} \in A/pA$ be the images of $a_{ij}$. Then we have $$\sum \bar{a}_{ij} (\dd^j T)^{p^i}=0 \in (A/pA)[T,\dd T,\ldots,\dd^r T]\, ,$$ which clearly implies $\bar{a}_{ij}=0$ for all $i,j$.
The $A$-module $\bX_{\di \ddi}^r(G)$ of $\D$-characters or order $r$ is finitely generated of rank at most $$\frac{(r+1)(r+2)}{2}\, .$$
[*Proof*]{}. By Lemma \[xxzz\], we have that $\bX_{\di \ddi}^r(G)$ embeds into a finitely generated $A$-module of rank at most $\frac{(r+1)(r+2)}{2}$. The result follows because $A$ is Noetherian.
In light of Lemma \[xxzz\], if $\psi$ is a $\D$-character of $G$ then, as an element of the ring $A[[T]]DT,\ldots,D^rT]\wh$, $\psi$ can be identified with the series $$\label{mamscu} \psi=\frac{1}{p} \sigma(\phip,\dd) l(T)\, ,$$ where $\sigma=\sigma(\xi_p,\xi_q) \in A[\xi_p,\xi_q]$ is a polynomial.
\[piiccfu\] The polynomial $\sigma$ is the [*Picard-Fuchs symbol*]{} of $\psi$ with respect to $T$.
Cf. [@char] for comments on the terminology. In a more general context, we will later define what we call the [*Fréchet symbol*]{}, and will explain the relation between these two symbol notions.
The following Lemmas will also be needed later.
\[caiin\] $$\left( L^{[a,b]} \circ e(pT) \right)_{|T=0}=
\left( p^{1+\epsilon(b)} \phip^a \dd^b T \right)_{|T=0}\, .$$
[*Proof*]{}. We have $$\begin{array}{rcl}
\left( L^{[a,b]} \circ e(pT) \right)_{|T=0} & = &
p^{\epsilon(b)}(\phip^a \dd^b
l)(0,\d(e(pT)),\dd(e(pT)),\ldots)_{|T=0} \vspace{1mm} \\
& = & p^{\epsilon(b)}(\phip^a \dd^b
l)(e(pT),\d(e(pT)),\dd(e(pT)),\ldots)_{|T=0} \vspace{1mm} \\
& = & (p^{\epsilon(b)} \phip^a \dd^b (l(e(pT))))_{|T=0} \vspace{1mm} \\
& = & \left( p^{1+\epsilon(b)} \phip^a \dd^b T \right)_{|T=0}\, .
\end{array}$$
\[fu\] The family $\{L^{[a,b]}\mid_{1 \leq a+b\leq r}\}$ is $A$-linearly independent.
[*Proof*]{}. By Lemma \[caiin\], it is enough to check that the family $$\{(\phip^a \dd^b T)\mid_{T=0, 1 \leq a+b \leq r}\}$$ is $A$-linearly independent. Let us assume that $$\sum_{a+b \geq 1} \lambda_{ab} (\phip^a \dd^b T)_{|T=0}=0\, .$$ This implies that $$\sum_{a+b \geq 1} \lambda_{ab} \phip^a \dd^b T \in TA[T,DT,D^2T,\ldots
] \subset TL[\phip^i \dd^j T\mid_{i\geq 0,j\geq 0}] \, ,$$ which clearly implies $\lambda_{ab}=0$ for all $a,b$.
Fréchet derivatives and symbols
===============================
We now develop arithmetic-geometric analogues of some classical constructions in the calculus of variations, including Fréchet derivatives and Euler-Lagrange equations. We use the Fréchet derivatives to define Fréchet symbols of $\D$-characters, and we relate Fréchet symbols to the previously defined Picard-Fuchs symbols.
Fréchet derivative
------------------
We recall that for a smooth scheme $X$ over $A$, we denote by $$T(X): ={\rm Spec}\, {\rm Symm}(\Omega_{X/A})$$ the tangent scheme of $X$. Also, we set $$\cO_{\di
\ddi}^{\infty}(X)=\lim_{\ra} \cO_{\di \ddi}^r(X)\, .$$ Let $\pi:T(X) \ra X$ be the canonical projection. Using ideas in [@book], we construct a natural compatible sequence of $A$-derivations $$\label{classsoon} \Theta:\cO_{\di
\ddi}^r(X) \ra \cO_{\di \ddi}^r(T(X))$$ inducing an $A$-derivation $$\label{air1} \Theta:\cO_{\di \ddi}^{\infty}(X) \ra
\cO^{\infty}_{\di \ddi}(T(X)),$$ respecting the filtration by orders. We call $\Theta f$ the $\D$-[*tangent map*]{} or [*Fréchet derivative*]{} of $f$. (In the ordinary case treated in [@book] $\Theta$ was denoted by $T$.) The construction is local and natural, so it provides a global concept. Thus, we may assume that $X$ is affine.
For any ring $S$, we denote by $S[\epsilon]$ (where $\epsilon^2=0$) the ring $S \oplus \epsilon S$ of [*dual numbers*]{} over $S$. Note that any prolongation sequence $S^*=\{S^r\}$ can be uniquely extended to a prolongation sequence $S^*[\epsilon]=\{S^r[\epsilon]\}$ where $\d \epsilon=\epsilon$, and so $\phip(\epsilon)=p \epsilon$, and $\dd \epsilon=0$, respectively. In particular, we have a $\D$-prolongation sequence $\cO_{\di \ddi}^*(T(X))[\epsilon]=
\{\cO_{\di \ddi}^r(T(X))[\epsilon]\}$.
On the other hand, we have a natural inclusion $\cO(X) \subset \cO(T(X))$, and a natural derivation $$\label{irinushe} d:\cO(X) \ra
\cO(T(X))={\rm Symm}(\Omega_{\cO(X)/A})$$ induced by the universal Kähler derivation $d:\cO(X) \ra \Omega_{\cO(X)/A}$. Hence, we have an $A$-algebra map $$\begin{array}{rcl}
\cO(X) & \ra & \cO(T(X))[\epsilon] \\ f & \mapsto & f+\epsilon \cdot df
\end{array}\, .$$ By the universality property of $\cO_{\di \ddi}^*(X)$, there are naturally induced ring homomorphisms $$\label{air3} \cO_{\di \ddi}^r(X) \ra \cO_{\di
\ddi}^r(T(X))[\epsilon]$$ whose composition with the first projection $$\begin{array}{ccc}
\cO_{\di \ddi}^r(T(X))[\epsilon] & \stackrel{pr_1}{\ra } &
\cO_{\di \ddi}^r(T(X)) \\ a+\epsilon b & \mapsto & a
\end{array}$$ is the identity. Composing the morphism (\[air3\]) with the second projection $$\begin{array}{ccc}
\cO_{\di \ddi}^r(T(X))[\epsilon] & \stackrel{pr_2}{\ra } &
\cO_{\di \ddi}^r(T(X)) \\ a+\epsilon b & \mapsto & b
\end{array}\, ,$$ we get $A$-derivations as in (\[classsoon\]), which agree with each other as $r$ varies, hence induce an $A$-derivation as in (\[air1\]). Note that $\Theta$ restricted to $\cO(X)$ equals $d$. Also the map $f \mapsto f+ \epsilon \Theta f$ in (\[air3\]) commutes with $\phip$ and $\dd$ (by universality). In particular we get that $$\label{iriplea}
\begin{array}{rcl}
\Theta \circ \phip & = & p \cdot \phip \circ \Theta\, ,\\
\Theta \circ \dd & = & \dd \circ \Theta \, .
\end{array}$$ Clearly we have the following:
The mapping $\Theta$ is the unique $A$-derivation $$\cO^{\infty}_{\di \ddi}(X) \ra \cO^{\infty}_{\di \ddi}(T(X))$$ extending $d$ in [(\[irinushe\])]{} and satisfying the commutation relations in [(\[iriplea\])]{}.
\[bibby\] For any $A$-derivation $\partial:\cO_X \ra \cO_X$, there exists a unique derivation $$\label{parst}
\partial_{\infty}:\cO^{\infty}_{\di \ddi}(X) \ra
\cO^{\infty}_{\di \ddi}(X)$$ extending $\partial$ and satisfying the commutation relations $$\begin{array}{rcl}
\partial_{\infty} \circ \phip & = & p \cdot \phip \circ
\partial_{\infty}\, ,\\
\partial_{\infty} \circ \dd & = & \dd \circ
\partial_{\infty}\, .
\end{array}$$
The first of the two conditions above says that $\partial_{\infty}$ is a $\d$-derivation. The derivation $\partial_{\infty}$ extends the derivation $\partial_*$ on $\cO^{\infty}_p(X)$ in [@book], Definition 3.40.
[*Proof*]{}. The uniqueness is clear. For the proof of existence, we may assume that $X$ is affine. By the universality property of the Kähler differentials, $\partial$ induces an $\cO(X)$-module map $$\<\partial,\ \>:\Omega_{\cO(X)/A} \ra \cO(X)$$ such that $\<\partial,df\>=\partial f$ for all $f \in \cO(X)$. By the universality property of the symmetric algebra, we get an induced $\cO(X)$-algebra map $$\cO(T(X)) \ra \cO(X)\, .$$ Composing this map with the inclusion $\cO(X) \subset
\cO^{\infty}_{\di \ddi}(X)$, we get a homomorphism $$\cO(T(X)) \ra \cO^{\infty}_{\di \ddi}(X)\, .$$ By the universality property of $\D$-jet spaces, we get a $\D$-ring homomorphism $$\<\partial,\ \>_{\infty}:\cO^{\infty}_{\di \ddi}(T(X)) \ra
\cO^{\infty}_{\di \ddi}(X)\, ,$$ which is an $\cO^{\infty}_{\di
\ddi}(X)$- algebra homomorphism. Composing the latter with the Fréchet derivative $$\Theta:\cO_{\di \ddi}^{\infty}(X) \ra \cO_{\di
\ddi}^{\infty}(T(X))\, ,$$ we get an $A$-derivation $\partial_{\infty}$ as in (\[parst\]), $$\partial_{\infty} f=\<\partial,\Theta f\>_{\infty}\, , \quad
f \in \cO^{\infty}_{\di \ddi}(X)\, ,$$ which clearly satisfies all the required properties.
The derivation $\partial_{\infty}$ in Corollary \[bibby\] is the [*prolongation*]{} of $\partial$. An [*infinitesimal $\infty$-symmetry*]{} of an element $f \in \cO^{\infty}_{\di \ddi}(X)$ is an $A$-derivation $\partial:\cO_X \ra \cO_X$ such that $\partial_{\infty} f=0$.
The concept introduced above is an analogue of that of an infinitesimal symmetry in differential geometry [@olver]. Note that the concept of infinitesimal $\infty$-symmetry does not reduce in the ordinary arithmetic case to the concept in [@book], Definition 3.48. Again, the difference lies in the powers of $p$ occurring in these two definitions.
Next we examine the image of the Fréchet derivative. For any affine $X$, let us define $$\label{air4} \cO_{\di \ddi}^r(T(X))^+:=\sum_{i+j=0}^r \sum_{f \in
\cO(X)} \cO_{\di \ddi}^r(X) \cdot \phip^i \dd^j (df) \subset
\cO^r(T(X))\, .$$ Note that if $\omega_1,\ldots,\omega_d$ is a basis of $\Omega_{X/A}$ (for instance, if $T_1,\ldots,T_d \in \cO(X)$ is a system of étale coordinates and $\omega_i=dT_i$), then $\cO^r(T(X))^+$ is a free $\cO_{\di \ddi}^r(X)$-module with basis $\{\phip^i \dd^j \omega_m\mid_{1 \leq m \leq d, 0 \leq i+j \leq r}\}$.
\[air5\] The image of the map $\Theta:\cO_{\di \ddi}^r(X) \ra
\cO_{\di \ddi}^r(T(X))$ is contained in $\cO_{\di
\ddi}^r(T(X))^+$.
[*Proof*]{}. Since $\Theta$ is an $A$-derivation and $\cO_{\di \ddi}^r(X)$ is topologically generated by $A$ and elements of the form $\d^i
\dd^j f$ with $i+j \leq r$, $f \in \cO(X)$, it suffices to check that for any such $i,j, f$, we have that $\Theta (\d^i \dd^j f) \in \cO_{\di \ddi}^r(T(X))^+$.
But $$\begin{array}{rcl}
\Theta (\d^i \dd^j f) & = & pr_2(\d^i \dd^j (f+\epsilon df))\\
& = & pr_2(\d^i (\dd^j f+\epsilon \dd^j(df)))\, .
\end{array}$$ By [@book], Lemma 3.34, the latter has the form $$\sum_{l=0}^i \Lambda_{il}(\dd^j f,\d \dd^j f,\ldots,\d^{i-1}\dd^j
f) \phip^l\dd^j(df)$$ where $\Lambda_{il}$ are polynomials with $A$-coefficients, and we are done.
\[dablr\] The Fréchet derivative is functorial with respect to pull back in the following sense. Let $f:\hat{Y} \ra
\hat{X}$ be a morphism of $p$-adic formal schemes between the $p$-adic completions of two schemes, $X$ and $Y$, of finite type over $A$. Cf. Remark \[ginge\]. Then we have a natural commutative diagram $$\label{shaa}
\begin{array}{ccc}
\cO^r_{\di \ddi} (X) & \stackrel{f^*}{\ra} & \cO^r_{\di \ddi} (Y)
\vspace{1mm} \\
\Theta \downarrow & \ & \downarrow \Theta\\
\cO^r_{\di \ddi} (T(X)) & \stackrel{f^*}{\ra} & \cO^r_{\di \ddi}
(T(Y))
\end{array} \, .$$
Fréchet symbol
--------------
We assume in what follows that $X$ has relative dimension $1$ over $A$. Let $f \in \cO^r_{\di \ddi}(X)$, and let $\omega$ be a basis of $\Omega_{X/A}$. Then we can write $$\label{numst} \Theta f=\theta_{f,\omega}(\phip,\dd) \omega=\sum
a_{ij} \phip^i \dd^j \omega \, ,$$ where $\theta_{f,\omega}$ is a polynomial $$\theta_{f,\omega} =\theta_{f,\omega} (\xi_p,\xi_q)=\sum a_{ij}
\xi_p^i \xi_q^j \in \cO^r_{\di \ddi}(X)[\xi_p,\xi_q]\, .$$ If, in addition, we have given an $A$-point $P \in X(A)$, by the universality property of $\D$-jet spaces, we obtain a naturally induced lift $P^r \in
J^r_{\di \ddi}(X)(A)$ of $P$. For any $g \in \cO^r_{\di
\ddi}(X)=\cO(J^r_{\di \ddi}(X))$, we denote by $g(P) \in A$ the image of $g$ under the “evaluation” homomorphism $\cO^r_{\di
\ddi}(X) \ra A$ induced by $P^r$. Then we may consider the polynomial $$\theta_{f,\omega,P}(\xi_p,\xi_q)=\sum a_{ij}(P)
\xi_p^i \xi_q^j\in A[\xi_p,\xi_q]\, .$$
\[centeicit\] The polynomial $\theta_{f,\omega}$ is the [*Fréchet symbol*]{} of $f$ with respect to $\omega$. The polynomial $\theta_{f,\omega,P}$ is the [*Fréchet symbol*]{} of $f$ at $P$ with respect to $\omega$.
The polynomials $\theta_{f,\omega}$ and $\theta_{f,\omega,P}$ have a certain covariance property with respect to $\omega$, which we explain next.
Let $B$ be any $\D$-ring in which $p$ is a non-zero divisor. We define the (right) action of the group $B^{\times}$ on the ring $B[\xi_p,\xi_q]$ as follows. If $b \in B^{\times}$ and $\theta \in B[\xi_p,\xi_q]$, then $\theta \cdot b \in B[\xi_p,\xi_q]$ is the unique polynomial with the property that for any $\D$-ring extension $C$ of $B$, and any $x \in C$, we have $$((\theta \cdot b)(\phip,\dd))(x)=\theta(\phip,\dd) \cdot (bx)\, .$$
Now taking $B$ to be either $\cO^{\infty}_{\di \ddi}(X)$ or $A$, it is easy to see that $$\label{iksplod}
\begin{array}{rcl}
\theta_{f,g \omega} & = & \theta_{f,\omega} \cdot g^{-1} \\
\theta_{f,g \omega,P} & = & \theta_{f,\omega,P} \cdot (g(P))^{-1}
\end{array}.$$ In particular, $\theta_{f,\omega,P}$ only depends on the image $\omega(P)$ of $\omega$ in the [*cotangent space of $X$ at $P$*]{}, $\Omega_{X/A} \otimes_{\cO(X),P} A$, where $A$ here is viewed as an $\cO(X)$-module via the evaluation map $P:\cO(X) \ra A$.
The proof of Corollary \[bibby\] and (\[numst\]) show that if $\omega$ is a basis of $\Omega_{X/A}$ and $\partial:\cO_X
\ra \cO_X$ is an $A$-derivation, then for any $f \in
\cO^{\infty}_{\di \ddi}(X)$ we have that $$\label{illle}
\partial_{\infty} f=\<
\partial,\theta_{f,\omega}(\phip,\dd)\omega\>_{\infty}=
\theta_{f,\omega}(\phip,\dd)(\<\partial,\omega\>),$$ where $\theta_{f,\omega}$ is the Fréchet symbol of $f$ with respect to $\omega$.
Our next proposition relates Fréchet symbols of $\D$-characters to the Picard-Fuchs symbol; cf. Definition \[piiccfu\]. This will be useful later, as in the applications to come, the Picard-Fuchs symbols will be easy to compute.
We let $G$ be either $\bG_a$, or $\bG_m$, or an elliptic curve over $A$.
\[inquests\] Assume that $\omega$ is an $A$-basis for the invariant $1$-forms on $G$, and let $\psi$ be a $\D$-character of $G$. Let $\theta=\theta_{\psi,\omega} \in \cO_{\di
\ddi}^{\infty}[\xi_p,\xi_q]$ be the Fréchet symbol of $\psi$ with respect to $\omega$, and let $\sigma \in A[\xi_p,\xi_q]$ be the Picard-Fuchs symbol of $\psi$ with respect to an étale coordinate $T$ at the origin $0$ such that $\omega(0)=dT$. Then $$\sigma^{(p)}=p \theta\, .$$
1. The relation $\sigma^{(p)}=p \theta$ reads $\sigma(p\xi_p,\xi_q)=p\theta(\xi_p,\xi_q)$.
2. The expression above implies that, in particular, $\theta \in
A[\xi_p,\xi_q]$, and $\Theta \psi=\frac{1}{p} \sigma(p \phip,\dd) \omega$.
3. If ${\rm Lie}(G)$ denotes the $A$-module of invariant $A$-derivations of $\cO_G$, and if $\partial \in {\rm Lie}(G)$, by (\[illle\]) we then have that $$\label{durre}
\partial_{\infty} \psi=\frac{1}{p} \sigma(p
\phip,\dd)(\<\partial,\omega\>) \in A\, .$$ In particular, $\partial$ is an infinitesimal $\infty$-symmetry of $\psi$ if, and only if, the element $u:=\<\partial,\omega\> \in
A=\bG_a(A)$ is a solution to the $\D$-character of $\bG_a={\rm Spec}\,
A[y]$ defined by $\frac{1}{p} \sigma(p \phip,\dd) y \in A\{y\}$.
[*Proof of Proposition \[inquests\]*]{}. Consider the diagram in (\[shaa\]) with $X=G$, $Y=\bG_a$, and $f=e(pT)$. By the definition of the Picard-Fuchs symbol, if $\sigma=\sum a_{ij} \xi_p^i \xi_q^j$ then $$e(pT)^* \psi=\sum a_{ij} \phip^i \dd^j T\, .$$ Applying the Fréchet derivative to this identity, and using the commutativity of the diagram (\[shaa\]), we get that $$\label{eric} e(pT)^* \Theta \psi = \Theta(e(pT)^*
\psi)=\Theta(\sum a_{ij} \phip^i \dd^j T)=\sum a_{ij} p^i \phip^i
\dd^j (dT)\, .$$ On the other hand, we may write $\theta_{\psi,\omega}=\sum b_{ij} \xi_p^i \xi_q^j$, with $b_{ij} \in \cO^r_{\di \ddi}(G)$. By the definition of the Fréchet symbol, we have $\Theta \psi=\sum b_{ij} \phip^i \dd^j \omega$, and since $\omega(0)=dT$, we have that $\omega=d l(T)$, where $l(T)$ is the compositional inverse of $e(T)$. Therefore, $e(pT)^* \omega=
d(l(e(pT)))=pdT$. We obtain that $$\label{erric} e(pT)^* \Theta \psi=e(pT)^* \left( \sum b_{ij}
\phip^i \dd^j \omega \right)=p \sum b_{ij} \phip^i \dd^j (dT)\, .$$ The identities (\[eric\]) and (\[erric\]) imply that $\left( \sum a_{ij}p^i \phip^i \dd^j \right) (dT)=
p \left( \sum b_{ij} \phip^i \dd^j \right) (dT)$, and this finishes the proof.
Euler-Lagrange equations
------------------------
The Fréchet symbol can be also used to introduce an Euler-Lagrange formalism in our setting. In order to explain this, we begin by making the following definition.
Let $B$ be any $\D$-ring in which $p$ is a non-zero divisor. For any $r \geq 1$, we denote by $B[\xi_p,\xi_q]_r$ the submodule of the polynomial ring $B[\xi_p,\xi_q]$ consisting of all polynomials of degree $\leq r$. The [*adjunction map*]{} $$Ad^r:B[\xi_p,\xi_q]_r \ra B[\xi_p,\xi_q] [1/p]$$ is defined by $$Ad^r(\sum_{ij}
b_{ij} \xi_p^i \xi_q^j):= \sum_{ij} (-1)^j p^{-ij} \xi_p^{r-i}
\xi_q^j \cdot b_{ij}\, .$$ This map induces an [*adjunction map*]{} $$ad^r:B[\xi_p,\xi_q]_r \ra B [1/p]$$ by $$ad^r(Q):=(Ad^r(Q)(\phip,\dd))(1)\, ,$$ which is explicitly given by $$\label{defadsus} ad^r(\sum_{ij}
b_{ij} \xi_p^i \xi_q^j):= \sum_{ij} (-1)^j p^{-ij} \phip^{r-i}
\dd^j b_{ij}\, .$$ For any $Q \in B[\xi_p,\xi_q]_r$, we have $$ad^{r+1}(Q)=(ad^r(Q))^{\phi}\, .$$
This adjunction map is a hybrid between the “familiar” adjunction map for usual linear differential operators with respect to $\dd$ (as encountered after the usual integration by parts argument in the calculus of variations), and the adjunction map for $\d$-characters (as defined in [@book]). The definition of the adjunction map above might seem ad hoc, and somewhat complicated, but it is justified by the following covariance property.
\[laiz\] For any $Q \in B[\xi_p,\xi_q]_r$ and $b \in B$, we have that $$ad^r(Q \cdot b)=b^{\phi^r} \cdot ad^r(Q)\, .$$
[*Proof*]{}. This follows from a direct computation.
The mapping $ad^r$ is related (but does not coincide) with the mapping $ad_r$ defined in [@book], p. 92. Indeed, the powers of $p$ in the definitions of $ad^r$ and $ad_r$ are different.
Returning to our geometric setting, let $X/A$ be a smooth scheme of relative dimension $1$, let $f \in \cO^r_{\di \ddi}(X)$, and let $\partial:\cO_X \ra \cO_X$ be an $A$-derivation. We define an element $\epsilon^r_{f,\partial} \in \cO^{\infty}_{\di \ddi}(X)[1/p]$ as follows. Let us assume first that $X$ is affine and $\Omega_{X/A}$ is free with basis $\omega$. We let $B$ in the discussion above be the $\D$-ring $\cO^{\infty}_{\di \ddi}(X)$. We consider the Fréchet symbol $\theta_{f,\omega} \in \cO^{\infty}_{\di \ddi}(X)[\xi_p,\xi_q]_r$, and its image under $ad^r$, $ad^r(\theta_{f,\omega}) \in \cO^{\infty}_{\di \ddi}(X)
[1/p]$. Then we set $$\epsilon^r_{f,\partial}:=\<\partial,\omega\>^{\phi^r} \cdot
ad^r(\theta_{f,\omega})\in \cO^{\infty}_{\di \ddi}(X) [1/p]\, .$$ By Lemma \[laiz\] and (\[iksplod\]), $\epsilon^r_{f,\partial}$ does not depend on the choice of $\omega$. Therefore, this definition globalizes to one in the case when $X$ is not necessarily affine and $\Omega_{X/A}$ is not necessarily free.
We say that the element $\epsilon^r_{f,\partial}$ is the [*Euler-Lagrange equation*]{} attached to the [*Lagrangian*]{} $f$ and the [*vector field*]{} $\partial$.
An [*energy function*]{} of order $r$ on $G$ is a $\D$-morphism $H:G \ra \bA^1$ that can be written as $$\label{biir}H=\sum_{ij} h_{ij} \psi_i \psi_j \, ,$$ where $h_{ij} \in A$ and $\psi_i$ are $\D$-characters of $G$ of order $r$.
If $H$ is an energy function of order $r$ on $G$ and $\partial:\cO_G \ra \cO_G$ is an $A$-derivation that constitutes a basis of ${\rm Lie}(G)$, then the Euler-Lagrange equation $\epsilon^r_{H,\partial}$ is a $K$-multiple of a $\D$-character.
[*Proof*]{}. Let us assume that $H$ is an in (\[biir\]), and let $\omega$ be an $A$-basis of the invariant $1$-forms on $G$. Let $$\label{altazi} \theta_{\psi_i,\omega}=\sum_{mn} a_{imn} \xi_p^m
\xi_q^n\, .$$ Then $$\begin{array}{rcl}
\Theta H & = & \sum_{ij} h_{ij} ( \psi_i \Theta \psi_j+\psi_j
\Theta \psi_i)\\
& = & \sum_{mnij} [h_{ij}(a_{jmn}\psi_i+a_{imn} \psi_j)]\phip^m \dd^n
\omega \, .
\end{array}$$ Hence $$\label{ele} \epsilon^r_{H,\partial}=\<\partial,\omega\>^{\phi^r}
\cdot \sum_{mnij} (-1)^n p^{-mn} \phip^{r-m} \dd^n[h_{ij}(a_{jmn}
\psi_i+a_{imn} \psi_j)]\, ,$$ and we are done.
A [*boundary element*]{} in $\cO_{\di \ddi}^{\infty}(X) \otimes K$ is an element of the form $\dd a +\phip b -b$, for some $a,b \in \cO_{\di \ddi}^{\infty}(X) \otimes K$.
The following can be interpreted as an analogue of Noether’s Theorem in mechanics [@olver]. It is a hybrid between the usual Noether Theorem and the “arithmetic Noether Theorem” in [@book]. We state it in the affine case.
Let $X$ be an affine smooth scheme over $A$ of relative dimension $1$, and let $\partial:\cO_X \ra \cO_X$ be an $A$-derivation that is a basis for the $\cO(X)$-module of all $A$-derivations. Then, for any $f \in \cO^r_{\di \ddi}(X)$ we that $\epsilon^r_{f,\partial}-\partial_{\infty}f \in
\cO_{\di \ddi}^{\infty}(X) \otimes K$ is a boundary element. In particular, if $\partial$ is an infinitesimal $\infty$-symmetry of $f$, then $\epsilon^r_{f,\partial}$ is a boundary element.
[*Proof*]{}. Let $\omega$ be a basis of $\Omega_{X/A}$, and let $\theta_{f,\omega}=\sum b_{ij} \xi_p^i \xi_q^j$ for $b_{ij} \in
\cO^{\infty}_{\di \ddi}(X)$. We set $v:=\<\partial,\omega\> \in \cO(X)^{\times}$ (in what follows we might assume that $v=1$, but this would not simplify the computation), and denote by $\equiv$ the congruence relation in $\cO^{\infty}_{\di
\ddi}(X) \otimes K$ modulo the group of boundary elements. Then we have $$\begin{array}{rcl}
\partial_{\infty}f & = & \sum b_{ij} \phip^i \dd^j v = \sum
b_{ij}p^{-ij} \dd^j \phip^i v\\
& \equiv & \sum (-1)^j p^{-ij} (\dd^j b_{ij})(\phip^i v)\, .
\end{array}$$ We also we have that $$\begin{array}{rcl}
\epsilon^r_{f,\partial} & = & \sum (-1)^j p^{-ij} (\phip^r
v)(\phip^{r-i}\dd^j b_{ij})\\
& = & \sum (-1)^j p^{-ij} \phip^{r-i}[(\dd^j
b_{ij})(\phip^i v)]\, .
\end{array}$$ Consequently, $$\epsilon^r_{f,\partial}-\partial_{\infty} f \equiv \sum (-1)^j
p^{-ij} (\phip^{r-i}-1)[(\dd^j b_{ij})(\phip^i v)] \equiv
0$$ because $\phip^k-1$ is divisible by $\phip-1$ in $A[\phip]$.
Additive group
==============
In this section we prove our main results about $\D$-characters and their space of solutions in the case where $G$ is the additive group.
Let $\bG_a={\rm Spec}\, A[y]$ be the additive group over our fixed $\D$-ring $A$. We equip $\bG_a$ with the invariant $1$-form $$\omega:=dy\, .$$
\[09876\] The $A$-module $\bX_{\di \ddi}^r(\bG_a)$ of $\D$-characters of order $r$ on $\bG_a$ is free with basis $$\{\phip^i \dd^j y \mid_{0 \leq i+j
\leq r}\}\, .$$ Hence the $A[\phip,\dd]$-module $\bX_{\di
\ddi}^{\infty}(\bG_a)$ of $\D$-characters of $\bG_a$ is free of rank one with basis $y$.
[*Proof*]{}. Same argument as in the proof of Lemma \[xxzz\].
Throughout the rest of this section, we let $A=R$. In particular, the notation and discussion in Example \[cucurigu\] applies to $\bG_a$ over $R$.
By Proposition \[09876\], any $\D$-character of $\bG_a$ can be written uniquely as $$\label{psia} \psi_a:=\psi_a^{\mu} :=\mu(\phip,\dd)y \in
R[y,Dy,\ldots,D^n y]\h,$$ where $$\mu(\xi_p,\xi_q)=\sum \mu_{ij} \xi_p^i\xi_q^j
\in R[\xi_p,\xi_q]$$ is a polynomial. Note that the Picard-Fuchs symbol $\sigma(\xi_p,\xi_q)$ of $\psi_a$ with respect to the étale coordinate $T=y$ is given by $$\sigma(\xi_p,\xi_q)=p\mu(\xi_p,\xi_q).$$ The Fréchet symbol of $\psi_a$ with respect to $\omega=dy$ is $$\theta(\xi_p,\xi_q)=\mu(p\xi_p,\xi_q).$$
We say that $\mu(\xi_p,\xi_q)$ is the [*characteristic polynomial*]{} of the character $\psi_a$. We say that the $\D$-character $\psi_a$ is [*non-degenerate*]{} if $\mu(0,0) \in
R^{\times}$. Given a non-degenerate character $\psi_a$, we say that ${\kappa} \in \bZ$ is a [*characteristic integer*]{} of $\psi_a$ if $\mu(0,\kappa)=0$. (Note that any characteristic integer of a non-degenerate $\D-$ character $\psi_a$ must be coprime to $p$.) We say that $\kappa \in \bZ$ is a [*totally non-characteristic*]{} integer if $\kappa \not\equiv 0$ mod $p$ and $\mu(0,\kappa) \not\equiv 0$ mod $p$. We denote by ${\mathcal K}$ the set of all characteristic integers of $\psi_a$, and set $\cK_{\pm}:=\cK \cap \bZ_{\pm}$. We denote by $\cK'$ the set of all totally non-characteristic integers. For all $0 \neq \kappa
\in \bZ$ and $\alpha \in R$, we define the [*basic series*]{} of $\psi_a$ by $$\begin{array}{rcl}
\label{ua} u_{a,\kappa,\alpha}:= u_{a,\kappa,
\alpha}^{\mu} & := & \sum_{n \geq 0} b_{n,\kappa}\phip^n(\alpha
q^{\kappa})\\
\ & = & \sum_{n \geq 0} b_{n,\kappa} \alpha^{\phi^n} q^{\kappa
p^n}\\
\ & = & \alpha q^{\kappa}+\cdots \in q^{\pm 1}R[[q^{\pm 1}]],\end{array}$$ where $\{b_{n,\kappa}\}_{n \geq 0}$ is the sequence of elements in $R$ defined inductively by $b_{0,\kappa}=1$, $$\label{aniinvatza}
b_{n,\kappa}:=-\frac{\sum_{s=1}^n \left(\sum_{j \geq 0}\mu_{sj} \kappa^j
p^{j(n-s)}\right) b_{n-s,\kappa}^{\phi^s}}{\sum_{j \geq 0}
\mu_{0j} \kappa^j p^{jn}}\, ,\; n \geq 1\, .$$ In this last expression, the denominator is congruent to $\mu(0,0)$ mod $p$ (and is therefore an element of $R^{\times}$).
The next Lemma intuitively says that the two collections of series $\{u_{a,\kappa,1}\ |\ \kappa \neq 0\}$ and $\{q^{\kappa}\ |\
\kappa \neq 0\}$ “diagonalize” $\psi_a$.
\[unidul\] For all $0 \neq \kappa \in \bZ$ and $\alpha \in R$ we have $\psi_a u_{a,\kappa,\alpha}=\mu(0,\kappa) \cdot \alpha
q^{\kappa}$.
[*Proof*]{}. The desired result follows from the following computation: $$\label{inima} \begin{array}{rcl} \psi_a u_{a,\kappa,\alpha} & = &
\sum_{i,j,m \geq 0} \mu_{ij} \phip^i \dd^j(b_{m,\kappa}
\phip^m(\alpha q^{\kappa}))
\vspace{1mm} \\
& = & \sum_{i,j,m \geq 0} \mu_{ij} b_{m,\kappa}^{\phi^i}
\kappa^j
p^{jm} \phip^{i+m}(\alpha q^{\kappa}) \vspace{1mm} \\
& = & \sum_{n \geq 0} \left( \sum_{s \geq 0} \left( \sum_{j
\geq 0} \mu_{sj} \kappa^j p^{j(n-s)} \right)
b_{n-s,\kappa}^{\phi^s} \right)
\phip^n(\alpha q^{\kappa}) \vspace{1mm} \\
\ & = & \left(\sum_{j \geq 0} \mu_{0j} \kappa^j\right) \alpha q^{\kappa}\\
\ & = & \mu(0,\kappa) \alpha q^{\kappa}\, .
\end{array}$$
For any series $u \in R[[q^{\pm 1}]]$ let $\bar{u} \in k[[q^{\pm
1}]]$ denote the reduction of $u$ mod $p$; we recall that $k=R/pR$. Also we denote by $$\bar{\mu}(\xi_p,\xi_q)=\sum
\bar{\mu}_{ij} \xi_p^i \xi_q^j \in k[\xi_p,\xi_q]$$ the reduction mod $p$ of the characteristic polynomial $\mu(\xi_p,\xi_q)$. It is convenient to introduce the following terminology.
We say that a polynomial $\mu \in R[\xi_p,\xi_q]$ is [*unmixed*]{} if $\bar{\mu}_{ij}=0$ for $ij\neq 0$ and there exists $i \neq 0$ such that $\bar{\mu}_{i0}\neq 0$. Equivalently, $\mu$ is unmixed if $$\bar{\mu}(\xi_p,\xi_q)=\bar{\mu}(\xi_p,0)+\bar{\mu}(0,\xi_q)-
\bar{\mu}(0,0)\, ,$$ and $$\bar{\mu}(\xi_p,0) \neq \bar{\mu}(0,0)\, .$$
We say that $S \subset \bZ_+ \backslash \{0\}$ is [*short*]{} if $$\frac{\max S}{\min S} < \frac{p}{2}\, .$$ We say that $S \subset \bZ_-\backslash\{0\}$ is [*short*]{} if the set $-S$ is short. In particular, a set $S \subset \bZ \backslash \{0\}$ consisting of a single element is short.
\[irinusescoala\] Let $S \subset \bZ_{\pm} \backslash p\bZ_{\pm}$ be a non-empty finite set of either positive or negative integers, and let $$u:=\sum_{\kappa \in S} u_{a,\kappa,\alpha_{\kappa}}\, ,$$ where $\alpha_{\kappa} \in R$, not all of them in $pR$. Then the following hold:
1. $\bar{u}$ is integral over $k[q^{\pm 1}]$, and the field extension $k(q) \subset k(q,\bar{u})$ is Abelian with Galois group killed by $p$.
2. If $\mu$ is unmixed, then $\bar{u} \not\in
k(q)$.
3. If $\mu$ is unmixed and $S$ is short, then $u$ is transcendental over $K(q)$.
[*Proof*]{}. Let us assume that $S \subset \bZ_+$. The case $S \subset \bZ_-$ is treated in a similar manner.
We prove assertion 1. By (\[aniinvatza\]), for $n \geq 1$, we have that $$\begin{array}{rcl}
\bar{b}_{n,\kappa} & = & -(\bar{\mu}_{00})^{-1} \left(
\sum_{s=1}^{n-1} \bar{\mu}_{s0} \bar{b}_{n-s,\kappa}^{p^s}
+\sum_{j \geq 0} \bar{\mu}_{nj} \bar{\kappa}^j \right)\\
& = & -(\bar{\mu}_{00})^{-1} \left( \sum_{s\geq 1}
\bar{\mu}_{s0} \bar{b}_{n-s,\kappa}^{p^s} +\sum_{j \geq 1}
\bar{\mu}_{nj} \bar{\kappa}^j \right)\, .
\end{array}$$ Also $$\bar{u}=\sum_{\kappa \in S} \sum_{n \geq 0} \bar{b}_{n,\kappa}
\bar{\alpha}_{\kappa}^{p^n} q^{\kappa p^n}\, .$$ Let us denote by $F_p:k \ra
k$ the $p$-th power Frobenius map. Consider the polynomial $g(q) \in
k[q]$ given by $$\begin{array}{rcl}
g(q) & = & \sum_{\kappa \in S}[\bar{\mu}(F_p,\kappa)-
\bar{\mu}(F_p,0)-\bar{\mu}(0,\kappa)]
(\alpha q^{\kappa})\\
& = & \sum_{\kappa \in S} \sum_{n \geq 0} \sum
_{j \geq 1} \bar{\mu}_{nj}
\bar{\kappa}^j\bar{\alpha}_{\kappa}^{p^n} q^{\kappa p^n}-
\sum_{\kappa \in S} \sum_{j \geq 0} \bar{\mu}_{0j}
\bar{\kappa}^j \bar{\alpha}_{\kappa}q^{\kappa}\, ,
\end{array}$$ and define the polynomial $G(t) \in k(q)[t]$ by $$G(t):=\sum_{s \geq 0} \bar{\mu}_{s0}t^{p^s}+g(q)\, .$$ We have that $G(t)$ has an invertible leading coefficient, is separable ($dG/dt=\bar{\mu}_{00}$), and has $\bar{u}$ as a root: $$\begin{array}{rcl}
G(\bar{u}) & = & \sum_{\kappa \in S} \sum_{s \geq 0} \sum_{n \geq
0} \bar{\mu}_{s0}
\bar{b}_{n,\kappa}^{p^s} \bar{\alpha}_{\kappa}^{p^{n+s}} q^{\kappa p^{n+s}}\\
& & + \sum_{\kappa \in S} \sum_{n \geq 0} \sum_{j \geq 1}
\bar{\mu}_{nj} \bar{\kappa}^j \bar{\alpha}_{\kappa}^{p^n}
q^{\kappa p^n}-\sum_{\kappa \in S} \sum_{j\geq 0}
\bar{\mu}_{0j}\bar{\kappa}^j \bar{\alpha}_{\kappa} q^{\kappa}\\
& = & \sum_{\kappa \in S} \sum_{n \geq 0} \left( \sum_{s \geq
0} \bar{\mu}_{s0} \bar{b}_{n-s,\kappa}^{p^s}+\sum_{j \geq 1}
\bar{\mu}_{nj}\bar{\kappa}^j \right)
\bar{\alpha}_{\kappa}^{p^n} q^{\kappa p^n}\\
& & - \sum_{\kappa \in S} \sum_{j\geq 0}
\bar{\mu}_{0j}\bar{\kappa}^j \bar{\alpha}_{\kappa} q^{\kappa}
\\ & = & 0\, .
\end{array}$$ The difference of any two roots of $G(t)$ is in $k$. Hence, $k(q,\bar{u})$ is Galois over $k(q)$ and its Galois group $\Sigma$ embeds into $k$ via the map $$\begin{array}{rcl}
\Sigma & \ra & k\\
\sigma & \mapsto & \sigma \bar{u}-\bar{u}\, .
\end{array}$$
We prove assertion 2. In this case we have $$g(q)=-\bar{\mu}(0,0) \cdot \sum_{\kappa \in S}
\bar{\alpha}_{\kappa} q^{\kappa}\, ,$$ and $G(t)$ has degree $p^e$ in $t$ for some $e \geq 1$. Assume that $\bar{u} \in k(q)$. Since $\bar{u}$ is integral over $k[q]$ it follows that $\bar{u} \in
k[q]$. Let $d$ be the degree of $\bar{u}$. Since $G(\bar{u})=0$, we see that $d \geq 1$. Since the integers in $S$ are not divisible by $p$, the coefficient of $q^{dp^e}$ in the polynomial $G(\bar{u})\in k[q]$ is non-zero, a contradiction.
We prove assertion 3. Let $\kappa_1<\kappa_2< \cdots <\kappa_s$ be the integers in $S$. By our assumption, there is a real $\epsilon > 0$ such that $(2+\epsilon)\kappa_s<\kappa_1 p$. We will show that $$u(q^{-1})=\sum_{j=1}^s \sum_{n \geq 0} b_{n,\kappa_j}
\alpha_{\kappa_j}^{\phi^n}
q^{-\kappa_j p^n} \in K((q^{-1}))$$ is transcendental over $K(q)$.
For any $\varphi=\sum_{n=-\infty}^f c_n q^n \in K((q^{-1}))$ with $c_f \geq 0$, set $|\varphi|:=e^f$. Also, set $|0|=0$. Roth’s theorem for characteristic zero function fields [@uchi] states that if $\varphi$ is algebraic over $K(q)$ then, for any $\epsilon >0$, the inequality $$0<|\varphi-P/Q|<|Q|^{-2-\epsilon}$$ has only finitely many solutions $P/Q$ with $P,Q \in K[q]$. On the other hand, for any $n$, we have $$\begin{array}{rcl}
0 & < & |u(q^{-1})-\sum_{j=1}^s \sum_{n=0}^N b_{n,\kappa_j}
\alpha_{\kappa_j}^{\phi^n}
q^{-\kappa_j p^n}|\\
& \leq & e^{-\kappa_1 p^{n+1}}\\
& < & (e^{\kappa_s p^n})^{-2-\epsilon}\\
& = & |q^{\kappa_s p^n}|^{-2-\epsilon}\, .
\end{array}$$ It follows that $u(q^{-1})$ is transcendental over $K(q)$, and this completes the proof.
1. If $\mu_{ij} \in pR$ for $i \geq 1$, then $u_{a,\kappa,\alpha}
\in R[q,q^{-1}]\h$.
2. The mapping $$\begin{array}{rcc}
R & \ra & R[[q^{\pm 1}]] \\
\alpha & \mapsto & u_{a,\kappa,\alpha}^{\rs}
\end{array}$$ is an injective group homomorphism.
3. Recall that, attached to a $\dd$-character $\psi_q$ we defined in (\[vocea\]) operators $B_{\kappa}^0$ and $B_{\pm}^0$. Let $\psi_q$ be, in our case, the identity; hence, for $0 \neq \kappa
\in \bZ$, $$\begin{array}{ccl}
R[[q^{\pm 1}]] & \stackrel{B_{\kappa}^0}{\ra} & R \\
B_{\kappa}^0\left(\sum a_n q^n \right) & = & a_{\kappa},
\end{array}$$ and $$\begin{array}{ccl}
R[[q^{\pm 1}]] & \stackrel{B_{\pm}^0}{\ra} & R^{\rho_{\pm}} \\
B_{\pm}^0\left(\sum a_n q^n \right) & = & (a_{\kappa})_{\kappa \in
\cK_{\pm}}.
\end{array}$$ Note that if $\kappa_1, \kappa_2 \in \bZ \backslash p\bZ$, we have $$\label{meah}
B_{\kappa_1}^0 u_{a,\kappa_2,\alpha}=\alpha \cdot
\delta_{\kappa_1 \kappa_2},$$ where $\delta_{\kappa_1 \kappa_2}$ is the Kronnecker delta.
4. We have the identity $$\label{sasesi} \dd u_{a,\kappa,\alpha}^{\mu}=\kappa \cdot
u_{a,\kappa,\alpha}^{\mu^{(p)}}\, ,$$ where we recall that $\mu^{(p)}(\xi_p,\xi_q):=\mu(p\xi_p,\xi_q)$.
5. We have the identity $$\label{picc} u_{a,\kappa,\zeta^{\kappa}
\alpha}(q)=u_{a,\kappa,\alpha}(\zeta q)$$ for all $\zeta \in \bmu(R)$; this holds because $\zeta^{\phi}=\zeta^p$. In particular, if $\alpha=\sum_{i=0}^{\infty} m_i \zeta_i^{\kappa}$, $\zeta_i \in \bmu(R)$, $m_i \in \bZ$, $v_p(m_i) \ra \infty$, then $$u_{a,\kappa,\alpha}(q)
= \sum_{i=0}^{\infty} m_i u_{a,\kappa,1}(\zeta_i q)\, .$$ Thus, if $f \in \bZ \bmu(R) \h$ is such that $(f^{[\kappa]})^{\sh}=\alpha
\in R$, then $u_{a,\kappa,\alpha}$ can be expressed using convolution: $$u_{a,\kappa,\alpha}=f \star
u_{a,\kappa,1}\, .$$ Note that $\{u_{a,\kappa,\alpha}\ |\ \alpha \in
R\}$ is a $\bZ \bmu(R)\h$-module (under convolution). If $\kappa
\in \bZ \backslash p\bZ$, this module structure comes from an $R$-module structure, still denoted by $\star$, by a base change via the surjective homomorphism $$\bZ \bmu(R)\h \stackrel{[\kappa]}{\ra} \bZ \bmu(R)\wh
\stackrel{\sh}{\ra} R$$ (see (\[notinjjj\])), and the $R$-module $\{u_{a,\kappa,\alpha}\ |\ \alpha \in R\}$ is free with basis $u_{a,\kappa,1}$. Hence, for $g \in \bZ \bmu(R)\wh$, $\beta=(g^{[\kappa]})^{\sh}$, we have that $$\beta \star u_{a,\kappa,\alpha}=g \star u_{a,\kappa,\alpha}\, .$$ In particular, $$u_{a,\kappa,\alpha} = \alpha \star u_{a,\kappa,1}\, .$$
6. We have the following “rationality” property: if $\alpha,
\mu_{ij} \in \bZ_{(p)}$, then $u_{a,\kappa,\alpha} \in
\bZ_{(p)}[[q^{\pm 1}]]$.
We say that the mapping $$\begin{array}{rcc}
R & \ra & R[[q]] \\ \alpha & \mapsto & v_{\alpha}
\end{array}$$ is a [*pseudo $\d$-polynomial map*]{} if for any integer $n \geq 0$ there exists an integer $r_n \geq 0$ and a polynomial $P_n \in R[x_0,x_1,\ldots,
x_{r_n}]$ such that, for all $\alpha \in R$ we have that $$v_{\alpha}=\sum P_n(\alpha,\d \alpha,\ldots,\d^{r_n}
\alpha) q^n\, . \label{tempo}$$ If $X$ is a scheme over $R[[q]]$, then a map $R \ra X(R[[q]])$ is said to be a [*pseudo $\d$-polynomial map*]{} if there exists an open subscheme $U \subset X$, and a closed embedding $U \subset
\bA^n$ such that the image of $R \ra X(R[[q]])$ is contained in $U(R[[q]])$, and the maps $$R \ra U(R[[q]]) \subset
\bA^N(R[[q]])=R[[q]]^N \stackrel{pr_i}{\ra} R[[q]],$$ are pseudo $\d$-polynomial. Here $pr_i$ are the various projections.
Similarly, a mapping $$\begin{array}{rcc}
R & \ra & R[[q^{-1}]] \\ \alpha & \mapsto & v_{\alpha}
\end{array}$$ is said to be a [*pseudo $\d$-polynomial map*]{} if for any integer $n \leq 0$ there exists an integer $r_n \geq 0$ and a polynomial $P_n \in
R[x_0,x_1,\ldots,x_{r_n}]$ such that (\[tempo\]) holds for all $\alpha \in R$, and given a scheme $X$ over $R[[q^{-1}]]$, a [*pseudo $\d$-polynomial*]{} map $R \ra
X(R[[q^{-1}]])$ is defined as above, with the rôle of $R[[q]]$ now being played by $R[[q^{-1}]]$.
The prefix [*pseudo*]{} was included in order to suggest an analogy with “differential operators of infinite order.” This is not to be confused with the pseudo-differential operators in micro-local analysis.
The basic series mappings $$\begin{array}{rcc}
R & \ra & R[[q^{\pm 1}]] \\ \alpha & \mapsto & u_{a,\kappa,\alpha}
\end{array}$$ are pseudo $\d$-polynomial maps. In particular, when interpreted as mapping $R \ra \bG_a(R[[q^{\pm
1}]])$, they are pseudo $\d$-polynomial maps.
The following example is as elementary as they come. More interesting ones will be given later on, while studying $\bG_m$ and elliptic curves.
\[addeq\] Let $\psi_a$ be a non-degenerate $\D$-character of $\bG_a$, and let $\cU_*$ be the corresponding groups of solutions. Let ${\mathcal K}$ be the set of characteristic integers, and $u_{a,\kappa,\alpha}$ be the basic series. Then the following hold:
1. If ${\mathcal K} =\emptyset$, then $\cU_{\la}=\cU_{\ra}=\cU_0$.
2. We have $$\begin{array}{rcl}
\cU_{\pm 1} & = & \bigoplus_{\kappa \in {\mathcal K}_{\pm}}
\{u_{a,\kappa,\alpha}\, | \; \alpha \in R\}\, ,
\end{array}$$ where $\oplus$ denotes internal direct sum. In particular, $\cU_{\pm 1}$ are free $R$-modules under convolution, with bases $\{u_{a,\kappa,1}\, |\; \kappa \in \cK_{\pm}\}$, respectively.
3. $\cU_{\ra}+\cU_{\la}=\cU_+ +\cU_-$.
[*Proof*]{}. Let $u=\sum_{n=-\infty}^{\infty}a_n q^n$ be either an element of $R((q))\wh$ or of $R((q^{-1}))\wh$. We express the $\D$-character $\psi_a$ as $\psi_a=\mu(\phip,\dd)y$, for some polynomial $\mu(\xi_p,\xi_q)=\sum_{i,j \geq 0}
\mu_{ij} \xi_p^i \xi_q^j \in R[\xi_p,\xi_q]$. Then $\psi^{\rs}_a u=0$ if, and only if, $$\label{tutu} \mu(0,n) a_n+\sum_{j \geq 0} \sum_{i \geq 1} \mu_{ij}
(n/p^i)^j a_{n/p^i}^{\phi^i}=0.$$ for all $n \in \bZ$. In this last expression, $a_{n/p^i}=0$ if $n/p^i \not \in \bZ$. Thus, if $\mu(0,n) \neq 0$ for all $n \in \bZ$, we derive by induction that $a_n=0$ for all $n \neq 0$, which proves the first part.
In order to prove 2), we first note that, by Lemma \[unidul\], $u_{a,\kappa,\alpha} \in \cU_{\pm 1}$ according as $\kappa \in
{\mathcal K}_{\pm}$ respectively. Now if $u^* \in \cU_{1}$, that is to say, if $u^*=\sum_{n \geq 1} a_n q^n$, we set $$u^{**}:=u^*-\sum_{\kappa \in {\mathcal K}_+}u_{a,\kappa,a_{\kappa}}
\in \cU_{1}\, .$$ Set $\rho_+:=\sharp \cK_+$. Using (\[tutu\]), one easily checks that the map $B_+^0:\cU_+ \ra R^{\rho_+}$ defined by $$B_+^0(\sum a_n q^n)=(a_{\kappa})_{\kappa \in \cK_+}$$ is injective. On the other hand, by (\[meah\]), we have $B_+^0 u^{**}=0$. Thus, $u^{**}=0$, and $u^{*}=\sum u_{a,\kappa.a_{\kappa}}$. A similar argument holds for $\cU_{-1}$. This completes the proof of the second part.
In order to prove 3), let us note that if $$u=\sum_{n=-\infty}^{\infty} a_n q^n \in \cU_{\ra}\, ,$$ it is then clear that $$\psi_a\left( \sum_{n<0} a_n q^n \right)=0\, , \; \text{and}\;
\psi_a\left( \sum_{n\geq 0} a_n q^n
\right)=0\, .$$ Thus, $\sum_{n<0} a_n q^n \in \cU_-$ and $\sum_{n\geq 0}
a_n q^n \in \cU_+$. Therefore, $u \in \cU_-+\cU_+$, and so $\cU_{\ra}
\subset \cU_-+\cU_+$. A similar argument shows that $\cU_{\la} \subset
\cU_-+\cU_+$.
\[duclacasino\] \[rupdi\] Let us examine a special case of Theorem \[addeq\]. For integers $r,s \geq 1$ and $\lambda \in R^{\times}$, we consider the $\D$-character $$\label{gilmore} \psi_a:=(\dd^r+\lambda \phip^s-\lambda)y\, ,$$ If $(r,s)$ is any one of the pairs $(1,1), (1,2), (2,1), (2,2)$, then $\psi_a$ can be viewed as an analogue of the convection equation, heat equation, sideways heat equation, or wave equation, respectively. The characteristic polynomial of $\psi_a$ is $$\mu(\xi_p,\xi_q)=\xi_q^r+\lambda \xi_p^s -\lambda\, .$$ Clearly $\mu$ is unmixed. The characteristic integers are the integer roots of the equation $$\xi_q^r-\lambda=0\, .$$ Thus, if $\lambda \not\in \{n^r\ |\ n \in \bZ\}$, there are no characteristic integers, and $\cU_{\ra}=\cU_{\la}=\cU_0=R^{\phi^s}$.
Assume in what follows that $\lambda=n^r$ for some $n \in \bZ$. For even $r$, we may assume further that $n>0$. Then $\cK=\{n\}$ for $r$ odd, and $\cK=\{-n,n\}$ for $r$ even. The basic series for $\kappa \in \cK$ are $$\label{maomor} u_{a,\kappa,\alpha}= \sum_{j \geq 0} (-1)^j
\frac{1}{F_j(p^{sr})} \phip^{sj}(\alpha q^{\kappa})\, ,$$ where $F_j(x) \in \bZ[x]$ are the polynomials $F_0(x)=1$, $$F_j(x):=\prod_{i=1}^j (x^i-1)\, , \; j \geq 1\, .$$ Notice that the integers $F_j(p^{sr})$ have a nice simple interpretation in terms of flags: $$F_j(p^{sr})=(p^{sr}-1)^j \cdot \sharp(GL_j(\bF_{p^{sr}})/
B_j(\bF_{p^{sr}}))\, ,$$ where $B_j(\bF_{p^{sr}})$ is the subgroup of $GL_j(\bF_{p^{sr}})$ consisting of all upper triangular matrices. Also notice that, for $\kappa \in \cK$, we have that $\bar{u}_{a,\kappa,\alpha} \in k[[q^{\pm 1}]]$, the reduction mod $p$ of $u_{a,\kappa,\alpha}$, is given by $$\bar{u}_{a,\kappa,\alpha}=\sum_{n \geq 0} \bar{\alpha}^{p^n} q^{\kappa p^n}\, ,$$ so $\bar{u}_{a,\kappa,\alpha}$ is a root of the Artin-Schreier polynomial $$t^p-t+\bar{\alpha} q^{\kappa} \in k(q)[t]\, .$$
We have the following:
1. For $n>0$ and odd $r$, $$\begin{array}{ccl}
\cU_{-1} & = & 0\, ,\\
\cU_{1} & = & \{u_{a,n,\alpha}\ |\ \alpha \in R\}\, .
\end{array}$$
2. For $n <0$ and odd $r$, $$\begin{array}{ccl}
\cU_{-1} & = & \{u_{a,n,\alpha}\, |\;\alpha \in R\}\, ,\\
\cU_{1} & = & 0\, .
\end{array}$$
3. For $r$ even, $$\begin{array}{ccl}
\cU_{-1} & = & \{u_{a,-n,\alpha}\, | \; \alpha \in R\}\, , \\
\cU_{1} & = & \{u_{a,n,\alpha}\, | \;\alpha \in R\}\, .
\end{array}$$
4. $\cU_{\ra}=\cU_+$, $\cU_{\la}=\cU_-$.
Indeed, (a), (b) and (c) follow directly by Theorem \[addeq\]. The two families of solutions in (c) should be viewed as analogues of the two waves traveling in opposite directions in the case of the classical wave equation. In contrast to this, we have only one “wave” in (a), which is the case of the “convection” equation.
We prove the first equality in (d). The second follows by a similar argument. Let $$u=\sum_{n=-\infty}^{\infty} a_n q^n \in \cU_{\ra}\, .$$ It is clear that $$\psi_a\left( \sum_{n<0} a_n q^n \right)=0\, .$$ By (a), (b) and (c), we must have that $$\sum_{n<0} a_n q^n=u_{a,-|\kappa|,\alpha}$$ for some $\alpha \in R$. But $a_n \ra 0$ as $n \ra -\infty$, and this is the case for $u_{a,-|\kappa|,\alpha}$ only when $\alpha=0$ (see (\[maomor\])). Thus, $\alpha=0$, and $u \in \cU_+$.
We derive here some consequences of Theorem \[addeq\].
Under the hypotheses of Theorem [\[addeq\]]{} let $u \in
\cU_{\pm 1}$. Then the following hold:
1. The series $\bar{u} \in k[[q^{\pm 1}]]$ is integral over $k[q^{\pm 1}]$ and the field extension $k(q) \subset k(q,\bar{u})$ is Abelian with Galois group killed by $p$.
2. If the characteristic polynomial of $\psi_a$ is unmixed and $\cK_{\pm}$ is short then $u$ is transcendental over $K(q)$.
[*Proof*]{}. This follows directly from Theorem \[addeq\] and Lemma \[irinusescoala\].
\[narrenume\] Under the hypotheses of Theorem [\[addeq\]]{} the maps $B_{\pm}^0:\cU_{\pm 1} \ra R^{\rho_{\pm}}$ are $R$-module isomorphisms. Furthermore, for any $u \in \cU_{\pm 1}$ we have $$u=\sum_{\kappa \in {\mathcal K}_{\pm}} (B_{\kappa}^0 u) \star
u_{a,\kappa,1}\, .$$
In particular the “boundary value problem at $q^{\pm 1}=0$” is well posed.
The next Corollary says that the “boundary value problem at $q
\neq 0$” is well posed.
\[cosi\] Under the hypotheses of Theorem [\[addeq\]]{}, assume further that $\cK_+=\{\kappa \}$. Then for any $q_0 \in
p^{\nu}R^{\times}$ with $\nu \geq 1$, and any $g \in p^{\kappa
\nu}R$, there exists a unique $u \in \cU_{1}^{\rs}$ such that $u(q_0)=g$.
[*Proof*]{}. We need to show that the map $$\begin{array}{ccl}
R & \ra & p^{\kappa \nu}R \\ \alpha & \mapsto & \sum_{n \geq 0}
b_{n,\kappa} \alpha^{\phi^n} q_0^{\kappa p^n}
\end{array}$$ is bijective. This follows by Lemma \[ajutator\] below.
\[ajutator\] Let $c_0 \in R^{\times}$, $c_1,c_2,c_3,\ldots \in pR$ and $c_n \ra 0$ $p$-adically as $n \ra \infty$. Then the map $$\begin{array}{ccl}
R & \ra & R \\
\alpha & \mapsto & \sum_{n \geq 0} c_n
\alpha^{\phi^n}
\end{array}$$ is bijective.
[*Proof*]{}. The injectivity is clear. And surjectivity follows by a Hensel-type argument.
The following Corollary is concerned with the inhomogeneous equation $\psi_a u=\varphi$. Recall that for $\varphi
\in q^{\pm 1}R[[q^{\pm 1}]]$ we define the [*support*]{} of $\varphi$ as the set $\{n ; c_n \neq 0\}$. This notion of support is standard for series but note that it is not a direct analogue of the notion of support in real analysis if one pursues the analogy according to which $q$ is an analogue of the exponential of complex time. Also, for any series $v \in R((q^{\pm 1}))\h$ we denote by $\bar{v} \in k((q^{\pm 1}))$ the reduction of $v$ mod $p$.
Let $\psi_a$ be a non-degenerate $\D$-character of $\bG_a$ and let $\varphi \in q^{\pm 1}R[[q^{\pm 1}]]$ be a series whose support is contained in the set $\cK'$ of totally non-characteristic integers of $\psi_a$. Then the following hold:
1. The equation $\psi_a u=\varphi$ has a unique solution $u \in \bG_a(q^{\pm
1}R[[q^{\pm 1}]])$ such that $u$ has support disjoint from the set $\cK$ of characteristic integers.
2. If $\bar{\varphi} \in k[q^{\pm
1}]$ then $\bar{u} \in k[[q^{\pm 1}]]$ is integral over $k[q^{\pm
1}]$ and the field extension $k(q) \subset k(q,\bar{u})$ is Abelian with Galois group killed by $p$.
3. If the characteristic polynomial of $\psi_a$ is unmixed and the support of $\varphi$ is short then $u$ is transcendental over $K(q)$.
The existence in assertion 1 follows from Lemma \[unidul\]. Uniqueness follows from Lemma \[narrenume\]. Assertions 2 and 3 follows from \[irinusescoala\].
Corollary \[cosi\] implies that if $\psi_a$ is non-degenerate and $\cK_+=\{1\}$, for any $q_0 \in pR^{\times}$ the group homomorphism $$\begin{array}{ccl}
R & \stackrel{S_{q_0}}{\ra} & \bG_a(R)=R \\
\alpha & \mapsto &
\frac{1}{p}u_{a,1,\alpha}(q_0)
\end{array}$$ is an isomorphism. Thus, for any $q_1,q_2 \in pR^{\times}$, we have an isomorphism $$S_{q_1,q_2}:=S_{q_2} \circ S_{q_1}^{-1}:\bG_a(R) \ra \bG_a(R)\, .$$ mapping that can be viewed as the “propagator” attached to $\psi_a$. Note that if $\zeta \in \bmu(R)$ and $q_0 \in p
R^{\times}$, then by (\[picc\]) we have that $$S_{\zeta q_0}(\alpha)=\frac{1}{p} u_{a,1,\alpha}(\zeta
q_0)=\frac{1}{p} u_{a,1,\zeta \alpha}(q_0)=S_{q_0}(\zeta \alpha)\, ,$$ so $$S_{\zeta q_0}=S_{q_0} \circ M_{\zeta}\, ,$$ where $M_{\zeta}:R \ra R$ is the mapping defined by $M_{\zeta}(\alpha):=\zeta \alpha$. Thus, for $\zeta_1, \zeta_2 \in \bmu(R)$, we get that $$S_{\zeta_1 q_0,\zeta_2 q_0}=S_{q_0} \circ M_{\zeta_2/\zeta_1}
\circ S_{q_0}^{-1}\, .$$ In particular, $$S_{q_0,\zeta_1 \zeta_2 q_0}=S_{q_0,\zeta_2 q_0} \circ
S_{q_0,\zeta_1 q_0}\, .$$ This latter equality can be interpreted as a (weak) incarnation of “Huygens principle” ([@rauch], p. 104).
Multiplicative group
====================
In this section we prove our main results about $\D$-characters and their space of solutions in the case where $G$ is the multiplicative group.
Let $\bG_m:={\rm Spec}\, A[y,y^{-1}]$ be the multiplicative group over our fixed $\D$-ring $A$. We equip $\bG_m$ with the invariant $1$-form $$\omega:=\frac{dy}{y}\, .$$
Let us consider the $\D$-characters $$\psi_{\di}, \psi_{\ddi} \in \bX^1_{\di \ddi}(\bG_m) \subset
A[y,y^{-1}, \d y, \dd y]\wh$$ defined by $$\begin{array}{rcl}
\psi_{\di} = \psi_{m,\di} & := & {\displaystyle \frac{1}{p}\log{ \left(
\frac{\phip(y)}{y^p} \right)}= \frac{1}{p}\log{ \left( 1+p \frac{\d
y}{y^p} \right)}=\frac{\d y}{y^p}-\frac{p}{2}\left(\frac{\d
y}{y^p}\right)^2+\cdots }\, , \vspace{2mm} \\
\psi_{\ddi} = \psi_{m,\ddi} & := & {\displaystyle \dd \log{y}:= \frac{\dd
y}{y}}\, .
\end{array}$$ Here, if $y=1+T$, then $$\log{y}:=l(T)=T-\frac{T^2}{2}+\frac{T^3}{3}-\cdots$$ is the logarithm of the formal group of $\bG_m$. We clearly have $\psi_{\di} \in \bX_{\di}^1(\bG_m)$, and $\psi_{\ddi} \in
\bX_{\ddi}^1(\bG_m)$. The images of $\psi_{\di}$ and $\psi_{\ddi}$ in $A[[T]][\d T, \dd T]\h$ are $$\begin{array}{rcl}
\psi_{\di} & = & {\displaystyle \frac{1}{p}(\phip-p)l(T)}\, , \vspace{1mm} \\
\psi_{\ddi} & = & \dd l(T)\, .
\end{array}$$
\[999\] We have that $\dd \psi_{\di}=(\phip-1)\psi_{\ddi}$ in $\bX_{\di
\ddi}^2(\bG_m)$.
[*Proof*]{}. By a direct calculation, $$\dd \psi_{\di}=\dd \left( \frac{1}{p} \left( \phip-p\right)
l(T) \right)=(\phip-1)\dd l(T)=(\phip-1) \psi_{\ddi}\, .$$
\[roset\] For each $r \geq 1$, the $L$-vector space $\bX_{\di \ddi}^r(\bG_m)
\otimes_A L$ has basis $$\{\phip^i \psi_{\di}\mid_{0 \leq i \leq r-1}\} \cup
\{\phip^i \dd^j \psi_{\ddi}\mid_{0 \leq i+j \leq r-1}\}\, .$$ In particular, $\psi_{\di}$ and $\psi_{\ddi}$ generate the $L[\phip,\dd]$-module $\bX^{\infty}_{\di \ddi}(\bG_m) \otimes L$.
[*Proof*]{}. There is an exact sequence of homomorphisms of groups in the category of $p$-adic formal schemes $$0={\rm Hom}(\hat{\bG}_m,\hat{\bG}_a) \stackrel{\pi_r^*}{\ra}
{\rm Hom}(J^r_{\di \ddi}(\bG_m),\hat{\bG}_a) \stackrel{\rho}{\ra}
{\rm Hom}(N^r,\hat{\bG}_a)\, ,$$ where $\pi_r:J^r_{\di \ddi}(\bG_m) \ra
\hat{\bG}_a$ is the natural projection, $N^r={\rm ker}\, \pi_r$, and $\rho$ is defined by restriction. We recall that ${\rm Hom}(J^r_{\di
\ddi}(\bG_m),\hat{\bG}_a)$ identifies with the module of $\D$-characters $\bX^r_{\di \ddi}(\bG_m)$. Looking at the level of Lie algebras, we see that the rank of ${\rm Hom}(N^r,\hat{\bG}_a)$ over $A$ is at most equal to its dimension, $r(r+3)/2$. Since $\rho$ is injective, it is enough to show that the family in the statement of the Proposition is $A$-linearly independent. Thus, it is enough to show that the image of this family via the map (\[prost\]) is $A$-linearly independent. But $$\begin{array}{rcl}
(\phip^i \psi_{\di}) \circ e(pT) & = & {\displaystyle
\phip^i \left( \frac{1}{p} (\phip-p) l(e(pT)) \right)=\phip^{i+1}T-p
\phip^i T}\, ,\vspace{1mm} \\
(\phip^i \dd^j \psi_{\ddi}) \circ e(pT) & = & \phip^i \dd^{j+1}
l(e(pT))=p\phip^i
\dd^{j+1} T\, ,
\end{array}$$ and it is rather clear that these elements are $A$-linearly independent.
From now on, we let $A=R$, hence $L=K$, and we use the notation and discussion in Example \[cucurigu\] applied to $\bG_m$ over $R$. We have a natural embedding $$\begin{array}{ccc}
\iota:q^{\pm 1}R[[q^{\pm 1}]] & \ra & \bG_m(q^{\pm 1}R[[q^{\pm 1}]]) \\
u & \mapsto & \iota(u)=1+u
\end{array}$$
By Proposition \[roset\], any $\D$-character of $\bG_m$ is a $K$-multiple of a $\D$-character of the form $$\label{rosett} \psi_m:= \nu(\phip,\dd)\psi_{\ddi}+\lambda(\phip)
\psi_{\di}\, ,$$ where $\nu(\xi_p,\xi_q) \in R[\xi_p,\xi_q]$, $\lambda(\xi_p) \in
R[\xi_p]$. The Picard-Fuchs symbol of $\psi_m$ with respect to the étale coordinate $T=y-1$ is trivially seen to be $$\sigma(\xi_p,\xi_q)=p\nu(\xi_p,\xi_q)\xi_q+\lambda(\xi_p)(\xi_p-p).$$ Hence, the Fréchet symbol with respect to the invariant form $\omega=\frac{dy}{y}=\frac{dT}{1+T}$ is given by $$\theta(\xi_p,\xi_q)=
\frac{\sigma(p\xi_p,\xi_q)}{p}
=\nu(p\xi_p,\xi_q)\xi_q+\lambda(p\xi_p)(\xi_p-1)\, .$$
Let $\psi_m$ be a $\D$-character of $\bG_m$ of the form [(\[rosett\])]{}. We define the [*characteristic polynomial*]{} $\mu(\xi_p,\xi_q)$ of $\psi_m$ to be the Fréchet symbol $\theta(\xi_p,\xi_q)$ of $\psi_m$ with respect to $\omega$. We say that the $\D$-character $\psi_m$ is [*non-degenerate*]{} if $\mu(0,0) \in R^{\times}$, or equivalently, if $\lambda(0) \in
R^{\times}$. For a non-degenerate character $\psi_m$ with symbol $\mu(\xi_p,\xi_q)$, we define the [*characteristic integers*]{} to be the integers $\kappa$ that are solutions of the equation $\mu(0,\kappa)=0$. Any such characteristic integer $\kappa$ must be coprime to $p$. We say that $\kappa \in \bZ$ is [*totally non-characteristic*]{} if $\kappa \not\equiv 0$ mod $p$ and $\mu(0,\kappa) \not\equiv 0$ mod $p$. We denote by $\cK$ the set of all characteristic integers and set $\cK_{\pm}:=\cK \cap
\bZ_{\pm}$. Also we denote by $\cK'$ the set of totally non-characteristic integers. For any $0 \neq \kappa \in \bZ$ and $\alpha \in R$, we define the [*basic series*]{} $$\label{val}
u_{m,\kappa,\alpha}:=\exp{ \left(
\int u_{a,\kappa,\alpha}^{\mu} \frac{dq}{q} \right)}
\in 1+ \alpha \frac{q^{\kappa}}{\kappa} +\cdots \in K[[q^{\pm 1}]]\, ,$$ where $u_{a,\kappa,\alpha}^{\mu}$ is as in (\[ua\]), $$\exp{(T)}=1+T+\frac{T^2}{2!}+\frac{T^3}{3!}+\cdots\, ,$$ and $$v \mapsto \int vdq$$ is the usual indefinite integration $K[[q]] \ra qK[[q]]$ or the integration $q^{-2}K[[q^{-1}]] \ra q^{-1}K[[q^{-1}]]$, according to the cases $\kappa$ positive or negative, respectively.
Consider the $\D$-characters $$\psi_m:=\dd^{r-1} \psi_{\ddi}+(\lambda_{s-1}\phip^{s-1}+\cdots
+\lambda_1 \phip + \lambda_0) \psi_{\di}\, ,$$ where $\lambda_0
\in R^{\times}$, $\lambda_1,\ldots ,\lambda_{s-1} \in R$. If $(r,s)$ is any one of the pairs $(1,1)$, $(1,2)$, $(2,1)$, $(2,2)$, then $\psi_m$ can be viewed as an analogue of the convection equation, heat equation, sideways heat equation, or wave equation respectively. The characteristic polynomial of $\psi_m$ is $$\mu(\xi_p,\xi_q)=\xi_q^r+(p^{s-1}\lambda_{s-1}\xi_p^{s-1}+\cdots +p \lambda_1
\xi_p+ \lambda_0)(\xi_p-1)\, ,$$ hence $\mu$ is unmixed, and the characteristic integers are the integer roots of the equation $$\xi_q^r-\lambda_0=0\, .$$ In particular, for $$\label{speetro}
\psi_m:=\dd^{r-1} \psi_q+\lambda \psi_p\, ,$$ the characteristic polynomial equals $$\mu(\xi_p,\xi_q)=\xi_q^r+\lambda \xi_p-\lambda\, ,$$ which is the case $s=1$ of the Example \[duclacasino\]. So in this case $u_{a,\kappa,\alpha}^{\mu}$ in (\[val\]) is given by $$\label{spetro}
u_{a,\kappa,\alpha}^{\mu}=\sum_{n \geq 0} (-1)^n
\frac{1}{F_n(p^r)} \phip^n(\alpha q^{\kappa})\, .$$
\[disom\] Let us consider the “simplest” possible energy function $$H=a(\psi_q)^2+2b\psi_p \psi_q +c(\psi_p)^2\, ,$$ for $a,b,c \in R$. Using the obvious equalities $$\begin{array}{rcl}
\theta_{\psi_q,\omega} & = & \xi_q\, ,\\
\theta_{\psi_p,\omega} & = & \xi_p-1\, ,
\end{array}$$ and (\[ele\]), we obtain the following formula for the Euler-Lagrange equation associated to $H$ and the vector field $\partial=y \partial_y$: $$\epsilon^1_{H,\partial}=(-2a^{\phi} \phip \dd -2 b^{\phi}
\phip^2+2b) \psi_q+(-2c^{\phi}\phip +2c)\psi_p\, .$$ This $\D$-character $\epsilon^1_{H,\partial}$ is non-degenerate if, and only if, $c \in R^{\times}$, and its characteristic polynomial is given by $$\mu(\xi_p,\xi_q)=(-2b^{\phi}p^2\xi_p^2-2a^{\phi}p\xi_p\xi_q+2b)\xi_q+
(-2c^{\phi}p \xi_p+2c)(\xi_p-1)\, .$$ The set of characteristic integers of $\epsilon^1_{H,\partial}$ is $$\cK=\left\{\frac{c}{b}\right\} \cap \bZ\, .$$
\[ol\] Let $\kappa \in \bZ \backslash p\bZ$. Then, for any $\alpha \in R$, we have that $u_{m,\kappa,\alpha} \in 1+q^{\pm
1}R[[q^{\pm 1}]]$. Furthermore, the mapping $$\begin{array}{rcl}
R & \ra & R[[q^{\pm 1}]]^{\times}=\bG_m(R[[q^{\pm 1}]]) \\
\alpha & \mapsto & u_{m,\kappa,\alpha}
\end{array}$$ is a pseudo $\d$-polynomial map.
[*Proof*]{}. We let $u:=u_{m,\kappa,\alpha}^{\rs}=\exp{h}$, where $h:=\int u_{a,\kappa,\alpha}^{\mu} \frac{dq}{q}$. By Dwork’s Lemma \[dwor\], in order to show that $u_{m,\kappa,\alpha}^{\rs} \in 1+q^{\pm 1}R[[q^{\pm 1}]]$ it is enough to prove that $u^{\phi}/u^p \in 1+pq^{\pm 1}R[[q^{\pm 1}]]$. Since $u^{\phi}/u^p=\exp((\phi-p)h)$, we just need to show that $$\label{cevreu} (\phip-p)h \in pq^{\pm 1}R[[q^{\pm 1}]]\, .$$
Let us start by observing that $$\label{tatty}
\dd
h=u_{a,\kappa,\alpha}^{\mu}\, ,$$ and consider the $\D$-character of $\bG_a$ defined by $$\psi_a:=\psi_a^{\mu}:=\mu(\phip,\dd)y\, .$$ By Lemma \[unidul\], we obtain that $$\begin{array}{rcl}
\dd(p\mu(0,\kappa)\alpha \kappa^{-1} q^{\kappa}) & = & p
\mu(0,\kappa) \alpha q^{\kappa}\\
& = & p
\psi_a^{\mu} u_{a,\kappa,\alpha}^{\mu}\\
\ & = & p[\nu(p
\phip,\dd)\dd+\lambda(p \phip)(\phip -1)]\dd h \vspace{1mm} \\
& = & \dd[p \nu(\phip,\dd) \dd+\lambda(\phip)(\phip -p)]h\, .
\end{array}$$ We deduce that $$[p \nu(\phip,\dd)\dd+\lambda(\phip)(\phip-p)]h -
p \mu(0,\kappa)\alpha \kappa^{-1}q^{\kappa}
\in q^{\pm 1} K[[q^{\pm
1}]] \cap K=0\, ,$$ and consequently, $$\begin{array}{rcl}
\lambda(\phip)(\phip -p)h & = & -p \nu(\phip,\dd)\dd h
+p \mu(0,\kappa)\alpha \kappa^{-1}q^{\kappa}\\
& = & -p \nu(\phip,\dd) u_{a,\kappa,\alpha}^{\mu}
+p \mu(0,\kappa)\alpha \kappa^{-1}q^{\kappa}\\
& \in & pq^{\pm 1} R[[q^{\pm 1}]]\, .
\end{array}$$ By Lemma \[floricica\], (\[cevreu\]) then follows.
In order to prove the remaining assertion, let us assume that $\kappa >0$. The opposite case can be argued similarly. Note that we can find polynomials $Q_n \in R[z_0,z_1,\ldots,z_{r_n}]$, and integers $m_n \geq 0$, such that $$u_{m,\kappa,\alpha}=\sum_{n \geq 1}
p^{-m_n} Q_n(\alpha,\alpha^{\phi},\ldots,\alpha^{\phi^{r_n}})q^n$$ for all $\alpha \in R$. By the part of the Lemma already proven, we have $$p^{-m_n}Q_n(\alpha,\alpha^{\phi},\ldots,\alpha^{\phi^{r_n}})\in R$$ for all $\alpha \in R$. We then apply Corollary 3.21 in [@book], and conclude that there exists a polynomial $P_n \in R[x_0,x_1,\ldots,x_{r_n}]$ such that $$p^{-m_n}Q_n(\alpha,\alpha^{\phi},\ldots,\alpha^{\phi^{r_n}})=
P_n(\alpha,\alpha^{\phi},\ldots,\alpha^{\phi^{r_n}})$$ for all $\alpha \in R$. Thus, the mapping $$\begin{array}{rcl}
R & \ra & R[[q]] \\ \alpha & \mapsto & u_{m,\kappa,\alpha}
\end{array}$$ is pseudo $\d$-polynomial. Then $$\begin{array}{rcl}
R & \ra & R[[q]] \\ \alpha & \mapsto &
\frac{1}{u_{m,\kappa,\alpha}}
\end{array}$$ is pseudo $\d$-polynomial also. By considering the embedding $$\begin{array}{rcc}
\bG_m & \ra & \bA^2 \\
x & \mapsto & (x,x^{-1})
\end{array}\, ,$$ the last assertion of the Lemma follows.
As in the additive case, we have the following “diagonalization” result.
\[gainuse\] \[saduslacaf\] For all $\kappa \in \bZ \backslash p\bZ$ and $\alpha \in
R$, we have that $$\begin{array}{rcl}
\psi_q u_{m,\kappa,\alpha} & = & u_{a,\kappa,\alpha}^{\mu}\\
\psi_m u_{m,\kappa,\alpha} & = & \mu(0,\kappa) \cdot \alpha
\kappa^{-1} q^{\kappa}.\end{array}$$
[*Proof*]{}. The first equality is clear. Let us assume now that $\kappa >0$. A similar argument can be used to handle the case $\kappa<0$. By Lemma \[999\], we obtain that $$\begin{array}{rcl}
\dd \psi^{\rs}_m & = & \dd \left[\nu(\phip,\dd)\psi_{\ddi} +
\lambda(\phip) \psi_{\di} \right]\\ & = &
[\nu(p\phip,\dd)\dd+\lambda(p\phip)(\phip-1)]\psi_{\ddi}
\\ & = & \psi_a^{\mu} \psi_{\ddi}\, ,
\end{array}$$ where $\psi_a^{\mu}:=\mu(\phip,\dd)y$. In particular, by Lemma \[unidul\] we have that $$\begin{array}{rcl} \dd \psi_m u_{m,\kappa,\alpha}
& = & \psi_a^{\mu} \psi_{\ddi} u_{m,\kappa,\alpha}\\
& = & \psi_a^{\mu} u_{a,\kappa,\alpha}^{\mu}\\
& = & \mu(0,\kappa) \alpha q^{\kappa}\\
& = & \dd(\mu(0,\kappa) \alpha
\kappa^{-1}q^{\kappa})\, ,
\end{array}$$ and so $v:=\psi_m
u_{m,\kappa,\alpha} -\mu(0,\kappa)\alpha \kappa^{-1} q^{\kappa}
\in R$. On the other hand, $$v(0)=(\psi_m u_{m,\kappa,\alpha})(0)=\psi_m(u_{m,\kappa,\alpha}(0))=
\psi_m(1)=0\, ,$$ hence $v=0$, and we are done.
1. For all $\kappa \in \bZ \backslash p\bZ$ the map $$\begin{array}{rcc}
R & \ra & R[[q^{\pm 1}]]^{\times} \\
\alpha & \mapsto & u_{m,\kappa,\alpha}^{\rs}
\end{array}$$ is an injective homomorphism.
2. We recall (see (\[vocea\])) the natural group homomorphisms attached to $\psi_q$, $$\begin{array}{ccc}
B_{\kappa}^0:R[[q^{\pm 1}]]^{\times} & \ra & R \\
B_{\kappa}^0 u & = & \Gamma_{\kappa} \psi_q u,
\end{array}$$ where $\kappa \in \bZ \backslash p\bZ$ and the homomorphism $$\begin{array}{ccc}
B_{\pm}^0:R[[q^{\pm 1}]]^{\times} & \ra & R^{\rho_{\pm}} \\
B_{\pm}^0 u & = & (\Gamma_{\kappa} \psi_q u)_{\kappa \in
\cK_{\pm}}.
\end{array}$$ For integers $\kappa_1,\kappa_2 \in \bZ \backslash p\bZ$ we get that $$\label{pueah} B_{\kappa_1}^0
u_{m,\kappa_2,\alpha} =\Gamma_{\kappa_1}
u^{\mu}_{a,\kappa_2,\alpha}=
\alpha \cdot \delta_{\kappa_1 \kappa_2}\, .$$
3. For $\kappa \in \bZ \backslash p\bZ$ we see that $$\label{pim} u_{m,\kappa,\zeta^{\kappa}
\alpha}(q)=u_{m,\kappa,\alpha}(\zeta q)$$ for all $\zeta \in \bmu(R)$. In particular, if $\alpha=\sum_{i=0}^{\infty} m_i \zeta^{\kappa}_i$, $\zeta_i \in
\bmu(R)$, $m_i \in \bZ$, $v_p(m_i) \ra \infty$, we have $$u_{m,\kappa,\alpha}(q)
=\prod_{i=0}^{\infty}(u_{m,\kappa,1}(\zeta_i q))^{m_i}\, .$$ (Note that the right hand side of the equality above converges in the $(p,q^{\pm 1})$-adic topology of $R[[q^{\pm 1}]]$.) Thus, if $f
\in \bZ \bmu(R)\h$ is such that $(f^{[\kappa]})^{\sh}=\alpha \in
R$, then $u_{m,\kappa,\alpha}$ can be expressed via convolution by $$u_{m,\kappa,\alpha}=f \star u_{m,\kappa,1}\, .$$ Under convolution, the set $\{u_{m,\kappa,\alpha}\ |\ \alpha \in
R\}$ is a $\bZ \bmu(R)\h$-module, whose module structure arises from an $R$-module structure (still denoted by $\star$) induced by base change via the morphism $$\bZ \bmu(R)\h \stackrel{[\kappa]}{\ra}
\bZ \bmu(R)\h \stackrel{\sh}{\ra} R\, .$$ (Cf. to (\[notinjjj\])). The $R$-module $\{u_{m,\kappa,\alpha}\, | \;
\alpha \in R\}$ is free with basis $u_{m,\kappa,1}$. So, for $g
\in \bZ \bmu(R)\h$, $\beta=(g^{[\kappa]})^{\sh}$, we have that $$\beta \star u_{m,\kappa,\alpha}=g \star u_{m,\kappa,\alpha}\, ,$$ and in particular $$u_{m,\kappa,\alpha} = \alpha \star u_{m,\kappa,1}\, .$$
4. We have the following “rationality” property: if $\alpha \in
\bZ_{(p)}$, then $u_{m,\kappa,\alpha} \in \bZ_{(p)}[[q^{\pm 1}]]$ for $\kappa \in \bZ \backslash p\bZ$.
\[666\] Let $\psi_m$ be a non-degenerate $\D$-character of $\bG_m$, $\cU_*$ the corresponding groups of solutions, $\cK$ the set of characteristic integers, and $u_{m,\kappa,\alpha}$ the basic series. We set $$\begin{array}{rclll}
\cU_{tors} & := & \bmu(R) & \subset & \cU_0\\
\cU_{\sim} & := & (R^{\times} \cdot q^{\bZ}) \cap \cU & \subset &
\cU_{\da}\, .
\end{array}$$ Then the following hold:
1. $\cU_{\ra}=\cU_{\sim} \cdot \cU_{1}$, and $\cU_{\la}=\cU_{\sim} \cdot \cU_{-1}$.
2. We have that $$\begin{array}{ccc}
\cU_{\pm 1} & = & \prod_{\kappa \in \cK_{\pm}}
\{u_{m,\kappa,\alpha}\, |\; \alpha \in R\}\, ,
\end{array}$$ where $\prod$ stands for internal direct product. In particular, $\cU_{\pm 1}$ are free $R$-modules under convolution, with bases $\{u_{m,\kappa,1}\, |\; \kappa \in
\cK_{\pm}\}$ respectively.
3. We have that $\cU_0=\cU_{tors}$.
\[funnyterm\] The elements $u=bq^n$ of $R^{\times} \cdot
q^{\bZ}$ deserve to be called [*plane waves*]{} with $$\begin{array}{ll}
\textit{frequency} & n
\in \bZ\, ,\\
\textit{wave number} & \gamma:=\psi_{m,\di}(b)={\displaystyle \frac{1}{p}\log
\frac{\phip(b)}{b^p}}\in R\, ,\\
\textit{wave length} & 1/\gamma \in K \cup \{\infty\}\, ,\\
\textit{propagation speed}
& n /\gamma \in K \cup \{\infty\}\, .
\end{array}$$ This can be justified by the analogy behind the chosen terminology. Indeed, if $u=bq^n$, then $$n=\frac{\dd u}{u}\, ,$$ and so $n$ is to be interpreted as the “logarithmic derivative” of $u$ with respect to $\dd$. Also, we have that $$\gamma=\frac{1}{p}\log \frac{\phip(u)}{u^p}\, ,$$ and so $\gamma$ is the analogue of a “logarithmic derivative of $u$ with respect to $\d$.” These observations suffice to justify the assertion.
For in the classical theory, the complex valued function $u(t,x)$ of real variables $t$ and $x$ defined by $$u(t,x):=a(x)e^{-2 \pi i \nu t}$$ is viewed as a plane wave with frequency $\nu$, wave number (at $x$) $\gamma=\gamma(x)=\frac{1}{2 \pi i}
\frac{a'(x)}{a(x)}$, wave length $1/\gamma$, and propagation speed $\nu /\gamma$. (The standard situation is that in which $a(x)=e^{2
\pi i \gamma x}$, where $\gamma \neq 0$ is a constant.) Notice that these $\nu$ and $\gamma$ are equal to $$-(2 \pi i)^{-1} \frac{\partial_t u}{u}\quad \text{and} \quad
(2 \pi i)^{-1} \frac{\partial_x u}{u}\, ,$$ respectively. Thus, our terminology correspond to these classical definitions of frequency and wave number provided that $\dd$ and $\d$ are viewed as analogues of $-(2 \pi i)^{-1}
\partial_t$ and $(2 \pi i)^{-1}\partial_x$, respectively.
The solutions in $\cU_{\sim}$ correspond to those solutions that in the classical case are obtained by “separation of variables.”
[*Proof of Theorem \[666\]*]{}. By Lemma \[saduslacaf\] we have $$\psi_m u_{m,\kappa,\alpha}=0$$ for $\kappa \in \cK$. We now show that if $u_m \in
\cU_{\ra}$ or $u_m \in \cU_{\la}$, then $u_m \in \cU_{\sim} \cdot
\cU^{\rs}_{\pm 1}$, respectively, which will end the proof of assertion 1). Indeed, let us just treat the case where $u_m \in
\cU_{\ra}$. The case $u_m \in \cU_{\la}$ follows by a similar argument.
Note that $$0=\dd \psi_m^{\rs} u_m=\psi^{\mu}_a \psi_{\ddi} u_m\, ,$$ and so by Theorem \[addeq\], there exist $\alpha_1,\ldots,\alpha_{s}
\in R$ and $\lambda \in R$ such that $$q \frac{du_m/dq}{u_m}=\psi_{\ddi} u_m=\lambda+\sum_{i=1}^s u_{a,\kappa_i,
\alpha_i}^{\mu}
=\lambda+ \sum_{i=1}^s q \frac{du_{m,\kappa_i,
\alpha_i}/dq}{u^{\rs}_{m,\kappa_i,
\alpha_i}}\, ,$$ where $\cK_+=\{\kappa_1,\ldots,\kappa_s\}$. Hence, if $$v:=\frac{u_m}{u_{m,\kappa_1,\alpha_1}\ldots u_{m,\kappa_s,\alpha_s}}
=\sum_{n=-\infty}^{\infty}
b_nq^n\, ,$$ then $$q\frac{dv/dq}{v}=\lambda \, ,$$ and so $nb_n=\lambda b_n$ for all $n$. Thus, $v=bq^n$ with $n \in \bZ$ and $b \in R^{\times}$. This shows that $u_m \in \cU_{\sim} \cdot
\cU_{1}$, which completes the argument.
We now prove assertion 2) for the case where $u_m \in \cU_{1}$. The case $u_m \in \cU_{-1}$ is treated by a similar argument. We write $$u_m=bq^n \prod u_{m,\kappa_i,\alpha_i}$$ as above. Since $u_m(0)=1=\prod u_{m,\kappa_i,\alpha_i}(0)$, it follows that $n=0$ and $b=1$ so $u_m=\prod
u_{m,\kappa_i,\alpha_i}$. This representation is unique due to formula (\[pueah\]).
The last assertion of the theorem is clear.
\[numagidila\] Under the hypotheses of Theorem [\[666\]]{}, let $u \in \cU_{\pm}$. Then the following hold:
1. The series $\overline{\psi_q u} \in k[[q^{\pm 1}]]$ is integral over $k[q^{\pm 1}]$ and the field extension $k(q)\subset
k(q,\overline{\psi_q u})$ is Abelian with Galois group killed by $p$.
2. If the characteristic polynomial of $\psi_m$ is unmixed and $\cK_{\pm}$ is short then $u$ is transcendental over $K(q)$.
[*Proof*]{}. Assume $u \in \cU_+$; the case $u \in \cU_-$ is similar. By Theorem \[666\] we may write $$u=a \prod_{\kappa \in \cK_+} u_{m,\kappa,\alpha_{\kappa}}\, ,$$ with $\alpha_{\kappa} \in R$, $a \in R^{\times}$. So by Lemma \[gainuse\], $$\psi_q u=\sum_{\kappa \in \cK_+} u^{\mu}_{a,\kappa,\alpha_{\kappa}}\, .$$ By Lemma \[irinusescoala\], assertion 1 follows. To check assertion 2 note that if $u$ were algebraic over $K(q)$ then the same would hold for $\psi_q u=\dd u/u$ and we would get a contradiction by Lemma \[irinusescoala\].
\[niciel\] Under the hypotheses of Theorem [\[666\]]{}, the maps $B_{\pm}^0:\cU_{\pm 1} \ra R^{\rho_{\pm}}$ are $R$-module isomorphisms. Furthermore, for any $u \in \cU_{\pm 1}$ we have $$u=\sum_{\kappa \in {\mathcal K}_{\pm}} (B_{\kappa}^0 u) \star
u_{m,\kappa,1}\, .$$
So, in particular, the “boundary value problem at $q^{\pm 1}=0$” is well posed. We now address the “boundary value problem at $q
\neq 0$”, and the “limit at $q=0$” issue:
\[maisus\] Under the hypotheses of Theorem [\[666\]]{}, the following hold:
1. If $\cK_+=\{\kappa\}$, then for any $q_0 \in
p^{\nu}R^{\times}$ with $\nu \geq 1$, and any $g \in 1+p^{\kappa
\nu}R$, there exists a unique $u \in \cU_{1}$ such that $u(q_0)=g$.
2. If $\cK_+=\{1\}$, then for any $q_0 \in pR^{\times}$, and any $g \in R^{\times}$, there exists a unique $u \in \cU_{tors} \cdot \cU_{+}$ such that $u(q_0)=g$. Furthermore, $u(0)$ is the unique root of unity in $R$ that is congruent to $u(q_0)$ mod $p$.
[*Proof*]{}. We recall that $\log: 1+p^NR \ra p^N R$ is a bijection for all $N \geq 1$. So, in order to prove assertion 1), we need to check that there is a unique $\alpha \in R$ such that $$\log u^{\rs}_{m,\kappa,\alpha}(q_0)=\log\ g\, .$$ By the proof of Proposition \[666\], we have that $$\log\ u_{m,\kappa,\alpha}^{\rs}(q_0)=\sum_{n \geq 0} c_n
\alpha^{\phi^n}\, ,$$ where the $p$-adic valuation of $c_n$ is $\kappa \nu p^n-n$. Hence, by Lemma \[ajutator\], the mapping $$\begin{array}{rcl}
R & \ra & p^{\kappa \nu}R \\ \alpha & \mapsto &
\sum c_n \alpha^{\phi^n}
\end{array}$$ is bijective, and we are done.
In order to prove assertion 2) let $g=\gamma_0 \cdot v_0$ with $\gamma_0$ be a root of unity, and $v_0 \in 1+pR$. By assertion 1), there exists $v \in \cU_{1}$ such that $v(q_0)=v_0$. Hence, if $u:=\gamma_0 \cdot v$, then $u(q_0)=g$, which proves the existence part. Now, if $\gamma_1 \in \cU_{tors}$, $v_1 \in \cU_{1}$, and $\gamma_1 v_1 (q_0)=\gamma_0 v_0$, we get that $\gamma_1 \equiv
\gamma_0$ mod $p$, so $\gamma_1=\gamma_0$ and $v_1(q_0)=v_0$. By the uniqueness in the first part above, we see that $v_1=v$, which proves the uniqueness part of the assertion.
The claim about $u(0)$ is clear.
The following Corollary is concerned with the inhomogeneous equation $\psi_m u=\varphi$.
\[magidila\] Let $\psi_m$ be a non-degenerate $\D$-character of $\bG_m$, and let $\varphi \in q^{\pm 1}R[[q^{\pm 1}]]$ be a series whose support is contained in the set $\cK'$ of totally non-characteristic integers of $\psi_m$. Then the following hold:
1. The equation $\psi_m u=\varphi$ has a unique solution $u \in
\bG_m(q^{\pm 1}R[[q^{\pm 1}]])$ such that the support of $\psi_q
u$ is disjoint from the set $\cK$ of characteristic integers.
2. If $\bar{\varphi} \in k[q^{\pm 1}]$, the series $\overline{\psi_q u} \in k[[q^{\pm 1}]]$ is integral over $k[q^{\pm 1}]$, and the field extension $k(q) \subset
k(q,\overline{\psi_q u})$ is Abelian with Galois group killed by $p$.
3. If the characteristic polynomial of $\psi_m$ is unmixed and the support of $\varphi$ is short then $u$ is transcendental over $K(q)$.
[*Proof*]{}. Let us assume that $\varphi \in qR[[q]]$. The case $\varphi \in q^{\pm 1}R[[q^{\pm 1}]]$ is similar. We express $\varphi$ as $$\varphi=\sum_{\kappa \in S} a_{\kappa}q^{\kappa}\, ,$$ where $S$ is the support of $\varphi$ and set $\alpha_{\kappa}:=\kappa (\mu(0,\kappa))^{-1} a_{\kappa}$. We define $$u:=\prod_{\kappa \in S} u_{m,\kappa,\alpha_{\kappa}}
\in \bG_m(qR[[q]])\, ,$$ which converges $q$-adically. By Lemma \[saduslacaf\], $\psi_m
u=\varphi$. And observe that $u$ is the unique solution in $\bG_m(qR[[q]])$ subject to the condition that $\psi_qu$ has support disjoint from $\cK$; cf. Lemma \[niciel\]. By Lemma \[saduslacaf\], $$\psi_q u=\sum_{\kappa \in S} u_{a,\kappa,\alpha_{\kappa}}\, ,$$ so we may reach the desired conclusion by Lemma \[irinusescoala\].
The second part of Corollary \[maisus\] implies that, if $\psi_m$ is a non-degenerate $\D$-character of $\bG_m$, and $\cK_+=\{1\}$, then for any $q_0 \in pR^{\times}$ the group homomorphism $$\begin{array}{ccl}
S_{q_0}:\bmu(R) \times R & \ra & \bG_m(R)=R^{\times} \\
(\xi,\alpha) & \mapsto & \xi \cdot u_{m,1,\alpha}(q_0)
\end{array}$$ is an isomorphism. Thus, for any $q_1,q_2 \in p R^{\times}$, we have an isomorphism $$S_{q_1,q_2}:=S_{q_2} \circ S_{q_1}^{-1}:\bG_m(R) \ra \bG_m(R)\, ,$$ which should be viewed as the “propagator” associated to $\psi_m$. Note that if $\zeta \in \bmu(R)$ and $q_0 \in p R^{\times}$, by (\[pim\]), $$S_{\zeta q_0}(\xi,\alpha)=\xi \cdot u_{m,1,\alpha}(\zeta
q_0)=\xi \cdot u_{m,1,\zeta \alpha}(q_0)=S_{q_0}(\xi,\zeta
\alpha)\, ,$$ and so $$S_{\zeta q_0}=S_{q_0} \circ M_{\zeta}\, ,$$ where $$\begin{array}{ccl}
M_{\zeta}:\bmu(R) \times R & \ra & \bmu(R) \times R \\
M_{\zeta}(\xi,\alpha) & = & (\xi,\zeta \alpha)
\end{array}\, .$$ Hence, as for the case of $\bG_a$, if $\zeta_1, \zeta_2 \in \bmu(R)$, we get that $$S_{\zeta_1 q_0,\zeta_2 q_0}=S_{q_0} \circ M_{\zeta_2/\zeta_1}
\circ S_{q_0}^{-1}\, .$$ In particular, $$S_{q_0,\zeta_1 \zeta_2 q_0}=S_{q_0,\zeta_2 q_0} \circ
S_{q_0,\zeta_1 q_0}\, ,$$ which can be interpreted as a (weak) “Huygens principle.”
Elliptic curves
===============
We begin this section by proving some general results about the space of $\D$-characters of an elliptic curve over a general $\D$-ring $A$. We determine bases for these spaces; the answer depends on whether certain Kodaira-Spencer classes (an arithmetic one, and a geometric one) vanish or not. We then prove our main results about $\D$-characters and their solution spaces for Tate curves over $R((q))$, and for elliptic curves over $R$.
General results
---------------
We have as before a fixed $\D$-ring ring $A$ that is $p$-adically complete Noetherian integral domain of characteristic zero. For technical reasons, we assume hereafter that the prime $p$ is greater than $3$. We consider an elliptic curve $E$ over $A$, equipped with an invertible $1$-form $\omega$. All the definitions and constructions below apply to the pair $(E,\omega)$.
Since $p \geq 5$, the integer $6$ is invertible in $A$, and therefore we have unique elements $a_4,a_6 \in A$ such that $4a_4^3+27a_6^2 \in
A^{\times}$, and such that, up to isomorphism, the pair $(E, \omega)$ consists of the closure in the projective plane over $A$ of the affine plane curve $$\label{cubic} y^2=x^3+a_4x+a_6\, ,$$ endowed with the $1$-form $$\label{cubomega} \omega=\frac{dx}{y}\, .$$
We set $$\label{Nsusr} N^r:={\rm ker}(J^r_{\di \ddi}(E)
\stackrel{\pi_r}{\ra} \hat{E}).$$ Exactly as in the theory of algebraic group extensions in the last Chapter of [@serre], we have an exact sequence $$\label{papica} 0={\rm Hom}(\hat{E},\hat{\bG}_a) \stackrel{\pi_r^*}{\ra}
{\rm Hom}(J^r_{\di \ddi}(E),\hat{\bG}_a) \stackrel{\rho}{\ra}
{\rm Hom}(N^r,\hat{\bG}_a) \stackrel{\partial}{\ra} H^1(E,\cO)\, ,$$ where $\rho$ is the restriction. Let us recall that ${\rm Hom}(J^r_{\di \ddi}(E),\hat{\bG}_a)$ identifies with the module $\bX_{\di \ddi}^r(E)$ of $\D$-characters. We review and use this sequence by closely following [@book], pp. 191-196, where the case of $J^r_{\di}(E)$ was considered.
First of all, note that the projection $\pi_r:J^r_{\di \ddi}(E) \ra \hat{E}$ has a natural structure of a principal homogeneous space under $\hat{E} \times N^r \ra \hat{E}$. By Proposition \[local\], $\pi_r$ has local sections in the Zariski topology. We consider a Zariski open covering and a corresponding family of sections of $\pi_r$; $$\label{sec}
E=\bigcup U_{\mu},\ \ s_{\mu}:\hU_{\mu} \ra \pi_r^{-1}(\hU_{\mu})\, .$$ We may and assume that there exists an index $\mu_0$ such that the zero section $0 \in E(A)$ belongs to $U_{\mu_0}(A)$, and $s_{\mu_0}(0)=0$. Then the morphism $$\label{noapte adinca}
\tau_{\mu}:\hU_{\mu} \times N^r \ra \pi_r^{-1}(\hU_{\mu})\, ,$$ which at the level of $S$-points ($S$ any $A$-algebra) is given by $(A,B) \mapsto s_{\mu}(A)+B$, is an isomorphism of principal homogeneous spaces under $\hU_{\mu} \times N^r \ra \hU_{\mu}$. The isomorphism $\tau_{\mu_0}$ induces the identity $0 \times N^r \ra \pi_r^{-1}(0)=N^r$, and if $T \in \cO(U_{\mu_0})$ is an étale coordinate on $U_{\mu_0}$, then we have an induced $\cO(\hat{U}_{\mu_0})$-isomorphism $$\label{extradoo}
\begin{array}{c}
\tau_{\mu_0}^*: \cO(\hat{U}_{\mu_0})[T^{(i,j)}\mid_{1 \leq
i+j \leq r} ]\wh
\stackrel{can}{\simeq} \cO^r(\pi_r^{-1}(\hat{U}_{\mu_0}))
\stackrel{\circ \tau_{\mu_0}}{\ra} \vspace{2mm} \\
\stackrel{\circ \tau_{\mu_0}}{\ra} \cO^r(\hat{U}_{\mu_0}
\times N^r)=
\cO^r(\hat{U}_{\mu_0})[T^{(i,j)}\mid_{1 \leq i+j \leq r}]\wh \, ,
\end{array}$$ where $can$ is the unique isomorphism sending $T^{(i,j)}$ into $\d^i \dd^j T$. Furthermore, $\tau_{\mu_0}^*$ is the identity modulo $T$.
Let $\hU_{\mu \nu}=\hU_{\mu} \cap \hU_{\nu}$. The sections (\[sec\]) induce maps $s_{\mu}-s_{\nu} \, : \, \hU_{\mu \nu} \ra J^r_{\di \ddi}(E)$ that clearly factor through maps $$\label{siminussj}
s_{\mu \nu}:\hU_{\mu \nu}\ra N^r.$$ In particular, at the level of $S$-points we have $(\tau_{\nu}^{-1} \circ \tau_{\mu})(A,B)=(A,s_{\mu \nu}(A)+B)$.
We define the map $\partial$ in (\[papica\]) by attaching to any homomorphism $\Theta:N^r \ra \hat{\bG}_a$ the cohomology class $\varphi:=\partial(\Theta) \in H^1(\hat{E}, \cO)=H^1(E,\cO)$ that is represented by the cocycle $\varphi_{\mu \nu}:=\Theta \circ s_{\mu \nu} \in \cO(\hU_{\mu \nu})$. Then, as in [@book], p. 192, we check that the sequence (\[papica\]) is exact.
\[indi\] The rank of the $A$-module $\bX_{\di \ddi}^r(E)$ is $r(r+3)/2$ if $\partial=0$, and $r(r+3)/2-1$ if $\partial \neq 0$.
[*Proof*]{}. Looking at the Lie algebras, it is clear that ${\rm Hom}(N^r,\hat{\bG}_a)$ is an $A$-module of rank at most ${\rm dim}\, N^r=r(r+3)/2$. By Lemma \[fu\], the homomorphisms $L^{[a,b]} \in {\rm Hom}(N^r,\hat{\bG}_a)$, $1 \leq a+b \leq r$, are $A$-linearly independent. Thus, ${\rm Hom}(N^r,\hat{\bG}_a)$ has rank $r(r+3)/2$ over $A$, and the conclusion follows.
Since $H^1(E,\cO)$ has rank $1$ over $A$, by the exact sequence (\[papica\]) we conclude that
The $A$-module $\bX_{\di \ddi}^1(E)$ is non-zero.
This result contrasts deeply with the “ode” story for both, the geometric [@ajm] and arithmetic [@char] case. Indeed, for “general” $E$ we then have that $$\bX_{\di}^1(E)=\bX_{\ddi}^1(E)=0\, .$$
In the sequel, we construct and analyze $\D$-characters of $E$ in more detail. We use the invertible $1$-form $\omega$ on $E$, and assume that the closed subscheme of $U_{\mu_0}$ defined by $T$ is the zero section $0$. We identify the $T$-adic completion of $\cO(U_{\mu_0})$ with the ring of power series $A[[T]]$, and furthermore, choose $T$ such that $\omega \equiv dT$ mod $(T)$. Using the cubic (\[cubic\]), we may take, for instance, $T=-\frac{x}{y}$. We set $W:=-\frac{1}{y}$.
This affine coordinate $T$, and $W$ around the zero section, are mapped into $T \in A[[T]]$ and $$\label{dub} W(T)=T^3+a_4T^7+\cdots \in A[[T]]\, ,$$ respectively; cf. [@sil2], p. 111.
For all integers $a,b \in \bZ_+$ with $a+b \geq 1$, we define [*$\D$-Kodaira-Spencer classes*]{} $f^{[a,b]} \in A$ as follows. For $r \geq a+b$, we have a natural isomorphism $N^r \simeq (\hat{\bA}^{\frac{r(r+3)}{2}},[+])$, where $\hat{\bA}^{\frac{r(r+3)}{2}}=Spf\ A[T^{(i,j)}\mid_{1 \leq i+j
\leq r}]\wh$, which we view hereafter as an identification. Thus, we obtain the identification $$\begin{array}{c}
s_{\mu \nu}=(\alpha_{\mu \nu}^{i,j}\mid_{1 \leq i+j \leq r}) \in
\cO(\hU_{\mu \nu})^{\frac{r(r+3)}{2}}\, , \\
\alpha_{\mu \nu}^{i,j}:=T^{(i,j)} \circ s_{\mu \nu}\, .
\end{array}$$ By (\[romanu\]), we have the series $L^{[a,b]} \in A[T^{(i,j)}\mid_{1 \leq i+j \leq r}]\wh$ defining homomorphisms $$L^{[a,b]}:N^r \simeq (\hat{\bA}^{\frac{r(r+3)}{2}},[+]) \ra \hat{{\mathbb G}
}_a=(\hat{\bA}^1,+)\, .$$ We define elements $$\varphi^{[a,b]}_{\mu \nu}:=L^{[a,b]}(\alpha_{\mu \nu}^{i,j}\mid_{1 \leq i+j
\leq r}) \in \cO(\hU_{\mu \nu})\, .$$ As we vary $\mu \nu$, the collection of such sections defines a cohomology class $$\varphi^{[a,b]} \in H^1(\hat{E},\cO_{\hat{E}})=H^1(E,\cO_E) \, ,$$ which is, of course, the class $\gamma(L^{[a,b]})$ defined by the exact sequence (\[papica\]). Let $$\<\ ,\ \>:H^1(E,\cO) \times H^0(E,\Omega^1) \ra A$$ denote the Serre duality pairing, and define $$\label{def of fr}
f^{[a,b]}:=\<\varphi^{[a,b]},\omega\> \in A\, .$$ It is clear that $f^{[a,b]}$ depends only on $a$ and $b$ but not on $r$. It is also clear that $\partial$ in the exact sequence (\[papica\]) is $0$ if and only if $f^{[a,b]}=0$ for all $1 \leq
a+b \leq r$. Proceeding verbatim as in [@book], we check that $f^{[a,b]}$ depends on the pair $(E,\omega)$, and not on the choice of $T$, as long as $T$ satisfies the condition that $\omega
\equiv dT$ mod $(T)$.
Now let $a,b,c,d \in \bZ_+$ with $1 \leq a+b,c+d \leq r$, and and consider the homomorphism $$\Theta:=\Theta^{[a,b]}_{[c,d]}
:N^r \ra \hat{\bG}_a$$ given by $$\Theta:=f^{[a,b]} L^{[c,d]}-f^{[c,d]}L^{[a,b]}\in A[T^{(i,j)}\mid_{1
\leq i,j \leq r}]\wh\, .$$ Then we have that $\partial(\Theta)=0$, for $\partial(\Theta)$ is the class $[\gamma_{\mu \nu}]$ in $H^1(\hat{E},\cO)$ of the cocycle $$\gamma_{\mu \nu}:=
f^{[a,b]}\varphi_{\mu \nu}^{[c,d]}-f^{[c,d]}\varphi_{\mu \nu}^{[a,b]}
\in \cO(\hU_{\mu \nu})\, ,$$ and $[\gamma_{\mu \nu}]=0$ because $$\<[\gamma_{\mu \nu}],
\omega\>=f^{[a,b]} \cdot \<\varphi^{[c,d]},\omega\> -
f^{[c,d]}
\cdot
\<\varphi^{[a,b]},\omega\>=0\, .$$ By the exactness of the sequence (\[papica\]), $\Theta$ lifts to a unique homomorphism $$\psi=\psi^{[a,b]}_{[c,d]}:J^r_{\di \ddi}(E) \ra \hat{\bG}_a$$ which we interpret as a $\D$-character $$\psi \in \bX^r_{\di \ddi}(E)\, .$$ This character $\psi$ depends only on $a,b,c,d$, but not on $r$ in the sense that if we change $r$ to $r+s$, then the new $\psi$ is obtained from the old one by composition with the projection $J^{r+s}_{\di \ddi}(E) \ra J^r_{\di \ddi}(E)$. Incidentally, $\psi$ is obtained by gluing functions $$\psi_{\mu} \circ \tau_{\mu}^{-1} \in \cO(\pi_r^{-1}(\hat{U}_{\mu}))\, ,$$ $$\label{prea mult de munca}
\psi_{\mu}:=f^{[a,b]} \cdot L^{[c,d]} - f^{[c,d]}
\cdot L^{[a,b]} +\gamma_{\mu} \in
\cO(\hat{U}_{\mu})[T^{(i,j)}\mid_{1 \leq i+j \leq r}]\wh\, ,$$ with $\gamma_{\mu} \in \cO(\hU_{\mu})$, and we have the identities $$\begin{array}{rcl}
\psi^{[a,b]}_{[a,b]} & = & 0 \, , \vspace{1mm}\\
\psi^{[a,b]}_{[c,d]}+\psi^{[c,d]}_{[a,b]} & = & 0\, , \vspace{1mm} \\
f^{[a_1,b_1]} \psi^{[a_2,b_2]}_{[a_3,b_3]}+
f^{[a_2,b_2]} \psi^{[a_3,b_3]}_{[a_1,b_3]}+
f^{[a_3,b_3]} \psi^{[a_1,b_1]}_{[a_2,b_2]} & = & 0\, ,
\end{array}$$ which follow from the very same identities that are obtained when the $\psi$s are replaced by the $\Theta$s.
Since $\tau_{\mu_0}$ is the identity modulo $T$, we have that $\psi \circ e(pT)$ is congruent to $\psi_{\mu_0} \circ e(pT)$ modulo $T$. By Lemmas \[caiin\] and \[xxzz\], we obtain $$\label{supitza} \left( \psi^{[a,b]}_{[c,d]} \right) \circ e(pT)=
p^{1+\epsilon(d)} f^{[a,b]} \phip^c \dd^d T - p^{1+\epsilon(b)}
f^{[c,d]} \phip^a \dd^b T +\tf^{[a,b]}_{[c,d]} T\, ,$$ where $\tf^{[a,b]}_{[c,d]} \in A$. So $\psi^{[a,b]}_{[c,d]}$, viewed as an element of $A[[T]][\d^i \dd^j T\mid_{1 \leq i+j \leq r}]\wh$, has the form $$\label{incaosupitza} \psi^{[a,b]}_{[c,d]}=\frac{1}{p}
\left[p^{1+\epsilon(d)} f^{[a,b]} \phip^c \dd^d -
p^{1+\epsilon(b)} f^{[c,d]} \phip^a \dd^b +\tf^{[a,b]}_{[c,d]}
\right] l(T)\, .$$
We will use the following notation: $$\begin{array}{rclrclrcl}
f^a_{\di} & := & f^{[a,0]}\, , & f^a_{\ddi} & := & f^{[0,a]}\, , & & \\
\tf^2_{\di} & := & \tf^{[1,0]}_{[2,0]}\, , & \tf^2_{\ddi} & := &
\tf^{[0,1]}_{[0,2]}\, , & \tf^1_{\di \ddi} & := & \tf^{[1,0]}_{[0,1]}\, .
\end{array}$$ The elements $f^a_{\di} \in A$ are (the images of) the elements $f_{jet}^a$ in [@book]. The elements $\tf_{\di}^2$ are (the images of) the elements $pf^{1,2}_{jet}$ in [@book]. By [@book], Proposition 7.20 (and Remark 7.21) and Corollary 8.84 (and Remark 8.85), we have $$\label{4res} \tf_{\di}^2=p(f^1_{\di})^{\phi}\, .$$
The element $f^1_{\di}$ was interpreted in [@book] as an [*arithmetic Kodaira-Spencer class*]{} of $E$. The element $f^1_{\ddi}$ is easily seen to be (an incarnation of) the usual Kodaira-Spencer class of $E$; cf. [@difmod]. The element $f^1_{\ddi} \in A$, and the reduction mod $p$ of $f^1_{\di}$, were explicitly computed in [@hurl]. The $\dd$-character $$\psi^2_{\ddi}:=\psi^{[0,1]}_{[0,2]}\in \bX_{\ddi}^2(E)$$ is (an “incarnation” of) the [*Manin map*]{} of $E$ [@man], constructed as in [@annals]. If $f^1_{\ddi} \neq 0$, then $\psi^2_{\ddi} \neq 0$. (Unlike the construction in [@annals] that was done over a field, our construction here is carried over the ring $A$.) The $\d$-character $$\psi^2_{\di}:=\psi^{[1,0]}_{[2,0]}\in \bX_{\di}^2(E)$$ is the [*arithmetic Manin map*]{} in [@char]. And if $f^1_{\di}
\neq 0$, then $\psi^2_{\di} \neq 0$. Both of these Manin maps are “ode” maps with respect to the geometric and the arithmetic direction separately. On the other hand, we can consider the $\D$-character $$\psi^1_{\di \ddi}:=\psi^{[1,0]}_{[0,1]}\in \bX_{\di \ddi}^1(E)\, .$$ If either $f^1_{\di} \neq 0$ or $f^1_{\ddi} \neq 0$, then $\psi_{\di \ddi}^1 \neq 0$. If both $f^1_{\di} \neq 0$ and $f^1_{\ddi} \neq 0$, then $\psi^1_{\di \ddi}$ is a “purely pde” operator (in the sense that it is not a sum of a $\d$-character and a $\dd$-character).
Indeed, we have the following consequence of Proposition \[indi\].
If $f^1_{\di} \neq 0$ and $f^1_{\ddi} \neq 0$, then:
1. $\psi^1_{\di \ddi}$ is an $L$-basis of $\bX^1_{\di \ddi}(E)\otimes
L$.
2. $\bX_{\di}^1(E)=\bX_{\ddi}^1(E)=0$.
We may view $\psi^1_{\di \ddi}$ as a canonical “convection equation” on $E$.
Using the notation above and the commutation relations in $A[\phip,\dd]$, we have the following equalities: $$\label{nh}
\begin{array}{rcl}
\psi^1_{\di \ddi} & = & \frac{1}{p} \left[ pf^1_{\di} \dd-f^1_{\ddi}
\phip +\tf^1_{\di \ddi} \right] l(T)\, , \vspace{1mm} \\
\dd \psi^1_{\di \ddi} & = & \frac{1}{p} \left[ p(\dd f^1_{\di})
\dd+pf^1_{\di}\dd^2 - (\dd f^1_{\ddi}) \phip -f^1_{\ddi} p \phip
\dd +
\dd \tf_{\di \ddi}^1 + \tf_{\di \ddi}^1 \dd \right] l(T)\, ,\vspace{1mm} \\
\phip \psi^1_{\di \ddi} & = & \frac{1}{p} \left[ p
(f^1_{\di})^{\phi} \phip \dd-(f^1_{\ddi})^{\phi}\phip^2+ (\tf_{\di
\ddi}^1)^{\phi} \phip \right] l(T)\, , \vspace{1mm} \\
\psi^2_{\di} & = & \frac{1}{p} \left[f^1_{\di} \phip^2-f^2_{\di}
\phip
+\tf^2_{\di} \right] l(T)\, ,\vspace{1mm} \\
\psi_{\ddi}^2 & = & \frac{1}{p} \left[ pf^1_{\ddi} \dd^2-pf^2_{\ddi}
\dd+\tf_{\ddi}^2 \right] l(T)\, .
\end{array}$$ Thus, if we represent the ordered elements $\dd \psi^1_{\di \ddi},\ \phip \psi^1_{\di \ddi}, \ \psi^2_{\di},
\ \psi^1_{\di \ddi},\ \psi^2_{\ddi}$ as $L$-linear combinations of the series $p \dd^2 l(T),\ p \phip \dd l(T),\ \phip^2 l(T),\ p \dd l(T), \
\phip l(T),\ l(T)$, then the matrix of $L$-coefficients is equal to $\frac{1}{p}M$, where $$M:=\left(
\begin{array}{cccccc}
f^1_{\di} & -f^1_{\ddi} & 0 & \dd f^1_{\di}+\frac{\tf^1_{\di
\ddi}}{p} & - \dd f^1_{\ddi} & \dd \tf^1_{\di \ddi} \vspace{2mm} \\
0 & (f^1_{\di})^{\phi} &
-(f^1_{\ddi})^{\phi} & 0 & (\tf^1_{\di
\ddi})^{\phi} & 0 \vspace{2mm} \\
0 & 0 & f^1_{\di} & 0 & -f_{\di}^2 & \tf_{\di}^2 \vspace{2mm} \\
0 & 0 & 0 & f^1_{\di} & -f^1_{\ddi} & \tf^1_{\di \ddi}\vspace{2mm} \\
f^1_{\ddi} & 0 & 0 & -f^2_{\ddi} & 0 & \tf_{\ddi}^2
\end{array}
\right)\, .$$
\[narnia\] Let us assume that $f^1_{\di} \neq 0$ and $f^1_{\ddi}\neq 0$. Then the following hold:
1. The elements $$\psi_{\di \ddi}^1,\ \dd \psi^1_{\di \ddi},\
\phip \psi^1_{\di \ddi},\ \psi^2_{\di}$$ form an $L$-basis of $\bX_{\di \ddi}^2(E) \otimes L$.
2. The elements $$\psi_{\di \ddi}^1,\ \phip \psi^1_{\di \ddi},\
\psi^2_{\ddi},\ \psi^2_{\di}$$ form an $L$-basis of $\bX_{\di \ddi}^2(E) \otimes L$.
3. There exists a $5$-tuple $(\alpha,\beta,\gamma,\nu,\lambda) \in
A^5$, which is unique up to scaling by an element of $A$, satisfying $$(\alpha \dd+\beta \phip +\gamma)\psi^1_{\di \ddi}=\nu
\psi^2_{\ddi}+\lambda \psi_{\di}^2,\quad \alpha \neq 0, \; \nu
\neq 0\, .$$
4. All $5 \times 5$ minors of the matrix $M$ vanish.
We may view the character $\nu \psi^2_{\ddi}+\lambda \psi_{\di}^2$ as a canonical “wave equation” on $E$.
[*Proof*]{}. By the form of the matrix $M$, each of the $4$-tuples in assertions 1) and 2) are $L$-linearly independent. By Proposition \[indi\], $\bX_{\di \ddi}^2(E)$ has rank $4$ over $A$. All the statements in the Proposition then follow.
Similar arguments (cf. also to the case of $\bG_m$) yield:
Let us assume that $f^1_{\di} \neq 0$ and $f^1_{\ddi}\neq 0$. Then, for any $r
\geq 2$, the $L$-vector space $\bX_{pq}^r(E) \otimes L$ has basis $$\{\phip^i \dd^j \psi^1_{pq}\mid_{0 \leq i+j \leq r-1}\}
\cup \{\phip^i \psi^2_p\mid_{0 \leq i \leq r-2}\}\, .$$
When either $f^1_{\di}=0$ or $f^1_{\ddi}=0$, the picture above changes, and in fact, it simplifies. For if with more generality we assume that $f^{[a,b]}=0$ for some $a,b$ with $a+b \leq r$, then $0=\varphi^{[a,b]}=\gamma(L^{[a,b]})$, and by the exact sequence (\[papica\]), $L^{[a,b]}$ lifts uniquely to a homomorphism $\psi^{[a,b]}:J^r_{\di \ddi}(E) \ra \hat{\bG}_a$, which we interpret as a $\D$-character $\psi^{[a,b]} \in \bX_{\di \ddi}^r(E)$. Let us observe in passing that $\psi^{[a,b]}$ is obtained by gluing functions $$\begin{array}{c}
\psi_{\mu} \circ \tau_{\mu}^{-1} \in
\cO(\pi_r^{-1}(\hat{U}_{\mu}))\, , \\
\psi_{\mu}:=L^{[a,b]} +\gamma_{\mu} \in
\cO(\hat{U}_{\mu})[T^{(i,j)}\mid_{1 \leq i+j \leq r}]\wh \, ,
\end{array}$$ with $\gamma_{\mu} \in \cO(\hU_{\mu})$. As before, we obtain $$\psi^{[a,b]} \circ e(pT)=p^{1+\epsilon(b)} \phip^a \dd^b
T+\tf^{[a,b]}\,$$ where $\tf^{[a,b]} \in A$. So $\psi^{[a,b]}$, viewed as a series, has the form $$\psi^{[a,b]}=\frac{1}{p}\left[ p^{1+\epsilon(b)} \phip^a \dd^b
+\tf^{[a,b]} \right] l(T)\, .$$
Incidentally, if $f^1_{\ddi}=0$, then $$\label{psi1q} \psi^1_{\ddi}:=\psi^{[0,1]} \in \bX^1_{\ddi}(E)$$ is (an “incarnation" of) the [*Kolchin logarithmic derivative*]{} of $E$ [@kolchin]; cf. Proposition \[logder\] below.
On the other hand, if $f^1_{\di}=0$ we set $$\label{psi1p} \psi^1_{\di}:=\psi^{[1,0]} \in \bX_{\di}(E)\, .$$
As before, we find the following bases for the set of $\D$-characters:
Let us assume that $f^1_{\di} = 0$ and $f^1_{\ddi}\neq 0$. Then, for each $r
\geq 2$, the $L$-vector space $\bX_{pq}^r(E) \otimes L$ has basis $$\{\phip^i \dd^j \psi^2_q\mid_{\ 0 \leq i+j \leq r-2}\} \cup
\{\phip^i \psi^1_p\mid_{0 \leq i \leq r-1}\}
\cup \{\phip^i \dd \psi^1_p\mid_{0 \leq i \leq r-2}\}\, .$$
\[ileana1\] Let us assume that $f^1_{\di} \neq 0$ and $f^1_{\ddi}=0$. Then, for each $r \geq 2$, the $L$-vector space $\bX^r_{\di \ddi}(E)
\otimes L$ has basis $$\{\phip^i \psi_p^2\mid_{0 \leq i \leq r-2}\} \cup \{\phip^i \dd^j
\psi^1_{\ddi}\mid_{0 \leq i+j \leq r-1}\}\, .$$
\[ileana2\] Let us assume $f^1_{\di} = 0$ and $f^1_{\ddi}= 0$. Then, for each $r \geq 1$, the $L$-vector space $\bX^r_{\di \ddi}(E)
\otimes L$ has basis $$\{\phip^i \psi_p^1\mid_{0 \leq i \leq r-1}\} \cup \{\phip^i \dd^j
\psi^1_{\ddi}\mid_{0 \leq i+j \leq r-1}\}\, .$$
Tate curves
-----------
Let $E_q$ be the [*Tate curve with parameter*]{} $q$ over $A:=R((q))\wh$, equipped with its canonical $1$-form $\omega_q$. This curve $E_q$ is defined as the elliptic curve in the projective plane over $A$ whose affine plane equation is $$y^2=x^3-\frac{1}{48}E_4(q)x+\frac{1}{864}E_6(q)\, ,$$ where $E_4$ and $E_6$ are the Eisenstein series $$\begin{array}{rcl}
E_4(q) & = & 1+240 \cdot s_3(q)\, ,\\
E_6(q) & = & 1-504 \cdot s_5(q)\, .
\end{array}$$ In here, for $m \geq 1$, we follow the usual convention and write $$s_m(q):=\sum_{n \geq 1} \frac{n^mq^n}{1-q^n} \in R[[q]]\, .$$ Also, the canonical form is defined by $$\omega_q=\frac{dx}{y}\, .$$
More generally, let $\beta \in R^{\times}$ be an invertible element that we shall view as a varying parameter, and let $(E_{\beta q},\omega_{\beta q})$ be the pair obtained by base change from $(E_q,\omega_q)$ via the isomorphism $$\begin{array}{rcl}
R((q))\h & \stackrel{\sigma_{\beta}}{\ra} & R((q))\wh \\
\sigma_{\beta}(\sum a_nq^n)
& = & \sum a_n \beta^n q^n
\end{array}\, .$$ Thus, the pair $(E_{\beta q},\omega_{\beta
q})$ is the elliptic curve $$y^2=x^3-\frac{1}{48}E_4(\beta q)x+\frac{1}{864}E_6(\beta q)$$ equipped with the $1$-form $$\omega_{\beta q}=\frac{dx}{y}\, .$$ (Let us observe that $\sigma_{\beta}$ is a $\dd$-ring homomorphism, but not a $\d$-ring homomorphism.) We shall refer to $(E_{\beta q},\omega_{\beta q})$ as the [*Tate curve with parameter $\beta q$*]{}.
Let us observe that the discussion and notation in Example \[cucurigu\] that concerns groups of solutions does not apply to $E_{\beta q}$ over $A$, as $A=R((q))\wh \neq R$.
From now on, we assume that all our quantities in the general theory (like the $f$’s, the $\tilde{f}$’s, and the $\psi$’s) are associated to the pair $(E_{\beta q},\omega_{\beta q})$.
\[tate1\] There exists $c \in \bZ_p^{\times}$ such that, for any $\beta \in R^{\times}$, we have $$f^1_{\di}=\eta,\quad f^2_{\di}=\eta^{\phi}+p\eta,\quad \tf_{\di}^2=p
\eta^{\phi}\, ,$$ where $$\eta:=\frac{c}{p}\log \frac{\phi(\beta)}{\beta^p}\, .$$
[*Proof*]{}. We consider the ring $R((q))\h[q',q'',\ldots]\wh$ with its unique $\d$-ring structure such that $\d q=q'$, $\d q'=q''$, etc. Let us note that $(E_{\beta q},\omega_{\beta q})$ is obtained by base change from $(E_q,\omega_q)$ via the composition $$\label{tzutz} R((q))\h \subset R((q))\h[q',q'',\ldots]\h
\stackrel{\tilde{\sigma}_{\beta}}{\ra} R((q))\h[q',q'',\ldots]\h
\stackrel{\pi}{\ra}R((q))\wh \, ,$$ where $\tilde{\sigma}_{\beta}(q)=\beta q$, $\tilde{\sigma}_{\beta}(q')=\d(\beta q)$, $\tilde{\sigma}_{\beta}(q'')=\d^2(\beta q),\ldots $, and $\pi(q)=q$, $\pi(q')=0$, $\pi(q'')=0, \ldots$ And let us note also that $\tilde{\sigma}_{\beta}$ and $\pi$ are $\d$-ring homomorphisms. By the functoriality of $f^r_{\di}$ plus [@difmod], Corollary 7.26, [@barcau], Corollary 6.1, and [@fermat], Lemma 6.14, there exists $c \in \bZ_p^{\times}$ such that, for any $\beta$, we have $$\begin{array}{rcl}
f^1_{\di} & = & {\displaystyle \frac{c}{p} \cdot\log\frac{\phip(\beta q)}
{(\beta q)^p}=\frac{c}{p}\log \frac{\phip(\beta)}{\beta^p}}\, , \vspace{1mm} \\
f^2_{\di} & = & (f^1_{\di})^{\phi}+pf^1_{\di}\, .
\end{array}$$ Thus, $f^1_{\di}$ and $f^2_{\di}$ have the desired values. The value of $\tf_{\di}^2$ follows by (\[4res\]).
\[tate2\] The following hold:
1. $f^1_{\ddi}$ does not depend on $\beta$.
2. $f^1_{\ddi} \in \bZ_p^{\times}$, and $f^2_{\ddi}=\tf^2_{\ddi}=0$.
[*Proof*]{}. We start by noting that $(E_{\beta q},\omega_{\beta q})$ is obtained by a base change from $(E_q,\omega_q)$ via the isomorphism $\sigma_{\beta}:R((q))\h \ra R((q))\wh$, and recall that $\sigma_{\beta}$ is a $\dd$-ring homomorphism. By functoriality, the elements $f^1_{\ddi}$, $f^2_{\ddi}$, $\tilde{f}^2_{\ddi}$ corresponding to a given $\beta$ are the images via $\sigma_{\beta}$ of the corresponding quantities for $\beta=1$. Thus, proving assertion 2) for $\beta=1$ suffices to conclude both assertions 1) and 2) for arbitrary $\beta$.
We prove next assertion 2) for $\beta=1$.
The fact that $f^1_{\ddi} \in \bZ_p^{\times}$ follows from [@difmod], Corollary 7.26. In order to prove that $f^2_{\ddi}=\tf^2_{\ddi}=0$, it is enough to show that there exists $\psi \in \bX_{\ddi}^2(E)$ such that $\psi=\dd^2 l(T)$. This is because $\bX^2_{\ddi}(E)$ has rank $1$ over $A$ [@ajm].
The existence of $\psi$ can be argued analytically, and we briefly sketch the argument next. (A purely algebraic argument is also available but the analytic one is simpler and classical, going back to Fuchs and Manin [@man].) For in our problem we can replace $R((q))$ by $\bQ((q))$, and then $\bQ((q))$ by $\bC((q))$. We view the Tate curve over $\bC((q))$ as arising from an analytic family $E_q^{an} \ra \Delta^*$ of elliptic curves over the punctured disk $\Delta^*$. The fiber $E_{q_{\tau}}$ of this family over a point $q_{\tau}=e^{2 \pi i \tau} \in \Delta^*$ identifies with $\bC/\<1,\tau\>$. Let $z$ be the coordinate on $\bC$. Then, for any local analytic section $P$, $q_{\tau} \mapsto P(q_{\tau})$, of $E_q^{an} \ra \Delta^*$ that is close to the zero section $q_{\tau} \mapsto P_0(q_{\tau})=0$, we have that $$\label{su} l(P(q_{\tau}))=
\int_{P_0(q_{\tau})}^{P(q_{\tau})} \omega_{q_{\tau}}$$ where $\omega_{q_{\tau}}$ is the $1$-form on $E_{q_{\tau}}$ whose pull back to $\bC$ is $dz$. The periods of $\omega_{q_{\tau}}$ on $E_{q_{\tau}}$ are $1$ and $\tau$, so they are annihilated by $$\left( \frac{d}{d \tau} \right)^2=-4 \pi^2 \left( q_{\tau}
\frac{d}{dq_{\tau}}\right)^2\, .$$ Hence the map $$P \mapsto \left( q_{\tau}
\frac{d}{dq_{\tau}}\right)^2 \left(
\int_{P_0(q_{\tau})}^{P(q_{\tau})} \omega_{q_{\tau}} \right)$$ is well defined for all local analytic sections $P$ of $E_q^{an}
\ra \Delta^*$ (not necessarily close to the zero section $P_0$). This map arises from a $\dd$-character $\psi$ of $E_q$ over $\bC((q))$ [@man]. By (\[su\]), $\psi$ coincides with $P \mapsto \dd^2 l(P)$ for $P$ close to the zero section, and this completes the argument.
\[tate1.5\] From now on, we choose $c$ as in Lemma \[tate1\], and we set $$\begin{array}{rcl}
\eta & := & f^1_{\di}=\frac{c}{p}\log \frac{\phip(\beta)}{\beta^p} \in R\,
, \vspace{1mm} \\
\gamma & := & f^1_{\ddi} \in
\bZ^{\times}_p\, .
\end{array}$$ Notice that we have $\eta=0$ if, and only if, $\beta$ is a root of unity. Later on, the quotient of these Kodaira-Spencer classes, $$\frac{\eta}{\gamma}=\frac{f^1_{\ddi}}{f^1_{\di}}\, ,$$ will play a key rôle.
\[tate3\] $\tf^1_{\di \ddi}=p \gamma$.
[*Proof*]{}. Let us assume first that $\beta$ is not a root of unity, and so, $\eta \neq 0$. The $5 \times 5$ minor of the matrix $M$ above obtained by removing the $6$-th column has determinant zero (cf. Proposition \[narnia\]), and using Lemma \[tate1\], Lemma \[tate2\], and Remark \[tate1.5\], we get the condition $$pf^1_{\di}(\tf^1_{\di \ddi} -p\gamma)^{\phi}+(f^1_{\di})^{\phi}
(\tf^1_{\di \ddi}-p\gamma)=0\, .$$ We take the valuation $v_p:R((q))\h \ra \bZ_+ \cup \{\infty\}$ in this identity, and use the fact that $v_p \circ \phi=v_p$ to get $\tf^1_{\di \ddi}-p\gamma=0$, completing the proof of this case.
If instead now $\beta \in R^{\times}$ is arbitrary, proceeding as in the proof of Lemma \[tate1\], we have that $\tilde{f}^1_{\di \ddi}$ is the image of an element $F \in R((q))\h[q',q'',\ldots ]\wh$ via the map $\pi \circ \tilde{\sigma}_{\beta}$ in (\[tzutz\]). We know that $$\label{tzinee} \pi \tilde{\sigma}_{\beta} F=p \gamma$$ for all $\beta \in R^{\times} \backslash \bmu(R)$. Since the map $$\begin{array}{rcl}
R^{\times} & \ra & R((q))\wh \\ \beta & \mapsto & \pi
\tilde{\sigma}_{\beta} F
\end{array}$$ is $p$-adically continuous, we have that that (\[tzinee\]) holds for all $\beta \in R^{\times}$. This completes the proof.
Combining Lemma \[tate1\], Lemma \[tate2\], and (\[nh\]), we derive the following.
\[pomu\] Let $E=E_{\beta q}$ be the Tate curve over $A=R((q))\h$ with parameter $\beta q$, $\beta \in R^{\times}$. When viewed as elements of the ring $$A[[T]][\d T, \dd T, \d^2 T, \d \dd T,
\dd^2 T]\wh\, ,$$ the characters $\psi^1_{\di \ddi}$, $\psi^2_{\ddi}$, $\psi_{\di}^2$ are equal to $$\begin{array}{rcl}
\psi^1_{\di \ddi} & = & {\displaystyle \frac{1}{p} [p \eta \dd -\gamma
\phip+p\gamma ]l(T)}\, ,
\vspace{2mm} \\ \psi^2_{\ddi} & = & \gamma \dd^2 l(T)\, , \vspace{2mm} \\
\psi^2_{\di} & = & {\displaystyle \frac{1}{p} [\eta
\phip^2-(\eta^{\phi}+p\eta)\phip+p\eta^{\phi}]l(T)}\, .
\end{array}$$ In particular, we have the relations $$\begin{array}{rcl}
(\gamma \eta^{\phi+1}\dd+ \gamma^2 \eta \phip-\gamma^2
\eta^{\phi}) \psi^1_{\di \ddi} & = & \eta^{\phi+2}
\psi^2_{\ddi}- \gamma^3 \psi_{\di}^2 \, , \vspace{2mm} \\
\gamma \dd^2 \psi^1_{\di \ddi} & = & (\eta \dd -p \gamma
\phip+\gamma) \psi^2_{\ddi}\, .
\end{array}$$
Of course, if $\beta$ is a root of unity, the first of the last two relations in the Proposition above reduces to the identity $0=0$. In this case, we also have that $$\label{fome} \psi^1_{\di \ddi}=-\gamma \psi^1_{\di}\, .$$
\[spallate\] By the Proposition above, the Fréchet symbol of $\psi^1_{\di \ddi}$ with respect to $\omega$ is $\theta_{\psi^1_{\di \ddi}, \omega}=\eta \xi_q-\gamma
\xi_p+\gamma$. Thus, if we consider the “simplest” of the energy functions on $E$ given by $$H:=(\psi^1_{\di \ddi})^2\, ,$$ a direct computation shows the following expression for the Euler-Lagrange equation attached to $H$ and the vector field $\partial$ dual to $\omega$: $$\epsilon^1_{H,\partial}=(-2 \eta^{\phi} \phip \dd+2 \gamma
\phip-2 \gamma)\psi^1_{\di \ddi}\, .$$ In particular, any solution of $\psi^1_{\di \ddi}$ is a solution of the Euler Lagrange equation $\epsilon^1_{H,\partial}$.
In the sequel, we will need to use some other facts about Tate’s curves that we now recall. Indeed, note that the cubic defining $E_{\beta q}$ has coefficients in $R[[q]]$, and hence, $E_{\beta q}$ has a natural projective (non-smooth) model ${\mathcal E}={\mathcal E}_{\beta
q}$ over $R[[q]]$ equipped with a “zero” section defined by $T=-\frac{x}{y}$. The completion of ${\mathcal E}_{\beta q}$ along this section is naturally isomorphic to $Spf\ R[[q]][[T]]$, where $R[[q,T]]=R[[q]][[T]]$ is viewed with its $T$-adic topology. This isomorphism induces a natural embedding $$\iota:qR[[q]] \ra {\mathcal E}(R[[q]]) \subset E(A)\, ,$$ that is explicitly given by sending any $u \in qR[[q]]$ into the point of ${\mathcal E}$ whose $(T,W)$-coordinates are $$(u,u^3-\frac{1}{48}E_4(\beta q)u^7+\cdots) \, ,$$ cf. (\[dub\]). We denote by $E_{1}(A)$ the image of this embedding, that is to say, $E_{1}(A)=\iota(qR[[q]])$. When no confusion can arise, we will view $\iota$ as an inclusion, and we will identify $\iota(qR[[q]])$ with $qR[[q]]$.
The formal group law $\cF=\cF_E=\cF(T_1,T_2)$ of $E$ with respect to $T=-\frac{x}{y}$ has coefficients in $R[[q]]$, and is isomorphic over $R[[q]]$ to the formal group $\cF_{\bG_m}=t_1+t_2+t_1t_2$ of $\bG_m$ via an isomorphism $\sigma(t)=t+\cdots\in t+t^2R[[q,t]])$.
For convenience, we recall how $\sigma$ arises. We first notice that for any variable $v$, the series $$\label{uglys}
\begin{array}{rcl} X(q,v) & := &
\frac{v}{(1-v)^2}+ \sum_{n \geq 1} \left( \frac{\beta^n
q^nv}{(1-\beta^n q^n v)^2}+
\frac{\beta^n
q^nv^{-1}}{(1-\beta^n q^n v^{-1})^2}-2 \frac{\beta^n
q^n}{(1-\beta^n q^n)^2}
\right)\, , \vspace{1mm} \\
Y(q,v) & := & \frac{v^2}{(1-v)^3}+ \sum_{n \geq 1} \left(
\frac{(\beta^n
q^nv)^2}{(1-\beta^n q^n v)^3} - \frac{\beta^n
q^nv^{-1}}{(1-\beta^n q^n v^{-1})^3}+ \frac{\beta^n
q^n}{(1-\beta^n q^n)^2} \right)\, ,
\end{array}$$ make sense as elements of the ring $\bZ[v,v^{-1}][[q]][\frac{1}{1-v}]$. Cf. [@sil2], p. 425. So, if we specialize $v \mapsto 1+t \in
\bZ[[t]]$ where $t$ is a variable, then $X$ and $Y$ become elements in $\bZ[[q]]((t))$, that is to say, Laurent series in $t$ with coefficients in $\bZ[[q]]$. Since $p \geq 5$, the series $$\begin{array}{rcl}
x & = & X + \frac{1}{12} \vspace{1mm} \\
y & = & -Y-\frac{X}{2}
\end{array}$$ belong to $\bZ_{(p)}[[q]]((t))$. Note that $X=t^{-2}+\cdots$ and $Y=t^3+\cdots$, hence, $x=t^{-2}+\cdots$ $y=-t^3+\cdots$ and so $T=-\frac{x}{y}=t+\cdots=:\sigma(t) \in t+t^2\bZ_{(p)}[[q,t]]$. It turns out that $t \mapsto T=\sigma(t)$ defines an isomorphism $\cF_{\bG_m} \simeq \cF_E$; cf. [@sil2], p. 431.
The [*characteristic polynomial*]{} $\mu(\xi_p,\xi_q)$ of a character $\psi^1_{pq}$ is the Fréchet symbol of $\psi^1_{\di \ddi}$ with respect to $\omega$: $$\mu(\xi_p,\xi_q):=\eta \xi_q - \gamma \xi_p
+\gamma\, .$$ Note that $\mu(0,0) \in R^{\times}$. The [*characteristic integers*]{} of $\psi^1_{pq}$ are the integers $\kappa$ such that $\mu(0,\kappa)=0$, that is to say, solutions of $\eta \kappa =-\gamma$. The set $\cK$ of characteristic integers has at most one element and is given by $$\cK=\{-\gamma/\eta\} \cap \bZ\, .$$ A [*totally non-characteristic*]{} integer $\kappa \in \bZ$ is an integer such that $\kappa \not\equiv 0$ mod $p$, and $\mu(0,\kappa)
\not\equiv 0$ mod $p$, that is, $$\eta \kappa^2+\gamma \kappa \not\equiv 0\quad \text{mod}\; p\, .$$ We denote by $\cK'$ the set of totally non-characteristic integers. For any $0 \neq \kappa \in \bZ$ and any $\alpha \in R$, the [*basic series*]{} is $$\label{filmm} u_{E,\kappa,\alpha}:=e_E \left( \int
u_{a,\kappa,\alpha}^{\mu} \frac{dq}{q} \right) \in qK[[q]]\, ,$$ where $e_E(T) \in TK[[q]][[T]]$ is the exponential of the formal group law $\cF_E$, and $u_{a,\kappa,\alpha}^{\mu}$ is as in Equation \[ua\]. If in addition $\kappa$ is characteristic (i.e. $\frac{\eta}{\gamma}=-\frac{1}{\kappa}$) then $$\label{nuilacasino}
u_{a,\kappa,\alpha}^{\mu}:=\sum_{j \geq 0} (-1)^j
\frac{\alpha^{\phi^j}}{F_j(p)} q^{\kappa p^j}\, .$$ Cf. Example \[rupdi\].
\[tate4\] Let us assume that $\kappa \in \bZ \backslash p\bZ$. Then we have $u_{E,\kappa,\alpha} \in qR[[q]]$. Moreover, the map $R \ra
R[[q]]$, $\alpha \mapsto u_{E,\kappa,\alpha}$ is a pseudo $\d$-polynomial map.
[*Proof*]{}. In order to prove the first assertion, let $$h:=l_E(u_{E,\kappa,\alpha})=\int u_{a,\kappa,\alpha}^{\mu}
\frac{dq}{q}\, ,$$ where $l_E(T)=l(T)\in K[[q]]$ is the logarithm of the formal group law $\cF_E$. The isomorphism $\sigma(T)=T+\cdots \in
R[[q]][[T]]$ between $\cF_E$ and $\cF_{\bG_m}$ clearly has the property that $$e_E(T)=\sigma(e_{\bG_m}(T))\, ,$$ where $$e_{\bG_m}(T)=\exp{(T)}-1=T+\frac{T^2}{2!}+\frac{T^3}{3!}+\cdots$$ So in order to check that $e_E(h) \in qR[[q]]$, it is enough to show that $e_{\bG_m}(h) \in qR[[q]]$, that is to say, show that $\exp{(h)} \in
1+qR[[q]]$. By Dwork’s Lemma \[dwor\], it is enough to show that $(\phi-p)h \in pqR[[q]]$. Now, by Lemma \[unidul\] we have that $$\begin{array}
{rcl} p(\eta \kappa+\gamma)\alpha q^{\kappa} & = & p(\eta
\dd-\gamma \phip+\gamma)u_{a,\kappa,\alpha}^{\mu}\\
& = & p(\eta \dd-\gamma \phip+\gamma)\dd h\\
& = & \dd(p \eta \dd -\gamma \phip +p \gamma)h \, .
\end{array}$$ Hence $$\label{tarre} (p\eta \dd-\gamma \phip+p\gamma)h-p(\eta
\kappa+\gamma)\alpha \kappa^{-1}q^{\kappa}
\in R \cap qK[[q]]=0\, ,$$ which gives $$\begin{array}{rcl}
\gamma(\phip-p)h & = & p \eta \dd h-p(\eta \kappa+\gamma)\alpha
\kappa^{-1} q^{\kappa}\\
& = & p \eta u_{a,\kappa,\alpha}^{\mu}-p(\eta \kappa +\gamma)
\alpha \kappa^{-1}q^{\kappa}\\
& \in & pqR[[q]]\, ,
\end{array}$$ and we are done.
The second assertion is proved exactly as in Lemma \[ol\].
We have the following “diagonalization” result.
\[oashteptpei\] \[gainuli\] Let us assume that $\kappa \in \bZ \backslash p\bZ$, and that $\alpha \in R$. Then $$\begin{array}{rcl}
\psi^2_q u_{E,\kappa,\alpha} & = & \gamma
\kappa u^{\mu^{(p)}}_{a,\kappa,\alpha}\\
\psi^1_{pq}u_{E,\kappa,\alpha}& = & (\eta \kappa+\gamma)\alpha
\kappa^{-1} q^{\kappa}\, .
\end{array}$$
[*Proof*]{}. We have that $$\begin{array}{rcl}
\psi^2_q u_{E,\kappa,\alpha} & = &
\gamma \dd^2
l_E \left (e_E \left (
\int u_{a,\kappa,\alpha}^{\mu} \frac{dq}{q} \right) \right) \vspace{1mm} \\
\ & = & \gamma \dd^2 \left(\int u_{a,\kappa,\alpha}^{\mu}
\frac{dq}{q} \right) \vspace{1mm} \\
\ & = & \gamma \dd u^{\mu}_{a,\kappa,\alpha} \vspace{1mm} \\
\ & = & \gamma \kappa u^{\mu^{(p)}}_{a,\kappa,\alpha}.
\end{array}$$ Consequently we have $$\begin{array}{rcl}
\dd^2 \psi^1_{\di \ddi} u_{E,\kappa,\alpha} & = & \frac{1}{\gamma}
(\eta \dd -p \gamma\phip +\gamma) \psi^2_{\ddi}(u_{E,\kappa,\alpha})
\vspace{1mm} \\
& = & (\eta \dd -p \gamma\phip +\gamma)
\kappa u_{a,\kappa,\alpha}^{\mu^{(p)}} \vspace{1mm} \\
& = & \kappa \mu^{(p)}(\phip,\dd)
u_{a,\kappa,\alpha}^{\mu^{(p)}} \vspace{1mm} \\
& = & \kappa \mu^{(p)}(0,\kappa) \alpha q^{\kappa} \vspace{1mm} \\
& = & \kappa(\eta \kappa+\gamma) \alpha q^{\kappa} \vspace{1mm} \\
& = & \dd^2((\eta \kappa+\gamma) \alpha \kappa^{-1} q^{\kappa})\, .
\end{array}$$ Hence $$\dd(\psi^1_{pq}u_{E,\kappa,\alpha}-(\eta\kappa+\gamma)\alpha
\kappa^{-1}q^{\kappa}) \in R \cap qR[[q]]=0,$$ and, therefore, $$\psi^1_{pq}u_{E,\kappa,\alpha}-(\eta\kappa+\gamma)\alpha
\kappa^{-1}q^{\kappa} \in R \cap qR[[q]]=0,$$ completing the proof.
1. For $\kappa \in \bZ \backslash p\bZ$ the map $$\begin{array}{rcl}
R & \ra & {\mathcal E}(R[[q]]) \\ \alpha & \mapsto &
\iota(u_{E,\kappa,\alpha})
\end{array}$$ is a group homomorphism.
2. In analogy with (\[vocea\]), if $\kappa \in
\bZ \backslash p\bZ$, we may define [*boundary value operators*]{} $$\begin{array}{rcl}
B_{\kappa}^0:{\mathcal E}(qR[[q]]) & \ra & R\, ,\\
B_{\kappa}^0 u & = &
\Gamma_{\kappa} \psi^2_q u\, .
\end{array}$$ Then, for $\kappa_1,
\kappa_2 \in \bZ \backslash p\bZ$, we have $$\begin{array}{rcl}
B_{\kappa_1}^0 u_{E,\kappa_2,\alpha} & = & \Gamma_{\kappa_1}
\psi^2_q u_{E,\kappa_2,\alpha} \vspace{1mm} \\
& = & \Gamma_{\kappa_1} \gamma \kappa_2
u^{\mu^{(p)}}_{a,\kappa_2,\alpha} \vspace{1mm} \\
& = & \gamma \alpha \kappa_2 \delta_{\kappa_1,
\kappa_2}\, .
\end{array}$$
3. For $\kappa \in \bZ \backslash p\bZ$ we have $$\label{pit} u_{E,\kappa,\zeta^{\kappa} \alpha}(q)
=u_{E,\kappa,\alpha}(\zeta q)$$ for all $\zeta \in \bmu(R)$. In particular, if $\alpha=\sum_{i=0}^{\infty} m_i \zeta_i^{\kappa}$, $\zeta_i \in
\bmu(R)$, $m_i \in \bZ$, $v_p(m_i) \ra \infty$, then $$u_{E,\kappa,\alpha}(q)=\left[ \sum_{i=0}^{\infty}
\right] [m_i](u_{E,1,1}(\zeta_i^{\kappa} q^{\kappa}))= \left[
\sum_{i=0}^{\infty} \right] [m_i](u_{E,\kappa,1}(\zeta_i q))\, .$$ Here $[ \sum ]$ is the sum taken in the formal group law, and $[m_i](T)\in R[[q]][[T]]$ is the series induced by “multiplication by $m_i$” in the formal group. It is known that the sequence $[m_i](T)$ converges to $0$ in $R[[q]][[T]]$ in the $(p,T)$-adic topology, so the series in the right hand side of the equality above converges in the $(p,q)$-adic topology of $R[[q]]$. Morally, $u_{E,\kappa,\alpha}$ is obtained from $u_{E,\kappa,1}$ via “convolution.” (We have not defined “convolution” for groups not defined over $R$. This is possible but we will not do it here.)
4. We have the following “rationality” property: if $\alpha,
\beta \in \bZ_{(p)}$ then $u_{E,\kappa,\alpha} \in
\bZ_{(p)}[[q]]$ for $\kappa \in \bZ \backslash p\bZ$.
For the next Proposition, we denote by $$\label{lalalay} \cU^1_{\ra}:=\{u \in E(A)\ |\ \psi^1_{\di
\ddi}u=0\}$$ the group of solutions of $\psi^1_{\di \ddi}$ in $A=R((q))\h$, and we set $$\label{poponeata} \cU^1_1:=\cU^1_{\ra}
\cap E_1(A).$$ Note that there is no natural analogue of $\cU_{\la}$ in this case.
We let $E_{0}(A)$ be the subgroup of all $u \in E(A)$ that are solutions to all $\dd$-characters of $E$. By [@man] or [@ajm], $E_{0}(A)$ is actually the set of all solutions of the Manin map $\psi^2_{\ddi}$. We set $$\label{tedrav}
\cU^1_{0}:=\cU^1_{\ra} \cap E_{0}(A).$$ We should view $\cU^1_{0}$ as a substitute for the groups of stationary solutions in the cases of $\bG_a$ and $\bG_m$, respectively.
\[unde\] Let $E_{\beta q}$ be the Tate curve with parameter $\beta q$, where $\beta \in R^{\times}$. Let $\cU^1_{\ra}$ be the group of solutions in $R((q))\h$ of the $\D$-character $\psi_{\di
\ddi}^1$. Then the following hold:
1. If $\frac{\eta}{\gamma} =-\frac{1}{\kappa}$ for some integer $\kappa \geq 1$ coprime to $p$, and if $u_{E,\kappa,\alpha}$ is the basic series in [(\[filmm\])]{}, then $$\begin{array}{rcl}
\cU^1_{\ra} & = & \cU^1_{0}+\cU^1_1\, ,\\
\cU^1_1 & = & \{
u_{E,\kappa,\alpha}\ |\ \alpha \in R\}\, .
\end{array}$$
2. If $\frac{\eta}{\gamma}$ is not of the form $\frac{\eta}{\gamma}=-\frac{1}{\kappa}$ for some integer $\kappa
\geq 1$ coprime to $p$, then $$\cU^1_1=0\, .$$
[*Proof*]{}. It is sufficient to prove the following claims:
1. If $\frac{\eta}{\gamma}=-\frac{1}{\kappa}$ for some integer $\kappa \geq 1$ coprime to $p$, then $\psi^1_{\di \ddi}
u_{E,\kappa,\alpha}=0$ for all $\alpha$.
2. Assume $\psi^1_{\di \ddi} u=0$ for some $0 \neq u
\in qR[[q]]$. Then $\frac{\eta}{\gamma}=-\frac{1}{\kappa}$ for some integer $\kappa \geq 1$ coprime to $p$, and $u=u_{E,\kappa,\alpha}$ for some $\alpha$.
3. If $\frac{\eta}{\gamma}=-\frac{1}{\kappa}$ for some integer $\kappa \geq 1$ coprime to $p$, and if $u \in E(A)$ is such that $\psi^1_{\di \ddi}u=0$, then there exists $\alpha_u$ such that $\psi^2_{\ddi}(u-u_{E,\kappa,\alpha_u})=0$.
The first of these claims follows directly by Lemma \[oashteptpei\].
For the second claim, we write $$0=\dd^2 \psi^1_{\di
\ddi}u=\left(\eta \dd-p \gamma \phip+\gamma \right)
\dd^2(l_E(u))\, .$$ Now $$0 \neq v:=\dd^2(l_E(u)) \in qR[[q]]\, ,$$ so, by Theorem \[addeq\], there exists an integer $\kappa$ such that $\eta \kappa+\gamma=0$ and $v=u_{a,\kappa,\alpha}^{\mu^{(p)}}$ for some $\alpha \in R$. By (\[sasesi\]), $\dd l_E(u)=u_{a,\kappa,\alpha}^{\mu}$ so $$u=e_E\left( \int u_{a,\kappa,\alpha}^{\mu} \frac{dq}{q}
\right)=u_{E,\kappa,\alpha}\, ,$$ and the result follows.
For the third claim note that, as before, we have $$0=\dd^2 \psi^1_{\di \ddi} u=
\frac{1}{\gamma} (\eta \dd-p \gamma \phip+\gamma) \psi^2_q u\, .$$ By Theorem \[addeq\], $\psi^2_{\ddi}
u=a_0+u_{a,\kappa,\alpha}^{\mu^{(p)}}$ for some $\alpha \in R$ and some $a_0 \in R$, with $(\eta \dd-p \gamma \phip+\gamma) a_0=0$. The latter simply says $p \phi_p(a_0)=a_0$, and since $v_p(\phi_p(a_0))=v_p(a_0)$, we must have $a_0=0$. Hence $\psi^2_{\ddi} u=u_{a,\kappa,\alpha}^{\mu^{(p)}}$. On the other hand, $$\psi^2_{\ddi} u_{E,\kappa,\alpha/\gamma \kappa}=
\gamma \kappa u^{\mu^{(p)}}_{a,\kappa,\alpha/\gamma \kappa}=
u^{\mu^{(p)}}_{a,\kappa,\alpha}.$$ and so $\psi^2_{\ddi}(u-u_{E,\kappa,\alpha/\gamma \kappa})=0$, completing the proof.
The condition in Theorem \[unde\] that the quotient of the Kodaira-Spencer classes has the form $$\frac{\eta}{\gamma} =-\frac{1}{\kappa}$$ for some integer $\kappa \geq 1$ coprime to $p$ has a nice interpretation in terms of wave lengths (in the sense of Remark \[funnyterm\]). The condition is equivalent to saying that the wave length of the parameter $\beta q$ of the Tate curve belongs to $$\left\{ -\frac{c}{\gamma},-\frac{2 c}{\gamma},-\frac{3
c}{\gamma},\ldots\right\} \cap \bZ_p^{\times}\, .$$ This is, again, a “quantization” condition.
Under the hypotheses of Theorem [\[unde\]]{} and the assumption that $\frac{\eta}{\gamma}=-\frac{1}{\kappa}$ for some integer $\kappa$, let $u \in \cU_1^1$. Then we have that the series $\overline{\psi_q^2 u} \in k[[q]]$ is integral over $k[q]$, and the field extension $k(q) \subset k(q,\overline{\psi_q^2 u})$ is Abelian with Galois group killed by $p$.
[*Proof*]{}. This follows immediately by Theorem \[unde\] and Lemmas \[irinusescoala\] and \[gainuli\], respectively.
The next Corollary says that the “boundary value problem at $q=0$" is well posed.
\[vainiciel\] Under the hypotheses of Theorem [\[unde\]]{} and the assumption that $\frac{\eta}{\gamma}=-\frac{1}{\kappa}$ for some integer $\kappa$, then for any $\alpha \in R$ there exists a unique $u \in \cU_{1}^1$ such that $B_\kappa^0 u=\alpha$.
Next we study $\D$-characters of order $2$ of the form $\psi_{\ddi}^2+\lambda \psi_{\di}^2$.
Let $E_{\beta q}$ be the Tate curve with parameter $\beta q$ for $\beta$ not a root of unity. Let $\cU^2_{\ra}$ be the set of solutions of the $\D$-character $\psi_{\ddi}^2+\lambda
\psi_{\di}^2$ in $A=R((q))\h$, where $\lambda \in R$. We set $$\begin{array}{rcl}
\cU^2_{0} & := & \cU^2_{\ra} \cap E_{0}(A)\, , \\
\cU^2_{1} & := & \cU^2_{\ra} \cap E_{1}(A)\, .
\end{array}$$ Then the following hold:
1. If $\frac{\eta}{\gamma}=-\frac{1}{\kappa}$ and $\lambda=\kappa^3$ for some integer $\kappa \geq 1$ coprime to $p$, and if $u_{E,\kappa,\alpha}$ is the basic series of $\psi^1_{pq}$ (cf. (\[filmm\])), then $$\begin{array}{rcl}
\cU^2_{\ra} & = & \cU^2_{0}+\cU^2_{1}\, , \\
\cU^2_{1} & = & \{u_{E,\kappa,\alpha}\, | \; \alpha \in R\} \, .
\end{array}$$
2. If $\frac{\lambda \eta^{\phi}}{\gamma} \not\in \{-n^2\, | \; n \in
\bZ\}$, then $\cU^2_{1}=0$.
[*Proof*]{}. We prove assertion 1). The fact that $\cU^2_{1}$ consists of exactly the $u_{E,\kappa,\alpha}$’s follows directly by Theorem \[unde\] and Proposition \[pomu\]. We just need to observe that, by Theorem \[addeq\], the map $$(\gamma \eta^2\dd+\gamma^2 \eta\phip-\gamma^2\eta):qR[[q]]\ra qR[[q]]$$ is injective. Now if $u \in \cU^2_{\ra}$ then, by Proposition \[pomu\], we have $$(\gamma \eta^2 \dd+\gamma^2 \eta \phip-\gamma^2 \eta) \psi^1_{\di \ddi}u=0
\, .$$ By Theorem \[addeq\], $\psi^1_{\di \ddi}u \in R$. Again, by Proposition \[pomu\], we get $$0=\gamma \dd^2 \psi^1_{\di \ddi} u=(\eta \dd-p\gamma
\phip+\gamma) \psi^2_{\ddi}u\, .$$ By Theorem \[addeq\] there is an $\alpha \in R$ such that $$\psi^2_{\ddi}u =u_{a,\kappa,\alpha}^{\mu^{(p)}}\, .$$ On the other hand, as in the proof of Theorem \[unde\], we have that $u_{a,\kappa,\alpha}^{\mu^{(p)}}=\psi^2_{\ddi}
u_{E,\kappa,\alpha/\gamma \kappa}$. Hence $\psi^2_{\ddi}(u-u_{E,\kappa,\alpha/\gamma \kappa})=0$, and the assertion is proved.
In order to prove assertion 2), let us assume $u \in qR[[q]]$ is such that $(\psi^2_{\ddi}+\lambda \psi^2_{\di})u=0$. Then $$\begin{array}{rcl}
0 = \dd(\psi^2_{\ddi}+\lambda \psi^2_{\di})u & = & \dd\left[ \gamma
\dd^2+\frac{\lambda}{p} \left( \eta
\phip^2-(\eta^{\phi}+p\eta)\phip+p\eta^{\phi} \right) \right]l(u)\\
& = & [\gamma \dd^2+p \lambda \eta
\phip^2-\lambda(\eta^{\phi}+p\eta)\phip+\lambda\eta^{\phi}]\dd
l(u)\, .
\end{array}$$ By our assumption, $\frac{\lambda
\eta^{\phi}}{\gamma}+n^2\neq 0$ for all $n \in \bZ$. By Theorem \[addeq\], $$\gamma \dd^2+p \lambda \eta
\phip^2-\lambda(\eta^{\phi}+p\eta)\phip+\lambda\eta^{\phi}$$ has no non-zero solution in $qR[[q]]$. Since $\dd l(u) \in qR[[q]]$, we get $\dd l(u)=0$ so $l(u) \in K \cap qK[[q]]=0$ so $u=0$.
Next we address the boundary value problem at $q \neq 0$ for $\psi^1_{\di \ddi}$. For $q_0 \in pR$ and $\beta \in R^{\times}$ not a root of unity, we let ${\mathcal E}_{\beta q_0}$ be the (non-smooth) plane projective curve that in the affine plane is given by the equation $$y^2=x^3-\frac{1}{48}E_4(\beta q_0)x+\frac{1}{864}E_6(\beta q_0)\, .$$ There is a commutative diagram of groups $$\label{pichiu}
\begin{array}{ccc}
qR[[q]] & \stackrel{\iota}{\ra} & {\mathcal E}_{\beta q}(R[[q]])\\
\da & & \da \pi_{q_0}\\
pR & \stackrel{\iota_{q_0}}{\ra} & {\mathcal E}_{\beta q_0}(R)
\end{array}\, ,$$ where the vertical arrows are induced by the ring homomorphism $$\begin{array}{rcl}
R[[q]] & \ra & R \\ u(q) & \mapsto & u(q_0)
\end{array}\, ,$$ $\iota_{q_0}$ sends $pa$ into $(pa,(pa)^3-\frac{1}{48}E_4(\beta q_0)(pa)^7+
\cdots)$ for $a \in R$, and the image of $\iota_{q_0}$ is ${\mathcal
E}_{\beta q_0}(pR)$, the preimage of $0$ via the reduction modulo $p$ mapping $$\label{petro} {\mathcal E}_{\beta q_0}(R) \ra {\mathcal E}_{\beta
q_0}(k)\, .$$ So we have $\iota_{q_0}(pR)={\mathcal E}_{\beta q_0}(pR)$. We note that we have usually identified $\iota(qR[[q]])$ with $qR[[q]]$, but in order to avoid the potential confusion produced by the choice of $q_0$, we will [**not**]{} identify $\iota_{q_0}(pR)$ with $pR$.
Let ${\mathcal E}_{\beta q_0}'(R)$ be the pull-back by the map (\[petro\]) of the locus ${\mathcal E}^{reg}_{\beta q_0}(k)$ of regular points on the cubic ${\mathcal E}_{\beta q_0}(k)$. So we have inclusions of groups $${\mathcal E}_{\beta q_0}(pR) \subset {\mathcal E}'_{\beta q_0}(R)
\subset {\mathcal E}_{\beta q_0}(R)\, .$$ Let us recall some facts about torsion points on Tate curves. There is a natural injective homomorphism $$\label{paru}
\begin{array}{rcl}
\tau:\bmu(R) & \ra & {\mathcal E}_{\beta q}(R[[q]])\\
v & \mapsto & (X(q,v)+\frac{1}{12},-Y(q,v)-\frac{1}{2}X(q,v))\, ,
\quad v \neq 1\, ,\\
1 & \mapsto & \infty\, ,
\end{array}$$ given by the series in (\[uglys\]). (The formula makes sense because if $1 \neq v \in \bmu(R)$ then $1-v \in
R^{\times}$.) The composition of the homomorphism in (\[paru\]) with the specialization map $\pi_{q_0}$ in (\[pichiu\]) gives a homomorphism $$\tau_{q_0}:\bmu(R) \ra {\mathcal E}_{\beta q_0}(R)\,$$ The composition of $\tau_{q_0}$ with the reduction mod $p$ mapping ${\mathcal E}_{\beta q_0}(R) \ra {\mathcal E}_{\beta q_0}(k)$ is the map $$\begin{array}{rcl}
\bmu(R) & \ra & {\mathcal E}_{\beta q_0}(k)\\
\zeta & \mapsto & \left( \frac{\zeta}{(1-\zeta)^2}+\frac{1}{12}\
\text{mod}\ p, -\frac{\zeta^2}{(1-\zeta)^3}-
\frac{\zeta}{2(1-\zeta)^2}
\ \text{mod}\ p \right)\, , \; \zeta \neq 1\, , \\
1 & \mapsto & \infty\, ,
\end{array}$$ which is an isomorphism of $\bmu(R) \simeq k^{\times}$ onto ${\mathcal E}_{\beta q_0}^{reg}(k)$. We conclude that the mapping $\tau_{q_0}$ above is an injective map, and any point $P \in {\mathcal E}'_{\beta q_0}(R)$ can be written uniquely as $$P=\iota_{q_0}(pa)+\tau_{q_0}(\zeta)\, ,$$ where $a \in R$ and $\zeta \in \bmu(R)$.
\[titoo\] Let $\cU^1_{\ra}$ be the set of solutions of $\psi^1_{\di \ddi}$ in $R((q))\wh$, and let $$\cU'_{\ra}:=\tau(\bmu(R)) \cdot \cU^1_{1} \subset \cU^1_{\ra}\, .$$ Assume $\frac{\eta}{\gamma}=-\frac{1}{\kappa}$ with $\kappa \geq 1$ an integer coprime to $p$. Then the following hold:
1. For any $q_0 \in p^{\nu}R^{\times}$ with $\nu \geq 1$ and any $g \in \iota_{q_0}(p^{\kappa \nu} R) \subset
\iota_{q_0}(pR)={\mathcal E}_{\beta q_0}(pR)$, there exists a unique $u \in \cU^1_{1}=\iota(qR[[q]])=qR[[q]]$ such that $u(q_0)=g$.
2. Assume $\kappa=1$. Then for any $q_0 \in pR^{\times}$ and any $g \in {\mathcal E}'_{\beta q_0}(R)$, there exists a unique $u \in
\cU'_{\ra}$ such that $u(q_0)=g$.
[*Proof*]{}. Let us prove assertion 1). By Theorem \[unde\] and Equations (\[filmm\]) and (\[nuilacasino\]), it is enough to show that the mapping $R \ra p^{\kappa
\nu}R$ defined by $$\alpha \mapsto e_E \left( \sum_{j \geq 0} (-1)^j
\frac{\alpha^{\phi^j}}{\kappa p^j F_j(p)} q_0^{\kappa p^j}
\right)$$ is a bijection. Let us recall that $e_E:p^N R \ra
\iota_{q_0}(p^N R)$ is an isomorphism for all $N$. So it is enough to show that the map $R \ra p^{\kappa \nu}R$ given by $$\alpha \mapsto \sum_{j \geq 0} (-1)^j
\frac{\alpha^{\phi^j}}{\kappa p^j F_j(p)} q_0^{\kappa p^j}\, ,$$ is a bijection. But this is clear by Lemma \[ajutator\].
For the proof of assertion 2), we let $g \in {\mathcal E}_{\beta
q_0}'(R)$, and write $g=\tau_{q_0}(\zeta_0)+w_0$, $w_0 \in
{\mathcal E}_{\beta q_0}(pR)$. By assertion 1), there exists $w \in
\cU_{1}^1$ such that $w(q_0)=w_0$. Let us set $u:=\tau(\zeta_0)+w$. Then $u(q_0)=\tau_{q_0}(\zeta_0)+w_0=g$, which completes the proof of the existence part of the assertion. In order to prove uniqueness, we let $u_1= \tau(\zeta_1)+w_1$ with $w_1 \in \cU^1_{1}$ such that $u_1(q_0)=g$. Then $$\tau_{q_0}(\zeta_1)+w_1(q_0)=\tau_{q_0}(\zeta_0)+w_0\, .$$ This implies that $\zeta_1=\zeta_0$ and $w_1(q_0)=w_0$. By the uniqueness in assertion 1), we get $w_1=w$. We conclude that $u_1=u$, and we are done.
The following Corollary is concerned with the inhomogeneous equation $\psi^1_{pq} u=\varphi$, and it is an immediate consequence of Lemmas \[oashteptpei\], \[vainiciel\], and \[irinusescoala\]
Under the hypotheses of Theorem [\[unde\]]{}, assume that $\frac{\eta}{\gamma}=-\frac{1}{\kappa}$ for some integer $\kappa$, and that $\varphi \in qR[[q]]$ is a series whose support is contained in the set $\cK'$ of totally non-characteristic integers of $\psi^1_{pq}$. Then the equation $\psi^1_{pq} u=\varphi$ has a unique solution $u \in E_1(A)$ such that the support of $\psi^2_q
u$ does not contain $\kappa$. Moreover, if $\bar{\varphi} \in
k[q]$ the series $\overline{\psi^2_q u} \in k[[q]]$ is integral over $k[q]$ and the field extension $k(q) \subset
k(q,\overline{\psi^2_q u})$ is Abelian with Galois group killed by $p$.
Let us assume that $\frac{\eta}{\gamma}=-1$. By Corollary \[titoo\], for any $q_0 \in pR^{\times}$ the group homomorphism $$\begin{array}{rcl}
S_{q_0}:\bmu(R) \times R & \ra & {\mathcal E}'_{\beta q_0}(R) \\
(\xi,\alpha) & \mapsto &\tau_{q_0}(\xi)+\iota_{q_0}(u_{E,1,\alpha}(q_0))
\end{array}$$ is an isomorphism. So, for any $q_1,q_2 \in pR^{\times}$ we have a group isomorphism $$S_{q_1, q_2}:=S_{q_2} \circ S_{q_1}^{-1}:{\mathcal E}'_{\beta q_1}(R)
\ra {\mathcal E}'_{\beta q_2}(R)\, .$$ The latter mapping can be viewed as the “propagator” attached to $\psi^1_{\di \ddi}$.
It is natural to make the propagator act as an endomorphism of a given group. We accomplish this by defining a new [*propagator*]{} $$S_{q_1,q_2}^{q_0}:=\Gamma_{q_0} \circ \Gamma_{q_2}^{-1} \circ
S_{q_1,q_2} \circ \Gamma_{q_1} \circ \Gamma_{q_0}^{-1}: {\mathcal
E}'_{\beta q_0}(R) \ra {\mathcal E}'_{\beta q_0}(R)\, ,$$ where $\Gamma_{q_i}$ is the group isomorphism $$\begin{array}{rcl}
\bmu(R) \times R & \stackrel{\Gamma_{q_i}}{\ra } &
{\mathcal E}'_{\beta q_i}(R) \\
(\xi,a) & \mapsto & \iota_{q_i}(pa)+\tau_{q_i}(\xi)
\end{array}\, .$$ Then it is easy to see that for $\zeta_1,\zeta_2 \in \bmu(R)$ we have that $$S_{q_0,\zeta_1 \zeta_2 q_0}^{q_0}=S_{q_0,\zeta_2 q_0}^{q_0}
\circ S_{q_0,\zeta_1 q_0}^{q_0}\, ,$$ which, once again, we view as a (weak) form of the “Huygens principle.”
Elliptic curves over $R$
------------------------
In this section we consider an elliptic curve $E$ over $A:=R$ defined by $$f=y^2-(x^3+a_4x+a_6)\, ,$$ where $a_4,a_6 \in R$, and equip it with the $1$-form $$\omega=\frac{dx}{y}\, .$$ We use the notation and discussion in Example \[cucurigu\] that applies to $E$ over $R$.
Clearly, $f^1_{\ddi}=0$. Let us fix in what follows the étale coordinate $T=-\frac{x}{y}$, and denote by $l(T)=l_E(T) \in K[[T]]$ the logarithm of the formal group $\cF_E$ of $E$ with respect to $T$.
\[logder\] The following holds:
1. The image of $\psi^1_{\ddi}$ in $$\cO\left(J^1_{\di \ddi}\left(\frac{A[x,y]}{(f)} \right) \right)=
\frac{A[x,y,\d x, \d y, \dd x, \dd y]\h}{(f,\d f,\dd f)}$$ is equal to ${\displaystyle \frac{\dd x}{y}}$.
2. The image of $\psi^1_{\ddi}$ in $A[[T]][\d T, \dd T]\h$ is equal to ${\displaystyle \dd l(T)=\frac{dl}{dT}(T) \cdot \dd T}$.
The first assertion above says that the $\D$-character $\psi^1_{\ddi}$ attached to $(E,\omega)$ coincides with the [*Kolchin’s logarithmic derivative*]{} [@kolchin].
[*Proof*]{}. We first check that $\frac{\dd x}{y}$ comes from a $\D$-character. For that, it is enough to show that its image $\left( \frac{\dd x}{y}\right) (T,\dd T)$ in $A[[T]][\d T,\dd T]\h$ defines a homomorphism from the formal group of $J^1_{\di \ddi}(E)$ to the formal group of $\bG_a$. This is so because “a partially defined map, which generically is a homomorphism, is an everywhere defined homomorphism.” We now recall that $\omega$ and $l$ are related by the equation $$\omega=\frac{dl}{dT} (T) \cdot dT\, .$$ Consequently, if we denote by $x(T)$ and $y(T)$ the images of $x$ and $y$ in $R((T))$, respectively, we have that $$\frac{dl}{dT} (T) \cdot dT=\left(\frac{dx}{y} \right) (T)=
(y(T))^{-1} \frac{dx}{dT} (T)\cdot dT\, .$$ Hence $$\label{stelutza} \left( \frac{\dd x}{y} \right) (T,\dd
T)=(y(T))^{-1} \frac{dx}{dT}(T) \cdot \dd T=\frac{dl}{dT}(T) \cdot
\dd T=\dd l(T)\, .$$ But clearly $\dd l(T)$ defines a homomorphism at the level of formal groups. Hence $\frac{\dd x}{y}$ is a $\D$-character of $E$. Now, (\[stelutza\]) also shows that $$\psi^1_{\ddi}-\frac{\dd x}{y} \in A[[T]]\, ,$$ and thus, $\psi^1_{\ddi}-\frac{\dd x}{y}$ defines a homomorphism $\hat{E} \ra \hat{\bG}_a$. Therefore, $\psi^1_{\ddi}-\frac{\dd x}{y}=0$ and assertion 1) is proved.
The second assertion follows from the first in combination with (\[stelutza\]).
As in the previous subsection, we identify $q^{\pm
1}R[[q^{\pm 1}]]$ with its image $E(q^{\pm 1}R[[q^{\pm 1}]])$ in $E(R[[q^{\pm 1}]])$ under the embedding $$\iota:q^{\pm 1} R[[q^{\pm 1}]] \ra E(q^{\pm 1}R[[q^{\pm 1}]])$$ given in $(T,W)$- coordinates by $$u \mapsto (u,u^3+a_4 u^7+\cdots)\, .$$
In the sequel, we fix an elliptic curve $E/R$. We define next the [*characteristic polynomial*]{} of a $\D$-character, [*non-degenerate*]{} $\D$-characters, their [*characteristic integers*]{}, and [*basic series*]{}. We distinguish the two cases $f^1_{\di} \neq 0$ and $f^1_{\di} = 0$, respectively.
If $f^1_{\di} \neq 0$, by [@book], p. 197, $v_p(f^2_{\di}) \geq v_p(f^1_{\di})$. We may therefore consider the $\D$-character $$\psi^2_{\di,nor} :=\frac{1}{f^1_{\di}} \psi^2_{\di} \in
\bX^2_{\di \ddi}(E)\, .$$ Its image in $R[[T]][\d T, \d^2 T]\h$ is $$\psi^2_{\di,nor}=\frac{1}{p}(\phip^2+\gamma_1 \phip+p
\gamma_0)l(T),$$ where $$\gamma_1:=-\frac{f^2_{\di}}{f^1_{\di}}\in
R,\ \ \gamma_0:= \frac{(f^1_{\di})^{\phi}}{f^1_{\di}}\in
R^{\times}.$$ By Proposition \[ileana1\], any $\D$-character of $E$ is a $K$-multiple of a $\D$-character of the form $$\label{mau} \psi_E:=\nu(\phip,\dd) \psi^1_{\ddi}+\lambda(\phip)
\psi^2_{\di,nor}\, ,$$ where $\nu(\xi_p,\xi_q) \in R[\xi_p,\xi_q]$ and $\lambda(\xi_p) \in R[\xi_p]$. The Picard-Fuchs symbol of $\psi_E$ with respect to $T$ is easily easily seen to be $$\sigma(\xi_p,\xi_q)=p \nu(\xi_p,\xi_q)\xi_q
+\lambda(\xi_p)(\xi_p^2+\gamma_1 \xi_p+p\gamma_0)\, .$$ Hence, the Fréchet symbol of $\psi_E$ with respect to $\omega$ is $$\theta(\xi_p,\xi_q)=\frac{\sigma(p\xi_p,\xi_q)}{p}=
\nu(p\xi_p,\xi_q)\xi_q+\lambda(p\xi_p)(p\xi_p^2+\gamma_1
\xi_p+\gamma_0)\, .$$
Let us assume that $E/R$ has $f^1_{\di} \neq 0$, and let $\psi_E$ be a $\D$-character of $E$ of the form in (\[mau\]). We define the [*characteristic polynomial*]{} $\mu(\xi_p,\xi_q)$ of $\psi_E$ to be the Fréchet symbol $\theta(\xi_p,\xi_q)$ of $\psi_E$ with respect to $\omega$. We say that the $\D$-character $\psi_E$ is [*non-degenerate*]{} if $\mu(0,0)\in R^{\times}$, or equivalently, if $\lambda(0)\in
R^{\times}$. For a non-degenerate character $\psi_E$, we define its [*characteristic integers*]{} to be the integers $\kappa$ such that $\mu(0,\kappa) =0$ (so any such $\kappa$ is coprime to $p$), and denote by $\cK$ the set of all such. We call $\kappa$ a [*totally non-characteristic*]{} integer if $\kappa \not\equiv 0$ mod $p$ and $\mu(0,\kappa) \not\equiv 0$ mod $p$. The set of all totally non-characteristic integers is denoted by $\cK'$. For $0 \neq \kappa \in \bZ$ and $\alpha \in R$, we define the [*basic series*]{} by $$\label{caino} u_{E,\kappa,\alpha}=e_E\left( \int
u_{a,\kappa,\alpha} \frac{dq}{q} \right) \in q^{\pm 1}K[[q^{\pm
1}]]\, ,$$ where $$\label{cain} u_{a,\kappa,\alpha}:=u_{a,\kappa,\alpha}^{\mu};$$ Cf. (\[ua\]).
Let us now consider now the case where $f^1_{\di} =0$. We may then look at the $\D$-character $\psi^1_{\di} \in \bX^1_{\di}(E)$, cf. (\[psi1p\]). Its image in $R[[T]][\d T]\h$ is $$\psi^1_{\di}=\frac{1}{p}(\phip +p \gamma_0) l(T)\, ,$$ where $\gamma_0 \in R$; cf. [@book], Remark 7.21. (We actually have $\gamma_0 \in R^{\times}$ if the cubic defining $E$ has coefficients in $\bZ_p$; cf. [@frob], Theorem 1.10.) By Proposition \[ileana2\], any $\D$-character of $E$ is a $K$-multiple of a $\D$-character of the form $$\label{drrr}
\psi_E:=\nu(\phip,\dd) \psi^1_{\ddi}+\lambda(\phip)
\psi^1_{\di}\, ,$$ where $\nu(\xi_p,\xi_q) \in R[\xi_p,\xi_q]$ and $\lambda(\xi_p)
\in R[\xi_p]$. We easily see that the Picard-Fuchs symbol of $\psi_E$ is given by $$\sigma(\xi_p,\xi_q)=p\nu(\xi_p,\xi_q)\xi_q+\lambda(\xi_p)
(\xi_p+p\gamma_0)\, .$$ Thus, the Fréchet symbol at the origin (with respect to $dT$) is $$\theta(\xi_p,\xi_q)=
\frac{\sigma(p\xi_p,\xi_q)}{p}=\nu(p\xi_p,\xi_q)\xi_q
+\lambda(p\xi_p)(\xi_p+\gamma_0)\, .$$
Let us assume that $E/R$ has $f^1_{\di}=0$. (Recall that then $E/R$ has ordinary reduction mod $p$; cf. [@book], Corollary 8.89.), and let us fix a $\D$-character of $E$ of the form $\psi_E$ as in [(\[drrr\])]{}. We define the [*characteristic polynomial*]{} $\mu(\xi_p,\xi_q)$ of $\psi_E$ to be the Fréchet symbol $\theta(\xi_p,\xi_q)$ of $\psi_E$ with respect to $\omega$. We say that the $\D$-character $\psi_E$ is [*non-degenerate*]{} if $\mu(0,0) \in R^{\times}$, or equivalently, if $\lambda(0)\in R^{\times}$ and $\gamma_0 \in
R^{\times}$. For a non-degenerate character $\psi_E$, we define the [*characteristic integers*]{} to be the integers $\kappa$ such that $\mu(0,\kappa) =0$ (so any such $\kappa$ is coprime to $p$), and denote by $\cK$ the set of all such. We say that $\kappa$ is a [*totally non-characteristic*]{} integers is $\kappa \not\equiv 0$ mod $p$ and $\mu(0,\kappa) \not\equiv 0$ mod $p$. The set of all totally non-characteristic integers is denoted by $\cK'$. For $0
\neq \kappa \in \bZ$ and $\alpha \in R$, we define the [*basic series*]{} $$\label{errr} u_{E,\kappa,\alpha}=e_E\left( \int
u_{a,\kappa,\alpha} \frac{dq}{q} \right) \in q^{\pm 1}K[[q^{\pm 1}]]\, ,$$ where $$\label{rrrr} u_{a,\kappa,\alpha}:=u_{a,\kappa,\alpha}^{\mu};$$ cf. (\[ua\]).
In the sequel, we consder an elliptic curve $E/R$ without imposing any a priori restriction on the vanishing of $f^1_{\di}$.
Let us assume that $\nu=1$, $\lambda \in R^{\times}$, that is to say, $\psi_E$ is either $\psi^1_q+\lambda \psi^2_{p,nor}$ or $\psi^1_q+\lambda \psi^1_p$ according as $f^1_p \neq 0$ or $f^1_p
=0$, respectively. Then the characteristic polynomial is unmixed (provided that $\gamma_1 \in R^{\times}$ in case $f^1_p \neq 0$), $$\cK=\{-\lambda \gamma_0\} \cap \bZ\, ,$$ and $\cK \neq \emptyset$ if, and only if, $\lambda \gamma_0 \in
\bZ$. In this case, $\psi_E$ should be viewed as an analogue of either the heat equation or the convection equation according as $f^1_{\di} \neq 0$ or $f^1_{\di}=0$, respectively.
Let us now assume that $\nu=\xi_q$ and $\lambda \in R^{\times}$, that is to say, $\psi_E$ is either $\dd \psi^1_q+\lambda
\psi^2_{p,nor}$ or $\dd \psi^1_q+\lambda \psi^1_p$ according as $f^1_p \neq 0$ or $f^1_p =0$, respectively. Then the characteristic polynomial is unmixed (provided that $\gamma_1 \in R^{\times}$ in case $f^1_p \neq 0$), $$\cK=\{\pm \sqrt{-\lambda \gamma_0}\} \cap \bZ\, ,$$ and $\cK \neq
\emptyset$ if, and only if, $-\lambda \gamma_0$ is a perfect square. If this the case, $\psi_E$ should be viewed as analogue of either the wave equation or the sideways heat equation according as $f^1_{\di} \neq 0$ or $f^1_{\di}=0$, respectively.
Let us consider the “simplest” of the energy functions in the case where $f^1_p=0$: $$H=a(\psi^1_q)^2+2b \psi^1_p \psi^1_q +c(\psi_p^1)^2 \, .$$ Here, $a,b,c \in R$.
A computation essentially identical to the one in Example \[disom\] leads to the following formula for the Euler-Lagrange equation $\epsilon^1_{H,\partial}$ attached to $H$ and the vector field $\partial$ dual to $\omega$: $$\epsilon^1_{H,\partial}=(-2a^{\phi}\phip \dd-2b^{\phi}
\phip^2+2b)\psi^1_q+ (2c^{\phi}\gamma_0^{\phi} \phip
+2c)\psi^1_p\, .$$ Hence, the characteristic polynomial of $\epsilon^1_{H,\partial}$ is $$\mu(\xi_p,\xi_q)=(-2a^{\phi}p \xi_p \xi_q-2b^{\phi}p^2
\xi_p^2+2b)\xi_q+ (2c^{\phi}\gamma_0^{\phi}p
\xi_p+2c)(\xi_p+\gamma_0)\, ,$$ so the $\D$-character $\epsilon^1_{H,\partial}$ is non-degenerate if, and only if, $c \in R^{\times}$ and $\gamma_0 \in R^{\times}$. Moreover, the set of characteristic integers is $$\cK=\{-\frac{c \gamma_0}{b} \}\cap \bZ\, .$$
On the other hand, when $f^1_p \neq 0$ we consider the “simplest” energy function $$H=a(\psi^1_q)^2+2b \psi^1_q \psi^2_{p,nor} +c(\psi_{p,nor}^2)^2\, ,$$ for $a,b,c \in R$. Then we have the following values for the Fréchet symbols: $$\begin{array}{rcl}
\theta_{\psi^1_q,\omega} & = & \xi_q\\
\theta_{\psi^2_{p,nor},\omega} & = & p \xi_p^2+\gamma_1 \xi_p
+\gamma_0
\end{array}\, ,$$ and we get the following formula for the Euler-Lagrange equation attached to $H$ and the vector field $\partial$ dual to $\omega$: $$\begin{array}{rcl}
\epsilon^2_{H,\partial} & = & (-2b^{\phi^2}p \phip^3 -2a^{\phi^2}
\phip^2 \dd)\psi^1_q \\
& \ & + [2b^{\phi^2}(\gamma_0^{\phi^2}-\gamma_1^{\phi})
\phip^2+ (2b^{\phi}\gamma_1^{\phi}-2b^{\phi^2}
\gamma_0^{\phi})\phip+ 2bp] \psi^1_q\\
& & + (2c^{\phi^2} \gamma_0^{\phi^2} \phip^2
+2c^{\phi}\gamma_1^{\phi} \phip +2cp) \psi^2_{p,nor}\, .
\end{array}$$ In particular $\epsilon^2_{H,\partial}$ is degenerate for all values of $a,b,c$. More is true, actually: $\epsilon^2_{H,\partial}$ is not a $K$-multiple of a non-degenerate $\D$-character.
Let $\kappa \in \bZ\backslash p\bZ$. Then we have $u_{E,\kappa,\alpha} \in q^{\pm 1}R[[q^{\pm 1}]]$ for all $\alpha
\in R$, and $$\begin{array}{rcl}
R & \ra & R[[q^{\pm 1}]] \\ \alpha & \mapsto & u_{E,\kappa,\alpha}
\end{array}$$ is a pseudo $\d$-polynomial mapping. Consequently, $$\begin{array}{rcl}
R & \ra & E(R[[q^{\pm 1}]]) \\
\alpha & \mapsto & \iota(u_{E,\kappa,\alpha})
\end{array}$$ is also a pseudo $\d$-polynomial mapping.
[*Proof*]{}. As an element of $R[[T]][\d T, \dd T,\d^2 T,\d \dd T,\dd^2T]\wh$, the $\D$-character $\psi_E$ coincides with $$\psi_E=\left( \nu(\phip,\dd)\dd + \frac{\lambda(\phip)}{p}
\phip^2 + \frac{\lambda(\phip)}{p}
\gamma_1 \phip+\lambda(\phip) \gamma_0 \right)l(T)$$ if $f^1_{\di} \neq 0$, and it coincides with $$\psi_E=(\nu(\phip,\dd)\dd+\frac{\lambda(\phip)}{p}
\phip+\lambda(\phip) \gamma_0)l(T)$$ if $f^1_{\di}=0$. In the first case, by Proposition \[logder\] we have that $$\label{casc}
\begin{array}{rcl} \dd \psi_E & = & (
\nu(p\phip,\dd)\dd
+ \lambda(p\phip) p
\phip^2 +\lambda(p\phip) \gamma_1 \phip
+ \lambda(p\phip) \gamma_0) \dd l(T)\\
\ & = & (\nu(p\phip,\dd)\dd + \lambda(p\phip) p \phip^2
+\lambda(p\phip) \gamma_1 \phip + \lambda(p\phip) \gamma_0)
\psi^1_{\ddi}\, .
\end{array}$$ Similarly, if $f^1_{\di}=0$ we have $$\label{mz23}
\begin{array}{rcl} \dd \psi_E & = & (\nu(p\phip,\dd)\dd +
\lambda(p\phip) \phip
+ \lambda(p\phip) \gamma_0) \dd l(T)\\
& = & (\nu(p\phip,\dd)\dd + \lambda(p\phip) \phip +
\lambda(p\phip) \gamma_0) \psi^1_{\ddi}\, .
\end{array}$$ Thus, if $\psi_E u=0$, we have $\psi^1_{\ddi}u$ a solution of $$\nu(p\phip,\dd)\dd+\lambda(p\phip)
p \phip^2+ \lambda(p\phip) \gamma_1 \phip +\lambda(p\phip) \gamma_0$$ when $f^1_{\di}\neq 0$, or a solution of $$\nu(p\phip,\dd)\dd+ \lambda(p\phip) \phip +\lambda(p\phip) \gamma_0$$ when $f^1_{\di}= 0$.
Let $\alpha \in R$, and set $$h:=\int u_{a,\kappa,\alpha} \frac{dq}{q} \in qK[[q]]\, ,$$ with $u_{a,\kappa,\alpha}$ as in (\[cain\]) or (\[rrrr\]), respectively. Note that $$\dd h=u_{a,\kappa,\alpha}\, .$$ By Theorem \[addeq\], when $f^1_{\di} \neq 0$ we get that $$\begin{array}{rcl}
\dd(\mu(0,\kappa)\alpha \kappa^{-1}q^{\kappa}) \! \! \! & = & \! \! \!
\mu(0,\kappa)\alpha q^{\kappa} \vspace{1mm} \\
& = & \! \! \!
(\nu(p\phip,\dd)\dd \! + \! \lambda(p\phip) p \phip^2+ \lambda(p\phip)
\gamma_1 \phip +\lambda(p\phip)
\gamma_0) u_{a,\kappa,\alpha} \vspace{1mm} \\
& = & \! \! \! (\nu(p\phip,\dd)\dd+\lambda(p\phip) p \phip^2+
\lambda(p\phip) \gamma_1 \phip +\lambda(p\phip)
\gamma_0) \dd h \vspace{1mm} \\
& = &\! \! \! \dd \left(\nu(\phip,\dd)\dd+ \frac{\lambda(\phip)}{p}
\phip^2+ \frac{\lambda(\phip)}{p} \gamma_1 \phip +\lambda(\phip)
\gamma_0 \right)h
\end{array}\, .$$ Hence $$\label{davidstar} \begin{array}{c} \left(\nu(\phip,\dd)\dd+
\frac{\lambda(\phip)}{p} \phip^2+ \frac{\lambda(\phip)}{p}
\gamma_1 \phip + \lambda(\phip)\gamma_0 \right)h
-\mu(0,\kappa)\alpha \kappa^{-1}q^{\kappa}\\
\in R \cap q^{\pm 1}K[[q^{\pm 1}]]=0,\end{array}$$ and we have that $$\begin{array}{rcl}
\lambda(\phip) \left(\frac{1}{p} \phip^2+ \frac{1}{p} \gamma_1
\phip + \gamma_0 \right)h & = & -\nu(\phip,\dd)\dd h
+\mu(0,\kappa)\alpha \kappa^{-1}q^{\kappa}
\\ & = & -\nu(\phip,\dd)u_{a,\kappa,\alpha}
+\mu(0,\kappa)\alpha \kappa^{-1}q^{\kappa}\\
& \in & q^{\pm 1}R[[q^{\pm 1}]]\, .
\end{array}$$ By Lemma \[floricica\], we get $$\label{onestar} \left( \frac{1}{p} \phip^2+\frac{1}{p} \gamma_1
\phip+\gamma_0 \right) h \in q^{\pm 1}R[[q^{\pm 1}]] \subset
R[[q^{\pm 1}]]\, .$$
Similarly, when $f^1_{\di}=0$, we get $$\label{oozz} \left( \frac{1}{p} \phip+ \gamma_0 \right) h \in
R[[q^{\pm 1}]]\, .$$
We claim that that if $f^1_{\di} \neq 0$ we have $$\label{twostars} \left(\frac{1}{p} \phip^2+ \frac{1}{p} \gamma_1
\phip +\gamma_0 \right)l_E(q) \in R[[q^{\pm 1}]]\, ,$$ where $l_E(q) \in K[[q^{\pm 1}]]$ is obtained from $l(T)=l_E(T)
\in K[[T]]$ by substitution of $q^{\pm 1}$ for $T$, and $\phip:R[[q^{\pm 1}]] \ra R[[q^{\pm 1}]]$ is defined by $\phip(q^{\pm 1})=q^{\pm p}$. In order to check that this holds, it is sufficient to check that $$\label{fourstars} (\phip^2+ \gamma_1 \phip + p \gamma_0)l_E(q^{\pm
1}) \in pR[[q^{\pm 1}]]\, .$$ Now recall that $$\label{threestars} (\phip^2+ \gamma_1 \phip + p \gamma_0)l_E(T)
\in pR[[T]][\d T, \d^2 T]\h\, ,$$ where, as usual, the mappings $R[[T]] \stackrel{\phip}{\ra}
R[[T]][\d T]\h \stackrel{\phip}{\ra}R[[T]][\d T, \d^2 T]\h$ are defined by $\phip(T)=T^p+p \d T$, $\phip(\d T)=(\d T)^p+p\d^2 T$. Taking the image (\[threestars\]) under the unique $\d$-ring homomorphism $R[[T]][\d T, \d^2 T]\h \ra R[[q^{\pm 1}]]$ that sends $T$ into $q^{\pm 1}$, we get equality (\[fourstars\]), completing the verification that (\[twostars\]) holds.
Similarly, when $f^1_{\di}=0$ we have that $$\label{zzzt} \left( \frac{1}{p} \phip + \gamma_0 \right) l_E(q)
\in R[[q^{\pm 1}]]\, .$$
If $f^1_{\di} \neq 0$, then by (\[onestar\]), (\[twostars\]), and Hazewinkel’s Functional Equation Lemma \[haze\], it follows that $e(h) \in R[[q^{\pm 1}]]$. Since $e(h)
\in q^{\pm 1}K[[q^{\pm 1}]]$, we get $$u_{E,\kappa,\alpha}=e_E(h) \in q^{\pm 1}R[[q^{\pm 1}]]\, .$$
Similarly, if $f^1_{\di}=0$, by (\[oozz\]) and (\[zzzt\]), we get that $$u_{E,\kappa,\alpha} \in q^{\pm 1}R[[q^{\pm 1}]]\, .$$
As in the proof of Lemma \[ol\], we see that $$\begin{array}{rcl}
R & \ra & R[[q^{\pm 1}]] \\
\alpha & \mapsto & u_{E,\kappa,\alpha}
\end{array}$$ is pseudo $\d$-polynomial mapping.
We have the following diagonalization result.
\[gainuseli\] \[viyne\] Let $\kappa \in \bZ \backslash p\bZ$, and $\alpha
\in R$. Then $$\begin{array}{rcl}
\psi^1_q u_{E,\kappa,\alpha} & = & u_{a,\kappa,\alpha}\\
\psi_E u_{E,\kappa,\alpha} & = & \mu(0,\kappa)\alpha \kappa^{-1}
q^{\kappa}.\end{array}$$
[*Proof*]{}. The first equality is clear. Now, by (\[davidstar\]), if $f^1_{\di}\neq 0$ we have that $$\begin{array}{rcl}
\psi_E u_{E,\kappa,\alpha} & = & \left( \nu(\phip,\dd)\dd+
\frac{\lambda(\phip)}{p} \phip^2+\frac{\lambda(\phip)}{p} \gamma_1
\phip +\lambda(\phip) \gamma_0 \right) l_E(e_E(h))\\
& = & \mu(0,\kappa) \alpha \kappa^{-1} q^{\kappa}\, .
\end{array}$$ A similar argument can be given in the case $f^1_p=0$.
1. For $\kappa \in \bZ\backslash p\bZ$ we have $$\label{pie}
u_{E,\kappa,\zeta^{\kappa} \alpha}(q)=u_{E,\kappa,\alpha}(\zeta q)$$ for all $\zeta \in \bmu(R)$. So if $\alpha=\sum_{i=0}^{\infty} m_i
\zeta_i^{\kappa}$, $\zeta_i \in \bmu(R)$, $m_i \in \bZ$, $v_p(m_i)
\ra \infty$, then $$u_{E,\kappa,\alpha}(q)
= \left[ \sum_{i=0}^{\infty} \right] [m_i](u_{E,\kappa,1}(\zeta_i
q))\, .$$ If $f \in \bZ \bmu(R)\h$ is such that $(f^{[\kappa]})^{\sh}=\alpha \in R$, then $$u_{E,\kappa,\alpha}=f \star u_{E,\kappa,1}\, .$$ Note that $\{u_{E,\kappa,\alpha}\ |\ \alpha \in R\}$ is a $\bZ
\bmu(R)\h$-module (under convolution). This module structure comes from an $R$-module structure (still denoted by $\star$) by base change via the morphism $$\bZ \bmu(R)\h \stackrel{[\kappa]}{\ra}
\bZ \bmu(R)\h \stackrel{\sh}{\ra} R$$ (cf. (\[notinjjj\])), and the $R$-module $\{u_{E,\kappa,\alpha}\, | \;
\alpha \in R\}$ is free, with basis $u_{E,\kappa,1}$. Thus, for $g \in \bZ \bmu(R)\h$, $\beta=(g^{[\kappa]})^{\sh}$, we have that $$\beta \star u_{E,\kappa,\alpha}=g \star u_{E,\kappa,\alpha}\, .$$ In particular $$u_{E,\kappa,\alpha} = \alpha \star u_{E,\kappa,1}\, .$$
2. We recall (see (\[vocea\])) the natural group homomorphisms attached to $\psi_q^1$, $$\begin{array}{ccc}
B_{\kappa}^0:E(R[[q^{\pm 1}]]) & \ra & R \\
B_{\kappa}^0 u & = & \Gamma_{\kappa} \psi^1_q u
\end{array}\, .$$ For $\kappa_1,\kappa_2 \in \bZ \backslash p\bZ$, we obtain that $$\label{blleah} B_{\kappa_1}^0 u_{E,\kappa_2,\alpha}
=\Gamma_{\kappa_1}
u^{\mu}_{a,\kappa_2,\alpha}=
\alpha \cdot \delta_{\kappa_1 \kappa_2}\, .$$
3. Assume that $E$ is defined over $\bZ_p$ and $f^1_{\di}\neq 0$. Then $f^1_{\di} \in \bZ_p$, so $\gamma_0=1$. Also, by the Introduction of [@frob], $\gamma_1 \in \bZ$. Thus, if in addition to $\alpha,
\lambda \in \bZ_{(p)}$ we have that the cubic defining $E$ has coefficients in $\bZ_{(p)}$, then $u_{E,\kappa,\alpha} \in \bZ_{(p)}[[q^{\pm
1}]]$ for $\kappa \in \bZ \backslash p\bZ$.
4. Assume $E$ is defined over $\bZ_p$ and that $f^1_{\di}= 0$. Then the Introduction in [@frob], $\gamma_0$ is an integer in a quadratic extension $F$ of $\bQ$. We view $F$ as embedded into $\bQ_p$, and set $\cO_{(p)}:=F \cap \bZ_p$. Then, if in addition to $\alpha, \lambda
\in \bZ_{(p)}$ we have that the cubic defining $E$ has coefficients in $\bZ_{(p)}$, we obtain that $u_{E,\kappa,\alpha} \in \cO_{(p)}[[q^{\pm
1}]]$ for $\kappa \in \bZ \backslash p\bZ$.
\[bloo\] Let $E$ be an elliptic curve over $R$, $\psi_E$ be a non-degenerate $\D$-character of $E$, and $\cU_*$ be the corresponding groups of solutions. If $\cK$ is the set of characteristic integers of $\psi_E$, and $u_{E,\kappa,\alpha}$ be the basic series, then we have that $$\cU_{\pm 1}=\oplus_{\kappa \in \cK_{\pm}} \{u_{E,\kappa,\alpha}\ |\
\alpha \in R\}\, ,$$ where $\oplus$ stands for the internal direct sum. In particular, $\cU_{\pm 1}$ are free $R$-modules under convolution, with bases $
\{u_{E,\kappa,1}\ |\ \kappa \in \cK_{\pm}\}\, ,
$ respectively.
[*Proof*]{}. By Lemma \[viyne\], $u_{E,\kappa,\alpha} \in
\cU_{1}$. Conversely, if $u \in \cU_{1}$, that is to say, if $u \in
E(qR[[q]])$ and $\psi_E u=0$, we have that $\psi^1_{\ddi} u \in
qR[[q]]$ is a solution of $$\nu(p\phip,\dd)\dd+\lambda(p\phip) p
\phip^2+\lambda(p\phip) \gamma_1 \phip+\lambda(p\phip) \gamma_0\, ,$$ or a solution of $$\nu(p\phip,\dd)\dd+\lambda(p\phip)
\phip+\lambda(p\phip) \gamma_0 \, ,$$ if $f^1_{\di}\neq 0$ or $f^1_{\di}=0$, respectively. Cf. (\[casc\]) and (\[mz23\]), respectively. By Theorem \[addeq\], we have $$\psi^1_{\ddi} u=\sum_{i=1}^s u^{\mu}_{a,\kappa_i,\alpha_i}$$ for some $\alpha_i \in R$, where $\cK_+=\{\kappa_1,\ldots,\kappa_s\}$. Therefore, $$u=e_E \left( \int \sum_{i=1}^s u^{\mu}_{a,\kappa_i,\alpha_i}
\frac{dq}{q} \right)=\sum_{i=1}^s u_{E,\kappa_i,\alpha_i}\, .$$ This representation is unique because of (\[blleah\]).
A similar argument works for $u \in \cU_{-1}$.
Under the hypotheses of Theorem [\[bloo\]]{}, let $u \in
\cU_{\pm}$. Then the following hold:
1. The series $\overline{\psi^1_q u} \in k[[q^{\pm1}]]$ is integral over $k[q^{\pm 1}]$, and the field extension $k(q) \subset
k(q,\overline{\psi^1_q u})$ is Abelian with Galois group killed by $p$.
2. If the characteristic polynomial of $\psi_E$ is unmixed and $\cK_{\pm}$ is short then $u$ is transcendental over $K(q)$.
[*Proof*]{}. Assertion 1 follows immediately by Theorem \[bloo\] and Lemmas \[irinusescoala\] and \[gainuseli\], respectively. In order to check assertion 2, note that by the above results, $\psi^1_q u$ is transcendental over $K(q)$. If $u$ were algebraic over $K(q)$, the point $$(T(q),W(q))=(u,u^3+a_4u^7+\cdots )$$ would have algebraic coordinates over $K(q)$. Hence, the same would be true about the point $$(x(q),y(q))=\left( \frac{T(q)}{W(q)}, -\frac{1}{W(q)} \right)\, ,$$ and therefore, $\psi^1_q u=\dd x(q)/y(q)$ would be algebraic over $K(q)$, a contradiction. Thus, $u$ is transcendental over $K(q)$, and assertion 2 is proved.
\[dacaniciel\] Under the hypotheses of Theorem [\[bloo\]]{}, the maps $B_{\pm}^0:\cU_{\pm 1} \ra R^{\rho_{\pm}}$ are $R$-module isomorphisms. Furthermore, for any $u \in \cU_{\pm 1}$, we have $$u=\sum_{\kappa \in {\mathcal K}_{\pm}} (B_{\kappa}^0 u) \star
u_{E,\kappa,1}\, .$$
In particular the “boundary value problem at $q^{\pm 1}=0$” is well posed.
Next we address the “boundary value problem at $q \neq 0$” for $\psi_E$.
If $E$ is an elliptic curve over $R$, we denote by $E(pR)$ the kernel of the reduction modulo $p$ map $E(R) \ra E(k)$. As usual, the group $E(pR)$ will be identified with $pR$ via the bijection $$\begin{array}{rcl}
pR & \ra & E(pR) \\ pa & \mapsto &
(pa, (pa)^3+a_4(pa)^7+\cdots)
\end{array}\, .$$
We denote by $E'(k)$ the group of all points in $E(k)$ of order prime to $p$. And we denote by $E'(R)$ the subgroup of all points in $E(R)$ whose image in $E(k)$ lie in $E'(k)$. There is a split exact sequence $0 \ra E(pR) \ra E'(R) \ra E'(k) \ra 0$, hence $E'(R)=E(pR) \oplus E'(R)_{tors}$.
\[haicasiel\] Under the hypotheses of Theorem [\[bloo\]]{}, and letting $$\cU'_+:=E'(R)_{tors} \cdot \cU_{1} \subset \cU_+\, ,$$ the following hold:
1. Assume $\cK_+=\{\kappa\}$. For any $q_0 \in p^{\nu}R^{\times}$ with $\nu \geq 1$, and any $g \in p^{\nu \kappa}R \subset
pR=E(pR)$ there exists a unique $u \in \cU_{1}$ such that $u(q_0)=g$.
2. Assume $\cK_+=\{1\}$. Then for any $q_0 \in pR^{\times}$ and any $g \in E'(R)$ there exists a unique $u \in \cU'_+$ such that $u(q_0)=g$.
[*Proof*]{}. The first assertion follows exactly as in the case of $\bG_m$; cf. Corollary \[maisus\].
In order to check the second assertion, note that we can write $g$ uniquely as $g=g_{1}+g_{2}$, where $g_{1} \in
E'(R)_{tors}$ and $g_{2} \in E(pR)$. By the first part, there exists $u_2 \in \cU_{1}$ such that $u_2(q_0)=g_{2}$. We set $u=g_{1}+u_2$. Then, $u(q_0)=g$.
The uniqueness of $u$ is also checked easily.
The following Corollary is concerned with the inhomogeneous equation $\psi_E u=\varphi$, and it is an immediate consequence of Lemmas \[viyne\], \[dacaniciel\], and \[irinusescoala\].
Let $\psi_E$ be a non-degenerate $\D$-character of $E$, and let $\varphi \in q^{\pm 1}R[[q^{\pm 1}]]$ be a series whose support is contained in the set $\cK'$ of totally non-characteristic integers of $\psi_E$. Then the following hold:
1. The equation $\psi_E u=\varphi$ has a unique solution $u \in E(q^{\pm
1}R[[q^{\pm 1}]])$ such that $\psi^1_q u$ has support disjoint from the set $\cK$ of characteristic integers.
2. If $\bar{\varphi} \in k[q^{\pm 1}]$ the series $\overline{\psi^1_q u}
\in k[[q^{\pm 1}]]$ is integral over $k[q^{\pm 1}]$ and the field extension $k(q) \subset k(q,\overline{\psi^1_q u})$ is Abelian with Galois group killed by $p$.
3. If the characteristic polynomial of $\psi_E$ is unmixed and the support of $\varphi$ is short then $u$ is transcendental over $K(q)$.
Corollary \[haicasiel\] implies that if $\cK_+=\{1\}$ and $q_0 \in
pR^{\times}$, the group homomorphism $$\begin{array}{rcl}
S_{q_0}:E'(R)_{tors} \times R & \ra & E'(R) \\
(P,\alpha) & \mapsto & P+u_{E,1,\alpha}(q_0)
\end{array}$$ is an isomorphism. So for any $q_1, q_2
\in pR^{\times}$, we have an isomorphism $$S_{q_1, q_2}:=S_{q_2} \circ S_{q_1}^{-1}:E'(R) \ra E'(R)\, .$$ The latter map should be viewed as the “propagator” attached to $\psi_E$.
As in the case of $\bG_a$ and $\bG_m$, respectively, if $\zeta \in
\bmu(R)$ and $q_0 \in p R^{\times}$, then, by (\[pie\]), $$S_{\zeta q_0}=S_{q_0} \circ M_{\zeta} \\$$ where $$\begin{array}{rcl}
M_{\zeta}:E'(R)_{tors} \times R & \ra & E'(R)_{tors} \times R \\
M_{\zeta}(P,\alpha) & := & (P,\zeta \alpha)
\end{array}$$ Hence, for $\zeta_1, \zeta_2 \in \bmu(R)$, we obtain that $$S_{\zeta_1 q_0,\zeta_2 q_0}=S_{q_0} \circ M_{\zeta_2/\zeta_1}
\circ S_{q_0}^{-1}\, .$$ In particular, $$S_{q_0,\zeta_1 \zeta_2 q_0}=S_{q_0,\zeta_2 q_0} \circ
S_{q_0,\zeta_1 q_0}\, ,$$ which can be interpreted as a (weak) “Huygens principle.”
[AAA]{}
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---
abstract: 'In this paper, the benefits of distributed energy resources (DERs) are considered in an energy management scheme for a smart community consisting of a large number of residential units (RUs) and a shared facility controller (SFC). A non-cooperative Stackelberg game between RUs and the SFC is proposed in order to explore how both entities can benefit, in terms of achieved utility and minimizing total cost respectively, from their energy trading with each other and the grid. From the properties of the game, it is shown that the maximum benefit to the SFC in terms of reduction in total cost is obtained at the unique and strategy proof Stackelberg equilibrium (SE). It is further shown that the SE is guaranteed to be reached by the SFC and RUs by executing the proposed algorithm in a distributed fashion, where participating RUs comply with their best strategies in response to the action chosen by the SFC. In addition, a charging-discharging scheme is introduced for the SFC’s storage device (SD) that can further lower the SFC’s total cost if the proposed game is implemented. Numerical experiments confirm the effectiveness of the proposed scheme.'
author:
- 'Wayes Tushar, Bo Chai, Chau Yuen, David B. Smith, Kristin L. Wood, Zaiyue Yang, and H. Vincent Poor, [^1] [^2] [^3] [^4] [^5] [^6]'
title: 'Three-Party Energy Management With Distributed Energy Resources in Smart Grid'
---
Smart grid, shared facility, Stackelberg game, energy management, distributed energy resources.
Introduction {#sec:introduction}
============
There has been an increasing interest in deploying distributed energy resources (DERs) because of their ability to reduce greenhouse gas emissions and alleviate global warming [@Georgilakis-JTPS:2013]. Moreover, DERs can assist consumers in reducing their dependence on the main electricity grid as their primary source of energy [@Tham-JTSMCS:2013], and consequently can lower their cost of electricity purchase. The smart grid with enhanced communication and sensing capabilities [@Sauter-TIE:2011] offers a suitable platform for exploiting the use of DERs to assist different energy entities in effectively managing their energy with reduced dependence on the main grid.
Most literature on energy management, as we will see in the next section, has considered scenarios where users with DERs are also equipped with a storage device (SD)[@Justo-J-RSER:2013; @Guerrero-JTIE:2013; @Guerrero-JTIE:2013-2]. However, in some cases it is also likely that the users are not interested in storing energy for future use, due to the start-up and required size of a storage device. Rather, they are more concerned with consuming/trading energy as soon as it is generated, e.g., grid-integrated solar without a battery back up system [@wt_battery_solar:2013]. Nevertheless, little has been done to study this kind of system. In fact, one key challenge to exploit the real benefit of using DERs in such settings is to develop appropriate system models and protocols, which are not only feasible in real world environments but, at the same time, also beneficial for the associated energy users in terms of their derived cost-benefit tradeoff. Such development further enables understanding the in-depth properties of the system and facilitates the design of suitable real time platforms for next generation power system control functions [@Westermann-TIE:2010].
To this end, we propose an energy management scheme in this paper for a smart community consisting of multiple residential units (RUs) with DERs and a shared facility controller[^7] (SFC) using non-cooperative game theory [@Ekneligoda-JTIE:2014]. To the best of our knowledge, ours is the first work to introduce the concept of a shared facility and consider a three-party energy management problem in smart grid applications. With the development of modern residential communities, shared facilities provide essential public services to the RUs, e.g., maintenance of lifts in community apartments. Hence, it is necessary to study the energy demand management of a shared facility for the benefit of the community as a whole. This is particularly necessary in the considered setting where each RU has DERs that can trade energy with both the grid and the SFC, and constitutes an important energy management problem, as we will see later, for both the SFC and RUs. Here, on the one hand, to obtain revenue, each RU would be interested in selling its energy either to the SFC or to the grid based on the prices offered by them, i.e., sell to the party with the higher price. On the other hand, the SFC wants to minimize its cost of energy purchase from the grid by making an offer to RUs such that the RUs would be more encouraged to sell their energy to the SFC instead of the grid. Thus, the SFC would need to buy less energy at a higher price from the grid. Because of the different properties and objectives of each party, the problem is more likely to handle heterogeneous customers than homogeneous ones.
Due to the heterogeneity of the distributed nodes in the system, and considering the independent decision making capabilities of the SFC and RUs, we are motivated to use a Stackelberg game [@Book_Dynamicgame-Basar:1999] to design their behavior. Distinctively, we develop a distributed protocol for the SFC, which is the leader of the game, to determine the buying price from the RUs, such that its total cost of buying energy from the grid and RUs is minimized. Meanwhile, we also show how the followers, i.e., RUs without storage facilities, react in response to the buying price set by the SFC to optimize their payoffs. We extend the study by considering the case when the SFC possesses an SD[^8], and further propose a charging-discharging scheme for the SFC that can be implemented in-line with the proposed Stackelberg game.
To this end, the main contributions of the paper are as follows: 1) A system model is proposed to facilitate energy management for the SFC and RUs in the community. Novel cost and utility models are proposed to achieve a good balance between reflecting practical requirements and providing mathematical tractability; 2) A non-cooperative Stackelberg game is proposed to capture the interaction between the SFC and the RUs. The proposed game requires limited communication between the SFC and each RU to solve the energy management problem in a decentralized fashion; 3) The properties of the game are analyzed, and the existence of a unique and strategy-proof solution is proven; 4) An algorithm is proposed that is guaranteed to reach the Stackelberg equilibrium, which can be adopted by the SFC and the RUs in a distributed fashion; and 5) A charging-discharging strategy is proposed for the SFC’s storage device based on the price offered by the main grid. The introduced strategy can be implemented, along with the proposed Stackelberg game in each time slot, to further improve the SFC’s benefit in terms of its total cost of energy purchase during a day.
The remainder of the paper is organized as follows. We discuss the state-of-the art of energy management research using DERs in Section \[sec:state-of-the-art\] followed by a description of the considered system model in Section \[sec:system-model\]. The energy management problem is formulated as a Stackelberg game in Section \[sec:problem-formulation\], where we also analyze the properties of the game and design a distributed algorithm. In Section \[sec:with-storage\], the proposed scheme is extended to the case where the SFC possesses an SD. Numerical examples are discussed in Section \[sec:case-study\], and some concluding remarks are contained in Section \[sec:conclusion\].
State-of-The Art {#sec:state-of-the-art}
================
Recently, there has been considerable research effort to understand the potential of DERs in smart grid [@Vasques-TIE:2010]. This is mainly due to their capability in reducing greenhouse gas emissions, as well as lowering the cost of electricity [@Georgilakis-JTPS:2013]. This literature can be divided into two general categories, where work such as [@Justo-J-RSER:2013; @Guerrero-JTIE:2013; @Guerrero-TIE:2011] and [@Guerrero-TIE:2009] that has studied the feasibility and controls of integrating DERs in smart grid is in the first category. In [@Justo-J-RSER:2013], a comprehensive literature review is provided discussing the connection and controls of AC and DC microgrid systems with DERs and energy storage systems. In [@Guerrero-JTIE:2013] advanced control techniques are studied, including decentralized and hierarchical controls for microgrids with distributed generation. A three-level hierarchical control process and electrical dispatching standards are presented in [@Guerrero-TIE:2011] with a view to integrating DERs with distributed storage systems in smart grid. A control scheme, using a droop control function for managing battery levels of domestic photovoltaic-uninterruptable power supplies (PV-UPS), is proposed in [@Guerrero-TIE:2009]. Other control schemes for efficient use of DERs in smart grid can be found in [@Georgilakis-JTPS:2013; @Hill-JTSG:2012], [@Liu-STSP:2014], [@Naveed-Energies:2013], [@Balaguer-TIE:2011] and [@Liu-ISGT:2013].
The second category of work in this area comprises various energy management/scheduling schemes that have exploited the use of DERs in smart grid. For instance, the authors in [@Zhang-J_ECM:2013] study an efficient energy consumption and operation management scheme for a smart building to reduce energy expenses and gas emissions by utilizing DERs. To provide flexibility to distribution system operators, a deterministic energy management scheme is designed in [@Kanchev-TIE:2011] for PV generators with embedded storage. An interesting smart grid management system is explored in [@Cecati-TIE:2011] that uses DERs to minimize the cost of power delivery including the cost of distributed generators, the cost of power provided by the primary substation, and the cost associated with grid power losses while delivering power to the consumers. In order to minimize the operational cost of renewable integration to distributed generation systems, a forecast based optimization scheme is developed in [@Chakraborty-TIE:2007]. Saber *et al*. propose a scheduling and controlling scheme for electric vehicles batteries in [@Saber-TIE:2011] so that batteries can be used and integrated with DERs for reducing emissions from electricity production. Further studies of optimization and scheduling techniques that exploit the use of DERs are available in [@Ramachandran-TIE:2011] and [@Angelis-TII:2013].
As can be seen from the above discussion, the scope of research on the use of DERs in smart grid is not limited to power and energy research communities such as in [@Georgilakis-JTPS:2013; @Tham-JTSMCS:2013; @Justo-J-RSER:2013] and [@Zhang-J_ECM:2013], but also extends to other research communities including those in smart grid [@Hill-JTSG:2012; @Fang-J-CST:2012], and *industrial electronics* (IE) [@Guerrero-JTIE:2013; @Vasques-TIE:2010; @Guerrero-TIE:2011; @Guerrero-TIE:2009; @Balaguer-TIE:2011; @Kanchev-TIE:2011; @Cecati-TIE:2011; @Chakraborty-TIE:2007; @Saber-TIE:2011; @Ramachandran-TIE:2011; @Angelis-TII:2013; @Yu-TIE:2011]. However, the majority of these research papers have considered the case in which all the entities with DERs also possess SDs. But this might not always be the case as we have argued in Section \[sec:introduction\]. In this regard, unlike the discussed literature, this paper investigates the case in which entities having DERs do not have SDs, by introducing the SFC. We use a noncooperative Stackelberg game to model the energy management scheme considering the distributed and rational nature of the nodes in the smart grid system, and thus complement the discussed previous work in the topic area. The work here has the potential to open new research opportunities for the IE and smart grid communities in terms of control of energy dispatch, size of storage devices and determination of suitable location and size of DERs that might support both the SFC and RUs to further attain different operational objectives in smart grid networks.
We stress that recent work has shown Stackelberg games to be very effective and suitable for designing energy management schemes. For example, in [@Maharjan-JTSG:2013], Maharjan *et al.* propose a Stackelberg game between multiple utility companies and consumers to maximize both the revenue of each utility company and the pay-off to each user. A Stackelberg game approach, using a genetic algorithm to obtain the Stackelberg solution, to maximize the profit of a electricity retailer and to minimize the payment bills of its customers, is proposed in [@Meng-JSpringer:2013]. A consumer-centric energy management scheme for smart grids is proposed in [@Tushar-TSG:2013] that prioritizes consumers’ benefits by reforming their energy trading with a central power station whereby the consumers receive their socially optimal benefits at the Stackelberg equilibrium. A four-stage Stackelberg game is studied in [@Bu-JTETC:2013], and analytical results are obtained via a backward induction process for electricity retailers using real-time pricing. The same authors also study the dynamics of the smart grid in designing green wireless cellular networks in [@Bu-JTWC:2012] using a similar game formulation. A bi-level programming technique is used in [@Asimakopoulou-JTSG:2013] to design a Stackelberg game for energy management of multiple micro-grids. However, we remark that the players and their respective strategies in games significantly differ from one game to another according to the system models, design objectives and algorithms that are used. To this end, we propose a suitable system model in the next section, which can facilitate the considered energy management between the SFC, RUs and the grid through a Stackelberg game.
System Model {#sec:system-model}
============
![System model for energy management in a smart community consisting of residential units, main power grid and a shared facility controller.[]{data-label="fig:system_model"}](systemmodel){width="0.8\columnwidth"}
Consider a smart community consisting of a large number of RUs and an SFC. The SFC controls the electricity for equipment and machines such as lifts, water pumps, parking lot gates and lights, which are shared and used on a daily basis by the residents of the community. Here, on the one hand, the SFC does not have any electricity generation capability and, hence, needs to buy all its energy either from the main grid or from the RUs in the network. On the other hand, each RU is assumed to have a DER without any SD that is capable of generating its own energy. An RU can be a single residential unit or a group of units, connected via an aggregator, which can act as a single entity. We assume that each RU can decide on the amount of electricity that it wants to consume, and hence the excess energy, if there is any, that it wants sell to the SFC or to the main grid for making revenue. All RUs and the SFC are assumed to be connected to one another and to the main grid by means of power and communication lines. A schematic diagram of this system is given in Fig. \[fig:system\_model\].
To this end, let us assume that there are $N$ RUs in the community and they belong to the set $\mathcal{N}$. Each RU $n\in\mathcal{N}$ is equipped with DERs, e.g., solar panels or wind turbines (or both), that can generate energy $E_n^\text{gen}$ at certain times during the day. We assume that each RU wants to manage its consumption $e_n$ such that it can sell the remainder of its generated energy $(E_n^\text{gen} - e_n)$ to the SFC or the grid to make revenue. Clearly, if $E_n^\text{gen} \leq E_n^\text{min}$, where $E_n^\text{min}$ is the essential load for RU $n$, the RU cannot take part in the energy management program as it cannot afford to sell any energy. Otherwise, as for the considered case, the RU adjusts its energy consumption $e_n, \text{s.t.,} e_n\geq E_n^\text{min}$, for its own use, and thus sells the remainder $(E_n^\text{gen}-e_n)$ to the SFC or to the main grid.
![Effect of the change of price per unit of energy paid by the SFC to each RU on the change of maximum utility that each RU receives from its energy consumption.[]{data-label="fig:effect-p-utility"}](changeofutility){width="\columnwidth"}
In general, the buying price $p_g^b$ set by the grid is considerably lower than its selling price $p_g^s$ [@McKenna-JIET:2013]. In this regard, we assume that the price per unit of energy that the SFC pays to each RU is set between the buying and selling price of the grid. Therefore, each RU can sell at a higher price, and the SFC can buy at a lower price and they trade energy with each other instead of trading with the main grid. Under such a setting, it is reasonable to assume that all RUs would be more interested in selling $(E_n^\text{gen}-e_n)~\forall n$ to the SFC instead of the grid. Now, let us assume that the SFC sets a price $p_\text{sf}^s$ per unit of energy to pay to each RU for buying its required energy $E_\text{sf}^\text{req}$. To this end, we propose that the total utility achieved by RU $n$ from its energy consumption $e_n$ and from its trading of energy $(E_n^\text{gen}-e_n)$ with the SFC is given by $$\begin{aligned}
U_n = k_n\ln(1 + e_n) + p_\text{sf}^s\left(E_n^\text{gen} - e_n\right),~k_n>0.\label{eqn:utility-ru}\end{aligned}$$ In , $k_n\ln(1 + e_n)$ is the utility that the RU $n$ achieves from consuming energy $e_n$, where $k_n$ is a preference parameter [@Wayes-J-TSG:2012; @Samadi-C-Smartgridcomm:2010]. It is clear from that an RU with higher $k_n$ would be more interested in consuming more $e_n$ to attain its maximum utility level compared to an RU with lower preference. $p_\text{sf}^s\left(E_n^\text{gen} - e_n\right)$ is the revenue that the RU receives from selling the rest of its energy to the SFC. We note that the natural logarithm $\ln(\cdot)$ has been extensively used for designing the utility [@Pavlidou-JCN:2008], and has also recently been shown to be suitable for designing the utility for power consumers [@Maharjan-JTSG:2013]. We can see that is a concave function and its relationship with $p_\text{sf}^s$ is shown in Fig. \[fig:effect-p-utility\]. As shown in Fig. \[fig:effect-p-utility\], as $p_\text{sf}^s$ increases, the maximum utility of the RU shifts towards the left. That is the RU tends to sell more energy to the SFC, e.g., by scheduling the use of its flexible devices [@Zhang-J_ECM:2013] to a later time, and thus becomes more interested in making further revenue.
By contrast, the SFC, having no generation capability, needs to buy all of its required energy from RUs and the main grid. In typical cases, the main grid sells energy at a higher price compared to the price from owners of renewable energy sources as for, e.g., feed-in tariff schemes [@McKenna-JIET:2013]. Hence, it is reasonable to assume that the SFC would mainly be interested in buying energy from the RUs at $p_\text{sf}^s$ to meet its requirement, and procuring the rest, if there is any, from the main grid. Nonetheless, if $p_\text{sf}^s$ is too low, a RU would sell less, or no, energy to the SFC and consume more for its own purposes instead. For example, the resident may want to start washing clothes rather than schedule it at a later time, and thus use their energy instead of selling it at a very low price. Consequently, the SFC will have to buy a larger fraction of its requirement at a higher price from the main grid. Conversely, if $p_\text{sf}^s$ is too high, e.g., close to the grid’s selling price $p_g^s$, it would cost the SFC significantly. Hence, the choice of $p_\text{sf}^s$ should be within a *reasonable* range to encourage the RUs to sell their energy to the SFC, but at the same time keeping the cost to the SFC at a minimum. However, if the energy from RUs is not enough to meet its requirement, the SFC needs to buy the remainder from the main grid with the price $p_g^s$. In this regard, we define a cost function to capture the total cost to the SFC for buying energy from RUs and the grid as follows: $$\begin{aligned}
C_\text{sf} = \sum_n e_{n,\text{sf}}^s p_{\text{sf}}^s + \left(E_\text{sf}^\text{req} - \sum_n e_{n,\text{sf}}^s\right)p_g^s,
\label{eqn:cost-sfc}\end{aligned}$$ where $e_{n,^\text{sf}}^s$ is the amount of energy that the SFC buys from the RU $n$. In , the first term captures the total cost of buying energy from the RUs. Meanwhile, the second term not only describes the cost of buying energy from the grid, but also satisfies the constraint on the demand of total required energy by the SFC, i.e., a SFC does not buy more than it requires.
Now, to decide on energy trading parameters $e_{n}$ and $p_\text{sf}^s$, on the one hand, the SFC interacts with each RU $n\in\mathcal{N}$ to minimize by choosing a suitable price to pay to each RU. On the other hand, each RU decides on the amount of energy it wants to sell to the SFC by controlling its energy consumption $e_n$ so as to maximize . To this end, we design the interaction and energy trading behavior of each energy entity in the next section.
Energy Management Between RUs and the SFC via Game Theory {#sec:problem-formulation}
=========================================================
Objective of the RU {#sec:objective-ru}
-------------------
First we note that and are coupled through $E_n^\text{gen}, p_\text{sf}^s$ and $e_n$. Since the RUs do not have any storage capacity, each RU would desire to sell all its excess energy, $$\begin{aligned}
e_{n,\text{sf}}^s = E_n^\text{gen} - e_n.\label{eqn:sell-with-generation}\end{aligned}$$ at a suitable price $p_\text{sf}^s$ to the SFC after adjusting for their consumption $e_n$. To that end, the objective of each RU $n$ can be defined as $$\begin{aligned}
\max_{e_n}~ U_n,\nonumber\\\text{s.t.,}~e_n\geq E_n^\text{min}.\label{eqn:obj-ru}\end{aligned}$$ Now from and , the first-order-differential condition for maximum utility is $$\begin{aligned}
\frac{k_n}{1 + e_n} - p_{\text{sf}}^s = 0,\end{aligned}$$ and hence $$\begin{aligned}
e_n = \frac{k_n}{p_\text{sf}^s} - 1,\label{eqn:relation-en-pn}\end{aligned}$$ which clearly relates the decision making process of each RU to the price set by the SFC. Here, $k_n$ should be sufficiently large such that always possesses a positive value for all resulting values of $e_n$ and $p_\text{sf}^s$, and $e_n$ is at least as large as its essential load. From , the amount of energy $e_n$ chosen to be consumed by each RU is inversely proportional to the price per unit of energy paid by the SFC to the RU. As a result, for a higher $p_\text{sf}^s$, the RU $n$ would be more inclined to sell to the SFC by reducing its consumption and vice-versa.
Objective of the SFC {#sec:objective-sfc}
--------------------
In contrast, the objective of the SFC is to minimize its total cost of buying energy. Since the SFC does not have any control over the pricing of the grid, it can only set its own buying price $p_\text{sf}^s$ to minimize . Hence, the objective of the SFC is $$\begin{aligned}
\min_{p_\text{sf}^s}C_\text{sf}.
\label{eqn:obj-sfc}\end{aligned}$$ Now, from the first order optimality condition of the SFC’s objective function , $$\begin{aligned}
\frac{\delta C_\text{sf}}{\delta p_\text{sf}^s} = 0.\label{eqn:cond-sfc-1}\end{aligned}$$ By replacing $e_{n,\text{sf}}^s$ in with $(E_n^\text{gen} - e_n)$, and considering the relationship between $e_n$ and $p_\text{sf}^s$ from , we obtain $$\begin{aligned}
\frac{\delta}{\delta p_\text{sf}^s}\Bigg(\sum_n(E_n^\text{gen} &-& k_n + p_\text{sf}^s) + E_\text{sf}^\text{req}p_g^s\nonumber\\ &-& p_g^s\sum_n\left(E_n^\text{gen} - \frac{k_n}{p_\text{sf}^s} + 1\right)\Bigg) = 0.\label{eqn:cond-sfc-2}\end{aligned}$$ And from , we derive $$\begin{aligned}
p_\text{sf}^s = \sqrt{\frac{p_g^s\sum_n k_n}{N + \sum_n E_n^\text{gen}}},\label{eqn:price-relation}\end{aligned}$$ which emphasizes that the optimal price set by the SFC is influenced by the total number of RUs that wish to sell their energy and the generation of their DERs during the considered time. It is also established from that $p_\text{sf}^s$ is affected by the grid’s price, which consequently influences the SFC to change its per unit price for the RUs. However, as discussed in Section \[sec:system-model\], to encourage the RUs to always sell their excess energy to the SFC we propose that the choice of price by the SFC is $$\begin{aligned}
p_\text{sf}^s = \begin{cases}
\sqrt{\frac{p_g^s\sum_n k_n}{N + \sum_n E_n^\text{gen}}}, & \text{if $p_\text{sf}^s>p_g^b$}\\
p_g^b + \alpha, & \text{otherwise}.
\end{cases}
\label{eqn:price-relation-2}\end{aligned}$$ Here, $\alpha>0$ is a small value to keep $p_\text{sf}^s$ higher than $p_g^b$. It is obvious from that the SFC can optimize its price in a centralized fashion to minimize its total cost of purchasing energy from the RUs and the grid, if it has full access to the private information of each RU $n$, such as $E_n^\text{gen}$ and $k_n$ . However, in reality, it might not be possible for the SFC to access this information in order to protect the users’ privacy, and hence a distributed mechanism is necessary to determine the parameters $p_\text{sf}^s$ and $e_n,~\forall n$. To that end, we propose a scheme based on a non-cooperative Stackelberg game in the following section.
Non-cooperative Stackelberg game {#sec:stackelberg-game}
--------------------------------
A Stackelberg game formally studies the multi-level decision making processes of a number of *independent* decision makers (i.e., followers) in response to the decision taken by the *leading* player (leader) of the game [@Book_Dynamicgame-Basar:1999]. In this section, we formulate a non-cooperative Stackelberg game, where the SFC is the leader, and RUs are the followers, to capture the interaction between the SFC and the RUs. The game is formally defined by its strategic form as $$\begin{aligned}
\Gamma = \{(\mathcal{N}\cup\{\text{SFC}\}), \{\mathbf{E}_n\}_{n\in\mathcal{N}}, \{U_n\}_{n\in\mathcal{N}}, p_\text{sf}^s, C_\text{sf}\},
\label{eqn:definition-game}\end{aligned}$$ which consists of the following components:
i) The RUs in set $\mathcal{N}$ act as followers and choose their strategies in response to the price set by the SFC, i.e., the leader of the game.
ii) $\mathbf{E}_n$ is the set of strategies of each RU $n\in\mathcal{N}$ from which it selects its strategy, i.e., the amount of energy $e_n\in\mathbf{E}_n$ to be consumed during the game.
iii) $U_n$ is the *utility function* of each RU $n$ as explained in that captures the RU’s benefit from consuming energy $e_n$ and selling energy $(E_n^\text{gen} - e_n)$ to the SFC.
iv) $p_\text{sf}^s$ is the price set by the SFC to buy energy from the RUs.
v) The *cost function* $C_\text{sf}$ of the SFC captures the total cost incurred by the SFC for trading energy with RUs and the main grid.
As discussed previously, the objectives of each RU and the SFC are to maximize the utility in and to minimize the cost in respectively by their chosen strategies. For this purpose, one suitable solution for the proposed game is the Stackelberg equilibrium (SE) at which the leader obtains its optimal price given the followers’ best responses. At this equilibrium, neither the leader nor any follower can benefit, in terms of total cost and utility respectively, by *unilaterally* changing their strategy.
Consider the game $\Gamma$ defined in , where $U_n$ and $C_\text{\emph{sf}}$ are determined by and respectively. A set of strategies $\left(\mathbf{e}^{*}, p_\text{sf}^{s*}\right)$ constitutes an SE of this game, if and only if it satisfies the following set of inequalities: $$\begin{aligned}
U_n(\mathbf{e}^{*}, p_\text{\emph{sf}}^{s*})\geq U_n(e_n,\mathbf{e}_{-n}^{*}, p_\text{\emph{sf}}^{s*}),~\forall n\in\mathcal{N},~\forall e_n\in\mathbf{E}_n,\label{eqn:definition-1}\end{aligned}$$ and $$\begin{aligned}
C_\text{\emph{sf}}(\mathbf{e}^{*}, p_\text{\emph{sf}}^{s*})\leq C_\text{\emph{sf}}(\mathbf{e}^{*}, p_\text{\emph{sf}}^s),\label{eqn:definition-2}\end{aligned}$$ where $\mathbf{e}_{-n}^{*} = \left[e_{1}^{*}, e_{2}^{*},\hdots, e_{n-1}^{*}, e_{n+1}^{*},\hdots, e_{N}^{*}\right]$ and $\mathbf{e}^{*} = \left[e_n^*, \mathbf{e}_{-n}^*\right]$. \[definition:1\]
Therefore, when all the players in $\left(\mathcal{N}\cup\{\text{SFC}\}\right)$ are at an SE, the SFC cannot reduce its cost by reducing its price from the SE price $p_\text{sf}^{s*}$, and similarly, no RU $n$ can improve its utility by choosing a different energy to $e_n^*$ for consumption.
Existence and Uniqueness of SE {#sec:existence-SE}
------------------------------
In non-cooperative games, an equilibrium in pure strategies is not always guaranteed to exist [@Book_Dynamicgame-Basar:1999]. Therefore, we need to investigate as to whether there exists an SE in the proposed Stackelberg game.
A unique SE always exists in the proposed Stackelberg game $\Gamma$ between the SFC and RUs in the set $\mathcal{N}$. \[theorem:1\]
First, we note that the utility function $U_n$ in is strictly concave with respect to $e_n~\forall n\in\mathcal{N}$, i.e., $\frac{\delta^2 U_n}{\delta e_n^2}<0$, and hence for any price $p_\text{sf}^s>0$, each RU $n$ will have a unique $e_n$, chosen from a bounded range $\left[E^\text{min}_n, E_n^\text{gen}\right]$, that maximizes $U_n$. We also note that the game $\Gamma$ reaches the SE when all the players in the game, including each participating RU and the SFC, have their optimized payoff and cost respectively, considering the strategies chosen by all players in the game. Thereby, it is evident that the proposed game $\Gamma$ reaches an SE as soon as the SFC is able to find an optimized price $p_\text{sf}^{s*}$ while the RUs choose their unique energy vector $\mathbf{e}^{*}$.
Now from , given the choices of energy by each RU $n$ in the network, the second derivative of $C_\text{sf}$ is $$\begin{aligned}
\frac{\delta^2 C_\text{sf}}{\delta p_\text{sf}^{s^2}} = \frac{2\sum_n k_n}{(p_\text{sf}^s)^3}>0,
\end{aligned}$$ and therefore, $C_\text{sf}$ is strictly convex with respect to $p_\text{sf}^s$. Hence, the SFC would be able to find an optimal unique per-unit price $p_\text{sf}^{s*}$ for buying its energy from the RUs based on their strategies. Therefore, there exists a unique SE in the proposed game, and thus Theorem \[theorem:1\] is proved.
In the next section, we propose an algorithm that all the RUs and the SFC can implement in a distributed fashion to reach the unique SE. We note that it is also possible to solve the energy management problem in a centralized fashion if we have global information such as $E_n^\text{gen}$ and $k_n$ available at the SFC. However, in order to protect the privacy of each RU and also to reduce the demand on communications bandwidth, a distributed algorithm is desired where the optimization can be performed by each RU and the SFC without the need for any private information to be available at the SFC.
Distributed Algorithm {#sec:algorithm}
---------------------
Initialization: $p_\text{sf}^{s*}=0$ $C_\text{sf}^*=p_g^s*E_\text{sf}^\text{req}$ RU $n$ adjusts its energy consumption $e_n$ according to $$\label{eqn:alg-1}
e_n^* = {\rm{arg}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop {\max }\limits_{0 \le {e_n} \le E_n^\text{gen}} {\kern 1pt} {\kern 1pt} {\kern 1pt} [{k_n}\ln (1 + {e_n}) + p_\text{sf}^s(E_n^\text{gen} - {e_n})].$$\
The SFC computes the cost according to $$\label{eqn:alg-2}
{C_\text{sf}} = p_\text{sf}^s\sum\limits_{n \in \mathcal{N}} {\left(E_n^\text{gen} - {e_n}\right)} + p_g^s\left(E_\text{sf}^\text{req} -
\sum\limits_{n \in \mathcal{N}} {\left(E_n^\text{gen} - {e_n}\right)} \right).$$\
The SFC records the optimal price and minimum cost $$\label{eqn:alg-3}
p_\text{sf}^{s*}=p_\text{sf}^{s}, C_\text{sf}^*=C_\text{sf}$$\
**The SE $(\mathbf{e}^*, p_\text{sf}^{s*})$ is achieved.**
In order to attain the SE, the SFC needs to communicate with each RU. We propose an algorithm that all the RUs and the SFC can implement in a distributed fashion to iteratively reach the unique SE of the proposed game. In each iteration, firstly the RU $n$ chooses its best energy consumption amount $e_n$ in response to the price set by the SFC, calculating $e_{n,\text{sf}}^s = (E_n^\text{gen} - e_n)$, and sending this information to the SFC. Secondly, having the information about the choices of energy $\mathbf{e}_\text{sf}^s = [e_{1,\text{sf}}^s, e_{2,\text{sf}}^s,\hdots, e_{N,\text{sf}}^s]$ by all RUs, the SFC decides on its best price that minimizes its total cost according to . The interaction continues until the conditions in and are satisfied, and therefore the Stackelberg game reaches the SE. Details are given in Algorithm \[alg:1\].
In the proposed algorithm, the conflict between the RUs’ choices of strategies stem from their impact on the choice of $p_\text{sf}^s$ by the SFC. Due to the strict convexity of $C_\text{sf}$, the choice of $p_\text{sf}^{s*}>0$ lowers the cost of the SFC to the minimum. Now, as the algorithm is designed, in response to the $p_\text{sf}^{s*}$, each RU $n$ chooses its strategy $e_n$ from the bounded range $\left[E_n^\text{min}, E_n^\text{gen}\right]$ to maximize its concave utility function $U_n$. Hence, due to the bounded strategy set and the continuity of the utility function $U_n$ with respect to $e_n$, each RU $n$ also reaches a fixed point at which its utility is maximized for the given price $p_\text{sf}^{s*}$ [@Maharjan-JTSG:2013]. As a consequence, the proposed algorithm is always guaranteed to converge to the unique SE of the game.
### Strategy-Proof Property {#sec:strategy-proof}
Since, each RU plays its best response in Algorithm \[alg:1\], it is important to investigate whether RUs can choose a different strategy or cheat other players in $\Gamma$ once they reach the SE. In this regard, we would like see whether it is possible for an RU to change the amount of energy that it offers to the SFC, i.e., $e_{n,\text{sf}}^s = (E_n^\text{gen}-e_n)$ by changing the energy consumption $e_n$ while using Algorithm \[alg:1\].
It is not possible for any RU $n\in\mathcal{N}$ to be untruthful about its strategy, i.e., sell more or less than what it promises to give the SFC, when all other players including the SFC and the RUs in $\mathcal{N}/\{n\}$ are adopting Algorithm \[alg:1\]. \[theorem:2\]
To prove Theorem \[theorem:2\], first we consider that $p_\text{sf}^{s*}$ and $\mathbf{e}^{*} = \left[e_1^*,\hdots,e_n^*,\hdots, e_N^*\right]$ are the SE solutions of the proposed game obtained via Algorithm \[alg:1\]. Let us assume that RU $n$ is untruthful, and chooses $e_n^{'}$ instead of $e_n^{*}$ to consume after reaching the SE. Therefore, from , $$\begin{aligned}
e_n^{'} = \frac{k_n}{p_\text{sf}^{s*}} - 1,\label{eqn:strategy-proof}\end{aligned}$$ which is impossible. This is due to the fact that, as the scheme is formulated, $p_\text{sf}^{s*}$ results from Algorithm \[alg:1\] only if all the RUs in $\mathcal{N}$ consume the SE amount of energy $e_n^{*}~\forall n$. In this regard, is only true if $e_n^{'} = e_n^{*}$, which successively proves the *strategy proof* property of the proposed algorithm.
Energy Management with Storage {#sec:with-storage}
==============================
We note that the proposed scheme in Section III determines the best price for the SFC to minimize its total cost of energy purchase at any given time. The scheme also benefits the RUs in terms of their energy consumption and trading with the SFC. However, DERs do not provide a stable supply. Sometimes there could be an abundant supply of energy whereas at other times there could be a scarcity. In other words, sometimes the SFC might need to buy less energy from the grid whereas at other times it might need to buy a larger amount. Following from these characteristics, we propose a storage scheme for the SFC in this section that can further reduce its total cost, if the scheme is implemented in conjunction with the Stackelberg game.
We assume that the SFC is equipped with an SD, and the charging and discharging of the SD at different times of the day is carried out based on the time of use (ToU) price [@Wang-JTSG:2013] announced by the grid. The intuition behind considering a ToU price as the baseline for the SD’s charging-discharging can be explained as follows: 1) Since the SFC does not know the private information of RUs, such as their energy generation and preferences, it cannot determine how much energy it can buy from them (and the associated cost) ahead of time. Hence, by allowing a ToU price to decide its charging and discharging, the SD can leverage the flexibility of the SFC in trading energy with the grid in the event of energy scarcity at the RU at any time of the day. And 2) It is reasonable and practical to assume that the grid’s ToU price is announced ahead of time [@Wang-JTSG:2013]. Hence, it would be more practical for the SFC to decide on the charging and discharging pattern of its SD based on a known price that gives a mathematically tractable solution.
To this end, we consider that the total time of energy management during a day is divided into $T$ time slots where each time slot $t$ has a duration of one hour [@Jin-J-TVT:2013]. At $t$, the total requirement of energy $E_\text{sf}^\text{req}(t)$ by the SFC has two components: $$\begin{aligned}
E_\text{sf}^\text{req}(t) = E_\text{sf}^\text{eqp}(t) + e_\text{sf}^\text{SD}(t),\label{eqn:requirement-withSD}\end{aligned}$$ where $E_\text{sf}^\text{eqp}(t)$ is the amount of energy exclusively required to run equipment of the shared facility, and $e_\text{sf}^\text{SD}(t)$ is the energy charged-to/discharged-from the SFC’s storage at $t$.
![Choice of charging and discharging duration of the SFC’s SD based on the price announced by the grid.[]{data-label="fig:charging-discharging-duration"}](pricingthreshold){width="\columnwidth"}
It is assumed that the announced price per time slot is available from the main grid ahead of time [@Wang-JTSG:2013]. We suppose that the SFC selects two price levels $p_g^\text{min}$ and $p_g^\text{max}$ as the minimum and maximum price thresholds from the announced price list. The SFC charges its battery at time $t$ if $p_g^s(t)<p_g^\text{min}$, and discharges the battery if $p_g^s(t)>p_g^\text{max}$. The duration of time at which these two conditions are satisfied are characterized as the charging duration $T_\text{chg}$ and the discharging duration $T_\text{dis}$ respectively. The choice of $T_\text{chg}$ and $T_\text{dis}$ based on $p_g^s(t)~\forall t\in T, p_g^\text{min}$ and $p_g^\text{max}$ are shown graphically in Fig. \[fig:charging-discharging-duration\]. To that end, the charging and discharging process of the SFC’s SD can be implemented as follows:
**Charging of the SD:**
1. The SFC, with an initial state of charge (SOC) at the SD $Q_\text{ini}$, determines $T_\text{chg}$ according to the price announced by the grid and selected $p_g^\text{min}$.
2. For each $t\in T_\text{chg}$, the SFC derives $p_g^\text{min} - p_g^s(t)$, and checks the total $\sum_{t\in T_\text{chg}}(p_g^\text{min} - p_g^s(t))$ for the full charging duration.
3. The SFC sets a target SOC at the end of the charging duration, i.e., $Q_\text{tar}^\text{ch}$, and charges its SD at each $t$ based on the proportion of price difference between time slots, $Q_\text{tar}^\text{ch}$, and $Q_\text{ini}$ through $$\begin{aligned}
e_\text{sf}^\text{SD}(t) = \frac{\left(p_g^\text{min}-p_g^s(t)\right)\left(Q_\text{tar}^\text{ch} - Q_\text{ini}\right)\sigma}{\sum_{t\in T_\text{chg}}\left(p_g^\text{min}-p_g^s(t)\right)},~\forall t\in T_\text{chg},\label{eqn:charging-sd}\end{aligned}$$ where $\sigma$ is the efficiency of SFC’s SD.
We stress that the SFC cannot charge its SD at a rate more than its maximum allowed charging rate [@Jin-J-TVT:2013]. Hence, can be modified as $$\begin{aligned}
e_\text{sf}^\text{SD}(t) = \min\left(\frac{\left(p_g^\text{min}-p_g^s(t)\right)\left(Q_\text{tar}^\text{ch} - Q_\text{ini}\right)\sigma}{\sum_{t\in T_\text{chg}}\left(p_g^\text{min}-p_g^s(t)\right)}, e_\text{sf}^\text{max}\right),\label{eqn:charge-sd}\end{aligned}$$ where $e_\text{sf}^\text{max}$ is the maximum charging/discharging rate of the SFC’s SD.
**Discharging of the SD:** The SFC discharges its SD at each time slot $t\in T_\text{dis}$ following a similar process to that described in the previous paragraph. Therefore, at each $t\in T_\text{dis}$: 1) the SFC derives $(p_g^s(t) - p_g^\text{max})$ and determines the overall $\sum_{t\in T_\text{dis}}\left(p_g^s(t) - p_g^\text{max}\right)$ for the whole duration of $T_\text{dis}$; and then 2) based on the proportion of price difference between discharging time slots, achieved SOC $Q_\text{tar}^\text{ch}$ during $T_\text{chg}$, and the target SOC $Q_\text{tar}^\text{dis}$ at the end of discharging period $T_\text{dis}$, the SFC discharges its SD using $$\begin{aligned}
e_\text{sf}^\text{SD}(t) = - \min\Bigg(\frac{(p_g^s(t) - p_g^\text{max})(Q_\text{tar}^\text{ch}-Q_\text{tar}^\text{dis})\sigma}{\sum_{t\in T_\text{dis}}(p_g^s(t) - p_g^\text{max})},\nonumber\\ e_\text{sf}^\text{max}, E_\text{sf}^\text{eqp}(t)\Bigg),\label{eqn:discharge-sd}\end{aligned}$$ for all $t\in T_\text{dis}$. As shown in , during discharge, the SFC cannot drain its SD by more than what is required by equipment as it would result in a negative requirement in . The negative sign in emphasizes that the SD is discharging during $T_\text{dis}$.
By adopting and the SFC is enabled to charge its SD during lower price periods and discharge it during higher price periods, which consequently reduces the cost of energy trading of the SFC. We note that a similar idea has been used previously to reduce the energy consumption cost of different energy entities by using batteries [@Fang-J-CST:2012]. However, in this work the inclusion of a Stackelberg game with this charging-discharging scheme in each time slot makes the RUs with DER part of the system, and thus significantly further reduces the costs to the SFC, as will be shown via numerical experiments in the next section. The choice of two thresholds provides the SFC with the flexibility to choose different ranges of charging, discharging and idle durations. For example, if $p_g^\text{max} = p_g^\text{min}$, the threshold of the proposed scheme would merge with the choice of threshold proposed in [@Wang-JTSG:2013]. However, the decision making mechanism of the charging and discharging amount at each time as proposed in this paper is completely different from that in [@Wang-JTSG:2013].
Case Study {#sec:case-study}
==========
![Convergence of the proposed scheme to the SE.[]{data-label="fig:convergence"}](Convergence){width="\columnwidth"}
For numerical case studies, we consider a number of RUs in the smart community that are interested in selling their energy to the SFC. Typical energy generation of each RU $n$ from its DERs is assumed to be 10 kWh [@NREL_wind_generation:2009], and is considered to be the same for all RUs in the network. The required energy by the SFC is assumed to be 50 kWh during the considered time. As shown in , the minimum requirement of each RU $n$ depends on its preference parameter $k_n$, which is chosen separately for different RUs, and is considered to be sufficiently large[^9] such that does not possess any negative values and $e_n$ is at least equal to $E_n^\text{min}$. The grid’s per-unit sale price is assumed to be 60 cents [@Jin-J-TVT:2013], whereby the SFC sets its initial price to be $8.45$ cents per kWh[^10] to pay to each RU. It is very important to highlight that all parameter values are particular to this study and may vary according to the need of the SFC, power generation of the grid, weather conditions of the day, time of the day/year, and the country.
In Fig. \[fig:convergence\], the convergence of the SFC’s total cost to the SE by following Algorithm \[alg:1\] is shown for a network of five RUs. Here we see that although the SFC wants to minimize its total cost, it cannot manage to do so with its initial choice of price for payment to the RUs. In fact, through interaction with each RU of the network, the SFC eventually increases its price in each iteration to encourage the RUs to sell more, and consequently the cost continuously reduces. As can be seen from Fig. \[fig:convergence\], the SFC’s choice of equilibrium price, and consequently the minimum total cost, reaches its SE after the $34^\text{th}$ iteration.
![Utility achieved by each RU at the SE.[]{data-label="fig:utility-each-RU"}](EnergyatSE){width="\columnwidth"}
As the SFC’s total cost of energy purchase reaches its SE, the RUs in the network also reach their best utilities by playing their best strategies in response to the price offered by the SFC. We show the utility achieved by each RU at the SE in Fig. \[fig:utility-each-RU\]. As discussed in Definition \[definition:1\], it is shown in Fig. \[fig:utility-each-RU\] that any deviation from the choice of energy consumption at the SE assigns a lower utility to the RU. In Fig. \[fig:utility-each-RU\] we compare both of the cases: 1) choice of energy more than the SE amount and 2) choice of energy lower than the SE amount, with the SE energy choice by each RU. It shows that only the SE assigns the maximum utility to each RU, and thus establishes a stable solution of the game.
Number of RU 5 10 15 20 25
----------------- ------- ------- ------- ------- -------
Cost (Baseline) 105 105 105 105 105
Cost (Proposed) 84.23 64.38 44.20 23.79 2.78
$\%$ Reduction 19.78 38.68 57.89 77.34 97.34
: Effect of the number of RUs on the total cost (in dollar) incurred by the SFC ($E^\text{req}_\text{sf} = 150$ kWh, $p_g^s = 70$ cents/kWh).
\[tab:cost-vs-rus\]
$E^\text{req}_\text{sf}$ 60 70 80 90 100
-------------------------- ------- ------- ------- ------- -------
Cost (Baseline) 42.0 49.0 56.0 63.0 70.0
Cost (Proposed) 1.384 8.384 15.38 22.38 29.38
$\%$ Reduction 96.70 82.89 72.53 64.47 58.02
: Effect of change of SFC’s required energy on its total cost in dollars ($N = 10, p_g^s = 70$ cents/kWh).
\[tab:cost-vs-req\]
In Tables \[tab:cost-vs-rus\] and \[tab:cost-vs-req\], we investigate how the proposed scheme captures the change in total cost to the SFC as different parameters, such as the number of RUs and the SFC’s energy requirement, change in the system. We compare the results with a baseline approach that does not have any DER facility, i.e., the SFC depends on the grid for all its energy. First, in Table \[tab:cost-vs-rus\], the cost to the SFC is shown to gradually decrease for the proposed case as the total number of RUs increases in the network. This is due to the fact that as the number of RUs increases in the system, the SFC can buy more energy at a cheaper rate from more RUs, and consequently becomes less dependent on the grid’s more expensive energy. Hence, the cost reduces eventually. However, due to the absence of any DERs, the cost to the SFC does not change with number of RUs in the network in the baseline approach, and the cost is shown to be significantly higher than for the proposed scheme. From Table. \[tab:cost-vs-rus\], on average the cost reduction is $58.2\%$ for the proposed case compared to the baseline approach, with the considered parameter values, as the number of RUs varies in the system.
Whereas the cost to the SFC decreases with an increase in RUs in the system, we observe the opposite effect on cost while the SFC’s energy requirement increases. As shown in Table \[tab:cost-vs-req\], the cost to the SFC increases for both the proposed and baseline approaches as the energy required by the SFC increases. In fact, it is trivial to observe that needing more energy leads the SFC to spend more on buying energy, which consequently increases the cost. Nonetheless, the proposed scheme needs to spend less to buy the same amount of energy due to the presence of DERs of the RUs, and thus noticeably benefits from its energy trading, in terms of total cost, when compared to the baseline scheme. From Table \[tab:cost-vs-req\], the SFC’s average cost is $74.9\%$ lower than that of the baseline approach for the considered changes in SFC’s energy requirements.
![Comparison of social cost obtained by the proposed distributed scheme with respect to the centralized scheme as the number of RUs varies in the network.[]{data-label="fig:CentralVsPropose"}](CentralVsPropose){width="\columnwidth"}
As we have discussed above, it is also possible to optimally manage energy between RUs and the SFC via a centralized control system to minimize the social cost[^11] if private information such as $k_n$ and $E_n^\text{gen}~\forall n$ is available to the controller. In this regard, we observe the performance in terms of social cost for both the centralized and proposed distributed schemes for two different price schemes in Fig. \[fig:CentralVsPropose\]. As can be seen from the figure, the social cost attained by adopting the distributed scheme is close to the optimal centralized scheme at the SE of the game in both the cases. However, the centralized scheme has access to the private information of each RU. Hence, the controller can optimally manage the energy, and as a result there is better performance in terms of reducing the SFC’s cost compared to the proposed scheme. According to Fig. \[fig:CentralVsPropose\], as the number of RUs increases in the network from $5$ to $25$, the average social cost for the proposed distributed scheme is only $7.07\%$ and $6.75\%$ higher than for the centralized scheme for $p_g^s = 85$ and $60$ cents/kWh respectively. This is a very promising result considering that the system is distributed.
![Comparison of the cost incurred by the SFC at different times of the day with and without an SD.[]{data-label="fig:CostVsTime"}](CostVsTime){width="\columnwidth"}
Having insight into the properties of the proposed Stackelberg game, we now show the performance of the proposed scheme when the SFC is equipped with an SD. For this purpose, we assume that the SFC has a 100 kWh SD with an efficiency of $0.9$ and a maximum charging-discharging rate of 24 kWh [@Sousa-J-TSG:2012]. The price at different times of the day is obtained from [@Jin-J-TVT:2013], and the maximum and minimum price thresholds are considered to be $p_g^\text{min} = 40$ and $p_g^\text{max} = 45$ cents per kWh respectively. The demand of the SFC at different times of the day is chosen randomly from $\left[300, 700\right]$ kWh. For $5$ RUs in the system, we show the cost of the SFC at different times of the day for three different cases in Fig. \[fig:CostVsTime\]. These cases are 1) when the SFC does not have any SD and does not take part in the game, 2) when the SFC has an SD but does not take part in the game, and finally, 3) when the SFC has an SD and also plays the game with the RUs following Algorithm \[alg:1\].
As can be seen from Fig. \[fig:CostVsTime\], during the period when the grid price is low, the cost to the SFC is higher for cases 2 and 3 compared to case 1. In fact, due to the lower price, the SFC is more interested in charging its SD during this time so as to use it in peak hours. Hence, its required energy is more than the case without the SD. As a result, the cost is higher. However, the cost without the proposed game is considerably higher than the cost when the SFC and RUs interact with each other via Algorithm \[alg:1\]. The reason is that without playing the game, the SFC needs to buy all its energy from the grid including the energy for its SD. By contrast, the proposed game allows the SFC to pay the RUs a lower price than the grid’s price to buy some of its required energy. Consequently, the SFC benefits in terms of its reduced total cost of energy purchase.
During peak hours, the cost to the SFC is significantly higher for case 1. In this case, the SFC needs to buy all its required energy from the grid at a significantly higher price. However, for the case when the SFC possesses an SD, the cost is lower as the stored energy allows the SFC to buy less from the grid compared to the previous case. Nevertheless, the most impressive performance is observed for case 3 when the SFC with an SD plays the Stackelberg game with the RUs following Algorithm \[alg:1\]. On the one hand, the stored energy allows the SFC to buy a lower amount of energy during the peak hour like in case 2. On the other hand, unlike the other two cases, by taking part in the Stackelberg game the SFC manages to buy a certain fraction of its requirement from the RUs at a cheaper rate, compared to the grid’s price, which minimizes its total cost of energy purchase noticeably. As Fig. \[fig:CostVsTime\] shows, on average, the cost reduction of the proposed case is $53.8\%$ compared to the case in which the SFC does not take part in the game. As can be seen from Fig. \[fig:CostVsTime\], the performance is even more impressive when compared to case 1.
![Change in cost savings with the change of the capacity and charging rate of the SD.[]{data-label="fig:CostSavings"}](CostVsCapacity){width="\columnwidth"}
However, cost savings are greatly affected by the SD’s characteristics such as its capacity. In fact, for a particular charging rate, higher capacity can considerably assist the SFC to improve its performance in terms of average cost savings during a day, as shown in Fig. \[fig:CostSavings\]. As the figure illustrates, the cost saving for an SFC by using an SD increases with an increase in the capacity of the SD. According to Fig. \[fig:CostSavings\], the savings in daily cost are, on average, $54.02\%$ and $58.1\%$ per day for the proposed case compared to the cases when the SFC does not play the game and when the SFC neither plays the game nor has any SDs respectively.
Conclusion {#sec:conclusion}
==========
In this paper, we have presented an energy management scheme for a smart community using a non-cooperative Stackelberg game. We have designed a system model suitable for applying the game, and have shown the existence of a strategy proof, unique Stackelberg equilibrium (SE), by exploring the properties of the game. We have shown that the use of distributed energy resources (DERs) is beneficial for both the shared facility controller (SFC) and residential units (RUs) at the SE. We have proposed a distributed algorithm, which is guaranteed to reach the SE of the game. Further, we have extended the scheme to the case in which the SFC has a storage device (SD). We have designed an effective charging-discharging scheme, for the SFC’s SD based on the grid’s price, which has been shown to have considerable influence on the cost incurred by the SFC. By the proposed charging-discharging scheme, the average cost to the SFC during a day has been shown to be reduced markedly compared to the case without a SD.
The proposed work can be extended in various ways. An interesting extension would be to check the impact of discriminate pricing among the RUs on the outcome of the scheme. Another compelling addition would be to determine how to set the threshold on the grid’s price. Furthermore, quantifying the inconvenience that the SFC/RUs face during their interaction is another possible future investigation based on this work.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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[Wayes Tushar]{}(S’06, M’13) received the B.Sc. degree in Electrical and Electronic Engineering from Bangladesh University of Engineering and Technology (BUET), Bangladesh, in 2007 and the Ph.D. degree in Engineering from the Australian National University (ANU), Australia in 2013. Currently, he is a postdoctoral research fellow at Singapore University of Technology and Design (SUTD), Singapore. Prior joining SUTD, he was a visiting researcher at National ICT Australia (NICTA) in ACT, Australia. He was also a visiting student research collaborator in the School of Engineering and Applied Science at Princeton University, NJ, USA during summer 2011. His research interest includes signal processing for distributed networks, game theory and energy management for smart grids. He is the recipient of two best paper awards, both as the first author, in Australian Communications Theory Workshop (AusCTW), 2012 and IEEE International Conference on Communications (ICC), 2013.
[Bo Chai]{} received the B.Sc degree in Automation from Zhejiang University, Hangzhou, China, in 2010, when he also graduated from Chu Kochen Honors College. He was a visiting student at Simula Research Laboratory, Oslo, Norway in 2011. Currently, he is a member of the Group of Networked Sensing and Control (IIPC-nesC) in the State Key Laboratory of Industrial Control Technology, Zhejiang University, and he is also a visiting student in Singapore University of Technology and Design from 2013 to 2014. His research interests include smart grid and cognitive radio.
[Chau Yuen]{} received the B. Eng and PhD degree from Nanyang Technological University, Singapore in 2000 and 2004 respectively. Dr Yuen was a Post Doc Fellow in Lucent Technologies Bell Labs, Murray Hill during 2005. He was a Visiting Assistant Professor of Hong Kong Polytechnic University in 2008. During the period of 2006 2010, he worked at the Institute for Infocomm Research (Singapore) as a Senior Research Engineer. He joined Singapore University of Technology and Design as an assistant professor from June 2010. He serves as an Associate Editor for IEEE Transactions on Vehicular Technology. On 2012, he received IEEE Asia-PaciÞc Outstanding Young Researcher Award.
[David Smith]{} is a Senior Researcher at National ICT Australia (NICTA) and is an adjunct Fellow with the Australian National University (ANU), and has been with NICTA and the ANU since 2004. He received the B.E. degree in Electrical Engineering from the University of N.S.W. Australia in 1997, and while studying toward this degree he was on a CO-OP scholarship. He obtained an M.E. (research) degree in 2001 and a Ph.D. in 2004 both from the University of Technology, Sydney (UTS), and both in Telecommunications Engineering. His research interests are in technology and systems for wireless body area networks; game theory for distributed networks; mesh networks; disaster tolerant networks; radio propagation and electromagnetic modeling; MIMO wireless systems; coherent and non-coherent space-time coding; and antenna design, including the design of smart antennas. He also has research interest in distributed optimization for smart grid. He has also had a variety of industry experience in electrical engineering; telecommunications planning; radio frequency, optoelectronic and electronic communications design and integration. He has published over 80 technical refereed papers and made various contributions to IEEE standardization activity; and has received four conference best paper awards.
[Kristin L. Wood]{} joined the faculty at the University of Texas in September 1989 after completing his doctoral work and established a computational and experimental laboratory for research in engineering design and manufacturing, in addition to a teaching laboratory for prototyping, reverse engineering measurements, and testing. During the 1997-98 academic year, Dr Wood was a Distinguished Visiting Professor at the United States Air Force Academy where he worked with USAFA faculty to create design curricula and research within the Engineering Mechanics / Mechanical Engineering Department. Through 2011, Dr Wood was a Professor of Mechanical Engineering, Design & Manufacturing Division at The University of Texas at Austin. He was a National Science Foundation Young Investigator, the “Cullen Trust for Higher Education Endowed Professor in Engineering", “University Distinguished Teaching Professor", and the Director of the Manufacturing and Design Laboratory (MaDLab) and MORPH Laboratory.
Dr Wood has published more than 300 commentaries, refereed articles and books, and has received three ASME Best Research Paper Awards, two ASEE Best Paper Awards, an ICED Best Research Paper Award, the Keck Foundation Award for Excellence in Engineering Education, the ASEE Fred Merryfield Design Award, the NSPE AT&T Award for Excellence in Engineering Education, the ASME Curriculum Innovation Award, the Engineering Foundation Faculty Excellence Award, the Lockheed Martin Teaching Excellence Award, the Maxine and Jack Zarrow Teaching Innovation Award, the Academy of University Distinguished Teaching Professors’ Award, and the Regents’ Outstanding Teacher Award.
[Zaiyue Yang]{} (M’10) received the B.S. and M.S. degrees from Department of Automation, University of Science and Technology of China, Hefei, China, in 2001 and 2004, respectively, and the Ph.D. degree from Department of Mechanical Engineering, University of Hong Kong, Hong Kong, in 2008. He then worked as Postdoctoral Fellow and Research Associate in the Department of Applied Mathematics, Hong Kong Polytechnic University before joining Zhejiang University, Hangzhou, China, in 2010. He is currently an associate professor there. His current research interests include smart grid, signal processing and control theory.
[H. Vincent Poor]{} (S’72, M’77, SM’82, F’87) received the Ph.D. degree in EECS from Princeton University in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the faculty at Princeton, where he is the Michael Henry Strater University Professor of Electrical Engineering and Dean of the School of Engineering and Applied Science. Dr. Poor?s research interests are in the areas of stochastic analysis, statistical signal processing, and information theory, and their applications in wireless networks and related fields such as social networks and smart grid. Among his publications in these areas are the recent books *Principles of Cognitive Radio* (Cambridge University Press, 2013) and *Mechanisms and Games for Dynamic Spectrum Allocation* (Cambridge University Press, 2014).
Dr. Poor is a member of the National Academy of Engineering and the National Academy of Sciences, and is a foreign member of Academia Europaea and the Royal Society. He is also a fellow of the American Academy of Arts & Sciences, the Royal Academy of Engineering (UK) and the Royal Society of Edinburgh. In 1990, he served as President of the IEEE Information Theory Society, and in 2004-07 he served as the Editor-in-Chief of the *IEEE Transactions on Information Theory*. He received a Guggenheim Fellowship in 2002 and the IEEE Education Medal in 2005. Recent recognition of his work includes the 2014 URSI Booker Gold Medal, and honorary doctorates from Aalborg University, the Hong Kong University of Science and Technology and the University of Edinburgh.
[^1]: W. Tushar, C. Yuen and K. L. Wood are with the Engineering Product Development at Singapore University of Technology and Design (SUTD), Dover Drive, Singapore 138682. (Email: {wayes\_tushar, yuenchau, kristinwood}@sutd.edu.sg).
[^2]: B. Chai and Z. Yang are with the State Key Laboratory of Industrial Control Technology at Zhejiang University, Hangzhou, China. (Email: chaibozju@gmail.com, yangzy@zju.edu.cn).
[^3]: David B. Smith is with the National ICT Australia (NICTA)$^\dag$, ACT 2601, Australia. D. Smith is also with the Australian National University. (Email: david.smith@nicta.com.au).
[^4]: H. Vincent Poor is with the School of Engineering and Applied Sciences at Princeton University, Princeton, NJ, USA. (Email: poor@princeton.edu).
[^5]: This work is supported by the Singapore University of Technology and Design (SUTD) through the Energy Innovation Research Program (EIRP) Singapore NRF2012EWT-EIRP002-045.
[^6]: $^\dag$NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.
[^7]: A dedicated authority responsible for managing shared equipments in a community.
[^8]: Please note that no RU possesses any SD.
[^9]: For this case study, each $k_n$ is generated as a uniformly distributed random variable from the range $\left[90, 150\right]$.
[^10]: Which is the buy back price of the grid [@Tushar-TSG:2013].
[^11]: Which is the difference between the total cost incurred by the SFC and total utility achieved by all RUs in the system.
|
---
address: 'University of Rome and INFN, I-00185, Rome, Italy'
author:
- 'M.Cirilli [^1]'
title: 'The precise determination of ${\Re e\,({\varepsilon^\prime/\varepsilon})}$ '
---
[\^0]{}[\^[0]{}]{}
Introduction
============
The violation of symmetry was first reported in 1964 by J.H.Christenson, J.W. Cronin, V. Fitch and R. Turlay, who detected a clean signal of violating $\KL\rightarrow{\pi^+\pi^-}$ decays [@disco]. conservation implies that the $\KS$ and $\KL$ particles are pure eigenstates and that $\KL$ decays only into $=-1$ and $\KS$ into $=+1$ final states. The observed signal of the forbidden $\KL\rightarrow {\pi\pi}$ decays ($=+1$) indicates that is not a conserved symmetry.
violation can occur via the mixing of eigenstates, called [*indirect*]{} violation, represented by the parameter $\eps$. violation can also occur in the decay process itself, through the interference of final states with different isospins. This is represented by the parameter $\eprime$ and is called [*direct*]{} violation. L. Wolfenstein in 1964 [@wolf] proposed a super-weak force responsible for $\Delta S=2$ transitions, so that all observed violation phenomena come from mixing and $\eprime =0$. In 1973, Kobayashi and Maskawa proposed a matrix representation of the coupling between fermion families [@kob]. In the case of three fermion generations, both direct and indirect violation are naturally accommodated in their model, via an irreducible phase.
The parameters $\eps$ and $\eprime$ are related to the amplitude ratios $$\begin{aligned}
\epm = \frac{ {\mathrm{A}({\KL \rightarrow {\pi^+\pi^-}})} }{ {\mathrm{A}({\KS \rightarrow
{\pi^+\pi^-}})} } = \eps + \eprime \end{aligned}$$ and $$\begin{aligned}
\eoo = \frac{ {\mathrm{A}({\KL \rightarrow {\pi^0\pi^0}})} }{ {\mathrm{A}({\KS \rightarrow
{\pi^0\pi^0}})} } = \eps - 2\eprime\end{aligned}$$ which represent the strength of the violating amplitude with respect to the conserving one, in each mode. By the mid-1970s, experiments had demonstrated that violation in the neutral kaon system is dominated by mixing, with the limit ${\Re e\,({\varepsilon^\prime/\varepsilon})}\le
10^{-2}$[@pioneer]. On the other hand, theoretical work showed that direct violation in the Standard Model could be large enough to be measurable [@firstthe]. This stimulated experimental effort with sophisticated detectors to measure ${\Re e\,({\varepsilon^\prime/\varepsilon})}$. The first evidence for the existence of a direct component of violation was published in 1988 [@evidence]. In 1993, two experiments published their final results without a conclusive answer on the existence of this component. NA31 [@na31] measured ${\Re e\,({\varepsilon^\prime/\varepsilon})}=(23.0\pm6.5){\mathrm{\times 10^{-4}}}$, indicating a $3.5\sigma$ effect. The result of E731 [@e731], ${\Re e\,({\varepsilon^\prime/\varepsilon})}=(7.4\pm5.9){\mathrm{\times 10^{-4}}}$, was instead compatible with no effect.
The controversial results from NA31 and E731 called for the realization of more precise experiments, to measure ${\Re e\,({\varepsilon^\prime/\varepsilon})}$ with a precision of $\cal O$$(10^{-4}$. Presently, there are three experiments in different laboratories working on the precise measurement of ${\Re e\,({\varepsilon^\prime/\varepsilon})}$: two of these, namely NA48 [@na48prop] at CERN and KTeV [@ktevreport] at Fermilab, represent the “evolution” of NA31 and E731 respectively; the third one is KLOE [@kloetp] at the Laboratori Nazionali di Frascati and its conceptual design is radically different from the other experiments.
This paper is devoted to a comparative presentation of KTeV and NA48, since these experiments have already published results. Detector designs and analysis techniques will be discussed, together with the results announced just recently from these two collaborations [@na48:eprime2001][@ktev:eprime2001].
The experimental method {#sec:method}
=======================
Experimentally, it is convenient to measure the double ratio $\R$, which is related to the ratio ${\Re e\,({\varepsilon^\prime/\varepsilon})}$: $$R = \frac{ {\mathrm{\Gamma}({\KL \rightarrow {\pi^0\pi^0}})} }{ {\mathrm{\Gamma}({KS \rightarrow {\pi^0\pi^0}})}
} / \frac{ {\mathrm{\Gamma}({\KL \rightarrow {\pi^+\pi^-}})} }{ {\mathrm{\Gamma}({\KS \rightarrow
{\pi^+\pi^-}})} } \approx 1 - 6 \times {\Re e\,({\varepsilon^\prime/\varepsilon})}\label{doubleratio}$$ The double ratio $\R$ is experimentally measured by *counting* the number of decays detected in each of the four modes in equation \[doubleratio\]. The statistical error is dominated by the events collected in the most suppressed decay, namely $\KL\rightarrow{\pi^0\pi^0}$ ($BR\sim 0.09$%). The value $\R$$_{true}$ is then deduced correcting the measured value $\R$$_{meas}$ for the kaon beam fluxes, detector acceptances, trigger efficiencies, backgrounds evaluations, etc., i.e. for all the possible biases in the counting process. It is now evident that the difficulty of ${\Re e\,({\varepsilon^\prime/\varepsilon})}$ measurements lies in the necessity to disentangle the violating $\KL$ modes from the dominant environment of conserving 3-body decays of both $\KL$ and $\KS$.
Advantages of the double ratio technique {#subsec:adv}
----------------------------------------
The main advantage of the double ratio measurement, when performed under the adequate data taking conditions, is that the corrections to $\R$$_{meas}$ can cancel out at first order.Let us consider the beam fluxes and trigger/reconstruction efficiencies corrections as an example:
- **Beam fluxes:** the knowledge of the kaon flux in the $\KS$ and $\KL$ beams is a priori needed for normalization purposes. However, if the charged and neutral decay modes of either the $\KS$ or the $\KL$ are simultaneously collected, then the ratio of ${\pi^+\pi^-}$ and ${\pi^0\pi^0}$ events in each beam is independent from the absolute flux. Hence, under these conditions, beam fluxes cancel out in the double ratio at first order.
- **Efficiencies:** the trigger scheme is conceived to minimise any loss of good events. However, a small correction usually has to be applied to $\R$$_{meas}$ to account for trigger inefficiencies. A first order cancellation of this correction can be achieved for the charged/neutral trigger efficiency if both $\KL$ and $\KS$ decays into the charged/neutral final state are simultaneously collected. The same principle also holds for any instability of a given detector, which could affect the reconstruction efficiency of the charged or neutral modes.
The best strategy to exploit the cancellation of eventual biases is to collect all the four modes simultaneously. This allows to evaluate only second order effects to get the true value $\R$$_{true}$ from $\R$$_{meas}$. Even in this ideal situation, there will still be some leftover corrections that do not cancel out. This is the case for the physical background, which comes only from $\KL$ decays and is clearly final-state-dependent. Also, acceptance corrections do not a priori cancel out in the four modes: this is related to the huge lifetime difference between $\KL$ and $\KS$, which causes very different longitudinal decay vertex distributions for the two beams, and to the different topologies of ${\pi^0\pi^0}\rightarrow 4\gamma$ and ${\pi^+\pi^-}$ events. Both the physical background and the acceptance correction must be carefully studied, and different solutions can be envisaged to handle them.
All the above considerations have been thoroughly taken into account while conceiving KTeV and NA48. The design of the experiments and the analysis methods focus on making the inevitable systematic biases in the event counting symmetric between at least two of the four components of the double ratio. In this way, most of the important systematic effects cancel to first order, and only the differences between two components need to be considered in detail in the analysis. This allows the systematic uncertainties to be kept sufficiently low.
KTeV and NA48: overview {#sec:overview}
=======================
Both KTeV and NA48 are fixed target experiments designed to simultaneously collect all the four decay modes in \[doubleratio\]. Measuring ${\Re e\,({\varepsilon^\prime/\varepsilon})}$ to a precision of $\sim 10^{-4}$ requires several millions of $\KL$ and $\KS \rightarrow {\pi\pi}$ decays: this implies taking data with high-intensity kaon beams and running for several years to achieve the desired statistics. A number of challenges had to be faced during the design phase. Stable detectors were needed to sustain the long data taking periods. The trigger electronics had to be fast enough to cope with the high flux of particles in the decay region and a powerful data acquisition was needed to handle the high trigger rates. The overwhelming $3{\pi^0}$ background set the requirement of an extremely precise electromagnetic calorimeter. The whole detector had to be radiation-hard, to cope with the beam intensities. Long R&Ds were necessary to meet these very demanding requirements, and this effort has also been profitable in view of future experiments in high-intensity environments (*e.g.*, the Tevatron and LHC).
KTeV collected $\sim 7$M events in the most suppressed channel $\KL
\rightarrow {\pi^0\pi^0}$ during the 96, 97 and 99 runs. A first ${\Re e\,({\varepsilon^\prime/\varepsilon})}$ measurement [@ktev:eprime99] was announced in February, 1999, based on $10$% of the total sample. In June, 2001, KTeV presented a new result on the 97 data sample, together with an update of the already published result: the combined measurement [@ktev:eprime2001] is obtained from $\sim 50$% of the available data. NA48 took data in 97, 98 and 99, collecting almost $4$M events in the neutral $\KL$ decay mode. The first ${\Re e\,({\varepsilon^\prime/\varepsilon})}$ measurement [@na48:eprime99] was reported in June, 1999, and was based on the statistics collected during the 97 run; a preliminary result on the 98 sample was presented in February, 2000. The final result on the 98 and 99 data [@na48:eprime2001] was announced in May, 2001. A slight increase in statistics for NA48 is expected from the data collected in the 2001 run.
KTeV: Detector and beam lines {#subsec:ktev}
-----------------------------
A schematic view of KTeV detectors and beams is shown in figure \[fig:ktevsetup\]. KTeV exploits the $800$ GeV proton beam delivered by the Tevatron: two nearly parallel kaon beams are produced by the protons hitting a $50$ cm long beryllium target at $4.8$ mrad angle. The beams are cleaned up and let fly for roughly $120$ m, so that only the $\KL$ component survives. The beams¹ direction defines the longitudinal $z$ axis. The decay region begins at the end of the last collimator; here the two beams are $10$ cm apart, and one of them hits a $1.8$ m long regenerator made of plastic scintillators. The regenerator beam is a coherent superposition $\KL +\rho \KS$ of long- and short-lived kaons. The regenerated fraction $\rho$ is proportional to the amount of matter traversed by the previously pure $\KL$ beam, and its value $0.03$ is sufficient to ensure that the regenerator $2\pi$ decays are dominated by $\KS \rightarrow {\pi\pi}$. The regenerator technique ensures that the $\KS$ are produced with an energy spectrum similar to that of $\KL$. The decay region extends up to $159$ m from the primary target.
=9.5cm
A distinctive feature of KTeV is the fact that the two beams are parallel and hit the detectors at separate points (left and right). This allows to easily identify $\KL$ and $\KS$ decays reconstructing the transverse decay vertex position and comparing it with the known regenerator position. The regenerator is fully instrumented, and switches beam line once per minute, in order to reduce the effects of possible left-right asymmetries of the detectors.
Charged kaon decays are detected by a spectrometer consisting of a central magnet with a $411$ MeV/$c$ kick in the horizontal plane and of four drift chambers with wires along the $x$ and $y$ directions. The spectrometer has a position resolution of $100 \mu$m and a momentum resolution $\sigma_{p}/p = 0.17\% \oplus [0.008 \times
p]\%$, where $p$ is in GeV/$c$ units.
Neutral decays are detected by a crystal calorimeter [@ktev:calo] consisting of $3\,100$ pure CsI blocks. The crystals cover $27
\mathrm{X}_{0}$ in length ($50$ cm) and have a transverse section of $2.5\times 2.5 \mathrm{cm}^{2}$ in the central region, where the density of photons is higher; in the outer region of the calorimeter, the granularity is of $5\times 5 \mathrm{cm}^{2}$. The main advantage of this calorimeter lies in its excellent stochastic term in the energy resolution, which allows to reach an overall resolution of $0.7$% for a $15$ GeV photon (as shown in figure \[calor\] left). The longitudinal light collection is equalised within $5$% by means of a meticulous crystal wrapping. In addition, the stability of the response for each crystal is continuously checked using a $\mathrm{Cs}^{137}$ source for calibration. The overall energy response is linear within $0.4$%.
=4.cm =5.5cm
The main apparatus is surrounded by circular vetoes to detect escaping photons, and a muon veto placed at the end of the line is used to identify ${\mathrm{K}_{\mu3}}$ decays.
NA48: Detector and beam lines {#subsec:na48}
-----------------------------
A schematic view of the NA48 beam lines is given in figure \[na48beams\]. The primary $450$ GeV proton beam is delivered from the SPS and impinges on a $40$ cm beryllium target with an incidence angle of $2.4$ mrad relative to the $\KL$ beam axis. The charged component of the outgoing particles is swept away by bending magnets, while the neutral beam component passes through three stages of collimation. The fiducial region starts at the exit of the “final” collimator, $126$ m downstream of the target. At this point, the neutral beam is dominated by long-lived kaons. The non-interacting protons from the $\KL$ target are directed onto a mechanically bent mono-crystal of silicon. A small fraction ($10^{-5}$) of protons satisfies the conditions for channelling and is deflected following the crystalline planes. Use of the crystal allows a deflection of $9.6$ mrad to be obtained in only $6$ cm length, corresponding to a bending power of $14.4$ Tm. The transmitted protons pass through the tagging station (or *tagger*), which precisely registers their time of passage. They are then deflected back onto the $\KL$ beam axis, transported through a series of quadrupoles and finally directed to the $\KS$ target (same size as $\KL$) located 72 mm above the $\KL$ beam axis. A combination of collimator and sweeping magnet defines a neutral beam at 4.2 mrad to the incoming protons. The decay spectrum of kaons at the exit of the collimator is similar to that in the $\KL$ beam, with an average energy of 110 GeV. The fiducial region begins 6 m downstream of the $\KS$ target, such that decays are dominated by short lived particles. At this point, the $\KS$ and $\KL$ beams emerge from the aperture of the final collimators into the common decay region. The whole $\KS$ target and collimator system is aligned along an axis pointing to the centre of the detector 120 m away, such that the two beams intersect at this point with an angle of 0.6 mrad.
=10.cm
Since the two beams are not separated at the detector position, as it is in KTeV, the identification of $\KL$ and $\KS$ decays must be accomplished in a different way. This is done using the tagging station, which consists of two scintillator ladders, crossing the beam horizontally and vertically. The coincidence between the proton time and the event time in the detectors assigns the decay to the $\KS$ beam. Two close pulses can be resolved down to 4–5 ns.
The reconstruction of charged decays is performed by a magnetic spectrometer, with a central magnet giving a $250 \mathrm{MeV}/c$ transverse kick and four chambers with plane wires oriented along four different directions $x$, $y$, $u$ and $v$. The redundancy of the planes allows to resolve the possible track ambiguities.
NA48 has chosen a liquid Krypton ionization calorimeter with a depth of $27 \mathrm{X}_{0}$, corresponding to $125$ cm. The read-out is performed by Cu-Be-Co ribbons defining $\sim 13\,000$ cells, in a structure of longitudinal projective towers pointing to the centre of the decay region. The cross section of a cell is about 2 cm $\times$ 2 cm, and the electrodes are guided longitudinally through precisely machined holes in five spacer plates. The planes also apply a $\pm$48 mrad zig-zag to the electrodes, in order to maintain the mechanical stability and to decrease the sensitivity of the energy resolution to the impact position. Good energy response is further guaranteed by the initial current readout technique which also provides a high rate capability. The overall energy resolution is $1.5$% at $10$ GeV. The energy response is linear to about 0.1% in the range 5–100 GeV. A precise time measurement is mandatory for the NA48 calorimeter, since it must be used together with the proton time from the tagger to distinguish $\KS$ from $\KL$. The neutral event time is reconstructed with a precision of $\sim 220$ ps.; tails coming from misreconstructed times are below the level of 10$^{-4}$ (see figure \[calor\] right).
A muon veto system is used to reject muons from ${\mathrm{K}_{\mu3}}$ decays.
KTeV and NA48: analysis techniques {#sec:anal}
==================================
Once the events are collected, all corrections that do not cancel in the double ratio must be applied. The long list of residual effects that must be studied includes physical backgrounds, $\KS -\KL$ misidentifications, trigger efficiencies, Monte Carlo correction, geometrical acceptances, detector biases (calorimeter energy scale, drift chamber alignment, etc.), accidental effects. In the following, I will focus on just a few of these effects, highlighting the important differences between KTeV and NA48 approach.
Selection of the ${\pi^+\pi^-}$ sample {#subsec:char}
--------------------------------------
Both experiments use the magnetic spectrometer to reconstruct the kaon mass, vertex and momentum. The resolution on the kaon mass in the charged mode is $1.5$ MeV in KTeV and $2.5$ MeV in NA48. The better KTeV resolution is due to the higher transverse kick of their magnet and to the choice of having only $x$ and $y$ planes in the drift chambers. This choice implies lighter chambers with respect to NA48, and so a lower multiple scattering term in the resolution. However, having only two views reduces the capability of resolving ambiguities in the track reconstruction: for this reason, KTeV needs additional information from the calorimeter to perform a reliable reconstruction of ${\pi^+\pi^-}$ events.
[|c|c|c|c|]{}\
& & &\
${\mathrm{K}_{\mu3}}+ {\mathrm{K}_{e3}}$ & $0.9\times 10^{-3}$ & $0.03\times
10^{-3}$ & $1.69\times 10^{-3}$\
& $0.10\times 10^{-3}$ & $0.10\times 10^{-3}$ & -\
& - & $0.73\times 10^{-3}$ & -\
${\mathrm{K}_{\mu3}}$ decays are rejected using the identified muon in the dedicated vetoes, while electrons from ${\mathrm{K}_{e3}}$ events are identified comparing the track momentum in the spectrometer with the corresponding energy in the calorimeter. Additional cuts are imposed on the reconstructed mass and transverse momentum. The leftover background contributions are then evaluated studying high-statistics samples of the identified 3-body decays. The background fractions in the two experiments are summarized in table \[tbl:bkgchar\], including the components due to scattering in the collimator and (only for KTeV) regenerator.
Selection of the ${\pi^0\pi^0}$ sample {#subsec:neu}
--------------------------------------
Both experiments base the reconstruction of neutral events on the information from the electromagnetic calorimeter. In addition to the energies and positions of the four photons, a mass constraint must be imposed in order to reconstruct the decay vertex position. KTeV method imposes the ${\pi^0}$ mass to all photon pairs combinations, computing the vertex position for each pairing; only the two closest solutions are kept, and they are combined to produce the most probable value for the kaon decay vertex. NA48 method imposes the kaon invariant mass on the $4\gamma$ event, thus constraining the decay vertex position. The two $pio$ are reconstructed choosing the best of all the possible pairings between the photons.
[|c|c|c|c|]{}\
& & &\
$\KL \rightarrow 3{\pi^0}$ & $1.1\times 10^{-3}$ & $0.3\times
10^{-3}$ & $0.59\times 10^{-3}$\
& $1.2\times 10^{-3}$ & $0.9\times
10^{-3}$ & $0.96\times 10^{-3}$\
& $2.5\times 10^{-3}$ & $11.3\times 10^{-3}$ & -\
& - & $0.1\times 10^{-3}$ & -\
Both experiment define a $\chi^{2}$ variable that states the compatibility of each event with the $\KZ \rightarrow {\pi^0\pi^0}$ hypothesis. Background events from $\KL \rightarrow 3{\pi^0}$ decays with lost or merged photons have a high value of the $\chi^{2}$ and are rejected. The amount of remaining background is evaluated from a high-statistic sample of $3{\pi^0}$ events. Background fractions for KTeV and NA48 are summarized in table \[tbl:bkgneu\].
$\KS$ and $\KL$ identification {#subsec:tagging}
------------------------------
As already described in sections \[subsec:ktev\] and \[subsec:na48\], the two experiments use different techniques to distinguish $\KS$ from $\KL$. KTeV takes advantage of having two parallel beams: the $10$ cm separation allows to disentangle $\KS$ and $\KL$ by looking at the reconstructed decay vertex position in the transverse plane, in the case of charged events. In the case of neutral events, the energy centroid of the four photons is used, as shown in figure \[tagging\] left. The halo surrounding one of the two beams is due to events scattered in the regenerator before decaying: this effect is accurately studied in the $p_{\mathrm{T}}^{2}$ distribution of charged events, and is then introduce into a detailed simulation to evaluate the contribution in the neutral case.
=4.6cm =5.cm
NA48 uses the tagging method: a decay is identified as $\KS$ if its event time is within a $\pm2$ ns coincidence with a proton time measured by the tagger. The principle can be clearly illustrated for charged events, which can be identified as $\KS$ or $KL$ also on the basis of the vertical separation between the two beams at the drift chamber position. This is shown in figure \[tagging\] right, where the difference between the event time and the time of the nearest proton in the tagger is plotted. It is evident that the inefficiencies in identifying true $\KS$ decays is very small ($10^{-4}$ level). On the other hand, there is a sizeable mistagging probability in the case of $\KL$: the high rate of events in the $\KL$ beam causes accidental coincidences, and it turns out that $\sim 10$% of the true $\KL$ events are misidentified as $\KS$. The final data samples are corrected for both inefficiency and accidental tagging.
From event counting to $\mathcal{R}$ {#subsec:rcalc}
------------------------------------
Having identified $\KS$ and $\KL$, as well as charged and neutral decays, both experiments end up with four samples of events. In the case of KTeV, the samples correspond to the Vacuum beam ($\KL$ decays) and to Regenerator beam (mostly $\KS$) into ${\pi^+\pi^-}$ and ${\pi^0\pi^0}$ final modes. The striking difference in the decay vertex distributions for $\KS$ and $\KL$ translates in a large acceptance correction, which therefore must be precisely known. The correction is implemented using a highly detailed Monte Carlo simulation which includes all known effects, as trigger and detector efficiencies, regeneration, $\KS -\KL$ interference, detector apertures, etc. The value of ${\Re e\,({\varepsilon^\prime/\varepsilon})}$ is then obtained fitting the data in $10$ GeV bins in the kaon energy, in order to minimise residual differences in the energy spectra.
NA48 has two charged/neutral samples of tagged events (essentially $\KS$) and two other charged/neutral samples of untagged events ($\KL$ with a $\sim 10$% contamination of $\KS$). All samples are corrected for mistagging and trigger inefficiencies. The final result is computed by dividing the data into 20 bins of kaon energy from 70 to 170 GeV, and calculating the double ratio for each bin. To cancel the contribution from the different lifetimes to the acceptance, $\KL$ events are weighted with the $\KS$ lifetime as a function of the reconstructed proper decay time. After weighting, the $\KL$ and $\KS$ decay distributions become nearly identical and the size of the acceptance correction is drastically reduced. The weighting technique avoids the need of an extremely sophisticated simulation, although it results in a $\sim 35$% increase of the statistical error on $\mathrm{R}$. All corrections are applied to each bin separately, and the results are averaged using an unbiased estimator.
=5.cm
Results from KTeV and NA48 {#sec:results}
==========================
The latest results from KTeV [@ktev:eprime2001] and NA48 [@na48:eprime2001] are summarized in figure \[result\], together with the final results from NA31 and E731. The new world average value of ${\Re e\,({\varepsilon^\prime/\varepsilon})}$ is $(17.2\pm 1.8)\times 10^{-4}$. This result confirms the existence of direct violation in the neutral kaon system. Whether the measured size of ${\Re e\,({\varepsilon^\prime/\varepsilon})}$ is compatible with Standard Model expectations or is a hint that new physics is at work, this is still matter of debate: theoretical calculations suffers from big uncertainties in the determination of the hadronix mass matrix elements, thus their predictive power on ${\varepsilon^\prime/\varepsilon}$ is rather poor.
Establishing beyond doubt the existence of the direct violation mechanism has been a long experimental adventure. Both KTeV and NA48 have still other data samples to analyse, and hopefully KLOE will provide also its measurement of ${\Re e\,({\varepsilon^\prime/\varepsilon})}$ with a different method. We recently witnessed the first observation of violation in a system other than the neutral kaon system, namely in ${\rm B^0}
-{\rm \overline{B}^0}$ oscillations. violation studies are also being performed in the sector of $\mathrm{B}$ and $\mathrm{K}$ rare decays. Considering all these constraints together, there is reasonable hope that a deeper understanding of the violation mechanism will be achieved in the forthcoming years.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to warmly thank the organisers of the $10^{th}$ Lomonosov Conference on Elementary Particle Physics for the interesting meeting.
References {#references .unnumbered}
==========
[99]{}
J.H. Christenson et al., Phys. Rev. Lett. **13**, 138 (1964)
L. Wolfenstein, Phys. Rev. Lett. **13**, 562 (1964).
M. Kobayashi and K. Maskawa, Prog. Theor. Phys. **49**, 652 (1973).
M. Holder et al., Phys. Lett. B **40**, 141 (1972); M. Banner et al., Phys. Rev. Lett. **28**, 1957 (1972).
J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B **109**, 213 (1976); F.J. Gilman and M.B. Wise, Phys. Lett. B **83**, 83 (1979).
H. Burkhardt et al., Phys. Lett. B **206**, 169 (1988).
G. Barr et al., Phys. Lett. B **317**, 233 (1993).
L.K. Gibbons et al., Phys. Rev. Lett. **70**, 1203 (1993).
G. D. Barr *et al.*, CERN/SPSC/90-22 (1990).
K. Arisaka *et al.*, FN-580 (1992).
The KLOE Collaboration, LNF-93/002 (1993).
A. Lai [*et al.*]{} \[NA48 Collaboration\], CERN-EP-2001-067.
J. Graham \[KTeV Collaboration\], FNAL Seminar; R. Kessler, hep-ex/0110020.
A. Alavi *et. al.*, \[KTeV Collaboration\], Phys. Rev. Lett. **83**, 22 (1999).
V. Fanti et al., Phys. Lett. **B** 465, 335-348 (1999).
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[^1]: e-mail: Manuela.Cirilli@cern.ch
|
---
abstract: 'We propose a communication- and computation-efficient distributed optimization algorithm using second-order information for solving empirical risk minimization (ERM) problems with a nonsmooth regularization term. Our algorithm is applicable to both the primal and the dual ERM problem. Current second-order and quasi-Newton methods for this problem either do not work well in the distributed setting or work only for specific regularizers. Our algorithm uses successive quadratic approximations of the smooth part, and we describe how to maintain an approximation of the (generalized) Hessian and solve subproblems efficiently in a distributed manner. When applied to the distributed dual ERM problem, unlike state of the art that takes only the block-diagonal part of the Hessian, our approach is able to utilize global curvature information and is thus magnitudes faster. The proposed method enjoys global linear convergence for a broad range of non-strongly convex problems that includes the most commonly used ERMs, thus requiring lower communication complexity. It also converges on non-convex problems, so has the potential to be used on applications such as deep learning. Computational results demonstrate that our method significantly improves on communication cost and running time over the current state-of-the-art methods.'
author:
- |
\
Department of Mathematics\
National University of Singapore\
Singapore 119076\
\
Wisconsin Institute for Discovery\
University of Wisconsin-Madison\
Madison, Wisconsin 53715\
\
Department of Computer Sciences\
University of Wisconsin-Madison\
Madison, Wisconsin 53706
bibliography:
- '../../inexactprox.bib'
- '../distributederm.bib'
title: '[[A Distributed Quasi-Newton Algorithm for Primal and Dual Regularized Empirical Risk Minimization]{}]{}'
---
Introduction {#sec:intro}
============
We consider using multiple machines to solve the following regularized problem $$\min_{{{\boldsymbol w}}\in {\mathbb{R}}^d} \quad P\left ({{\boldsymbol w}}\right) \coloneqq
\xi\left( X^\top {{\boldsymbol w}}\right) + g\left( {{\boldsymbol w}}\right),
\label{eq:primal}$$ where $X$ is a $d$ by $n$ real-valued matrix, and $g$ is a convex, closed, and extended-valued proper function that can be nondifferentiable, or its dual problem $$\min_{{{\boldsymbol \alpha}}\in {\mathbb{R}}^n} \quad D\left({{\boldsymbol \alpha}}\right) \coloneqq g^*\left(X{{\boldsymbol \alpha}}\right) +
\xi^*\left(-{{\boldsymbol \alpha}}\right),
\label{eq:dual}$$ where for any given function $f(\cdot)$, $f^*$ denotes its convex conjugate $$f^*\left(z\right) \coloneqq \max_{y}\quad z^\top y - f\left(y\right).$$ Each column of $X$ represents a single data point or instance, and we assume that the set of data points is partitioned and spread across $K > 1$ machines (i.e. distributed *instance-wise*). We write $X$ as $$X \coloneqq \left[X_1, X_2,\dotsc, X_K \right]
\label{eq:storage}$$ where $X_k$ is stored exclusively on the $k$th machine. The dual variable ${{\boldsymbol \alpha}}$ is formed by concatenating ${{\boldsymbol \alpha}}_1,{{\boldsymbol \alpha}}_2,\dotsc,{{\boldsymbol \alpha}}_K$ where ${{\boldsymbol \alpha}}_k$ is the dual variable corresponding to $X_k$. We let ${\mathcal{I}^X}_1,\dotsc,{\mathcal{I}^X}_K$ denote the indices of the columns of $X$ corresponding to each of the $X_k$ matrices. We further assume that $\xi$ shares the same block-separable structure and can be written as follows: $$\xi\left( X^\top {{\boldsymbol w}}\right) = \sum_{k=1}^K \xi_k\left(X_k^\top {{\boldsymbol w}}\right),
\label{eq:finitesum}$$ and therefore in , we have $$\xi^*\left( -{{\boldsymbol \alpha}}\right) = \sum_{k=1}^K \xi^*_k\left( -{{\boldsymbol \alpha}}_k
\right).
\label{eq:finitesumdual}$$ For the ease of description and unification, when solving the primal problem, we also assume that there exists some partition ${\mathcal{I}^g}_1,\dotsc,{\mathcal{I}^g}_K$ of $\{1,\dotsc,d\}$ and $g$ is block-separable according to the partition: $$g\left({{\boldsymbol w}}\right) = \sum_{k=1}^K g_k\left({{\boldsymbol w}}_{{\mathcal{I}^g}_k}\right),
\label{eq:blockseparable}$$ though our algorithm can be adapted for non-separable $g$ with minimal modification, see the preliminary version [@LeeLW18a].
When we solve the primal problem , $\xi$ is assumed to be a differentiable function with Lipschitz continuous gradients, and is allowed to be nonconvex. On the other hand, when the dual problem is considered, for recovering the primal solution, we require strong convexity on $g$ and convexity on $\xi$, and $\xi$ can be either nonsmooth but Lipschitz continuous (within the area of interest), or Lipschitz continuously differentiable. Note that strong convexity of $g$ implies that $g^*$ is Lipschitz-continuously differentiable [@HirL01a Part E, Theorem 4.2.1 and Theorem 4.2.2], making have the same structure as such that both problems have one smooth and one nonsmooth term. There are several reasons for considering the alternative dual problem. First, when $\xi$ is nonsmooth, the primal problem becomes hard to solve as both terms are nonsmooth, meanwhile in the dual problem, $\xi^*$ is guaranteed to be smooth. Second, the number of variables in the primal and the dual problem are different. In our algorithm whose spatial and temporal costs are positively correlated to the number of variables, when the data set has much higher feature dimension than the number of data points, solving the dual problem can be more economical.
The bottleneck in performing distributed optimization is often the high cost of communication between machines. For or , the time required to retrieve $X_k$ over a network can greatly exceed the time needed to compute $\xi_k$ or its gradient with locally stored $X_k$. Moreover, we incur a delay at the beginning of each round of communication due to the overhead of establishing connections between machines. This latency prevents many efficient single-core algorithms such as coordinate descent (CD) and stochastic gradient and their asynchronous parallel variants from being employed in large-scale distributed computing setups. Thus, a key aim of algorithm design for distributed optimization is to improve the communication efficiency while keeping the computational cost affordable. Batch methods are preferred in this context, because fewer rounds of communication occur in distributed batch methods.
When the objective is smooth, many batch methods can be used directly in distributed environments to optimize them. For example, Nesterov’s accelerated gradient (AG) [@Nes83a] enjoys low iteration complexity, and since each iteration of AG only requires one round of communication to compute the new gradient, it also has good communication complexity. Although its supporting theory is not particularly strong, the limited-memory BFGS (LBFGS) method [@LiuN89a] is popular among practitioners of distributed optimization. It is the default algorithm for solving $\ell_2$-regularized smooth ERM problems in Apache Spark’s distributed machine learning library [@Men16a], as it is empirically much faster than AG (see, for example, the experiments in @WanLL16a). Other batch methods that utilize the Hessian of the objective in various ways are also communication-efficient under their own additional assumptions [@ShaSZ14a; @ZhaL15a; @LeeWCL17a; @ZhuCJL15a; @LinTLL14a].
However, when the objective is nondifferentiable, neither LBFGS nor Newton’s method can be applied directly. Leveraging curvature information from the smooth part ($\xi$ in the primal or $g^*$ in the dual) can still be beneficial in this setting. For example, the orthant-wise quasi-Newton method OWLQN [@AndG07a] adapts the LBFGS algorithm to the special nonsmooth case in which $g(\cdot) \equiv \|\cdot\|_1$ for , and is popular for distributed optimization of $\ell_1$-regularized ERM problems. Unfortunately, extension of this approach to other nonsmooth $g$ is not well understood, and the convergence guarantees are only asymptotic, rather than global. Another example is that for , state of the art distributed algorithms [@Yan13a; @LeeC17a; @ZheXXZ17a] utilize block-diagonal entries of the real Hessian of $g^*(X{{\boldsymbol \alpha}})$. To the best of our knowledge, for ERMs with *general* nonsmooth regularizers in the instance-wise storage setting, proximal-gradient-like methods [@WriNF09a; @BecT09a; @Nes13a] are the only practical distributed optimization algorithms with convergence guarantees for the primal problem . Since these methods barely use the curvature information of the smooth part (if at all), we suspect that proper utilization of second-order information has the potential to improve convergence speed and therefore communication efficiency dramatically. As for algorithms solving the dual problem , computing $X{{\boldsymbol \alpha}}$ in the instance-wise storage setting requires communicating a $d$-dimensional vector, and only the block-diagonal part of $\partial^2_{{{\boldsymbol \alpha}}} g^*(X{{\boldsymbol \alpha}})$ can be obtained easily. Therefore, global curvature information is not utilized in existing algorithms, and we expect that utilizing global second-order information of $g^*$ can also provide substantial benefits over the block-diagonal approximation approaches. We thus propose a practical distributed inexact variable-metric algorithm that can be applied to both and . Our algorithm uses gradients and updates information from previous iterations to estimate curvature of the smooth part in a communication-efficient manner. We describe construction of this estimate and solution of the corresponding subproblem. We also provide convergence rate guarantees, which also bound communication complexity. These rates improve on existing distributed methods, even those tailor-made for specific regularizers.
More specifically, We propose a distributed inexact proximal-quasi-Newton-like algorithm that can be used to solve both and under the instance-wise split setting that share the common structure of having a smooth term $f$ and a nonsmooth term $\Psi$. At each iteration with the current iterate $x$, our algorithm utilizes the previous update directions and gradients to construct a second-order approximation of the smooth part $f$ by the LBFGS method, and approximately minimizes this quadratic term plus the nonsmooth term $\Psi$ to obtain an update iteration $p$. $$p \approx \arg\min_{p}\, Q_H(p;x),
\label{eq:quadratic}$$ where $H$ is the LBFGS approximation of the Hessian of $f$ at $x$, and $$Q_H(p;x)\coloneqq \nabla f(x)^\top p + \frac12 p^\top H p +
\Psi(x+p) - \Psi(x).
\label{eq:Q}$$
For the primal problem , we believe that this work is the first to propose, analyze, and implement a practically feasible distributed optimization method for solving with general nonsmooth regularizer $g$ under the instance-wise storage setting. For the dual problem , our algorithm is the first to suggest an approach that utilizes global curvature information under the constraint of distributed data storage. This usage of non-local curvature information greatly improves upon state of the art for the distributed dual ERM problem which uses the block-diagonal parts of the Hessian only. An obvious drawback of the block-diagonal approach is that the convergence deteriorates with the number of machines, as more and more off-block-diagonal entries are ignored. In the extreme case, where there are $n$ machines such that each machine stores only one column of $X$, the block-diagonal approach reduces to a scaled proximal-gradient algorithm and the convergence is expected to be extremely slow. On the other hand, our algorithm has convergence behavior independent of number of machines and data distribution over nodes, and is thus favorable when many machines are used. Our approach has both good communication and computational complexities, unlike certain approaches that focus only on communication at the expense of computation (and ultimately overall time).
Contributions
-------------
We summarize our main contributions as follows.
- The proposed method is the first real distributed second-order method for the dual ERM problem that utilizes global curvature information of the smooth part. Existing second-order methods use only the block-diagonal part of the Hessian and suffers from asymptotic convergence speed as slow as proximal gradient, while our method enjoys fast convergence throughout. Numerical results show that our inexact proximal-quasi-Newton method is magnitudes faster than state of the art for distributed optimizing the dual ERM problem.
- We propose the first distributed algorithm for primal ERMs with general nonsmooth regularizers under the instance-wise split setting. Prior to our work, existing algorithms are either for a specific regularizer (in particular the $\ell_1$ norm) or for the feature-wise split setting, which is often impractical. In particular, it is usually easier to generate new data points than to generate new features, and each time new data points are obtained from one location, one needs to distribute their entries to different machines under the feature-wise setting.
- The proposed framework is applicable to both primal and dual ERM problems under the same instance-wise split setting, and the convergence speed is not deteriorated by the number of machines. Existing methods that applicable to both problems can deal with feature-wise split for the primal problem only, and their convergence degrades with the number of machines used, and are thus not suitable for large-scale applications where thousands of or more machines are used. This unification also reduces two problems into one and facilitates future development for them.
- Our analysis provides sharper convergence guarantees and therefore better communication efficiency. In particular, global linear convergence for a broad class of non-strongly convex problems that includes many popular ERM problems are shown, and an early linear convergence to rapidly reach a medium solution accuracy is proven for convex problems.
Organization
------------
We first describe the general distributed algorithm in Section \[sec:alg\]. Convergence guarantee, communication complexity, and the effect of the subproblem solution inexactness are analyzed in Section \[sec:analysis\]. Specific details for applying our algorithm respectively on the primal and the dual problem are given in Section \[sec:primaldual\]. Section \[sec:related\] discusses related works, and empirical comparisons are conducted in Section \[sec:exp\]. Concluding observations appear in Section \[sec:conclusions\].
Notation and Assumptions
------------------------
We use the following notation.
- $\|\cdot\|$ denotes the 2-norm, both for vectors and for matrices.
- Given any symmetric positive semi-definite matrix $H \in
{\mathbb{R}}^{d\times d}$ and any vector $p \in {\mathbb{R}}^d$, $\|p\|_H$ denotes the semi-norm $\sqrt{p^\top H p}$.
In addition to the structural assumptions of distributed instance-wise storage of $X$ in and the block separability of $\xi$ in , we also use the following assumptions throughout this work. When we solve the primal problem, we assume the following.
\[assum:primal\] The regularization term $g({{\boldsymbol w}})$ is convex, extended-valued, proper, and closed. The loss function $\xi(X^\top {{\boldsymbol w}})$ is $L$-Lipschitz continuously differentiable with respect to ${{\boldsymbol w}}$ for some $L>0$. That is, $$\left\|X^\top \xi'\left( X^\top {{\boldsymbol w}}_1 \right) - X^\top \xi' \left( X^\top
{{\boldsymbol w}}_2 \right)\right\| \leq L \left\|{{\boldsymbol w}}_1 -
{{\boldsymbol w}}_2\right\|,
\forall {{\boldsymbol w}}_1, {{\boldsymbol w}}_2 \in {\mathbb{R}}^d.
\label{eq:Lipschitz}$$
On the other hand, when we consider solving the dual problem, the following is assumed.
\[assum:dual\] Both $g$ and $\xi$ are convex. $g^*(X{{\boldsymbol \alpha}})$ is $L$-Lipschitz continuously differentiable with respect to ${{\boldsymbol \alpha}}$. Either $\xi^*$ is $\sigma$-strongly convex for some $\sigma > 0$, or the loss term $\xi(X^\top {{\boldsymbol w}})$ is $\rho$-Lipschitz continuous for some $\rho$.
Because a function is $\rho$-Lipschitz continuously differentiable if and only if its conjugate is $(1/\rho)$-strongly convex [@HirL01a Part E, Theorem 4.2.1 and Theorem 4.2.2], Assumption \[assum:dual\] implies that $g$ is $\|X^\top X\|/L$-strongly convex. From the same reasoning, $\xi^*$ is $\sigma$-strongly convex if only if $\xi$ is $(1 / \sigma)$ Lipschitz continuously differentiable. Convexity of the primal problem in Assumption \[assum:dual\] together with Slater’s condition guarantee strong duality @BoyV04a [Section 5.2.3], which then ensures is indeed an alternative to . Moreover, from KKT conditions, any optimal solution ${{\boldsymbol \alpha}}^*$ for gives us a primal optimal solution ${{\boldsymbol w}}^*$ for through $${{\boldsymbol w}}^* = \nabla g^*(X{{\boldsymbol \alpha}}^*).
\label{eq:wopt}$$
Algorithm {#sec:alg}
=========
We describe and analyze a general algorithmic scheme that can be applied to solve both the primal and dual problems under the instance-wise distributed data storage scenario . In Section \[sec:primaldual\], we discuss how to efficiently implement particular steps of this scheme for and .
Consider a general problem of the form $$\min_{x\in {\mathbb{R}}^N}\quad F(x) \coloneqq f(x) + \Psi(x),
\label{eq:f}$$ where $f$ is $L$-Lipschitz continuously differentiable for some $L >
0$ and $\Psi$ is convex, closed, proper, extended valued, and block-separable into $K$ blocks. More specifically, we can write $\Psi(x)$ as $$\Psi(x) = \sum_{k=1}^K \Psi_k(x_{{\mathcal{I}}_k}).
\label{eq:psiblockseparable}$$ where ${\mathcal{I}}_1,\dotsc,{\mathcal{I}}_K$ partitions $\{1,\dotsc,N\}$. We assume as well that for the $k$th machine, $\nabla_{{\mathcal{I}}_k} f(x)$ can be obtained easily after communicating a vector of size $O(d)$ across machines, and postpone the detailed gradient calculation until we discuss specific problem structures in later sections. Note that this $d$ is the primal variable dimension in and is independent of $N$.
The primal and dual problems are specific cases of the general form . For the primal problem we let $N =d$, $x = {{\boldsymbol w}}$, $f(\cdot) = \xi(X^\top
\cdot)$, and $\Psi(\cdot) = g(\cdot)$. The block-separability of $g$ gives the desired block-separability of $\Psi$ , and the Lipschitz-continuous differentiability of $f$ comes from Assumption \[assum:primal\]. For the dual problem , we have $N = n$, $x = {{\boldsymbol \alpha}}$, $f(\cdot) = g^*(X\cdot)$, and $\Psi(\cdot) = \xi^*(-\cdot)$. The separability follows from , where the partition reflects the data partition in and Lipschitz continuity from Assumption \[assum:dual\].
Each iteration of our algorithm has four main steps – (1) computing the gradient $\nabla f(x)$, (2) constructing an approximate Hessian $H$ of $f$, (3) solving a quadratic approximation subproblem to find an update direction $p$, and finally (4) taking a step $x + \lambda p$ either via line search or trust-region approach. The gradient computation step and part of the line search process is dependent on whether we are solving the primal or dual problem, and we defer the details to Section \[sec:primaldual\]. The approximate Hessian $H$ comes from the LBFGS algorithm [@LiuN89a]. To compute the update direction, we approximately solve , where $Q_H$ consists of a quadratic approximation to $f$ and the regularizer $\Psi$ as defined in . We then use either a line search procedure to determine a suitable stepsize $\lambda$ and perform the update $x \leftarrow x + \lambda
p$, or use some trust-region-like techniques to decide whether to accept the update direction with unit step size.
We now discuss the following issues in the distributed setting: communication cost in distributed environments, the choice and construction of $H$ that have low cost in terms of both communication and per machine computation, procedures for solving , and the line search and trust-region procedures for ensuring sufficient objective decrease.
Communication Cost Model
------------------------
For the ease of description, we assume the [*allreduce*]{} model of MPI [@MPI94a] throughout the work, but it is also straightforward to extend the framework to a master-worker platform. Under this [*allreduce*]{} model, all machines simultaneously fulfill master and worker roles, and for any distributed operations that aggregate results from machines, the resultant is broadcast to all machines.
This can be considered as equivalent to conducting one map-reduce operation and then broadcasting the result to all nodes. The communication cost for the allreduce operation on a $d$-dimensional vector under this model is $$\log \left( K \right) T_{\text{initial}} + d T_{\text{byte}},
\label{eq:commcost}$$ where $T_{\text{initial}}$ is the latency to establish connection between machines, and $T_{\text{byte}}$ is the per byte transmission time (see, for example, @ChaHPV07a [Section 6.3]).
The first term in also explains why batch methods are preferable. Even if methods that frequently update the iterates communicate the same amount of bytes, it takes more rounds of communication to transmit the information, and the overhead of $\log (K) T_{\text{initial}}$ incurred at every round of communication makes this cost dominant, especially when $K$ is large.
In subsequent discussion, when an allreduce operation is performed on a vector of dimension $O(d)$, we simply say that a round of $O(d)$ communication is conducted. We omit the latency term since batch methods like ours tend to have only a small constant number of rounds of communication per iteration. By contrast, non-batch methods such as CD or stochastic gradient require number of communication rounds per epoch equal to data size or dimension, and therefore face much more significant latency issues.
Constructing a good $H$ efficiently {#subsec:products}
-----------------------------------
We use the Hessian approximation constructed by the LBFGS algorithm [@LiuN89a] as our $H$ in , and propose a way to maintain it efficiently in a distributed setting. In particular, we show that most vectors involved can be stored perfectly in a distributed manner in accord with the partition ${\mathcal{I}}_k$ in , and this distributed storage further facilitates parallelization of most computation. Note that the LBFGS algorithm works even if the smooth part is not twice-differentiable, see Lemma \[lemma:sparsa\]. In fact, Lipschitz continuity of the gradient implies that the function is twice-differentiable almost everywhere, and generalized Hessian can be used at the points where the smooth part is not twice-differentiable. In this case, the LBFGS approximation is for the generalized Hessian.
Using the compact representation in @ByrNS94a, given a prespecified integer $m > 0$, at the $t$th iteration for $t > 0$, let $m(t) \coloneqq \min(m,t)$, and define $${{\boldsymbol s}}_i \coloneqq x^{i+1} - x^i, \quad
{{\boldsymbol y}}_i \coloneqq \nabla f (x^{i+1}) - \nabla f(x^i),
\quad \forall i.$$ The LBFGS Hessian approximation matrix is $$H_t = \gamma_t I - U_t M_t^{-1} U_t^\top,
\label{eq:Hk}$$ where $$\label{eq:M}
U_t \coloneqq \left[\gamma_t S_t, Y_t\right],\quad
M_t \coloneqq \left[\begin{array}{cc}
\gamma_t S_t^\top S_t, & L_t\\
L_t^\top & - D_t \end{array}\right],\quad
\gamma_t \coloneqq \frac{{{\boldsymbol y}}_{t-1}^\top {{\boldsymbol y}}_{t-1}}{{{\boldsymbol s}}_{t-1}^\top {{\boldsymbol y}}_{t-1}},
$$ and
\[eq:updates\] $$\begin{aligned}
S_t &\coloneqq \left[{{\boldsymbol s}}_{t-m(t)}, {{\boldsymbol s}}_{t-m(t)+1},\dotsc, {{\boldsymbol s}}_{t-1}\right],\\
Y_t &\coloneqq \left[{{\boldsymbol y}}_{t-m(t)}, {{\boldsymbol y}}_{t-m(t)+1},\dotsc, {{\boldsymbol y}}_{t-1}\right],\\
D_t &\coloneqq {\text{diag}}\left({{\boldsymbol s}}_{t-m(t)}^\top {{\boldsymbol y}}_{t-m(t)}, \dotsc,
{{\boldsymbol s}}_{t-1}^\top {{\boldsymbol y}}_{t-1}\right),\\
\left(L_t\right)_{i,j} &\coloneqq
\begin{cases}
{{\boldsymbol s}}_{t - m(t) - 1 + i}^\top {{\boldsymbol y}}_{t - m(t) - 1 + j}, &\text{
if } i > j,\\
0, &\text{ otherwise.}
\end{cases}\end{aligned}$$
For $t=0$ where no ${{\boldsymbol s}}_i$ and ${{\boldsymbol y}}_i$ are available, we either set $H_0 \coloneqq a_0 I$ for some positive scalar $a_0$, or use some Hessian approximation constructed using local data. More details are given in Section \[sec:primaldual\] when we discuss the primal and dual problems individually. If $f$ is not strongly convex, it is possible that is only positive semi-definite, making the subproblem ill-conditioned. In this case, we follow @LiF01a, taking the $m$ update pairs to be the most recent $m$ iterations for which the inequality $${{\boldsymbol s}}_i^\top {{\boldsymbol y}}_i \geq \delta {{\boldsymbol s}}_i^\top {{\boldsymbol s}}_i
\label{eq:safeguard}$$ is satisfied, for some predefined $\delta > 0$. It can be shown that this safeguard ensures that $H_t$ are always positive definite and the eigenvalues are bounded within a positive range. For a proof in the case that $f$ is twice-differentiable, see, for example, the appendix of @LeeW17a. For completeness, we provide a proof without the assumption of twice-differentiability of $f$ in Lemma \[lemma:sparsa\].
To construct and utilize this $H_t$ efficiently, we store $(U_t)_{{\mathcal{I}}_k,:}$ on the $k$th machine, and all machines keep a copy of the whole $M_t$ matrix as usually $m$ is small and this is affordable. Under our assumption, on the $k$th machine, the local gradient $\nabla_{{\mathcal{I}}_k} f$ can be obtained, and we will show how to compute the update direction $p_{{\mathcal{I}}_k}$ locally in the next subsection. Thus, since ${{\boldsymbol s}}_i$ are just the update direction $p$ scaled by the step size $\lambda$, it can be obtained without any additional communication. All the information needed to construct $H_t$ is hence available locally on each machine. We now consider the costs associated with the matrix $M_t^{-1}$. The matrix $M_t$, but not its inverse, is maintained for easier update. In practice, $m$ is usually much smaller than $N$, so the $O(m^3)$ cost of inverting the matrix directly is insignificant compared to the cost of the other steps. On contrary, if $N$ is large, the computation of the inner products ${{\boldsymbol s}}_i^\top {{\boldsymbol y}}_j$ and ${{\boldsymbol s}}_i^\top {{\boldsymbol s}}_j$ can be the bottleneck in constructing $M_t^{-1}$. We can significantly reduce this cost by computing and maintaining the inner products in parallel and assembling the results with $O(m)$ communication cost. At the $t$th iteration, given the new ${{\boldsymbol s}}_{t-1}$, because $U_t$ is stored disjointly on the machines, we compute the inner products of ${{\boldsymbol s}}_{t-1}$ with both $S_t$ and $Y_t$ in parallel via the summations $$\sum_{k=1}^K \left( (S_t)_{{\mathcal{I}}_k,:}^\top ({{\boldsymbol s}}_{t-1})_{{\mathcal{I}}_k} \right),\quad
\sum_{k=1}^K \left( (Y_t)_{{\mathcal{I}}_k,:}^\top ({{\boldsymbol s}}_{t-1})_{{\mathcal{I}}_k} \right),$$ requiring $O(m)$ communication of the partial sums on each machine. We keep these results until ${{\boldsymbol s}}_{t-1}$ and ${{\boldsymbol y}}_{t-1}$ are discarded, so that at each iteration, only $2m$ (not $O(m^2)$) inner products are computed.
Solving the Quadratic Approximation Subproblem to Find Update Direction {#subsec:sparsa}
-----------------------------------------------------------------------
The matrix $H_t$ is generally not diagonal, so there is no easy closed-form solution to . We will instead use iterative algorithms to obtain an approximate solution to this subproblem. In single-core environments, coordinate descent (CD) is one of the most efficient approaches for solving [@YuaHL12a; @KaiYDR14a; @SchT16a]. When $N$ is not too large, instead of the distributed approach we discussed in the previous section, it is possible to construct $H_t$ on all machines. In this case, a local CD process can be applied on all machines to save communication cost, in the price that all machines conduct the same calculation and the additional computational power from multiple machines is wasted. The alternative approach of applying proximal-gradient methods to may be more efficient in distributed settings, since they can be parallelized with little communication cost for large $N$.
The fastest proximal-gradient-type methods are accelerated gradient (AG) [@BecT09a; @Nes13a] and SpaRSA [@WriNF09a]. SpaRSA is a basic proximal-gradient method with spectral initialization of the parameter in the prox term. SpaRSA has a few key advantages over AG despite its weaker theoretical convergence rate guarantees. It tends to be faster in the early iterations of the algorithm [@YanZ11a], thus possibly yielding a solution of acceptable accuracy in fewer iterations than AG. It is also a descent method, reducing the objective $Q_H$ at every iteration, which ensures that the solution returned is at least as good as the original guess $p = 0$
In the rest of this subsection, we will describe a distributed implementation of SpaRSA for , with $H$ as defined in . The major computation is obtaining the gradient of the smooth (quadratic) part of , and thus with minimal modification, AG can be used with the same per iteration cost. To distinguish between the iterations of our main algorithm (i.e. the entire process required to update $x$ a single time) and the iterations of SpaRSA, we will refer to them by *main iterations* and *SpaRSA iterations* respectively.
Since $H$ and $x$ are fixed in this subsection, we will write $Q_H(\cdot;x)$ simply as $Q(\cdot)$. We denote the $i$th iterate of the SpaRSA algorithm as $p^{(i)}$, and we initialize $p^{(0)}
= 0$ whenever there is no obviously better choice. We denote the smooth part of $Q_H$ by $\hat f(p)$, and the nonsmooth $\Psi(x+p)$ by $\hat \Psi(p)$. $$\hat f(p) \coloneqq \nabla f(x)^\top p + \frac12 p^\top H p,\quad
\hat \Psi(p) \coloneqq \Psi(x+p) - \Psi(x).
\label{eq:quadfg}$$ At the $i$th iteration of SpaRSA, we define $$u^{(i)}_{\psi_i} \coloneqq p^{(i)} - \frac{\nabla \hat
f(p^{(i)})}{ \psi_i},
\label{eq:udef}$$ and solve the following subproblem: $$p^{(i+1)} = \arg \min_{p} \, \frac{1}{2} \left\|p -
u^{(i)}_{\psi_i} \right\|^2 + \frac{\hat \Psi(p)}{\psi_i},
\label{eq:dk}$$ where $\psi_i$ is defined by the following “spectral” formula: $$\psi_i = \frac{\left(p^{(i)} - p^{(i-1)}\right)^\top \left(\nabla
\hat f(p^{(i)}) - \nabla \hat
f(p^{(i-1)})\right)}{\left\|p^{(i)} - p^{(i-1)}\right\|^2}.
\label{eq:psi}$$ When $i=0$, we use a pre-assigned value for $\psi_0$ instead. (In our LBFGS choice for $H_t$, we use the value of $\gamma_t$ from as the initial estimate of $\psi_0$.) The exact minimizer of can be difficult to compute for general $\Psi$. However, approximate solutions of suffice to provide a convergence rate guarantee for solving [@SchRB11a; @SchT16a; @GhaS16a; @LeeW18a]. Since it is known (see Lemma \[lemma:sparsa\]) that the eigenvalues of $H$ are upper- and lower-bounded in a positive range after the safeguard is applied, we can guarantee that this initialization of $\psi_i$ is bounded within a positive range; see Section \[sec:analysis\]. The initial value of $\psi_i$ defined in is increased successively by a chosen constant factor $\beta>1$, and $p^{(i+1)}$ is recalculated from , until the following sufficient decrease criterion is satisfied: $$Q\left(p^{(i+1)}\right) \leq Q\left(p^{(i)}\right) -
\frac{\sigma_0 \psi_i}{2} \left\|p^{(i+1)} -
p^{(i)}\right\|^2,
\label{eq:accept}$$ for some specified $\sigma_0 \in (0,1)$. Note that the evaluation of $Q(p)$ needed in can be done efficiently through a parallel computation of $$\sum_{k=1}^K \frac12 \left(\nabla_{{\mathcal{I}}_k} \hat f\left( p \right) +
\nabla_{{\mathcal{I}}_k} f\left( x \right)\right)^\top p_{{\mathcal{I}}_k} + \hat
\Psi_k\left( p_{{\mathcal{I}}_k}\right).$$ From the boundedness of $H$, one can easily prove that is satisfied after a finite number of increases of $\psi_i$, as we will show in Section \[sec:analysis\]. In our algorithm, SpaRSA runs until either a fixed number of iterations is reached, or when some certain inner stopping condition for optimizing is satisfied.
For general $H$, the computational bottleneck of $\nabla \hat f$ would take $O(N^2)$ operations to compute the $Hp^{(i)}$ term. However, for our LBFGS choice of $H$, this cost is reduced to $O(mN + m^2)$ by utilizing the matrix structure, as shown in the following formula: $$\begin{aligned}
\nabla \hat f\left(p\right) = \nabla f\left( x \right)
+ H p
= \nabla f(x) + \gamma p - U_t \left(M_t^{-1} \left(U_t^\top
p\right)\right).
\label{eq:u}\end{aligned}$$ The computation of can be parallelized, by first parallelizing computation of the inner product $U_t^\top p^{(i)}$ via the formula $$\sum_{k=1}^K \left(U_t\right)_{{\mathcal{I}}_k,:}^\top p^{(i)}_{{\mathcal{I}}_k}$$ with $O(m)$ communication. (We implement the parallel inner products as described in Section \[subsec:products\].) We let each machine compute a subvector of $u$ in according to . From the block-separability of $\Psi$, the subproblem for computing $p^{(i)}$ can be decomposed into independent subproblems partitioned along ${\mathcal{I}}_1,\dotsc,{\mathcal{I}}_K$. The $k$th machine therefore locally computes $p^{(i)}_{{\mathcal{I}}_k}$ without communicating the whole vector. Then at each iteration of SpaRSA, partial inner products between $(U_t)_{{\mathcal{I}}_k,:}$ and $p^{(i)}_{{\mathcal{I}}_k}$ can be computed locally, and the results are assembled with an allreduce operation of $O(m)$ communication cost. This leads to a round of $O(m)$ communication cost per SpaRSA iteration, with the computational cost reduced from $O(mN)$ to $O(mN/K)$ per machine on average. Since both the $O(m)$ communication cost and the $O(mN/K)$ computational cost are inexpensive when $m$ is small, in comparison to the computation of $\nabla f$, one can afford to conduct multiple iterations of SpaRSA at every main iteration. Note that the total latency incurred over all allreduce operations as discussed in can be capped by setting a maximum iteration limit for SpaRSA. The distributed implementation of SpaRSA for solving is summarized in Algorithm \[alg:sparsa\].
Given $\beta, \sigma_0 \in (0,1)$, $M_t^{-1}$, $U_t$, $\gamma_t$, and ${\mathcal{I}}_k$; Set $p^{(0)}_{{\mathcal{I}}_k} \leftarrow 0$; $\psi = \gamma_t$; Compute $\psi$ in through $$\sum_{j=1}^K \left(p^{(i)}_{{\mathcal{I}}_j} -
p^{(i-1)}_{{\mathcal{I}}_j}\right)^\top
\left(\nabla_{{\mathcal{I}}_j} \hat f \left(p^{(i)} \right) - \nabla_{{\mathcal{I}}_j}
\hat f \left(p^{(i-1)} \right)\right),\quad\text{ and }\quad
\sum_{j=1}^K
\left\|p^{(i)}_{{\mathcal{I}}_j} - p^{(i-1)}_{{\mathcal{I}}_j}\right\|^2;$$ Obtain $$U_t^\top p^{(i)} = \sum_{j=1}^K \left(U_t\right)_{{\mathcal{I}}_j,:}^\top
p^{(i)}_{{\mathcal{I}}_j};$$ Compute $$\nabla_{{\mathcal{I}}_k} \hat
f\left(p^{(i)}\right) = \nabla_{{\mathcal{I}}_k} f \left(x\right) + \gamma
p^{(i)}_{{\mathcal{I}}_k} -
\left(U_t\right)_{{\mathcal{I}}_k,:}\left(M_t^{-1} \left(U_t^\top
p^{(i)}\right)\right)$$ by ; Solve on coordinates indexed by ${\mathcal{I}}_k$ to obtain $p_{{\mathcal{I}}_k}$; $p^{(i+1)}_{{\mathcal{I}}_k} \leftarrow p_{{\mathcal{I}}_k}$; $\psi_i \leftarrow \psi$; Break; $\psi \leftarrow \beta^{-1} \psi$; Re-solve with the new $\psi$ to obtain a new $p_{{\mathcal{I}}_k}$; Break if some stopping condition is met;
Sufficient Function Decrease
----------------------------
After obtaining an update direction $p$ by approximately solving , we need to ensure sufficient objective decrease. This is usually achieved by some line-search or trust-region procedure. In this section, we describe two such approaches, one based on backtracking line search for the step size, and one based on a trust-region like approach that modifies $H$ repeatedly until an update direction is accepted with unit step size.
For the line-search approach, we follow @TseY09a by using a modified-Armijo-type backtracking line search to find a suitable step size $\lambda$. Given the current iterate $x$, the update direction $p$, and parameters $\sigma_1, \theta \in (0,1)$, we set $$\Delta \coloneqq \nabla f\left( x \right)^\top p + \Psi\left(x + p
\right) - \Psi\left(x \right)
\label{eq:Delta}$$ and pick the step size as the largest of $\theta^0, \theta^1,\dotsc$ satisfying $$F\left(x + \lambda p \right) \leq F\left( x \right) + \lambda
\sigma_1 \Delta.
\label{eq:armijo}$$ The computation of $\Delta$ is negligible as all the terms are involved in $Q(p;x)$, and $Q(p;x)$ is evaluated in the line search procedure of SpaRSA. For the function value evaluation, the objective values of both and can be evaluated efficiently if we precompute $Xp$ or $X^\top p$ in advance and conduct all reevaluations through this vector but not repeated matrix-vector products. Details are discussed in Section \[sec:primaldual\]. Note that because $H_t$ defined in attempts to approximate the real Hessian, empirically the unit step $\lambda=1$ frequently satisfies , so we use the value $1$ as the initial guess. For the trust-region-like procedure, we start from the original $H$, and use the same $\sigma_1, \theta \in (0,1)$ as above. Whenever the sufficient decrease condition $$F\left( x + p \right) - F\left( x \right) \leq \sigma_1 Q_H(p;x)
\label{eq:decrease}$$ is not satisfied, we scale up $H$ by $H \leftarrow H/\theta$, and resolve , either from $0$ or from the previously obtained solution $p$ if it gives an objective better than $0$. We note that when $\Psi$ is not present, both the backtracking approach and the trust-region one generate the same iterates. But when $\Psi$ is incorporated, the two approaches may generate different updates. Similar to the line-search approach, the evaluation of $Q_H(p;x)$ comes for free from the SpaRSA procedure, and usually the original $H$ generates update steps satisfying . Therefore, solving multiple times per main iteration is barely encountered in practice.
The trust-region procedure may be more expensive than line search because solving the subproblem again is more expensive than trying a different step size, although both cases are empirically rare. But on the other hand, when there are additional properties of the regularizer such as sparsity promotion, a potential benefit of the trust-region approach is that it might be able to identify the sparsity pattern earlier because unit step size is always used.
Our distributed algorithm for is summarized in Algorithm \[alg:proximalbfgs\]. We refer to the line search and trust-region variants of the algorithm as DPLBFGS-LS and DPLBFGS-TR respectively, and we will refer to them collectively as simply DPLBFGS.
Given $\theta, \sigma_1 \in (0,1)$, $\delta > 0$, an initial point $x=x^0$, a partition $\{{\mathcal{I}}_k\}_{k=1}^K$ satisfying ; Obtain $F(x)$; Compute $\nabla f(x)$; Initialize $H$; Update $U_{{\mathcal{I}}_k,:}$, $M$, and $\gamma$ by -; Compute $M^{-1}$; Implicitly form a new $H$ from ; Solve using some existing distributed algorithm to obtain $p_{{\mathcal{I}}_k}$; Solve using Algorithm \[alg:sparsa\] in a distributed manner to obtain $p_{{\mathcal{I}}_k}$; Compute $\Delta$ defined in ; $\lambda = \theta^i$; Compute $F(x + \lambda p)$; Break; $\lambda = 1$; Compute $Q_H(p;x)$; $H \leftarrow H / \theta$; Re-solve to obtain update $p_{{\mathcal{I}}_k}$; Compute $Q_H(p;x)$; $x_{{\mathcal{I}}_k} \leftarrow x_{{\mathcal{I}}_k} +\lambda p_{{\mathcal{I}}_k}$, $F(x)
\leftarrow F(x + \lambda p)$; $x^{t+1} \coloneqq x$; $({{\boldsymbol s}}_t)_{{\mathcal{I}}_k} \leftarrow x^{t+1}_{{\mathcal{I}}_k} -
x^t_{{\mathcal{I}}_k}$, $({{\boldsymbol y}}_t)_{{\mathcal{I}}_k} \leftarrow \nabla_{{\mathcal{I}}_k}
f(x^{t+1} - \nabla_{{\mathcal{I}}_k} f (x^t)$;
Cost Analysis {#subsec:cost}
-------------
We now describe the computational and communication cost of our algorithms. The computational cost for each machine depends on which $X_k$ is stored locally and the size of $|{\mathcal{I}}_k|$, and for simplicity we report the computational cost *averaged over all machines*. The communication costs do not depend on $X_k$.
For the distributed version of Algorithm \[alg:sparsa\], each iteration costs $$O\left(\frac{N}{K} + \frac{mN}{K} + m^2\right) =
O\left(\frac{mN}{K} + m^2\right)
\label{eq:costsparsa}$$ in computation, where the $N/K$ term is for the vector additions in , and $$O\left(m + \text{number of times \eqref{eq:accept} is
evaluated}\right)$$ in communication. In the next section, we will show that is accepted within a fixed number of times and thus the overall communication cost is $O(m)$.
For DPLBFGS, we will give details in Section \[sec:primaldual\] that for both and , each gradient evaluation for $f$ takes $O(\#\text{nnz} / K)$ per machine computation in average and $O(d)$ in communication, where \#nnz is the number of nonzero elements in the data matrix $X$. As shown in the next section, in one main iteration, the number of function evaluations in the line search is bounded, and its cost is negligible if we are using the same $p$ but just different step sizes; see Section \[sec:primaldual\]. For the trust region approach, the number of times for modifying $H$ and resolving is also bounded, and thus the asymptotical cost is not altered. In summary, the computational cost per main iteration is therefore $$O\left(\frac{\#\text{nnz}}{K} + \frac{mN}{K} +
m^3 + \frac{N}{K}\right) = O\left( \frac{\#\text{nnz}}{K} +
\frac{mN}{K} + m^3 \right),
\label{eq:costmain}$$ and the communication cost is $$O\left( 1 + d \right) = O\left( d \right),$$ where the $O(1)$ part is for function value evaluation and checking the safeguard . We note that the costs of Algorithm \[alg:sparsa\] are dominated by those of DPLBFGS if a fixed number of SpaRSA iterations is conducted every main iteration.
Convergence Rate and Communication Complexity Analysis {#sec:analysis}
======================================================
The use of an iterative solver for the subproblem generally results in an inexact solution. We first show that running SpaRSA for any fixed number of iterations guarantees a step $p$ whose accuracy is sufficient to prove overall convergence.
\[lemma:sparsa\] Consider optimizing by DPLBFGS. By using $H_t$ as defined in with the safeguard mechanism in , we have the following.
We have $L^2/\delta \geq \gamma_t \geq \delta$ for all $t > 0$, where $L$ is the Lipschitz constant for $\nabla f$. Moreover, there exist constants $c_1 \geq c_2 > 0$ such that $c_1 I
\succeq H_t \succeq c_2 I$ for all $t>0$.
At every SpaRSA iteration, the initial estimate of $\psi_i$ is bounded within the range of $$\left[\min\left\{c_2, \delta\right\}, \max\left\{c_1,
\frac{L^2}{\delta}\right\}\right],$$ and the final accepted value $\psi_i$ is upper-bounded.
SpaRSA is globally Q-linear convergent in solving . Therefore, there exists $\eta \in [0,1)$ such that if we run at least $S$ iterations of SpaRSA for all main iterations for any $S>0$, the approximate solution $p$ satisfies $$-\eta^S Q^* = \eta^S\left(Q\left(0\right) - Q^*\right) \geq
Q\left(p\right) - Q^*,
\label{eq:approx}$$ where $Q^*$ is the optimal objective of .
Lemma \[lemma:sparsa\] establishes how the number of iterations of SpaRSA affects the inexactness of the subproblem solution. Given this measure, we can leverage the results developed in @LeeW18a [@PenZZ18a] to obtain iteration complexity guarantees for our algorithm. Since in our algorithm, communication complexity scales linearly with iteration complexity, this guarantee provides a bound on the amount of communication. In particular, our method communicates $O(d+mS)$ bytes per iteration (where $S$ is the number of SpaRSA iterations used, as in Lemma \[lemma:sparsa\]) and the second term can usually be ignored for small $m$.
We show next that the step size generated by our line search procedure in DPLBFGS-LS is lower bounded by a positive value.
\[lemma:delta\] Consider such that $f$ is $L$-Lipschitz differentiable and $\Psi$ is convex. If SpaRSA is run at least $S$ iterations in solving , the corresponding $\Delta$ defined in satisfies $$\Delta \leq -\frac{c_2 \left\|p\right\|^2}{
1 + \eta^{\frac{S}{2}}},
\label{eq:deltabound}$$ where $\eta$ and $c_2$ are the same as that defined in Lemma \[lemma:sparsa\]. Moreover, the backtracking subroutine in DPLBFGS-LS terminates in finite number of steps and produces a step size $$\lambda \geq \min\left\{1, \frac{2\theta\left(1 -
\sigma_1\right) c_2}{L\left(1 + \eta^{{S}/{2}}\right)}\right\}
\label{eq:linesearchbound2}$$ satisfying .
We also show that for the trust-region technique, at one main iteration, the number of times we solve the subproblem until a step is accepted is upper-bounded by a constant.
\[lemma:Hbound\] For DPLBFGS-TR, suppose each time when we solve we have guarantee that the objective value is no worse than $Q(0)$. Then when is satisfied, we have that $$\|H_t\| \leq c_1 \max\left\{1, \frac{L
}{c_2 \theta}\right\}.
\label{eq:m1}$$ Moreover, at each main iteration, the number of times we solve with different $H$ is upper-bounded by $$\max\left\{ 1, \left\lceil\log_{\theta} \frac{c_2}{L}\right\rceil \right\}$$
Note that the bound in Lemma \[lemma:Hbound\] is independent to the number of SpaRSA iterations used. It is possible that one can incorporate the subproblem suboptimality to derive tighter but more complicated bounds, but for simplicity we use the current form of Lemma \[lemma:Hbound\].
The results in Lemmas \[lemma:delta\]-\[lemma:Hbound\] are just worst-case guarantees; in practice we often observe that the line search procedure terminates with $\lambda=1$ for our original choice of $H$, as we see in our experiments. This also indicates that in most of the cases, is satisfied with the original LBFGS Hessian approximation without scaling $H$. We now lay out the main theoretical results in Theorems \[thm:Fconvex\] to \[thm:Fnonconvex\], which describe the iteration and communication complexity under different conditions on the function $F$. In all these results, we assume the following setting:
> We apply DPLBFGS to solve the main problem , running Algorithm \[alg:sparsa\] for $S$ iterations in each main iteration. Let $x^t$, $\lambda_t$, and $H_t$ be respectively the $x$ vector, the step size, and the final accepted quadratic approximation matrix at the $t$th iteration of DPLBFGS for all $t\geq 0$. Let $M$ be the supremum of $\|H_t\|$ for all $t$ (which is either $c_1$ or $c_1 L / (c_2
> \theta)$ according to Lemmas \[lemma:sparsa\] and \[lemma:Hbound\]), and $\bar\lambda$ be the infimum of the step sizes over iterations (either $1$ or the bound from Lemma \[lemma:delta\]). Let $F^*$ be the optimal objective value of , $\Omega$ the solution set, and $P_{\Omega}$ the (convex) projection onto $\Omega$.
\[thm:Fconvex\] If $F$ is convex, given an initial point $x^0$, assume $$R_0 \coloneqq \sup_{x: F\left( x \right) \leq F\left( x^0
\right)}\quad\left\|x - P_\Omega(x)\right\|
\label{eq:R0}$$ is finite, we obtain the following expressions for rate of convergence of the objective value.
When $$F(x^t) - F^* \geq \left( x^t - P_{\Omega}\left( x^t
\right) \right)^\top H_t \left( x^t - P_{\Omega}\left(
x^t\right)\right),$$ the objective converges linearly to the optimum: $$\frac{F(x^{t+1}) - F^*}{F(x_t)^- F^*} \leq 1 - \frac{\left(
1 - \eta^S \right)\sigma_1 \lambda_t}{2}.$$
For any $t \geq t_0$, where $$t_0 \coloneqq \arg \min\{t\mid M R_0^2 > F\left( x^t
\right) - F^*\},$$ we have $$\begin{aligned}
F\left(x^t\right) - F^* &\leq \frac{2 M R_0^2}{\sigma_1 (1 -
\eta^S)\sum_{i=t_0}^{t-1} \lambda_t + 2}.\end{aligned}$$ Moreover, $$t_0 \leq \max\left\{ 0, 1 + \frac{2}{\sigma_1 (1 - \eta^s)
\bar{\lambda}} \log \frac{f\left( x^0 \right) - f^*}{M R_0^2}
\right\}.$$
Therefore, for any $\epsilon > 0$, the number of rounds of $O(d)$ communication required to obtain an $x^t$ such that $F(x^t) - F^* \leq
\epsilon$ is at most $$\begin{cases}
O\left(\max\left\{ 0, 1 + \frac{2}{\sigma_1 (1 - \eta^s)
\bar{\lambda}} \log \frac{F\left( x^0 \right) - F^*}{M
R_0^2}\right\}
+ \frac{2 M R_0^2}{\sigma_1 \bar{\lambda} \left( 1 - \eta^S \right)\epsilon}
\right)&\text{ if } \epsilon < M R_0^2,\\
O\left(\max\left\{ 0, 1 + \frac{2}{\sigma_1 \left( 1 - \eta^S \right)
\bar{\lambda}} \log \frac{F\left( x^0 \right) -
F^*}{\epsilon} \right\}\right)
&\text{ else}.
\end{cases}$$
When $F$ is convex and the quadratic growth condition $$F\left( x \right) - F^* \geq \frac{\mu}{2} \left\| x -
P_{\Omega}\left( x \right) \right\|^2, \quad \forall x \in {\mathbb{R}}^N
\label{eq:qg}$$ holds for some $\mu > 0$, we get a global Q-linear convergence rate: $$\frac{F\left(x^{t+1}\right) - F^*}{F\left( x^t \right) - F^*}
\leq 1 - \lambda_t \sigma_1 \left(1 - \eta^S\right)\cdot
\begin{cases}
\frac{\mu}{4\|H_t\|}, &\text{ if } \mu \leq 2 \|H_t\|,\\
1 - \frac{\|H_t\|}{\mu}, &\text{ else.}
\end{cases}
\label{eq:qlinear}$$ Therefore, the rounds of $O(d)$ communication needed for getting an $\epsilon$-accurate objective is $$\begin{cases}
O\left(\max\left\{ 0, 1 + \frac{2}{\sigma_1 (1 - \eta^s)
\bar{\lambda}} \log \frac{F\left( x^0 \right) - F^*}{M
R_0^2}\right\}
+ \frac{4M}{\mu \bar{\lambda}\sigma_1 \left( 1 - \eta^S \right)}
\log\frac{M R_0^2}{\epsilon}
\right)&\text{ if } \epsilon < M R_0^2, \mu \leq 2 M,\\
O\left(\max\left\{ 0, 1 + \frac{2}{\sigma_1 (1 - \eta^s)
\bar{\lambda}} \log \frac{F\left( x^0 \right) - F^*}{M
R_0^2}\right\}
+ \frac{\mu}{(\mu - M) \bar{\lambda}\sigma_1 \left( 1 - \eta^S \right)}
\log\frac{M R_0^2}{\epsilon}
\right)&\text{ if } \epsilon < M R_0^2, \mu > 2 M,\\
O\left( 0, 1 + \frac{2}{\sigma_1 \left( 1 - \eta^S \right)
\bar{\lambda}} \log \frac{F\left( x^0 \right) - F^*}{\epsilon} \right)
&\text{ if } \epsilon \geq MR_0^2.
\end{cases}$$
Suppose that the following relaxation of strong convexity holds: There exists $\mu > 0$ such that for any $x\in {\mathbb{R}}^N$ and any $a
\in [0,1]$, we have $$F\left(a x + \left(1 - a\right)
P_{\Omega}\left(x\right)\right)
\leq a F\left(x\right) +
\left(1 - a\right) F^* - \frac{\mu a\left( 1 - a \right)}{2}
\left\|x - P_{\Omega}\left(x\right)\right\|^2.
\label{eq:strong}$$ Then DPLBFGS converges globally at a Q-linear rate faster than . More specifically, $$\begin{aligned}
\frac{F\left(x^{t+1}\right) - F^*}{F\left( x^t \right) - F^*}
\leq
1 - \frac{\lambda_t \sigma_1 \left(1 -\eta^S \right)\mu}{\mu +
\|H_t\|}.
$$ Therefore, to get an approximate solution of that is $\epsilon$-accurate in the sense of objective value, we need to perform at most $$\begin{cases}
O\left(\max\left\{ 0, 1 + \frac{2}{\sigma_1 (1 - \eta^s)
\bar{\lambda}} \log \frac{F\left( x^0 \right) - F^*}{M
R_0^2}\right\} +
\frac{\mu + M}{\mu \sigma_1 \bar{\lambda} \left( 1 -
\eta^S \right)} \log\frac{M R_0^2}{\epsilon} \right)
&\text{ if } \epsilon < M R_0^2,\\
O\left( 0, 1 + \frac{2}{\sigma_1 \left( 1 - \eta^S \right)
\bar{\lambda}} \log \frac{F\left( x^0 \right) - F^*}{\epsilon} \right)
&\text{ else}.
\end{cases}
$$ rounds of $O(d)$ communication.
\[thm:Fnonconvex\] If $F$ is non-convex, the norm of $$G_t \coloneqq \arg\min_p \quad \nabla f\left(x^t\right)^\top p +
\frac{\|p\|^2}{2}
+ \Psi\left(x + p\right)$$ converges to zero at a rate of $O(1 / \sqrt{t})$ in the following sense: $$\begin{aligned}
\min_{0 \leq i \leq t }\left\|G_i\right\|^2 \leq \frac{F\left( x^0
\right) - F^*}{\sigma_1 \left( t+1 \right)} \frac{M^2\left(1 +
\frac{1}{c_2} + \sqrt{1 - \frac{2}{M} +
\frac{1}{c_2^2}}\right)^2}{2 c_2(1 - \eta^S) \min_{0\leq i \leq t}
\lambda_i }.\end{aligned}$$ Moreover, if there are limit points in the sequence $\{x^0,
x^1,\dotsc\}$, then all limit points are stationary.
Note that it is known that the norm of $G_t$ is zero if and only if $x^t$ is a stationary point, so this measure serves as an indicator for the first-order optimality condition. The class of quadratic growth includes many non-strongly-convex ERM problems. Especially, it contains problems of the form $$\min_{x\in \mathcal X}\quad g\left( Ax \right) + b^\top x,
\label{eq:qgform}$$ where $g$ is strongly convex, $A$ is a matrix, $b$ is a vector, and $\mathcal{X}$ is a polyhedron. Commonly seen non-strongly-convex ERM problems including $\ell_1$-regularized logistic regression, LASSO, and the dual problem of support vector machines all fall in the form and therefore our algorithm enjoys global linear convergence on them.
Solving the Primal and the Dual Problem {#sec:primaldual}
=======================================
Now we discuss details on how to apply DPLBFGS described in the previous section to the specific problems and respectively. We discuss how to obtain the gradient of the smooth part $f$ and how to conduct line search efficiently under distributed data storage. For the dual problem, we additionally describe how to recover a primal solution from our dual iterates.
Primal Problem {#subsec:primal}
--------------
Recall that the primal problem is $
\min_{{{\boldsymbol w}}\in {\mathbb{R}}^d} \xi\left( X^\top {{\boldsymbol w}}\right) + g\left( {{\boldsymbol w}}\right),
$ and is obtained from the general form by having $N =d$, $x = {{\boldsymbol w}}$, $f(\cdot) = \xi(X^\top \cdot)$, and $\Psi(\cdot) =
g(\cdot)$. The gradient of $\xi$ with respect to ${{\boldsymbol w}}$ is $$X \nabla \xi(X^\top {{\boldsymbol w}}) = \sum_{k=1}^K \left(X_k
\nabla \xi_k (X_k^\top {{\boldsymbol w}})\right).$$ We see that, except for the sum over $k$, the computation can be conducted locally provided ${{\boldsymbol w}}$ is available to all machines. Our algorithm maintains $X_k^\top {{\boldsymbol w}}$ on the $k$th machine throughout, and the most costly steps are the matrix-vector multiplications between $X_k$ and $\nabla \xi_k(X_k^\top {{\boldsymbol w}})$. Clearly, computing $X_k^\top {{\boldsymbol w}}$ and $X_k \nabla
\xi_k(X_k^\top {{\boldsymbol w}})$ both cost $O(\#\text{nnz} / K)$ in average among the $K$ machines. The local $d$-dimensional partial gradients are then aggregated through an allreduce operation using a round of $O(d)$ communication.
To initialize the approximate Hessian matrix $H$ at $t=0$, we set $H_0
\coloneqq a_0 I$ for some positive scalar $a_0$. In particular, we use $$a_0 \coloneqq \frac{\left|\nabla f({{\boldsymbol w}}_0)^\top \nabla^2 f({{\boldsymbol w}}_0) \nabla
f({{\boldsymbol w}}_0)\right|}{\left\|\nabla f({{\boldsymbol w}}_0)\right\|^2},
\label{eq:a0}$$ where $\nabla^2 f({{\boldsymbol w}}_0)$ denotes the generalized Hessian when $f$ is not twice-differentiable.
For the function value evaluation part of line search, each machine will compute $\xi_k(X_k^\top{{\boldsymbol w}}+ \lambda X_k^\top) +
g_k({{\boldsymbol w}}_{{\mathcal{I}^g}_k} + \lambda p_{{\mathcal{I}^g}_k})$ (the left-hand side of ) and send this scalar over the network. Once we have precomputed $ X_k^\top {{\boldsymbol w}}$ and $X_k^\top p$, we can quickly obtain $X_k^\top({{\boldsymbol w}}+ \lambda p)$ for any value of $\lambda$ without having to performing matrix-vector multiplications. Aside from the communication needed to compute the summation of the $f_k$ terms in the evaluation of $f$, the only other communication needed is to share the update direction $p$ from subvectors $p_{{\mathcal{I}^g}_k}$. Thus, two rounds of $O(d)$ communication are incurred per main iteration.
Dual Problem {#subsec:dual}
------------
Now consider applying DPLBFGS to the dual problem . To fit it into the general form , we have $N = n$, $x = {{\boldsymbol \alpha}}$, $f(\cdot) = g^*(X\cdot)$, and $\Psi(\cdot)
= -\xi^*(-\cdot)$. In this case, we need a way to efficiently obtain the vector $${{\boldsymbol z}}\coloneqq X{{\boldsymbol \alpha}}$$ on each machine in order to compute $g^*\left( X{{\boldsymbol \alpha}}\right)$ and the gradient $X^\top \nabla g^*(X{{\boldsymbol \alpha}})$.
Since each machine has access to some columns of $X$, it is natural to split ${{\boldsymbol \alpha}}$ according to the same partition. Namely, we set ${\mathcal{I}}_k$ as described in to ${\mathcal{I}^X}_k$. Every machine can then individually compute $X_k {{\boldsymbol \alpha}}_k$, and after one round of $O(d)$ communication, each machine has a copy of ${{\boldsymbol z}}= X{{\boldsymbol \alpha}}=
\sum_{k=1}^K X_k {{\boldsymbol \alpha}}_k$. After using ${{\boldsymbol z}}$ to compute $\nabla_{{{\boldsymbol z}}}
g^*({{\boldsymbol z}})$, we can compute the gradient $\nabla_{{\mathcal{I}^X}_k} g^*(X{{\boldsymbol \alpha}}) =
X_k^\top \nabla g^*(X{{\boldsymbol \alpha}})$ at a computation cost of $O(\#\text{nnz}/K)$ in average among the $K$ machines, matching the cost of computing $X_k
{{\boldsymbol \alpha}}_k$ earlier.
To construct the approximation matrix $H_0$ for the first main iteration, we make use of the fact that the (generalized) Hessian of $g^*(X{{\boldsymbol \alpha}})$ is $$X^\top \nabla^2 g^*({{\boldsymbol z}}) X.
\label{eq:Hess}$$ Each machine has access to one $X_k$, so we can construct the block-diagonal proportion of this Hessian locally for the part corresponding to ${\mathcal{I}^X}_k$. Therefore, the block-diagonal part of the Hessian is a natural choice for $H_0$. Under this choice of $H_0$, the subproblem is decomposable along the ${\mathcal{I}^X}_1,\dotsc,{\mathcal{I}^X}_K$ partition and one can apply algorithms other than SpaRSA to solve this. For example, we can apply CD solvers on the independent local subproblems, as done by [@LeeC17a; @Yan13a; @ZheXXZ17a]. As it is observed in these works that the block-diagonal approaches tend to converge fast at the early iterations, we use it for initializing our algorithm. In particular, we start with the block-diagonal approach, until $U_t$ has $2m$ columns, and then we switch to the LBFGS approach. This turns out to be much more efficient in practice than starting with the LBFGS matrix.
For the line search process, we can precompute the matrix-vector product $Xp$ with the same $O(d)$ communication and $O(\#\text{nnz}/K)$ per machine average computational cost as computing $X{{\boldsymbol \alpha}}$. With $X{{\boldsymbol \alpha}}$ and $Xp$, we can now evaluate $X{{\boldsymbol \alpha}}+ \lambda Xp$ quickly for different $\lambda$, instead of having to perform a matrix-vector multiplication of the form $X({{\boldsymbol \alpha}}+ \lambda p)$ for every $\lambda$. For most common choices of $g$, given ${{\boldsymbol z}}$, the computational cost of evaluating $g^*({{\boldsymbol z}})$ is $O(d)$. Thus, the cost of this efficient implementation per backtracking iteration is reduced to $O(d)$, with an overhead of $O(\#\text{nnz}/K)$ per machine average per main iteration, while the naive implementation takes $O(\#\text{nnz}/K)$ per backtracking iteration. After the sufficient decrease condition holds, we locally update ${{\boldsymbol \alpha}}_k$ and $X{{\boldsymbol \alpha}}$ using $p_{{\mathcal{I}^X}_k}$ and $Xp$. For the trust region approach, the two implementations take the same cost.
### Recovering a Primal Solution
In general, the algorithm only gives us an approximate solution to the dual problem , which means the formula $${{\boldsymbol w}}\left({{\boldsymbol \alpha}}\right) \coloneqq \nabla g^*\left( X {{\boldsymbol \alpha}}\right).
\label{eq:walpha}$$ used to obtain a primal optimal point from a dual optimal point (equation , derived from KKT conditions) is no longer guaranteed to even return a feasible point without further assumptions. Nonetheless, this is a common approach and under certain conditions (the ones we used in Assumption \[assum:dual\]), one can provide guarantees on the resulting point.
It can be shown from existing works [@Bac15a; @ShaZ12a] that when ${{\boldsymbol \alpha}}$ is not an optimum for , for , certain levels of primal suboptimality can be achieved, which depend on whether $\xi$ is Lipschitz-continuously differentiable or Lipschitz continuous. This is the reason why we need the corresponding assumptions in Assumption \[assum:dual\]. A summary of those results is available in [@LeeC17a]. We restate their results here for completeness but omit the proof.
\[thm:dualitygap\] Given any $\epsilon > 0$, and any dual iterate ${{\boldsymbol \alpha}}\in {\mathbb{R}}^n$ satisfying $$D({{\boldsymbol \alpha}}) - \min_{\bar{{\boldsymbol \alpha}}\in {\mathbb{R}}^n} \quad D(\bar{{\boldsymbol \alpha}}) \leq \epsilon.$$ If Assumption \[assum:dual\] holds, then the following results hold.
If the part in Assumption \[assum:dual\] that $\xi^*$ is $\sigma$-strongly convex holds, then ${{\boldsymbol w}}({{\boldsymbol \alpha}})$ satisfies $$P\left( {{\boldsymbol w}}\left( {{\boldsymbol \alpha}}\right) \right) - \min_{{{\boldsymbol w}}\in
{\mathbb{R}}^d}\quad P\left( {{\boldsymbol w}}\right) \leq \epsilon \left( 1 +
\frac{L}{\sigma} \right).$$
If the part in Assumption \[assum:dual\] that $\xi$ is $\rho$-Lipschitz continuous holds, then ${{\boldsymbol w}}({{\boldsymbol \alpha}})$ satisfies $$P\left( {{\boldsymbol w}}\left( {{\boldsymbol \alpha}}\right) \right) - \min_{{{\boldsymbol w}}\in
{\mathbb{R}}^d}\quad P\left( {{\boldsymbol w}}\right) \leq \max\left\{
2 \epsilon, \sqrt{8 \epsilon \rho^2 L} \right\}.$$
One more issue to note from recovering the primal solution through is that our algorithm only guarantees monotone decrease of the dual objective but not the primal objective. To ensure the best primal approximate solution, one can follow [@LeeC17a] to maintain the primal iterate that gives the best objective for up to the current iteration as the output solution. The theorems above still apply to this iterate and we are guaranteed to have better primal performance.
Related Works {#sec:related}
=============
The framework of using the quadratic approximation subproblem to generate update directions for optimizing has been discussed in existing works with different choices of $H$, but always in the single-core setting. @LeeSS14a focused on using $H = \nabla^2 f$, and proved local convergence results under certain additional assumptions. In their experiment, they used AG to solve . However, in distributed environments, for or , using $\nabla^2 f$ as $H$ needs an $O(d)$ communication per AG iteration in solving , because computation of the term $\nabla^2 f(x) p$ involves either $X D X^\top p$ or $X^\top D
X p$ for some diagonal matrix $D$, which requires one [*allreduce*]{} operation to calculate a weighted sum of the columns of $X$.
@SchT16a and @GhaS16a showed global convergence rate results for a method based on with bounded $H$, and suggested using randomized coordinate descent to solve . In the experiments of these two works, they used the same choice of $H$ as we do in this paper, with CD as the solver for , which is well suited to their single-machine setting. Aside from our extension to the distributed setting and the use of SpaRSA, the third major difference between their algorithm and ours is how sufficient objective decrease is guaranteed. When the obtained solution with a unit step size does not result in sufficient objective value decrease, they add a multiple of the identity matrix to $H$ and solve again starting from $p^{(0)} = 0$. This is different from how we modify $H$ and in some worst cases, the behavior of their algorithm can be closer to a first-order method if the identity part dominates, and more trials of different $H$ might be needed. The cost of repeatedly solving from scratch can be high, which results in an algorithm with higher overall complexity. This potential inefficiency is exacerbated further by the inefficiency of coordinate descent in the distributed setting. Our method can be considered as a special case of the algorithmic framework in @LeeW18a [@BonLPP16a], which both focus on analyzing the theoretical guarantees under various conditions for general $H$. In the experiments of @BonLPP16a, $H$ is obtained from the diagonal entries of $\nabla^2 f$, making the subproblem easy to solve, but this simplification does not take full advantage of curvature information. Although most our theoretical convergence analysis follows directly from @LeeW18a and its extension [@PenZZ18a], these works do not provide details of experimental results or implementation, and their analyses focus on general $H$ rather than the LBFGS choice we use here. For the dual problem , there are existing distributed algorithms under the instance-wise storage scheme (for example, [@Yan13a; @LeeC17a; @ZheXXZ17a; @DunLGBHJ18a] and the references therein). As we discussed in Section \[subsec:dual\], it is easy to recover the block-diagonal part of the Hessian under this storage scheme. Therefore, these works focus on using the block-diagonal part of the Hessian and use to generate update directions. In this case, only blockwise curvature information is obtained, so the update direction can be poor if the data is distributed nonuniformly. In the extreme case in which each machine contains only one column of $X$, only the diagonal entries of the Hessian can be obtained, so the method reduces to a scaled version of proximal gradient. Indeed, we often observe in practice that these methods tend to converge quickly in the beginning, but after a while the progress appears to stagnate even for small $K$.
@ZheXXZ17a give a primal-dual framework with acceleration that utilizes a distributed solver for to optimize . Their algorithm is essentially the same as applying the Catalyst framework [@LinMH15a] on a strongly-convex primal problem to form an algorithm with an inner and an outer loop. In particular, their approach consists of the following steps per round to optimize a strongly-convex primal problem with the additional requirement that $g$ being Lipschitz-continuously differentiable.
Add a quadratic term centered at a given point $y$ to form a subproblem with better condition.
Approximately optimize the new problem by using a distributed dual problem solver, and
find the next $y$ through extrapolation techniques similar to that of accelerated gradient [@Nes13a; @BecT09a].
A more detailed description of the Catalyst framework (without requiring both terms to be differentiable) is given in Appendix \[app:catalyst\]. We consider one round of the above process as one outer iteration of their algorithm, and the inner loop refers to the optimization process in the second step. The outer loop of their algorithm is conducted on the primal problem and a distributed dual solver is simply considered as a subproblem solver using results similar to Theorem \[thm:dualitygap\]. Therefore this approach is more a primal problem solver than a dual one, and it should be compared with other distributed primal solvers for smooth optimization but not with the dual algorithms. However, the Catalyst framework can be applied directly on the dual problem directly as well, and this type of acceleration can to some extent deal with the problem of stagnant convergence appeared in the block-diagonal approaches for the dual problem. Unfortunately, those parameters used in acceleration are not just global in the sense that the coordinate blocks are considered all together, but also global bounds for all possible ${{\boldsymbol w}}\in {\mathbb{R}}^d$ or ${{\boldsymbol \alpha}}\in {\mathbb{R}}^n$. This means that the curvature information around the current iterate is not considered, so the improved convergence can still be slow. By using the Hessian or its approximation as in our method, we can get much better empirical convergence.
A column-wise split of $X$ in the dual problem corresponds to a primal problem where $X$ is split row-wise. Therefore, existing distributed algorithms for the dual ERM problem can be directly used to solve in a distributed environment where $X$ is partitioned feature-wise (i.e. along rows instead of columns). However, there are two potential disadvantages of this approach. First, new data points can easily be assigned to one of the machines in our approach, whereas in the feature-wise approach, the features of all new points would need to be distributed around the machines. Second, as we mentioned above, the update direction from the block-diagonal approximation of the Hessian can be poor if the data is distributed nonuniformly across machines, and data is more likely to be distributed evenly across instances than across features. Thus, those algorithms focusing on feature-wise split of $X$ are excluded from our discussion and empirical comparison.
Numerical Experiments {#sec:exp}
=====================
We investigate the empirical performance of DPLBFGS for solving both the primal and dual problems and on binary classification problems with training data points $({{\boldsymbol x}}_i, y_i)
\in {\mathbb{R}}^d \times \{-1,1\}$ for $i=1,\dotsc,n$. For the primal problem, we consider solving $\ell_1$-regularized logistic regression problems: $$P({{\boldsymbol w}}) = C\sum_{i=1}^n \log\left(1 + e^{-y_i {{\boldsymbol x}}_i^\top
{{\boldsymbol w}}}\right) + \|{{\boldsymbol w}}\|_1,
\label{eq:logistic}$$ where $C > 0$ is a parameter prespecified to trade-off between the loss term and the regularization term. Note that since the logarithm term is nonnegative, the regularization term ensures that the level set is bounded. Therefore, within the bounded set, the loss function is strongly convex with respect to $X^\top {{\boldsymbol w}}$ and the regularizer can be reformulated as a polyhedron constrained linear term. One can thus easily show that satisfies the quadratic growth condition . Therefore, our algorithm enjoys global linear convergence on this problem.
For the dual problem, we consider $\ell_2$-regularized squared-hinge loss problems, which is of the form $$D({{\boldsymbol \alpha}}) = \frac12 \left\| Y X{{\boldsymbol \alpha}}\right\|_2^2 + \frac{1}{4C} \|{{\boldsymbol \alpha}}\|_2^2
- \mathbf{1}^\top {{\boldsymbol \alpha}}+ {\mathbb{1}}_{{\mathbb{R}}_+^n}\left({{\boldsymbol \alpha}}\right),
\label{eq:l2svm}$$ where $Y$ is the diagonal matrix consists of the labels $y_i$, $\mathbf{1} = (1,\dots,1)$ is the vector of ones, given a convex set $\mathbf{X}$, ${\mathbb{1}}_{\mathbf{X}}$ is its indicator function such that $${\mathbb{1}}_{\mathbf{X}}(x) = \begin{cases}
0 &\text{ if } x \in X,\\
\infty &\text{ else},
\end{cases}$$ and ${\mathbb{R}}_+^n$ is the nonnegative orthant in ${\mathbb{R}}^n$. This strongly convex quadratic problem is considered for easier implementation of the Catalyst framework in comparison.
We consider the publicly available binary classification data sets listed in Table \[tbl:data\],[^1] and partitioned the instances evenly across machines. $C$ is fixed to $1$ in all our experiments for simplicity. We ran our experiments on a local cluster of $16$ machines running MPICH2, and all algorithms are implemented in C/C++. The inversion of $M$ defined in is performed through LAPACK [@And99a]. The comparison criteria are the relative objective error $$\left|\frac{F(x) - F^*}{F^*}\right|$$ versus either the amount communicated (divided by $d$) or the overall running time, where $F^*$ is the optimal objective value, and $F$ can be either the primal objective $P({{\boldsymbol w}})$ or the dual objective $D({{\boldsymbol \alpha}})$, depending on which problem is being considered. The former criterion is useful in estimating the performance in environments in which communication cost is extremely high.
The parameters of our algorithm were set as follows: $\theta = 0.5$, $\beta = 2$, $\sigma_0 = 10^{-2}$, $\sigma_1 = 10^{-4}$, $m=10$, $\delta = 10^{-10}$. The parameters in SpaRSA follow the setting in [@WriNF09a], $\theta$ is set to halve the step size each time, the value of $\sigma_0$ follows the default experimental setting of [@LeeWCL17a], $\delta$ is set to a small enough value, and $m=10$ is a common choice for LBFGS. The code used in our experiments is available at <http://github.com/leepei/dplbfgs/>.
In all experiments, we show results of the backtracking variant only, as we do not observe significant difference in performance between the line-search approach and the trust-region approach in our algorithm.
Data set $n$ (\#instances) $d$ (\#features) \#nonzeros
------------ ------------------- ------------------ --------------- --
news 19,996 1,355,191 9,097,916
epsilon 400,000 2,000 800,000,000
webspam 350,000 16,609,143 1,304,697,446
avazu-site 25,832,830 999,962 387,492,144
: Data statistics.[]{data-label="tbl:data"}
In the subsequent experiments, we first use the primal problem to examine how inexactness of the subproblem solution affects the communication complexity, overall running time, and step sizes. We then compare our algorithm with state of the art distributed solvers for . Finally, comparison on the dual problem is conducted.
Effect of Inexactness in the Subproblem Solution
------------------------------------------------
We first examine how the degree of inexactness of the approximate solution of subproblems affects the convergence of the overall algorithm. Instead of treating SpaRSA as a steadily linearly converging algorithm, we take it as an algorithm that sometimes decreases the objective much faster than the worst-case guarantee, thus an adaptive stopping condition is used. In particular, we terminate Algorithm \[alg:sparsa\] when the norm of the current update step is smaller than $\epsilon_1$ times that of the first update step, for some prespecified $\epsilon_1 > 0$. From the proof of Lemma \[lemma:sparsa\], the norm of the update step bounds the value of $Q(p) - Q^*$ both from above and from below (assuming exact solution of , which is indeed the case for the selected problems), and thus serves as a good measure of the solution precision. In Table \[tbl:stop\], we compare runs with the values $\epsilon_1 =
10^{-1}, 10^{-2}, 10^{-3}$. For the datasets news20 and webspam, it is as expected that tighter solution of results in better updates and hence lower communication cost, though it may not result in a shorter convergence time because of more computation per round. As for the dataset epsilon, which has a smaller data dimension $d$, the $O(m)$ communication cost per SpaRSA iteration for calculating $\nabla \hat f$ is significant in comparison. In this case, setting a tighter stopping criterion for SpaRSA can incur higher communication cost and longer running time.
In Table \[tbl:steps\], we show the distribution of the step sizes over the main iterations, for the same set of values of $\epsilon_1$. As we discussed in Section \[sec:analysis\], although the smallest $\lambda$ can be much smaller than one, the unit step is usually accepted. Therefore, although the worst-case communication complexity analysis is dominated by the smallest step encountered, the practical behavior is much better. This result also suggests that the difference between DPLBFGS-LS and DPLBFGS-TR should be negligible, as most of the times, the original $H$ with unit step size is accepted.
Data set $\epsilon_1$ Communication Time
---------- -------------- --------------- ------
$10^{-1}$ 28 11
$10^{-2}$ 25 11
$10^{-3}$ 23 14
$10^{-1}$ 144 45
$10^{-2}$ 357 61
$10^{-3}$ 687 60
$10^{-1}$ 452 3254
$10^{-2}$ 273 1814
$10^{-3}$ 249 1419
: Different stopping conditions of SpaRSA as an approximate solver for . We show required amount of communication (divided by $d$) and running time (in seconds) to reach $F({{\boldsymbol w}}) - F^* \leq 10^{-3} F^*$.[]{data-label="tbl:stop"}
Data set $\epsilon_1$ percent of $\lambda = 1$ smallest $\lambda$
---------- -------------- -------------------------- --------------------
$10^{-1}$ $95.5\%$ $2^{-3}$
$10^{-2}$ $95.5\%$ $2^{-4}$
$10^{-3}$ $95.5\%$ $2^{-3}$
$10^{-1}$ $96.8\%$ $2^{-5}$
$10^{-2}$ $93.4\%$ $2^{-6}$
$10^{-3}$ $91.2\%$ $2^{-3}$
$10^{-1}$ $98.5\%$ $2^{-3}$
$10^{-2}$ $97.6\%$ $2^{-2}$
$10^{-3}$ $97.2\%$ $2^{-2}$
: Step size distributions.[]{data-label="tbl:steps"}
Comparison with Other Methods for the Primal Problem
----------------------------------------------------
Now we compare our method with two state-of-the-art distributed algorithms for . In addition to a proximal-gradient-type method that can be used to solve general in distributed environments easily, we also include one solver specifically designed for $\ell_1$-regularized problems in our comparison. These methods are:
- DPLBFGS-LS: our Distributed Proximal LBFGS approach. We fix $\epsilon_1 = 10^{-2}$.
- SpaRSA [@WriNF09a]: the method described in Section \[subsec:sparsa\], but applied directly to but not to the subproblem .
- OWLQN [@AndG07a]: an orthant-wise quasi-Newton method specifically designed for $\ell_1$-regularized problems. We fix $m=10$ in the LBFGS approximation.
All methods are implemented in C/C++ and MPI. As OWLQN does not update the coordinates $i$ such that $-X_{i,:}\nabla \xi(X^T {{\boldsymbol w}}) \in \partial g_i({{\boldsymbol w}}_i)$ given any ${{\boldsymbol w}}$, the same preliminary active set selection is applied to our algorithm to reduce the subproblem dimension and the computational cost, but note that this does not reduce the communication cost as the gradient calculation still requires communication of a full $d$-dimensional vector.
The AG method [@Nes13a] can be an alternative to SpaRSA, but its empirical performance has been shown to be similar to SpaRSA [@YanZ11a] and it requires strong convexity and Lipschitz parameters to be estimated, which induces an additional cost.
A further examination on different values of $m$ indicates that convergence speed of our method improves with larger $m$, while in OWLQN, larger $m$ usually does not lead to better results. We use the same value of $m$ for both methods and choose a value that favors OWLQN.
The results are provided in Figure \[fig:compare\]. Our method is always the fastest in both criteria. For epsilon, our method is orders of magnitude faster, showing that correctly using the curvature information of the smooth part is indeed beneficial in reducing the communication complexity.
It is possible to include specific heuristics for $\ell_1$-regularized problems, such as those applied in @YuaHL12a [@KaiYDR14a], to further accelerate our method for this problem, and the exploration on this direction is an interesting topic for future work.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Communication Time
![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:compare"}](fig/newscomm.png "fig:"){width=".40\linewidth"} ![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:compare"}](fig/newstime.png "fig:"){width=".40\linewidth"}
![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:compare"}](fig/epscomm.png "fig:"){width=".40\linewidth"} ![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:compare"}](fig/epstime.png "fig:"){width=".40\linewidth"}
![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:compare"}](fig/webspamcomm.png "fig:"){width=".40\linewidth"} ![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:compare"}](fig/webspamtime.png "fig:"){width=".40\linewidth"}
![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:compare"}](fig/avazucomm.png "fig:"){width=".40\linewidth"} ![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:compare"}](fig/avazutime.png "fig:"){width=".40\linewidth"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Comparison on the Dual Problem {#subsec:dualexp}
------------------------------
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Communication Time
![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual"}](fig/news20dualcomm.png "fig:"){width=".40\linewidth"} ![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual"}](fig/news20dualtime.png "fig:"){width=".40\linewidth"}
![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual"}](fig/epsilondualcomm.png "fig:"){width=".40\linewidth"} ![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual"}](fig/epsilondualtime.png "fig:"){width=".40\linewidth"}
![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual"}](fig/webspamdualcomm.png "fig:"){width=".40\linewidth"} ![Comparison between different methods for in terms of relative objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual"}](fig/webspamdualtime.png "fig:"){width=".40\linewidth"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Communication Time
![Comparison between different methods for in terms of relative *primal* objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual-primal"}](fig/news20dualprimalcomm "fig:"){width=".40\linewidth"} ![Comparison between different methods for in terms of relative *primal* objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual-primal"}](fig/news20dualprimaltime "fig:"){width=".40\linewidth"}
![Comparison between different methods for in terms of relative *primal* objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual-primal"}](fig/epsilondualprimalcomm.png "fig:"){width=".40\linewidth"} ![Comparison between different methods for in terms of relative *primal* objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual-primal"}](fig/epsilondualprimaltime.png "fig:"){width=".40\linewidth"}
![Comparison between different methods for in terms of relative *primal* objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual-primal"}](fig/webspamdualprimalcomm.png "fig:"){width=".40\linewidth"} ![Comparison between different methods for in terms of relative *primal* objective difference to the optimum. Left: communication (divided by $d$); right: running time (in seconds).[]{data-label="fig:dual-primal"}](fig/webspamdualprimaltime.png "fig:"){width=".40\linewidth"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Now we turn to solve the dual problem, considering the specific example . We compare the following algorithms.
- BDA [@LeeC17a]: a distributed algorithm using Block-Diagonal Approximation of the real Hessian of the smooth part with line search.
- BDA with Catalyst: using the BDA algorithm within the Catalyst framework [@LinMH15a] for accelerating first-order methods.
- ADN [@DunLGBHJ18a]: a trust-region approach where the quadratic term is a multiple of the block-diagonal part of the Hessian, scaled adaptively as the algorithm progresses.
- DPLBFGS-LS: our Distributed Proximal LBFGS approach. We fix $\epsilon_1 = 10^{-2}$ and limit the number of SpaRSA iterations to $100$. For the first ten iterations when $m(t) < m$, we use BDA to generate the update steps instead.
For BDA, we use the C/C++ implementation in the package MPI-LIBLINEAR.[^2] We implement ADN by modifying the above implementation of BDA. In both BDA and ADN, following [@LeeC17a] we use random-permutation coordinate descent (RPCD) for the local subproblems, and for each outer iteration we perform one epoch of RPCD. For the line search step in both BDA and DPLBFGS-LS, since the objective is quadratic, we can find the exact minimizer efficiently (in closed form). The convergence guarantees still holds for exact line search, so we use this here in place of the backtracking approach described earlier.
We also applied the Catalyst framework [@LinMH15a] for accelerating first-order methods to BDA to tackle the dual problem, especially for dealing with the stagnant convergence issue. This framework requires a good estimate of the convergence rate and the strong convexity parameter $\sigma$. From , we know that $\sigma = 1 / (2C)$, but the actual convergence rate is hard to estimate as BDA interpolates between (stochastic) proximal coordinate descent (when only one machine is used) and proximal gradient (when $n$ machines are used). After experimenting with different sets of parameters for BDA with Catalyst, we found the following to work most effectively: for every outer iteration of the Catalyst framework, $K$ iterations of BDA is conducted with early termination if a negative step size is obtained from exact line search; for the next Catalyst iteration, the warm-start initial point is simply the iterate at the end of the previous Catalyst iteration; before starting Catalyst, we run the unaccelerated version of BDA for certain iterations to utilize its advantage of fast early convergence. Unfortunately, we do not find a good way to estimate the $\kappa$ term in the Catalyst framework that works for all data sets. Therefore, we find the best $\kappa$ by a grid search. We provide a detailed description of our implementation of the Catalyst framework on this problem and the related parameters used in this experiment in Appendix \[app:catalyst\].
We focus on the combination of Catalyst and BDA (instead of with ADN) for a few reasons. Since both BDA and ADN are distributed methods that use the block-diagonal portion of the Hessian matrix, it should suffice to evaluate the application of Catalyst to the better performing of the two to represent this class of algorithms. In addition, dealing with the trust-region adjustment of ADN becomes complicated as the problem changes through the Catalyst iterations.
The results are shown in Figure \[fig:dual\]. We do not present results on the avazu data set in this experiment as all methods take extremely long time to converge. We first observe that, contrary to what is claimed in [@DunLGBHJ18a], BDA outperforms ADN on news20 and webspam, though the difference is insignificant, and the two are competitive on epsilon. This also justifies that applying the Catalyst framework on BDA alone suffices. Comparing our DPLBFGS approach to the block-diagonal ones, it is clear that our method performs magnitudes better than the state of the art in terms of both communication cost and time. For webspam and epsilon, the block-diagonal approaches are faster at first, but the progress stalls after a certain accuracy level. In contrast, while the proposed DPLBFGS approach does not converge as rapidly initially, the algorithm consistently makes progress towards a high accuracy solution.
As the purpose of solving the dual problem is to obtain an approximate solution to the primal problem through the formulation , we are interested on how the methods compare in terms of the primal solution precision. This comparison is presented in Figure \[fig:dual-primal\]. Since these dual methods are not descent methods for the primal problem, we apply the pocket approach [@Gal90a] suggested in [@LeeC17a] to use the iterate with the smallest primal objective so far as the current primal solution. We see that the primal objective values have trends very similar to the dual counterparts, showing that our DPLBFGS method is also superior at generating better primal solutions.
A potentially more effective approach is a hybrid one that first uses a block-diagonal method and then switches over to our DPLBFGS approach after the block-diagonal method hits the slow convergence phase. Developing such an algorithm would require a way to determine when we reach such a stage, and we leave the development of this method to future work. Another possibility is to consider a structured quasi-Newton approach to construct a Hessian approximation only for the off-block-diagonal part so that the block-diagonal part can be utilized simultaneously.
We also remark that our algorithm is partition-invariant in terms of convergence and communication cost, while the convergence behavior of the block-diagonal approaches depend heavily on the partition. This means when more machines are used, these block-diagonal approaches suffer from poorer convergence, while our method retains the same efficiency regardless of the number of machines begin used and how the data points are distributed (except for the initialization part).
Conclusions {#sec:conclusions}
===========
In this work, we propose a practical and communication-efficient distributed algorithm for solving general regularized nonsmooth ERM problems. The proposed approach is the first one that can be applied both to the primal and the dual ERM problem under the instance-wise split scheme. Our algorithm enjoys fast performance both theoretically and empirically and can be applied to a wide range of ERM problems. Future work for the primal problem include active set identification for reducing the size of the vector communicated when the solution exhibits sparsity, and application to nonconvex applications; while for the dual problem, it is interesting to further exploit the structure so that the quasi-Newton approach can be combined with real Hessian entries at the block-diagonal part to get better convergence.
Proofs
======
In this appendix, we provide proof for Lemma \[lemma:sparsa\]. The rest of Section \[sec:analysis\] directly follows the results in @LeeW18a [@PenZZ18a], and are therefore omitted. Note that implies , and implies because $R_0^2$ is upper-bounded by $2(F(x^0) - F^*)/\mu$. Therefore, we get improved communication complexity by the fast early linear convergence from the general convex case.
We prove the three results separately.
We assume without loss of simplicity that is satisfied by all iterations. When it is not the case, we just need to shift the indices but the proof remains the same as the pairs of $({{\boldsymbol s}}_t, {{\boldsymbol y}}_t)$ that do not satisfy are discarded.
We first bound $\gamma_t$ defined in . From Lipschitz continuity of $\nabla f$, we have that for all $t$, $$\frac{\|{{\boldsymbol y}}_t\|^2}{{{\boldsymbol y}}_t^\top {{\boldsymbol s}}_t} \leq \frac{L^2
\|{{\boldsymbol s}}_t\|^2}{{{\boldsymbol y}}_t^\top {{\boldsymbol s}}_t} \leq \frac{L^2}{\delta},
\label{eq:upper2}$$ establishing the upper bound. For the lower bound, implies that $$\|{{\boldsymbol y}}_t\| \geq \delta \|{{\boldsymbol s}}_t\|, \quad \forall t.$$ Therefore, $$\frac{{{\boldsymbol y}}_{t}^\top {{\boldsymbol s}}_{t}}{{{\boldsymbol y}}_{t}^\top {{\boldsymbol y}}_{t}} \leq
\frac{\|{{\boldsymbol s}}_{t}\|}{\|{{\boldsymbol y}}_{t}\|} \leq \frac{1}{\delta}, \quad \forall t.$$
According to [@ByrNS94a], the representation is equivalent to the inverse of $$\begin{aligned}
B_t &\coloneqq V^\top_{t-1}\cdots V^\top_{t-m(t))} B_t^0 V_{t-
m(t)}\cdots
V_{t-1}
+ \rho_{t-1} {{\boldsymbol s}}_{t-1} {{\boldsymbol s}}_{t-1}^\top +\nonumber\\
&\qquad \sum_{j=t-m(t)}^{t-2} \rho_{j} V_{t-1}^\top\cdots
V_{j+1}^\top {{\boldsymbol s}}_{j} {{\boldsymbol s}}_{j}^\top V_{j+1} \cdots V_{t-1},
\label{eq:Bt}\end{aligned}$$ where for $j\geq 0$, $$\begin{gathered}
V_j \coloneqq I - \rho_j {{\boldsymbol y}}_j{{\boldsymbol s}}_j^\top,\quad
\rho_j \coloneqq \frac{1}{ {{\boldsymbol y}}_j^\top{{\boldsymbol s}}_j},
\label{eq:v}\end{gathered}$$ and $$B_t^0 = \gamma_t I.$$ We will show that there exists $c_1 \geq c_2 > 0$ such that $$\frac{1}{c_2} I\succeq B_t \succeq \frac{1}{c_1} I, \quad \forall
t.$$
Following [@LiuN89a], $H_t$ can be obtained equivalently by $$\begin{aligned}
H_t^{(0)} &= \gamma_t I,\nonumber\\
H_t^{(k+1)} &= H_t^{(k)} -
\frac{H_t^{(k)}{{\boldsymbol s}}_{t-m(t) + k} {{\boldsymbol s}}_{t-m(t) + k}^\top H_t^{(k)}}{{{\boldsymbol s}}_{t-m(t) + k}^\top
H_t^{(k)}{{\boldsymbol s}}_{t-m(t) + k}}
+\frac{{{\boldsymbol y}}_{t-m(t) + k}{{\boldsymbol y}}_{t-m(t) + k}^\top }{{{\boldsymbol y}}_{t-m(t) + k}^\top {{\boldsymbol s}}_{t-m(t) + k}}, \;\;
k=0,\dots, m(t)-1,
\label{eq:inverse}\\
H_t &= H_t^{(m(t))}.
\nonumber\end{aligned}$$ Therefore, we can bound the trace of $H_t^(k)$ and hence $H_t$ through . $$\begin{aligned}
{\mbox{\rm trace}}\left(H_t^{(k)}\right) \leq {\mbox{\rm trace}}\left(H_t^{(0)}\right) +
\sum_{j=t - m(t) }^{t-m(t) + k} \frac{{{\boldsymbol y}}_{j}^\top{{\boldsymbol y}}_{j}}{{{\boldsymbol y}}_{j}^\top
{{\boldsymbol s}}_{j}} \leq \gamma_t N + \frac{k L^2}{\delta},
\quad \forall t,
\label{eq:uppB}\end{aligned}$$ where $N$ is the matrix dimension. According to [@ByrNS94a], the matrix $H^{(k)}_t$ is equivalent to the inverse of $$\begin{aligned}
B^{(k)}_t &\coloneqq V^\top_{t-m(t) + k}\cdots V^\top_{t-m(t))} B_t^0 V_{t-
m(t)}\cdots
V_{t-m(t) + k}
+ \rho_{t-m(t) + k} {{\boldsymbol s}}_{t-m(t) + k} {{\boldsymbol s}}_{t-m(t) + k}^\top +\nonumber\\
&\qquad \sum_{j=t-m(t)}^{t-m(t)-1 + k} \rho_{j} V_{t-m(t) + k}^\top\cdots
V_{j+1}^\top {{\boldsymbol s}}_{j} {{\boldsymbol s}}_{j}^\top V_{j+1} \cdots V_{t-m(t) + k},
\label{eq:Bt}\end{aligned}$$ where for $j\geq 0$, $$V_j \coloneqq I - \rho_j {{\boldsymbol y}}_j{{\boldsymbol s}}_j^\top,\quad
\rho_j \coloneqq \frac{1}{ {{\boldsymbol y}}_j^\top{{\boldsymbol s}}_j},\quad
B_t^0 = \frac{1}{\gamma_t} I.$$ From the form , it is clear that $B_t^{(k)}$ and hence $H_t$ are all positive-semidefinite because $\gamma_t \geq 0, \rho_j >
0$ for all $j$ and $t$. Therefore, from positive semidefiniteness, implies the existence of $c_1 > 0$ such that $$H_t^{(k)}\preceq c_1 I,\quad k = 0,\dotsc,m(t), \quad \forall t.$$ Next, for its lower bound, from the formulation for in [@LiuN89a], and the upper bound $\|H_t^{(k)}\| \leq c_1$, we have $$\begin{aligned}
\det\left(H_t\right)
=\det\left(H_t^{(0)}\right) \prod_{k=t - m(t)}^{t-1}
\frac{{{\boldsymbol y}}_{k}^\top{{\boldsymbol s}}_{k}}{{{\boldsymbol s}}_{k}^\top {{\boldsymbol s}}_{k}}
\frac{{{\boldsymbol s}}_{k}^\top{{\boldsymbol s}}_{k}}{{{\boldsymbol s}}_{k}^\top H_t^{(k-t + m(t))}
{{\boldsymbol s}}_{k}}
\geq \gamma_t^N \left(\frac{\delta}{c_1} \right)^{m(t)}
\geq M_1.\end{aligned}$$ for some $M_1>0$. From that the eigenvalues of $H_t$ are upper-bounded and nonnegative, and from the lower bound of the determinant, the eigenvalues of $H_t$ are also lower-bounded by a positive value $c_2$, completing the proof.
By directly expanding $\nabla \hat f$, we have that for any $p_1, p_2$, $$\begin{aligned}
\nabla \hat f(p_1) - \nabla \hat f(p_2)
= \nabla f(x) + H p_1 - \left(\nabla f(x) +
H p_2\right)
= H (p_1 - p_2).
\end{aligned}$$ Therefore, we have $$\frac{\left(\nabla \hat f(p_1) - \nabla \hat
f(p_2)\right)^\top \left( p_1 - p_2 \right)}{\left\|p_1 -
p_2\right\|^2}
= \frac{\left\|p_1 - p_2\right\|_H^2}{\left\|p_1 -
p_2\right\|^2}
\in \left[c_2, c_1\right]$$ for bounding $\psi_i$ for $i > 0$, and the bound for $\psi_0$ is directly from the bounds of $\gamma_t$. The combined bound is therefore $[\min\{c_2, \delta\}, \max \{c_1,
L^2/\delta\}]$. Next, we show that the final $\psi_i$ is always upper-bounded. The right-hand side of is equivalent to the following: $$\arg \min_{{\boldsymbol d}}\, \hat Q_{\psi_i}\left({{\boldsymbol d}}\right) \coloneqq \nabla
\hat f\left(p^{(i)}\right)^\top {{\boldsymbol d}}+
\frac{\psi_i \left\|{{\boldsymbol d}}\right\|^2}{2} + \hat \Psi\left({{\boldsymbol d}}+
p\right) - \hat \Psi\left(p\right).
\label{eq:hatQ}$$ Denote the solution by ${{\boldsymbol d}}$, then we have $p^{(i+1)}
= p^{(i)} + {{\boldsymbol d}}$. Note that we allow ${{\boldsymbol d}}$ to be an approximate solution. Because $H$ is upper-bounded by $c_1$, we have that $\nabla \hat
f$ is $c_1$-Lipschitz continuous. Therefore, $$\begin{aligned}
Q\left(p^{(i+1)}\right) - Q\left(p^{(i)}\right)
\nonumber
\leq&~ \nabla \hat f(p^{(i)})^\top \left( p^{(i+1)} - p^{(i)}
\right) + \frac{c_1}{2} \left\|p^{(i+1)} - p^{(i)}\right\|^2
+ \hat \Psi\left(p^{(i+1)}\right) - \hat \Psi\left( p^{(i)}
\right) \\
\label{eq:tmp}
\stackrel{\eqref{eq:hatQ}}{=}&~ \hat Q_{\psi_i}({{\boldsymbol d}}) -
\frac{\psi_i}{2} \left\| {{\boldsymbol d}}\right\|^2 + \frac{c_1}{2}
\left\|{{\boldsymbol d}}\right\|^2.
\end{aligned}$$ As $\hat Q_{\psi_i}(0) = 0$, provided that the approximate solution ${{\boldsymbol d}}$ is better than the point $0$, we have $$\hat Q ({{\boldsymbol d}}) \leq \hat Q(0) = 0.
\label{eq:Qbound}$$ Putting into , we obtain $$Q\left(p^{(i+1)}\right) - Q\left(p^{(i)}\right)
\leq
\frac{c_1 - \psi_i}{2}\|{{\boldsymbol d}}\|^2.$$ Therefore, whenever $$\frac{c_1 - \psi_i}{2} \leq -\frac{\sigma_0 \psi_i}{2},$$ holds. This is equivalent to $$\psi_i \geq \frac{c_1}{1 - \sigma_0},$$ Note that the initialization of $\psi_i$ is upper-bounded by $c_1$ for all $i > 1$, so the final $\psi_i$ is indeed upper-bounded. Together with the first iteration where we start with $\psi_0 =
\gamma_t$, we have that $\psi_i$ for all $i$ are always bounded from the boundedness of $\gamma_t$.
From the results above, at every iteration, SpaRSA finds the update direction by constructing and optimizing a quadratic approximation of $\hat f(x)$, where the quadratic term is a multiple of identity, and its coefficient is bounded in a positive range. Therefore, the theory developed by [@LeeW18a] can be directly used to show the desired result even if is solved only approximately. For completeness, we provide a simple proof for the case that is solved exactly.
We note that since $Q$ is $c_2$-strongly convex, the following condition holds. $$\frac{\min_{{{\boldsymbol s}}\in \nabla \hat f\left(p^{(i + 1)}\right) + \partial \hat
g\left( p^{(i+ 1)} \right)}
\left\|{{\boldsymbol s}}\right\|^2}{2 c_2} \geq Q\left(p^{(i + 1)}\right) -
Q^*.
\label{eq:kl}$$ On the other hand, from the optimality condition of , we have that for the optimal solution ${{\boldsymbol d}}^*$ of , $$-\psi_i {{\boldsymbol d}}^* = \nabla \hat f\left(p^{(i)}\right) + {{\boldsymbol s}}_{i+1},
\label{eq:dopt}$$ for some $${{\boldsymbol s}}_{i+1} \in \partial \hat \Psi\left( p^{(i+1)}\right).$$ Therefore, $$\begin{aligned}
\nonumber
Q\left( p^{(i+1)} \right) - Q^*
\stackrel{\eqref{eq:kl}}{\leq}&~ \frac{1}{2 c_2}
\left\|\nabla \hat f\left( p^{(i+1)} \right) - \nabla \hat
f\left( p^{(i)} \right) + \nabla \hat f\left( p^{(i)}
\right) + {{\boldsymbol s}}_{i+1}\right\|^2\\
\nonumber
\stackrel{\eqref{eq:dopt}}{\leq}&~ \frac{1}{c_2} \left\|
\nabla \hat f\left( p^{(i+1)} \right)
- \nabla \hat f\left( p^{(i)} \right) \right\|^2 + \left\|\psi_i
{{\boldsymbol d}}^*\right\|^2\\
\label{eq:dbound}
\leq&~ \frac{1}{c_2} \left( c_1^2 + \psi_i^2 \right)
\left\|{{\boldsymbol d}}^*\right\|^2.
\end{aligned}$$ By combining and , we obtain $$\begin{aligned}
Q\left( p^{(i+1)} \right) - Q\left( p^{(i)} \right)
\leq -\frac{\sigma_0 \psi_i}{2}\left\|{{\boldsymbol d}}^*\right\|^2
\leq -\frac{\sigma_0 \psi_i}{2} \frac{c_2}{c_1^2 + \psi_i^2}
\left(Q\left( p^{(i+1)} \right) - Q^*\right).
\end{aligned}$$ Rearranging the terms, we obtain $$\left(1 + \frac{c_2\sigma_0 \psi_i}{2 ( c_1^2 +
\psi^2)}\right) \left(Q\left( p^{(i+1)} \right) - Q^*\right)
\leq Q\left( p^{(i)} \right) - Q^*,$$ showing Q-linear convergence of SpaRSA, with $$\eta = \sup_{i=0,1,\dotsc}\quad \left( 1 + \frac{c_2 \sigma_0 \psi_i}{2 \left( c_1^2
+ \psi_i^2
\right)} \right)^{-1} \in [0,1).$$ Note that since $\psi_i$ are bounded in a positive range, we can find this supremum in the desired range.
Implementation Details and Parameter Selection for the Catalyst Framework {#app:catalyst}
=========================================================================
We first give an overview to the version of Catalyst framework for strongly-convex problems [@LinMH15a] for accelerating convergence rate of first-order methods, then describe our implementation details in the experiment in Section \[subsec:dualexp\]. The Catalyst framework is described in Algorithm \[alg:catalyst\].
\[alg:catalyst\]
Input: $x^0 \in {\mathbb{R}}^N$, a smoothing parameter $\kappa$, the strong convexity parameter $\mu$, an optimization method ${\mathcal{M}}$, and a stopping criterion for the inner optimization. Initialize $y^0 = x^0$, $q = \mu / (\mu + \kappa)$, $\beta = (1
- \sqrt{q}) / (1 + \sqrt{q})$. Use ${\mathcal{M}}$ with the input stopping condition to approximately optimize $$\min_x\quad F(x) + \frac{\kappa}{2}\|x - y^{k-1}\|^2
\label{eq:catalyst}$$ from a warm-start point $x^k_0$ to obtain the iterate $x^k$. $y_k = x^k + \beta (x^k - x^{k-1})$. Output $x^k$.
According to [@LinMH15a], when ${\mathcal{M}}$ is the proximal gradient method, the ideal value of $\kappa$ is $\max(L - 2\mu, 0)$, and when $L > 2\mu$, the convergence speed can be improved to the same order as accelerated proximal gradient (up to a logarithm factor difference). Similarly, when ${\mathcal{M}}$ is stochastic proximal coordinate descent with uniform sampling, by taking $\kappa = \max(L_{\max} - 2\mu, 0)$, where $L_{\max}$ is the largest block Lipschitz constant, one can obtain convergence rate similar to that of accelerated coordinate descent. Since when using proximal coordinate descent as the local solver, both BDA and ADN interpolate between proximal coordinate descent and proximal gradient,[^3] depending on the number of machines, it is intuitive that acceleration should work for them.
Considering , the problem is clearly strongly convex with parameter $1 / (2C)$, thus we take $\mu = 1 / (2C)$. For the stopping condition, we use the simple fixed iteration choice suggested in [@LinMH15a] (called (C3) in their notation). Empirically we found a very effective way is to run $K$ iterations of BDA with early termination whenever a negative step size is obtained from exact line search. For the warm-start part, although is a regularized problem, the objective part is smooth, so we take their suggestion for smooth problem to use $x^k_0 = x^{k-1}$. Note that they suggested that for general regularized problems, one should take one proximal gradient step of the original $F$ at $x^{k-1}$ to obtain $x^k_0$. We also experimented with this choice, but preliminary results show that using $x^{k-1}$ gives better initial objective value for .
The next problem is how to select $\kappa$. We observe that for webspam and epsilon, the convergence of both BDA and ADN clearly falls into two stages. Through some checks, we found that the first stage can barely be improved. On the other hand, if we pick a value of $\kappa$ that can accelerate convergence at the later stage, the fast early convergence behavior is not present anymore, thus it takes a long time for the accelerated approach to outperform the unaccelerated version. To get better results, we take an approach from the hindsight: first start with the unaccelerated version with a suitable number of iterations, and then we switch to Catalyst with $\kappa$ properly chosen by grid search for accelerating convergence at the later stage. The parameters in this approach is recorded in Table \[tbl:catalyst\]. We note that this way of tuning from the hindsight favors the accelerated method unfairly, as it takes information obtained through running other methods first. In particular, it requires the optimal objective (obtained by first solving the problem through other methods) and running the unaccelerated method to know the turning point of the convergence stages (requires the optimal objective to compute). Parameter tuning for $\kappa$ is also needed. These additional efforts are not included in the running time comparison, so our experimental result does not suggest that the accelerated method is better than the unaccelerated version. The main purpose is to show that our proposed approach also outperforms acceleration methods with careful parameter choices.
Data set \#BDA iterations before starting Catalyst $\kappa$
---------- ------------------------------------------- ----------
news $0$ $17$
epsilon $2,000$ $12,000$
webspam $400$ $2,000$
: Catalyst parameters.[]{data-label="tbl:catalyst"}
[^1]: Downloaded from <https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/>.
[^2]: <http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/distributed-liblinear/>.
[^3]: Although we used RPCD but not stochastic coordinate descent, namely sampling with replacement, it is commonly considered that RPCD behaves similar to, and usually outperforms slightly, the variant that samples without replacement; see, for example, analyses in [@LeeW16a; @WriL17a] and experiment in [@ShaZ13a].
|
---
abstract: 'We present a conspicuous number of indefinite integrals involving Heun functions and their products obtained by means of the Lagrangian formulation of a general homogeneous linear ordinary differential equation. As a by-product we also derive new indefinite integrals involving the Gauss hypergeometric function and products of hypergeometric functions with elliptic functions of the first kind. All integrals we obtained cannot be computed using Maple and Mathematica.'
author:
- 'D. Batic'
- 'O. Forrest'
- 'M. Nowakowski'
title: New indefinite integrals of Heun functions
---
Introduction
============
Given the ordinary second-order linear differential equation $$\label{0}
y^{''}(x)+P(x)y^{'}(x)+Q(x)y(x)=0,$$ [@con1; @con2] derived the indefinite integral $$\label{1}
\int\! f(x)\left[h^{''}(x)+P(x)h^{'}(x)+Q(x)h(x)\right]y(x)\,\mathrm{d}x=f(x)W(y,h)(x)+c,$$ where a prime denotes differentiation with respect to the independent variable and $$\label{f}
f(x):=e^{\int\! P(x)\,\mathrm{d}x},$$ $h$, $P$, and $Q$ are complex-valued differentiable functions in the variable $x\in\mathbb{R}$ with $h$ at least twice continuously differentiable, and $W(y,h)(x)=y(x)h^{'}(x)-h(x)y^{'}(x)$ denotes the Wronskian. Possible methods allowing to obtain an indefinite integral involving the solution of (\[0\]) by using (\[1\]) are
1. a choice of $h$ in terms of a simple elementary function such as $$\label{scelta_h}
h(x)=x^m e^{\rho x^\ell}\left\{\begin{array}{c}
\sin{(k_1 x)}\\
\cos{(k_2 x)}
\end{array}
\right\}$$ with $m,\ell\in\mathbb{N}_0:=\mathbb{N}\cup\{0\}$, $\rho,k_1,k_2\in\mathbb{C}$ and so on. Note that the above choices contain the more simple cases $h(x)=1$ and $h(x)=x^m$. It is worth to mention that when $h$ is a constant function, (\[1\]) takes the simpler form $$\label{zero}
\int\! f(x)Q(x)y(x)\,\mathrm{d}x=-f(x)y^{'}(x)+c.$$
2. If we want to integrate a solution $y(x)$ of (\[0\]), we need to choose $h$ so that $$\label{y_alone}
h^{''}(x)+P(x)h^{'}(x)+Q(x)h(x)=\frac{1}{f(x)}.$$
3. A specification of $h$ as a solution of the differential equation (\[0\]) with one or two terms in (\[0\]) deleted. For instance, we might require that $h$ satisfies one of the following differential equations $$\begin{aligned}
h^{''}(x)+P(x)h^{'}(x)&=&0,\label{2}\\
P(x)h^{'}(x)+Q(x)h(x)&=&0,\label{3}\\
h^{''}(x)+Q(x)h(x)&=&0.\label{4}\end{aligned}$$ In the case that $P$ and $Q$ consists of multiple terms we might also try to specify $h$ with $P$ or $Q$ with some of their subterms removed.
4. Take $h$ to be the solution of the equation conjugate to (\[0\]), that is $h^{''}(x)+P(x)h^{'}(x)+\overline{Q}(x)h(x)=0$. This equation has the same $P$ as in (\[0\]) but different $Q$. Since we can always construct a transformation of the dependent variable to make any two differential equations conjugate, this method allows to construct indefinite integrals of products of the solutions of the two conjugate equations according to the formula [@con1; @con2] $$\label{intcon}
\int\! f(x)\left[Q(x)-\overline{Q}(x)\right]h(x)y(x)\,\mathrm{d}x=f(x)\left[h^{'}(x)y(x)-h(x)y^{'}(x)\right]+c.$$
Here, we derive new indefinite integrals involving solutions of the Heun equation [@ronv] which is represented by (\[0\]) with $$\label{pq}
P(x)=\frac{\gamma}{x}+\frac{\delta}{x-1}+\frac{\epsilon}{x-a},\quad
Q(x)=\frac{\alpha\beta x-q}{x(x-1)(x-a)}$$ where $\alpha,\beta,\gamma,\delta,\epsilon\in\mathbb{C}$ satisfy the Fuchsian condition $\epsilon=\alpha+\beta+1-\gamma-\delta$, $a\in\mathbb{R}\backslash\{0,1\}$, and $q\in\mathbb{C}$ is the so-called accessory parameter.
Construction of indefinite integrals
====================================
Since the function $h$ appearing in (\[1\]) is not fixed, we have in practice an unlimited number of cases for any given special function. In the following we treat only those choices of $h$ such that equation (\[1\]) allows to derive new and interesting indefinite integrals. Last but not least, we checked that Maple and Mathematica software packages are unable to evaluate the indefinite integrals we computed in this paper. As a by-product we also obtain new indefinite integrals involving the Gauss hypergeometric function. These integrals do not appear to be listed in [@Grad]. As in [@HR] we restrict the local solutions to the analytic ones around the origin, here denoted by $H_l(a,q;\alpha,\beta,\gamma,\delta;x)$ but extensions are possible for other solutions around other singularities. Let us choose $h$ according to (\[scelta\_h\]). Moreover, (\[f\]) gives $f(x)=x^\gamma (x-1)^\delta(x-a)^\epsilon$. Hence, from (\[1\]) we obtain $$\int\!x^{\gamma+m-2}(x-1)^{\delta-1}(x-a)^{\epsilon-1}e^{\rho x^\ell}
F(x)
H_l(a,q;\alpha,\beta,\gamma,\delta;x)\,\mathrm{d}x=$$ $$\label{Heun2}
x^{\gamma+m-1}(x-1)^{\delta}(x-a)^{\epsilon}e^{\rho x^\ell}
\left\{
\begin{array}{c}
\mathfrak{q}(x)\sin{(k_1 x)}+k_1 x\cos{(k_1 x)}H_l(a,q;\alpha,\beta,\gamma,\delta;x)\\
\mathfrak{q}(x)\cos{(k_2 x)}-k_2 x\sin{(k_2 x)}H_l(a,q;\alpha,\beta,\gamma,\delta;x)
\end{array}
\right\}+c$$ with $$\begin{aligned}
F(x)&=&\left\{
\begin{array}{c}
x\mathfrak{p}_1(x,k_1)\cos{(k_1 x)}+\mathfrak{p}_2(x,k_1)\sin{(k_1 x)}\\
\mathfrak{p}_2(x,k_2)\cos{(k_2 x)}-x\mathfrak{p}_1(x,k_2)\sin{(k_2 x)}
\end{array}
\right\},\\
\mathfrak{q}(x)&=&(m+\rho\ell x^\ell)H_l(a,q;\alpha,\beta,\gamma,\delta;x)-xH^{'}_l(a,q;\alpha,\beta,\gamma,\delta;x),\\
\mathfrak{p}_1(x,k)&=&k\sum_{i=0}^2\mathfrak{a}_i x^i+2k\rho\ell x^\ell(x-1)(x-a),\\
\mathfrak{p}_2(x,k)&=&\sum_{i=0}^4\mathfrak{b}_i x^i+\rho\ell x^\ell\left[\sum_{i=0}^2\mathfrak{c}_i x^i+\rho\ell x^\ell(x-1)(x-a)\right]\end{aligned}$$ and $$\begin{aligned}
\mathfrak{a_2}&=&\alpha+\beta+2m+1,~ \mathfrak{a}_1=\mathfrak{a}_2+\delta(1-a)-\mathfrak{a}_0,~\mathfrak{a}_0=a(\gamma+2m),\\
\mathfrak{b}_4&=&-k^2,\quad\mathfrak{b}_3=k^2(a+1),~
\mathfrak{b}_2=-ak^2+\alpha\beta+m(\alpha+\beta+m),\\ \mathfrak{b}_1&=&m[\delta(1-a)-\alpha-\beta-m]-\mathfrak{b}_0-q,~\mathfrak{b}_0=am(m+\gamma-1),\\
\mathfrak{c}_2&=&\alpha+\beta+\ell+2m,\quad
\mathfrak{c}_1=\delta+a(1-\gamma-\delta+\ell+2m)-\mathfrak{c}_2,\\
\mathfrak{c}_0&=a&(\ell+\gamma+2m-1).\end{aligned}$$ From (\[Heun2\]) we can readily obtain new indefinite integrals for the Gauss hypergeometric function. To this purpose, we only need to observe that choosing appropriately the parameters of the Heun equation as in [@HR; @Maier] yields $$\begin{aligned}
H_l(2,\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-2\gamma+1;x)&=&{}_{2}F_{1}\left(\frac{\alpha}{2},\frac{\beta}{2};\gamma;\mathfrak{h}(x)\right),\label{L1}\\
H_l\left(4,\alpha\beta;\alpha,\beta,\frac{1}{2},\frac{2}{3}(\alpha+\beta);x\right)&=&{}_{2}F_{1}\left(\frac{\alpha}{3},\frac{\beta}{3};\frac{1}{2};\mathfrak{f}(x)\right),\label{L2}\\
H_l\left(2,\alpha\beta;\alpha,\beta,\frac{\alpha+\beta+2}{4},\frac{\alpha+\beta}{2};x\right)&=&{}_{2}F_{1}\left(\frac{\alpha}{4},\frac{\beta}{4};\frac{\alpha+\beta+2}{4};\mathfrak{g}(x)\right)\label{L3}\end{aligned}$$ with $$\label{hfg}
\mathfrak{h}(x)=x(2-x),\quad
\mathfrak{f}(x)=\frac{x}{4}(x-3)^2,\quad
\mathfrak{g}(x)=-4x(x-1)^2(x-2).$$ Furthermore, 15.2.1 in [@abra] gives $$\begin{aligned}
&&\frac{H^{'}_l(2,\alpha\beta;\alpha,\beta,\gamma,\alpha+\beta-2\gamma+1;x)}{t(x)}={}_{2}F_{1}\left(\frac{\alpha}{2}+1,\frac{\beta}{2}+1;\gamma+1;\mathfrak{h}(x)\right),\label{L4}\\
&&\frac{H^{'}_l\left(4,\alpha\beta;\alpha,\beta,\frac{1}{2},\frac{2}{3}(\alpha+\beta);x\right)}{s(x)}={}_{2}F_{1}\left(\frac{\alpha}{3}+1,\frac{\beta}{3}+1;\frac{3}{2};\mathfrak{f}(x)\right),\label{L5}\\
&&\frac{H^{'}_l\left(2,\alpha\beta;\alpha,\beta,\frac{\alpha+\beta+2}{4},\frac{\alpha+\beta}{2};x\right)}{r(x)}=
{}_{2}F_{1}\left(\frac{\alpha}{4}+1,\frac{\beta}{4}+1;\frac{\alpha+\beta+6}{4};\mathfrak{g}(x)\right)\label{L6}\end{aligned}$$ with $t(x)=\alpha\beta(1-x)/(2\gamma)$, $s(x)=\gamma t(x)(3-x)/3$, $r(x)=\kappa(1-x)(2x^2-4x+1)$, and $\kappa=2\alpha\beta/(\alpha+\beta+2)$. Moreover, by means of (\[zero\]) we obtain immediately the following result $$\int\!x^{\gamma-1}(x-1)^{\delta-1}(x-a)^{\epsilon-1}(\alpha\beta x-q)H_\ell\left(a,q;\alpha,\beta,\gamma,\delta;x\right)\,\mathrm{d}x=$$ $$\label{F12}
-x^{\gamma}(x-1)^{\delta}(x-a)^{\epsilon}H^{'}_\ell\left(a,q;\alpha,\beta,\gamma,\delta;x\right)+c.$$ As a verification of the correctness of (\[F12\]), we observe that if we define $z=\mathfrak{h}(x)$ from which $x=1+\sqrt{1-z}$, and let $a=\alpha/2$, $b=\beta/2$, and $c=\gamma$, then (\[F12\]) together with (\[L1\]) and (\[L4\]) reproduces 1.15.3.9 in [@Prud], namely $$\label{PrudF}
\int\!z^{c-1}(1-z)^{a+b-c}{}_{2}F_{1}\left(a,b;c;z\right)\,\mathrm{d}z=
\frac{z^c}{c}(1-z)^{a+b-c+1}{}_{2}F_{1}\left(a+1,b+1;c+1;z\right)+\widetilde{c}.$$ As a further independent check, if we let $z=\mathfrak{f}(x)$ from which $$x=2+\frac{1}{g(z)}+g(z),\quad
g(z)=\sqrt[3]{2\sqrt{z^2-z}+2z-1},$$ and define $a=\alpha/3$, $b=\beta/3$, and $c=1/2$, then (\[F12\]) together with (\[L2\]) and (\[L5\]) reproduces again (\[PrudF\]). We also arrive at the same conclusion if we consider $z=\mathfrak{g}(x)$ with $x=1+(\sqrt{2+2\sqrt{1-z}}/2)$, make the identification $a=\alpha/4$, $b=\beta/4$, and $c=a+b+1/2$, and then, use (\[L3\]) and (\[L6\]). An integral involving $H_l(a,q;\alpha,\beta,\gamma,\delta;x)$ alone can be constructed by making the choice $h(x)=\alpha^{-1}(1-\alpha)^{-1}$ with $\alpha\neq 0,1$, $q=\alpha(1-\alpha)$, $\gamma=\epsilon=1$ and $\delta=0$. In this case, we find $$\label{HFe}
\int\! H_l(a,\alpha-\alpha^2;\alpha,1-\alpha,1,0;x)\,\mathrm{d}x=\frac{x(a-x)}{\alpha(1-\alpha)}H^{'}_l(a,\alpha-\alpha^2;\alpha,1-\alpha,1,0;x)+c.$$ Note that even in this specialized case Maple/Mathematica is not able to solve the above indefinite integral. As a verification of the correctness of our method, we let $a=\alpha$, $b=1-\alpha$, and $c=1$, then (\[HFe\]) together with (\[L1\]) and (\[L4\]) gives $$\int\! {}_{2}F_{1}\left(\frac{a}{2},\frac{1}{2}-\frac{a}{2};1;\mathfrak{h}(x)\right)\,\mathrm{d}x=\frac{1}{2}x(1-x)(2-x){}_{2}F_{1}\left(\frac{a}{2}+1,\frac{3}{2}-\frac{a}{2};2;\mathfrak{h}(x)\right)+c.$$ If we further define $z=\mathfrak{h}(x)$ from which $x=1+\sqrt{1-z}$, the above integral becomes $$\int\! (1-z)^{-\frac{1}{2}}{}_{2}F_{1}\left(\frac{a}{2},\frac{1}{2}-\frac{a}{2};1;z\right)\,\mathrm{d}z=z\sqrt{1-z}{}_{2}F_{1}\left(\frac{a}{2}+1,\frac{3}{2}-\frac{a}{2};2;z\right)+c$$ which can be obtained as a special case of 1.15.3.9 in [@Prud]. A new indefinite integral involving a product of a Heun function with an incomplete elliptic integral of the first kind can be obtained by taking the function $h$ to be a solution of (\[2\]) with $P$ given as in (\[pq\]) and $\gamma=\delta=\epsilon=1/2$. Then, 3.131(3) in [@Grad] implies that $$h(x)=\int_0^x\!\,\frac{\mathrm{d}u}{\sqrt{u(1-u)(a-u)}}=\frac{2}{\sqrt{a}}F\left(\varphi(x),\frac{1}{\sqrt{a}}\right),\quad
\varphi(x)=\arcsin{\sqrt{x}},$$ where $F$ is the incomplete elliptic integral of the first kind. Furthermore, note that the above choice of the parameters $\gamma$, $\delta$ and $\epsilon$ requires that $\alpha+\beta=1/2$. Finally, we obtain the result $$\int\!\frac{\alpha(1-2\alpha)x-2q}{\sqrt{x(x-1)(x-a)}}F\left(\varphi(x),\frac{1}{\sqrt{a}}\right)H_\ell\left(a,q;\alpha,\frac{1}{2}-\alpha,\frac{1}{2},\frac{1}{2};x\right)\,\mathrm{d}x=$$ $$\label{Hn}
\sqrt{a}H_\ell\left(a,q;\alpha,\frac{1}{2}-\alpha,\frac{1}{2},\frac{1}{2};x\right)-
2\mathfrak{r}(x)F\left(\varphi(x),\frac{1}{\sqrt{a}}\right)H^{'}_\ell\left(a,q;\alpha,\frac{1}{2}-\alpha,\frac{1}{2},\frac{1}{2};x\right)+c.$$ with $\mathfrak{r}(x)=\sqrt{x(x-1)(x-a)}$. If we let $z=\mathfrak{h}(x)$ with $x=1-\sqrt{1-z}$, make the identification $a=\alpha/2$, $b=1/4-a$, and $c=1/2$, and then, use (\[L1\]) and (\[L4\]), we obtain $$\int\,z^{-\frac{1}{2}}(1-z)^{-\frac{1}{4}}F\left(\psi(z),\frac{1}{\sqrt{2}}\right){}_{2}F_{1}\left(a,\frac{1}{4}-a;\frac{1}{2};z\right)\,\mathrm{d}z=$$ $$\label{laut}
\frac{\sqrt{2}}{a(4a-1)}{}_{2}F_{1}\left(a,\frac{1}{4}-a;\frac{1}{2};z\right) +2z^{\frac{1}{2}}(1-z)^\frac{3}{4}F\left(\psi(z),\frac{1}{\sqrt{2}}\right){}_{2}F_{1}\left(a+1,\frac{5}{4}-a;\frac{3}{2};z\right)+c$$ provided that $a\neq 0,1/4$ and $\psi(z)=\arcsin{\sqrt{1-\sqrt{1-z}}}$. To the best of our knowledge, the integral (\[laut\]) seems to be new. We can also find indefinite integrals of products of hypergeometric functions with Heun functions by considering again (\[2\]) with $\delta=0$ or $\epsilon=0$. The case $\delta$=0 yields $h(x)=x^{1-\gamma}{}_{2}F_{1}\left(\epsilon,1-\gamma;2-\gamma;x/a\right)$ and (\[1\]) gives $$\int\!\frac{(x-a)^{\epsilon-1}(\alpha\beta x-q)}{x-1}{}_{2}F_{1}\left(\epsilon,\tau;1+\tau;\frac{x}{a}\right)H_\ell\left(a,q;\alpha,\beta,\gamma,0;x\right)\,\mathrm{d}x=(x-a)^\epsilon\cdot$$ $$\left\{\tau\left[{}_{2}F_{1}\left(\epsilon,\tau;1+\tau;\frac{x}{a}\right)+\frac{\epsilon x}{a(1+\tau)}{}_{2}F_{1}\left(\epsilon+1,1+\tau;2+-\tau;\frac{x}{a}\right)\right]H_\ell\left(a,q;\alpha,\beta,\gamma,0;x\right)\right.$$ $$\label{auf1}
\left.-x{}_{2}F_{1}\left(\epsilon,\tau;1+\tau;\frac{x}{a}\right)H^{'}_\ell\left(a,q;\alpha,\beta,\gamma,0;x\right)\right\}+c$$ with $\epsilon=\alpha+\beta+\tau$ and $\tau=1-\gamma$. Finally, for $\epsilon$=0 we have $h(x)=x^{1-\gamma}{}_{2}F_{1}\left(\delta,1-\gamma;2-\gamma;x\right)$ and from (\[1\]) we obtain $$\int\!\frac{(x-1)^{\delta-1}(\alpha\beta x-q)}{x-a}{}_{2}F_{1}\left(\delta,\tau;1+\tau;x\right)H_\ell\left(a,q;\alpha,\beta,\gamma,\delta;x\right)\,\mathrm{d}x=(x-1)^\delta\cdot$$ $$\left\{\tau\left[{}_{2}F_{1}\left(\delta,\tau;1+\tau;x\right)+\frac{\delta x}{1+\tau}{}_{2}F_{1}\left(\delta+1,1+\tau;2+\tau;x\right)\right]H_\ell\left(a,q;\alpha,\beta,\gamma,\delta;x\right)\right.$$ $$\label{auf2}
\left.-x{}_{2}F_{1}\left(\delta,\tau;1+\tau;x\right)H^{'}_\ell\left(a,q;\alpha,\beta,\gamma,\delta;x\right)\right\}+c$$ with $\delta=\alpha+\beta+\tau$ and $\tau=1-\gamma$. If we consider the special cases (\[L2\]) and (\[L3\]) for (\[auf1\]) and (\[auf2\]) together with relations (\[L5\]) and (\[L6\]), it can be shown after a lengthy but straightforward computation that the corresponding integrals are special cases of $1.15.2(4)$ in [@Prud]. This result can be interpreted as a further validation of formulae (\[auf1\]) and (\[auf2\]).\
Furthermore, we can also take $h$ to be a solution of the ODE (\[3\]) with $P$ and $Q$ as given in (\[pq\]), i.e. $$\begin{aligned}
h(x)&=&\mbox{exp}\left(-\int\!\frac{Q(x)}{P(x)}\,\mathrm{d}x\right)=\mbox{exp}\left(-\int\!\frac{\alpha\beta x-q}{K(x)}\,\mathrm{d}x\right),\quad
K(x)=\sum_{i=0}^2\mathfrak{k}_i x^i,\label{K}\\
\mathfrak{k}_2&=&\alpha+\beta+1,\quad
\mathfrak{k}_1=-[a(\gamma+\delta)+\alpha+\beta+1-\delta],\quad
\mathfrak{k}_0=a\gamma.\end{aligned}$$ Let $$\label{Delta}
\Delta=\mathfrak{k}_0\mathfrak{k}_2-\frac{1}{4}\mathfrak{k}_1^2.$$ Then, the integral in terms of which the function $h$ is expressed, can be explicitly computed by means of $2.103.5$ in [@Grad] yielding $$\label{ccc}
h(x)=\left\{
\begin{array}{ccc}
K^{-\frac{\alpha\beta}{2\mathfrak{k}_2}}(x)\mbox{exp}\left[\frac{\alpha\beta\mathfrak{k}_1+2q\mathfrak{k}_2}{2\mathfrak{k}_2\sqrt{\Delta}}\arctan{\left(\frac{2\mathfrak{k}_2 x+\mathfrak{k}_1}{2\sqrt{\Delta}}\right)}\right]&\mbox{if}~\Delta>0,\\
(x-x_0)^{-\frac{\alpha\beta}{\mathfrak{k}_2}}e^{\frac{C}{x-x_0}} & \mbox{if}~\Delta=0,\\
K^{-\frac{\alpha\beta}{2\mathfrak{k}_2}}(x)\left[\frac{2\mathfrak{k}_2 x+\mathfrak{k}_1-2\sqrt{-\Delta}}{2\mathfrak{k}_2 x+\mathfrak{k}_1+2\sqrt{-\Delta}}\right]^{\frac{\alpha\beta\mathfrak{k}_1+2q\mathfrak{k}_2}{4\mathfrak{k}_2\sqrt{-\Delta}}}&\mbox{if}~\Delta<0,
\end{array}
\right.$$ with $C=(\alpha\beta x_0-q)/\mathfrak{k}_2$. Note that we need to require that the singularity at $x_0=-\mathfrak{k}_1/2\mathfrak{k}_2$ lies outside the interval where the local solution of the Heun equation is defined. At this point (\[1\]) leads to the following indefinite integral $$\int\!x^\gamma(x-1)^\delta(x-a)^\epsilon h(x) \frac{Q^2(x)+W(Q,P)(x)}{P^2(x)}H_\ell(a,q;\alpha,\beta,\gamma,\delta;x)\,\mathrm{d}x=$$ $$\label{hh1}
-x^\gamma(x-1)^\delta(x-a)^\epsilon h(x)\left[\frac{Q(x)}{P(x)}H_\ell(a,q;\alpha,\beta,\gamma,\delta;x)+H^{'}_\ell(a,q;\alpha,\beta,\gamma,\delta;x)\right]+c,$$ where $P$ and $Q$ are given as in (\[pq\]) and $W$ denotes the Wronskian. Equation (\[hh1\]) gives rise to new indefinite integrals for the hypergeometric function. To this purpose let $a=2$, $q=\alpha\beta$, $\delta=\alpha+\beta-2\gamma+1$ which imply $\epsilon=\gamma$. Furthermore, if we define $z=2x-x^2$ from which $x=1-\sqrt{1-z}$, we get $$\int\!z^c(1-z)^{a+b-c}(1-\rho_1 z)(1-\rho_2 z)^{-2-\omega} {}_{2}F_{1}\left(a,b;c;z\right)\,\mathrm{d}z=$$ $$\label{hhh1}
\rho_3 z^c(1-z)^{a+b+1-c}(1-\rho_2 z)^{-\omega}\left[\frac{{}_{2}F_{1}\left(a,b;c;z\right)}{1-\rho_2 z}-{}_{2}F_{1}\left(a+1,b+1;c+1;z\right)\right]+\widetilde{c},$$ where $$\begin{aligned}
\rho_1&=&\frac{1+2(a+b+2ab)}{2[1-c+2(a+b+ab)]},\quad\rho_2=\frac{1+2(a+b)}{2c},\\
\rho_3&=&\frac{2c}{2(a+b+ab)-c+1},\quad\omega=\frac{2ab}{1+2(a+b)},\end{aligned}$$ and $a=\alpha/2$, $b=\beta/2$, $c=\gamma$. Note that for $a=2$, $q=\alpha\beta$, $\delta=\alpha+\beta-2\gamma+1$ the integral (\[hhh1\]) holds for both cases $\Delta>0$ and $\Delta<0$ because independently of the sign of the discriminant we have $$h(x)=\left[(\alpha+\beta+1)(x^2-2x)+2\gamma\right]^{-\frac{\alpha\beta}{2(\alpha+\beta+1)}}.$$ Let us consider the case $\Delta=0$ under the assumptions $a=2$, $q=\alpha\beta$, $\delta=\alpha+\beta-2\gamma+1$. Then, $$\label{lepanto}
\Delta=(\alpha+\beta+1)(\alpha+\beta+1-2\gamma).$$ The equation $\Delta=0$ admits the following solutions
1. $\gamma=(\alpha+\beta+1)/2$ if we look at $\Delta=0$ as an equation for $\gamma$.
2. $\alpha_1=-\beta-1$ or $\alpha_2=2\gamma-\beta-1$ if we consider $\Delta=0$ as a quadratic equation in the parameter $\alpha$.
3. $\beta_1=-\alpha-1$ or $\beta_2=2\gamma-\alpha-1$ if we consider $\Delta=0$ as a quadratic equation in the parameter $\beta$.
Since $3.$ can be obtained from $2.$ by interchanging the parameters $\alpha$ and $\beta$, we will not consider this case. If $\gamma=(\alpha+\beta+1)/2$, then $\delta=0$ and $$h(x)=(x-1)^{-\frac{\alpha\beta}{\alpha+\beta+1}}.$$ Furthermore, we find that $$\frac{Q^2(x)+W(Q,P)(x)}{P^2(x)}=\frac{\alpha\beta(\alpha\beta+\alpha+\beta+1)}{(\alpha+\beta+1)^2(x-1)^2},\quad
\frac{Q(x)}{P(x)}=\frac{\alpha\beta}{(\alpha+\beta+1)(x-1)}.$$ Finally, if we define $z=2x-x^2$ from which $x=1-\sqrt{1-z}$ and make use of (\[L1\]) and (\[L4\]), we hand up with following known indefinite integral for the hypergeometric function $$\int\!z^{a+b+\frac{1}{2}}(1-z)^{-\frac{3}{2}-\xi}{}_{2}F_{1}\left(a,b;a+b+\frac{1}{2};z\right)\,\mathrm{d}z=\lambda z^{a+b+\frac{1}{2}}(1-z)^{-\xi}\cdot$$ $$\label{hhh1n}
\left[\frac{{}_{2}F_{1}\left(a,b;a+b+\frac{1}{2};z\right)}{\sqrt{1-z}}-\sqrt{1-z}{}_{2}F_{1}\left(a+1,b+1;a+b+\frac{3}{2};z\right)\right]+c,$$ where $$\lambda=\frac{2(2a+2b+1)}{2(2ab+a+b)+1},\quad\xi=\frac{2ab}{2a+2b+1},\quad a=\frac{\alpha}{2},\quad b=\frac{\beta}{2}.$$ If we consider instead the case $\alpha=-\beta-1$, then $\delta=-2\gamma$ and $\epsilon=\gamma$. Moreover, we have $$h(x)=\mbox{exp}\left(\frac{\beta(\beta+1)}{4\gamma}(x^2-2x)\right).$$ Furthermore, we find that $$\frac{Q^2(x)+W(Q,P)(x)}{P^2(x)}=\frac{\beta(\beta+1)\left[\beta(\beta+1)(x-1)^2+2\gamma\right]}{4\gamma^2},~
\frac{Q(x)}{P(x)}=\frac{\beta(\beta+1)(1-x)}{2\gamma}.$$ Finally, if we define $z=2x-x^2$ from which $x=1-\sqrt{1-z}$ and make use of (\[L1\]) and (\[L4\]), we hand up with the following new indefinite integral for the hypergeometric function $$\int\!z^{c}(1-z)^{-c-\frac{1}{2}}(1-p_1 z)e^{-p_2 z}{}_{2}F_{1}\left(-b-\frac{1}{2},b;c;z\right)\,\mathrm{d}z=$$ $$\label{hhh1nT}
\lambda_1 z^c(1-z)^{\frac{1}{2}-c}e^{-p_2 z}\left[{}_{2}F_{1}\left(-b+\frac{1}{2},b+1;c+1;z\right)-{}_{2}F_{1}\left(-b-\frac{1}{2},b;c;z\right)\right]+\widetilde{c},$$ where $$\lambda_1=\frac{2c}{b(2b+1)+c},\quad p_1=\frac{b(2b+1)}{b(2b+1)+c},\quad p_2=\frac{b(2b+1)}{2c},\quad b=\frac{\beta}{2},\quad c=\gamma.$$ The case $\alpha=2\gamma-\beta-1$ will not be treated here because it gives rise to an integral similar to (\[hhh1n\]). Further indefinite integrals can be obtained by choosing the function $h$ to be a particular solution to (\[4\]). There are several possibilities. For instance, if we make a s-homotopic transformation followed by a transformation of the independent variable in (\[4\]), we find two linearly independent solutions of the form $$h_i(x)=x^{-\alpha_i}(x-a)H_\ell\left(\frac{1}{a},q_i;\alpha_i,\beta_i,\gamma_i,0;\frac{1}{x}\right),\quad i=1,2$$ with $$\begin{aligned}
&&q_1=\frac{q-\alpha\beta}{a}+\omega-\rho,~
\alpha_1=\frac{1+\rho}{2},~\beta_1=\alpha_1+1,~\gamma_1=\frac{2(\omega-\alpha\beta+\rho)}{1+\rho},\\
&&q_2=\frac{\rho[\rho^2(q-\alpha\beta)-\alpha\beta(4a\omega+3)+3q]-4[\alpha^2\beta^2(a-3)+(3q-a+1)]}{a(1+\rho)^3},\\
&&\alpha_2=\frac{2\alpha\beta}{1+\rho},\quad
\beta_2=\alpha_2+1,\quad\gamma_2=2\alpha_2,\quad\rho=\sqrt{1-4\alpha\beta},\quad\omega=1-\alpha\beta\label{ro}\end{aligned}$$ and (\[1\]) gives rise to a couple of indefinite integral involving products of Heun functions and its derivatives, more precisely, for each $i=1,2$ we find $$\int\!x^{\gamma-1}(x-1)^{\delta-1}(x-a)^{\epsilon-1}K(x)h^{'}_i(x)
H_l(a,q;\alpha,\beta,\gamma,\delta;x)\,\mathrm{d}x=$$ $$\label{HeunHH}
x^{\gamma}(x-1)^{\delta}(x-a)^{\epsilon}
\left[h_i^{'}(x)H_l(a,q;\alpha,\beta,\gamma,\delta;x)-h_i(x)H^{'}_l(a,q;\alpha,\beta,\gamma,\delta;x)\right]+c,$$ where $K$ has been defined in (\[K\]). It is interesting to observe that if $q=0$ in (\[4\]), the two linearly independent solutions are given by $$h_1(x)=\frac{x-a}{(x-1)^{b_1}}{}_{2}F_{1}\left(a_1,b_1;c_1;\frac{a-1}{x-1}\right),\quad
h_2(x)=\frac{x-a}{(x-1)^{a_1}}{}_{2}F_{1}\left(a_2,b_2;c_2;\frac{a-1}{x-1}\right),$$ where $$a_1=\frac{3-\rho}{2},\quad b_1=a_1-1,\quad c_1=2b_1,\quad
a_2=\frac{1+\rho}{2},\quad b_2=a_2+1,\quad c_2=2a_1$$ with $\rho$ defined in (\[ro\]), and from (\[HeunHH\]) we can derive indefinite integrals of products of hypergeometric and Heun functions. Another possibility is to look at (\[4\]) as a Heun equation with $\gamma=\delta=\epsilon=0$ and $\beta=-1-\alpha$. Then, we obtain an integral similar to (\[HeunHH\]) but with $h_i^{'}(x)$ replaced by $H_l^{'}(a,q;\alpha,-1-\alpha,0,0;x)$. Furthermore, let us consider the particular solution $y(x)=H_l(a,q;\alpha,\beta,\gamma,\delta;x)$ to the Heun equation and a conjugate ODE to the Heun equation with $\overline{Q}(x)=(\alpha\beta x+q)/x(x-1)(x-a)$ for which we pick the particular solution $h(x)=H_l(a,-q;\alpha,\beta,\gamma,\delta;x)$. Then, (\[intcon\]) yields the following indefinite integral involving products of Heun functions $$\int\!x^{\gamma-1}(x-1)^{\delta-1}(x-a)^{\epsilon-1}H_l(a,q;\alpha,\beta,\gamma,\delta;x)H_l(a,-q;\alpha,\beta,\gamma,\delta;x)\,\mathrm{d}x=$$ $$\frac{x^{\gamma}(x-1)^{\delta}(x-a)^{\epsilon}}{2q}W\left(H_l(a,-q;\alpha,\beta,\gamma,\delta;x),H_l(a,q;\alpha,\beta,\gamma,\delta;x)\right)+c$$ provided that $q\neq 0$. We conclude this section by constructing an indefinite integral involving products of Heun functions with complete elliptic integrals. To this purpose, we consider the Heun equation with $\gamma=1$ and $\delta=\epsilon=0$. Then, one particular solution is $$\begin{aligned}
y(x)&=&\left( x-1 \right)^{\alpha}\psi(x),\label{y}\\
\psi(x)&=&H_l\left(1-a, \alpha^2(1-a)-\alpha-q,-\alpha,-\alpha+1,-2\,\alpha+1,0,
\frac{1-a}{1-x}\right).\end{aligned}$$ As a conjugate equation we take $$h^{''}(x)+\frac{1}{x}h^{'}(x)+\frac{1}{1-x^2}h(x)=0$$ having a particular solution expressed in terms of complete elliptic integrals [@Grad], namely $h(x)=\mathbf{E}(x^{'})$ with complementary modulus $x^{'}=\sqrt{1-x^2}$. Then, (\[intcon\]) together with the functional relation 8.123(4) in [@Grad] between elliptic integrals $$\frac{d\mathbf{E}(x^{'})}{dx^{'}}=\frac{\mathbf{E}(x^{'})-\mathbf{K}(x^{'})}{x^{'}}$$ gives $$\int\!\frac{(x-1)^{\alpha-1}\mathfrak{Q}(x)}{(x+1)(x-a)}\mathbf{E}(x^{'})\psi(x)\,\mathrm{d}x=$$ $$x(x-1)^\alpha\left[\frac{(1-\alpha)x-\alpha}{x^2-1}\mathbf{E}(x^{'})\psi(x)-\frac{x}{x^2-1}\mathbf{K}(x^{'})\psi(x)-\mathbf{E}(x^{'})\psi^{'}(x)\right]$$ with $\mathfrak{Q}(x)=(1-\alpha^2)x^2-(a+q+\alpha^2)x-q$ and $\psi$ given by (\[y\]).
Comments and conclusions
========================
We applied the so-called Lagrangian method to obtain indefinite integrals of functions belonging to the family of Heun confluent functions. This approach allowed us to derive several novel indefinite integrals for the confluent, biconfluent, doubly confluent, and triconfluent Heun functions for which sample results have been provided. Our findings only scratch the surface of the wealth of new integral formulae one may obtain by using the aformentioned method.
[99]{} Conway JT. A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015;26:1-12.
Conway JT. Indefinite integrals of some special functions from a new method. Integral Transforms Spec. Funct. 2015;26:1-14.
Ronveaux A. Heun’s Differential Equations. New York (NY): Oxford University Press; 1995.
Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products.San Diego: Elsevier Academic Press; 2007.
Hounkonnou MN, Ronveaux A. About derivatives of Heun’s functions from polynomial transformations of hypergeometric equations. Appl. Math. Comput. 2009;209:421-24.
Maier RS. On reducing the Heun equation to hypergeometric equation. J. Differ. Equat. 2005;213:171–203.
Abramowitz M, Stegun IA. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications; 1964.
Prudnikov AP, Brychkov YuA, Marichev OI. Integrals and Series, Vol.3, More special functions. New York (NY): Gordon and Breach; 1990
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---
abstract: 'We investigate the Anderson transition found in the spectrum of the Dirac operator of Quantum Chromodynamics (QCD) at high temperature, studying the properties of the critical quark eigenfunctions. Applying multifractal finite-size scaling we determine the critical point and the critical exponent of the transition, finding agreement with previous results, and with available results for the unitary Anderson model. We estimate several multifractal exponents, finding also in this case agreement with a recent determination for the unitary Anderson model. Our results confirm the presence of a true Anderson localization-delocalization transition in the spectrum of the quark Dirac operator at high-temperature, and further support that it belongs to the 3D unitary Anderson model class.'
author:
- 'L[á]{}szl[ó]{}'
- Matteo
- Ferenc
- 'Tam[á]{}s G.'
- Imre
title: Anderson transition and multifractals in the spectrum of the Dirac operator of Quantum Chromodynamics at high temperature
---
Introduction {#sec:qcd_intro}
============
The Anderson metal-insulator transition is a genuine quantum phase transition, which has been widely investigated in condensed matter physics since the seminal paper of Anderson [@Anderson58]. In the past years Anderson transitions were found in a wide range of physical systems, such as ultrasound in disordered elastic networks [@Hu08; @Faez09], light in disordered photonic lattices in the transverse direction [@Segev13], or in an ultracold atomic system in a disordered laser trap [@AspectBouyer12].
A characteristic feature of Anderson transitions is the rich multifractal structure of critical eigenstates, which has been the subject of intense research activity in recent years (see Ref. for a review). Direct signs of multifractals at the metal-insulator transition point have been observed experimentally in dilute magnetic semiconductors [@Richardella10]. Multifractality can moreover influence the behavior of various systems near criticality in different ways. For example, the large overlap of multifractal wave-functions can increase the superconducting critical temperature [@Feigelman10Burmistrov12]. The multifractality of the local density of states may induce a new phase because of the presence of local Kondo effects induced by local pseudogaps at the Fermi energy [@Kettemann12].
The simplest model displaying a metal-insulator transition is the Anderson model, which describes non-interacting fermions in a disordered crystal. Disorder is usually introduced through a random on-site potential, while hopping elements are fixed (up to a random phase or $SU(2)$ rotation). In this case the system belongs to one of the Wigner-Dyson (WD) symmetry classes depending on the global symmetries of the system. On the other hand, systems with vanishing on-site terms and random hopping terms, if the lattice is bipartite, possess an additional chiral symmetry and belong to one of the chiral WD classes [@EversMirlin08].
Quite surprisingly, an Anderson transition has been shown to take place also in the spectrum of the Dirac operator in Quantum Chromodynamics (QCD) at high temperature [@GarciaGarcia:2006gr; @Pittler; @KP; @Kovacs12; @Giordano14] (see Ref. for a review). QCD is the quantum field theory governing strong interactions at the microscopic level, and operates on length and energy scales vastly different from the ones usually encountered in condensed matter physics. QCD is a non Abelian gauge theory, describing the interactions of quarks, which are fermions, and gluons, which are the vector bosons of the $SU(3)$ gauge symmetry. Although these are the fundamental degrees of freedom, they do not appear in the spectrum of the theory at low temperatures, which contains only hadrons, i.e., bound states of quarks and gluons, due to the phenomenon of confinement. However, at a (pseudo)critical temperature, $T_c$, strongly interacting matter undergoes a crossover to the so-called quark-gluon-plasma phase, and at high temperatures quarks and gluons are deconfined. This transition is accompanied by the restoration of the approximate chiral symmetry that is spontaneously broken at low temperatures [@Aoki:2006we].
[c c c]{}
[eigvec\_L56\_ins.png]{} (5,95)[(a)]{}
&
[eigvec\_L56\_crit.png]{} (5,95)[(b)]{}
&
[eigvec\_L56\_met.png]{} (5,95)[(c)]{}
\
\
Contributions of quarks to observables, as well as all quark correlation functions, are entirely encoded in the eigenvalues and the eigenvectors of the Dirac operator in the background of a non Abelian gauge field. Physical quantities are then obtained after averaging over the gauge field configurations with the appropriate path-integral measure. In this respect, the eigenmodes of the Dirac operator can be formally treated as the eigenstates of a random “Hamiltonian”, with disorder provided by the gauge field fluctuations. Among the eigenmodes, a prominent role is played by the low-lying ones: for example, they give the most important contributions to the quark correlation functions, and determine the fate of chiral symmetry through the Banks-Casher relation [@Banks:1979yr]. The low end of the spectrum looks completely different in the hadronic and in the quark-gluon-plasma phase. At low temperatures, the density of states is finite near the origin, and both low-lying and bulk eigenmodes are extended throughout the system. In contrast, at high temperatures, above $T_c$, the density of states vanishes at the origin, and the low-lying eigenmodes are localized on the scale of the inverse temperature, while higher up in the spectrum, beyond a temperature-dependent critical “energy”, $E_c(T)$, the eigenmodes are again extended [@KP; @Kovacs12]. The temperature dependence of the “mobility edge” was investigated in Ref. , in which it was found that $E_c(T)$ extrapolates to zero at a temperature compatible with $T_c$. Typical Dirac eigenmodes in the localized, critical and delocalized regimes are shown in Fig. \[fig:qcd\_eigvecs\]. The transition in the spectrum from localized to delocalized eigenmodes has been shown to be a second-order phase transition, with critical exponent compatible with the one found in the three-dimensional unitary Anderson model [@Giordano14].
It is rather surprising at first that the Anderson transition in the high-temperature QCD Dirac spectrum seems to belong to the same universality class as that of the three-dimensional unitary Anderson model. From the point of view of statistical systems, QCD at a finite temperature $T$ is in fact a four-dimensional Euclidean model, with the “temporal” dimension compactified on a circle of length $1/T$. However, it has been argued that high-temperature QCD is an effectively three-dimensional disordered system with on-site disorder, the strength of which is set by the temperature [@Giordano:2015vla; @Bruckmann:2011cc]. While this makes it more plausible that the two models actually belong to the same universality class, it does not make it less important to look for further evidence. In this respect, finding the same multifractal structure in the critical eigenstates would give strong support to the claim of Ref. , and so to the broader universality of the critical properties of Anderson transitions. The study of this multifractal structure is precisely the aim of this work.
The plan of the paper is the following. In section \[sec:MFSS\] we give a brief discussion of multifractality, and of the method of multifractal finite-size scaling (MFSS). In section \[sec:qcd\_numerics\] we describe in some detail the Dirac operator and the numerical simulations of QCD employed in this paper. In section \[sec:qcd\_corrE\] we study the correlations between eigenvectors of the Dirac operator around the critical energy, both for comparison to the 3D unitary Anderson model, and for their appropriate treatment in the statistical analysis. In section \[sec:qcd\_MFSS\] we discuss the results of MFSS for the eigenvectors of the Dirac operator. Finally, in section \[sec:concl\] we state our conclusions.
Finite-size scaling laws for generalized multifractal exponents {#sec:MFSS}
===============================================================
In this section we briefly review wave-function multifractality and the technique of multifractal finite-size scaling. The wave-function $\psi(\vec x)$ of a particle in $\mathbb{R}^d$ is naturally associated to a local probability distribution, namely $|\psi(\vec x)|^2$, giving the probability to find the particle in an infinitesimal neighborhood of $\vec x$. For smooth wave functions, the probability to find the particle in a small but finite neighborhood of $\vec x$ of size $r$ scales as $\sim r^d$. For fractal wave functions, this probability scales as $\sim r^\alpha$, where $\alpha<d$ is called the fractal dimension. For strongly fluctuating wave-functions, however, this probability scales in general as $\sim
r^{\alpha(\vec x)}$, with an $\vec x$-dependent power $\alpha(\vec x)$ called the local dimension. In turn, points with the same local dimension, $\alpha(\vec x)=\alpha$, constitute a subset of $\mathbb{R}^d$ characterized by its own fractal dimension, which generally depends on $\alpha$. The wave-function therefore defines not one, but many different fractals, and is therefore said to be multifractal. Multifractal wave-functions are strongly fluctuating on every length-scale, and their characterization requires an infinite number of fractal dimensions, called multifractal exponents (MFEs). An example of a multifractal wave-function is shown in Fig. \[fig:qcd\_eigvecs\](b).
Multifractality is a known feature of critical eigenfunctions at the Anderson metal-insulator transition [@EversMirlin08], that can be studied by means of multifractal finite-size scaling (MFSS) [@Rodriguez_prl]. In recent high-precision calculations [@Rodriguez11; @Ujfalusi15; @Ujfalusi14], MFSS has been successfully employed to determine the MFEs of critical eigenfunctions, as well as to obtain a more precise estimate of the critical disorder and of the critical exponents, for Anderson models in different symmetry classes. In this work we want to perform a similar MFSS analysis to study the Anderson localization-delocalization transition in the spectrum of the Dirac operator in QCD.
In the remainder of this section we describe MFSS in some detail. Our methods and notations are essentially the same as in Ref. , to which we refer the reader for a more detailed discussion. There is however one important difference, concerning the way in which the transition is approached. In Ref. the transition was studied by looking at wave functions at the band center and varying the amount of disorder, $W$. In QCD the amount of disorder is effectively set by the temperature, and it is more convenient to keep it fixed and study the transition as a function of energy, $E$, by looking at wave-functions near the mobility edge, $E_c$. Therefore, $W$ has been replaced by $E$ in the expressions of Ref. .
Let us consider a $d$-dimensional cubic lattice of linear size $L$, and a critical eigenfunction of a random Hamiltonian, $\psi(\vec
x)$, defined on the lattice sites $\vec x$ and normalized to 1. We can divide the lattice into smaller boxes of linear size $\ell$, and compute the probability corresponding to the $k$-th box as $$\label{eq:multifractal_mu}
\mu_k=\sum_{\vec x \in {\rm box}_k} |\psi(\vec x)|^2,$$ where the sum runs over the lattice sites contained in the $k$-th box. The generalized inverse participation ratios (GIPRs) are the moments of the box probability. The GIPRs and their derivatives read $$\label{eq:multifractals_SqRq}
R_q=\sum_{k=1}^{\lambda^{-d}} \mu_k^q\qquad
S_q=\frac{dR_q}{dq}=\sum_{k=1}^{\lambda^{-d}} \mu_k^q \ln \mu_k,$$ where $\lambda=\frac{\ell}{L}$, and the sum runs over all the $\lambda^{-d}$ boxes of size $\ell$. For small $\lambda$, the averages of $R_q$ and $S_q$ over disorder realizations follow a power-law behavior as a function of $\lambda$, which leads one to define the following exponents: $$D_q=\lim_{\lambda\to 0}\frac{1}{q-1}\frac{\ln {\left< R_q \right>}}{\ln
\lambda}\qquad \alpha_q=\lim_{\lambda\to
0}\frac{{\left< S_q \right>}}{{\left< R_q \right>}\ln\lambda}.
\label{eq:multifractals_D_alpha}$$ $D_q$ and $\alpha_q$ are generalized fractal dimensions, usually referred to as multifractal exponents (MFEs). One can similarly define MFEs for localized and delocalized states by substituting critical eigenfunctions with localized or delocalized eigenfunctions in Eq. . In the delocalized/metallic part of the spectrum, states extend over the whole lattice, so their effective size grows proportionally to the volume, thus leading to $D_q^{met}\equiv d$. On the other hand, in the localized/insulating regime, states are exponentially localized, so that their effective size does not change with the system size, resulting in $D_q^{ins}\equiv 0$ for $q> 0$, and $D_q^{ins}\equiv
\infty$ for $q<0$. At criticality, $E=E_c$, the eigenstates are instead expected to be multifractal, with nontrivial, $q$-dependent $D_q$ and $\alpha_q$.
This jump of the MFEs at the critical point happens only in an infinite system. The main idea of MFSS is to use the MFEs as order parameters for finite size-scaling. In order to do that we have to define the finite size version of the MFEs at a given energy, $$\begin{aligned}
\label{eq:alphaD_ens}
\tilde{\alpha}_q^{ens}(E,L,\ell) &=& \frac{{\left< S_q \right>}}{{\left< R_q \right>}\ln\lambda}\,,\\
\tilde{D}_q^{ens}(E,L,\ell) &=& \frac{1}{q-1}\frac{\ln{\left< R_q \right>}}{\ln\lambda}\,,\end{aligned}$$ where it is understood that wave-functions of energy around $E$ are used on the right-hand side, and where the superscript [*ens*]{} is to remind the reader that one has to perform [*ensemble*]{} averaging over the different disorder realizations. $\tilde{\alpha}_q$ and $\tilde{D}_q$ are called generalized multifractal exponents (GMFEs). Every GMFE approaches the value of the corresponding MFE at the critical point, $E=E_c$, only in the limit $\lambda\to 0$. One can also define [*typical*]{} MFEs, $$\begin{aligned}
\tilde{\alpha}_q^{typ}(E,L,\ell) &=&
{\left< \frac{S_q}{R_q} \right>}\frac{1}{\ln\lambda}\,,\\
\tilde{D}_q^{typ}(E,L,\ell) &=& \frac{1}{q-1}\frac{{\left< \ln
R_q \right>}}{\ln\lambda}\,,
\label{eq:alphaD_typ}\end{aligned}$$ which can be used as well in a finite-size scaling analysis. However, as we said above, MFEs are defined through [*ensemble*]{} averaging in principle \[see Eq. (\[eq:multifractals\_D\_alpha\])\], and when computing MFEs in Sec. \[sec:qcd\_MFSS\] we use [*ensemble*]{} averaged quantities only.
In the renormalization group language, the Anderson transition is characterized by a single relevant operator [@AALR], and so in the vicinity of the critical point one can derive scaling laws for the GMFEs, which can be summarized in a single equation, using a common letter, $G$, for the GMFEs: $$\tilde{G}_q(E,L,\ell) =
G_q+\frac{1}{\ln\lambda}\mathcal{G}_q\left(\frac{L}{\xi},
\frac{\ell}{\xi}\right) \,.
\label{eq:fss_scalinglaw_Ll}$$ At the critical point, the localization length diverges as $\xi\sim[\varrho(E-E_c)]^{-\nu}$, where $\varrho(E-E_c)\approx E-E_c$ for $(E-E_c)/E_c\ll 1$. The system sizes employed in this paper, however, are not big enough to justify the use of one-parameter scaling, and so we included the contribution of an irrelevant operator, $\eta=\eta(E-E_c)$, which leads us to write $$\begin{aligned}
\tilde{G}_q(E,L,\ell) =
G_q &+& \frac{1}{\ln\lambda} \left[\mathcal{G}^{r}_q\left(\varrho
L^{\frac{1}{\nu}}, \varrho
\ell^{\frac{1}{\nu}}\right)+\right. \nonumber\\
&+&\left. \eta \ell^{-y}\mathcal{G}^{ir}_q\left(\varrho
L^{\frac{1}{\nu}},\varrho \ell^{\frac{1}{\nu}}\right)\right].
\label{eq:fss_anderson_scalinglaw_Ll}\end{aligned}$$ A second irrelevant term, proportional to $L^{-y'}$, is expected to be less important and will be neglected in the analysis [@Rodriguez11; @Ujfalusi15].
Fits to the numerical data are performed by expanding $\mathcal{G}^{r}$ and $\mathcal{G}^{ir}$ in the variables $\varrho
L^{\frac{1}{\nu}}$ and $\varrho \ell^{\frac{1}{\nu}}$ up to order $n_{r}$ and $n_{ir}$, respectively. The number of parameters therefore grows as $\sim n_{r}^2+n_{ir}^2$. Moreover, $\varrho$ and $\eta$ must also be expanded in powers of $E-E_c$ up to order $n_\varrho$ and $n_\eta$, which further increases the number of fitting parameters. The fit provides all the physically interesting quantities, namely the critical point, $E_c$, the critical exponent, $\nu$, the irrelevant exponent $y$, and the MFE, $G_q$.
A simpler fit can be performed by setting $\lambda=\ell/L$ to a fixed value. In this case, dropping $\lambda$ from the notation, we can write $$\tilde{G}_q(E,L) = {\cal G}_q\left(\frac{L}{\xi}\right) =
{\cal G}^{r}_q\left(\varrho L^{\frac{1}{\nu}}\right) + \eta
L^{-y}{\cal G}^{ir}_q\left(\varrho L^{\frac{1}{\nu}}\right),
\label{eq:fss_anderson_scalinglaw_lambda}$$ having absorbed $G_q$ and the factor $\lambda^{-y}/\ln \lambda$ into new functions ${\cal G}_q^{r/ir}$. The main advantage is that since ${\cal G}_q^{r/ir}$ are now single-variable functions, the number of expansion parameters grows only as $\sim n_{r}+n_{ir}$. On the other hand, with this method one can determine only $E_c$, $\nu$, and $y$, while the value of the MFE, $G_q$, cannot be obtained.
Properties of the Dirac operator and details of the simulations {#sec:qcd_numerics}
===============================================================
In this section we give the relevant details about the Dirac operator and QCD, and about how the QCD Dirac spectrum can be studied by means of numerical simulations. The continuum Euclidean Dirac operator is $$D(A)=\sum_{\mu=1}^4 \gamma_\mu(\partial_\mu+igA_\mu),$$ where $\gamma_\mu$ are the Euclidean Dirac matrices, $g$ is the coupling constant, and $A_\mu$ is the non Abelian gauge field. More precisely, $A_\mu=\sum_a A_\mu^a t^a$ is a Hermitian $3\times 3$ matrix, where $A_\mu^a=A_\mu^a(x)=A_\mu^a(\vec {x},t)$ is real and the sum runs over the generators $t^a$ of $SU(3)$. The Dirac operator is thus anti-Hermitian, so admitting a straightforward interpretation as ($i$ times) the Hamiltonian of a quantum system. The Dirac operator is a chiral operator with the following structure in spinor space, $$D=
i \left(\begin{matrix}
0 &W\\
W^\dagger & 0\\
\end{matrix}\right),$$ with $W$ a complex matrix with no further symmetry [@Verbaarschot00]. As a random matrix model, the Dirac operator in a random gauge field belongs therefore to the chiral unitary class. Chiral symmetry is expressed by the anticommutation relation $\{\gamma_5,D\}=0$, which implies that the nonzero eigenvalues come in pairs $\pm iE_n$. It is thus sufficient to consider the positive part of the spectrum only.
The partition function of QCD at temperature $T$ can be expressed as a functional integral, $$\label{eq:qcd_part}
Z_{\rm QCD} = \int [dA]\,e^{-S_{\rm g}[A]} \prod_f\det [D(A) + m_f] \,,$$ with the constraint $A_\mu(\vec {x},1/T)=A_\mu(\vec {x},0)$. The product is over the six different types of quarks (“flavors”), with $m_f$ the mass of quark $f$. Here $S_{\rm g}[A]$ is a positive functional of the gauge field, which together with the determinants provides the probability distribution of the disorder, i.e., of the gauge field configurations. Numerical simulations of QCD require the discretization of Eq. on a finite lattice. For a review of lattice QCD see, e.g., Ref. . While the discretization of the gauge fields poses no particular problem, and can be performed preserving exact gauge invariance [@Wilson:1974sk], fermion fields are known to be more problematic, and the discretization of the Dirac operator spoils some of the properties of its continuum counterpart. Nevertheless, the discretization that we employed, namely staggered fermions [@Susskind:1976jm], preserves the anti-Hermiticity and the symmetry of the spectrum with respect to the origin, and moreover preserves the chiral unitary symmetry class [@Verbaarschot00].
It must be noted at this point that in the case of the Anderson model, chiral and non-chiral symmetry classes differ only in their properties near the band center [@Evangelou03], i.e., $E=0$, while the properties of the bulk of the spectrum are similar. For example, the authors of Ref. found Wigner-Dyson statistics in the bulk spectrum of a three-dimensional chiral orthogonal disordered model. Moreover, even the critical exponent of the orthogonal and of the chiral orthogonal class turn out to be the same, up to very high numerical precision [@Biswas00]. We expect the same to be true for the multifractal exponents.
Let us now describe the numerical setting in some detail. QCD is discretized on a periodic hypercubic lattice $x_\mu\in\mathbb{Z}$, of spatial extent $L$ in each direction and temporal extent $L_t$. The gauge fields $A_\mu$ are replaced by corresponding gauge links, i.e., parallel transporters along each link of the lattice, which are elements of the gauge group, $SU(3)$. The functional $S_{\rm g}$ is discretized and expressed in terms of the gauge links, and the integration over gauge fields is replaced by the integration with the Haar measure over gauge links, i.e., over the gauge-group valued variables on the links. Finally, the continuum Dirac operator is replaced by the staggered Dirac operator, which reads $$\label{eq:stag_op}
D^{\rm stag}_{xy} = {\frac{1}{2}}\sum_{\mu=1}^4 \eta_\mu(x)
\left[ \delta_{x+\hat\mu,y}U_\mu(x)
- \delta_{x-\hat\mu,y}U_\mu^\dag(x-\hat\mu)\right]\,,$$ with $\eta_\mu(x)=(-1)^{\sum_{\nu<\mu} x_\nu}$, and $U_\mu(x)\in SU(3)$ the gauge link connecting the lattice site $x$ to the neighboring site along direction $\hat\mu$. The staggered Dirac operator carries only spacetime and color indices, i.e., it has no spinorial structure. The eigenvalue equation $ D^{\rm
stag}\chi = iE\chi$ must be supplemented with the antiperiodic boundary condition $\chi(\vec x,L_t) = -\chi(\vec x,0)$ for the quark eigenfunction.
As we have already remarked, the Dirac operator can be viewed as a random Hamiltonian, with disorder provided by the fluctuations of the gauge fields, and distributed according to the Boltzmann weight appearing in the partition function. In its discretized version, the Dirac operator is a large sparse matrix, with nonzero random elements only in the off-diagonal, nearest-neighbor hopping terms, which depend on the parallel transporter on the corresponding link of the lattice. This resembles an Anderson model with off-diagonal disorder, although here the fluctuations of the gauge links are correlated, rather than independent. However, since the theory has a mass gap, correlations decrease exponentially with the distance. Moreover, the strong correlation between the different time-slices makes the model effectively three-dimensional, with the fluctuations of the temporal links acting effectively as a three-dimensional diagonal disorder [@Bruckmann:2011cc; @Giordano:2015vla]. The size of the gauge field fluctuations are determined by the temperature, which therefore is expected to play the same role as the amount of disorder in the Anderson model. This is confirmed by the fact that the temperature governs the position of the mobility edge.
In the present work we have studied the spectrum of the Dirac operator by generating gauge link configurations, i.e., realizations of disorder, by means of Monte-Carlo methods. Numerical calculations were done on a GPU cluster. In our simulations we have included only the three lightest flavors (up, down, and strange), with equal masses for the up and down quark. For many purposes, this is a good approximation of the real world. The lattice spacing in physical units was set to $a=0.125~{\rm fm}$ and the temporal size was fixed to $L_t=4$, resulting in the temperature $T\approx 2.6\, T_c$, well above the crossover temperature (see Refs. for more details). Technical details about the numerical implementation and the scale-setting procedure can be found in Refs. . We have computed the eigenpairs of the Dirac operator from the smallest eigenvalue up to the upper end of the critical region, on lattices of spatial sizes in the range $L=24-56$ (in lattice units). A detailed list is reported in Tab. \[tab:qcd\_fss\_systemsize\] along with the corresponding number of samples.
The three-dimensional box probability, Eq. , required for the multifractal analysis, was constructed as follows. To have a gauge-invariant description we summed over the color components, labelled by $c$. Moreover, due to the strong correlation between the lattice time-slices, the eigenvectors of the Dirac operator look qualitatively the same on each of them, so we can also sum over the time-slices, $t$. The squared amplitude $|\psi(\vec
x)|^2$ is then defined as $|\psi(\vec x)|^2 \equiv \sum_{t,c}
|\chi_c(\vec x,t)|^2$, and provides the basic three-dimensional spatial probability distribution, from which the box probability distribution is then obtained in the usual way.
system size $(L)$ number of samples
------------------- -------------------
24 41517
28 20548
32 19250
36 14869
40 8812
44 5242
48 7008
56 3107
: System sizes and corresponding number of gauge configurations used in this work.[]{data-label="tab:qcd_fss_systemsize"}
Correlations between eigenvectors {#sec:qcd_corrE}
=================================
In this section we investigate the correlations between different eigenvectors of the Dirac operator in a given gauge configuration. Our motivation is twofold. On the one hand, we want to compare the eigenvector correlations in QCD with the ones found in the unitary Anderson model. On the other hand, these correlations have to be properly taken into account when fitting the numerical data to determine the various critical quantities, as we do in Sec. \[sec:qcd\_MFSS\].
[c c]{}
[corr\_and\_log\_L10\_W18.370000]{} (-2,72)[(a)]{}
[corr\_log\_032\_035]{} (-2,72)[(b)]{}
Cuevas and Kravtsov [@CuevasKravtsov07] showed that in the Anderson model there are non-negligible correlations between eigenvectors. Similar correlations are therefore expected also in other disordered systems, like the one under consideration. The relevant quantities are the density-density correlations, which are defined in terms of the overlap integral, which for the $i$-th and $j$-th eigenfunctions reads $$\label{eq:qcd_K2}
K_2^{ij}=\int d^3x {\left| \psi_i \right|}^2 {\left| \psi_j \right|}^2.$$ In the case of QCD, ${\left| \psi_i \right|}^2$ has the meaning discussed above at the end of Sec. \[sec:qcd\_numerics\]. One then defines the joint probability distribution of $K_2^{ij}$ and of the energy difference between eigenstates, $$P(\omega,k)={\left< \sum_{i,j}\delta(E_i-E_j-\omega)\delta(K_2^{ij}-k) \right>}.$$ To characterize the average behavior of the overlap integral as a function of energy, its conditional expectation value is the natural choice, $$\label{eq:correlations}
C(\omega)= {\frac{\int dk\,kP(\omega,k)}{\int dk\,
P(\omega,k)}}.$$ The quantity $C(\omega)$ is expected to be of order $1/N$ along the whole spectrum, where $N=L^3$ is the volume of the system. Indeed, for two delocalized states $K_2^{ij}\approx 1/N$, while for two localized states $K_2^{ij}$ is nonzero only if they happen to be in the same region, in which case it is of order 1, and the probability that this happens is of order $1/N$. Fig. \[fig:qcd\_corrE\](a) shows the eigenvector correlation $C(\omega)$ in the unitary Anderson model at criticality. One can see a large enhancement of the correlation at small $\omega$, and decreasing behavior with growing energy separations, which is similar to the results of Ref. for the orthogonal Anderson model. Examining the same correlation for critical eigenfunctions in QCD, we also find an enhancement at small energy separations, see Fig. \[fig:qcd\_corrE\](b). In the critical regime the behavior of the two systems is very similar, and even the approximate exponent of the power-law decay is close to $0.5$ in both cases. This is a nice example of the similarity of the two models, and in Sec. \[sec:qcd\_MFSS\] we present further similarities in more detail.
MFSS for the eigenvectors of the Dirac operator {#sec:qcd_MFSS}
===============================================
In this section we would like to characterize the Anderson phase transition in the spectrum of the Dirac operator of QCD in the frame of the MFSS, described in Sec. \[sec:MFSS\]. As discussed at the end of Sec. \[sec:qcd\_numerics\], a three-dimensional spatial probability distribution was calculated from the eigenvectors. From that, the GMFEs $\tilde{\alpha}_q$ and $\tilde{D}_q$ were then computed according to Eqs. (\[eq:alphaD\_ens\])–(\[eq:alphaD\_typ\]). More precisely, we chose $26$ values of energy, $E_i$, in the range $E\in[0.32,0.35]$, and for the $i$-th energy value and the $k$-th gauge configuration we computed $R_{qi}^k$ and $S_{qi}^k$ according to Eq. . In order to decrease the numerical noise we averaged over all the eigenvectors in an energy range of width $\Delta E=0.0012$ around $E_i$. The GMFEs $\tilde{\alpha}_q(E_i,L,\ell)$ and $\tilde{D}_q(E_i,L,\ell)$ are then obtained by averaging $R_{qi}^k$ and $S_{qi}^k$ over the index $k$, i.e., over configurations, or in other words, over different realizations of disorder.
An example of the resulting GMFEs at fixed $\lambda=0.125$ is depicted in Fig. \[fig:qcd\_lambda0125\_alphaqDq\_raw\]. As the system size grows, the curves shift to opposite directions on the two sides of the transition. At low energy they shift down, indicating a localized phase, while at high energy they shift up, suggesting a metallic phase, as expected. In between, the curves should cross at a common point, corresponding to the critical energy, but due to finite size effects originating from the irrelevant terms this is true only approximately.
[c c]{}
[ens\_D\_lambda0.125000\_q0.100000]{} (52,18)[{width=".18\linewidth"} (-91,56)[(a)]{} ]{}
&
[typ\_alpha\_lambda0.125000\_q1.000000]{} (52,18)[{width=".18\linewidth"} (-91,56)[(b)]{} ]{}
\
Data were then fitted with the scaling laws Eqs. (\[eq:fss\_anderson\_scalinglaw\_Ll\]) and (\[eq:fss\_anderson\_scalinglaw\_lambda\]), minimizing the quantity $\chi^2/(N_{df}-1)$, using the MINUIT library [@James75]. Here $N_{df}$ is the number of degrees of freedom and $\chi^2$ is the distance between the numerical data, $y_i$, and the fitting function, $f_i$, in the appropriate metric, i.e., $$\label{eq:chisqu}
\chi^2 = \sum_{i,j} (y_i-f_i) (C^{-1})_{ij} (y_j-f_j),$$ where $C$ is the covariance matrix of the data points. In the light of the results of Sec. \[sec:qcd\_corrE\], which show that there are strong correlations between eigenvectors in a given gauge configuration, strong correlations are also expected among GMFEs at different energies, and so the inclusion of correlations in the fitting procedure is necessary to obtain accurate results. The error bars of the best fit parameters were estimated by Monte-Carlo simulation, generating $N_{\rm MC}=100$ sets of synthetic data, distributed according to a Gaussian distribution with means equal to the raw data points and covariance matrix equal to the covariance matrix of the sample. We then determined the error bars from the distributions of the resulting fit parameters, choosing the $95\%$ confidence level.
In order to perform best fits, the scaling laws Eqs. (\[eq:fss\_anderson\_scalinglaw\_Ll\]) and (\[eq:fss\_anderson\_scalinglaw\_lambda\]) need to be expanded in powers of $E-E_c$, and this requires to set the expansion orders $n_{r/ir}$ of the relevant/irrelevant scaling term $\mathcal{G}^{r/ir}_q$, as well as the expansion orders $n_{\varrho}$ and $n_{\eta}$ of $\varrho$ and $\eta$. Since the relevant operator is more important than the irrelevant one we always used $n_{r}\geq n_{ir}$ and $n_{\varrho}\geq n_{\eta}$. We then repeated the fit for several choices of the expansion orders.
The quality of the best fits was judged according to two criteria. The first criterion was how close the ratio $\chi^2/(N_{df}-1)$ approached unity, and only fits with $\chi^2/(N_{df}-1)\approx 1$ were considered acceptable. The second criterion was stability against changing the expansion orders, in order to keep under control the systematic effects due to the truncation of the scaling function. We estimated the systematic error due to truncation as twice the standard deviation of the critical parameters, in the sample comprising the stable fits and the essentially equivalent ones obtained by increasing or lowering the expansion orders. The factor of 2 is required by consistency with the 95% confidence level chosen for the statistical error.
We first performed the MFSS at fixed $\lambda$, as described in Sec. \[sec:MFSS\], both for ensemble and typical averaging. We used $\lambda=0.125$, as this value is compatible with several of the system sizes listed in Tab. \[tab:qcd\_fss\_systemsize\]. The fixed $\lambda$ method is more stable, since the number of parameters to fit grows only linearly with the expansion orders. Stability was a serious issue, because the largest system size available, $L=56$, was only about half of the one used in Refs. . Due to this limitation, fits were stable for adding or removing an expansion parameter only in the range $0\leq q \leq 1$. The reason for this is that, for large $|q|$, the $q$-th power in Eq. (\[eq:multifractals\_SqRq\]) strongly enhances the contribution of the few spatial points with very large (if $q>0$) or very small (if $q<0$) wave-function amplitude squared, which therefore dominate the sum. This results in an effectively reduced statistics and so in a noisy dataset, and leads to a regime $0\leq q \leq 1$, where GMFEs behave numerically the best. Notice that, by construction, $D_0=d=3$ and $D_1=\alpha_1$, and moreover $\alpha_{0.5}=d$ due to a symmetry relation derived in Ref. .
The resulting critical parameters are listed in Tab. \[tab:qcd\_fitres\_lambda0125\] and shown in Fig. \[fig:qcd\_fitres\_lambda0125\]. The results are essentially independent of $q$ and the type of averaging, as expected. We also checked that the critical parameters do not depend on the width of the energy window, $\Delta E$, used in the computation of the GMFEs. As we show in Fig. \[fig:qcd\_DDeltaE\], the results for $E_c$ and $\nu$ are independent of $\Delta E$ within errors. Moreover, the choice $\Delta E=0.0012$ is optimal, as it leads to the best accuracy.
----------------------------------------------------- ----------------------------------------------------- ----------------------------------------------------
{width=".325\textwidth"} {width=".325\textwidth"} {width=".325\textwidth"}
----------------------------------------------------- ----------------------------------------------------- ----------------------------------------------------
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[$\scriptstyle q$]{} [exp]{} $\scriptstyle{E_c}$ $\scriptstyle{\varepsilon^{\rm syst}_{E_c}}$ $\scriptstyle{\nu}$ $\scriptstyle{\varepsilon^{\rm syst}_{\nu}}$ $\scriptstyle{y}$ $\scriptstyle{\varepsilon^{\rm syst}_{y}}$ $\scriptstyle{N_{df}}$ $\scriptstyle{\chi^2}$ $\scriptstyle{n_r
n_{ir} n_\varrho n_\eta}$
---------------------- ---------------------------------- ------------------------------------------- ---------------------------------------------- --------------------------------------- ---------------------------------------------- --------------------------------------- -------------------------------------------- ------------------------ ------------------------ -----------------------------
$\scriptstyle{0}$ $\scriptstyle{\alpha^{ens/typ}}$ $\scriptstyle{0.3353\ (0.3340..0.3363)}$ $\scriptstyle{0.0004}$ $\scriptstyle{1.443\ (1.421..1.478)}$ $\scriptstyle{0.056}$ $\scriptstyle{3.069\ (2.382..4.010)}$ $\scriptstyle{0.278}$ $\scriptstyle{118}$ $\scriptstyle{120}$ $\scriptstyle{4\ 2\ 2\ 0}$
$\scriptstyle{D^{ens}}$ $\scriptstyle{0.3355\ (0.3345..0.3364)}$ $\scriptstyle{0.0003}$ $\scriptstyle{1.449\ (1.429..1.481)}$ $\scriptstyle{0.048}$ $\scriptstyle{3.130\ (2.509..4.094)}$ $\scriptstyle{0.240}$ $\scriptstyle{118}$ $\scriptstyle{119}$ $\scriptstyle{4\ 2\ 2\ 0}$
$\scriptstyle{D^{typ}}$ $\scriptstyle{0.3354\ (0.3344..0.3365)}$ $\scriptstyle{0.0007}$ $\scriptstyle{1.456\ (1.425..1.478)}$ $\scriptstyle{0.048}$ $\scriptstyle{3.322\ (2.564..4.301)}$ $\scriptstyle{0.412}$ $\scriptstyle{118}$ $\scriptstyle{120}$ $\scriptstyle{4\ 2\ 2\ 0}$
$\scriptstyle{\alpha^{ens}}$ $\scriptstyle{0.3359\ (0.3342..0.3368)}$ $\scriptstyle{0.0001}$ $\scriptstyle{1.470\ (1.437..1.521)}$ $\scriptstyle{0.026}$ $\scriptstyle{3.380\ (2.217..4.683)}$ $\scriptstyle{0.056}$ $\scriptstyle{118}$ $\scriptstyle{118}$ $\scriptstyle{4\ 2\ 2\ 0}$
$\scriptstyle{\alpha^{typ}}$ $\scriptstyle{0.3358\ (0.3341..0.3365)}$ $\scriptstyle{0.0001}$ $\scriptstyle{1.485\ (1.457..1.539)}$ $\scriptstyle{0.026}$ $\scriptstyle{3.736\ (2.443..4.896)}$ $\scriptstyle{0.148}$ $\scriptstyle{117}$ $\scriptstyle{121}$ $\scriptstyle{4\ 2\ 2\ 1}$
$\scriptstyle{D^{ens}}$ $\scriptstyle{0.3355\ (0.3340..0.3366)}$ $\scriptstyle{0.0002}$ $\scriptstyle{1.457\ (1.426..1.494)}$ $\scriptstyle{0.048}$ $\scriptstyle{3.190\ (2.258..4.134)}$ $\scriptstyle{0.188}$ $\scriptstyle{118}$ $\scriptstyle{117}$ $\scriptstyle{4\ 2\ 2\ 0}$
$\scriptstyle{D^{typ}}$ $\scriptstyle{0.3354\ (0.3333..0.3362)}$ $\scriptstyle{0.0004}$ $\scriptstyle{1.488\ (1.448..1.567)}$ $\scriptstyle{0.054}$ $\scriptstyle{3.228\ (1.971..4.058)}$ $\scriptstyle{0.334}$ $\scriptstyle{117}$ $\scriptstyle{116}$ $\scriptstyle{4\ 3\ 2\ 0}$
$\scriptstyle{D^{ens}}$ $\scriptstyle{0.3357\ (0.3346..0.3369)}$ $\scriptstyle{0.0001}$ $\scriptstyle{1.466\ (1.433..1.510)}$ $\scriptstyle{0.040}$ $\scriptstyle{3.220\ (2.416..4.504)}$ $\scriptstyle{0.118}$ $\scriptstyle{118}$ $\scriptstyle{117}$ $\scriptstyle{4\ 2\ 2\ 0}$
$\scriptstyle{D^{typ}}$ $\scriptstyle{0.3356\ (0.3324..0.3368)}$ $\scriptstyle{0.0001}$ $\scriptstyle{1.450\ (1.416..1.496)}$ $\scriptstyle{0.036}$ $\scriptstyle{3.356\ (1.666..4.845)}$ $\scriptstyle{0.148}$ $\scriptstyle{116}$ $\scriptstyle{117}$ $\scriptstyle{4\ 3\ 2\ 1}$
$\scriptstyle{\alpha^{ens}}$ $\scriptstyle{0.3356\ (0.3339..0.3366)}$ $\scriptstyle{0.0002}$ $\scriptstyle{1.462\ (1.424..1.517)}$ $\scriptstyle{0.044}$ $\scriptstyle{3.221\ (2.154..4.364)}$ $\scriptstyle{0.184}$ $\scriptstyle{118}$ $\scriptstyle{119}$ $\scriptstyle{4\ 2\ 2\ 0}$
$\scriptstyle{\alpha^{typ}}$ $\scriptstyle{0.3355\ (0.3330..0.3366)}$ $\scriptstyle{0.0001}$ $\scriptstyle{1.465\ (1.443..1.543)}$ $\scriptstyle{0.032}$ $\scriptstyle{3.453\ (1.955..4.937)}$ $\scriptstyle{0.194}$ $\scriptstyle{117}$ $\scriptstyle{122}$ $\scriptstyle{4\ 2\ 2\ 1}$
$\scriptstyle{D^{ens}}$ $\scriptstyle{0.3361\ (0.3348..0.3371)}$ $\scriptstyle{0.0001}$ $\scriptstyle{1.468\ (1.428..1.507)}$ $\scriptstyle{0.038}$ $\scriptstyle{3.264\ (2.392..4.563)}$ $\scriptstyle{0.118}$ $\scriptstyle{118}$ $\scriptstyle{117}$ $\scriptstyle{4\ 2\ 2\ 0}$
$\scriptstyle{D^{typ}}$ $\scriptstyle{0.3360\ (0.3340..0.3371)}$ $\scriptstyle{0.0001}$ $\scriptstyle{1.449\ (1.425..1.529)}$ $\scriptstyle{0.034}$ $\scriptstyle{3.394\ (2.127..5.271)}$ $\scriptstyle{0.130}$ $\scriptstyle{117}$ $\scriptstyle{119}$ $\scriptstyle{4\ 2\ 2\ 1}$
$\scriptstyle{D^{ens}}$ $\scriptstyle{0.3363\ (0.3342..0.3374)}$ $\scriptstyle{0.0002}$ $\scriptstyle{1.465\ (1.422..1.573)}$ $\scriptstyle{0.036}$ $\scriptstyle{3.313\ (1.984..4.770)}$ $\scriptstyle{0.128}$ $\scriptstyle{118}$ $\scriptstyle{118}$ $\scriptstyle{4\ 2\ 2\ 0}$
$\scriptstyle{D^{typ}}$ $\scriptstyle{0.3361\ (0.3344..0.3372)}$ $\scriptstyle{0.0002}$ $\scriptstyle{1.437\ (1.412..1.538)}$ $\scriptstyle{0.036}$ $\scriptstyle{3.298\ (2.145..4.711)}$ $\scriptstyle{0.256}$ $\scriptstyle{117}$ $\scriptstyle{118}$ $\scriptstyle{4\ 2\ 2\ 1}$
$\scriptstyle{1}$ $\scriptstyle{\alpha^{ens/typ}}$ $\scriptstyle{0.3364\ (0.3346..0.3376)}$ $\scriptstyle{0.0001}$ $\scriptstyle{1.464\ (1.425..1.535)}$ $\scriptstyle{0.034}$ $\scriptstyle{3.334\ (2.175..5.018)}$ $\scriptstyle{0.108}$ $\scriptstyle{118}$ $\scriptstyle{118}$ $\scriptstyle{4\ 2\ 2\ 0}$
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Dependence of the fitted critical point and critical exponent, as obtained from $D_{0.1}^{ens}$ at fixed $\lambda=0.125$, for various energy windows $\Delta E$. Error bars correspond to the $95\%$ confidence band. Only statistical errors are shown.[]{data-label="fig:qcd_DDeltaE"}](DDeltaE.pdf "fig:"){width=".49\columnwidth"} ![Dependence of the fitted critical point and critical exponent, as obtained from $D_{0.1}^{ens}$ at fixed $\lambda=0.125$, for various energy windows $\Delta E$. Error bars correspond to the $95\%$ confidence band. Only statistical errors are shown.[]{data-label="fig:qcd_DDeltaE"}](DDeltanu.pdf "fig:"){width=".49\columnwidth"}
To quote a final result for the critical parameters, we have averaged the values of $E_c$, $\nu$ and $y$, and of the corresponding errors, obtained with the various GMFEs. (A weighted average, using the inverse of the error band as weight, yields similar numbers.) Our result for the critical point, $E_c=0.3357\ (0.3340..0.3368)$, is compatible with the value reported in Ref. at the 2-$\sigma$ level. On average, the systematic error on $E_c$ is $\varepsilon_{E_c}^{\rm syst}=0.0002$, so negligible compared to the statistical error. Our result for the critical exponent, $\nu=1.461\
(1.429..1.519)$, agrees at the 1-$\sigma$ level with the result of Ref. , and with previous results for the critical exponent of the unitary Anderson model [@Ujfalusi15; @SO]. For this quantity, one has also to take into account that on average the systematic error due to truncation, $\varepsilon_{\nu}^{\rm syst}=0.040$, is of the same size as the statistical error. On the other hand, our value for the irrelevant exponent, $y=3.307\
(2.210..4.572)$, is significantly different from the value of Ref. , $y^{\rm UV}=1.651\
(1.601..1.707)$. It is well known that it is very difficult to determine irrelevant exponents accurately, and to explain this discrepancy further work and higher-quality data are needed. It is possible that for the system sizes presently available, more than one irrelevant term gives important contributions, so that our result for $y$ would be a sort of “effective” irrelevant exponent. In any case this point requires further analysis.
As a final remark, we note that since results for different $q$ are strongly correlated, there is no significance in the fact that our values for the critical point are systematically lower, and the ones for the critical exponent are systematically higher than the reference values.
[c c]{}
[MFEs\_alpha]{} (0,74)[(a)]{}
&
[MFEs\_D]{} (0,74)[(b)]{}
\
The convergence of the fixed-$\lambda$ MFSS confirms the presence of a critical point in the QCD Dirac spectrum where the system undergoes a true localization-delocalization transition, employing completely different observables than the ones used in Ref. . The results of our analysis also provide further evidence that the transition in the QCD Dirac spectrum belongs to the universality class of the 3D unitary Anderson model. Moreover, despite the fact that it does not provide the values of the MFEs, the convergence of this method also strongly indicates the presence of multifractality at the critical point.
We next procedeed to apply the variable-$\lambda$ method, in order to try and determine the multifractal exponents, and compare them to the ones obtained for the unitary Anderson model. However, this method requires small values of $\lambda$ to work properly, and is also more demanding as it is a two-variable fit. In practice, the $\chi^2/N_{df}$ ratio reached a value close to unity only if we left out the smallest system sizes, below $L_{min}=36$, and if we used data corresponding to $\ell=1$ and $2$ only. Although using $L_{min}=36$ and $\ell=1,2$ improved the convergence, the fits were still unstable against changing the expansion orders. This can be understood, as a similar amount of independent data is available as in the fixed-$\lambda$ method, but there are many more parameters to fit, as discussed in Sec. \[sec:MFSS\]. In order to be able to estimate the MFEs, we then fixed the critical energy and the critical exponent to the values obtained with the fixed-$\lambda$ method, $E_c=0.3357$ and $\nu=1.461$, in this way stabilizing the fits. The systematic uncertainty corresponding to this procedure was estimated by repeating the fits with $E_c$ and $\nu$ fixed to one of the four possible combinations of the values $E_c^{l,u}$ and $\nu^{l,u}$, which are the lower and upper boundaries of the confidence interval of $E_c$ and $\nu$, respectively. The largest and smallest values obtained in this way were then used as upper and lower error bar on the MFEs. We experienced that the main source of uncertainty comes from the choice of $E_c$, while fits are much less sensitive to the choice of $\nu$. Moreover, statistical errors (estimated by Monte-Carlo) and systematic errors due to truncation were comparatively negligible.
The results of this procedure are depicted in Fig. \[fig:qcd\_fitres\_MFEs\]. A set of nontrivial MFEs was obtained, thus providing direct evidence of the multifractality of the critical eigenfunctions of the QCD Dirac operator. Moreover, our results for the MFEs in QCD are compatible with the ones obtained in the unitary Anderson model, which further confirms that the transition belongs to the chiral unitary Anderson class.
Summary {#sec:concl}
=======
We investigated the Anderson transition in the spectrum of the Dirac operator of QCD at high temperature, found by the authors of Ref. . While that work made use of spectral statistics, our aim in this paper was to examine the transition by studying the eigenvectors, and their multifractal properties at the critical energy. The results of Ref. for the correlation length critical exponent suggested that the Anderson transition in QCD belongs to the same universality class as the three-dimensional unitary Anderson model. We therefore looked for more similarities between these models.
First we examined the correlations between eigenvectors of a given gauge configuration. We found strong correlations between eigenmodes of the QCD Dirac operator, decreasing with energy separation in a similar way as in the unitary Anderson model. We then performed two multifractal finite-size scaling (MFSS) analyses, one with fixed ratio $\lambda$ of the coarse-graining box size to the system size, and one with variable $\lambda$. MFSS with the fixed-$\lambda$ method allowed an alternative determination of the critical point and of the critical exponent, which is in agreement with the findings of Ref. , and, for the critical exponent, with those of Refs. for the unitary Anderson model. To perform MFSS with the variable-$\lambda$ method and determine the multifractal exponents (MFEs), we performed fits fixing the critical energy and the critical exponent to the values obtained with the fixed-$\lambda$ method. The resulting MFEs are compatible with the MFEs found in the unitary Anderson model.
In conclusion, our work confirms the presence of an Anderson metal-insulator phase transition in the spectrum of the Dirac operator in high-temperature QCD, and provides further evidence that this transition belongs to the three-dimensional unitary Anderson model class. Morever, we have shown that the critical wave-functions of the Dirac operator are multifractals. The physical consequences of the QCD Anderson transition and of multifractality still largely need to be explored, and may lead in particular to a better understanding of the QCD chiral transition. Further work along these lines might prove beneficial for condensed matter physics as well, as it approaches the subject of localization/delocalization transitions from a broader perspective.
We thank the Budapest-Wuppertal group for allowing us to use their code to generate the gauge configurations. Financial support to LU and IV from OTKA under Grant No. K108676, and from the Alexander von Humboldt Foundation is gratefully acknowledged. MG and TGK are supported by the Hungarian Academy of Sciences under “Lendület” grant No. LP2011-011. FP is supported by OTKA under the grant OTKA-NF-104034.
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---
abstract: |
As a result of the growing size of Deep Neural Networks (DNNs), the gap to hardware capabilities in terms of memory and compute increases. To effectively compress DNNs, quantization and connection pruning are usually considered. However, unconstrained pruning usually leads to unstructured parallelism, which maps poorly to massively parallel processors, and substantially reduces the efficiency of general-purpose processors. Similar applies to quantization, which often requires dedicated hardware.
We propose Parameterized Structured Pruning (PSP), a novel method to dynamically learn the shape of DNNs through structured sparsity. PSP parameterizes structures (e.g. channel- or layer-wise) in a weight tensor and leverages weight decay to learn a clear distinction between important and unimportant structures. As a result, PSP maintains prediction performance, creates a substantial amount of sparsity that is structured and, thus, easy and efficient to map to a variety of massively parallel processors, which are mandatory for utmost compute power and energy efficiency. PSP is experimentally validated on the popular CIFAR10/100 and ILSVRC2012 datasets using ResNet and DenseNet architectures, respectively.
author:
- Günther Schindler
- Wolfgang Roth
- Franz Pernkopf
- Holger Fröning
bibliography:
- 'egbib.bib'
title: Parameterized Structured Pruning for Deep Neural Networks
---
|
---
abstract: |
We firstly present definitions and properties in study of Maji [@maji-2013] on neutrosophic soft sets. We then give a few notes on his study. Next, based on Çağman [@cagman-2014], we redefine the notion of neutrosophic soft set and neutrosophic soft set operations to make more functional. By using these new definitions we construct a decision making method and a group decision making method which selects a set of optimum elements from the alternatives. We finally present examples which shows that the methods can be successfully applied to many problems that contain uncertainties.
Neutrosophic set; Soft set; Neutrosophic soft set; decision making.
author:
- Faruk Karaaslan
title: '**Neutrosophic soft sets with applications in decision making**'
---
Introduction
============
Many problems including uncertainties are a major issue in many fields of real life such as economics, engineering, environment, social sciences, medical sciences and business management. Uncertain data in these fields could be caused by complexities and difficulties in classical mathematical modeling. To avoid difficulties in dealing with uncertainties, many tools have been studied by researchers. Some of these tools are fuzzy sets [@zadeh-1965], rough sets [@paw-82] and intuitionistic fuzzy sets [@atanassov-1986]. Fuzzy sets and intuitionistic fuzzy sets are characterized by membership functions, membership and non-membership functions, respectively. In some real life problems for proper description of an object in uncertain and ambiguous environment, we need to handle the indeterminate and incomplete information. But fuzzy sets and intuitionistic fuzzy sets don’t handle the indeterminant and inconsistent information. Samarandache [@smar-95] defined the notion of neutrosophic set which is a mathematical tool for dealing with problems involving imprecise and indeterminant data.
Molodtsov introduced concept of soft sets [@molodtsov-1999] to solve complicated problems and various types of uncertainties. In [@maj-03sst], Maji et al. introduced several operators for soft set theory: equality of two soft sets, subsets and superset of soft sets, complement of soft set, null soft sets and absolute soft sets. But some of these definitions and their properties have few gaps, which have been pointed out by Ali et al.[@ali-09osnop] and Yang [@yang-08]. In 2010, Çağman and Enginoğlu [@cag-10sstuidcm] made some modifications the operations of soft sets and filled in these gap. In 2014, Çağman [@cagman-2014] redefined soft sets using the single parameter set and compared definitions with those defined before.
Maji [@maji-2013] combined the concept of soft set and neutrosophic set together by introducing a new concept called neutrosophic soft set and gave an application of neutrosophic soft set in decision making problem. Recently, the properties and applications on the neutrosophic sets have been studied increasingly [@bro-13gns; @bro-13ins; @deli-14ivnss; @deli-14nsm].The propose of this paper is to fill the gaps of the Maji’s neutrosophic soft set [@maji-2013] definition and operations redefining concept of neutrosophic soft set and operations between neutrosophic soft sets. First, we present Maji’s definitions and operations and we verify that some propositions are incorrect by a counterexample. Then based on Çağman’s [@cagman-2014] study we redefine neutrosophic soft sets and their operations. Also, we investigate properties of neutrosophic soft sets operations. Finally we present an application of a neutrosophic soft set in decision making.
Preliminaries
=============
In this section, we will recall the notions of neutrosophic sets [@smarandache-2005] and soft sets [@molodtsov-1999]. Then, we will give some properties of these notions. Throughout this paper $X$, $E$ and $P(X)$ denote initial universe, set of parameters and power set of $X$, respectively.
[@smarandache-2005] A neutrosophic set $A$ on the universe of discourse $X$ is defined as $$A=\big\{\langle x,T_A(x),I_A(x),F_A(x)\rangle: x\in X\big\}$$ where $T_A,I_A,F_A: X \to]^-0,1^+[$ and $^-0\leq
T_A(x)+I_A(x)+F_A(x)\leq3^+$. From philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of $]^-0,1^+[$. But in real life application in scientific and engineering problems it is difficult to use neutrosophic set with value from real standard or non-standard subset of $]^-0,1^+[$. Hence we consider the neutrosophic set which takes the value from the subset of $[0,1]$.
[@molodtsov-1999] Let consider a nonempty set $A$, $A\subseteq E$. A pair $(F,A)$ is called a soft set over $X$, where $F$ is a mapping given by $F:A\to P(X)$.
\[e-softset\] Let $X=\{x_l,x_2,x_3,x_4,x_5,x_6,x_7,x_8\}$ be the universe which are eight houses and $E=\{e_1,e_2,e_3,e_4,e_5,e_6\}$ be the set of parameters. Here, $e_i$ $(i=1,2,3,4,5,6)$ stand for the parameters “*modern*”, “*with parking*”, “*expensive*”, “*cheap*”, “*large*” and “*near to city*” respectively. Then, following soft sets are described respectively Mr. A and Mr. B who are going to buy $$\begin{aligned}
F & = & \big\{(e_1,\{x_1,x_3,x_4\}),(e_2,\{x_1,x_4,x_7,x_8\}),(e_3,\{x_1,x_2,x_3,x_8\})\big\}\\
G & = & \big\{(e_2\{x_1,x_3,x_6\}),(e_3,X),(e_5,\{x_2,x_4,x_4,x_6\})\big\}.\end{aligned}$$
From now on, we will use definitions and operations of soft sets which are more suitable for pure mathematics based on study of Çağman [@cagman-2014].
[@cagman-2014] A soft set $F$ over $X$ is a set valued function from $E$ to $P(X)$. It can be written a set of ordered pairs $$F=\big\{(e,F(e)):e\in E\big\}.$$ Note that if $F(e)=\emptyset$, then the element $(e, F(e))$ is not appeared in $F$. Set of all soft sets over $X$ is denoted by $\mathbb{S}$.
[@cagman-2014] Let $F,G\in\mathbb{S}$. Then,
1. If $F(e)=\emptyset$ for all $e\in E$, $F$ is said to be a null soft set, denoted by $\Phi$.
2. If $F(e)=X$ for all $e\in E$, $F$ is said to be absolute soft set, denoted by $\hat{X}$.
3. $F$ is soft subset of $G$, denoted by $F\tilde\subseteq G$, if $F(e)\subseteq G(e)$ for all $e\in E$.
4. $F=G$, if $F\tilde\subseteq G$ and $G\tilde\subseteq F$.
5. Soft union of $F$ and $G$, denoted by $F\tilde\cup G$, is a soft set over $X$ and defined by $F\tilde\cup G:E\to P(X)$ such that $(F\tilde\cup G)(e)=F(e)\cup G(e)$ for all $e\in E$.
6. Soft intersection of $F$ and $G$, denoted by $F\tilde\cap G$, is a soft set over $X$ and defined by $F\tilde\cap G:E\to P(X)$ such that $(F\tilde\cap G)(e)=F(e)\cap G(e)$ for all $e\in E$.
7. Soft complement of $F$ is denoted by $F^{\tilde c}$ and defined by $F^{\tilde c}:E\to P(X)$ such that $F^{\tilde c}(e)=X\setminus F(e)$ for all $e\in E$.
Let us consider soft sets $F$,$G$ in the Example \[e-softset\]. Then, we have $$\begin{aligned}
F\tilde\cup G & = & \big\{(e_1,\{x_1,x_3,x_4\}),(e_2,\{x_1,x_3,x_4,x_6,x_7,x_8\}),\\
& & (e_3,X),(e_5,\{x_2,x_4,x_4,x_6\})\big\}\\
F\tilde\cap G & = & \big\{(e_2\{x_1\}),(e_3,\{x_1,x_2,x_3,x_8\})\big\} \\
F^{\tilde c} & = & \big\{(e_1,\{x_2,x_5,x_6,x_7,x_8\}),(e_2,\{x_2,x_3,x_5,x_6\}),\\
& & (e_3,\{x_4,x_5,x_6,x_7\}),(e_4,X),(e_5,X),(e_6,X)\big\}.\end{aligned}$$
[@maji-2013] Let $X$ be an initial universe set and $E$ be a set of parameters. Consider $A\subset E$. Let $P(X)$ denotes the set of all neutrosophic sets of $X$. The collection $(F,A)$ is termed to be the soft neutrosophic set over $X$, where $F$ is a mapping given by $F:A \to P(X).$
For illustration we consider an example.
\[examplemaji\] Let $X$ be the set of houses under consideration and $E$ is the set of parameters. Each parameter is a neutrosophic word or sentence involving neutrosophic words. Consider $E=\{ \textrm{beautiful,
wooden, costly, very costly},$ $\textrm{moderate, green
surroundings, in good repair, in bad repair, cheap, expensive}\}$. In this case, to define a neutrosophic soft set means to point out beautiful houses, wooden houses, houses in the green surroundings and so on. Suppose that, there are five houses in the universe $X$ given by, $U = \{h_1, h_2, h_3, h_4, h_5\}$ and the set of parameters $A = \{e_1, e_2, e_3, e_4\}$, where $e_1$ stands for the parameter ’beautiful’, $e_2$ stands for the parameter ’wooden’, $e_3$ stands for the parameter ’costly’ and the parameter $e_4$ stands for ’moderate’. Suppose that, $$\begin{aligned}
F(beautiful) &=& \{\langle h_1, 0.5, 0.6, 0.3 \rangle,\langle h_2,
0.4, 0.7, 0.6 \rangle,\langle h_3, 0.6, 0.2, 0.3 \rangle,\nonumber\\
&& \langle h_4, 0.7, 0.3, 0.2 \rangle,\langle h_5, 0.8, 0.2, 0.3
\rangle\},\nonumber\\
F(wooden) &=& \{\langle h_1, 0.6, 0.3, 0.5 \rangle,\langle h_2, 0.7,
0.4, 0.3 \rangle, \langle h_3, 0.8, 0.1, 0.2 \rangle,\nonumber\\
&&\langle h_4, 0.7, 0.1, 0.3 \rangle,\langle h_5, 0.8, 0.3, 0.6
\rangle\},\nonumber\\
F(costly) &=& \{\langle h_1, 0.7, 0.4, 0.3 \rangle,\langle h_2, 0.6,
0.7, 0.2 \rangle, \langle h_3, 0.7, 0.2, 0.5 \rangle,\nonumber \\
&&\langle h_4, 0.5, 0.2, 0.6 \rangle, \langle h_5, 0.7, 0.3, 0.4
\rangle \},\nonumber\\
F(moderate) &=& \{\langle h_1, 0.8, 0.6, 0.4 \rangle,\langle h_2,
0.7, 0.9, 0.6 \rangle,\langle h_3, 0.7, 0.6, 0.4 \rangle,
\nonumber\\
&&\langle h_4, 0.7, 0.8, 0.6 \rangle,\langle h_5, 0.9, 0.5, 0.7
\rangle\}.\nonumber\end{aligned}$$ The neutrosophic soft set $(NSS)$ $(F,E)$ is a parameterized family $\{F(e_i); i = 1,2,...,10\} $of all neutrosophic sets of $X$ and describes a collection of approximation of an object.
Thus we can view the neutrosophic soft set $(NSS)$ $(F,A)$ as a collection of approximation as below: $$\begin{aligned}
(F,A)&=&\{ beautiful\; houses = \{\langle h_1, 0.5, 0.6, 0.3
\rangle,\langle h_2, 0.4, 0.7, 0.6 \rangle,\\
&& \langle h_3, 0.6, 0.2,
0.3 \rangle,\langle h_4, 0.7, 0.3, 0.2 \rangle,\langle h5, 0.8, 0.2,
0.3 \rangle\},\\
&& wooden\; houses = \{\langle h_1, 0.6, 0.3, 0.5 \rangle,\langle h_2, 0.7, 0.4, 0.3
\rangle,\\
&&\langle h_3, 0.8, 0.1, 0.2 \rangle, \langle h_4, 0.7, 0.1,
0.3 \rangle,\langle h_5, 0.8, 0.3, 0.6 \rangle\}, \\
&&costly \;houses = f\langle h_1,
0.7, 0.4, 0.3 \rangle ,\langle h_2, 0.6, 0.7, 0.2 \rangle, \\
&&\langle h_3, 0.7, 0.2, 0.5 \rangle,\langle h_4, 0.5, 0.2, 0.6
\rangle,\langle h_5, 0.7, 0.3, 0.4 \rangle\},\\
&& moderate \;houses=\langle h_1, 0.8, 0.6, 0.4 \rangle,\langle h_2, 0.7, 0.9, 0.6
\rangle,\\
&&\langle h_3, 0.7, 0.6, 0.4 \rangle, \langle h_4, 0.7, 0.8,
0.6 \rangle,
\langle h_5, 0.9, 0.5, 0.7 \rangle\}\}.\end{aligned}$$
[@maji-2013]\[ns-subset\] Let $(F,A)$ and $(G,B)$ be two neutrosophic sets over the common universe $X$. $(F,A)$ is said to be neutrosophic soft subset of $(G,B)$ is $A\subset B$, and $T_{F(e)}(x)\leq T_{G(e)}(x)$, $ I_{F(e)}(x)\leq I_{G(e)}(x)$ $F_{F(e)}(x) \geq F_{G(e)}(x)$, $\forall e\in A$, $\forall x\in U$. We denote it by $(F,A) \subseteq (G,B)$. $(F,A)$ is said to be neutrosophic soft super set of $(G,B)$ if $(G,B)$ is a neutrosophic soft subset of $(F,A)$. We denote it by $(F,A)\supseteq(G,B)$.
if $(F,A)$ is neutrosophic soft subset of $(G,B)$ and $(G,B)$ is neutrosophic soft subset of $(F,A)$. We denote it $(F,A)=(G,B).$
[@maji-2013]\[ns-notset\] NOT set of a parameters. Let $E=\{e_1,e_2,...e_n\}$ be a set of parameters. The NOT set of $E$, denoted by $\rceil E$ is defined by $\rceil E=\{\neg e_1, \neg
e_2,...\neg e_n\}$, where $\neg e_i=$not $e_i$ $\forall i$(it may be noted that $\rceil$ and $\neg$ are different operators).
[@maji-2013]\[ns-complement\] Complement of a neutrosophic soft set $(F,A)$ denoted by $(F,A)^c$ and is defined as $(F,A)^c=(F^c,\rceil
A)$, where $F^c. \, \rceil A\to P(X)$ is mapping given by $F^c(\alpha)=$ neutrosophic soft complement with $T_{F^c(x)}=F_{F(x)}$, $I_{F^c(x)}=I_{F(x)}$ and $F_{F^c(x)}=T_{F(x)}$.
[@maji-2013]\[ns-empty\] Empty or null neutrosophic soft set with respect to a parameter. A neutrosophic soft set $(H,A)$ over the universe $X$ is termed to be empty or null neutrosophic soft set with respect to the parameter $e$ if $T_{H(e)}(m)=0,
F_{H(e)}=0$ and $I_{H(e)}(m)=0$ $\forall m\in X$, $\forall e\in A$
In this case the null neutrosophic soft set $(NNSS)$ is denoted by $\Phi_A$
[@maji-2013]\[ns-union\] Union of two neutrosophic soft sets. Let $(H,A)$ and $(G,B)$ be two $NSSs$ over the common universe $X$. Then the union of $(H,A)$ and $(G,B)$ is defined by $(H,A)\cup (G,B)=(K,C)$, where $C=A\cup B$ and the truth-membership, indeterminacy-membership and falsity-membership of $(K,C)$ are as follow. $$\begin{aligned}
T_{K(e)}(m)&=&T_{H(e)}(m), if e\in A-B\nonumber\\
&=&T_{G(e)}(m), if e\in B-A\nonumber\\
&=&max(T_{H(e)}(m),T_{G(e)}(m)), if e\in A\cap B\nonumber\\
I_{K(e)}(m)&=&I_{H(e)}(m), if e\in A-B\nonumber\\
&=&I_{G(e)}(m), if e\in B-A\nonumber\\
&=&\frac{I_{H(e)}(m)+I_{G(e)}(m)}{2} , if e\in A\cap B.\nonumber\\
F_{K(e)}(m)&=&F_{H(e)}(m), if e\in A-B\nonumber\\
&=&F_{G(e)}(m), if e\in B-A\nonumber\\
&=&min(F_{H(e)}(m),F_{G(e)}(m)), if e\in A\cap B\nonumber\end{aligned}$$
[@maji-2013]\[ns-intersection\] Let $(H,A)$ and $(G,B)$ be two $NSSs$ over the common universe $X$. Then, intersection of $(H,A)$ and $(G,B)$ is defined by $(H,A)\cap (G,B)=(K,C)$, where $C=A\cap B$ and the truth-membership, indeterminacy-membership and falsity-membership of $(K,C)$ are as follow. $$\begin{aligned}
T_{K(e)}(m)&=&min(T_{H(e)}(m),T_{G(e)}(m)), \; if\; e\in A\cap B\nonumber\\
I_{K(e)}(m)&=&\frac{I_{H(e)}(m)+I_{G(e)}(m)}{2},\; if\; e\in A\cap B.\nonumber\\
F_{K(e)}(m)&=&max(F_{H(e)}(m),F_{G(e)}(m)),\; if \;e\in A\cap B\nonumber\end{aligned}$$
For any two $NSSs$ $(H,A)$ and $(G,B)$ over the same universe $X$ and on the basis of the operations defined above, we have the following propositions.
[@maji-2013]\[prop1\]
(1) $(H,A)\cup (H,A)=(H,A)$
(2) $(H,A)\cup (G,B)=(G,B)\cup (H,A) $
(3) $(H,A)\cap (H,A)=(H,A)$
(4) $(H,A)\cap (G,B)=(G,B)\cap (H,A)$
(5) $(H,A)\cup \Phi=(H,A)$ \[prop1-5\]
(6) $(H,A)\cap \Phi=\Phi$ \[prop1-6\]
(7) $[(H,A)^c]^c=(H,A)$
For any two $NSSs$ $(H,A)$, $(G,B)$ and $(K,C) $over the same universe $X$, we have the following propositions.
[@maji-2013]
(1) $(H,A)\cup [(G,B)\cup (K,C)]=[(H,A)\cup (G,B)]\cup (K,C).$
(2) $(H,A)\cap [(G,B)\cap (K,C)]=[(H,A)\cap (G,B)]\cap (K,C).$
(3) $(H,A)\cup [(G,B)\cap (K,C)]=[(H,A)\cup (G,B)]\cap [(H,A)\cup (K,C)].$
(4) $(H,A)\cap [(G,B)\cup (K,C)]=[(H,A)\cap (G,B)]\cup [(H,A)\cap (K,C)].$
[@maji-2013]\[ns-and\] Let $(H,A)$ and $(G,B)$ be two $NSSs$ over the common universe $X$. Then ’AND’ operation on them is denoted by ’$(H,A)\bigwedge(G,B)$’ and is defined by $(H,A)\bigwedge
(G,B)=(K,A\times B)$, where the truth-membership, indeterminacy-membership and falsity-membership of $(K,A\times B)$ are as follow. $$\begin{aligned}
T_{K(\alpha,\beta)}(m)&=&min(T_{H(e)}(m),T_{G(e)}(m))\nonumber\\
I_{K(\alpha,\beta))}(m)&=&\frac{I_{H(e)}(m)+I_{G(e)}(m)}{2} \nonumber\\
F_{K(\alpha,\beta))}(m)&=&max(F_{H(e)}(m),F_{G(e)}(m)), \forall
\alpha\in A, \forall b\in B\nonumber\end{aligned}$$
[@maji-2013]\[ns-or\] Let $(H,A)$ and $(G,B)$ be two $NSSs$ over the common universe $X$. Then ’OR’ operation on them is denoted by ’$(H,A)\bigvee(G,B)$’ and is defined by $(H,A)\bigvee(G,B)=(O,A\times B)$, where the truth-membership, indeterminacy-membership and falsity-membership of $(O,A\times B)$ are as follow. $$\begin{aligned}
T_{O(\alpha,\beta))}(m)&=&max(T_{H(e)}(m),T_{G(e)}(m)),\\
I_{O(\alpha,\beta))}(m)&=&\frac{I_{H(e)}(m)+I_{G(e)}(m)}{2},\\
F_{O(\alpha,\beta))}(m)&=&min(F_{H(e)}(m),F_{G(e)}(m)), \forall \alpha\in A, \forall b\in B\nonumber\\\end{aligned}$$
Notes on neutrosophic soft sets [@maji-2013] {#notes-on-neutrosophic-soft-sets .unnumbered}
--------------------------------------------
In this section, we verify that some propositions in the study of Maji [@maji-2013] are incorrect by counterexamples.
1. If Definition is true, then Definition is incorrect.
2. Proposition - and , $(F,A)\cap \Phi =
\Phi$ and $(F,A)\cup \Phi = (F,A)$ are incorrect.
We verify these notes by counterexamples.
Let us consider neutrosophic soft set $(F,A)$ in Example and null neutrosophic soft set $\Phi$. If Definition is true, it is required that null soft set is neutrosophic soft subset of all neutrosophic soft sets. But, since $ T_{\Phi(beautiful)}(h_1)\leq
T_{F(beautiful)}(h_1) $ and $I_{\Phi(beautiful)}(h_1)\leq
I_{F(beautiful)}(h_1)$ but $F_{\Phi(beautiful)}(h_1)\not\geq
F_{F(beautiful)}(h_1)$, $\Phi\not\subseteq (F,A)$.
Let us consider neutrosophic soft set $(F,A)$ in Example and null neutrosophic soft set $\Phi$. Then, $$\begin{aligned}
(F,A)\cap \Phi&=&\{e_1 =\{\langle h_1, 0, 0.3, 0.3 \rangle,\langle
h_2, 0, 0.35, 0.6 \rangle,\nonumber\\
&& \langle h_3, 0, 0.1,0.3 \rangle,\langle h_4, 0, 0.15, 0.2
\rangle,\langle h_5, 0, 0.1, 0.3 \rangle\},\nonumber\\
&&e_2=\{\langle h_1, 0, 0.15, 0.5 \rangle,\langle h_2, 0, 0.2, 0.3
\rangle,\langle h_3, 0, 0.05, 0.2 \rangle, \nonumber\\
&&\langle h_4, 0, 0.05, 0.3 \rangle, \langle h_5, 0, 0.15, 0.6
\rangle\},\nonumber\\
&&e_3=\{\langle h_1,0, 0.2, 0.3 \rangle ,\langle h_2, 0, 0.35, 0.2
\rangle,\langle h_3, 0, 0.1, 0.5 \rangle,\nonumber \\
&&\langle h_4, 0, 0.1, 0.6 \rangle,\langle h_5, 0, 0.15, 0.4
\rangle\}, \nonumber\\
&&e_5=\{\langle h_1, 0, 0.3, 0.4 \rangle,\langle h_2, 0, 0.45, 0.6
\rangle,\langle h_3, 0, 0.3, 0.4 \rangle,\nonumber\\
&&\langle h_4, 0, 0.4, 0.6 \rangle, \langle h_5, 0, 0.25, 0.7
\rangle\}\}.\nonumber\\
&\not =&\Phi\nonumber\end{aligned}$$ and $$\begin{aligned}
(F,A)\cup \Phi&=&\{e_1 =\{\langle h_1, 0.5, 0.3, 0 \rangle,\langle
h_2, 0.4 0.35, 0 \rangle,\nonumber\\
&& \langle h_3, 0.6, 0.1,0 \rangle,\langle h_4, 0.7, 0.15, 0
\rangle,\langle h_5, 0.8, 0.1, 0 \rangle\},\nonumber\\
&&e_2=\{\langle h_1, 0.6, 0.15, 0 \rangle,\langle h_2, 0.7, 0.2, 0
\rangle,\langle h_3, 0.8, 0.05, 0 \rangle, \nonumber\\
&&\langle h_4, 0.7, 0.05, 0 \rangle, \langle h_5, 0.8, 0.15, 0
\rangle\},\nonumber\\
&&e_3=\{\langle h_1,0.7, 0.2, 0 \rangle ,\langle h_2, 0.6, 0.35, 0
\rangle,\langle h_3, 0.7, 0.1, 0\rangle,\nonumber \\
&&\langle h_4, 0.5, 0.1, 0 \rangle,\langle h_5, 0.7, 0.15, 0
\rangle\}, \nonumber\\
&&e_5=\{\langle h_1, 0.8, 0.3, 0 \rangle,\langle h_2, 0.7, 0.45, 0
\rangle,\langle h_3, 0.7, 0.3, 0 \rangle,\nonumber\\
&&\langle h_4, 0.7, 0.4, 0 \rangle, \langle h_5, 0.9, 0.25, 0
\rangle\}\}.\nonumber\\
&\not =&(F,A)\nonumber\end{aligned}$$
Neutrosophic soft sets
======================
In this section, we will redefine the neutrosophic soft set based on paper of Çağman [@cagman-2014].
\[neutrosophicsoftset\] A neutrosophic soft set (or namely *ns*-set) $f$ over $X$ is a neutrosophic set valued function from $E$ to $N(X)$. It can be written as $$f=\Big\{\big(e,\{\langle x,T_{f(e)}(x),I_{f(e)}(x),
F_{f(e)}(x)\rangle:x\in X\}\big): e\in E\Big\}$$ where, $N(X)$ denotes all neutrosophic sets over $X$. Note that if $f(e)=\big\{\langle x,0,1,1\rangle: x\in X\big\}$, the element $(e,
f(e))$ is not appeared in the neutrosophic soft set $f$.Set of all *ns*-sets over $X$ is denoted by $\mathbb{NS}$.
\[neutrosophicssubset\]Let $f,g\in \mathbb{NS}$. $f$ is said to be neutrosophic soft subset of $g$, if $T_{f(e)}(x)\leq T_{g(e)}(x)$, $ I_{f(e)}(x)\geq I_{g(e)}(x)$ $F_{f(e)}(x) \geq F_{g(e)}(x)$, $\forall e\in E$, $\forall x\in U$. We denote it by $f \sqsubseteq g$. $f$ is said to be neutrosophic soft super set of $g$ if $g$ is a neutrosophic soft subset of $f$. We denote it by $f\sqsupseteq g$.
If $f$ is neutrosophic soft subset of $g$ and $g$ is neutrosophic soft subset of $f$. We denote it $f=g$
\[ns-empty\] Let $f\in \mathbb{NS}$. If $T_{f(e)}(x)=0$ and $I_{f(e)}(x)=F_{f(e)}(x)=1$ for all $e\in E$ and for all $x\in X$, then $f$ is called null *ns*-set and denoted by $\tilde\Phi$.
\[universal\] Let $f\in \mathbb{NS}$.If $T_{f(e)}(x)=1$ and $I_{f(e)}(x)=F_{f(e)}(x)=0$ for all $e\in E$ and for all $x\in X$, then $f$ is called universal *ns*-set and denoted by $\tilde X$.
\[uni-int\]Let $f,g\in \mathbb{NS}$. Then union and intersection of ns-sets $f$ and $g$ denoted by $f\sqcup g$ and $f\sqcap g$ respectively, are defined by as follow $$\begin{aligned}
f\sqcup g & = & \Big\{\big(e,\{\langle x,T_{f(e)}(x)\vee
T_{g(e)}(x),
I_{f(e)}(x)\wedge I_{g(e)}(x),\\
&&F_{f(e)}(x)\wedge F_{g(e)}(x)\rangle:x\in X\}\big):e\in E\Big\}.\end{aligned}$$ and *ns*-intersection of $f$ and $g$ is defined as $$\begin{aligned}
f\sqcap g & = & \Big\{\big(e,\{\langle x,T_{f(e)}(x)\wedge
T_{g(e)}(x),
I_{f(e)}(x)\vee I_{g(e)}(x),\\
&&F_{f(e)}(x)\vee F_{g(e)}(x)\rangle:x\in X\}\big):e\in E\Big\}.\end{aligned}$$
\[ns-complement\] Let $f,g\in \mathbb{NS}$. Then complement of ns-set $f$, denoted by $f^{\tilde c}$, is defined as follow $$f^{\tilde c}=\Big\{\big(e,\{\langle
x,F_{f(e)}(x),1-I_{f(e)}(x),T_{f(e)}(x)\rangle :x\in X\}\big):e\in
E\Big\}.$$
Let $f,g,h\in\mathbb{NS}$. Then,
1. $\tilde \Phi\sqsubseteq f$
2. $f\sqsubseteq \tilde X$
3. $f\sqsubseteq f $
4. $f\sqsubseteq g $ and $g\sqsubseteq h \Rightarrow$ $f\sqsubseteq h$
[*Proof.* ]{}The proof is obvious from Definition , and Definition .
Let $f\in\mathbb{NS}$. Then
1. $\tilde \Phi^{\tilde c}=\tilde X$
2. $\tilde X^{\tilde c}=\tilde\Phi $
3. $(f^{\tilde c})^{\tilde c}=f$.
[*Proof.* ]{}The proof is clear from Definition , and .
Let $f,g,h\in\mathbb{NS}$. Then,
1. $f\sqcap f=f$ and $f\sqcup f=f$
2. $f\sqcap g=g\sqcap f$ and $f\sqcup g=g\sqcup f$
3. $f\sqcap\tilde\Phi=\tilde\Phi$ and $f\sqcap\tilde X=f$
4. $f\sqcup\tilde\Phi=f$ and $f\sqcup\tilde X=\tilde X$
5. $f\sqcap(g\sqcap h)=(f\sqcap g)\sqcap h$ and $f\sqcup(g\sqcup h)=(f\sqcup g)\sqcup h$
6. $f\sqcap(g\sqcup h)=(f\sqcap g)\sqcup(f\sqcap h)$ and $f\sqcup(g\sqcap h)=(f\sqcup g)\sqcap(f\sqcup h).$
[*Proof.* ]{}The proof is clear from definition and operations of neutrosophic soft sets.
\[t-ns-demorgan\] Let $f,g\in\mathbb{NS}$. Then, De Morgan’s law is valid.
1. $(f\sqcup g)^{\tilde c}=f^{\tilde c}\sqcap g^{\tilde c}$
2. $(f\sqcup g)^{\tilde c}=f^{\tilde c}\sqcap g^{\tilde c}$
[*Proof.* ]{}$f,g\in\mathbb{NS}$ is given.
1. From Definition \[ns-complement\], we have $$\begin{aligned}
(f\sqcup g)^{\tilde c} & = & \Big\{\big(e,\{\langle
x,T_{f(e)}(x)\vee T_{g(e)}(x),
I_{f(e)}(x)\wedge I_{f(e)}(x),\\
&&F_{f(e)}(x)\wedge F_{f(e)}(x)\rangle:x\in X\}\big):e\in E\Big\}^{\tilde c}\\
& = & \Big\{\big(e,\{\langle x,F_{f(e)}(x)\wedge F_{f(e)}(x),
1-(I_{f(e)}(x)\wedge I_{f(e)}(x)),\\
&&T_{f(e)}(x)\vee T_{g(e)}(x)\rangle:x\in X\}\big):e\in E \Big\}\\
&&\langle x,F_{f(e)}(x), 1-I_{f(e)}(x),T_{f(e)}(x)\rangle:e\in E\Big\}\\
& = & \Big\{\big(e,\{X\}\big):e\in E\Big\}\\
&\sqcap&\Big\{\big(e,\{\langle x,F_{g(e)}(x), 1-I_{g(e)}(x),T_{g(e)}(x)\rangle:x\in X\}\big):e\in E\Big\}\\
& = & f^{\tilde c}\sqcap g^{\tilde c}.\end{aligned}$$
2. It can be proved similar way (*i.*)
Let $f,g\in\mathbb{NS}$. Then, difference of $f$ and $g$, denoted by $f\setminus
g$ is defined by the set of ordered pairs $$f\setminus g=\Big\{(e,\{\langle x,T_{f\setminus g(e)}(x),
I_{f\setminus g(e)}(x),F_{f \setminus g(e)}(x)\rangle: x\in X\}):
e\in E\Big\}$$ here, $T_{f\setminus g(e)}(x)$, $I_{f\setminus g(e)}(x)$ and $F_{f\setminus g(e)}(x)$ are defined by $$T_{f\setminus g(e)}(x)=\left\{
\begin{array}{ll}
T_{f(e)}(x)-T_{g(e)}(x), & T_{f(e)}(x)>T_{g(e)}(x) \\
0, & otherwise
\end{array}
\right.$$ $$I_{f\setminus g(e)}(x)=\left\{
\begin{array}{ll}
I_{g(e)}(x)-I_{f(e)}(x), & I_{f(e)}(x)<I_{g(e)}(x) \\
0, & otherwise
\end{array}
\right.$$ $$F_{f\setminus g(e)}(x)=\left\{
\begin{array}{ll}
F_{g(e)}(x)-F_{f(e)}(x), & G_{f(e)}(x)<G_{g(e)}(x) \\
0, & otherwise
\end{array}
\right.$$
\[or\] Let $f,g\in \mathbb{NS}$. Then ’OR’ product of ns-sets $f$ and $g$ denoted by $f\wedge g$, is defined as follow $$\begin{aligned}
f\bigvee g & = & \Big\{\big((e,e'),\{\langle x,T_{f(e)}(x)\vee
T_{g(e)}(x),
I_{f(e)}(x)\wedge I_{g(e)}(x),\\
&&F_{f(e)}(x)\wedge F_{g(e)}(x)\rangle:x\in X\}\big):(e,e')\in
E\times E\Big\}.\end{aligned}$$
\[and\] Let $f,g\in \mathbb{NS}$. Then ’AND’ product of ns-sets $f$ and $g$ denoted by $f\vee g$, is defined as follow $$\begin{aligned}
f\bigwedge g & = & \Big\{\big((e,e'),\{\langle x,T_{f(e)}(x)\wedge
T_{g(e)}(x),
I_{f(e)}(x)\vee I_{g(e)}(x),\\
&&F_{f(e)}(x)\vee F_{g(e)}(x)\rangle:x\in X\}\big):(e,e')\in E\times
E\Big\}.\end{aligned}$$
Let $f,g\in \mathbb{NS}$. Then,
1. $(f\bigvee g)^{\tilde c}=f^{\tilde c}\bigwedge g^{\tilde c}$
2. $(f\bigwedge g)^{\tilde c}=f^{\tilde c}\bigvee g^{\tilde c}$
[*Proof.* ]{}The proof is clear from Definition and .
Decision making method
======================
In this section we will construct a decision making method over the neutrosophic soft set. Firstly, we will define some notions that necessary to construct algorithm of decision making method.
\[rp-matris\] Let $X=\{x_1,x_2,...x_m\}$ be an initial universe, $E=\{e_1,e_2,...e_n\}$ be a parameter set and $f$ be a neutrosophic soft set over $X$. Then, according to the Table of “Saaty Rating Scale” relative parameter matrix $d_E$ is defined as follow $$d_E=\left[
\begin{array}{cccc}
1& d_E(e_1,e_2)& \ldots & d_E(e_1,e_n) \\
d_E(e_2,e_1)& 1& \ldots & d_E(e_2,e_n) \\
\vdots & \vdots& \vdots & \vdots \\
d_E(e_n,e_1)& d_E(e_n,e_2)& \ldots & 1 \\
\end{array}
\right]$$ If $d_E(e_i,e_j)=d_{12}$, we can write matrix $$d_E=\left[
\begin{array}{cccc}
1& d_{11}& \ldots & d_{1n} \\
d_{21}& 1& \ldots & d_{2n} \\
\vdots & \vdots& \vdots & \vdots \\
d_{n1}& d_{n2}& \ldots & 1 \\
\end{array}
\right]$$ Here, $d_{12}$ means that how much important $e_1$ by $e_2$. For example, if $e_1$ is much more important by $e_2$, then we can write $d_{12}=5$ from Table 1.
$$\begin{tabular}{|l|l|l|}
\hline Intensity importance & Definition & Explanation \\
\hline
\(1\)& Equal importance & Two factors contribute \\
& & equally to the objective \\
\hline \(3\)& Somewhat more important& Experience and judgement\\
& &slightly favour one over
the other \\
\hline \(5\)& Much more important & Experience and judgement\\
& &strongly favour one over
the other \\
\hline \(7\)& Very much more important& Experience and judgement\\
& &very strongly favour one over the other. \\
& & Its importance is demonstrated in
practice \\
\hline \(9\)& Absolutely more important & The evidence favouring one
over the other\\
& & is of the
highest possible validity. \\
\hline
\(2,4,6,8\)& Intermediate
values & When compromise is needed \\
\hline
\end{tabular}$$
\[satty\]
Let $f$ be a neutrosophic soft set and $d_E$ be a relative parameter matrix of $f$. Then, score of parameter $e_i$, denoted by $c_i$ and is calculated as follows $$c_i=\sum_{j=1}^nd_{ij}$$
Normalized relative parameter matrix ($nd_E$ for short) of relative parameter matrix $d_E$, denoted by $\hat d$, is defined as follow, $$nd_E=\left[
\begin{array}{cccc}
\frac{1}{c_1} & \frac{d_{12}}{c_1} & \ldots & \frac{d_{1n}}{c_1} \\
\frac{d_{21}}{c_2} & \frac{1}{c_2}& \ldots & \frac{d_{2n}}{c_2} \\
\vdots & \vdots & \ddots & \vdots \\
\frac{d_{n1}}{c_n} & \frac{d_{n2}}{c_n} & \ldots & \frac{1}{c_n}
\end{array}
\right]$$ if $\frac{d_{ij}}{c_i}=\hat d_{ij}$, we can write matrix $nd_E$
$$\hat d=\left[
\begin{array}{cccc}
\hat d_{11} & \hat d_{12} & \ldots & \hat d_{1n} \\
\hat d_{21} & \hat d_{22} & \ldots & \hat d_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
\hat d_{n1} & \hat d_{n2} & \ldots & \hat d_{nn}
\end{array}
\right]$$
Let $f$ be a neutrosophic soft set and $\hat d$ be a normalized parameter matrix of $f$. Then, weight of parameter $e_j\in E$, denoted by $w(e_j)$ and is formulated as follows. $$w(e_j)=\frac{1}{|E|}\sum_{i=1}^n\hat d_{ij}$$
Now, we construct compare matrices of elements of $X$ in neutrosophic sets $f(e)$, $\forall e\in E$.
Let $E$ be a parameter set and $f$ be a neutrosophic soft set over $X$. Then, for all $e\in E$, compression matrices of $f$, denoted $X_{f(e)}$ is defined as follow $$X_{f(e)}=\left[
\begin{array}{cccc}
x_{11} & x_{12} & \cdots & x_{1m} \\
x_{21} & x_{22} & \cdots & x_{2m} \\
\vdots & \vdots & \ddots & \vdots \\
x_{m1} & x_{m2} & \cdots & x_{mm} \\
\end{array}
\right]$$
$$x_{ij}=\frac{\Delta_{T(e)}(x_{ij})+\Delta_{I(e)}(x_{ij})+\Delta_{F(e)}(x_{ij})+1}{2}$$ such that $$\begin{aligned}
\Delta_{T(e)}(x_{ij})&=&T(e)(x_{i})-T(e)(x_{j})\\
\Delta_{I(e)}(x_{ij})&=&I(e)(x_{j})-I(e)(x_{i})\\
\Delta_{F(e)}(x_{ij})&=&F(e)(x_{j})-F(e)(x_{i})\\\end{aligned}$$
Let $X_{f(e)}$ be compare matrix for $e\in E$. Then, weight of $x_j\in X$ related to parameter $e\in E$, denoted by $W_{f(e)}(x_j)$ is defined as follow, $$W_{f(e)}(x_j)=\frac{1}{|X|}\sum_{i=1}^mx_{ij}$$
Let $E$ be a parameter set, $X$ be an initial universe and $w(e)$ and $W_{f(e)}(x_j)$ be weight of parameter $e$ and membership degree of $x_j$ which related to $e_j\in E$, respectively. Then, decision set, denoted $D_{E}$, is defined by the set of ordered pairs $$D_E=\{(x_j,F(x_j)): x_j\in X\}$$ where $$F(x_j)=\frac{1}{|E|}\sum_{j=1}^nw(e_j)\cdot W_{f(e)}(x_j)$$
Note that, $F$ is a fuzzy set over $X$.
Now, we construct a neutrosophic soft set decision making method by the following algorithm;
Algorithm 1 {#algorithm-1 .unnumbered}
===========
***Step 1:*** Input the neutrosophic soft set $f$,\
***Step 2:*** Construct the normalized parameter matrix,\
***Step 3:*** Compute the weight of each parameters,\
***Step 4:*** Construct the compare matrix for each parameter,\
***Step 5:*** Compute membership degree, for all $x_j\in X$\
***Step 6:*** Construct decision set $D_E$\
***Step 7:*** The optimal decision is to select $x_k=max F(x_j)$.\
Let $X$ be the set of blouses under consideration and $E$ is the set of parameters. Each parameters is a neutrosophic word or sentence involving neutrosophic words. Consider $E=\{\textrm{bright}, \textrm{cheap}, \textrm{colorful},
\textrm{cotton}\}$. Suppose that, there are five blouses in the universe $X$ given by $X=\{x_1,x_2,x_3,x_4,x_5\}$. Suppose that,\
**Step 1:** Let us consider the decision making problem involving the neutrosophic soft set in [@bro-13gns]\
**Step 2:** $$d_E=\left(
\begin{array}{cccc}
1& 1/3& 5 & 1/3 \\
3 & 1 & 2 & 3 \\
1/5 & 1/2 & 1 & 2 \\
3 & 1/3 & 1/2 & 1 \\
\end{array}
\right)$$ $c_1=6.67$, $c_2=9$, $c_3=3.7$ and $c_4=4.88$ and
$$\hat d_E=\left(
\begin{array}{cccc}
.15& .05 & .75 & .05 \\
.33 & .11 & .22 & .33 \\
.05 & .14 & .27 & .54 \\
.62 & .07 & .10 & .21 \\
\end{array}
\right)$$\
**Step 3:** From normalized matrix, weight of parameters are obtained as $w(e_1)=.29$, $w(e_2)=.09$, $w(e_3)=.34$ and $w(e_4)=.28$.\
**Step 4:** For each parameter, compare matrices and normalized compare matrices are constructed as follow\
Let us consider parameter “bright”. Then, $$X_{f(bright)}=\left[
\begin{array}{ccccc}
.50 & .10 & .25 & .20 & .15 \\
.45 & .50 & .20 & .15 & .10 \\
.75 & .80 & .50 & .45 & .40 \\
.80 & .85 & .55 & .50 & .45 \\
.85 & .90 & .60 & .55 & .50\\
\end{array}
\right], \quad X_{f(cheap)}=\left[
\begin{array}{ccccc}
.50 & .40 & .15 & .25 & .35 \\
.50 & .50 & .30 & .35 & .45 \\
.85 & .75 & .50 & .60 & .70 \\
.75 & .65 & .40 & .50 & .60 \\
.65 & .55 & .30 & .40 & .50\\
\end{array}
\right]$$ and $$X_{f(colorful)}=\left[
\begin{array}{ccccc}
.50 & .35 & .55 & .65 & .40 \\
.65 & .50 & .65 & .18 & .55 \\
.50 & .35 & .50 & .65 & .40 \\
.35 & .30 & .15 & .50 & .25 \\
.40 & .45 & .60 & .75 & .50 \\
\end{array}
\right], \quad
X_{f(cotton)}=\left[
\begin{array}{ccccc}
.50 & .25 & .35 & .50 & .15 \\
.75 & .50 & .60 & .75 & .40 \\
.65 & .40 & .50 & .65 & .30 \\
.50 & .25 & .35 & .50 & .15 \\
.85 & .60 & .70 & .85 & .50\\
\end{array}
\right]$$
**Step 5:** For all $x_j\in X$ and $e\in E$,\
$
W_{f(bright)}(x_1)=.67,\; W_{f(bright)}(x_2)=.63,\; W_{f(bright)}(x_3)=.42,\\
W_{f(bright)}(x_4)=.37, \; W_{f(bright)}(x_5)=.32
$\
$
W_{f(cheap)}(x_1)=.80, \;W_{f(cheap)}(x_2)=.57, \;W_{f(cheap)}(x_3)=.33,\\
W_{f(cheap)}(x_4)=.42, \;W_{f(cheap)}(x_5)=.52
$\
$
W_{f(colorful)}(x_1)=.48,\; W_{f(colorful)}(x_2)=.39, \; W_{f(colorful)}(x_3)=.49, \\
W_{f(colorful)}(x_4)=.55, \; W_{f(colorful)}(x_5)=.42
$\
$
W_{f(cotton)}(x_1)=.65, \; W_{f(cotton)}(x_2)=.40, \; W_{f(cotton)}(x_3)=.50, \\
W_{f(cotton)}(x_4)=.65, \; W_{f(cotton)}(x_5)=.30
$\
**Step 6:** By using step 3 and step 5, $D_E$ is constructed as follow $$D_E=\{(x_1,0.15),(x_2,0.12),(x_3,0.11),(x_4,0.13),(x_5,0.09)\}$$
**Step 7:** Note that, membership degree of $x_1$ is greater than the other. Therefore, optimal decision is $x_1$ for this decision making problem.
Group decision making
=====================
In this section, we constructed a group decision making method using intersection of neutrosophic soft sets and Algorithm 1.\
Let $X= \{x_1, x_2,..., x_n \}$ be an initial universe and let $d =
\{d^1, d^2,..., d^m\}$ be a decision maker set and $E=\{e_1,e_2,...,e_k\}$ be a set of parameters. Then, this method can be described by the following steps\
Algorithm 2 {#algorithm-2 .unnumbered}
===========
***Step 1:*** Each decision-maker $d^i$ construct own neutrosophic soft set, denoted by $f_{d_i}$, over $U$ and parameter set $E$.\
***Step 2:*** Let for $p,r\leq k$, $[d^i_{pr}]$ a relative parameter matrix of decision-maker $d^i\in D$ based on the Saaty Rating Scale. Decision-maker $d^i$ gives his/her evaluations separately and independently according to his/her own preference based on Saaty Rating Scale. In this way, each decision-maker $d^i$ presents a relative parameter matrix.
$$[d^i_{pr}]=\left(
\begin{array}{cccc}
d^i_{11} & d^i_{12} & \cdots & d^i_{1k} \\
d^i_{21} & d^i_{22} & \cdots & d^i_{2k} \\
\vdots & \vdots & \ddots & \vdots \\
d^i_{k1} & d^i_{k2} & \vdots & d^i_{kk} \\
\end{array}
\right)$$ here $d^i_{pr}$ is equal $d_E(e_p, e_r)$ that in Definition .\
***Step 3:*** Arithmetic mean matrix is constructed by using the the relative parameter matrix of each decision-maker $d^i$. It will be denoted by $[i_{pr}]$ and will be computed as in follow $$i_{pr}=\frac{1}{|d|}\sum_{i=1}^md^i_{pr}$$\
***Step 4:*** Normalized parameter matrix, is constructed using the arithmetic mean matrix $[i_{pr}]$, it will be shown $[\hat
i_{pr}]$ and weight of each parameter $e_i\in E$ ($w(e_i)$) is computed.\
***Step 5:*** Intersection of neutrosophic soft sets (it will be denoted by $I_{f_{d_i}}$) which are constructed by decision makers is found. $$I_{f_{d}}=\bigcap_{i=1}^mf_{d^i}$$\
***Step 6:*** Based on the matrix $I_{f_{d}}$, for each element of $e\in E$ compare matrix, denoted by $I_{f_{d}(e)}$ is constructed.\
***Step 7:*** By the $I_{f_{d}(e)}$, weight of each element of $X$ denoted by $W_{I_{f_{d}(e)}}(x_i)$, are computed.\
***Step 8:*** Decision set $D_E$ is constructed by using values of $w(e)$ and $W_{I_{f_{d^i}}}(x)$. Namely;
$$D_E=\{(x_i,F(x_i)): x_i\in X\}$$ and $$F(x_i)=\frac{1}{|E|}\sum_{j=1}^nw(e_j)\cdot W_{I_f(e)}(x_i)$$\
***Step 9:*** From the decision set, $x_k=max F(x_i)$ is selected as optimal decision.
Assume that a company wants to fill a position. There are 6 candidates who fill in a form in order to apply formally for the position. There are three decision makers; one of them is from the department of human resources and the others is from the board of directors. They want to interview the candidates, but it is very difficult to make it all of them. Let $d=\{d_1,d_2,d_3\}$ be a decision makers set, $X=\{x_1,x_2,x_3,x_4,x_5\}$ be set of candidates and $E =\{e_1,e_2,e_3,e_4\}$ be a parameter set such that parameters $e_1,e_2,e_3$ and $e_4$ stand for ”experience”,”computer knowledge”, ”higher education” and ”good health””, respectively.\
***Step:1*** Let each decision maker construct neutrosophic soft sets over $X$ by own interview:\
$ f_{d^1}=\left\{
\begin{array}{c}
f_{d^1}(e_1)=\{\langle x_1, .4,.2,.7\rangle,\langle x_2,
.5,.6,.2\rangle,\langle x_3, .7,.3,.3\rangle,\langle x_4,
.6,.5,.4\rangle,\langle x_5, .3,.5,.5\rangle\},\\
f_{d^1}(e_2)=\{\langle x_1, .3,.5,.2\rangle,\langle x_2,
.4,.4,.3\rangle,\langle x_3, .5,.7,.8\rangle,\langle x_4,
.7,.1,.3\rangle,\langle x_5, .6,.3,.2\rangle\},\\
f_{d^1}(e_3)=\{\langle x_1, .7,.4,.3\rangle,\langle x_2,
.6,.1,.5\rangle,\langle x_3, .5,.2,.4\rangle,\langle x_4,
.2,.2,.6\rangle,\langle x_5, .3,.3,.6\rangle\},\\
f_{d^1}(e_4)=\{\langle x_1, .7,.3,.5\rangle,\langle x_2,
.3,.5,.3\rangle,\langle x_3, .2,.4,.3\rangle,\langle x_4,
.4,.2,.5\rangle,\langle x_5, .5,.2,.6\rangle\}
\end{array}\right\}
$\
\
\
$f_{d^2}=\left\{
\begin{array}{c}
f_{d^2}(e_1)=\{\langle x_1, .5,.2,.3\rangle,\langle x_2,
.3,.5,.6\rangle,\langle x_3, .4,.3,.3\rangle,\langle x_4,
.2,.5,.4\rangle,\langle x_5, .5,.5,.5\rangle\},\\
f_{d^2}(e_2)=\{\langle x_1, .5,.4,.6\rangle,\langle x_2,
.7,.2,.5\rangle,\langle x_3, .6,.3,.5\rangle,\langle x_4,
.7,.2,.3\rangle,\langle x_5, .6,.4,.2\rangle\},\\
f_{d^2}(e_3)=\{\langle x_1, .6,.2,.5\rangle,\langle x_2,
.4,.4,.6\rangle,\langle x_3, .2,.5,.4\rangle,\langle x_4,
.3,.5,.4\rangle,\langle x_5, .3,.3,.6\rangle\},\\
f_{d^2}(e_4)=\{\langle x_1, .3,.4,.5\rangle,\langle x_2,
.4,.3,.2\rangle,\langle x_3, .4,.4,.3\rangle,\langle x_4,
.4,.2,.5\rangle,\langle x_5, .2,.5,.6\rangle\}
\end{array}\right\}
$ and\
$ f_{d^3}=\left\{
\begin{array}{c}
f_{d^3}(e_1)=\{\langle x_1, .4,.5,.7\rangle,
\langle x_2,.5,.3,.4\rangle,
\langle x_3, .7,.3,.5\rangle,
\langle x_4,.4,.5,.3\rangle,
\langle x_5, .7,.8,.6\rangle\},\\
f_{d^3}(e_2)=\{\langle x_1, .6,.2,.6\rangle,
\langle x_2,.4,.3,.5\rangle,
\langle x_3, .5,.4,.7\rangle,
\langle x_4,.3,.1,.5\rangle,
\langle x_5, .4,.3,.1\rangle\},\\
f_{d^3}(e_3)=\{\langle x_1, .4,.3,.2\rangle,
\langle x_2,.6,.7,.2\rangle,
\langle x_3, .3,.5,.2\rangle,
\langle x_4,.6,.6,.4\rangle,
\langle x_5, .6,.5,.5\rangle\},\\
f_{d^3}(e_4)=\{\langle x_1, .5,.3,.1\rangle,
\langle x_2,.2,.5,.2\rangle,
\langle x_3, .5,.5,.4\rangle,
\langle x_4,.5,.2,.5\rangle,
\langle x_5, .5,.3,.6\rangle\}
\end{array}\right \}$\
***Step 2:*** Relative parameter matrix of each decision maker are as in follow $$[d^1_{pr}]=\left[
\begin{array}{cccc}
1 & 3 & 1/5 & 2 \\
1/3 & 1 & 3 & 6 \\
5 & 1/3 & 1 & 1/5 \\
1/2 & 1/6 & 5 & 1 \\
\end{array}
\right]
\quad
[d^2_{pr}]=\left[
\begin{array}{cccc}
1 & 5 & 1/7 & 2 \\
1/5 & 1 & 1/2 & 6 \\
7 & 2 & 1 & 1/3 \\
1/2 & 1/6 & 3 & 1 \\
\end{array}
\right]$$ and $$[d^3_{pr}]=\left[
\begin{array}{cccc}
1 & 3 & 1/3 & 4 \\
1/3 & 1 & 1/3 & 1/6 \\
3& 3 & 1 & 1/2 \\
1/4 & 6 & 2 & 1 \\
\end{array}
\right]$$\
***Step 3:*** $[i_{pr}]$ can be obtained as follow
$$[i_{pr}]=\left[
\begin{array}{cccc}
1 & 3.67 &.23 &2.67 \\
.29 & 1 & 1.28 &4.06 \\
5& 1.78 & 1 &.34 \\
.42& 4.06 &3.33 & 1 \\
\end{array}
\right]$$\
***Step 4:*** $[\hat i_{pr}]$ and weight of each parameter can be obtained as follow
$$[\hat i_{pr}]=\left[
\begin{array}{cccc}
.13& .49 &.03 &.35 \\
.04& .15 & .19 & .61 \\
.62& .22 & .12 &.04 \\
.05& .46 &.38 & .11 \\
\end{array}
\right]$$ and $w(e_1)=.21$, $w(e_2)=0.33$, $w(e_3)=.18$ $w(e_4)=.28$.\
***Step 5:*** Intersection of neutrosophic soft sets $f_{d^1},f_{d^2}$ and $f_{d^3}$ is as follow;\
\
$ I_{f_d}=\left\{
\begin{array}{c}
I_{f_d}(e_1)=\{\langle x_1, .4,.5,.7\rangle,
\langle x_2,.3,.6,.6\rangle,
\langle x_3, .4,.3,.5\rangle,
\langle x_4,.2,.5,.5\rangle,
\langle x_5, .3,.8,.6\rangle\},\\
I_{f_d}(e_2)=\{\langle x_1, .3,.5,.6\rangle,
\langle x_2,.4,.4,.5\rangle,
\langle x_3, .5,.7,.8\rangle,
\langle x_4,.3,.2,.5\rangle,
\langle x_5, .4,.4,.2\rangle\},\\
I_{f_d}(e_3)=\{\langle x_1, .6,.5,.5\rangle,
\langle x_2,.4,.7,.6\rangle,
\langle x_3, .2,.5,.4\rangle,
\langle x_4,.2,.6,.6\rangle,
\langle x_5, .3,.5,.6\rangle\},\\
I_{f_d}(e_4)=\{\langle x_1, .3,.4,.5\rangle,
\langle x_2,.2,.5,.3\rangle,
\langle x_3, .2,.5,.4\rangle,
\langle x_4,.4,.2,.5\rangle,
\langle x_5, .2,.5,.6\rangle\}
\end{array}\right \}$\
***Step 6:*** For each parameter, compare matrices of elements of $X$ are obtained as in follow; $$I_{f_d(e_1)}=\left[
\begin{array}{ccccc}
.50 & .55 & .30 & .45 &.65 \\
.45 & .50 & .25 & .40 &.60 \\
.70& .75 & .50 & .35 & .85\\
.55 & .60 & .65 & .50 &.70\\
.65 & .40 & .15 & .30 &.70\\
\end{array}
\right],
\quad I_{f_d(e_2)}=\left[
\begin{array}{ccccc}
.50 & .35 & .60 & .30 &.20 \\
.65 & .50 & .75 & .35 &.35 \\
.40 & .25 & .50 & .20 &.10\\
.70 & .65 & .80 & .50 &.40\\
.80 & .65 & .90 & .60 &.50\\
\end{array}
\right]$$ and $$I_{f_d(e_3)}=\left[
\begin{array}{ccccc}
.50 & .75 & .65 & .80 &.70 \\
.25 & .50 & .40 & .55 &.45 \\
.35& .60 & .50 & .65 & .55\\
.20 & .45 & .35 & .50 &.45\\
.30 & .55 & .45 & .55 &.50\\
\end{array}
\right],
\quad I_{f_d(e_4)}=\left[
\begin{array}{ccccc}
.50 & .50 & .55 & .35 &.65 \\
.50 & .50 & .45 & .40 &.65 \\
.45 & .55 & .50 & .30 &.60\\
.65 & .60 & .70 & .50 &.80\\
.35 & .35 & .40 & .20 &.50\\
\end{array}
\right]$$\
***Step 7:*** Membership degrees of elements of $X$ related to each parameter $e\in E$ are obtained as follow;
$W_{f_{d}(e_1)}(x_1)=.57$, $W_{f_{d}(e_1)}(x_2)=.56$, $W_{f_{d}(e_1)}(x_3)=.37$, $W_{f_{d}(e_1)}(x_4)=.40$ and $W_{f_{d}(e_1)}(x_5)=.66$\
$W_{f_{d}(e_2)}(x_1)=.61$, $W_{f_{d}(e_2)}(x_2)=.48$, $W_{f_{d}(e_2)}(x_3)=.71$, $W_{f_{d}(e_2)}(x_4)=.39$ and $W_{f_{d}(e_2)}(x_5)=.31$\
$W_{f_{d}(e_3)}(x_1)=.32$, $W_{f_{d}(e_3)}(x_2)=.57$, $W_{f_{d}(e_3)}(x_3)=.47$, $W_{f_{d}(e_3)}(x_4)=.61$ and $W_{f_{d}(e_3)}(x_5)=.53$\
$W_{f_{d}(e_4)}(x_1)=.49$, $W_{f_{d}(e_4)}(x_2)=.50$, $W_{f_{d}(e_4)}(x_3)=.52$,$W_{f_{d}(e_4)}(x_4)=.35$ and $W_{f_{d}(e_4)}(x_5)=.64$\
***Step 8:*** $$\begin{aligned}
F(x_1)&=&\frac{1}{|E|}\sum_{j=1}^nw(e_j)\cdot W_{{f_{d}(e_j)}}(x_1)\\
&=&\frac{1}{4}(.21\cdot.57+.33\cdot.61+.18\cdot.32+.28\cdot.49)\\
&=&.126\end{aligned}$$ similarly $F(x_2)=.130$, $F(x_3)=.136$, $F(x_4)=.105$ and $F(x_5)=.129$. Then, we get $$D_E=\{(x_1,.126),(x_1,.130),(x_1,.136),(x_1,.105),(x_1,.129)\}$$\
***Step 9:*** Note that, membership degree of $x_3$ is greater than membership degrees of the others. Therefore, optimal decision is $x_3$ for this decision making problem.
Conclusion
==========
In this paper, we firstly investigate neutrosophic soft sets given paper of Maji [@maji-2013] and then we redefine notion of neutrosophic soft set and neutrosophic soft set operations. Finally, we present two applications of neutrosophic soft sets in decision making problem.
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|
---
abstract: 'Motivated by Kesten’s bridge decomposition for two-dimensional self-avoiding walks in the upper half plane, we show that the conjectured scaling limit of the half-plane SAW, the ${\textrm{SLE}}(8/3)$ process, also has an appropriately defined bridge decomposition. This continuum decomposition turns out to entirely be a consequence of the restriction property of ${\textrm{SLE}}(8/3)$, and as a result can be generalized to the wider class of *restriction measures*. Specifically we show that the restriction hulls with index less than one can be decomposed into a Poisson Point Process of *irreducible bridges* in a way that is similar to Itô’s excursion decomposition of a Brownian motion according to its zeros.'
address:
- |
Department of Mathematics\
University of Toronto\
Toronto, ON, Canada
- |
École Normale Supérieure\
Paris, France\
author:
- Tom Alberts
- 'Hugo Duminil-Copin'
bibliography:
- '../../Restrict.bib'
title: Bridge Decomposition of Restriction Measures
---
[^1]
Introduction
============
One of the greatest successes of the Schramm-Loewner Evolution (SLE), and the broader study of two-dimensional conformally invariant stochastic processes that it enabled, has been the ability to describe the scaling limits of two-dimensional lattice models that arise in statistical mechanics. There are many known examples: ${\textrm{SLE}}(2)$ as the scaling limit for loop erased random walk, ${\textrm{SLE}}(3)$ as the scaling limit of critical Ising interfaces, ${\textrm{SLE}}(6)$ as the limit of percolation exploration paths, etc. One of the most important open problems in the field is to prove that the scaling limit of the infinite *self-avoiding walk* in the upper half plane ${\mathbb{H}}$ is given by ${\textrm{SLE}}(8/3)$. It is known that *if* the scaling limit of half-plane SAWs exists *and* is conformally invariant, then the scaling limit must be SLE($8/3$). Both the existence and conformal invariance are widely believed to be true, yet proofs remain elusive. For an accessible and relatively recent source on the current status of this problem, we refer the reader to [@lsw:saw]. Even without formally establishing the scaling limit result, it is often still possible to independently check that the various well-studied properties of half-plane SAWs carry over to the SLE($8/3$) process. The main results of this paper should be seen in this context. In [@kesten:saw1] it is shown that half-plane SAWs admit what is called a *bridge decomposition*, which raised the question of finding a similar decomposition for SLE($8/3$). In this paper we will show that an appropriately defined continuum decomposition does exist, and we will describe some of its properties. A somewhat surprising aspect of the existence is that it depends only on the fact that SLE($8/3$) satisfies the restriction property, and not on the fine details of the process itself. Specifically, the decomposition has no explicit reliance on the Loewner equation. Using this fact we are able to extend the continuum bridge decomposition beyond SLE($8/3$) to a wider class of random sets whose laws are given by the so-called restriction measures. These probability measures were introduced and studied extensively in [@lsw:conformal_restriction], and they occupy an important position in the hierarchy of two-dimensional conformally invariant processes. We will give a more detailed description of restriction measures in Section \[PrelimSection\], but we emphasize that the reader who is uninterested in general restriction measures will lose nothing by focusing on SLE($8/3$) as the canonical one.
Motivation: Bridge Decomposition of SAWs
----------------------------------------
To motivate the continuum bridge decomposition, we first describe the corresponding decomposition for half-plane SAWs. This is thoroughly described in [@madras_slade:saw_book], along with many other interesting properties of the self-avoiding walk. In the discrete setting we will work exclusively on the lattice ${\mathbb{Z}}+ i {\mathbb{Z}}$. An $N$-step self-avoiding walk $\omega$ on ${\mathbb{Z}}+ i{\mathbb{Z}}$ is a sequence of lattice sites $[\omega(0), \omega(1), \ldots, \omega(N)]$ satisfying ${\left| \omega(j+1) - \omega(j) \right|} = 1$ and $\omega(i) \neq \omega(j)$ for $i \neq j$. We will write ${\left| \omega \right|} = N$ to denote the length of $\omega$. Given walks $\omega$ and $\omega'$ of length $N$ and $M$ (respectively), the *concatenation* of $\omega$ and $\omega'$ is defined by $$\begin{aligned}
\omega \oplus \omega' = \left[ \omega(0), \ldots, \omega(N), \omega'(1) + \omega(N), \ldots, \omega'(M) + \omega(N) \right].\end{aligned}$$ Letting $c_N$ denote the number of self-avoiding walks of length $N$, it is easy to see that $$\begin{aligned}
c_{N+M} \leq c_N c_M\end{aligned}$$ since any SAW of length $N+M$ can always be written as the concatenation of two SAWs of length $N$ and $M$. A standard submultiplicativity argument then proves the existence of a constant $\mu > 0$ such that $$\begin{aligned}
\label{connectiveConstant}
\lim_{N \to \infty} \frac{\log c_N}{N} = \log \mu,\end{aligned}$$ or $c_N \approx \mu^N$ in the common shorthand. The exact value of $\mu$ is not known, nor is it expected to be any special value, but numerically it has been shown that $\mu$ is close to $2.638$ (see [@madras_slade:saw_book Section 1.2]).
We will mostly deal with half-plane SAWs rooted at the origin, i.e. self-avoiding paths $\omega$ such that $\omega(0) = 0$ and ${\textrm{Im}}\, \omega(j) > 0$ for all $j > 0$. Let $\mathcal{H}$ denote the set of all such walks. The most commonly used probability measure on $\mathcal{H}$, and the one that we will consider throughout, is the weak limit of the uniform measure on $\{ \omega \in \mathcal{H} : {\left| \omega \right|} =N \}$, as $N \to \infty$. This limit is proven to exist in [@madras_slade:saw_book], and again in the appendix of [@lsw:saw]. The key element of both proofs is, in fact, the *bridge decomposition* of the walks in $\mathcal{H}$, the study of which was initiated by Kesten [@kesten:saw1; @kesten:saw2] and goes as follows. A *bridge of length $N$* is a self-avoiding walk $\omega$ such that ${\left| \omega \right|} = N$ and $$\begin{aligned}
{\textrm{Im}}\, \omega(0) < {\textrm{Im}}\, \omega(j) \leq {\textrm{Im}}\, \omega(N), \quad 1 \leq j \leq N.\end{aligned}$$ Note that the concatenation of any two bridges is still a bridge, but that not every bridge is the concatenation of two shorter ones. A bridge with the latter property is said to be *irreducible*, and such bridges are the basic building blocks of walks in $\mathcal{H}$. Indeed, given any $\omega \in \mathcal{H}$, one performs a bridge decomposition of $\omega$ by searching for the smallest time $j$ such that ${\textrm{Im}}\, \omega(k) \leq {\textrm{Im}}\, \omega(j)$ for $k \leq j$ and ${\textrm{Im}}\, \omega(k) > {\textrm{Im}}\, \omega(j)$ for $k > j$. By the minimality of $j$, the subpath $[w(0), w(1), \ldots, w(j)]$ is an irreducible bridge, and the shifted subpath $[0, w(j+1) - w(j), \ldots, w(k) - w(j), \ldots]$ for $k \geq j$ is a new element of $\mathcal{H}$ on which we may repeat this procedure. Iterating in this fashion produces the bridge decomposition of $\omega$ into a sequence of irreducible bridges, and the decomposition is clearly unique[^2].
Much of the study of the infinite self-avoiding walk in the upper half plane therefore reduces to the study of irreducible bridges. Let $\mathcal{B}$ be the set of all irreducible bridges rooted at the origin, and $\lambda_N$ be the number of length $N$ elements of $\mathcal{B}$. Using some clever tricks involving generating functions, Kesten was able to prove what is now called **Kesten’s relation**: $$\begin{aligned}
\label{kesten1}
\sum_{N \geq 1} \lambda_N \mu^{-N} = \sum_{\omega \in \mathcal{B}} \mu^{-{\left| \omega \right|}} = 1,\end{aligned}$$ for the same $\mu$ as in (for proofs see [@kesten:saw1] or [@madras_slade:saw_book Section 4.3]). Kesten’s relation shows that $\textbf{P}(\omega) := \mu^{-{\left| \omega \right|}}$ is a probability measure on $\mathcal{B}$, and by concatenating together an independent sequence of irreducible bridges each sampled from $\textbf{P}$, a probability measure is induced on $\mathcal{H}$. In [@madras_slade:saw_book] and [@lsw:saw], the latter measure is shown to be the only possible candidate for the weak limit of the uniform measure on $\{ \omega \in \mathcal{H} : {\left| \omega \right|} =N \}$, and therefore the question of existence of this weak limit is immediately settled.
The bridge decomposition shows that infinite half-plane SAWs have a renewal structure to them. At the end of each irreducible bridge the future path of the walk lies entirely in the half-plane above the horizontal line where the bridge ended. The future path is again a concatenation of a sequence of irreducible bridges, so that its law is the same as the law of the original path and the future path is independent of the past. In this sense the walk renews itself whenever it is at the end of an irreducible bridge, and it is appropriate to call such times renewal times. Note that the renewal times are functions of the *entire* half-plane SAW, since the algorithm for the bridge decomposition depends upon knowing the entire walk.
![A sample SLE(8/3) curve in the lighter colour, with the bridge points superimposed in black. The bridge heights are plotted on the vertical axis. The ${\textrm{SLE}}(8/3)$ curve is generated by Tom Kennedy’s algorithm and freely available graphics program; see [@kennedy:algo].[]{data-label="points-fig"}](points_lines){width="15cm" height="12cm"}
Statement of Results: The Continuum Bridge Decomposition
--------------------------------------------------------
In the continuum we will show that an analogue of bridge times exists for the so-called *restriction hulls* in ${\mathbb{H}}$, and that these times are also renewal times. Using this renewal structure, we proceed to decompose the restriction hulls into countably many continuum irreducible bridges. This continuum decomposition most closely resembles the discrete one in the case of ${\textrm{SLE}}(8/3)$, but we will see that it also holds for more general restriction hulls with parameter $\alpha < 1$. We will give a more in-depth description of the restriction hulls in Section \[PrelimSection\], but provide a brief summary here.
Roughly speaking, a restriction hull is a stochastic process taking values in the space of unbounded hulls in ${\mathbb{H}}$. An *unbounded hull* is a closed, connected subset $K \subset \overline{{\mathbb{H}}}$ such that ${\mathbb{H}}\backslash K$ consists of exactly two connected components. The unbounded hulls that we will consider are closed, connected subsets of $\overline{{\mathbb{H}}}$ that connect $0$ and $\infty$, and intersect ${\mathbb{R}}$ only at zero; moreover it will be possible to time parameterize them into a growing family $(K_t, t \geq 0)$ of *hulls* (closed, connected subsets $A$ of $\overline{{\mathbb{H}}}$ such that ${\mathbb{H}}\backslash A$ is simply connected with exactly one connected component) with $K_{\infty} = K$. This time parameterization is provided by the well-known construction of restriction hulls that was originally laid out in [@lsw:conformal_restriction] and [@lawler_werner:loop_soup]. Those papers show that attaching the filled-in loops from a realization of the Brownian Loop Soup to an independent SLE curve induces a restriction law on unbounded hulls in ${\mathbb{H}}$. By changing the $\kappa$ parameter for the SLE and the intensity parameter for the loop soup (in a specific way) an entire family ${{\mathbb{P}}_{\alpha}}$ of restriction measures on unbounded hulls is created. Here $\alpha$ is a real parameter with $\alpha \geq 5/8$.
The definition of a continuum bridge is motivated by the algorithm for decomposing half-plane SAWs into irreducible bridges, which essentially searches for horizontal lines that separate the future path from the past.
Let $K$ be a hull (unbounded or not).
- Call $L > 0$ a *bridge height* for $K$ if the horizontal line $y=L$ intersects $K$ at exactly one point, i.e. if $K \cap \{ y=L \}$ is a singleton.
- If $z \in {\mathbb{H}}$ is such a singleton then we call it a *bridge point*. Let $C$ be the set of bridge points of $K$, and let $D$ be the set of bridge heights (note that $D = \{ {\textrm{Im}}\, z : z \in C \}$).
- Let $G$ be the set of *bridge times* at which the hull is at a bridge point, which can be written as $G := \{ t \geq 0 : K_t \backslash K_{t-} \cap C \neq \emptyset \}$.
- A *continuum bridge* is a segment of the bridge between two bridge times, i.e. if $s, t \in G$ with $s < t$ then the hull $K_{t-} \backslash K_{s-}$ is a bridge. A continuum bridge is said to be *irreducible* if it contains no bridge points (other than the starting and ending points).
Note that bridge heights, points and times are all functions of the *entire* hull $K$. A subset of $K$ is, by itself, not enough to determine $C, D$ or $G$. At any fixed time $t \geq 0$ it is possible to determine what are the bridge points of the hull $K_t$, but not which of those are bridge points of the entire hull $K_{\infty} = K$, since some of the bridge points of $K_t$ may ultimately be destroyed by the future hull as it grows. There are two main steps behind the continuum bridge decomposition. The first is to show that bridge points actually exist for hulls with $\alpha < 1$, which is not a priori clear. We do this by calculating the almost sure Hausdorff dimensions of $C$ and $D$ and showing that they are strictly larger than zero (and in fact the same). Specifically we will show the following:
\[BridgeDimension\] Suppose $K$ has the law of ${{\mathbb{P}}_{\alpha}}$, then
1. \[prop:scaling\] the laws of $C$ and $D$ are scale invariant (i.e. $rC \equiv C$ and $rD \equiv D$ for all $r > 0$),
2. \[prop:perfect\] $C$ and $D$ are almost surely perfect (i.e. closed and without isolated points),
3. \[prop:constantdim\] the Hausdorff dimensions of both $C$ and $D$ are constant, ${{\mathbb{P}}_{\alpha}}-a.s.$,
4. \[prop:hdim\] ${\mathrm{dim_H} \,}{C} = {\mathrm{dim_H} \,}{D} = \max (2 - 2\alpha, 0), \,\, {{\mathbb{P}}_{\alpha}}-a.s.$,
5. \[prop:emptyC\] $C$ and $D$ are empty, ${{\mathbb{P}}_{\alpha}}-a.s.$ if and only if $\alpha\geq 1$.
The proof of Theorem \[BridgeDimension\] is taken up in Section \[BridgeSection\], but we will mention here that the key element is the **restriction formula:** $$\begin{aligned}
\label{restrictionFormula}
{{\mathbb{P}}_{\alpha} \left( K \cap A = \emptyset \right)} = \phi_A'(0)^{\alpha},\end{aligned}$$ where $A$ is a hull that does *not* contain zero, and $\phi_A$ is a conformal map from ${\mathbb{H}}\backslash A$ to ${\mathbb{H}}$ such that $\phi_A(z) \sim z$ as $z \to \infty$. Most of the proof of Theorem \[BridgeDimension\] is based on an analysis of $\phi_A'(0)$ for a specific choice of the hull $A$. The proof of part builds upon the $\alpha = 1$ case, which is related to Brownian excursions, and uses the fact that the vertical component of a Brownian excursion is a Bessel-$3$ process.
Given that bridge points exist for $\alpha < 1$, the next step is to prove an analogue of the renewal theory for half-plane SAWs. In Section \[RenewalSection\] we show that the restriction hulls have an extended Markov property with respect to the information gained by observing the hull as it grows along with the *global* bridge points of $K$ as they appear, and as a corollary we show that the bridge times are actually renewal times for the hull process. In Section \[LocalTimeSection\] we will use this Markov property and Theorem \[BridgeDimension\] to show the existence of a “local time” for the time spent by a restriction hull at its bridge points, and the local time can then be used to prove:
\[LocalTimeIntroTheorem\] There exists a local time $\lambda$ supported on bridge heights such that $\theta_{\lambda}(K_{\lambda}\setminus K_{\lambda-})$ is a Poisson Point Process, where $\theta_t$ is an operator that shifts back to the origin the part of the hull that comes after time $t$. Moreover, the local time is the inverse of a stable subordinator of index $2-2\alpha$.
The general theory of Poisson Point Processes then implies the existence of a sigma-finite measure $\nu_{\alpha}$ on continuum irreducible bridges that is the analogue of the measure $\textbf{P}$ on irreducible bridges for half-plane SAWs. In Section \[LocalTimeSection\] we mention some basic properties of this measure. We also show that the Poisson Point Process can be used to recover the restriction hull, so that as in the discrete case, the irreducible bridges are the building blocks of the restriction hull processes. We should mention that most of these ideas are similar in spirit to the excursion decomposition of a one-dimensional Brownian motion according to its zeros, as was first described by Itô. In recent years, similar two-dimensional conformally invariant decompositions of this type have also been considered by Dubédat [@dubedat:excursions] and Virág [@virag:beads]. They provide decompositions of unbounded hulls arising from certain variants of ${\textrm{SLE}}(\kappa, \rho)$ and Brownian excursions, respectively, although their decompositions are at cutpoints rather than bridge points (i.e. points that, if removed from the set, would disconnect it into two pieces). Clearly bridge points are cutpoints but not vice versa, and there does not appear to be any direct relationship between our decomposition and theirs. In one sense their decompositions are more involved than ours, since their hulls refresh at cutpoints only after conformally mapping away the past, whereas our hulls refresh at bridge points after a simple shift of the future hull back to the origin. This difference is mostly cosmetic, however, and in spirit all these decompositions are quite similar.
The paper is organized as follows: in Section \[PrelimSection\] we give the necessary background on restriction measures and introduce some notation. Section \[BridgeSection\] is devoted to proving the existence of bridge points and Theorem \[BridgeDimension\], while Section \[RenewalSection\] proves an extended Markov property and a refreshing property of the restriction hulls with respect to the filtration generated by bridge points as they appear. Section \[LocalTimeSection\] then uses these results to prove the decomposition of Theorem \[LocalTimeIntroTheorem\]. Finally, in Section \[Open\] we present a series of open questions that were raised by our work.
**Acknowledgements:** We are grateful to Wendelin Werner for initially suggesting this problem to us, for many helpful and encouraging discussions along the way, and for hosting the first author at the École Normale Supérieure where most of this work was completed. We also thank Vladas Sidoravicius for hosting us at IMPA, where this work was begun, and Bálint Virág for some enlightening conversations. Finally, we thank an anonymous referee for some very helpful suggestions which greatly improved the presentation of this work.
Restriction Measures \[PrelimSection\]
======================================
In this section we review the basic construction and properties of restriction measures. We include no proofs but give references to the appropriate sources. For thorough overviews of the subject see [@lsw:conformal_restriction; @lawler_werner:loop_soup; @lawler:book]. The reader interested only in the bridge decomposition for ${\textrm{SLE}}(8/3)$, and not for general restriction measures, can entirely ignore the presence of the loops in this section.
To begin with, consider a simply connected domain $D$ in the complex plane ${\mathbb{C}}$ (other than the whole plane itself) and two boundary points $z, w \in \partial D$. A *chordal restriction measure* corresponding to the triple $(D, z, w)$ is a probability measure ${\mathbb{P}}^{(D, z, w)}$ on closed subsets of $\overline{D}$. The measures are supported on closed, connected subsets of $K \subset \overline{D}$ such that $K \cap \partial D = \{z, w \}$ and $D \backslash K$ has exactly two components (for the triple $({\mathbb{H}}, 0, \infty)$ we call these sets unbounded hulls, for obvious reasons). The restriction measures satisfy the following properties, which essentially characterize them uniquely:
- **Restriction property:** for all simply connected subsets $D'$ of $D$ such that $D \backslash D'$ is also simply connected and bounded away from $z$ and $w$, the law of ${\mathbb{P}}^{(D, z, w)}$, conditioned on $K \subset D'$, is ${\mathbb{P}}^{(D', z, w)}$,
- **Conformal invariance:** if $f: D \to D'$ is conformal and $K$ has ${\mathbb{P}}^{(D, z, w)}$ as its law, then $f(K)$ is distributed according to ${\mathbb{P}}^{(f(D), f(z), f(w))}$.
It turns out that for a given triple $(D, z, w)$ there is only a one-parameter family of such laws, indexed by a real number $\alpha$. We denote the law by ${{\mathbb{P}}_{\alpha}}^{(D, z, w)}$, and due to the conformal invariance property it is enough to define the restriction measure for a single triple $(D, z, w)$. The canonical choice is $({\mathbb{H}}, 0, \infty)$, and for shorthand we will write ${{\mathbb{P}}_{\alpha}}$ for ${{\mathbb{P}}_{\alpha}}^{({\mathbb{H}}, 0, \infty)}$. In [@lsw:conformal_restriction] it is shown that these restriction measures exist only if the parameter $\alpha$ satisfies $\alpha \geq 5/8$, and that the measure is supported on simple curves only if $\alpha = 5/8$. In the latter case the restriction measure is simply the ${\textrm{SLE}}(8/3)$ law from $z$ to $w$ in $D$. For $\alpha = 1$ it turns out that the restriction measure coincides with the law of filled-in Brownian excursions in $D$ from $z$ to $w$.
For all $\alpha \geq 5/8$, one of the fundamental constructions of [@lsw:conformal_restriction] is that restriction measures can be realized by adding to an ${\textrm{SLE}(\kappa)}$ curve the filled-in loops that it intersects from an independent realization of the Brownian loop soup, for an appropriate choice of $\kappa$ for the curve and intensity parameter $\lambda$ for the loop soup. Let $$\begin{aligned}
\kappa = \frac{6}{2\alpha + 1}, \quad \lambda = (8-3\kappa)\alpha,\end{aligned}$$ and let $\gamma$ be a chordal ${\textrm{SLE}(\kappa)}$ and ${{\mathcal{L}}_{\lambda}}$ be an independent realization of the Brownian loop soup (in ${\mathbb{H}}$) with intensity parameter $\lambda$. The individual loops in ${{\mathcal{L}}_{\lambda}}$ will be generically denoted by $\eta$, they can be thought of as continuous curves $\eta : [0, t_{\eta}] \to {\mathbb{H}}$ such that $\eta(0) = \eta(t_{\eta})$. Throughout we will use $\gamma$ and $\eta$ to denote the curves as well as their traces, i.e. $\gamma[0, \infty)$ and $\eta[0, t_{\eta}]$, respectively. It will be clear from the context which we are referring to. Let $K$ be the hull generated by the union of $\gamma$ and all the (filled-in) $\eta \in {{\mathcal{L}}_{\lambda}}$ such that $\eta \cap \gamma \neq \emptyset$. Then [@lsw:conformal_restriction] (along with [@lawler_werner:loop_soup]) proves that $K$ is distributed according to ${{\mathbb{P}}_{\alpha}}$.
This construction allows us to identify restriction hulls with pairs $(\gamma, {\mathcal{L}})$, where $\gamma : [0, t_{\gamma}] \to {\mathbb{C}}$ is a continuous, simple curve and ${\mathcal{L}}$ is a set of loops. Furthermore, the curve plus loops structure gives a clean way of time parameterizing the hulls. Letting $K$ be a restriction hull, which we identify with $(\gamma, {\mathcal{L}})$, we define $K_t$ to be the hull generated by $\gamma[0,t]$ plus the union of all filled-in loops $\eta \in {{\mathcal{L}}_{\lambda}}$ such that $\eta \cap \gamma[0,t] \neq \emptyset$. Then $(K_t)_{t \geq 0}$ is a growing family of hulls that increases to $K_{\infty} = K$. It is important for us to have such a time parameterization so that we may properly describe the renewal theory for the restriction hulls, but the particular time parameterization is not especially important since we are mostly interested in the restriction hull as a topological object. We remark that this growing family is not continuous with respect to the time parametrization, since loops are added “all at once”, but again it does not really matter for our purposes (nevertheless, notice that the parameterization is right continuous). The only issue to point out is that the bridge points of a restriction hull will always be a subset of the underlying (simple) curve $\gamma$, and therefore to each bridge point there is a corresponding unique bridge time. Hence the set of bridge times $G$ is a well defined object.
The curve-plus-loops structure also makes it easy to define various operations on hulls. Given two pairs $(\gamma, {\mathcal{L}})$ and $(\gamma^*, {\mathcal{L}}^*)$ with $\gamma(0) = \gamma^*(0) = 0$, their *concatenation* is defined by $$\begin{aligned}
(\gamma, {\mathcal{L}}) \oplus (\gamma^*, {\mathcal{L}}^*) = \left(\gamma \oplus \gamma^*, {\mathcal{L}}\cup (\gamma(t_{\gamma})+{\mathcal{L}}^*)\right),\end{aligned}$$ where $\gamma \oplus \gamma^*$ is the usual concatenation of curves given by $$\begin{aligned}
\left( \gamma \oplus \gamma^* \right) (t) = \left\{
\begin{array}{ll}
\gamma(t), & 0 \leq t \leq t_{\gamma} \\
\gamma^*(t - t_{\gamma}) + \gamma(t_{\gamma}), & t_{\gamma} \leq t \leq t_{\gamma}+t_{\gamma^*}
\end{array}
\right.\end{aligned}$$ We also define a time shift for the hulls. For $t \leq s \leq t_{\gamma}$, define the curve $\gamma^{t,s}$ by $\gamma^{t,s}(t') := \gamma(t+t')$ for $0 \leq t' \leq s-t$, and let $${\mathcal{L}}^{t,s} := \{ \eta \in {\mathcal{L}}: \eta \cap \gamma^{t,s} \neq \emptyset, \eta \cap \gamma[0,t]=\emptyset\}.$$ Then we define $\Lambda_{t,s} K := (\gamma^{t,s}, {\mathcal{L}}^{t,s})$, which is the future hull between times $t$ and $s$, and $\theta_{t,s} K := \Lambda_{t,s} K - \gamma(t)$, which shifts the future hull to start at the origin. If $s = t_{\gamma}$, which usually for us means $s = \infty$, we write $\Lambda_t$ and $\theta_t$ for these operators. In the case that $K$ is an unbounded hull in ${\mathbb{H}}$ and $t$ is a bridge time for $K$, it is easy to see that $\theta_t K$ is also an unbounded hull in ${\mathbb{H}}$. At non-bridge times $\theta_t K$ does not remain in ${\mathbb{H}}$.
Imagine a walker moving along the hull that has discovered $K_t$ at time $t$. The information that is progressively revealed to the walker is encapsulated by the filtration $$\begin{aligned}
{\mathcal{F}}_t := \sigma(K_s; 0 \leq s \leq t).\end{aligned}$$ With respect to this filtration, the following Domain Markov property is true: $$\begin{aligned}
\label{DomainMarkov}
\textrm{The conditional law of } \Lambda_t K, \textrm{ given } {\mathcal{F}}_t, \textrm{ is } {{\mathbb{P}}_{\alpha}}^{({\mathbb{H}}\backslash \gamma[0,t], \gamma(t), \infty)}.\end{aligned}$$ This is similar to the Domain Markov property for regular SLE, where the future curve is an independent ${\textrm{SLE}(\kappa)}$ curve from $\gamma(t)$ to $\infty$ in ${\mathbb{H}}\backslash \gamma[0,t]$, except that in the case of restriction measures one also attaches to the curve the filled-in loops of an independent realization of the Brownian loop soup in the domain ${\mathbb{H}}\backslash \gamma[0,t]$. Note, however, that both the future curve and loops are sampled from the laws corresponding to the domains ${\mathbb{H}}\backslash \gamma[0,t]$, *not* the laws corresponding to ${\mathbb{H}}\backslash K_t$. In short, the future curve and future loops are allowed to intersect the past loops but *not* the past curve $\gamma[0,t]$.
For the domain $({\mathbb{H}}, 0, \infty)$ recall that the restriction measures satisfy the restriction formula : $$\begin{aligned}
{{\mathbb{P}}_{\alpha} \left( K \cap A = \emptyset \right)} = \phi_{A}'(0)^{\alpha},\end{aligned}$$ where $A$ is a hull in ${\mathbb{H}}$ that is a positive distance from zero, and $\phi_A$ is a conformal map from ${\mathbb{H}}\backslash A$ onto ${\mathbb{H}}$ satisfying $\phi_A(z) \sim z$ as $z \to \infty$. In fact, specifying the above probabilities for a sufficiently large class of hulls $A$ (so-called *smooth hulls*) uniquely determines ${{\mathbb{P}}_{\alpha}}$, see [@lsw:conformal_restriction] for a proof of this fact. For general triples $(D, z, w)$, the restriction formula is $$\begin{aligned}
\label{genRestrictionFormula}
{{\mathbb{P}}_{\alpha}}^{(D,z,w)} \left(K \cap A = \emptyset \right) = \phi_{f(A)}'(0)^{\alpha},\end{aligned}$$ where $A$ is a hull in $D$ not containing $z$, and $f$ is a conformal map from $D$ onto ${\mathbb{H}}$ that sends $z$ to $0$ and $w$ to $\infty$.
The restriction formula will be heavily used throughout this paper. For a given hull $A$ there are various techniques from both complex analysis and probability theory that can be used to compute $\phi_A'(0)$. We will exclusively use probabilistic techniques involving Brownian motion; these are described in the next section.
Bridge Lines and Bridge Points \[BridgeSection\]
================================================
The main focus of this section is proving Theorem ef[BridgeDimension]{}. Specifically, we establish the existence of bridge points and lines for restriction hulls with $\alpha < 1$, and also prove the non-existence for $\alpha \geq 1$.
First observe that part of Theorem \[BridgeDimension\] is trivial. The scale invariance of $C$ and $D$ follows immediately from the scale invariance of the restriction hulls (which itself follows from the scale invariance of SLE and of the loop soup). To prove part , first recall that bridge points of a restriction hull are always on the SLE curve itself and never on a loop, and that there is always a unique bridge time corresponding to every bridge point. We refer to the end of the section for the proof.
The most involved proofs are for calculating the Hausdorff dimensions of $C$ and $D$. The computation of the Hausdorff dimensions in Theorem \[BridgeDimension\] follows standard “one-point” and “two-point” arguments, as in, for example, [@alberts_sheff:dimension; @beffara:curvedim; @lawler:cutpoints; @schramm_zhou:dimension]. The idea behind this argument is to approximate $C$ and $D$ by “thickened” sets ${C_{\epsilon}}$ and ${D_{\epsilon}}$, and then obtain estimates on the probability that a given set of points belongs to the thickened sets. A specific bound on the probability that one point belongs to the thickened set gives an upper bound on the Hausdorff dimension, and a similar bound on the probability that two points are in the thickened sets, together with the order of magnitude of the one-point estimate, gives a lower bound on the dimension. We recall the result that we will use in the remainder; throughout this paper we use the notation $f(\epsilon) \asymp g(\epsilon)$ to indicate that there exists constants $C_1$ and $C_2$ independent of $\epsilon$ such that $C_1 g(\epsilon) \leq f(\epsilon) \leq C_2 g(\epsilon)$, for all $\epsilon$ sufficiently small.
\[HausdorffComputation\]Let $H$ be a random subset of ${\mathbb{C}}$ and ${H_{\epsilon}}$ be the set of points at distance less than $\epsilon$ from $H$. Suppose that the two following conditions are fulfilled for some $s \geq 0$ and constant $c > 0$:
- for all $z\in \mathbb{H}$, ${{\mathbb{P}}\left ( z \in {H_{\epsilon}} \right)} \asymp \epsilon^{s}$,
- for all distinct $w,z \in {\mathbb{H}}$, ${{\mathbb{P}}\left ( w,z \in {H_{\epsilon}} \right)}\leq c\epsilon^{s} \wedge c(\epsilon^{2s}/{\left| w-z \right|}^s)$.
Then ${\mathrm{dim_H} \,}H \leq 2-s$ with probability one, and with some strictly positive probability we also have ${\mathrm{dim_H} \,}H \geq 2-s$. If $H$ is a random subset of ${\mathbb{R}}$ then the same conclusion holds with $2-s$ replaced by $1-s$.
Note that Proposition \[HausdorffComputation\] by itself is not enough to conclude that the Hausdorff dimension of $H$ is a constant, since the lower bound only holds on some event of positive probability. In our situation we are able to conclude that the Hausdorff dimension of $C$ and $D$ is constant by using a $0$-$1$ law. The argument that follows uses the Blumenthal $0$-$1$ Law and is modified from [@lawler:cutpoints].
We will prove the result for $C$, a similar argument holds for $D$. For $0 \leq t \leq s$, define $C_t(s) := \{ \textrm{bridge points of } K_s \} \cap K_t$. For a fixed $d > 0$, let $W_t(s) := \{ {\mathrm{dim_H} \,}C_t(s) \geq d \}$. It is enough to show that ${{\mathbb{P}}_{\alpha} \left( W_{\infty}(\infty) \right)} = 0$ or $1$.
First note that for fixed $s$, both the sets $C_t(s)$ and $W_t(s)$ are increasing in $t$, while for fixed $t$ they are decreasing in $s$. Defining $$\begin{aligned}
V_s := \bigcap_{n=1}^{\infty} W_{\frac{1}{n}}(s) = \left \{ {\mathrm{dim_H} \,}C_t(s) \geq d \,\,\, \forall \,\, 0 < t \leq s \right \},\end{aligned}$$ it follows that $V_s$ is also decreasing in $s$. For each element of the event $V_s \backslash V_{\infty}$, there exists a $t_0$ such that $0 < t_0 \leq s$ and for all $0 < t \leq t_0$, $$\begin{aligned}
{\mathrm{dim_H} \,}C_t(\infty) < d \leq {\mathrm{dim_H} \,}C_t(s).\end{aligned}$$ But this can only happen if for every $0 < t \leq t_0$, the future hull $\Lambda_s K$ destroys bridge points of $K_s$ that are in $K_t$, and since this happens for every $0 < t \leq t_0$ and $K_t \to \{ 0 \}$ as $t \to 0$, this forces that the future hull comes arbitrarily close to the real axis. But this is clearly an event of measure zero. Hence for every $s > 0$, ${{\mathbb{P}}_{\alpha} \left( V_s \backslash V_{\infty} \right)} = 0$, from which it immediately follows that $$\begin{aligned}
{{\mathbb{P}}_{\alpha} \left( \bigcap_{n=1}^{\infty} V_{\frac{1}{n}} \right)} = {{\mathbb{P}}_{\alpha} \left( V_{\infty} \right)}.\end{aligned}$$ However, the intersection of the $V_{1/n}$ is ${\mathcal{F}}_{0+}$-measurable, and in the case of ${\textrm{SLE}}(8/3)$ it follows that ${{\mathbb{P}}_{5/8} \left( V_{\infty} \right)} = 0$ or $1$ by the Blumenthal $0$-$1$ Law, since the corresponding measure ${\mathbb{P}}_{5/8}$ is a pushforward of Wiener measure through the Loewner equation. For general $\alpha>5/8$, the same type of Blumenthal $0$-$1$ Law holds via the usual argument. Indeed, the Domain Markov property implies that $\phi_{K_t}(\Lambda_t K)$ is a restriction hull that is independent of ${\mathcal{F}}_t$, hence for $A \in {\mathcal{F}}_{0+}$ and $t > 0$ and any bounded, continuous function $f$ on hulls we have $$\begin{aligned}
{\mathrm{\textbf{E}} \left[ f \left( \phi_{K_t}(\Lambda_t K) \right) \mathbf{1}_A \right]} = {\mathrm{\textbf{E}} \left[ f \left( \phi_{K_t}(\Lambda_t K) \right) \right]} {{\mathbb{P}}_{\alpha} \left( A \right)}\end{aligned}$$ Taking a limit of both sides as $t \downarrow 0$ and using the fact that $f$ is continuous and $\phi_{K_t}$ goes continuously to the identity we get that $$\begin{aligned}
{\mathrm{\textbf{E}} \left[ f(K) \mathbf{1}_A \right]} = {\mathrm{\textbf{E}} \left[ f(K) \right]} {{\mathbb{P}}_{\alpha} \left( A \right)},\end{aligned}$$ which shows that $A$ is independent of all elements of ${\mathcal{F}}_{\infty}$, and therefore of itself.
We now use Proposition \[HausdorffComputation\] to prove part of Theorem \[BridgeDimension\]. We use the following events to define our thickened sets.
For $z \in {\mathbb{H}}$ and $\epsilon > 0$, let $I(z, \epsilon)$ be the horizontal line $y = {\textrm{Im}}\, z$ with the gap of width $2 \epsilon$ centered around $z$ removed. That is $$\begin{aligned}
I(z, \epsilon) := \left \{ w \in {\mathbb{H}}: {\textrm{Im}}\, w = {\textrm{Im}}\, z, \, \left| {\textrm{Re}}(w-z) \right| \geq \epsilon \right \}.\end{aligned}$$ Define the sets ${C_{\epsilon}}$ and ${D_{\epsilon}}$ by $$\begin{aligned}
{C_{\epsilon}} &:= \left \{ z \in {\mathbb{H}}: {I(z, \epsilon) \cap K = \emptyset}\right \}, \quad
{D_{\epsilon}} := \left \{ L > 0 : {I(n \epsilon + iL, \epsilon) \cap K = \emptyset} \textrm{ for some } n \in {\mathbb{Z}}\right \}.\end{aligned}$$
![The dotted point is $z$ and the two horizontal lines on either side form the set $I(z, \epsilon)$. This figure depicts the event that an SLE(8/3) avoids the hull $I(z, \epsilon)$.[]{data-label="compute-fig"}](bridge_compute){width="15cm" height="13cm"}
With the definitions above, the following is true ${{\mathbb{P}}_{\alpha}}$-a.s.: $$\begin{aligned}
C = \bigcap_{\epsilon > 0} {C_{\epsilon}}, \quad D = \bigcap_{\epsilon > 0} {D_{\epsilon}}.\end{aligned}$$
Recall that $C$ consists of $z \in {\mathbb{H}}$ for which $K \cap \{ y = {\textrm{Im}}\, z\} = \{ z \}$. Hence if $z \in C$ then $z \in {C_{\epsilon}}$ for all $\epsilon > 0$. To prove the converse, note that if $z \in {C_{\epsilon}}$ for every $\epsilon > 0$ then $z$ is the only possible element in the set $K \cap \{ y = {\textrm{Im}}\, z \}$. But the latter set is always non-empty, since restriction hulls are connected and their vertical component goes from zero to infinity (${{\mathbb{P}}_{\alpha}}$-a.s.), and therefore with ${{\mathbb{P}}_{\alpha}}$-probability $1$ the set $K \cap \{ y = L \}$ is non-empty for all $L > 0$. The proof for $D$ is exactly the same.
The restriction formula makes it easy to compute the probability that a point $z \in {\mathbb{H}}$ is in ${C_{\epsilon}}$. Indeed, by formula we have $$\begin{aligned}
{{\mathbb{P}}_{\alpha} \left( z \in {C_{\epsilon}} \right)} = {{\mathbb{P}}_{\alpha} \left( {I(z, \epsilon) \cap K = \emptyset}\right)} = {\phi_{I(z, \epsilon)}}'(0)^{\alpha},\end{aligned}$$ where ${\phi_{I(z, \epsilon)}}$ is a conformal map from ${\mathbb{H}}\backslash I(z, \epsilon)$ onto ${\mathbb{H}}$ such that ${\phi_{I(z, \epsilon)}}(w) \sim w$ as $w \to \infty$. Similarly, $$\begin{aligned}
{{\mathbb{P}}_{\alpha} \left( w,z \in {C_{\epsilon}} \right)} = \phi_{I(w,\epsilon) \cup I(z, \epsilon)}'(0)^{\alpha}.\end{aligned}$$ By Proposition \[HausdorffComputation\], the Hausdorff computation for $C$ and $D$ therefore comes down to an estimate of the derivative of these conformal maps at zero. We list three possible methods for these estimates. One deals only with conformal maps and is entirely analytic. The others use probabilitic techniques. We recall the analytic method but do not enter into details.
$ $\
**Analytic Method:** While it is not possible to write down ${\phi_{I(z, \epsilon)}}$ explicitly, one can write down the general form of its inverse. Let $$\begin{aligned}
{f_{z, \epsilon}}(w) := \lambda w + \frac{{\textrm{Im}}\, z}{\pi} \left( \log(w-a) - \log(w-b) + \pi i \right),\end{aligned}$$ where the imaginary part of the logarithm is zero along the positive real axis and $\pi$ on the negative real axis. For appropriate choices of real constants $\lambda, a$, and $b$ (with $a < b$, $\lambda > 0$), ${f_{z, \epsilon}}$ maps ${\mathbb{H}}$ onto ${\mathbb{H}}\backslash I(z, \epsilon)$. These constants implicitly depend on $z$ and $\epsilon$, although it is difficult to give closed-form expressions for them. Close analysis of the asymptotic behavior of $\lambda, a$, and $b$ could be used to get estimates on ${\phi_{I(z, \epsilon)}}'(0)$ as $\epsilon \downarrow 0$, but we will mostly avoid this strategy. We will, however, mention that $a$ and $b$ are determined mostly by $z$, while $\lambda$ is proportional to $\epsilon^{-2}$.
$ $\
**Brownian Excursion Method:** The first probabilistic method uses a well-known formula, due to Bálint Virág [@virag:beads], for *Brownian excursions* in the upper half plane. Recall that a Brownian excursion in $\mathbb{H}$ can be thought of as a Brownian motion that is started at zero and conditioned to have a positive imaginary part at all later times. Such excursions can be realized by a random path whose horizontal component is a one-dimensional Brownian motion and whose vertical component is an independent Bessel-$3$ process.
[([@virag:beads])]{}\[BELemma\] Let $A$ be a compact hull in the upper half plane such that ${\mathbb{H}}\backslash A$ is simply connected and ${\operatorname{dist}}(0, A) > 0$, and $\phi_A$ be a conformal map from ${\mathbb{H}}\backslash A$ into ${\mathbb{H}}$ such that $\phi_A(0) = 0$ and $\phi_A(z) \sim z$ as $z \to \infty$. If $BE$ denotes the path of a Brownian excursion in ${\mathbb{H}}$ from $0$ to $\infty$, then $$\begin{aligned}
\phi_A'(0) = {{\mathbb{P}}\left ( BE \textrm{ does not intersect } A \right)}.\end{aligned}$$
In particular, this lemma shows that the filling in of a Brownian excursion has the law of a restriction measure with index $1$. It can also be used to get the estimates of Proposition \[HausdorffComputation\], but we prefer the following method that produces asymptotic results (even if they are not necessary in our setting).
$ $\
**Brownian Motion Method:** Instead of using Brownian excursions to compute $\phi_A'(0)$, one can use Brownian motion directly. Oftentimes this is easier as it doesn’t require dealing with the conditioning. In an appropriate sense, $\phi_A'(0)$ is the exit density at zero (with respect to Lebesgue measure) of a Brownian motion in ${\mathbb{H}}\backslash A$, starting from $\infty$. This is also called the *excursion Poisson kernel* as seen from $\infty$. In what follows we let $B$ be a complex Brownian motion.
Given a simply connected domain $D$ with $z \in D$, $w \in \partial D$, let $H_D(z,w)$ denote the Poisson kernel. In the case $D = {\mathbb{H}}\backslash A$, we will often be interested in the “Poisson kernel as seen from infinity”, for which we introduce the notation $$\begin{aligned}
H_{{\mathbb{H}}\backslash A}(\infty, w) := \lim_{L \uparrow \infty} L H_{{\mathbb{H}}\backslash A}(iL, w).\end{aligned}$$
The following estimates will be useful when using Lemma \[BMLemma\] to estimate $\phi_A'(0)$. For $x > 0$, $H_{{\mathbb{H}}}(z, x) = \frac{1}{\pi} {\textrm{Im}}(z)/|z-x|^2$ and consequently $H_{{\mathbb{H}}}(\infty, x) = \frac{1}{\pi}$. Recall that under a conformal map $f : D \to D'$, $H_D(z,w)$ changes according to the scaling rule $H_D(z,w) = |f'(w)| H_{f(D)}(f(z), f(w)).$ In particular, we have the scaling rule $H_{{\mathbb{H}}\backslash A}(\infty, w) = H_{{\mathbb{H}}\backslash rA}(\infty, rw).$
The next lemma outlines how to use Brownian motion directly to estimate $\phi_A'(0)$. The method of proof is virtually identical to the one for Lemma \[BELemma\], so we refer the reader to [@virag:beads] for details.
\[BMLemma\] For a complex Brownian motion and a compact hull $A$ in the upper half-plane such that ${\mathbb{H}}\backslash A$ is simply connected and ${\operatorname{dist}}(0,A) > 0$, $$\begin{aligned}
\phi_A'(0) = H_{{\mathbb{H}}\backslash A}(\infty, 0).\end{aligned}$$
The computation of ${\phi_{I(z, \epsilon)}}'(0)$ is thus reduced to some estimates on the exit density of a Brownian motion in the domain ${\mathbb{H}}\backslash I(z, \epsilon)$. In order to simplify the computations, we first estimate exit densities for an intermediate set ${S_{\epsilon}}$.
\[BMStripLemma\] Let ${S_{\epsilon}} = {\mathbb{R}}\times [0, 2i] \backslash I(i, \epsilon)$. Then for $x \in {\mathbb{R}}$ and $\lambda \in [-1,1]$,
$$\begin{aligned}
\label{StripPoissonKernel}
H_{{S_{\epsilon}}}(\lambda \epsilon + i, x) \sim \frac{\pi \sqrt{1 - \lambda^2}}{ 8 \cosh^2 ( \pi x/2) } \epsilon\end{aligned}$$
as $\epsilon \downarrow 0$, where “$\sim$" means that the ratio of the two terms converges to $1$ uniformly with respect to $x$ and $\lambda$. In particular, the probability that the Brownian motion started at $i$ exits $S_{\epsilon}$ on ${\mathbb{R}}$ is of order $\epsilon$.
Let ${z_{\epsilon}} = \lambda \epsilon + i$. In this case, it is easy to find an explicit conformal map from ${S_{\epsilon}}$ onto ${\mathbb{H}}$. A simple one is given by $$\begin{aligned}
{f_{\epsilon}}(z) = \left(\frac{e^{\pi z}+e^{\pi \epsilon}}{e^{\pi z}+e^{-\pi \epsilon}}\right)^{1/2}.\end{aligned}$$ By the scaling rule for the Poisson kernel $$\begin{aligned}
H_{{S_{\epsilon}}}({z_{\epsilon}}, x) &= |{f_{\epsilon}}'(x)| H_{{\mathbb{H}}}( {f_{\epsilon}}({z_{\epsilon}}), {f_{\epsilon}}(x)) = \frac{|{f_{\epsilon}}'(x)|}{\pi} \frac{{\textrm{Im}}({f_{\epsilon}}({z_{\epsilon}}))}{|{f_{\epsilon}}({z_{\epsilon}}) - {f_{\epsilon}}(x)|^2}.\end{aligned}$$ It is straightforward to verify that $$\begin{aligned}
{f_{\epsilon}}(x) &\sim 1,\end{aligned}$$ as $\epsilon \downarrow 0$, and $$\begin{aligned}
{\left| {f_{\epsilon}}'(x) \right|} &= \frac{1}{2 f_{\epsilon}(x)} \frac{2\pi e^{\pi x} \sinh(\pi \epsilon)}{(e^{\pi x} + e^{-\pi \epsilon})^2} \\
&\sim \frac{\pi^2 \epsilon}{4 \cosh^2(\pi x/2)}\end{aligned}$$ Similarly $$\begin{aligned}
{f_{\epsilon}}({z_{\epsilon}}) &= \left(\frac{e^{\pi \epsilon}-e^{\pi \lambda \epsilon}}{e^{-\pi \epsilon}-e^{\pi \lambda \epsilon}}\right)^{1/2}\\
& \sim \left(\frac{1-\lambda}{-1-\lambda}\right)^{1/2}\\
&= i \left(\frac{1-\lambda}{1+\lambda}\right)^{1/2}.\end{aligned}$$ Assembling the pieces proves , and then integrating over $x$ proves the last statement.
\[BMStripLemma2\] Let $x \in {\mathbb{R}}$ and $\lambda \in [-1, 1]$. Then $$\begin{aligned}
H_{{\mathbb{H}}\backslash I(i, \epsilon)}(\lambda \epsilon + i, x) \sim H_{S_{\epsilon}} (\lambda \epsilon + i, x)\end{aligned}$$ as $\epsilon \downarrow 0$.
If a Brownian motion started at $\lambda \epsilon + i$ exits ${S_{\epsilon}}$ at $x$, then it also exits ${\mathbb{H}}\backslash I(i, \epsilon)$ at $x$. Consequently, the Poisson kernel on the left hand side is bigger than the one on the right. They are not the same because the Brownian motion in ${\mathbb{H}}\backslash I(i, \epsilon)$ can hit the line $y = 2i$ before hitting zero, which the Brownian motion in ${S_{\epsilon}}$ is not allowed to do. Asymptotically this event contributes nothing; indeed there is only an $O(\epsilon)$ chance that the Brownian motion even makes it up to $y = 2i$, and then another $O(\epsilon)$ chance that it passes back through the gap. Overall this makes the event of order $\epsilon^2$ (uniformly in $x$ and $\lambda$), which, by Lemma \[BMStripLemma\], is negligible compared to $H_{S_{\epsilon}}(\lambda \epsilon + i, x)$.
\[1PointBound\] For $z = y(x+i) \in {\mathbb{H}}$, $$\begin{aligned}
{\phi_{I(z, \epsilon)}}'(0) \sim U(z) \epsilon^2\end{aligned}$$ as $\epsilon \downarrow 0$, where $$\begin{aligned}
U(y(x+i)) = \frac{\pi}{16 y^2 \cosh^2(\pi x/2)}.\end{aligned}$$
It suffices to prove the result in the case $z = x + i$, for the general form use the scaling rule. We use Brownian motion coming down from infinity as in Lemma \[BMLemma\]. In order to reach $0$, the Brownian motion coming down from infinity must first pass through the gap of width $2 \epsilon$ centered at $z$, and then from the gap it must transition to zero while avoiding $I(z, \epsilon)$. The two events are independent by the Strong Markov property, and each one is $O(\epsilon)$. More precisely, by Lemmas \[BMStripLemma\] and \[BMStripLemma2\], $$\begin{aligned}
{\phi_{I(z, \epsilon)}}'(0) &= H_{{\mathbb{H}}\backslash I(x+i, \epsilon)}(\infty, 0) \\
&= \int_{[-\epsilon,\epsilon]} H_{{\mathbb{H}}}(\infty, x+y)H_{{\mathbb{H}}\backslash I(x+i, \epsilon)}(x+y+i, 0) \, dy \\
&= \int_{-\epsilon}^{\epsilon} \frac{1}{\pi} H_{{\mathbb{H}}\backslash I(i, \epsilon)}(y+i, -x) \, dy \\
&= \frac{\epsilon}{\pi} \int_{-1}^1 H_{{\mathbb{H}}\backslash I(i, \epsilon)}(\lambda \epsilon + i, -x) \, d \lambda \\
&\sim \frac{\epsilon^2}{8 \cosh^2(\pi x/2)} \int_{-1}^1 \sqrt{1-\lambda^2} \, d \lambda .\end{aligned}$$
From Proposition \[1PointBound\] and the restriction formula, it is easy to derive the probability that a bridge point is within distance $\epsilon$ of a given point $z$ decays like $\epsilon^{2\alpha}$. From this the first part of Proposition \[HausdorffComputation\] follows easily, but we need a last proposition in order to derive the two point estimate.
\[2PointBound\] Let $z, w \in {\mathbb{H}}$, with ${\textrm{Im}}(z) > {\textrm{Im}}(w)$, and $\epsilon_z, \epsilon_w > 0$. Let $A = I(z, \epsilon_z) \cup I(w, \epsilon_w)$. Then $$\begin{aligned}
\phi_A'(0) \asymp U(z-w)U(w) \epsilon_z^2 \epsilon_w^2 ,\end{aligned}$$ as $\epsilon_z, \epsilon_w \downarrow 0$.
The argument is virtually the same as for the one-point estimate in Proposition \[1PointBound\], the only difference being that the Brownian motion, after passing through the first gap at $z$ then has to pass through a second gap at $w$. The probability of the latter event can be estimated using Proposition \[1PointBound\]; indeed, after temporarily shifting $w$ to zero, there is a $U(z-w) \epsilon_z^2 \epsilon_w$ chance that the Brownian motion hits in an $\epsilon_w$ neighbourhood of $w$ (and therefore also the second gap). With some positive probability it hits in the middle of the second gap, where the probability of moving to zero is, up to a constant, given by $U(w) \epsilon_w$. These two probabilities multiply since, by the Strong Markov property, the path before the second gap is independent of the path after the second gap.
By carefully decomposing the path according to the points it passes through in the gaps and then integrating, the statement of Proposition \[2PointBound\] could be strengthened to an asymptotic result rather than just up to constants. For our purposes, however, this is not required.
Propositions \[1PointBound\] and \[2PointBound\] combine with Proposition \[HausdorffComputation\] to prove the result for $C$.
For $D$, the key observation is that if two gaps on a horizontal line do not overlap, then the curve can only avoid the line by going through one of them. Consequently, for $n \neq m$, the events ${I(n\epsilon+iL, \epsilon/2) \cap K = \emptyset}$ and ${I(m\epsilon+iL, \epsilon/2) \cap K = \emptyset}$ are disjoint, and therefore $$\begin{aligned}
{{\mathbb{P}}_{\alpha} \left( L \in {D_{\epsilon}} \right)} &= {{\mathbb{P}}_{\alpha} \left( \bigcup_{n \in {\mathbb{Z}}} \left \{ {I(n\epsilon + iL, \epsilon/2) \cap K = \emptyset} \right \} \right)} \\
& = \sum_{n \in {\mathbb{Z}}} {{\mathbb{P}}_{\alpha} \left( {I(n\epsilon+iL, \epsilon/2) \cap K = \emptyset} \right)} \\
& \sim \frac{1}{L^{2\alpha}} \sum_{n \in {\mathbb{Z}}} U \left( \frac{n \epsilon}{L} + i \right)^{\alpha} \left( \frac{\epsilon^{2\alpha}}{4^{\alpha}} \right)\\
& \sim \frac{\epsilon^{2\alpha-1}}{4^{\alpha} L^{2\alpha-1}} \int_{{\mathbb{R}}} U(x + iL)^{\alpha} \, dx \\
& \sim \frac{\pi^{\alpha} \epsilon^{2 \alpha - 1}}{32^{\alpha} L^{2 \alpha - 1}} \int_{{\mathbb{R}}} \cosh^{-2\alpha} \left( \pi x/2 \right) \, dx.\end{aligned}$$ The transition from sum to integral is a Riemann sum approximation. By $2 \alpha > 1$, the integral is a finite constant depending only on $\alpha$. This gives the one-point estimate for $D$.
Similarly, for $0 < L < L'$, $$\begin{aligned}
{{\mathbb{P}}_{\alpha} \left( L, L' \in {D_{\epsilon}} \right)} &= {{\mathbb{P}}_{\alpha} \left( \bigcup_{m,n \in {\mathbb{Z}}} \left \{ n\epsilon + iL, m\epsilon + iL' \in C_{\epsilon/2} \right \} \right)} \\
&= \sum_{m,n \in {\mathbb{Z}}} {{\mathbb{P}}_{\alpha} \left( n \epsilon + iL, m \epsilon + iL' \in C_{\epsilon/2} \right)} \\
&\asymp \sum_{m,n \in {\mathbb{Z}}} \epsilon^{4 \alpha} U \left[ (m-n)\epsilon + i(L'- L) \right]^{\alpha} U(n\epsilon + iL)^{\alpha} \\
&\asymp \epsilon^{4 \alpha - 2} \int_{{\mathbb{R}}} U(x + i(L'-L))^{\alpha} \, dx \int_{{\mathbb{R}}} U(x + iL)^{\alpha} \, dx \\
&\asymp \frac{\epsilon^{4 \alpha - 2}}{L^{2 \alpha - 1} (L'- L)^{2 \alpha - 1}}\end{aligned}$$ We use the same transition from sum to integral as in the one-point bound. Proposition \[HausdorffComputation\] now completes the proof.
We show that $C$ and $D$ are almost surely empty for $\alpha \geq 1$. For $\alpha<1$, the Haussdorff dimension is strictly positive and the set is non empty.
For $\alpha=1$, recall that the imaginary part of a Brownian excursion is a Bessel(3) process, and a bridge height for the hull necessarily corresponds to a point of increase for the Bessel(3) process. However, it is well known that Bessel(3) has no point of increase since, for example, a Bessel(3) process reversed from its last passage time of a level has the same law as a Brownian motion up to its first hitting time of zero, and Brownian motion is known to have no points of increase (see [@revuz_yor] for details of both facts).
For $\alpha > 1$ consider the rectangle $R = [-1,1] \times [1/2, 1]$. Cover it with $2^{2n}$ squares each of side length $2^{-n}$, and let $\{S_i\}_{1 \leq i \leq 2^{2n}}$ be the boxes and $z_i$ be their centers. Then, by Proposition \[1PointBound\], the expected number of squares containing a bridge point decays exponentially fast since $$\begin{aligned}
\mathbb{E}_{\alpha} \left[ \sum_{i=1}^{2^{2n}} {\mathbf{1} \left \{ C \cap S_i \neq \emptyset \right \}} \right] & \asymp \sum_{i=1}^{2^{2n}} {{\mathbb{P}}_{\alpha} \left( I(z_i, 2^{-n}) \cap K \neq \emptyset \right)} \\
& \asymp \sum_{i=1}^{2^{2n}} U(z_i) (2^{-n})^{2\alpha} \\
&= 2^{(2-2\alpha)n} 2^{-2n} \sum_{i=1}^{2^{2n}} U(z_i) \\
& \leq C 2^{(2-2\alpha)n},\end{aligned}$$ for some constant $C > 0$. The last inequality is a simple consequence of the fact that $U$ is Riemann integrable and hence $$\begin{aligned}
2^{-2n} \sum_{i=1}^{2^{2n}} U(z_i) \to \int_{R} U(z) \, dA(z) < \infty,\end{aligned}$$ where $dA(z)$ is two-dimensional Lebesgue measure. The Borel-Cantelli lemma then proves that $R$ almost surely contains no bridge points. By scale invariance any scaled version of $R$ also contains no bridge points. Translates of $R$ in the horizontal direction also contain no bridge points, since clearly the expected number of bridge points in translates of $R$ decreases as the rectangle is moved away from the imaginary axis. Finally, since the entire half-plane can be covered with countably many scaled and translated versions of $R$, the entire plane must almost surely be free of bridge points.
We end this section with the proof of part of Theorem \[BridgeDimension\]. The lack of isolated points in $C$ and $D$ is also a consequence of the renewal property of restriction hulls at bridge points, so we defer the proof of this fact until the end of Section \[RenewalSection\].
We prove the result for $D$; the proof for $C$ is similar. To prove that $D$ is closed, suppose that $L$ is a limit point of $D$. Without loss of generality we may assume that the limiting sequence of bridge heights $L_n$ that converges to $L$ is strictly increasing. If $t$ is the bridge time corresponding to $L$, then the restriction hull after time $t$ must reside in the domain ${{\textrm{Im}}\, z \geq L}$ (since each $L_n$ is a bridge height). Then $L$ is not in $D$ if and only if the future hull touches the line ${\textrm{Im}}\, z = L$ but does not cross it, which is clearly an event of probability zero. Indeed, for two points $z$ and $w$ on the same horizontal line let us define $A(z, \epsilon_z, w, \epsilon_w)$ to be the event that the hull goes through the balls $B(z,\epsilon_z)$ and $B(w,\epsilon_w)$ while avoiding $I(z,\epsilon_z)\cap I(w,\epsilon_w)$. The estimates of Proposition \[2PointBound\] can be used to show that the probability of $A(z,\epsilon_z,w,\epsilon_w)$ is of order $\epsilon_z^{2\alpha} \epsilon_w^{2\alpha}$, which easily implies the result since $\alpha > 1/2$.
Renewal at Bridge Lines \[RenewalSection\]
==========================================
In this section we show that the restriction hulls renew themselves at bridge heights. Most of the section is technical, so first we would like to give the intuition behind the renewal property. It is almost entirely a consequence of restriction. Suppose that $K$ is a restriction hull with the law ${{\mathbb{P}}_{\alpha}}$. Given ${\mathcal{F}}_t$, the Domain Markov property says that the future hull has the restriction law corresponding to the domain $({\mathbb{H}}\backslash \gamma[0,t], \gamma(t), \infty)$. But if we also know that $t$ is a bridge time, then the future hull is separated from the past by the bridge line that the hull is currently at. The future hull is therefore conditioned not to go below this bridge line, and this conditioning is, by the restriction property, “equivalent” to sampling the future hull from the restriction measure corresponding to the half plane above the bridge line. Shifting the bridge point back to the origin, this means that the shifted future hull $\theta_t K$ also obeys the law ${{\mathbb{P}}_{\alpha}}$ and is independent of ${\mathcal{F}}_t$.
There are two main technical obstacles to this intuition. The first is that the event that $t$ is a bridge time for $K$ is not measurable with respect to ${\mathcal{F}}_t$, since the set of bridge times is a function of the entire hull. To address this problem and still have a meaningful notion of renewal, we simply expand our filtration to a larger one ${\mathcal{G}}_t$ that tells us which bridge heights of $K_t$ are also bridge heights of $K$. The second and more problematic technicality is that $t$ being a bridge time is an event of measure zero, and so conditioning on it requires some care. Theorem \[DecompMarkov\] deals with this latter problem by showing that the restriction hulls obey a certain Domain Markov property with respect to ${\mathcal{G}}_t$, and from this concludes that they refresh themselves at ${\mathcal{G}}_t$-stopping times $\tau$ such that ${{\mathbb{P}}_{\alpha} \left( \tau \in G \right)} = 1$ (recall that $G$ is the set of bridge times).
We make the following definitions:
For $t \geq 0$, let $D_t$ be the set of bridge heights of $K_t$. Note that $D_t$ is ${\mathcal{F}}_t$-measurable and $D_{\infty} = D$. Observe that $D_t \cap D$ is the set of bridge heights of $K_t$ that are also bridge heights of $K$, and $D_t \backslash D$ is the set of bridge heights of $K_t$ that are *not* bridge heights of $K$. We also define $$\begin{aligned}
L_t := \sup D_t \cap D, \quad L_t' := \inf D_t \backslash D.\end{aligned}$$ Note that neither of these quantities, nor $D_t \cap D$ or $D_t \backslash D$, are ${\mathcal{F}}_t$-measurable. However, they are measurable with respect to the enlarged filtration $$\begin{aligned}
{\mathcal{G}}_t := \sigma \left(K_s, D_s \cap D; 0 \leq s \leq t \right).\end{aligned}$$ Clearly ${\mathcal{F}}_t \subset {\mathcal{G}}_t$, and in this larger filtration the bridge lines (and points, and times) of $K$ that belong to $K_t$ are measurable objects.
Notice that $D_t \cap D$ is almost surely closed, and therefore $L_t$ is actually a maximum rather than a supremum (i.e. $L_t \in D_t \cap D$). Hence $L_t$ is the largest bridge height of $K_t$ that is also a bridge height of $K$. Clearly $L_t \leq L_t'$. The next result follows easily from these definitions.
\[AlgebraEquality\] The $\sigma$-algebra ${{\mathcal{G}}_t}$ is generated by $K_t$ and $L_t$, i.e. $$\begin{aligned}
{\mathcal{G}}_t = \sigma \left({\mathcal{F}}_t, L_t \right).\end{aligned}$$
Clearly $\sigma \left({\mathcal{F}}_t, L_t \right) \subset {\mathcal{G}}_t$, since $L_t$ is determined by $D_t \backslash D$. For the other direction, it is clear that $D_t \cap D = \{ L \in D_t : L \leq L_t \}$. Hence $D_t \cap D$ is determined by both $D_t$ (which is itself determined by $K_t$) and $L_t$. This is sufficient because for $s < t$ we have $D_s \cap D \subset D_t \cap D$, and hence $D_s \cap D$ is the intersection of $D_s$, which is ${\mathcal{F}}_s$-measurable, and $D_t \cap D$, which we have just shown is $\sigma \left( {\mathcal{F}}_t, L_t \right)$-measurable.
\[RightIsolated\] For a fixed $t > 0$, $L_t < L_t'$ with probability one.
First observe that $t$ is almost surely not a bridge time. It is easy to see that the distance between $\gamma[t,\infty)$ and the last bridge line $\text{Im}(z)=L_t$ is strictly positive (for instance, there must exist another bridge height higher than $L_t$, and between, it is a continuous compact curve). But a bridge height for $\gamma[0,t)$ that is not a bridge height for the whole curve must be greater than $\inf \text{Im}(\gamma[t,\infty))$. We deduce that $L'_t$ is strictly greater than $L_t$.
Given a subset $K$ of ${\mathbb{C}}$, define $J(K) := \inf \left \{ {\textrm{Im}}\, z : z \in K \right \}$.
With this definition in hand we state the paper’s main technical theorem.
\[DecompMarkov\] Suppose $K = (\gamma, {\mathcal{L}})$ obeys the law ${{\mathbb{P}}_{\alpha}}$, and let $\tau$ be a ${{\mathcal{G}}_t}$-stopping time. On the event that $\tau$ is a bridge time the $\mathcal{G}_{\tau}$-conditional law of $\theta_{\tau}K$ is simply the law of a restriction hull in ${\mathbb{H}}$. If $\tau$ is not a bridge time then the conditional law of $\Lambda_{\tau}K$, given ${\mathcal{G}}_{\tau}$, is the same as the law of a restriction hull $K'$ in ${\mathbb{H}}\backslash \gamma[0,\tau]$ whose distribution is the restriction measure corresponding to the triple $({\mathbb{H}}\backslash \gamma[0,\tau], \gamma(\tau), \infty)$, but further conditioned on the event $L_{\tau} < J(K') \leq L_{\tau}'$.
Note that if $\tau$ is a bridge time then $L_{\tau} = {\textrm{Im}}\gamma(\tau)$ and $L_{\tau'} = \infty$. In this situation the notation $L_{\tau} < J(K') < L_{\tau'}$ can be interpreted as meaning that the future hull lies strictly above the bridge line, which is an event of measure zero. To fully emphasize this very important point we have handled this case with a separate statement at the beginning of the theorem.
Theorem \[DecompMarkov\] should be seen as the extension of the Domain Markov property to the enlarged filtration ${\mathcal{G}}_t$. In words, it simply says that the extra information in ${\mathcal{G}}_{\tau}$ forces the future restriction hull to go below the horizontal line $y = L_{\tau}'$ but stay above the horizontal line $y = L_{\tau}$. This extra conditioning stops $L_{\tau}'$ from being a bridge height for $K$ but preserves $L_{\tau}$ as a bridge height. A detailed proof of the theorem follows. It uses a standard procedure, which we modified from [@virag:beads], to bootstrap from the easy case of $\tau$ being a deterministic time to the general case that $\tau$ is a stopping time.
To simplify notation, we will write $$\begin{aligned}
{{\mathbb{P}}_{\alpha}}^t := {{\mathbb{P}}_{\alpha}}^{({\mathbb{H}}\backslash \gamma[0,t], \gamma(t), \infty)} \left( \, \cdot \, \left| L_{t} < J(K') \leq L_{t}' \right. \right)\end{aligned}$$ throughout this proof. The goal of the proof is to show that the ${\mathcal{G}}_{\tau}$-conditional law of $\Lambda_{\tau} K$ is ${{\mathbb{P}}_{\alpha}}^{\tau}$.
Consider first the case that $\tau$ is a deterministic time $t$. Recall that conditioning on ${{\mathcal{G}}_t}$ is the same as conditioning on ${\mathcal{F}}_t$ and $L_t$, by Proposition \[AlgebraEquality\]. Conditional on ${\mathcal{F}}_t$, the Domain Markov property says that $\Lambda_t K$ has the restriction law for the triple $({\mathbb{H}}\backslash \gamma[0,t], \gamma(t), \infty)$. Conditioning again on $L_t$ forces the future hull to stay above $y = L_t$ but to go below $y = L_t'$, and since $L_t < L_t'$ with positive probability this conditioning is well-defined. Hence the law conditioned on ${\mathcal{G}}_t$ is exactly ${{\mathbb{P}}_{\alpha}}^t$.
Another way of stating the above is as follows: let $X$ be a bounded, continuous[^3] function on hulls. Then $$\begin{aligned}
\label{MarkovEquiv}
{\mathrm{\textbf{E}}_{\alpha}^{} \left[ \left. X(\Lambda_t K) \right| {{\mathcal{G}}_t}\right]} = {\mathrm{\textbf{E}}_{\alpha}^{t} \left[ X \right]},\end{aligned}$$ where ${\mathrm{\textbf{E}}}_{\alpha}$ and ${\mathrm{\textbf{E}}}_{\alpha}^t$ denote expectations with respect to ${{\mathbb{P}}_{\alpha}}$ and ${{\mathbb{P}}_{\alpha}}^t$, respectively. To finish the proof we need to extend to ${{\mathcal{G}}_t}$-stopping times instead of just fixed times. First suppose that $\tau$ only takes values in some countable set $\mathcal{T}$. Then $$\begin{aligned}
{\mathrm{\textbf{E}}_{\alpha}^{} \left[ \left. X(\Lambda_{\tau} K) \right| \mathcal{G}_{\tau} \right]} &= \sum_{t \in \mathcal{T}} {\mathrm{\textbf{E}}_{\alpha}^{} \left[ \left. X(\Lambda_{\tau} K) {\mathbf{1} \left \{ \tau=t \right \}} \right| \mathcal{G}_{\tau} \right]} \\
&= \sum_{t \in \mathcal{T}} {\mathrm{\textbf{E}}_{\alpha}^{} \left[ \left. X(\Lambda_t K) {\mathbf{1} \left \{ \tau = t \right \}} \right| {{\mathcal{G}}_t}\right]} \\
&= \sum_{t \in \mathcal{T}} {\mathbf{1} \left \{ \tau = t \right \}} {\mathrm{\textbf{E}}_{\alpha}^{} \left[ \left. X(\Lambda_t K) \right| {{\mathcal{G}}_t}\right]} \\
&= \sum_{t \in \mathcal{T}} {\mathbf{1} \left \{ \tau = t \right \}} {\mathrm{\textbf{E}}_{\alpha}^{t} \left[ X \right]} \\
&= {\mathrm{\textbf{E}}_{\alpha}^{\tau} \left[ X \right]}.\end{aligned}$$ From this we can bootstrap up to the case of general $\tau$. Let $\tau_n$ be the smallest element of $2^{-n} {\mathbb{N}}$ that is greater than or equal to $\tau$. Then the last argument applies to $\tau_n$, so that $$\begin{aligned}
\label{DiscreteX}
{\mathrm{\textbf{E}}_{\alpha}^{} \left[ \left. X(\Lambda_{\tau_n} K) \right| \mathcal{G}_{\tau_n} \right]} = {\mathrm{\textbf{E}}_{\alpha}^{\tau_n} \left[ X \right]}.\end{aligned}$$ However, since $\tau_n$ is determined at time $\tau$ (i.e. $\tau_n$ is ${\mathcal{G}}_{\tau}$-measurable), $$\begin{aligned}
{\mathrm{\textbf{E}}_{\alpha}^{} \left[ \left. X(\Lambda_{\tau_n} K) \right| \mathcal{G}_{\tau_n} \right]} = {\mathrm{\textbf{E}}_{\alpha}^{} \left[ \left. X(\Lambda_{\tau_n} K) \right| \mathcal{G}_{\tau} \right]}.\end{aligned}$$ Since $\Lambda_{\tau_n} K \to \Lambda_{\tau} K$ as $n \to \infty$, and $X$ is bounded and continuous, it follows that the left hand side of converges to $$\begin{aligned}
{\mathrm{\textbf{E}}_{\alpha}^{} \left[ \left. X(\Lambda_{\tau} K) \right| \mathcal{G}_{\tau} \right]}.\end{aligned}$$ Hence, if we can show that ${\mathrm{\textbf{E}}_{\alpha}^{\tau_n} \left[ X \right]}$ converges to ${\mathrm{\textbf{E}}_{\alpha}^{\tau} \left[ X \right]}$ then we are done. Since $X$ is bounded and continuous, this is equivalent to showing that almost surely the law ${{\mathbb{P}}_{\alpha}}^{\tau_n}$ converges weakly to ${{\mathbb{P}}_{\alpha}}^{\tau}$, which we prove in the next lemma.
Let $\tau$ be a ${\mathcal{G}}_t$-stopping time and $\tau_n$ be the smallest element of $2^{-n} {\mathbb{N}}$ that is greater than or equal to $\tau$. Then ${{\mathbb{P}}_{\alpha}}^{\tau_n}$ converges weakly to ${{\mathbb{P}}_{\alpha}}^{\tau}$ with probability one, where we define ${{\mathbb{P}}_{\alpha}}^{\tau} \left( \cdot \right) := {{\mathbb{P}}_{\alpha}}\left( \theta_{\tau} \cdot \right)$ in the case that $\tau$ is a bridge time.
Throughout this proof we will let $H_t := ({\mathbb{H}}+ iL_t) \backslash \gamma[0,t]$.
As shown in [@lsw:conformal_restriction Lemma 3.2], a probability measure on unbounded hulls in the plane is uniquely determined by the collection of probabilities $$\begin{aligned}
{{\mathbb{P}}\left ( K \cap A = \emptyset \right)}\end{aligned}$$ that is indexed by a sufficiently large class of hulls $A$. Hence it is enough to show that $$\begin{aligned}
\label{probConvergence}
{{\mathbb{P}}_{\alpha}}^{\tau_n} \left(K' \cap A = \emptyset \right) \to {{\mathbb{P}}_{\alpha}}^{\tau} \left( K' \cap A = \emptyset \right)\end{aligned}$$ for all hulls $A$ in this class, with probability one. In our case, it is sufficient to prove that for each fixed restriction hull in ${\mathbb{H}}$, the convergence holds for all hulls $A$ in $H_{\tau}$ that are a positive distance from $\gamma(\tau)$. Note that since $\tau_n \downarrow \tau$ and $\gamma$ is continuous, for sufficiently large $n$ one must have that $A$ is at positive distance from $\gamma(\tau_n)$ also. Hence the probabilities on both sides are well defined. We prove in the two distinct cases that $\tau$ is and is not a bridge time.
$ $\
<span style="font-variant:small-caps;">Case 1: $\tau$ is not a bridge time</span>
First observe that in the definition of ${{\mathbb{P}}_{\alpha}}^t$, the conditioning $J(K') > L_t$ forces the hull $K'$ to avoid the region $\{{\textrm{Im}}\, z \leq L_t \}$, and by the restriction property this can equally be achieved by sampling $K'$ from the restriction measure corresponding to the triple $(H_t, \gamma(t), \infty)$. Thus we have the relation $$\begin{aligned}
{{\mathbb{P}}_{\alpha}}^{({\mathbb{H}}\backslash \gamma[0,t], \gamma(t), \infty)} \left( \, \cdot \, \left| L_{t} < J(K') \leq L_{t}' \right. \right) = {{\mathbb{P}}_{\alpha}}^{(H_t, \gamma(t), \infty)} \left( \, \cdot \, \left | J(K') \leq L_{t}' \right. \right).\end{aligned}$$ Let $g_t$ be the conformal map from $H_t$ onto ${\mathbb{H}}$ such that $g_t(\gamma(t)) = 0$ and $g_t(z) \sim z$ as $z \to \infty$. Let $R_t := \{ z \in H_t : {\textrm{Im}}\, z \leq L_t' \}$. Then $$\begin{aligned}
{{\mathbb{P}}_{\alpha}}^t \left( \cdot \right) = {{\mathbb{P}}_{\alpha}}^{(H_t, \gamma(t), \infty)} \left( \cdot \left | K' \cap R_t \neq \emptyset \right. \right).\end{aligned}$$ The first key observation is that for all $n$ sufficiently large we have that $L_{\tau_n} = L_{\tau}$. This equality is clear since $G$ is closed, and hence $\tau_n$ must belong to the same connected component of $G^c$ that $\tau$ belongs to, for $n$ sufficiently large. For these $n$ we have $L_{\tau_n} = L_{\tau}$. For $L_{\tau}'$ there are two distinct possibilities, which we now treat separately.
First note that necessarily $L_{\tau}' < \infty$. Indeed, the maximum of the imaginary part of ${\textrm{Im}}\, K_{\tau}$ is always an element of $D_{\tau}$, and since $\tau$ is not a bridge time this maximum cannot be in $D$. So first consider the case that $L'_{\tau} < {\textrm{Im}}\, \gamma(\tau)$. By formula , we have that $$\begin{aligned}
{{\mathbb{P}}_{\alpha}}^t \left( K' \cap A = \emptyset \right) &= \frac{{{\mathbb{P}}_{\alpha}}^{(H_t, \gamma(t), \infty)} \left(K' \cap A = \emptyset, K' \cap R_t \neq \emptyset \right)}{{{\mathbb{P}}_{\alpha}}^{(H_t, \gamma(t), \infty)} \left( K' \cap R_t \neq \emptyset \right)} \notag \\
&= \frac{\phi_{A_t}'(0)^{\alpha} - \phi_{A_t \cup S_t}'(0)^{\alpha}}{1 - \phi_{S_t}'(0)^{\alpha}}. \label{bigFormula}\end{aligned}$$ where $A_t = g_t(A)$ and $S_t = g_t(R_t)$ (this is justified since neither $A$ nor $R_{\tau}$ contains $\gamma(\tau)$). Equation shows that it is sufficient to prove $$\begin{aligned}
\label{threeConvergences}
\phi_{A_{\tau_n}}'(0) \to \phi_{A_{\tau}}'(0), \quad \phi_{A_{\tau_n} \cup S_{\tau_n}}'(0) \to \phi_{A_{\tau} \cup S_{\tau}}'(0), \quad \phi_{S_{\tau_n}}'(0) \to \phi_{S_{\tau}}'(0).\end{aligned}$$ For $n$ large enough, $L_{\tau_n}' = L_{\tau}'$ since for any neighborhood of ${\textrm{Im}}\, \gamma(\tau)$ there is an $n$ sufficiently large such that $D_{\tau_n} \backslash D_{\tau}$ is contained within this neighborhood. Since $L_{\tau}' < {\textrm{Im}}\, \gamma(\tau)$, by making the neighborhood sufficiently small we get that $D_{\tau_n} \backslash D$ and $D_{\tau} \backslash D$ must have the same infimum; that is $L_{\tau_n}' = L_{\tau}'$. Hence, $A_{\tau_n}$ and $S_{\tau_n}$ are only decreasing as $\gamma[0,\tau_n]$ decreases, and again since $\gamma[0,\tau_n]$ is a simple curve that shrinks to $\gamma[0, \tau]$ it follows that $g_{\tau_n}$ converges uniformly to $g_{\tau}$ on all subcompacts of $H_{\tau}$, from which the convergences of follow (by Cauchy’s derivative formula and the Schwarz reflection principle, see [@lsw:conformal_restriction]).
The second possibility is to have $L'_{\tau} = {\textrm{Im}}\, \gamma(\tau)$. On the one hand, the conditioning on $K'$ going below ${\textrm{Im}}(\gamma(\tau))$ is trivial so that ${{\mathbb{P}}_{\alpha}}^{\tau}={{\mathbb{P}}_{\alpha}}^{(H_{\tau},\gamma(\tau),\infty)}$. On the other hand, $L'_{\tau_n}$ is greater than $L'_{\tau}$ so that one can strengthen the conditioning of ${{\mathbb{P}}_{\alpha}}^{\tau_n}$ by requiring that the future hull goes below $L'_{\tau}$. Since $\gamma$ is a simple curve shrinking to 0, one again has that $g_{\tau_n}$ converges uniformly to $g_{\tau}$ on all subcompacts of $H_{\tau}$, which proves that the conditioning becomes trivial.
$ $\
<span style="font-variant:small-caps;">Case 2: $\tau$ is a bridge time</span>
In this case note that $A$ is a hull in the domain ${\mathbb{H}}+ i {\textrm{Im}}\, \gamma(\tau) = {\mathbb{H}}+ i L_{\tau}$; hence it is simply a translate of a hull in ${\mathbb{H}}$. Moreover $g_{\tau}$ is simply the shift map $z \to z - \gamma(\tau)$, from which it follows that $A_{\tau} = A - \gamma(\tau)$ and $S_{\tau} = {\mathbb{H}}$. Since ${{\mathbb{P}}_{\alpha}}^{\tau}(\cdot) = {{\mathbb{P}}_{\alpha}}(\theta_{\tau} \cdot)$, proving amounts to showing that $$\begin{aligned}
{{\mathbb{P}}_{\alpha}}^{\tau_n} \left( K' \cap A = \emptyset \right) \to \phi_{A_{\tau}}'(0).\end{aligned}$$ We use to rewrite the left hand side. Define $U_t = \phi_{A_t}(S_t \cap A_t^c)$ so that $$\begin{aligned}
\phi_{A_t \cup S_t} = \phi_{U_t} \circ \phi_{A_t},\end{aligned}$$ from which it follows that $$\begin{aligned}
\phi'_{A_t \cup S_t}(0) = \phi'_{U_t}(0) \phi'_{A_t}(0).\end{aligned}$$ Therefore $$\begin{aligned}
{{\mathbb{P}}_{\alpha}}^{\tau_n} \left( K' \cap A = \emptyset \right) = \phi'_{A_{\tau_n}}(0)^{\alpha} \frac{1 - \phi'_{U_{\tau_n}}(0)^{\alpha}}{1 - \phi'_{S_{\tau_n}}(0)^{\alpha}}.\end{aligned}$$ The convergence of $\phi_{A_{\tau_n}}'(0)$ to $\phi_{A_{\tau}}'(0)$ is simple since it only involves the map $g_{\tau_n}$. Note that $L_{\tau} \leq L_{\tau_n} \leq {\textrm{Im}}\, \gamma(\tau_n)$, so that the domains $H_{\tau_n}$ converge to $H_{\tau}$, and since $\gamma$ is a simple curve it once again follows that $g_{\tau_n}$ converges uniformly to $g_{\tau}$ on all subcompacts of $A_{\tau}$. As before, this implies the convergence of $\phi_{A_{\tau_n}}'(0)$ to $\phi_{A_{\tau}}'(0)$.
It remains to be shown that, as $n \to \infty$, $$\begin{aligned}
\frac{1 - \phi'_{U_{\tau_n}}(0)^{\alpha}}{1 - \phi'_{S_{\tau_n}}(0)^{\alpha}} = \frac{{{\mathbb{P}}_{\alpha}}\left ( K'' \cap U_{\tau_n} \neq \emptyset \right )}{{{\mathbb{P}}_{\alpha}}\left( K'' \cap S_{\tau_n} \neq \emptyset \right)} \to 1.\end{aligned}$$ Observe that $$\begin{aligned}
{{\mathbb{P}}_{\alpha}}\left( K \cap U_{\tau_n} \neq \emptyset \right) &= {{\mathbb{P}}_{\alpha}}\left( K \cap \phi_{A_{\tau_n}}(S_{\tau_n} \cap A_{\tau_n}^c) \neq \emptyset \right) \\
&= {{\mathbb{P}}_{\alpha}}^{({\mathbb{H}}\backslash A_{\tau_n}, 0, \infty)} \left( K \cap S_{\tau_n} \neq \emptyset \right) \\
&\sim {{\mathbb{P}}_{\alpha}}^{({\mathbb{H}}\backslash A_{\tau}, 0, \infty)} \left( K \cap S_{\tau_n} \neq \emptyset \right).\end{aligned}$$ The last relation follows since $g_{\tau_n}$ converges uniformly to $g_{\tau}$ on all subcompacts of $H_{\tau_n}$, to which $A$ eventually belongs, so that $A_{\tau_n}$ converges to $A_{\tau}$. Next recall that $S_{\tau_n} = g_{\tau_n} \left( R_{\tau_n} \right)$, and $$0 < \sup {\textrm{Im}}R_{\tau_n} \leq L_{\tau_n}' - {\textrm{Im}}\gamma(\tau),$$ with the right hand side going to zero as $n \to \infty$. Since the distance of $A_{\tau}$ from zero is positive, for $n$ sufficiently large the probability that a restriction hull intersects $S_{\tau_n}$ is of the order of $\sup {\textrm{Im}}R_{\tau_n}$ and dominated by hulls that intersect $S_{\tau_n}$ near zero. Since the set $S_{\tau_n}$ is the same near zero in both ${\mathbb{H}}$ and ${\mathbb{H}}\backslash A_{\tau}$, the ratio $$\begin{aligned}
\frac{ {{\mathbb{P}}_{\alpha}}^{({\mathbb{H}}\backslash A_{\tau}, 0, \infty)} \left( K \cap S_{\tau_n} \neq \emptyset \right) }{ {{\mathbb{P}}_{\alpha}}\left( K \cap S_{\tau_n} \neq \emptyset \right) }\end{aligned}$$ tends to $1$.
Theorem \[DecompMarkov\] is most useful when $\tau$ is a bridge time, meaning it almost surely takes values in $G$. In that case $\gamma(\tau)$ is a bridge point for $K$, and the corresponding bridge line separates the future hull from the past. Shifting the future hull back to the origin by subtracting off $\gamma(\tau)$, we have the following:
\[ShiftedMarkovCorollary\] At ${{\mathcal{G}}_t}$-stopping times $\tau$ that almost surely take values in $G$, the shifted future hull $\theta_{\tau}K$ obeys the law ${{\mathbb{P}}_{\alpha}}$.
Corollary \[ShiftedMarkovCorollary\] will be the key element in proving that the restriction hulls can be decomposed into a Poisson Point Process, which is the subject of the next section. Before doing that, we immediately apply the corollary to Theorem \[BridgeDimension\], part by showing that $C$ and $D$ almost surely have no isolated points.
We have already shown that $C$ and $D$ are closed, we prove that $C$ has no isolated points. Almost surely, zero is not isolated in $C$ because of the scale invariance and the fact that bridge points exist. For a rational number $r$, let $\tau_r$ be the first bridge time after time $r$. Then by the previous corollary, we deduce that the law of $\theta_{\tau_r}K$ obeys the law ${{\mathbb{P}}_{\alpha}}$. Since $\gamma(\tau_r)$ shifts to zero under $\theta_{\tau_r}$, the previous remark shows that $\gamma(\tau_r)$ is almost surely not isolated. From these facts we deduce that the event $\{ \gamma(\tau_r)$ is not isolated in $C$ for all rational $r\}$ has probability one. If a point $\gamma(t) \in C$ were isolated then there would have to be an interval of time around $t$ which contains no other bridge times, but since this interval contains a rational time we arrive at a contradiction.
Local Time of the Decomposition \[LocalTimeSection\]
====================================================
In this section we will show that there exists a natural local time on the bridge heights that we use to decompose the restriction hulls into a Poisson Point Process of irreducible bridges. All the results of this section derive from the theory of subordinators and regenerative sets, which is well described in [@bertoin:subordinators]. We briefly recall the definition of regenerative sets, which is taken from [@bertoin:subordinators Chapter 2].
A random subset $S$ of $[0, \infty)$ is a *regenerative set* with respect to a filtration ${\mathcal{F}}_t$ if for every $s \geq 0$, conditionally on $M_s = \inf \{ t > s : t \in S\} < \infty$, the shifted set $\left( S - M_s \right) \cap [0, \infty)$ has the same law as $S$ and is independent of $\mathcal{F}_{M_s}$.
Using the results of Sections \[BridgeSection\] and \[RenewalSection\], we can immediately prove:
\[DecompLocalTime\] The set $D$ of bridge heights is regenerative with respect to $\mathcal{D}_L := \sigma(D \cap [0,L])$.
Consider $L \geq 0$. Since $D$ is closed, $M_L \in D$ almost surely. Then $M_L$ is a bridge height, and the time $\tau_L$ at which the curve reaches this bridge height is a ${{\mathcal{G}}_t}$-stopping time taking values in $G$. By Corollary \[ShiftedMarkovCorollary\], the $\mathcal{G}_{\tau_L}$-law of $\theta_{\tau_L}K$ is the same as the original law of $K$. Consequently, the $\mathcal{G}_{\tau_L}$-law of $D(\theta_{\tau_L}K) = D - M_L$ is the same as the law of $D$. Since $\mathcal{D}_L \subset \mathcal{G}_{\tau_L}$ this completes the proof.
Proposition \[DecompLocalTime\] proved that the set $D$ is regenerative, and consequently by [@bertoin:subordinators Theorem 2.1] it is the closure of the image of some subordinator (and the subordinator is unique up to a linear change of its time scale). On the other hand, Theorem \[BridgeDimension\] showed that $D$ is scale invariant, and it is an easy step to deduce from this that the subordinator must be stable. Recall that there is a one-parameter family of stable subordinators, indexed by the real numbers between $0$ and $1$, and, as shown in [@bertoin:subordinators Chapter 5], the index of a stable subordinator is the same as the Hausdorff dimension of its image. Hence we have the following:
\[subordinatorCorollary\] Under the law ${{\mathbb{P}}_{\alpha}}$, the set $D$ is the closure of the image of a stable subordinator $(\sigma_{\lambda}, \lambda \geq 0)$ of index $2-2\alpha$.
The parameter $\lambda$ can be thought of as the local time corresponding to the subordinator. Recall that the local time for $\sigma$ is the function $\lambda : [0, \infty) \to [0, \infty)$ defined by $\lambda(s) := \inf \{ t \geq 0 : \sigma_t > s \}$, and it is well known in the subordinator literature that $\lambda$ is an increasing, continuous function which increases only on $D$. This means that if we run the restriction hulls on the $\lambda$ time scale, then the hull grows only when it is crossing bridge lines. For $\lambda \geq 0$ we define $$\begin{aligned}
\tau_h := \inf \{ t \geq 0 : \sup {\textrm{Im}}(K_t) = h \},\end{aligned}$$ and $$\begin{aligned}
t(\lambda) := \tau_{\sigma_{\lambda}}.\end{aligned}$$ Note that $\sigma_{\lambda}$ is the bridge height at which $\lambda$ units of local time are first accumulated, and then $t(\lambda)$ is the time, in the original parameterization of the restriction hull, at which the local time first reaches $\lambda$. It follows that $t(\lambda)$ is an increasing, right-continuous process for which the closure of its image is precisely the set of bridge times $G$. Intervals of $\lambda$ on which the process $t(\lambda)$ is flat correspond to times at which the restriction hull is between bridge heights. Using the $t(\lambda)$ time-scale, we are able to define a Poisson Point Process taking values in the space of irreducible bridges rooted at the origin. Let $\delta$ be the curve which starts and ends at zero in zero time (i.e. $\delta : \{0 \} \to \{ 0 \})$. For $\lambda \geq 0$, define $e_{\lambda}$ by $$\begin{aligned}
\label{eDecomp}
e_{\lambda} = \left\{
\begin{array}{ll}
\theta_{t(\lambda-), t(\lambda)}K, & t(\lambda) > t(\lambda-) \\
\delta, & t(\lambda) = t(\lambda-)
\end{array}
\right.\end{aligned}$$ From this we have the following:
$e_{\lambda}$ is an $\left( \mathcal{F}_{t(\lambda)} \right)_{\lambda \geq 0}$ Poisson Point Process on the space of irreducible bridges.
Take a subset $U$ of the set of irreducible bridges that doesn’t contain $\delta$, and an interval $I := [\lambda_1, \lambda_2]$. As in [@revuz_yor Chapter XII], one needs to show that the number of times that $e_{\lambda}$ belongs to $U$ for $\lambda \in I$ is independent of $\mathcal{F}_{t(\lambda_1)}$ and has the same law as the number of times that $e_{\lambda}$ belongs to $U$ for $\lambda \in [0, \lambda_2 - \lambda_1]$. But this is essentially a property of Corollary .
We denote by $\nu_{\alpha}$ the intensity measure of the Poisson Point Process $e_{\lambda}$, and we call it the **continuum irreducible bridge measure**. It conveniently encodes all the behavior of continuum irreducible bridges. For a set of irreducible bridges $E$, $\nu_{\alpha}(E)$ is simply the expected number of elements of $E$ that occur in $e[0,1]$, which may or may not be finite. For instance, if $E_L$ is the set of irreducible bridges with height greater than $L$, then a simple consequence of Corollary \[subordinatorCorollary\] is that $\nu_{\alpha}(E_L) = c_{\alpha} L^{2\alpha - 2}$ for some fixed constant $c_{\alpha}$, and furthermore, $$\begin{aligned}
\label{condHeightLaw}
\textbf{P}_{\alpha}^L(\cdot) := \frac{\nu_{\alpha}(\cdot \cap E_L)}{\nu_{\alpha}(E_L)}\end{aligned}$$ is exactly the law of the first irreducible bridge with height greater than $L$. To make the analogy with other well-known decompositions of stochastic processes, $\nu_{\alpha}$ is the equivalent of Itô’s measure on $1$-dimensional Brownian excursions, or Balint Virág’s measure on $2$-dimensional Brownian Beads. Compared to half-plane SAWs, $\nu_{\alpha}$ is the analogue of the measure $\textbf{P}(\omega) = \beta^{-{\left| \omega \right|}}$ on SAW irreducible bridges, although we point out that $\textbf{P}$ is a probability measure (by Kesten’s relation), whereas $\nu_{\alpha}$ is infinite but $\sigma$-finite.
In the case of half-plane SAWs, the measure on paths is realized by concatenating together an i.i.d. sequence of irreducible bridges, each distributed according to $\textbf{P}$, and in the continuum a similar statement holds. If $(e_{\lambda})_{\lambda \geq 0}$ is a Poisson Point Process of irreducible bridges with intensity measure $\nu_{\alpha}$, then the concatenation $$\begin{aligned}
K = \bigoplus_{\lambda \geq 0} e_{\lambda}\end{aligned}$$ has the law of an index $\alpha$ restriction hull. Note, however, that we are not attempting to show that the irreducible bridges can be concatenated together in such a way as to reconstruct the sequence of growing hulls $(K_t)_{t \geq 0}$, even though this should be possible with enough care. Recall though that the time parameterization we are using for the restriction hulls is completely artificial to begin with, and therefore attempting to reconstruct it would mostly be an uninteresting and unuseful exercise.
Open Questions \[Open\]
=======================
In this final section we present some open questions that were raised by our work.
What other properties of the irreducible bridge measure $\nu_{\alpha}$ can be derived?
Our work has essentially determined only one main property of bridges: that the distribution of their vertical height is the same as the jump distribution for a stable subordinator of index $2-2\alpha$ (up to a multiplicative constant). Ultimately we hope that much more can be said about irreducible bridges than this. It may be naturally difficult to say anything more, since even in the case of half-plane SAWs there is not much known about irreducible bridges (although in the “off-critical” case there are some results, see [@madras_slade:saw_book Chapter 4]). For other two-dimensional decompositions, notably Virág’s Brownian Beads, it appears similarly difficult to say anything about the bead measure.
Is there a constructive way of building irreducible bridges?
In the case of ${\textrm{SLE}}(8/3)$, for example, is there a driving term for the Loewner equation that outputs irreducible bridges (perhaps with at least some specified vertical height)? And for general restriction measures with $\alpha < 1$, can some driving term for the Loewner equation be combined with the Brownian loop soup to produce irreducible bridges for restriction hulls?
Is there a natural “length” that can be put on irreducible bridges?
For half-plane SAWs the length of the walk is simply the number of steps in it, and many results on SAWs are expressed in terms of this length. We expect that there is some way of defining a similar natural length on irreducible bridges, and that this length is somehow the scaling limit of the length for SAWs. However, because the irreducible bridges are fractal objects it is not an easy matter to define a non-trivial length on them. In the case of ${\textrm{SLE}}(8/3)$ specifically, this question is closely related to the problem of the “natural time parameterization” for ${\textrm{SLE}}$, which has recently been considered by Lawler and Sheffield [@lawler_sheff:time]. The key idea of their time parameterization is to build a length measure on the curve (that also has some other desirable properties), and then reparameterize in such a way that the length of the curve at time $t$ is $t$, as with the SAWs. Their length measure should also be a natural length measure for irreducible bridges.
Is there some sort of continuous analogue of Kesten’s relation?
This is closely related to the problem of the natural length on irreducible bridges described above. Supposing that $L(K)$ is the “natural length” of an irreducible bridge, and making an analogy with , we might expect that $$\begin{aligned}
\int_0^{\infty} \beta^{-l} \nu_{\alpha}\left( L(K) \in dl \right)\end{aligned}$$ is finite for $\beta < \mu$ but infinite for $\beta > \mu$, for some universal $\mu$, and then one can ask for the behavior at this critical $\mu$.
Can the restriction hulls be time parameterized in such a way that the time parameterization also refreshes itself at bridge points?
Presently we are only showing that the hulls refresh themselves as sets and *not* as time parameterized objects. But it is entirely plausible that there is some time parameterization which refreshes itself at bridge points along with the geometrical objects, especially considering that the counting parameterization for half-plane SAWs has this property (at each bridge point, one simply starts counting off the number of steps anew). It is possible that the natural time parameterization of Lawler and Sheffield will have this property for ${\textrm{SLE}}(8/3)$ but it is not immediately clear that this will be the case, since their time parameterization has no way of seeing that it is currently at a bridge point and therefore is unlikely to refresh at such bridge times.
Can some element of the bridge decomposition be used to prove the existence of, or at least heuristically deduce, critical exponents for half-plane SAWs or SAW bridges?
For example, it is conjectured that the number of $N$-step SAW bridges grows asymptotically like $N^{-\beta} u^N$ as $N \to \infty$, for the same $\mu$ as in and some unknown constant $\beta$. Recently, Neal Madras has privately communicated to us his conjecture that $\beta = 7/16$, although this quantity was likely known beforehand in the physics literature. He uses two different methods to derive this value, the first being based purely on some heuristics for half-plane SAWs, and the other making use of the relation and the conjecture that the scaling limit of half-plane SAWs is ${\textrm{SLE}}(8/3)$. Being able to answer further questions of this type would be extremely helpful for studying half-plane SAWs.
\[kappaQuestion\] Do bridge heights and lines exist for SLE($\kappa$) for values of $\kappa$ different from $8/3$. If so, what is the Hausdorff dimension of $C$ and $D$ and how does it depend on $\kappa$?
Currently we only know that at $\kappa = 0$ and $\kappa = 8/3$, the Hausdorff dimensions of $C$ and $D$ are $1$ and $3/4$, respectively (the $\kappa = 0$ result is clear from the fact that the corresponding SLE curve is a vertical line). We conjecture that the Hausdorff dimensions of $C$ and $D$ are always the same, and they are a strictly decreasing, continuous function of $\kappa$. When $\kappa = 4$ the Hausdorff dimension must certainly be zero since the SLE($4$) curve comes arbitrarily close to the real line, but we do not know if this is the smallest $\kappa$ for which the dimension is zero. We have no conjecture as to what that $\kappa$ might be, other than it is somewhere between $8/3$ and $4$.
We should briefly mention that, as a corollary of Theorem \[BridgeDimension\], we do have lower bounds on the Hausdorff dimension of $C$ and $D$ for $2 \leq \kappa \leq 8/3$. Since attaching loops to an SLE curve can only reduce the number of bridge points that the SLE curve has, we know
Let $C$ and $D$ be the set of bridge points and heights for an SLE($\kappa$) curve, with $2 \leq \kappa \leq 8/3$. Then the Hausdorff dimensions of $C$ and $D$ are both almost surely constant, with ${\mathrm{dim_H} \,}C \geq 3 - \frac{6}{\kappa}$.
This lower bound is probably far from sharp, since it is increasing with $\kappa$ rather than decreasing. To prove that the Hausdorff dimensions of $C$ and $D$ are almost surely constant, Theorem \[BridgeDimension\] part can be used without modification.
[^1]: Research of Tom Alberts supported in part by NSF Grant OISE 0730136, and a postdoctoral fellowship from the Natural Sciences and Engineering Research Council of Canada. Research of Hugo Duminil-Copin supported in part by project MRTN-CT-2006-035651, Acronym CODY, of the European Commission, and a grant from the Swiss National Science Foundation.
[^2]: There is a minor technicality to point out here: if the walk oscillates infinitely often in the vertical direction without approaching some limit (including infinity) the decomposition algorithm will terminate after finitely many iterations and the remaining part of the walk will not be a bridge. However, we will see in the next paragraph that this is a probability zero event under the standard measure on $\mathcal{H}$, and that the vertical component of the SAW always goes to infinity with probability one.
[^3]: The topology we consider is close to the Caratheodory topology and has been defined in [@lsw:conformal_restriction Lemma 3.5]
|
---
abstract: 'Quantum discord quantifies non-classical correlations going beyond the standard classification of quantum states into entangled and unentangled ones. Although it has received considerable attention, it still lacks any precise interpretation in terms of some protocol in which quantum features are relevant. Here we give quantum discord its first information-theoretic operational meaning in terms of entanglement consumption in an *extended quantum state merging* protocol. We further relate the asymmetry of quantum discord with the performance imbalance in quantum state merging and dense coding.'
author:
- 'D. Cavalcanti'
- 'L. Aolita'
- 'S. Boixo'
- 'K. Modi'
- 'M. Piani'
- 'A. Winter'
bibliography:
- 'omd.bib'
title: Operational interpretations of quantum discord
---
Introduction
============
The study of quantum correlations has mostly been focused on entanglement [@horod]. This is because entanglement has been identified as a key ingredient in quantum information processing, allowing to perform a number of tasks that are either impossible to realize or less efficient with only classical resources at disposal. However, entanglement does not account for all the non-classical properties of quantum correlations. Zurek [@zurek2000] (see also [@henderson; @ollivier]) identified *quantum discord* (QD) as a feature of quantum correlations that encapsulates entanglement but goes beyond it as it is also present even in separable states. Over the past decade, QD has been the focus of several theoretical and experimental studies addressing its formal characterization [@piani; @*CesarEtal07; @*modietal; @*paris10; @*Adesso10; @*dvb10; @ferraro], its behavior under dynamical processes [@ferraro; @werlangPRA; @*Maziero09; @*mazzola; @*fanchini; @*jsxu], and its connection with quantum computation [@dattashaji; @*Lanyon] and quantum phase transitions [@sarandy; @*werlangQPT]. QD was initially introduced in the context of the analysis of quantum measurements [@ollivier] and afterwards interpretations in terms of the difference in performance of quantum and classical Maxwell demons were given [@ZurekDemon; @*Terno]. Nevertheless, a large part of the quantum information community has always been skeptical towards QD as an information-theoretic quantiÞer. This is because QD has not a clear operational interpretation in this context. That is, we lack an information-theoretic task for which the QD provides a quantitative measure about the performance in the task. Thus, without this kind of operational interpretation, QD is very often considered simply a “quantumness parameter”.
In this Letter we give quantum discord its long sought operational interpretation. We relate QD to state merging (SM) [@merging], a well known task in quantum information. In SM a tripartite pure state is considered, i.e., Alice ($A$), Bob ($B$), and Charlie ($C$) share (many copies of) a pure state $\psi_{ABC}$. The goal in the task is that $A$ transfers her part of the state to $B$, $\psi_{ABC} \rightarrow \psi_{B'BC}$ (see Fig. \[triangle\]), by using classical communication and shared entanglement. Here we show that the minimal total entanglement consumed in a process we call “extended state merging” (ESM) from $A$ to $B$ is exactly equal to the QD between $B$ and $C$ (with measurements on $C$). We further unravel a connection between QD to a well-known protocol in quantum information processing: *dense coding* (DC) [@bennett92]. DC is a task that uses pre-established quantum correlations to send classical messages more efficiently than by classical mean.
We focus on the finite-dimensional case with the three parties $A$, $B$, and $C$ sharing a pure state $\psi_{ABC}$. All bipartite and single-party states are obtained by taking the appropriate partial traces of $\psi_{ABC}$. The *quantum (von Neumann) entropy* of a state $\rho$ is defined as $S(\rho) =-\trace \rho \log_2\rho$. It is the generalization to the quantum domain of the *classical (Shannon) entropy* of a probability distribution $\{p_i\}$ given by $H(\{p_i\})=-\sum_i p_i \log_2 p_i$. We write $S(X)$ to denote the entropy of the reduced state $\rho_X$. Similarly, we write $H(a)$ to denote the Shannon entropy of a classical random variable $a$ distributed according to some probability distribution $\{p^a_i\}$. The latter may be the marginal probability distribution $p^a_i=\sum_jp^{ab}_{ij}$ of a bivariate (in general, multivariate) probability distribution $\{p^{ab}_{ij}\}$ of two classical random variables $a$ and $b$.
Conditional entropy and coherent information
============================================
For a bipartite system $AB$, the quantum (von Neumann) conditional entropy is defined as $S(A|B):=S(AB)-S(B)$ [@nielsen]. It is the quantum version of the classical (Shannon) conditional entropy $H(a|b):= H(a,b)-H(b)$. Note that both are asymmetric quantities. $H(a|b)$ measures how much uncertainty is left—on average—about the value of $a$ given the value of $b$. It can be written as \[eq:classical\_cond\_entropy\] H(a|b)=\_jp\^b\_jH(a|b=j), where $H(a|b=j)$ is the entropy of the conditional probability distribution $p^a_{i|b=j} :=p^{ab}_{ij}/p^{b}_j$. It has a clear operational interpretation as the amount of classical information that $A$ has to give—on average—to $B$, who knows the value of $b$, so that the latter gains full knowledge also of the value of $a$ [@slepianwolf]. Given this interpretation for $H(a|b)$, it is always non-negative.
However, the situation changes drastically for quantum states, because $S(A|B)$ can take negative values, e.g. for pure entangled states. This fact was, for a long time, an obstacle to an operational interpretation of $S(A|B)$. On the other hand, its opposite was identified as an important quantity in the context of quantum information, and was even given a name of its own: [*coherent information*]{} ${{I(A\rangle B)}}:=-S(A|B)$. Coherent information was originally introduced to measure the amount of quantum information conveyable by a quantum channel [@Schumacher]; given that it is always non-positive in the classical case, one may say that it is a purely quantum quantity.
Quantum discord
===============
One remedy to negative quantum conditional entropy is to generalize the classical conditional entropy to quantum using Eq. , as was done in [@henderson; @ollivier] by defining $S(A|B_c) := \min_{\{N_{j}\}} \sum_j p^B_j S(A|B=j)$, where the minimization is over generalized measurements $\{N_{j}\}$ [^1], with $N_j\geq0$ for all $j$ and $\sum_j N_j=\openone_B$. We also have $S(A|B=j) = S(\rho_{A|j})$, where $\rho_{A|j} = \trace_B (\openone_A\otimes N_B^j\rho_{AB})/p^B_j$ with $p^B_j=\trace(\openone_A\otimes N^j_B\rho_{AB})$. $S(A|B_c)$ is always positive and can also be thought of as a measure of the uncertainty left on average about $A$ given that $B$ has been measured. For classical systems both $S(A|B)$ and $S(A|B_c)$ coincide with the classical conditional entropy, but in general $S(A|B_c)$ is strictly larger than $S(A|B)$. The difference in these two quantity is indeed the definition of the *quantum discord with measurements on $B$* [@ollivier] \[disco\] [[D(A | B)]{}]{}:=S(A|B\_c) - S(A|B). QD can be seen as the gap between the standard measure for total correlations present in a quantum state $\rho_{AB}$, given by *quantum mutual information* $I(A:B):=S(A)-S(A|B)$ [@groisman], and the *Henderson-Vedral measure of classical correlations* $I(A:B_c) := S(A) -S(A|B_c)$ [@henderson]. As ${{D(A | B)}}=I(A:B)-I(A:B_c)$, the QD can be considered a (asymmetric) quantifier of non-classical correlations present in a quantum state. We will refer to ${{D(X | Y)}}$ as to the “discord of $XY$ measured by $Y$”.
State merging and entanglement consumption
==========================================
A fully convincing operational interpretation of quantum conditional entropy and coherent information was given with the introduction of the task of *quantum state merging* (SM) [@merging]. SM, say from $A$ to $B$, is a process by which $A$ and $B$ transfer $A$’s part of the state to $B$ maintaining the coherence with the reference $C$. $A$ and $B$ both know the state they share, and they can apply arbitrary local operations coordinated by classical communication (LOCC). By acting on $n$ copies of $\psi_{ABC}$, their goal is to end up with a state close to $\psi_{B'BC}^{\otimes n}$, such that the subsystem $B'$ is in Bob’s hands and plays in the new state exactly the same role as $A$ played in the old one. Errors are allowed, but they must vanish in the limit $n\rightarrow\infty$. To achieve their goal, $A$ and $B$ are allowed to use extra, pre-established two-qubit maximally-entangled pairs (ebits), but these constitute a valuable resource they must pay for. It turns out that the value of $S(A|B)$ quantifies exactly the optimal amount—per copy of the state—of ebits spent in the process. A positive value means that entanglement must be consumed, while a negative amount means not only that no extra entanglement is needed, but also that $A$ and $B$ retain $-S(A|B)={{I(A\rangle B)}}$ ebits per copy merged. See Fig. \[triangle\] for an illustration of SM.
A useful way to think of the role played by the conditional entropy in SM is to imagine a hypothetic entanglement bank in which $A$ and $B$ possess a joint account: the entanglement balance after merging—in ebits, per copy merged—is given precisely by $-S(A|B)$. When $S(A|B) \geq 0$, $A$ and $B$ have to withdraw $S(A|B)$ from their account to perform SM. On the other hand, when $S(A|B) < 0$ then the process can be completed without any withdrawing. Moreover, after merging they end up sharing ${{I(A\rangle B)}} = -S(A|B)$ extra ebits of entanglement, which they deposit in their account for future use.
At the end of this process, the only correlations between A and B are those present in the bank account. In particular, there is no additional entanglement left between A and B. Given this, the bank-account picture suggests to consider a more comprehensive balance, that takes into account also the entanglement “lost" in the process. Indeed, coherent information is positive only if the state is entangled, and while A and B may end up with “leftover" Bell pairs after SM, they do not share anymore the starting entangled states. Thus, it is useful and sensible to define the *total entanglement consumption* as \[cost1\] [(A)]{}:= [E\_F(A:B)]{} + S(A|B), where ${E_F(A:B)}:=\min_{\{p_i,\psi^{AB}_i\}} \sum_i p_i S\big(\trace_{A} (\psi_i^{AB}) \big)$ is the *entanglement of formation* (EoF) of $\rho_{AB}$, with the minimum taken over pure-state ensembles $\{p_i,\psi^{AB}_i\}$ for $\rho_{AB}$ [@bennett]. EoF quantifies the minimum amount of pure-state entanglement that $A$ and $B$ need to consume to create $\rho_{AB}$ by LOCC with strategies where each pure-state member of the ensemble forming $\rho_{AB}$ is prepared independently. Thus, $\Gamma$ quantifies the total entanglement consumed in SM, by taking into account the amount of entanglement $A$ and $B$ would have needed to prepare $\rho_{AB}$ by LOCC—and “lost” during SM—plus the amount of entanglement used by the process of SM itself. In order to give a more precise operational interpretation, we consider a two-step process. In the first stage, Alice and Bob prepare the state $\rho_{AB}$. To this aim, they have to share classical information, and potentially use some other local ancillas. We demand that, in order to end up sharing $\rho_{AB}$ and not some larger state, after preparing the state and before the merging, they remove all ancillas. Then Eq. (3) indeed characterizes the entanglement cost of a two-stage process that we call extended state merg- ing (ESM): (i) state preparation through the (possibly non-optimal – see section Regularization below) protocol described before and (ii) merging.
Operational interpretations of quantum discord
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Quantum Discord and Extended State Merging
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Now we are in the position to give QD an operational interpretation. In Appendix 1 we prove the following: \[eq:opmeaning\] [[D(A | C)]{}]{}=[(A)]{}. This equation says that QD between $C$ and $A$ with measurements on $C$ is equal to the total entanglement consumption in ESM from $A$ to $B$. To the best of our knowledge, this yields the first information-theoretic scenario where the value of QD provides concrete quantitative information about a task’s performance or cost.
Asymmetry of quantum discord
----------------------------
One immediate exercise of the last equation is to give meaning to the asymmetry of QD, that is, the fact that in general $D(A|C)\neq D(C|A)$. Thanks to Eq. we can interpret the asymmetry of discord as the differences in the cost of ESM for $A$ versus $C$ to send their parts of the state to $B$, [[*[i.e. ]{}*]{}]{}: \[eq:discasym\] D(A|C)-D(C|A)=[(A)]{}-[(C)]{}.
Quantum Discord and Dense Coding
--------------------------------
Coherent information also describes the usefulness of a quantum state $\rho_{AB}$ as a resource for dense coding (DC) [@bennett92]. DC—say from a sender $A$ to a receiver $B$, initially sharing $\rho_{AB}$—is a procedure by which $A$ is able, by sending her subsystem to $B$, to transmit more classical information than she could if the system was classical; i.e., the maximal rate of classical information transmission per copy of $\rho_{AB}$ used can be larger. If $A$’s encoding is done by unitary rotations, the correction to the classical capacity that she could achieve by sending a classical system with dimension equal to that of her subsystem, $d_A$, is exactly the coherent information ${{I(A\rangle B)}}$ [@DC; @winterdense; @Bruss; @HoroPiani]. In the most general DC scenario [@DC; @winterdense; @HoroPiani], $A$ encodes her message by means of general quantum operations $\Lambda_A:M_{d_A}\rightarrow M_{d'_A}$, where $d_A$ is the dimension of the original subsystem in the hands of $A$, while $d'_A$ is the dimension of the subsystem sent to $B$, and $M_d$ denote the set of $d\times d$ complex matrices. If the encoding is applied at the level of single copies of the shared state $\rho_{AB}$, the DC single-copy capacity can be achieved by a unitary encoding after a pre-processing operation whose aim is exactly that of increasing coherent information. More precisely the capacity is equal to ${\chi_{\text{DC}}(A\rangle B)}:=\log_2 d'_A+\max_{\Lambda_A}{{I(A'\rangle B)}}$, where the maximization is over all quantum operations with output dimension $d'_A$ and ${{I(A'\rangle B)}}$ is the coherent information of $(\Lambda_{A}\otimes\openone_B)[\rho_{AB}]$. This capacity depends on the output dimension $d'_A$, but, given that $\log_2d'_A$ can be considered as a classical contribution, one can focus on the *quantum advantage of DC* \[eq:advantage\] [\_(A)]{}:=\_[\_A]{}[[I(A’B)]{}]{}. The maximization above has no restriction on the output dimension, which can anyway be taken to be less or equal to $d_A^2$ [@HoroPiani]. The maximization over $\Lambda_A$ ensures that the coherent information of the pre-processed state is non-negative.
In Appendix 2 we prove the following connection between QD and DC: \[eq:Discoop1\] [[D(A | C)]{}]{}-[[D(B | C)]{}]{}=[\_(C)]{}-[\_(C)]{}. Note that, if $C$ sends subsystems with the same dimension to $A$ and $B$ (in particular a dimension large enough to achieve the quantum advantage of DC with both receivers) this difference can be written as ${{D(A | C)}}-{{D(B | C)}}={\chi_{\text{DC}}(C\rangle A)}-{\chi_{\text{DC}}(C\rangle B)}$, i.e., in terms of the DC capacity itself. Eq. gives an operational meaning in terms of performance to the differences in QD: the difference in the QD of $AC$ and $BC$, both measured by $C$, is the same as the difference in the DC capacity from $C$ to either $A$ or $B$. The same difference in QD can be related to the coherent information, as can be seen using Eq. twice: ${{D(A | C)}}-{{D(B | C)}}={{I(A\rangle C)}}-{{I(B\rangle C)}}={{I(C\rangle A)}}$. Or, for measurements on different parties, ${{D(C | A)}}-{{D(C | B)}}=\Gamma(C\rangle B)-\Gamma(C\rangle A).$
Regularization
==============
All the relations we have found, although already meaningful in the form above, can be cast in their regularized version, so that they become, in the case of ESM, more consistent from an operational and information-theoretic point of view. To do so we note that the minimal amount of ebits needed to create $\rho_{AB}$ over all possible LOCC strategies is given by the *entanglement cost* $E_C(A:B) = \lim_{n\rightarrow\infty} \frac{1}{n}E_F(A:B)_{\rho_{AB}^{\otimes n}}$ [@regEf]. We can then define the *asymptotic total entanglement consumption of ESM* as the regularized version of Eq. , i.e, as $\Gamma^{\infty}(A\rangle B) := \lim_{n\rightarrow\infty} {\Gamma(A\rangle B)_{\rho_{AB}^{\otimes n}}}/{n}=E_C(A:B)+S(A|B),$ having used that conditional entropy is additive. As ESM is itself an asymptotic process, the regularized total cost $\Gamma^{\infty}$ is a quantity better motivated than the unregularized $\Gamma$ from an operational and information-theoretic point of view. It is worth remarking that both $\Gamma$ and $\Gamma^\infty$ are positive, because coherent information is a lower bound on distillable entanglement [@hashinginequality], and therefore on entanglement cost. By Eq. we have that ${{D^\infty(A | C)}}=\Gamma^{\infty}(A\rangle B)$.
Conclusions
===========
We have seen that the QD is intimately related to the tasks of ESM and DC. For a pure tripartite state, the QD reveals what is the entanglement consumption in ESM and in which direction more classical information can be sent through DC. Moreover the asymmetry of the QD can be given an operational interpretation, since it matches the asymmetry of the tasks to which we have related it, ESM and DC, which are inherently directional.
Finally, a recent paper has unraveled a different connection between QD and SM [@MD]. There, it was observed that the right-hand side of can be interpreted as the difference in quantum communication costs between performing SM with a partially measured version of $\rho_{AB}$ (first term) and with $\rho_{AB}$ (second term) directly. Such an interpretation of QD regards a relation between *different* states, one obtained from the other via measurement, while the one presented here refers to just one state (and its purification). On the other hand, since QD can be expressed also as the difference in mutual information between such two states (see the paragraph after (2)), an approach similar to that of [@MD] can lead to interpretations in terms of quantum locking [@divincenzo_locking_2003; @*hayden_randomizing_2004; @prep] and correlations erasure [@groisman].
We thank A. Acín, C. Bény, J. Calsamiglia, C. Caves, A. Datta, M. Gu, and V. Vedral for valuable comments. This work was supported by the National Research Foundation, the Ministry of Education of Singapore, the Spanish “Juan de la Cierva" Programme, NSERC, QuantumWorks, Ontario Centres of Excellence, the Royal Society, U.K. EPSRC and the European Commission.
Appendix 1: Proof of Eq. (4)
============================
We start by recalling the Koashi-Winter monogamy relation [@KoashiWinter] for quantum correlations within a pure tripartite state $\psi_{ABC}$: \[eq:KW\] S(B)=E\_F(A:B)+I(B:C\_c). This, together with the definition of $I(B:C_c)$, implies that E\_F(A:B)=S(B|C\_c)=S(A|C\_c), which we can substitute in the definition of ${{D(A | C)}}$ to get [@fanchini1] [[D(A | C)]{}]{}=E\_F(A:B)-S(A|C). Now, note that $S(A|C)=S(AC)-S(C)$ and, since $\psi_{ABC}$ is a pure state, we have $S(AC)=S(B)$ and $S(C)=S(AB)$. Hence $S(A|C)=S(B)-S(AB)=-S(A|B)$, so that [[D(A | C)]{}]{}=E\_F(A:B)+S(A|B)=[(A)]{}.
Appendix 2: Proof of Eq. (7)
============================
A monogamy equality similar to Eq. with regards to DC was given in [@HoroPiani]: \[eq:HP\] S(A)=E\_P(A:C)+[\_(B)]{}, where $E_P$ is the *entanglement of purification*, defined as [@IBMHor2002] $E_P(A:C):=\min_{\psi_{AA'CC'}} S\big( \trace_{CC'} (\psi_{AA'CC'}) \big)$, with the minimum taken over all pure states $\psi_{AA'CC'}$ such that $\trace_{A'C'}(\psi_{AA'CC'})=\rho_{AC}$. Using the fact that for a tripartite pure state ${{I(A\rangle C)}}=S(C)-S(B)$, and expressing $S(B)$ according to , from one obtains ${{D(A | C)}} = S(C) - {\Delta_{\text{DC}}(C\rangleB)} - \big( E_P(A:B) - {E_F(A:B)} \big)$. Applying this equivalence twice one gets [[D(A | C)]{}]{}-[[D(B | C)]{}]{}=[\_(C)]{}-[\_(C)]{}.
[^1]: In [@ollivier] the generalized measurement were actually restricted to complete von Neumann measurements.
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abstract: 'Two-dimensional Projected Entangled Pair States (PEPS) provide a unique framework giving access to detailed entanglement features of correlated (spin or electronic) systems. For a bi-partitioned quantum system, it has been argued that the Entanglement Spectrum (ES) is in a one-to-one correspondence with the physical edge spectrum on the cut and that the structure of the corresponding Entanglement Hamiltonian (EH) reflects closely bulk properties (finite correlation length, criticality, topological order, etc...). However, entanglement properties of systems with spontaneously broken continuous symmetry are still not fully understood. The spin-1/2 square lattice Heisenberg antiferromagnet provides a simple example showing spontaneous breaking of SU(2) symmetry down to U(1). The ground state can be viewed as a “quantum Néel state" where the classical (Néel) staggered magnetization is reduced by quantum fluctuations. Here I consider the (critical) Resonating Valence Bond state doped with spinons to describe such a state, that enables to use the associated PEPS representation (with virtual bond dimension $D=3$) to compute the EH and the ES for a partition of an (infinite) cylinder. In particular, I find that the EH is (almost exactly) a chain of a dilute mixture of heavy ($\downarrow$ spins) and light ($\uparrow$ spins) hardcore bosons, where light particles are subject to long-range hoppings. The corresponding ES shows drastic differences with the typical ES obtained previously for ground states with restored SU(2)-symmetry (on finite systems).'
author:
- Didier
title: Entanglement Hamiltonian of the quantum Néel state
---
Introduction
============
It is known from early Quantum Monte Carlo (QMC) simulations that the ground state (GS) of the spin-1/2 Heisenberg antiferromagnet (AFM) on the bipartite square lattice is magnetically ordered [@QMC_young] and, hence, breaks the hamiltonian SU(2) symmetry. The GS can be viewed as a “quantum Néel state" (QNS) where the maximum classical value $m_{\rm stag}=1/2$ of the staggered magnetization is reduced by (moderate) quantum fluctuations. More recent QMC simulations [@QMC_sandvik] have provided GS energy, staggered magnetization and spin-spin correlations with unprecedented accuracy. In particular, it has been established that the QNS exhibits power-law decaying spin-spin correlations characteristic of a [*critical state*]{} [@QMC_sandvik2].
Recently, a number of new powerful tools based on entanglement measures have emerged. The entanglement spectrum (ES) and its associated Entanglement Hamiltonian defined via the reduced density matrix (RDM) of a bi-partioned quantum system (see definitions later) provides new insights. In particular, it has been argued that the ES is in a one-to-one correspondence with the physical edge spectrum on the cut for topological ground states [@li_haldane] and low-dimensional quantum antiferromagnets [@didier] and that the structure of the corresponding Entanglement Hamiltonian (EH) reflects closely the bulk properties (holographic principle) [@PEPS_cirac]. However, new interesting features might arise in the entanglement properties of systems with spontaneously broken continuous symmetry, such as the QNS for which SU(2) symmetry is broken down to U(1). First, the entanglement entropy (the entropy associated to the RDM) have revealed anomalous additive (logarithmic) corrections [@ee; @ee2] to the area law – i.e. the linear (asymptotic) scaling of the entropy with the length of the cut. It was proposed afterwards that the origin of such corrections may lie in the existence of Goldstone modes [@metlitski] associated to the spontaneously broken continuous symmetry. Note however that, in any finite system (as in most “exact” simulations), the SU(2) symmetry is restored by quantum fluctuations and one has a unique GS instead of a degenerate manifold. In fact, recent state-of-the-art SU(2)-symmetric Density Matrix Renormalization Group (DMRG) studies established an interesting correspondence [@TowerStates], in the (singlet) ground state of two-dimensional antiferromagnets in their magnetically-ordered phases, between the SU(2) tower of states and the lower part of the ES below an “entanglement gap” (although DMRG does not provide information on the momenta of the ES). This suggests strongly that the above-mentioned corrections in the entropy should be associated with the tower of states structure, while the area law arises from ES levels above the entanglement gap [@TowerStates]. [*A priori*]{} important differences may occur in the entanglement properties of a Néel-like wave-function breaking the continuous SU(2) symmetry explicitly i.e. with a finite staggered magnetization. In particular, one expects the ES (and the EH) of a symmetry-broken QNS to differ qualitatively from the ones associated to the GS with restored SU(2) symmetry, computed on finite systems [@TowerStates; @EH_luitz]. Computing the entanglement properties of a (variational) state with a finite order parameter is the main goal of this paper.
![(a) The Néel state is represented as a spinon-doped RVB state : Singlets are oriented from the A to the B sublattice and doped spinons are polarized along $\hat z$ ($-\hat z$) on the A (B) sites. Implicitly, a sum over all singlet/spinon configurations is assumed, the average spinon density being controlled by a fugacity. (b) Under a $\pi$-rotation around $\hat y$ on all the B-sites, all spinons become oriented along $\hat z$ and singlets transform into on every NN bonds.[]{data-label="Fig:neel"}](neel_rot){width="0.9\columnwidth"}
The formalism of Projected Entangled Pair States (PEPS) [@verstraetewolf06; @PEPS_cirac] enables i) to easily construct symmetry broken variational states and ii) to compute the corresponding EH. Note that, for a given variational state and system size (one uses infinite cylinders with a finite perimeter), the calculation of the EH is [*fundamentally exact*]{} and provides a complete analytic expansion in terms of N-body interactions whose amplitudes are numerically computed. Here I therefore make use of a simple PEPS ansatz of the QNS in order to calculate its EH associated to a bi-partition of an infinite cylinder. The variational wave function used here is in fact the simplest PEPS (i.e. with the smallest bond dimension $D=3$) one can construct to capture the physics of the symmetry-broken Néel state. Ansätze with a larger bond dimension will not allow to consider a cylinder with a large enough perimeter. Note that the PEPS formalism provides also the momentum-resolved ES. This is to be contrasted to DMRG that also gives easily the ES but without the corresponding momenta of the Schmidt states. Also an analytic form of the EH cannot be obtained in DMRG.
As shown recently using PEPS, a EH with local interactions is expected in a gapped bulk phase (with short-range entanglement), whereas a diverging interaction length of the EH is the hallmark of critical behavior in the bulk [@PEPS_cirac]. One therefore expects to see fingerprints of the critical behavior of the QNS in its Entanglement Hamiltonian.
Doped-RVB ansatz for the Néel state
===================================
I start with the square lattice Resonating Valence Bond (RVB) wavefunction defined as an equal-weight superposition of nearest-neighbor (NN) hardcore singlet coverings [@RVB_anderson1; @RVB_anderson2]. The sign structure of the wave function is fixed by imposing that the singlets are all oriented from one A sublattice to the other B sublattice. Such a wave function is a global spin singlet – i.e. a SU(2)-invariant state – with algebraic (i.e. critical) dimer correlations (and short-range spin correlations) [@RVB_critical1; @RVB_critical2]. To construct a simple ansatz for the QNS, let us now assume that one breaks SU(2) symmetry down to U(1) by doping the NN RVB state with on-site spinons (i.e. spin-1/2 excitations) with [*opposite*]{} orientations on the two sublattices. For simplicity, I choose hereafter the staggered magnetization pointing along the $\hat z$-axis. Such a simple ansatz is schematically shown in Fig. \[Fig:neel\](a). The average density of spinons – identical on the two sublattices – directly gives the staggered magnetization $m_{\rm stag}$ ($\times 2$) and, as one will see later on, can be controlled by a fugacity $\gamma$.
Before going further, it is convenient to rewrite the Néel-RVB state in a translationally invariant form. Indeed, under a (spin) $\pi$-rotation around $\hat y$ on the B-sites, B-spinons transform as and . Under such a (unitary) transformation, the new Néel-RVB state acquires the same (average) polarization on the A and B sublattices as shown in Fig. \[Fig:neel\](b). The original NN singlets are also transformed into dimers which are now symmetric w.r.t. the bond centers.
PEPS construction and energetics
================================
Such a state can in fact be represented by a PEPS $|\Psi_{\rm PEPS}\big>$ with bond dimension $D=3$, where each lattice site is replaced by a rank-5 tensor ${\cal A}^{s}_{\alpha,\alpha';\beta,\beta'}$ labeled by one physical index, $s=0$ or $1$, and by four virtual bond indices (varying from 0 to 2) along the horizontal ($\alpha,\alpha'$) and vertical ($\beta,\beta'$) directions, as shown in Fig. \[Fig:peps\](a). Physically, the absence of singlet on a bond is encoded by the virtual index being “2” on that bond. I define : $${\cal A}={\cal R+\gamma S} \, ,$$ where $\cal R$ is the original RVB tensor [@RVB_norbert; @RVB_didier], $\cal S$ is a polarized spinon tensor and $\gamma\in \mathbb{R}$ is a fugacity controlling the average spinon density. To enforce the hardcore dimer constraint, one takes ${\cal R}^{s}_{\alpha,\alpha';\beta,\beta'}=1$ whenever three virtual indices equal 2 and the fourth one equals $s$, and ${\cal R}^{s}_{\alpha,\alpha';\beta,\beta'}=0$ otherwise. The spinon tensor has only one non-zero element, ${\cal S}^{1}_{2,2;2,2}=1$. The wave function amplitudes are then obtained by contracting all virtual indices (except the ones at the boundary of the system). Note that the above PEPS ansatz for the Néel state bares similarities with the one used to describe the honeycomb RVB spin liquid under an applied magnetic field [@RVB_magnet]. However, a crutial difference is that this new ansatz is, by construction, fully U(1)-invariant in contrast to the spinon-doped RVB state of Ref. [@RVB_magnet].
![(Color online) (a) Local (rank-5) PEPS tensor. (b) Tensors are placed on a square lattice wrapped on a cylinder of perimeter $N_v$ and (quasi-) infinite length $N_h\gg N_v$. $B_L$ and $B_R$ boundary conditions are realized by fixing the virtual variables going out of the cylinder ends. A bipartition of the cylinder generates two L and R edges along the cut. []{data-label="Fig:peps"}](peps){width="0.9\columnwidth"}
Following a usual procedure, I now place the square lattice of tensors on infinite cylinders with $N_v$ sites in the periodic (vertical) direction as shown in Fig. \[Fig:peps\](b) and use standard techniques (involving exact tensor contractions and iterations of the transfer operator) to compute relevant observables. In the PEPS formulation the boundary conditions $B_L$ and $B_R$ can be simply set by fixing the virtual states on the bonds “sticking out" at each cylinder end. E.g. open boundary conditions are obtained by setting the boundary virtual indices to “2". Generalized boundary conditions can be realized as in Fig. \[Fig:peps\](b) by setting some of the virtual indices on the ends to “0" or “1".
![(Color online) NN (a) and next-NN (b) correlators $2\big<{\bf S}_i\cdot{\bf S}_j\big>$ – corresponding to the energies per site in units of the coupling constants – plotted as a function of $m_{\rm stag}$. Computations are done on infinite cylinders of perimeter $N_v=4$ and $N_v=6$. []{data-label="Fig:energies"}](Ener_vsM){width="0.9\columnwidth"}
I have computed the (staggered) magnetization $m_{\rm stag}$ and the expectation values of the Heisenberg exchange interactions ${\bf S}_i\cdot{\bf S}_j$ between NN and next-NN sites, varying $\gamma$ from zero to large values (to approach the classical Néel state). The data (normalized as the energy per site of the corresponding Heisenberg model) are displayed as a function of $m_{\rm stag}$ in Fig. \[Fig:energies\](a,b). The NN energy shows a broad minimum around $m_{\rm stag}\sim 0.35$, a value a bit larger than the QMC extrapolation $\sim 0.307$ [@QMC_sandvik] for the pure NN quantum AFM. However, (i) the variational energy curve is rather flat around the minimum and (ii) the minimum energy is only within $\sim 1.5\%$ of the QMC estimate, a remarkable result considering the simplicity of the one-dimensional family of $D=3$ PEPS. Note also that the minimum energy agrees very well with optimized $D=3$ iPEPS [@bauer] and finite PEPS up to $D=6$ [@michael]. For completeness, I also show the next-NN energy in Fig. \[Fig:energies\](b). In fact, the pure (critical) RVB state provides the lowest next-NN exchange energy, suggesting the existence of a transition, upon increasing the next-NN coupling, from the Néel state to a [*gapless*]{} spin liquid [@wang; @sheng]. Note that a direct transition from the Néel state to a Valence Bond Crystal – with no intermediate gapless spin liquid phase – is also a realistic scenario [@vbc; @QMC_sandvik2].
Entanglement Hamiltonian on infinite cylinders {#Sec:boundary}
==============================================
Bipartition and reduced density matrix
--------------------------------------
To define an Entanglement Hamiltonian associated to the family of Néel-RVB wavefunctions, I partition the $N_v\times N_h$ cylinder into two half-cylinders of lengths $N_h/2$, as depicted in Fig. \[Fig:peps\](b). Partitioning the cylinder into two half-cylinders reveals two edges L and R along the cut. Ultimately, I aim to take the limit of infinite Néel-RVB cylinders, i.e. $N_h\rightarrow\infty$ as before.
The reduced density matrix of the left half-cylinder obtained by tracing over the degrees of freedom of the right half-cylinder, $\rho_L={\rm Tr}_R\{
|\Psi_{\rm PEPS}\big>\big<\Psi_{\rm PEPS}|\}$, can be simply mapped, via a spectrum conserving isometry $U$, onto an operator $\sigma_b^2$ acting only on the $D^{\otimes N_v}$ edge (virtual) degrees of freedom, i.e. $\rho_L=U^\dagger \sigma_b^2 \,U$ [@PEPS_cirac]. The [*Entanglement (or boundary) Hamiltonian*]{} $H_b$ introduced above is defined as $\sigma_b^2=\exp{(-H_b)}$. As $\sigma_b^2$, $H_b$ is one-dimensional and its spectrum – the [*entanglement spectrum*]{} (ES) – is the same as the one of $-\ln{\rho_A}$. Note that the left and the right half-cylinders give identical EH. For further details on the derivation and the procedure, the reader is kindly asked to refer to Ref. .
For a topological state, such as the $\gamma=0$ RVB state, the Entanglement Hamiltonian depends on the choice of the $B_L$ and $B_R$ cylinder boundaries that define “topological sectors" [@RVB_didier; @Topo_norbert]. Adding any staggered magnetization $m_{\rm stag}$ in the PEPS immediately breaks the gauge symmetry of the tensors which is responsible for the disconnected topological sectors, as also happens in the case of field-induced magnetized RVB states [@RVB_magnet]. Therefore, all topological sectors are mixed and $H_b$ become independent of the boundary conditions $B_L$ and $B_R$ provided $N_h\rightarrow\infty$. Note also that $H_b$ inherits the U(1) symmetry (associated to rotations around the direction of $m_{\rm stag}$) of the Néel state.
Expansion in terms of N-body operators
--------------------------------------
To have a better insight of the Entanglement Hamiltonian, I expand it in terms of a basis of $N$-body operators, $N=0,1,2,\cdots$ [@PEPS_cirac; @RVB_didier]. For this purpose, I use a local basis of $D^2=9$ (normalized) ${\hat x}_\nu$ operators, $\nu=0,\cdots,8$ which act on the local (i.e. at some site $i$) configurations $\{ |0\big>, |1\big>, |2\big> \}$, where $|2\big>$ is the vacuum or “hole" state and $|0\big>$ and $|1\big>$ can be viewed as spin down and spin up particles, respectively. More precisely, ${\hat x}_0=\mathbb{I}^{\otimes 3}$, ${\hat x}_1=\sqrt{\frac{3}{2}}(|0\big>\big<0| -|1\big>\big<1|)$ and ${\hat x}_2=\frac{1}{\sqrt{2}}(|0\big>\big<0|+|1\big>\big<1|-2|2\big>\big<2|)$, for the diagonal matrices, complemented by $\hat x_3=\hat x_4^\dagger=\sqrt{3}|0\big>\big<1|$ acting as (effective) spin-1/2 lowering/raising operators, and $\hat x_5=\hat x_7^\dagger=\sqrt{3} |2\big>\big<0|$ and $\hat x_6=\hat x_8^\dagger=\sqrt{3} |2\big>\big<1|$ acting as particle hoppings. In this basis $H_b$ reads [@RVB_didier], $$\begin{aligned}
H_b &=& c_0 N_v+\sum_{\nu,i} c_{\nu} {\hat x}_\nu^i
+ \sum_{\nu,\mu,r,i} d_{\nu\mu}(r) \, {\hat x}_\nu^i {\hat x}_\mu^{i+r}
\nonumber \\
&+& \sum_{\lambda,\mu,\nu,r,r',i} e_{\lambda\mu\nu}(r,r') \, {\hat x}_\lambda^i {\hat x}_\mu^{i+r} {\hat x}_\nu^{i+r'}
+ \cdots \, ,\label{Eq:Hb0}\end{aligned}$$ where site superscript indices have been added and only the first one-body, two-body and three-body terms are shown.
![(Color online) Weights of the Entanglement Hamiltonian $H_b$ expended in terms of N-body operators. Data of several Néel-RVB wavefunctions (whose $\gamma$ values are mentioned on the plot) are shown. Calculations are done on an infinite cylinder with perimeter $N_v=6$. As seen e.g. in Ref. [@RVB_didier], finite size effects for such integrated quantities are typically quite small. []{data-label="Fig:weights"}](weights_vs_order.pdf){width="0.9\columnwidth"}
The total weights corresponding to each order of the expansion of $H_{b}$ in terms of N-body operators are shown in Fig. \[Fig:weights\] as a function of the order $N$ using a semi-logarithmic scale. The data reveal clearly a fast decay of the weight with the order $N$. This decay is compatible with an exponential law although more decades in the variation of the weights (i.e. larger $N_v$) would be needed to draw a definite conclusion. In any case, $H_{b}$ is dominated by two-body contributions in addition to the normalization constant and subleading one-body terms. The quantum Néel state is believed to be critical with power-law decay of spin-spin correlations [@QMC_sandvik2]. Therefore, according to Ref. [@PEPS_cirac], one expects $H_b$ to be long-ranged to some degree. So, one still needs to refine the analysis and investigate further the r-dependence of the leading two-body contributions. In the next Subsection, I show that $H_b$ indeed possesses long-range two-body terms that I characterize.
Entanglement Hamiltonian: an effective one-dimensional t–J model
----------------------------------------------------------------
It is known that the EH of the $\gamma=0$ RVB PEPS belongs to the $1/2\oplus 0$ representation of SU(2) and its Hilbert space is the same as the one of a one-dimensional bosonic model [@RVB_didier], interpreting $|0\big>$ and $|1\big>$ states ($|2\big>$ states) as $\downarrow$ and $\uparrow$ spins (holes). In the presence of a finite (staggered) magnetization in the bulk, the SU(2) symmetry is broken but $H_b$ keeps the unbroken $U(1)$ symmetry corresponding to spin rotations around the direction of the staggered magnetization.
![Largest weights $|c_\nu|$ and $|d_{\nu\mu}(r)|^2$ of the one-body (i.e. $r=0$) and two-body operators in the expansion of $H_b$ of the Néel-RVB PEPS as a function of distance $r$, for increasing $\gamma$ values (corresponding to staggered magnetizations $m_{\rm stag}\sim 0.059, 0.327$ and $0.483$, respectively).[]{data-label="Fig:bham"}](BHam_weights){width="0.9\columnwidth"}
The (largest) non-zero real coefficients in (\[Eq:Hb0\]) computed on an infinitely-long cylinder of perimeter $N_v=6$ are shown in Fig. \[Fig:bham\](a-c) for small (a), intermediate (b) and large (c) (staggered) magnetizations. At large and intermediate values of $m_{\rm stag}$, one finds a dominant one-body (diagonal) term which can be interpreted as a chemical potential term (up to a multiplicative factor) : $${\cal H}_2=c_2\sum_i {\hat x}_2^i = \frac{3}{\sqrt{2}}c_2\sum_{i} (n_i-2/3) \, ,
\label{Eq:mu}$$ where $n_i$ counts the number of particles (i.e. “0" and “1" states) on site $i$. The subleading one-body operator takes the form of a Zeeman coupling : $$\begin{aligned}
{\cal H}_{1}&=&\sqrt{6}c_1\sum_{i} S_i^z\, ,
\label{Eq:nn}\end{aligned}$$ where $S_i^z$ is an effective spin-1/2 component (along $\hat z$) and $c_1\simeq c_2/10$.
The leading 2-body contributions are hopping terms [*at all distances*]{} for the majority “spins" ($|1\big>$ states) : $$\begin{aligned}
{\cal H}_{68}(r)&=&d_{68}(r)\sum_{i} ({\hat x}_8^i{\hat x}_6^{i+r} + {\hat x}_6^i{\hat x}_8^{i+r})\nonumber \\
&=&3d_{68}(r)\sum_{i} (b_{i+r,1}^\dagger b_{i,1} + b_{i,1}^\dagger b_{i+r,1}) \, ,
\label{Eq:hopping}\end{aligned}$$ where $b_{i,s}^\dagger$ ($b_{i,s}$) are the canonical bosonic creation (annihilation) operators of the virtual $s=0,1$ states. Note that the minority spins ($|0\big>$ states) only hop at [*even*]{} distances (weights at odd distances are negligable) with much weaker amplitudes, . The next subleasing corrections are diagonal 2-body density-density interactions $$\begin{aligned}
{\cal H}_{22}(r)&=&d_{22}(r)\sum_{i} {\hat x}_2^i{\hat x}_2^{i+r}\nonumber \\
&=&\frac{9}{2}d_{22}(r)\sum_{i} (n_i-2/3) (n_{i+r}-2/3)\, ,
\label{Eq:nn}\end{aligned}$$ which become dominant when $\gamma, m_{\rm stag}\rightarrow 0$.
Other generic operators allowed by the $U(1)$ symmetry, like the anisotropic XXZ chain ($d_{11}\ne d_{34}=d_{43}$) or mixed operators of the form ${\cal H}_{12}\propto
\sum_{i} S_i^z (n_{i\pm r}-2/3)$ are also present but their amplitudes turn out to be quite small. Interestingly, $H_b$ (approximately) conserves the hole “2-charge" and, hence, does not contain pair-field operators with sizable amplitudes, in contrast to previous studies of $D=3$ PEPS [@RVB_didier; @RVB_magnet]. If one restricts to the dominant contributions (\[Eq:mu\]) and (\[Eq:hopping\]), $H_b$ is exactly a chain of a dilute mixture of heavy ($\downarrow$ spins or $|0\big>$ states) and light ($\uparrow$ spins or $|1\big>$ states) hardcore bosons, where light particles are subject to long-range hopping.
\[sec:ES\] Entanglement spectrum
================================
0.3truecm ![Entanglement spectrum of a bipartitioned $N_v=6$ RVB-Néel cylinder as a function of the momentum along the cut, for different values of the spinon fugacity $\gamma=0.1, 0.6$ and $3$ corresponding to $m_{\rm stag}\sim 0.059, 0.327$ and $0.483$, respectively. Different symbols are used for different $S_z$ sectors of the edge. []{data-label="Fig:es"}](ES_neel "fig:"){width="0.9\columnwidth"}
It is also of high interest to examine the ES in the QNS and compare it to ES obtained for GS where SU(2)-symmetry is restored on finite size systems [@TowerStates]. By definition the ES is the spectrum of $-\ln{\rho_L}$. Since $\rho_L$ and $\sigma_b^2=\exp{(-H_b)}$ are related by an isometry, it is also the spectrum of the Entanglement Hamiltonian $H_b$. ES are shown in Fig. \[Fig:es\](a-c) for 3 values of the fugacity $\gamma$, as a function of the momentum along the cut. Since $\sigma_b^2$ conserves the total $S_z$ of the chain ($U(1)$ symmetry), it can be block-diagonalized using this quantum number and the eigenvalues of $-\ln{\sigma_b^2}$ are displayed in each $S_z$ sector separately. It can be seen from Fig. \[Fig:es\](a) that the $\gamma=0$ SU(2) spin multiplets are split by a small spinon density. For increasing $\gamma$ (i.e. staggered magnetization), the splittings of the Kramer’s multiplets increase (see Fig. \[Fig:es\](b,c)) due to the relative increase of the amplitudes of the SU(2)-symmetry breaking terms like (\[Eq:hopping\]) in the EH. In the limit of large $\gamma$ where the classical Néel state is approached, one finds separated bands of energy levels. It may be that the ES is gapped for all $\gamma$ but finite size effects remain too large to reach a definite conclusion. In any case, the ES of Fig. \[Fig:es\] are to be contrasted to the ES obtained in DMRG for GS with restored SU(2)-symmetry (due to the use of finite size systems). Obviously the two types of ES are very different with a SU(2) tower of states structure at low energy for the ES of the singlet GS [@TowerStates] and a U(1) symmetric ES in the (variational) symmetry-broken Néel state.
Summary and discussion
======================
In this paper, I have investigated entanglement properties of a simple one-dimensional family of PEPS designed to describe qualitatively the GS of the square lattice AFM. These ansätze exhibit a finite staggered magnetization i.e. they break explicitly the SU(2) symmetry down to U(1) and can be studied on infinite cylinders with a finite perimeter. The goal of this study is therefore to examine the effects of such a finite order parameter on various entanglement properties and compare them to (QMC or SU(2)-symmetric DMRG) studies where symmetry is restored in a finite system. Thanks to the PEPS structure, the Entanglement Hamiltonian associated to a bipartition of the cylinder can be derived exactly (for a fixed perimeter). It is found that the EH inherits the U(1) symmetry of the Néel state and possesses a very simple structure : (i) its Hilbert space is the same as the one of a one-dimensional bosonic model, interpreting the 3 virtual states on the edge as a $\uparrow$ spin, a $\downarrow$ spin and a hole, (ii) when expended using a local basis of operators, it shows dominant two-body interactions and (iii) higher-order operators (three-body terms and beyond) represent less than 10$\%$ of its total weight. Examining in details the form of the two-body interactions, I find that the dominant ones are long-range hoppings of the majority (let say $\uparrow$) spins. It is however not possible to distinguish a power-law versus an exponential decay of these hopping terms. In any case, the associated Entanglement Spectrum is found to be qualitatively very different from the ones obtained in GS with restored SU(2)-symmetry [@TowerStates] (no tower of states structure is found, as suspected). Whether the entropy exhibits additive logarithmic corrections as in Refs. [@ee; @ee2] is difficult to answer. The absence of the tower of states in the ES suggests a negative answer. However, an hypothetical power-law decay of the hopping terms in the EH (instead of exponential) might lead to some additive corrections to the entropy. It would be interesting to complement our calculation of the ES using “conceptually exact" numerical methods (such as QMC or SU(2)-symmetric DMRG) on large but finite systems, adding a small external staggered field (to produce a finite order parameter), taking the limit of infinite system size first.
I acknowledge fundings by the “Agence Nationale de la Recherche" under grant and support from the CALMIP supercomputer center (Toulouse). I thank Claire for her patience and I am indebted to Fabien Alet, Ignacio Cirac, Nicolas Laflorencie, Roger Melko, Anders Sandvik, Norbert Schuch and Frank Verstraete for numerous discussions and insightful comments.
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---
abstract: 'Numerous lines of evidence indicate that the matter content of the Universe is dominated by some unseen component. Determining the nature of this Dark Matter is one of the most important problems in cosmology. Weakly Interacting Massive Particles (WIMPs) are widely considered to be one of the best candidates which may comprise the Dark Matter. A brief overview of the different methods being used to search for WIMP Dark Matter is given, focusing on the technologies of several benchmark experiments.'
author:
- |
Nader Mirabolfathi\
[*University of California, Berkeley*]{}
title: '**Direct and Indirect Searches for Dark Matter in the Form of Weakly Interacting Massive Particles (WIMPs)**'
---
=14.5pt
Introduction
============
The nature of dark matter is one of the oldest and most important open questions in cosmology, dating back to the first observations of anomalous high kinetic energies in distant galaxy clusters made by Swiss cosmologist Fritz Zwicky in 1933 [@zwicky]. Since then, exciting developments in observations of the cosmic microwave background (CMB), large-scale structure (LSS), and type Ia supernovae have allowed more accurate measurement of the various parameters of the “standard model of cosmology”. In particular, the Universe appears to be dominated ($\sim70\%$) by the vacuum energy density, or Dark Energy, while the remainder ($\sim30\% $) is made up of matter. Furthermore, most of the matter of the Universe appears to be non-luminous (i.e. dark) and non-baryonic. The nature of the Dark Matter is still unknown, and remains the subject of intense research activity. Weakly Interacting Massive Particles (WIMPs), a generic name for heavy particles interacting at the weak scale with baryonic matter, are among the best candidates which may comprise the Dark Matter. If such particles are produced thermally in the early universe, their weak-scale couplings explains why their relic density is of the order of critical density today. Independently, supersymmetry theories predict a stable s-particle state whose properties are very similar to the hypothetical WIMPs.
This paper is a brief review and update of different experiments aiming to detect Dark Matter in the form of WIMPs. The experiments described in this paper are not meant to be a complete list, but are selected to represent broader classes of the similar experiments.
Direct WIMP Searches
====================
WIMPs can be detected either via (very rare) elastic scattering off the nucleus of ordinary matter (“direct detection”) or by measuring their annihilation products (“indirect detection”). The latter method is the subject of the section 3. In this section, the expected spectrum and rate of interaction of WIMPs with ordinary matter are estimated. After the experimental challenges for directly detecting WIMPs have been introduced, a brief review of the status and results from few important direct detection experiments will be presented.
WIMPs are expected to interact with the nucleons in ordinary matter [@goodman]. The WIMP-nucleon elastic-scattering cross-section $\sigma_{\mathrm{WIMP-p}}$ is SUSY model dependent [@jungman]. One of the goals of all WIMP direct detection experiments is to determine or to limit $\sigma_{\mathrm{WIMP-p}}$, and thus constrain the free parameter space available for SUSY models. If the WIMPs are bound to the galaxy by the gravitational force, we can assume that their distribution should follow (Boltzman):
$$f(\vec r , \vec v, \vec v_E)=e^{-\frac{M_W (\vec v_E+\vec v)^2+M_W \Phi (\vec r)}{k_B T}}
\label{equ1}$$
where $M_W$ and $T$ are the WIMP mass and the equivalent temperature ($kT=1/2 M_W v_{0}^{2}$), $\Phi(\vec r)$ is the local gravitational potential, $v$ is WIMP velocity with respect to the earth, and $v_E$ is the velocity of the Earth with respect to the center of the galaxy (sum of the Sun’s velocity with respect to the center of the galaxy and the Earth’s velocity with respect to the Sun):
$$v_E = 232 + 15 \cos{2\pi \frac{t-152.5_{\mathrm{days}}}{365.25_{\mathrm{days}}}} \;\;\;[\textrm{km/s}]
\label{equ2}$$
Since the sinusoidal behavior comes from the Earth’s motion around the sun, we expect its amplitude to be determined by the relationship between the Sun’s velocity vector and the plane of the Earth’s orbit. Thus we should expect an annual modulation of the WIMP flux of $\sim \pm 6\%$. The event rate is the product of the number of target nuclei ($mN_0/A$), the incoming flux of WIMPs, $v\cdot n$ (n determined by the eq. \[equ1\] from cosmology), and $\sigma_{\mathrm{WIMP-p}}$ (predicted from the SUSY model in play). One could integrate over the WIMP velocity distribution and obtain the overall interaction rate. However, we are more interested in deriving the differential recoil energy spectrum. We will see that such a spectrum will directly give $\sigma_{\mathrm{WIMP-p}}$ and the mass of the WIMPs. The recoil energy of a nucleus of mass $M_T$, which is hit by a WIMP of energy $E$, and which is recoiling at an angle $\theta$, is given by:
$$E_R = E \frac{4 M_W M_T}{(M_W + M_T)^2} \frac{(1-\cos{\theta})}{2}
\label{equ3}$$
Assuming a hard sphere scattering model (uniform $E_R$ distribution) we can calculate the differential rate as:
$$\frac{\partial{R}}{\partial{E_R}} = \frac{R_0}{rE_0} \frac{2\pi^{3/2}v_0}{K} \int_{v_{min}}^{v_{esc}} v e^{\frac{(v+v_E)^2}{v_{0}^{2}}}dv
\label{equ4}$$
$$r = \frac{4M_WM_T}{(M_W+M_T)^2}
\label{equ4b}$$
where $R_0 \sim n_0v_0 \sigma_{\mathrm{WIMP-p}}$ , $E_0=k_BT$, $K$ is a normalization factor, $v_{min}$ is the minimum WIMPs velocity necessary to produce a recoil of $E_R$ and $v_{esc}$ is the galactic escape velocity. At the limit conditions ($v_e=0$ and $v_{esc} =\infty$), eq. \[equ4\] becomes:
$$\frac{\partial{R}}{\partial{E_R}} = \frac{R_0}{rE_0} e^{-\frac{E_R}{rE_0}}
\label{equ5}$$
This form, although incorrect, illustrates the typical exponential behavior of the recoil energy spectrum. In particular, it shows that by observing the amplitude and the shape of the recoil energy spectrum, it is possible to constrain two physically important parameters: $M_W$ and $\sigma_{\mathrm{WIMP-p}}$.
So far, the discussion has been limited to zero-momentum transfer. When the momentum transfer $q=(2M_TE_R)^{1/2}$ is large enough so that the wavelength $h/q$ is comparable to the size of the nucleus, coherence is lost and the cross section begins to decrease. This can usually be described by including a multiplicative form factor which depends on the type of WIMP-nucleon interaction (spin-dependent or spin-independent) as well as the nuclear structure: $\sigma(q2)=\sigma_0 F^2(q^2)$. Figure \[fig1\] (left) shows the nuclear form factors calculated for various materials used in WIMP search experiments. Since different experiments use different target nuclei, it is preferable to report the recoil energy spectrum referred to nucleons (e.g. proton) rather than the nucleus, to allow easy comparison between experiments. If the WIMP-nucleon interaction is spin-independent, the contribution of various nucleons will be added coherently. Two corrections are important: The cross-section scales as $\mu_T=M_TM_W/(M_T+M_W)$ and the WIMP-nucleus coupling scales as $A^2$. Hence,
$$\sigma_{\mathrm{WIMP-Nucleus}} = \sigma_{\mathrm{WIMP-p}}\frac{\mu^{2}_{\mathrm{Nucleus}}}{\mu^{2}_{p}}A^2
\label{equ6}$$
If the WIMP-nucleon cross-section is spin-dependent, WIMP-nucleon amplitudes still add coherently but the contributions of spin-paired nucleons will cancel each other. It is clear that the $A^2$ factor usually makes the spin-independent interaction dominant over the spin-dependent. We can now summarize these descriptions in a single equation:
$$\frac{\partial{R} (v_e, v_{esc})}{\partial{E_R}}\bigg|_{(T,q^2)} =\frac{\partial{R} (v_e, v_{esc})}{\partial{E_R}}\bigg|_{(p,0)}
\times F^{2}(E_R)\times S
\label{equ7}$$
where the ($T,q^2$) subscripts denote the WIMP interaction with the target nucleus at non-zero momentum transfer, ($p,0$) denote the same with the proton at zero momentum transfer, $F^2$ is the nuclear form factor, and $S$ is the scaling factor (eq. \[equ6\]).
Figure \[fig1\] (right) shows the expected rates on various materials for a 100 $\mathrm{GeV/c}^2$ WIMP with $\sigma = 10^{-42}\, \mathrm{cm}^2$. Several important points are apparent from these plots. First, the expected event rates are very low: Even for a 10 keV experimental energy threshold, one expects $<$ 0.5 event/kg/day. Therefore, a very important goal of the direct detection experiments is to understand and to suppress various types of background. Second, due to the exponential nature of the spectrum, the majority of the signal is at very low energies. Therefore a low-energy threshold is essential. Third, despite the significant $A^2$ advantage of heavy nuclei (e.g. Xe), the nuclear form factor (Figure \[fig1\] left) suppression makes them less optimal at E$>$20 keV. Therefore, Xe-based experiments would be more advantageous only if they could decrease the experimental threshold very low (which is not easy, as described later in this paper).
Thus the main goal of a WIMP search experiment is to produce detectors with extremely low background and very low threshold ($<$15 keV) using materials with the best sensitivity to WIMPs. Due to the small interaction cross-section, a high mass - or, more accurately, a high exposure ($M\times T$) - is desirable. This can be obtained using large detector masses and long exposure times. This requires a high stability of the readout system (in particular, readout threshold) to various environmental conditions.
Currently we can classify direct WIMPs search experiments into two different categories: The first category, the first developed historically, focuses on building a detector with high mass and to passively reduce background by shielding the detector active region at deep underground sites. The hope is that after long exposures the sensitivity to the WIMP signal rises above background and eventually one can hope to detect the cosmological signature (eq. \[equ2\]). However, because the signal-to-background discrimination is of statistical nature, the sensitivity of this method only increases with $(M \times T)^{1/2}$. As representatives of this category of experiments, we discuss in more detail DAMA-NaI and ZEPLINI in the following sections. The second category of direct WIMP search experiments focuses mainly on an event-by-event discrimination of signal against background. As described at the beginning of this chapter, WIMPs interacting with the nucleus cause nuclear-recoils while most of the radioactive background (electromagnetic interaction) interact with electrons, giving electron-recoils. It has been shown (see section 2.3) that with a proper detector design one can distinguish the two types of events with a very high efficiency. Compared to the first category of experiments, the sensitivity is now enhanced in direct proportion to the exposure. As we will see, the experiments using this method obtained the best WIMP sensitivities.
DAMA
----
The DAMA project was begun in 1990 by an Italian group at Gran Sasso underground laboratory [@belli]. This elegant project is based on highly radiopure NaI(Tl) scintillator detectors shielded rigorously from radioactive background. The scintillation properties of NaI have been studied extensively for nuclear physics instruments. In particular, it has been shown that nuclear-recoil events can produce scintillation photons. It is possible to purify the crystals to achieve very low levels of background. The scintillation-yields are fairly low (0.3 for Na and 0.09 for I), leading to a recoil-energy threshold of $\sim$20 keV for I, which is the more interesting nucleus for the WIMP-search due to its large $A^2$ factor. The group also showed that there is a slight difference between the pulse shapes produced by nuclear-recoil events and those produced by electron-recoil events. Though the latter factor could help to statistically discriminate WIMPs against radioactive background, it has been ignored in the DAMA data analysis due to low efficiency.
![[*On the left*]{}: The DAMA experiment’s annual modulation of the residual rate (total rate minus constant) in the (2-4), (2-5), and (2-6) keV energy intervals as a function of the time over 7 annual cycles (total exposure 107731 kg $\times$ day); end of data taking July 2002. [*On the right*]{}: Power spectrum of the measured (2-6) keV modulation; the principal mode corresponds well to a 1 year period [@dama].[]{data-label="fig2"}](fig1a.eps "fig:"){height="9cm"} ![[*On the left*]{}: The DAMA experiment’s annual modulation of the residual rate (total rate minus constant) in the (2-4), (2-5), and (2-6) keV energy intervals as a function of the time over 7 annual cycles (total exposure 107731 kg $\times$ day); end of data taking July 2002. [*On the right*]{}: Power spectrum of the measured (2-6) keV modulation; the principal mode corresponds well to a 1 year period [@dama].[]{data-label="fig2"}](fig1b.eps "fig:"){height="6cm"}
The DAMA experiment is placed among the first category of dark matter experiments (see the end of section 2), which require a large detector exposure (107,731 kg-day over 7 years of operation). If WIMPs form a halo embedding our galaxy, as the Sun is moving through the WIMP halo, the Earth should feel a wind of WIMPs with strength a function of its position in its orbit. Hence, one would expect to observe an annual modulation of the interaction rates. In 2000, using a five-year exposure, the DAMA collaboration claimed to observe a 6.3 $\sigma$ C.L annual modulation in WIMP-proton elastic scattering. Recently, DAMA confirmed the observation [@dama] by adding the results from two more annual cycles (Figure \[fig2\]). While the DAMA evidence for the annual modulation is clear, its interpretation is more questionable. Most of the modulation signal comes from the lowest energy bins (2-6 keV), where understanding the efficiencies is particularly important. Although DAMA performed a study of the various possible systematic effects, some doubts remain that the signal may be caused by some other, less interesting, effects. The doubts are further fueled by the fact that there are three experiments (namely CDMS, Edelweiss, and ZEPLIN I) which have explored the same parameter space and found no signal.
Several upcoming experiments may help resolve the conflict: NaIAD (Boulby mine, UK)[@ahmed] has 65 kg of NaI crystals and is already acquiring data. ANAIS (Canfranc, Spain) [@morales] plans to use a detector mass of 107 kg of NaI, and has currently successfully tested a prototype. Both of these experiments should test the entire signal region claimed by DAMA in the coming 2-3 years.
Xe-based experiments
--------------------
Liquid Xe can also be used to search for a direct WIMP signal. Xe has an obvious advantage over other materials due to its large $A^2$ (A=131) factor. However, as shown in Figure \[fig1\], this advantage is partially offset by the form factor suppression. Xe-based detectors would be particularly advantageous if their nuclear-recoil energy threshold can be reduced below 20 keV.
The signature of an interaction in Xe is twofold. First, there is electron excitation of Xe atoms, which leads to scintillation. Second, there is ionization of Xe atoms. The two signals can be used to discriminate against the electron-recoil events. In the absence of an electric field, the electron and ionized Xe recombine, producing secondary scintillation. The timing of the two scintillation pulses (nanosecond scale) differs for electron and nuclear-recoils, so pulse-shape discrimination can be used to suppress the electron-recoil background. This technique is used by ZEPLIN I (Boulby mine, UK) [@hart]. Their preliminary result is comparable with the Ge-based experiments CDMS (SUF) and Edelweiss, and incompatible with the signal region claimed by DAMA (Figure \[fig4\]). However, these results are not based on an [*in situ*]{} neutron calibration.
Alternatively, an electric field may be used to extract the electrons of the ionization signal. Such techniques are being investigated for ZEPLIN II and ZEPLIN III (both at the Boulby mine, UK) [@xe]. Possibly the most important advantage of Xe detectors is their scalability to large detector masses. Such experiments (ZEPLIN IV and XENON [@aprile]) are in the proposal stage. However, the dependence of the basic parameters, the ionization-yield and the scintillation-yield, on the energy and on the type of recoil are yet to be demonstrated.
Low temperature detectors: Solutions to event-by-event discrimination
---------------------------------------------------------------------
The cryogenic calorimeters are, so far, the technologies best able to meet the two necessary requirements for a WIMP detector: Low threshold ($<$10keV) and good energy resolution ($<$100 eV). A cryogenic calorimeter consist of a dielectric crystal ($\mathrm{Al}_2\mathrm{O}_3$, Ge, Si, $\mathrm{CaWO}_4$, etc.) cooled to temperatures as low as 0.01 $^{\circ}\mathrm{K}$. Because the heat capacity $C$ of a crystal varies as $(T/\Theta_D)^3$, ($\Theta_D$ is the Debye temperature, e.g. 374 $^{\circ}\mathrm{K}$ for Ge) at very low T a small energy deposit from a particle interaction could significantly ($10^{-5}$ $^{\circ}\mathrm{K}$) change the temperature of the absorber ($\Delta T=E/C$). A properly-attached thermometer (Mott-Anderson insulator or superconductor at its $Tc$) could thus measure the deposited energy. This is the best calorimetric measurement because all primary excitations due to the particle interaction will eventually transform into thermal excitations. From another point of view, at very low temperatures the phonon (lattice vibration quanta) content of the crystal thermal bath is very low, and thus the out-of-equilibrium (athermal) phonons produced after an interaction may be easily distinguished and counted in order to measure the deposited energy. This combined with the fact that the excitation energy to create phonons is very low, $\sim 10^{-5}$ eV, compared to $\sim$1eV for conventional semiconductor detectors, makes cryogenic detectors the best calorimeters at low energies yet developed.
The above introduction suggests two distinct (but physically related) methods of calorimetric measurement. One method consists of measuring the detector temperature after it reaches the equilibrium state (at higher $T$ of course) after the interaction. In this case, $\Delta T =E/C$ will directly measure the energy of the interacting particle. The second method, which requires a more elaborated readout system, is based on measuring the energy content of the athermal phonons created after an interaction, under the assumption that athermal phonons are proportionally produced and detected. As the athermal phonons carry information about the history of the event, the second method is more advantageous when it becomes important to reconstruct the history of an event in the detector. For example, it is often possible to reconstruct the location of an event in the detector, which in some cases is very important as we will see later in this paper. The signal amplitude in the first method depends on the mass of the detector as $C \propto M$, which limits the mass of the detectors. The second method does not suffer from this limitation, as long as the lifetime of the athermal phonons in the detector is longer than the response time of the readout system.
Calorimetric measurement alone is not enough to discriminate nuclear recoils (WIMPs) from electron-recoils (radioactive background), as the energy deposited does not depend on the type of interaction. However, it has been shown [@shutt1] that in semiconductor crystals (Ge, Si, etc.), the ionization-yield (charge/recoil energy) differs significantly between an electron-recoil and a nuclear-recoil. In Ge crystals, for example, the ionization-yield is 3 times bigger for an electron-recoil than for a nuclear-recoil. Figure 3 (left) shows the calibration results for one of the CDMS detector. Therefore, by simultaneously measuring ionization and phonons signal, one can obtain an event-by-event discrimination between a WIMP signal and the background. The CDMS and Edelweiss experiments use this method of detection and they are currently presenting the best sensitivities to WIMPs. This discrimination based on the ionization-heat measurement fails when an event occurs very close to the detector surface, in the “ionization dead-layer” [@shutt2]. The charge collection for such event could be incomplete, which could cause electron-recoil misidentification. By measuring athermal phonons, CDMS is able to identify and reject events occurring very close to the surface based on timing parameters of the phonon pulse [@mandic1]. The Edelweiss experiment currently uses NTD-based heat sensors sensitive only to overall changes in temperature, and is unable to localize events in this manner. However, the group’s recent studies of position-sensitive ionization-heat detectors look very promising [@mirabol],[@marnieros].
Scintillation-yield (light/recoil energy) could also differ between electron-recoils and nuclear-recoils. The CRESST experiment is based on simultaneous measurement of scintillation and ionization. There are also experiments, such as CUORE and CUORCINO [@cuore], which are based on heat measurement alone. The complex techniques involved in low temperature devices make such detectors difficult to scale to large masses. The main challenge of the above mentioned experiments is to increase the mass of the detectors without compromising their sensitivity.
### CDMS
CDMS uses ZIP (Z-dependent Ionization Phonon) detector technology to detect WIMPs [@irwin]. ZIPs are disc-shaped (76 mm in diameter, 10 mm high) germanium or silicon crystals (absorber). One face of the disc is divided into four quadrants. Each quadrant is covered by a thin layer (350 nm) of lithographically-patterned aluminum fins (athermal phonon collectors) and 1024 tungsten transition-edge sensors ($1 \mu m \times 250\mu m \times 35nm$ TES) which are evenly distributed over the surface. The Al layer is directly sputtered on the surface of the absorber ($>80\%$ surface coverage) and provides a good phonon-phonon coupling between the two materials. The athermal phonons can pass the interface between the absorber and Al fins and directly relax their energy into the Al by breaking the Cooper pairs (creating quasi-particles) in the Al. These quasi-particles will tunnel into and become trapped in the TES’s (tungsten $Tc \sim 0.07$ $^{\circ}\mathrm{K}$). The tungsten TES’s are biased at the superconducting transition temperature, and thus a small variation in the TES temperature (due to the quasiparticle trapping) will cause a significant change in the TES resistance ($\sim$10 m$\Omega$), which is then read out by a SQUID amplifier. The other face of the detector, which is also covered by aluminum, allows an electric field to be applied in order to collect the charge.
CDMS uses two methods to solve the near-surface event problem. First, an amorphous Si layer is introduced between the absorber and electrodes, which reduces the effect of near-surface trapping processes [@shutt2]. Second, the timing parameters of athermal-phonon signals can be used to identify the near surface events [@mandic1]. Figure \[fig3\] (right) shows the effectiveness of using athermal-phonon signal timing parameters in rejecting the near-surface events.
Each group of six Ge (250 g) or Si (100 g) detectors is packed in a single “tower” with their corresponding cold readout electronic instruments. Five “towers” are currently installed in a $\mathrm{He}_3-\mathrm{He}_4$ dilution fridge (operating $T<0.05$ $^{\circ}\mathrm{K}$) at Soudan underground laboratory. An overburden of 780 m of rock reduces the surface muon flux by a factor of $5 \cdot 10^{-4}$. Furthermore, the detectors are shielded against ambient radioactivity by $\sim$0.5 cm of copper, 22.5 cm of lead, and 50 cm of polyethylene (to shield against neutrons). A 5-cm-thick scintillator muon veto enclosing the shielding identifies charged particles (and some neutral particles) that pass through it.
Recently, CDMS published [@cdms] the analysis of its first Ge WIMP-search data (from the first “tower”) taken at Soudan during the period October 11, 2003 through January 11, 2004. After excluding time for calibrations, cryogen transfers, maintenance, and periods of increased noise, they obtained 52.6 live days with the four Ge and two Si detectors of “Tower 1”. This analysis revealed no nuclear-recoil events in 52.6 kg-d raw exposure in the Ge detectors. The data was used to set an upper limit on the WIMP-nucleon cross-section of $4 \cdot 10^{-43} \, \mathrm{cm}^2$ at the 90$\%$ C.L. at a WIMP mass of 60 $\mathrm{GeV}/\mathrm{c}^2$ for coherent scalar interactions and a standard WIMP halo (Figure \[fig4\]). CDMS, which currently gives the best sensitivity to WIMPs yet attained, is now operating 2 detector towers (Tower 1+Tower 2) and plans to run 5 towers through the year 2005. The expected sensitivity reach for $\sigma_{\mathrm{WIMP-nucleon}}$ is $\sim 3 \cdot 10^{-44}\, \mathrm{cm}^2$ based on 1200 kg-d projected esposure. Also a 99$\%$-C.L. detection possibility is considered if $\sigma_{\mathrm{WIMP-nucleon}} \sim 6 \cdot 10^{-44}\, \mathrm{cm}^2$.
![[*On the left*]{}: Ionization-yield versus recoil energy for a typical CDMS neutron (here with ${}^{252}$Cf) and gamma calibration. Also shown on the figure are the $\pm2\sigma$ nuclear-recoil band (dashed curve) and $\pm2\sigma$ electron-recoil band (solid curve). [*On the right*]{}: Phonon start time versus ionization-yield for ${}^{133}$ Ba gamma-calibration events (diamonds) and ${}^{252}$Cf neutron-calibration events (dots) in the energy range 20-40 keV in a typical CDMS detector. The diamonds that spread from yield=1 to yield=0.3 are near-surface events. Lines indicate typical timing and ionization-yield cuts, resulting in a high nuclear-recoil efficiency and a low rate of misidentified surface events[@cdms].[]{data-label="fig3"}](Z235_Cf_yErPRL.eps "fig:"){height="6.5cm"} ![[*On the left*]{}: Ionization-yield versus recoil energy for a typical CDMS neutron (here with ${}^{252}$Cf) and gamma calibration. Also shown on the figure are the $\pm2\sigma$ nuclear-recoil band (dashed curve) and $\pm2\sigma$ electron-recoil band (solid curve). [*On the right*]{}: Phonon start time versus ionization-yield for ${}^{133}$ Ba gamma-calibration events (diamonds) and ${}^{252}$Cf neutron-calibration events (dots) in the energy range 20-40 keV in a typical CDMS detector. The diamonds that spread from yield=1 to yield=0.3 are near-surface events. Lines indicate typical timing and ionization-yield cuts, resulting in a high nuclear-recoil efficiency and a low rate of misidentified surface events[@cdms].[]{data-label="fig3"}](Z5pdelcvsyPRL.eps "fig:"){height="6.5cm"}
### Edelweiss
The Edelweiss experiment [@benoit1] is located at the LSM (French acronym for Modane Underground Laboratory). About 1700 m of rock protect the experiment from radioactive backgrounds generated by cosmic rays. In the laboratory, the muon flux is reduced by a factor $2\cdot 10^{-6}$ compared to the flux at sea level. The experiment is surrounded by passive shielding made of paraffin (30 cm), lead (15 cm), and copper (10 cm). Edelweiss uses the same principle as CDMS for WIMP detection: Ionization-heat discrimination. Unlike CDMS’s athermal phonon sensors, the tiny rise in temperature due to a particle event is measured by an NTD (Neutron Transmutation Doped) heat sensor glued onto one of the charge-collection electrodes.
In 2000 and 2002, 11.6 kg-day were recorded with two different detectors [@benoit2]. In 2003, three new detectors were placed in the cryostat and 20 kg-day were added to the previous published data. Three events compatible with nuclear-recoils have been observed. However, the recoil energy of one of the events is incompatible with a WIMP mass $<1\, \mathrm{TeV/c}^2$. The two other events have been used to set the upper limit for WIMP-nucleon spin-independent interaction shown in Figure \[fig4\]. The new limit is identical to the previous (11.7 kg-day), since the experiment is currently background-limited. The lack of an active surface-event rejection makes the distinction between nuclear-recoils and near-surface background events very difficult. Edelweiss is now implementing a new design based on NbSi thin-film Anderson insulator thermometers. The new detectors, which are sensitive to athermal phonons and have already demonstrated a high surface event rejection efficiency, will be functional during the Edelweiss II experimental stage [@broniat].
As of March 2004, the Edelweiss I experiment has been stopped to allow the installation of the second-stage Edelweiss II. The aim is a factor of 100 improvement in sensitivity. A new low-radioactivity cryostat (with a capacity of 50 liters), able to receive up to 120 detectors, is being tested in the CRTBT laboratory at Grenoble. The first runs will be performed with twenty-one 320 g Ge detectors equipped with NTD heat sensors and seven 400 g Ge detectors with NbSi thin film. With an improved polyethylene and lead shielding and an outer muon veto, the expected sensitivity for $\sigma_{\mathrm{WIMP-nucleon}}$ is about $10^{-44}\, \mathrm{cm}^2$.
### Scintillation-heat : CRESST
The simultaneous detection of scintillation light and phonons in cryogenic calorimeters using scintillating absorber crystals can give a background suppression similar to that provided by the simultaneous measurement of ionization and light. Very recently it was shown [@coron] that a large variety of scintillating crystals ($\mathrm{CaWO}_4$, BaF, $\mathrm{PbWO}_4$, etc.) can be used in this manner. This gives this method a big advantage in identification of WIMP signals. The experiments using this technique are CRESST II and Rosebud. This technique has an important advantage over the Ge-based detectors in that it does not have surface-event problems. However, the technique also has some difficulties. First, rather than using PMTs to observe the scintillation signal (due to their high radioactive background), the current approach (taken by CRESST II [@altman]) is to use a second, phonon-mediated detector adjacent to the primary detector. The light collection is relatively poor, resulting in an energy threshold of 15-20 keV. Second, there are three nuclei in the crystal, all of which could potentially interact with the WIMPs. The scintillation-yield produced by the three nuclei has yet to be studied carefully, making event interpretation difficult. The goal of CRESST II is to build a 10 kg detector consisting of 300 g crystals to reach a sensitivity for $\sigma_{\mathrm{WIMP-nucleon}}$ of the order of $10^{-44}\, \mathrm{cm}^2$.
Indirect WIMP Searches
======================
We now review the current state of the various WIMP indirect search methods. Indirect detection experiments search for products of WIMP annihilation in regions that are expected to have relatively large WIMP concentration. Examples of such regions are galactic centers, the center of the Sun, or the center of the Earth, where the WIMPs are expected to be gravitationally captured. Such searches assume that the WIMP is its own antiparticle (as predicted by SUSY models) or that equal numbers of WIMPs and anti-WIMPs are present. Higher WIMP density gives a larger annihilation signal, which can be manifested as a flux of $\gamma$-rays, neutrinos, or antimatter (positrons or anti-protons) produced in the WIMP-annihilation. We discuss these possibilities in some detail.
$\gamma$-rays
-------------
WIMP-annihilation can produce $\gamma$-rays in several different ways. First, a continuous spectrum is produced from the hadronization and decay of $\pi_0$’s produced in the cascading of the annihilation products. Second, $\gamma$-ray spectral lines are produced by the annihilation channels in which $\gamma$’s are directly produced, such as $XX \rightarrow \gamma \gamma$ (producing a line at $M_X$) and $XX \rightarrow \gamma Z$ (producing a line at $M_X(1-M^{2}_{Z}/M^{2}_{X})$). Observing such lines would be a clear detection of WIMP annihilation. Such $\gamma$-rays could be produced close to the galactic center. Although the production rates are relatively low, a large halo density may compensate sufficiently to make such signals observable.
Ground-based experiments rely on Atmospheric Cerenkov Telescopes (ACTs), which detect the Cerenkov light emitted by the shower produced by a $\gamma$-ray interacting at the top of the atmosphere. Some experiments, such as CELESTE (France) [@cenbg] and STACEE (New Mexico) [@stacee], use the large mirrored areas used by solar power plants. These two experiments are sensitive to 20-250 GeV $\gamma$-rays. A number of experiments use dedicated mirrors or arrays of mirrors with a detector in the focal point: CANGAROO (Australia) [@icrhp], VERITAS (Arizona) [@veritas], CAT (France) [@cat], HESS (Namibia) [@hess], HEGRA (Canary Islands, dismantled) [@hegra], and MAGIC (Canary Islands) [@magic]. Such experiments are typically sensitive to 100 GeV - 10 TeV $\gamma$-rays. They are also capable of distinguishing (usually at $>$ 99$\%$ efficiency) between the showers caused by $\gamma$-rays and those caused by cosmic rays (their dominant background). Finally, there are satellite-based experiments: EGRET [@hartman] completed its mission and observed $\gamma$-rays in the 20 MeV - 30 GeV energy range, and GLAST [@glast] is scheduled to launch in 2006 and observe $\gamma$-rays of energies 10 MeV - 100 GeV.
Recently, two experiments have observed an excess flux of $\gamma$-rays coming from the galactic center. VERITAS [@kosack], operating at the Whipple 10 m telescope on Mt. Hopkins, Arizona, observed an integral flux of $1.6\pm0.5\pm0.3\cdot 10^{-8} \mathrm{m}^{-2}\mathrm{s}^{-1}$ with the energy threshold of 2.8 TeV. CANGAROO [@tsuchi], with a lower threshold of 250 GeV, made a $\sim 10 \sigma$ detection of the $\gamma$-ray source in the galactic center over the range 250 GeV -2.5 TeV. However it is difficult to reconcile [@hooper] the results of CANGAROO and VERITAS. The spectrum measured by CANGAROO is consistent with a WIMP mass of 1-3 TeV, while VERITAS, with its energy threshold of 2.8 TeV, requires a much heavier WIMP. Moreover, very high annihilation rates are required for this signal to be explained by WIMP annihilation. This implies very high annihilation cross-section and very high dark matter concentration at the galactic center. We note the possibility that these observations could also be explained by astrophysical sources, in particular the black hole at the galactic center. Results from the HESS experiment are expected in the near future - with four telescopes - HESS is expected to be more sensitive in the direction of the galactic center and to have superior angular resolution.
Neutrinos
---------
Although WIMPs are expected to scatter very infrequently, they do scatter off of nuclei in the Sun or the Earth, lose, and become gravitationally bound. Hence, the density of WIMPs at the center of the Earth or the Sun can be considerably larger than in the halo, implying higher annihilation rates. Neutrinos produced in such WIMP-annihilations would penetrate to the Earth’s surface, or escape from the Sun. The neutrinos can be produced both directly $XX\rightarrow \nu \bar{\nu}$ and indirectly $XX\rightarrow f \bar{f}$, where the fermion $f$ can decay and emit a neutrino. Hence, the energy spectrum is expected to be continuous, rather than a line, but it is expected to extend up to the WIMP mass. If the neutrino interacts with rock sufficiently close to the Earth’s surface, the products of the interaction may be detectable. The muon neutrinos are, hence, of the most interest, because their interactions produce muons which can travel considerable distance through the rock and reach a detector (electrons are absorbed at very short distances). The muon neutrinos can be detected at the surface of the Earth, usually using dedicated solar or atmospheric neutrino detectors. In practice, one searches for upward-going muons - for the high-energy neutrinos, the muons produced are well-collimated with the original neutrino direction and carry much of the original neutrino’s energy. Hence, one can search for the upward-going muon signal with a high-energy-threshold detector. The only known background are atmospheric neutrinos produced in the cosmic rays interactions with the atmosphere at the opposite side of the Earth.
Experiments designed to study solar or atmospheric neutrinos can also be used to look for the WIMP-annihilation neutrino signal. At the moment, none of the experiments has observed excess neutrinos from the Earth or the Sun, but several experiments have determined upper bounds on their flux: Baksan (neutrino experiment in Caucasus, Russia) [@boliev], SuperKamiokande (atmospheric neutrino experiment in Japan) [@desai], MACRO (liquid scintillator neutrino experiment in Italy) [@ambrosio], and AMANDA II (ice Cerenkov detector at the South Pole) [@ahrens]. These experiments are just beginning to probe the theoretically-allowed regions in supersymmetric WIMP models. Future experiments, such as ANTARES [@antares] and Lake Baikal [@baikal], as well as future runs of AMANDA II and IceCube, are expected to improve the sensitivity to WIMP-annihilation neutrinos by $\sim$2 orders of magnitude.
Conclusion
==========
Direct detection experiments have already explored the regions of the most optimistic SUSY models. Despite their lower exposures ($\sim$50 kg-day, compared to 110,000 kg-day), event-by-event discrimination methods are currently giving the best sensitivities to the WIMP-nucleon scalar scattering cross-section. Extremely high discrimination combined with large mass seems to be the only solution for the next generation of direct detection experiments. The two-order-of-magnitude increase in the sensitivity of next-generation experiments will explore the core of many SUSY models in the next few years. Indirect detection will be complementary, but hardly competitive, for low-$\sigma$ scalar WIMP detection. When combined with accelerator (LHC) results, the next generation of direct detection experiments may soon let us pinpoint the nature of the dark matter.
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---
bibliography:
- 'prlSpectra2.bib'
title: 'Pion, Kaon, and Proton Production in Central Collisions at = 2.76 TeV'
---
The ALICE Collaboration {#app:collab}
=======================
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---
abstract: 'The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains bounds for the Castelnuovo-Mumford regularity of the Rees algebra and for the Artin-Rees numbers.'
author:
- '[José M. Giral and Francesc Planas-Vilanova]{}'
title: Integral degree of a ring and reduction numbers
---
Introduction
============
Let $A$ be a commutative noetherian ring with identity, let $I$ be an ideal of $A$ and let $M$ be a finitely generated $A$-module. An ideal $J\subset I$ is said to be a reduction of $I$ with respect to $M$ if $I^{n+1}M=JI^{n}M$ for some integer $n\geq 0$. The least such integer $n\geq 0$ is called the $J$-[*reduction number of*]{} $I$ [*with respect to*]{} $M$ and is denoted by ${\rm rn}_{J}(I;M)$. If $M=A$, the phrase “with respect to $M$” is omitted and one writes ${\rm
rn}_{J}(I)$. Clearly, if $J$ is a reduction of $I$, then $J$ is a reduction of $I$ with respect to $M$ and ${\rm rn}_{J}(I;M)\leq {\rm
rn}_{J}(I)$. If $I$ is regular (i.e. $I$ contains a non zero divisor) and $J$ is a principal reduction, then ${\mbox{$\mathrm{rn}$}}_{J}(I)$ is independent of the given principal reduction and is denoted by ${\mbox{$\mathrm{rn}$}}(I)$.
In the last decade there has been a great deal of attention to finding bounds on the reduction number (see e.g. [@aht], [@cpv], [@dgh2], [@dkv], [@hoa] [@rossi], [@vasconcelos1], [@vasconcelos2], [@vasconcelos3], by no means a complete list of references). If the ideals have principal reductions and the integral closure is finite, d’Anna, Guerrieri and Heinzer gave an absolute bound for the reduction number in terms of the minimal number of generators of the integral closure ([@dgh2], Corollary 5.2). In this paper we express the supremum of reduction numbers of ideals having principal reductions in terms of the following new invariant associated to the ring $A$, provided $A$ contains the field of rational numbers ${\mbox{$\mathbb{Q}$}}$. If $A\subset B$ is a ring extension and $b\in
B$ is integral over $A$, let the [*integral degree*]{} of $b$ over $A$ be $$\begin{aligned}
{\rm id}_{A}(b)={\rm min}\{ n\geq 1\mid b\mbox{ satisfies an integral
equation of degree }n\}.\end{aligned}$$ If $A\subset B$ is an integral extension, the [*integral degree*]{} of $B$ over $A$ is defined as $$\begin{aligned}
{\mbox{$\mathrm{d}$}}_{A}(B)={\rm sup}\{{\rm id}_{A}(b)\mid b\in B\},\end{aligned}$$ When $B$ is taken to be $\overline{A}$, the integral closure of $A$ in its total quotient ring, ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})$ is just called the [*integral degree*]{} of $A$. We will prove that if $A$ has finite integral closure then it has also finite integral degree and that the converse is not true in general. Our main result is:
Let $A$ be a noetherian ring, $A\supset{\mbox{$\mathbb{Q}$}}$. Then $$\begin{aligned}
{\mbox{$\mathrm{d}$}}_{A}(\overline{A})={\rm sup}\, \{ {\rm rn}(I)\mid I\mbox{ regular
ideal of }A\mbox{ having a principal reduction}\} +1.\end{aligned}$$
In Theorem \[id=rn\] it is possible to replace ${\mbox{$\mathrm{rn}$}}(I)+1$ by either ${\mbox{$\mathrm{reg}$}}({\mbox{$\mathcal{R}(I)$}})+1$ or ${\mbox{$\mathrm{rt}$}}(I)$, ${\mbox{$\mathrm{reg}$}}({\mbox{$\mathcal{R}(I)$}})$ and ${\mbox{$\mathrm{rt}$}}(I)$ being the Castelnuovo-Mumford regularity of the Rees algebra of $I$ and the relation type of $I$, respectively.
It is known that Artin-Rees numbers are bounded by the relation type and that, in some particular cases, the relation type can be bounded by reduction numbers. Having in mind this idea and as a consequence of Theorem \[id=rn\], we get the following results in the context of uniform Artin-Rees properties.
Let $A$ be a noetherian ring with finite integral degree ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})=d$. Suppose that $A\supset{\mbox{$\mathbb{Q}$}}$. Let $N\subset M$ be two finitely generated $A$-modules. Let $I$ be a regular ideal of $A$ having a principal reduction generated by a $d$-sequence with respect to $M/N$. Then, for every integer $n\geq d$, $$\begin{aligned}
I^{n}M\cap N=I^{n-d}(I^{d}M\cap N).\end{aligned}$$
In other words, ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})$ is a uniform Artin-Rees number for the pair $N\subset M$ and the whole set of regular ideals having principal reductions generated by a $d$-sequence with respect to $M/N$. Our ideal-theoretic version is the following.
Let $A$ be a noetherian ring, $A\supset{\mbox{$\mathbb{Q}$}}$. Let ${\mbox{$\mathfrak{a}$}}$ be an ideal of $A$ such that $A/{\mbox{$\mathfrak{a}$}}$ has finite integral degree ${\mbox{$\mathrm{d}$}}_{A/{\mbox{$\mathfrak{a}$}}}(\overline{A/{\mbox{$\mathfrak{a}$}}})=d$. Let $I$ be an ideal of $A$ such that $IA/{\mbox{$\mathfrak{a}$}}$ has an $A/{\mbox{$\mathfrak{a}$}}$-regular principal reduction. Then, for every integer $n\geq d$, $$\begin{aligned}
I^{n}\cap {\mbox{$\mathfrak{a}$}}=I^{n-d}(I^{d}\cap {\mbox{$\mathfrak{a}$}}).\end{aligned}$$
If the integral degree of $\overline{A/{\mbox{$\mathfrak{a}$}}}$ is not finite or if $IA/{\mbox{$\mathfrak{a}$}}$ has no principal reduction, then there may not exist such a uniform Artin-Rees number (Example \[ehe\] and Example \[wange\]). This will be seen by using an example of Eisenbud and Hochster in [@eh], the work where they raised the uniform Artin-Rees conjecture, and an example of Wang in [@wang] (see also [@ocarroll1], [@do], [@ocarroll2], [@huneke], [@planas] for more information). On the other hand, it is well known that there exists a uniform Artin-Rees number for the set of principal ideals of a noetherian ring and that, in general, there does not exist a uniform Artin-Rees number for the set of three generated ideals (see the work of O’Carroll in [@ocarroll2] and the just mentioned example of Wang in [@wang]). Therefore, it remained to study if there exists a uniform Artin-Rees number for the whole set of two-generated ideals (without any other assumption on the ideals). We obtain a slightly weaker uniform Artin-Rees property for the set of two-generated regular ideals (not true anymore for the set of three-generated ideals, see Example \[wange\]). Concretely,
Let $(A,{\mbox{$\mathfrak{m}$}})$ be a noetherian local ring with infinite residue field. Let ${\mbox{$\mathfrak{a}$}}$ be an ideal of $A$ such that $A/{\mbox{$\mathfrak{a}$}}$ has finite integral degree ${\mbox{$\mathrm{d}$}}_{A/{\mbox{$\mathfrak{a}$}}}(\overline{A/{\mbox{$\mathfrak{a}$}}})=d$. Let $I$ be a two-generated ideal of $A$ such that $IA/{\mbox{$\mathfrak{a}$}}$ is $A/{\mbox{$\mathfrak{a}$}}$-regular. Then, for every $n\geq d$, $$\begin{aligned}
I^{n}\cap {\mbox{$\mathfrak{a}$}}=I^{n-d}(I^{d}\cap {\mbox{$\mathfrak{a}$}}) +{\mbox{$\mathfrak{m}$}}I^{n}\cap {\mbox{$\mathfrak{a}$}}.\end{aligned}$$
The paper is organized as follows. Sections \[armj\], \[rtmj\], \[cmr\] and \[rtrn\] are devoted to the following four invariants and the relationship among them: Artin-Rees numbers modulo an ideal, relation type of a standard module, Castelnuovo-Mumford regularity and reduction number with respect to a module. Concretely, in Section \[armj\] we introduce the Artin-Rees number $s_{J}(N,M;I)$ of an ideal $I$, two finitely generated $A$-modules $N\subset M$ and modulo another ideal $J$. This number is the minimum integer $s\geq 0$ such that $I^{n}M\cap N=I^{n-s}(I^{s}M\cap N)+JI^{n}M\cap N$ for all $n\geq
s+1$, and thus it controls the weaker Artin-Rees property of Theorem \[ideals2\]. Following the ideas in [@planas], in Section \[rtmj\] we bound above the Artin-Rees number $s_{J}(N,M;I)$ by the relation type of the Rees module ${\mbox{$\mathcal{R}_{J}(I;M/N)$}}=(\oplus
_{n\geq 0}I^{n}M/N)\otimes A/J$. Section \[cmr\] is dedicated to recalling some definitions around Castelnuovo-Mumford regularity and formulating an extension to modules of some results of Trung in [@trung1] and [@trung2]. In Section \[rtrn\], we prove that the relation type of an ideal $I$ with respect to a module $M$, ${\rm
rt}(I;M)$, is bounded above by ${\mbox{$\mathrm{rn}$}}_{J}(I;M)+{\mbox{$\mathrm{rt}$}}(I;J^{r}M)$, where $J$ is a reduction of $I$ with respect to $M$ and $r:={\mbox{$\mathrm{rn}$}}_{J}(I;M)$ is the $J$-reduction number of $I$ with respect to $M$. If $J$ is generated by a complete $d$-sequence with respect to $I$ and $M$ (the terminology is explained in Section \[rtrn\]), then the relation type of $J$ with respect to $I^{r}M$ satisfies ${\rm rt}(J;I^{r}M)=1$ and ${\mbox{$\mathrm{reg}$}}({\mbox{$\mathcal{R}(I;M)$}})={\mbox{$\mathrm{rn}$}}_{J}(I;M)$. Thus one has the inequality ${\rm rt}(I;M)\leq {\rm rn}_{J}(I;M)+1$, which is well-known for the case $M=A$ and $J$ a principal reduction of a regular ideal $I$ (see the work of d’Anna, Guerrieri and Heinzer [@dgh1], Huckaba [@huckaba2], [@huckaba3], Schenzel [@schenzel] and Trung [@trung1],[@trung2]). In Section \[id\], we introduce and study ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})$, the integral degree of $A$, a new invariant associated to the ring $A$. We prove that if $A$ has finite integral closure then it also has finite integral degree. The ingenious example of Akizuki ([@akizuki], see also [@reid]) provides us with an example of a one-dimensional noetherian local domain $A$ with finite integral degree but infinite integral closure. In Section \[idrn\] we prove the main result of the paper, namely, ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})$ is equal to the supremum of the reduction numbers plus one (or else the Castelnuovo-Mumford regularity of the Rees algebra plus one or the relation type) of regular ideals having principal reductions. Finally, in Section \[uarp\] we prove all the results concerning Artin-Rees numbers.
All rings will be commutative and with identity. As usual, $M$-regular will mean not contained in the set of zero divisors of $M$ and $\mu$ will stand for the minimal number of generators.
Artin-Rees modulo an ideal {#armj}
==========================
Let us introduce a slight variant of the Artin-Rees Lemma which will be very useful. Let $A$ be a noetherian ring, $I$ an ideal of $A$ and $N\subseteq M$ two finitely generated $A$-modules. The Artin-Rees Lemma assures that there exists an integer $s\geq 0$, depending on $N$, $M$ and $I$, such that for all $n\geq s$, $$\begin{aligned}
I^{n}M\cap N=I^{n-s}(I^{s}M\cap N).\end{aligned}$$ In particular, for any ideal $J$ of $A$, one obtains what we will call [*Artin-Rees modulo*]{} $J$: $$\begin{aligned}
I^{n}M\cap N=I^{n-s}(I^{s}M\cap N)+JI^{n}M\cap N .\end{aligned}$$ For every integer $n\geq 1$, let $$\begin{aligned}
E_{J}(N,M;I)_{n}=\frac{I^{n}M\cap N}{I(I^{n-1}M\cap
N)+JI^{n}M\cap N}\, .\end{aligned}$$ For easy reference, and without proof, we state the Artin-Rees lemma modulo $J$.
\[sj\] Let $A$ be a ring, $I,J$ ideals of $A$ and $N\subseteq M$ two $A$-modules. Set $$\begin{aligned}
s_{J}(N,M;I)={\rm min}\{ s\geq 0\mid
E_{J}(N,M;I)_{n}=0 \mbox{ for all } n\geq s+1\}.\end{aligned}$$ Then, the following conditions are equivalent:
- $I^{n}M\cap N=I^{n-s}(I^{s}M\cap N)+JI^{n}M\cap N$ for all $n\geq s+1$.
- $s_{J}(N,M;I)\leq s$.
If $A$ is noetherian and $N\subseteq M$ are finitely-generated $A$-modules, then $s_{J}(N,M;I)$ is finite.
If $J=0$, we recover the standard notion of Artin-Rees and simply write $s(N,M;I)$. Remark that if $J_{1}\subset J_{2}$ are two ideals, and for $n\geq 1$, there is a natural epimorphism $E_{J_{1}}(N,M;I)_{n}\to E_{J_{2}}(N,M;I)_{n}$ and thus $s_{J_{2}}(N,M;I)\leq s_{J_{1}}(N,M;I)\leq s(N,M;I)$.
\[sjs0\][ If $A$ is noetherian, $J\subset I$ are two ideals of $A$ contained in the Jacobson radical of $A$, and $N\subset M$ are two finitely generated $A$-modules, then $s_{J}(N,M;I)=s(N,M;I)$. ]{}
Since $0\subset J\subset I$, then $s_{I}(N,M;I)\leq
s_{J}(N,M;I)\leq s(N,M;I)$. It is enough to see $s(N,M;I)\leq
s_{I}(N,M;I)$. Set $s=s_{I}(N,M;I)$, so $I^{n}M\cap
N=I^{n-s}(I^{s}M\cap N)+I^{n+1}M\cap N$ for all $n\geq s+1$. Then $I^{n+1}M\cap N=I^{n+1-s}(I^{s}M\cap N)+I^{n+2}M\cap N$ and substituting the second equality in the first, $I^{n}M\cap
N=I^{n-s}(I^{s}M\cap N)+I^{n+1-s}(I^{s}M\cap N)+I^{n+2}M\cap
N=I^{n-s}(I^{s}M\cap N)+I^{n+2}M\cap N$. Inductively, $I^{n}M\cap
N=\cap _{k\geq 1}(I^{n-s}(I^{s}M\cap N)+I^{n+k}M\cap N)\subset \cap
_{k\geq 1}(P+I^{n+k}M)$, where $P=I^{n-s}(I^{s}M\cap N)\subset
I^{n}M\cap N\subset M$. But, $$\begin{aligned}
\frac{\bigcap _{k\geq 1}(P+I^{n+k}M)}{P}= \bigcap _{k\geq
1}\left(\frac{P+I^{n+k}M}{P}\right)=\bigcap _{k\geq 1}I^{n+k}(M/P),\end{aligned}$$ which is zero by Krull’s intersection theorem. Therefore, $\cap
_{k\geq 1}(P+I^{n+k}M)=P$ and $I^{n}M\cap N=I^{n-s}(I^{s}M\cap N)$.
Relation type modulo an ideal {#rtmj}
=============================
A standard $A$-algebra is a commutative graded algebra $U=\oplus
_{n\geq 0}U_{n}$, with $U_{0}=A$ and $U$ generated by the elements of degree 1. The Rees algebra of $I$ is the standard $A$-algebra ${\mbox{$\mathcal{R}(I)$}}=\oplus _{n\geq 0}I^{n}$. For any ideal $J$ of $A$, the [*Rees algebra of*]{} $I$ [*modulo*]{} $J$ will be the standard $A/J$-algebra ${\mbox{$\mathcal{R}_{J}(I)$}}={\mbox{$\mathcal{R}(I)$}}\otimes A/J=\oplus _{n\geq 0}
I^{n}/JI^{n}$. Taking $J=I$, we recover the associated graded ring of $I$, ${\mbox{$\mathcal{R}_{I}(I)$}}={\mbox{$\mathcal{G}(I)$}}=\oplus _{n\geq 0}I^{n}/I^{n+1}$, and taking $J={\mbox{$\mathfrak{m}$}}$ a maximal ideal of $A$, we recover the fiber cone of $I$, ${\mbox{$\mathcal{R}_{{\mbox{$\mathfrak{m}$}}}(I)$}}={\mbox{$\mathcal{F}_{\mathfrak{m}}(I)$}}=\oplus _{n\geq 0}I^{n}/{\mbox{$\mathfrak{m}$}}I^{n}$.
A standard $U$-module will be a graded $U$-module $F=\oplus _{n\geq
0}F_{n}$ such that $F_{n}=U_{n}F_{0}$ for all $n\geq 0$. The Rees module of $I$ with respect to $M$ is the standard ${\mbox{$\mathcal{R}(I)$}}$-module ${\mbox{$\mathcal{R}(I;M)$}}=\oplus _{n\geq 0}I^{n}M$. For any ideal $J$ of $A$, the [*Rees module of*]{} $I$ [*with respect to*]{} $M$ [*and modulo*]{} $J$ will be the standard ${\mbox{$\mathcal{R}_{J}(I)$}}$-module ${\mbox{$\mathcal{R}_{J}(I;M)$}}={\mbox{$\mathcal{R}(I;M)$}}\otimes A/J=\oplus _{n\geq 0}
I^{n}M/JI^{n}M$. Taking $J=I$, we recover the associated graded module of $I$ with respect to $M$, ${\mbox{$\mathcal{R}_{I}(I;M)$}}={\mbox{$\mathcal{G}(I;M)$}}=\oplus
_{n\geq 0}I^{n}M/I^{n+1}M$ and taking $J={\mbox{$\mathfrak{m}$}}$ a maximal ideal of $A$, we recover the fiber cone of $I$ with respect to $M$, ${\mbox{$\mathcal{R}_{{\mbox{$\mathfrak{m}$}}}(I;M)$}}={\mbox{$\mathcal{F}_{\mathfrak{m}}(I;M)$}}=\oplus _{n\geq 0}I^{n}M/{\mbox{$\mathfrak{m}$}}I^{n}M$.
Given two standard $U$-modules $F$, $G$ and $\varphi :G\rightarrow F$, a surjective graded morphism of $U$-modules, put $E(\varphi )_{n}=
{\rm ker}\varphi _{n}/U_{1}{\rm ker}\varphi _{n-1}$ for $n\geq
2$. Consider $\gamma :{\bf S}(U_{1})\otimes F_{0}\buildrel \alpha
\otimes 1\over \rightarrow U\otimes F_{0}\rightarrow F$, where $\alpha
:{\bf S}(U_{1})\rightarrow U$ is the canonical symmetric presentation of $U$ and $U\otimes F_{0}\rightarrow F$ is the structural morphism. For $n\geq 2$, the [*module of effective $n$-relations of*]{} $F$ is $E(F)_{n}=E(\gamma )_{n}= {\rm ker}\gamma _{n}/U_{1}{\rm
ker}\gamma _{n-1}$. The [*relation type of*]{} $F$ is ${\rm
rt}(F)={\rm min}\{ r\geq 1\mid E(F)_{n}=0\mbox{ for all }n\geq r+1\}$, which is finite if $A$ is noetherian, $U$ is a finitely generated algebra and $F$ is a finitely generated $U$-module. It can be shown that the module of effective $n$-relations, $n\geq 2$, and the relation type do not depend on the chosen symmetric presentation ([@planas], Definition 2.4, see also [@dgh1], [@vasconcelos3], [@wang]). In particular, in order to find the effective relations of $F$ and its relation type, one can always take a presentation of $F$ as a quotient of a polynomial module with coeficients in $F_{0}$.
The [*module of effective*]{} $n$-[*relations of*]{} $I$ [*with respect to*]{} $M$ is $E(I;M)_{n}=E({\mbox{$\mathcal{R}(I;M)$}})_{n}$ and the relation type of $I$ with respect to $M$ is ${\rm rt}(I;M)={\rm
rt}({\mbox{$\mathcal{R}(I;M)$}})$. For any ideal $J$ of $A$, the [*module of effective $n$-relations of*]{} $I$ [*with respect to*]{} $M$ [*and modulo*]{} $J$ will be $E_{J}(I;M)_{n}=E({\mbox{$\mathcal{R}_{J}(I;M)$}})_{n}$ and the relation type of $I$ with respect to $M$ and modulo $J$ will be ${\rm
rt}_{J}(I;M)={\rm rt}({\mbox{$\mathcal{R}_{J}(I;M)$}})$. If $M=A$, then we omit the phrase “with respect to $M$” and simply write $E(I)_{n}$, ${\rm
rt}(I)$, $E_{J}(I)_{n}$ and ${\rm rt}_{J}(I)$.
\[coef-mod\]
Let $A$ be a ring, $J,I,{\mbox{$\mathfrak{a}$}}$, ideals of $A$ and $M$ an $A$-module.
- Then ${\rm rt}(I;A/{\mbox{$\mathfrak{a}$}})={\rm rt}(IA/{\mbox{$\mathfrak{a}$}})={\rm
rt}(IA/{\mbox{$\mathfrak{a}$}};A/{\mbox{$\mathfrak{a}$}})$.
- ${\rm rt}_{J}(I;M)\leq {\rm rt}(I;M)$.
- If $A$ is noetherian, $J\subset I$ and $M$ is finitely generated, then ${\rm rt}_{J}(I;M)={\rm rt}(I;M)$.
${\mbox{$\mathcal{R}(I;A/{\mbox{$\mathfrak{a}$}})$}}={\mbox{$\mathcal{R}(IA/{\mbox{$\mathfrak{a}$}})$}}={\mbox{$\mathcal{R}(IA/{\mbox{$\mathfrak{a}$}};A/{\mbox{$\mathfrak{a}$}})$}}$. Moreover, the relation type of the standard ${\mbox{$\mathcal{R}(I)$}}$-module ${\mbox{$\mathcal{R}(I;A/{\mbox{$\mathfrak{a}$}})$}}$, the relation type of the standard $A/{\mbox{$\mathfrak{a}$}}$-algebra ${\mbox{$\mathcal{R}(IA/{\mbox{$\mathfrak{a}$}})$}}$ and the relation type of the standard $A/{\mbox{$\mathfrak{a}$}}$-module ${\mbox{$\mathcal{R}(IA/{\mbox{$\mathfrak{a}$}};A/{\mbox{$\mathfrak{a}$}})$}}$ all coincide (see [@planas], Remark 2.5). This proves $(1)$. The proof of $(2)$ and $(3)$ follow from [@planas], Remark 2.7 (and in contrast to Remark \[sjs0\], here we do not need $I$ to be included in the Jacobson radical).
Next we show the relation between $E(I;M)_{n}$ and $E_{J}(I;M)_{n}$ and describe these modules for the two-generated regular case.
\[E(x,y)\] Let $A$ be a ring, $I$ and $J$ ideals of $A$ and $M$ an $A$-module. Then, for every integer $n\geq 2$, there exists an exact sequence of $A$-modules: $$\begin{aligned}
E(I;JM)_{n}\longrightarrow E(I;M)_{n}\longrightarrow
E_{J}(I;M)_{n}\rightarrow 0.\end{aligned}$$ In particular, if $I=(x,y)$ is two-generated and $x$ is $M$-regular, then, for every $n\geq 2$, $$\begin{aligned}
E_{J}(I;M)_{n}=\frac{(xI^{n-1}M:y^{n})}{ (xI^{n-1}JM:y^{n})\cap
JM+(xI^{n-2}M:y^{n-1})}.\end{aligned}$$
Let $f:P\rightarrow I$ be a presentation of $I$, with $P$ a free $A$-module, and, for every $n\geq 2$, consider the following commutative diagram:
(330,125)(0,0)
(75,100)[(0,0)]{} (210,100)[(0,0)]{} (310,100)[(0,0)]{} (370,100)[(0,0)[$0$]{}]{}
(130,100)[(1,0)[40]{}]{} (250,100)[(1,0)[40]{}]{} (330,100)[(1,0)[30]{}]{}
(75,60)[(0,0)]{} (210,60)[(0,0)]{} (310,60)[(0,0)]{} (370,60)[(0,0)[$0$]{}]{}
(130,60)[(1,0)[40]{}]{} (250,60)[(1,0)[40]{}]{} (330,60)[(1,0)[30]{}]{}
(75,20)[(0,0)]{} (210,20)[(0,0)[[ $P\otimes I^{n-1}M/I^{n-1}JM$]{}]{}]{} (310,20)[(0,0)]{} (370,20)[(0,0)[$0$]{}]{}
(140,20)[(1,0)[20]{}]{} (260,20)[(1,0)[20]{}]{} (340,20)[(1,0)[20]{}]{}
(75,50)[(0,-1)[20]{}]{} (75,50)[(0,-1)[16]{}]{} (210,50)[(0,-1)[20]{}]{} (210,50)[(0,-1)[16]{}]{} (310,50)[(0,-1)[20]{}]{} (310,50)[(0,-1)[16]{}]{} (75,90)[(0,-1)[20]{}]{} (210,90)[(0,-1)[20]{}]{} (310,88)[(0,0)[[$\lor$]{}]{}]{} (310,86)[(0,-1)[16]{}]{}
(150,110)[(0,0)]{} (150,70)[(0,0)]{} (150,30)[(0,0)]{} (270,110)[(0,0)]{} (270,70)[(0,0)]{} (270,30)[(0,0)]{}
The top, middle and bottom rows of these diagrams represent the last three nonzero terms of the $n$-th homogeneous part of the Koszul complexes induced by the ${\bf S}(P)$-linear forms $P\otimes
{\mbox{$\mathcal{R}(I;JM)$}}\rightarrow {\mbox{$\mathcal{R}(I;JM)$}}$, $P\otimes
{\mbox{$\mathcal{R}(I;M)$}}\rightarrow {\mbox{$\mathcal{R}(I;M)$}}$ and $P\otimes
{\mbox{$\mathcal{R}_{J}(I;M)$}}\rightarrow {\mbox{$\mathcal{R}_{J}(I;M)$}}$. The differentials are defined as usual: $\partial_{2,n}((x\wedge y)\otimes z)=y\otimes
xz-x\otimes yz$ and $\partial_{1,n}(x\otimes t)=xt$, $x,y\in P$, $z\in
I^{n-2}M$, $t\in I^{n-1}M$; $\partial^{\prime}_{i,n}$ and $\overline{\partial}_{i,n}$ are defined analogously (see e.g. [@bh], Definition 1.6.1). The vertical morphisms are induced by the obvious inclusions and quotients and define morphisms of complexes. By a similar reasoning to that in [@planas], Proposition 2.6, the first homology groups of these complexes are, respectively, ${\rm ker}\partial^{\prime}_{1,n}/{\rm
im}\partial^{\prime}_{2,n}=E(I;JM)_{n}$, ${\rm ker}\partial_{1,n}/{\rm
im}\partial_{2,n}=E(I;M)_{n}$ and ${\rm
ker}\overline{\partial}_{1,n}/{\rm
im}\overline{\partial}_{2,n}=E_{J}(I;M)_{n}$. The exact sequence we seek is nothing else but the short exact sequence induced in homology.
If $I=(x,y)$ with $x$ an $M$-regular element, take $P=A^{2}$ and $f:P\rightarrow I$ with $f(1,0)=x$ and $f(0,1)=y$. Then, the middle row becomes isomorphic to the complex: $$\begin{aligned}
I^{n-2}M\buildrel \partial_{2,n}\over\longrightarrow I^{n-1}M\oplus
I^{n-1}M\buildrel\partial_{1,n}\over\longrightarrow I^{n}M\rightarrow 0,\end{aligned}$$ with differentials $\partial_{2,n}(u)=(-yu,xu)$ and $\partial_{1,n}(z,t)=xz+yt$. Take $(z,t)=(\sum a_{i}u_{i},\sum
b_{i}v_{i})$, $(z,t)\in {\rm ker}\partial_{1,n}$, with $a_{i},b_{i}\in
I^{n-1}$, $u_{i},v_{i}\in M$ and $xz+yt=0$. Write $b_{i}=c_{i}y^{n-1}+d_{i}x$, $c_{i}\in A$ and $d_{i}\in I^{n-2}$. Then $$\begin{aligned}
y^{n}\sum c_{i}v_{i}=y\sum b_{i}v_{i}-y\sum d_{i}xv_{i}=yt-x\sum
d_{i}yv_{i}=-x(z-\sum d_{i}yv_{i}).\end{aligned}$$ Thus $\sum c_{i}v_{i}\in (xI^{n-1}M:y^{n})$. Consider $$\begin{aligned}
\varphi :{\rm ker}\partial_{1,n}\longrightarrow
\frac{(xI^{n-1}M:y^{n})}{(xI^{n-2}M:y^{n-1})} \; ,\end{aligned}$$ defined by $\varphi (z,t)=\overline{\sum c_{i}v_{i}}$. It is not difficult to see that $\varphi$ is well-defined, surjective and ${\rm
im}\partial_{2,n}\subset {\rm ker}\varphi$. Moreover, if $x$ is $M$-regular, then ${\rm ker}\varphi\subset {\rm
im}\partial_{2,n}$. Thus $$\begin{aligned}
E(I;M)_{n}=\frac{(xI^{n-1}M:y^{n})}{(xI^{n-2}M:y^{n-1})}.\end{aligned}$$ Using the former exact sequence of modules of effective relations, one deduces the expression of $E_{J}(I;M)_{n}$.
Next we compare the Artin-Rees number modulo $J$ with the relation type modulo $J$.
\[slrt\] Let $A$ be a ring, $I$ and $J$ two ideals of $A$ and $N\subset M$ two $A$-modules. Then $$\begin{aligned}
s_{J}(N,M;I)\leq {\rm rt}_{J}(I;M/N)\leq {\rm max}({\rm
rt}_{J}(I;M),s_{J}(N,M;I)).\end{aligned}$$
Take $F={\mbox{$\mathcal{R}_{J}(I;M/N)$}}$, $G={\mbox{$\mathcal{R}_{J}(I;M)$}}$ and $H={\bf
S}(I/JI)\otimes M$ and $\varphi :G\rightarrow F$ and $\gamma
:H\rightarrow G$ induced by the natural surjective graded morphisms ${\mbox{$\mathcal{R}(I;M)$}}\rightarrow {\mbox{$\mathcal{R}(I;M/N)$}}$ and ${\bf S}(I)\otimes
M\rightarrow {\mbox{$\mathcal{R}(I;M)$}}$. By [@planas], Lemma 2.3, for every integer $n\geq 2$, one has the short exact sequence of $A$-modules: $$\begin{aligned}
E(\gamma )_{n}\rightarrow E(\varphi\circ\gamma )_{n}\rightarrow
E(\varphi )_{n}\rightarrow 0.\end{aligned}$$ But $E(\gamma )_{n}=E_{J}(I;M)_{n}$ and $E(\varphi\circ\gamma
)_{n}=E_{J}(I;M/N)_{n}$ and a short computation shows that $E(\varphi
)_{n}=E_{J}(N,M;I)_{n}$. From the exact sequence we obtain the desired inequalities.
Castelnuovo-Mumford regularity {#cmr}
==============================
The purpose of this section is to recall some definitions and formulate, in order to use them subsequently, a generalization to modules of some results of Trung in [@trung1] and [@trung2]. Being natural extensions of his results, we omit or just sketch the proofs. Let $A$ be a noetherian ring and $U=\oplus
_{n\geq 0}U_{n}$ a finitely generated standard $A$-algebra. Let $F=\oplus _{n\geq 0}F_{n}$ be a standard $U$-module. Define $$\begin{aligned}
a(F)=\left\{ \begin{array}{ll} \mbox{\rm max}\{ n\geq 0\mid F_{n}\neq
0\}&\mbox{ if }F\neq 0.\\ -\infty &\mbox{ if }F=0.\end{array}\right.\end{aligned}$$ Let $U_{+}=\oplus_{n>0}U_{n}$ be the irrelevant ideal of $U$. If $i\geq 0$, denote by $$\begin{aligned}
a_{i}(F)=a(H^{i}_{U_{+}}(F)),\end{aligned}$$ where $H^{i}_{U_{+}}(\cdot )$ denotes the $i$-th local cohomology functor with respect to the ideal $U_{+}$. The [*Castelnuovo-Mumford regularity of $F$*]{} is defined to be $$\begin{aligned}
{\mbox{$\mathrm{reg}$}}{(F)}=\mbox{\rm max}\{ a_{i}(F)+i\mid i\geq 0\}\end{aligned}$$ (see e.g. [@bs], 15.2.9, [@schenzel], [@trung2]). We shall mainly be concerned with the case $U={\mbox{$\mathcal{R}(I)$}}$, the Rees algebra of an ideal $I$ of $A$, and $F={\mbox{$\mathcal{R}(I;M)$}}$, the Rees module of $I$ with respect to a finitely generated $A$-module $M$. In particular, if $M\neq 0$, then ${\mbox{$\mathrm{reg}$}}{(F)}\neq -\infty$ (see e.g. [@bs], 15.2.13).
A sequence ${\bf z}=z_{1},\ldots ,z_{s}$ of homogeneous elements of $U$ is called [*$n$-regular with respect to $F$*]{} if, for all $i=1,\ldots, s$, $$\begin{aligned}
((z_{1},\ldots ,z_{i-1})F:z_{i})_{n}=((z_{1},\ldots ,z_{i-1})F)_{n}.\end{aligned}$$ The least integer $m\geq 0$ such that ${\bf z}$ is $n$-regular for all $n\geq m+1$ is denoted by $a({\bf z})$ (see [@trung1], Section 2). In other words, $$\begin{aligned}
a({\bf z})={\rm max}\{ a((z_{1},\ldots ,z_{i-1})F:z_{i}/(z_{1},\ldots ,
z_{i-1})F) \mid i=1,\ldots ,s\}.\end{aligned}$$
A sequence ${\bf z}=z_{1},\ldots ,z_{s}$ of homogeneous elements of $U$ is called a [*$U_{+}$-filter-regular sequence with respect to $F$*]{} if $z_{i}\not\in {\mbox{$\mathfrak{p}$}}$ for any associated prime ideal ${\mbox{$\mathfrak{p}$}}$ of $F/(z_{1},\ldots ,z_{i-1})F$, ${\mbox{$\mathfrak{p}$}}\not\supseteq U_{+}$, for all $i=1,\ldots ,s$ (see [@trung1], Section 2, [@trung2], Section 2).
[([@bs], 18.3.8, [@trung1], 2.1)]{} Let ${\bf z}=z_{1},\ldots ,z_{s}$ be a sequence of homogeneous elements of $U$. Then ${\bf z}$ is a $U_{+}$-filter regular sequence with respect to $F$ if and only if $a({\bf z})<\infty$.
\[trung1/2.3\] [([@trung1], 2.3)]{} Let $z\in U_{1}$ be a homogeneous $U_{+}$-filter-regular element with respect to $F$. Then, for all $i\geq 0$, $$\begin{aligned}
a_{i+1}(F)+1\leq a_{i}(F/zF)\leq \mbox{\rm max}\{ a_{i}(F),
a_{i+1}(F)+1\}.\end{aligned}$$
\[trung2/2.2\] [([@trung2], 2.2)]{} Let ${\bf z}=z_{1},\ldots ,z_{s}$ be a $U_{+}$-filter-regular sequence with respect to $F$, $z_{i}\in U_{1}$ for all $i=1,\ldots ,s$. Then $$\begin{aligned}
&&a({\bf z})={\rm max}\{ a_{i}(F)+i\mid i=0,\ldots ,s-1\}\mbox{ and,
for all }0\leq t\leq s,\\ &&{\rm max}\{ a_{i}(F)+i\mid i=0,\ldots
,t\}= {\rm max}\{ a((z_{1},\ldots ,z_{i})F:U_{+}/(z_{1},\ldots
,z_{i})F) \mid i=0,\ldots, t\}.\end{aligned}$$
\[reg-a\] [([@trung2], 2.4)]{} Let ${\bf z}=z_{1},\ldots ,z_{s}$ be a $U_{+}$-filter-regular sequence with respect to $F$, $z_{i}\in U_{1}$, $i=1,\ldots ,s$, which generates a reduction $Q$ of $U_{+}$ with respect to $F$. Then $$\begin{aligned}
{\mbox{$\mathrm{reg}$}}{(F)}=\mbox{\rm max}\{ a({\bf z}),{\rm rn}_{Q}(U_{+};F)\}.\end{aligned}$$
By Lemma \[trung2/2.2\], $a({\bf z})={\rm max}\{ a((z_{1},\ldots
,z_{i})F:U_{+}/(z_{1},\ldots ,z_{i})F)\mid i=0,\ldots
,s-1\}$. Further, ${\rm rn}_{Q}(U_{+};F)=a(F/QF)=a((z_{1},\ldots
,z_{s})F:U_{+}/(z_{1},\ldots ,z_{s})F)$. Therefore, $$\begin{aligned}
&&{\rm max}\{ a({\bf z}),{\rm rn}_{Q}(U_{+};F)\}={\rm max}\{
a((z_{1},\ldots ,z_{i})F:U_{+}/(z_{1},\ldots ,z_{i})F) \mid
i=0,\ldots, s\}=\\&&{\rm max}\{ a_{i}(F)+i\mid i=0,\ldots ,s\}.\end{aligned}$$ Since ${\mbox{$\mathrm{reg}$}}{(F)}={\rm max}\{ a_{i}(F)+i\mid i\geq 0\}$, it is enough to show that $H^{i}_{U_{+}}(F)=0$ for all $i>s$. If $s=0$, then $0$ is a reduction of $U_{+}$ with respect to $F$ and $F_{n}$ for all large $n$. So $F$ is a $U_{+}$-torsion module and $H^{i}_{U_{+}}(F)=0$ for all $i>0$ (see e.g. [@bs], 2.1.7). If $s\geq 1$, by induction, $H_{U_{+}}^{i}(F/z_{1}F)=0$ for all $i>s-1$. So $a_{i}(F/z_{1}F)=-\infty$ for all $i>s-1$. By Lemma \[trung1/2.3\], $a_{i+1}(F)=-\infty$ and $H^{i+1}_{U_{+}}(F)=0$ for all $i>s$.
Now take $A$ a noetherian ring, $I$ an ideal of $A$ and $M$ a finitely generated $A$-module. Consider ${\mbox{$\mathcal{R}(I)$}}=\oplus _{n\geq
0}I^{n}t^{n}\subset A[t]$ as a subring of $A[t]$.
\[filter\] [([@trung2], 4.1)]{} Let $A$ be a noetherian ring, let $I$ be an ideal of $A$ and let $M$ be a finitely generated $A$-module. Let $x_{1},\ldots ,x_{s}$ be a sequence of elements in $I$. Then $x_{1}t,\ldots ,x_{s}t$ is a ${\mbox{$\mathcal{R}(I)$}}_{+}$-filter-regular sequence with respect to ${\mbox{$\mathcal{R}(I;M)$}}$ if and only if for all large $n\geq 1$, $$\begin{aligned}
[(x_{1},\ldots ,x_{i-1})I^{n}M:x_{i}]\cap I^{n}M=(x_{1},\ldots ,
x_{i-1})I^{n-1}M\mbox{ for }i=1,\ldots ,s. \quad (^{*})\end{aligned}$$ If that is the case, $a({\bf z})$ is the least integer $r$ such that $(^{*})$ holds for all $n\geq r+1$.
${\bf z}=x_{1}t,\ldots ,x_{s}t$ is a ${\mbox{$\mathcal{R}(I)$}}_{+}$-filter-regular sequence with respect to ${\mbox{$\mathcal{R}(I;M)$}}$ if and only if $[(x_{1}t,\ldots ,x_{i-1}t){\mbox{$\mathcal{R}(I;M)$}}:x_{i}t]_{n}$ is equal to $[(x_{1}t,\ldots ,x_{i-1}t){\mbox{$\mathcal{R}(I;M)$}}]_{n}$ for all large $n\geq 1$. But the first module is equal to $[(x_{1},\ldots
,x_{i-1})I^{n}M:x_{i}]\cap I^{n}M$ and the second is equal to $(x_{1},\ldots , x_{i-1})I^{n-1}M$.
\[regu\] [([@trung2], 4.2)]{} Let $A$ be a noetherian ring, let $I$ be an ideal of $A$ and let $M$ be a finitely generated $A$-module. Let $J=(x_{1},\ldots ,x_{s})$ be a reduction of $I$ with respect to $M$. Suppose that ${\bf
z}=x_{1}t,\ldots ,x_{s}t$ is a ${\mbox{$\mathcal{R}(I)$}}_{+}$-filter-regular sequence with repect to ${\mbox{$\mathcal{R}(I;M)$}}$. Then $$\begin{aligned}
{\mbox{$\mathrm{reg}$}}({\mbox{$\mathcal{R}(I;M)$}})= \mbox{\rm min}\{ r\geq 0\mid r\geq {\rm
rn}_{J}(I;M)\mbox{ and }(^{*})\mbox{ holds for all }n\geq r+1\}.\end{aligned}$$
Let $Q=({\bf z})$ denote the ideal generated by ${\bf
z}=x_{1}t,\ldots ,x_{s}t$, $U={\mbox{$\mathcal{R}(I)$}}$ the Rees algebra of $I$ and $F={\mbox{$\mathcal{R}(I;M)$}}$ the Rees module of $I$ with respect to $M$. Since $J$ is a reduction of $I$ with respect to $M$, then $Q$ is a reduction of $U_{+}$ with respect to $F$. Moreover, if $I^{r+1}M=JI^{r}M$, then $U_{+}^{r+1}F=QU_{+}^{r}F$ and ${\rm rn}_{Q}(U_{+};F)={\rm
rn}_{J}(I;M)$. By Proposition \[reg-a\], ${\mbox{$\mathrm{reg}$}}(F)={\rm max}\{ a({\bf
z}),{\rm rn}_{J}(I;M)\}$. The conclusion follows from Lemma \[filter\].
Relation type and reduction number {#rtrn}
==================================
The first result of the section suggests the relationship subsisting between the relation type and the reduction number (see [@vasconcelos3], page 63).
\[rtlrtrn\] Let $A$ be a ring, $I$ an ideal of $A$ and $M$ an $A$-module. Let $J\subset I$ be a reduction of $I$ with respect to $M$ and with reduction number ${\rm rn}_{J}(I;M)=r$. Then $$\begin{aligned}
{\rm rt}(I;M)\leq {\mbox{$\mathrm{rn}$}}_{J}(I;M)+{\mbox{$\mathrm{rt}$}}(J;I^{r}M).\end{aligned}$$
Let us prove that $E(I;M)_{n}=0$ for all $n\geq
r+{\mbox{$\mathrm{rt}$}}(J;I^{r}M)+1$. Write $n=r+k$, where $k\geq {\mbox{$\mathrm{rt}$}}(J;I^{r}M)+1(\geq
2)$. In particular, $I^{n}M=J^{k}I^{r}M=JI^{n-1}M$, $I^{n-1}M=J^{k-1}I^{r}M=JI^{n-2}M$ and $I^{n-2}M=J^{k-2}I^{r}M=JI^{n-3}M$ (where $I^{n-3}=A$ if $n=2$ and $r=0$). Consider the following diagram:
(330,85)(15,0)
(100,60)[(0,0)]{} (200,60)[(0,0)]{} (300,60)[(0,0)]{} (360,60)[(0,0)[$0$]{}]{}
(140,60)[(1,0)[30]{}]{} (155,68)[(0,0)]{} (230,60)[(1,0)[50]{}]{} (255,68)[(0,0)]{} (320,60)[(1,0)[30]{}]{}
(140,20)[(1,0)[30]{}]{} (155,12)[(0,0)]{} (230,20)[(1,0)[50]{}]{} (255,12)[(0,0)]{} (320,20)[(1,0)[30]{}]{}
(100,20)[(0,0)[[$\Lambda_{2}(I)\otimes I^{n-2}M$]{}]{}]{} (200,20)[(0,0)[[$I\otimes I^{n-1}M$]{}]{}]{} (300,20)[(0,0)[[$I^{n}M$]{}]{}]{} (360,20)[(0,0)[$0$]{}]{} (365,17)[(0,0)[.]{}]{}
(100,50)[(0,-1)[20]{}]{} (90,40)[(0,0)]{} (200,50)[(0,-1)[20]{}]{} (190,40)[(0,0)]{} (299,50)[(0,-1)[20]{}]{} (301,50)[(0,-1)[20]{}]{}
The top row represents the last three nonzero terms of the $k$-th homogeneous part of the Koszul complex induced by the ${\mbox{$\mathcal{R}(J)$}}$-linear form $J\otimes {\mbox{$\mathcal{R}(J;I^{r}M)$}}\rightarrow
{\mbox{$\mathcal{R}(J;I^{r}M)$}}$ and the bottom row represents the last three nonzero terms of the $n$-th homogeneous part of the Koszul complex induced by the ${\mbox{$\mathcal{R}(I)$}}$-linear form $I\otimes
{\mbox{$\mathcal{R}(I;M)$}}\rightarrow {\mbox{$\mathcal{R}(I;M)$}}$. The Koszul differentials are defined as usual (e.g. [@bh], Definition 1.6.1, see also the proof of Proposition \[E(x,y)\]). The vertical morphisms are induced by the inclusion $J\subset I$ and define a morphism of complexes. By [@planas], Proposition 2.6, the first homology groups of these complexes are ${\rm ker}\partial ^{\prime}_{1,k}/{\rm im}\partial
^{\prime}_{2,k}=E(J;I^{r}M)_{k}$ and ${\rm ker}\partial _{1,n}/{\rm
im}\partial _{2,n}=E(I;M)_{n}$. Thus we want to prove ${\rm
ker}\partial _{1,n}\subset {\rm im}\partial _{2,n}$. Take $u=\sum
_{i}x_{i}\otimes m_{i}\in I\otimes I^{n-1}M$ such that $\partial
_{1,n}(u)=\sum _{i}x_{i}m_{i}=0$. Write each $m_{i}=\sum
_{j}y_{i,j}m_{i,j}$, $y_{i,j}\in J$, $m_{i,j}\in I^{n-2}M$. Take $v=\sum _{i,j}y_{i,j}\wedge x_{i}\otimes m_{i,j}\in \Lambda
_{2}(I)\otimes I^{n-2}M$. Then $\partial _{2,n}(v)=u-w$, where $w=
\sum _{i,j}y_{i,j}\otimes x_{i}m_{i,j}\in I\otimes I^{n-1}M$. Consider $w^{\prime}= \sum _{i,j}y_{i,j}\otimes x_{i}m_{i,j}\in
J\otimes J^{k-1}I^{r}M$. Remark that $\partial
^{\prime}_{1,k}(w^{\prime})=\partial _{1,n}(f(w^{\prime}))=\partial
_{1,n}(w)=0$. Since $k\geq {\rm rt}(J;I^{r}M)+1$, then $E(J;I^{r}M)_{k}=0$ and $w^{\prime}\in {\rm im}\partial
^{\prime}_{2,k}$. Take $t^{\prime}\in \Lambda_{2}(J)\otimes
J^{k-2}I^{r}M$ such that $\partial
_{2,k}(t^{\prime})=w^{\prime}$. Then $\partial_{2,n}(v+g(t^{\prime}))=u-w+
f(\partial^{\prime}_{2,k}(t^{\prime}))=u-w+f(w^{\prime})=u$ and $u\in{\rm im}\partial _{2,n}$.
The purpose now is to control the relation type of the reduction $J$ with respect to $I^{r}M$. We will use the filter-regular conditions $(^{*})$ of Lemma \[filter\], which firstly appeared, to our knowledge, in a paper by Costa for $M=A$ and $J=I$ ([@costa], page 258).
\[costacond\] Let $A$ be a ring, let $I$ be an ideal of $A$ and let $M$ be an $A$-module. Let $J=(x_{1},\ldots,x_{s})\subset I$ be a reduction of $I$ with respect to $M$ and with reduction number ${\rm
rn}_{J}(I;M)=r$. Suppose that there exists $k\geq 1$ such that for all $n\geq r+k$ and all $i=1,\ldots ,s$, $$\begin{aligned}
[(x_{1},\ldots,x_{i-1})I^{n}M:x_{i}]\cap
I^{n}M=(x_{1},\ldots,x_{i-1})I^{n-1}M.\end{aligned}$$ Then ${\rm rt}(J;I^{r}M)\leq k$. Moreover, if $A$ is noetherian and $M$ is finitely generated, then ${\mbox{$\mathrm{rn}$}}_{J}(I;M)\leq
{\mbox{$\mathrm{reg}$}}({\mbox{$\mathcal{R}(I;M)$}})\leq {\mbox{$\mathrm{rn}$}}_{J}(I;M)+k-1$.
Write $J_{0}=0$ and $J_{i}=(x_{1},\ldots,x_{i})$ for $i=1,\ldots, s$. Let $m\geq k+1$ and consider the last three nonzero terms of the $m$-th homogeneous part of the Koszul complex induced by the ${\mbox{$\mathcal{R}(J_{i})$}}$-linear form $J_{i}\otimes {\mbox{$\mathcal{R}(J;I^{r}M)$}}
\rightarrow {\mbox{$\mathcal{R}(J;I^{r}M)$}}$: $$\begin{aligned}
\Lambda _{2}(J_{i})\otimes J^{m-2}I^{r}M\buildrel
\partial_{2,m-2}\over \longrightarrow J_{i}\otimes
J^{m-1}I^{r}M\buildrel \partial_{1,m-1}\over\longrightarrow
J^{m}I^{r}M\to 0.\end{aligned}$$ If $i=s$, then $J_{s}=J$ and one has the Koszul complex $$\begin{aligned}
\Lambda _{2}(J)\otimes J^{m-2}I^{r}M\buildrel
\partial_{2,m-2}\over \longrightarrow J\otimes
J^{m-1}I^{r}M\buildrel \partial_{1,m-1}\over\longrightarrow
J^{m}I^{r}M\to 0,\end{aligned}$$ whose first homology group ${\rm ker}\partial_{1,m-1}/{\rm
im}\partial_{2,m-2}$ is, by [@planas], Proposition 2.6, equal to the module of $m$-effective relations $E(J;I^{r}M)_{m}$. Thus, it is enough to prove by induction on $i=1,\ldots,s$, that ${\rm
ker}\partial _{1,m-1}\subset {\rm im}\partial _{2,m-2}$ for all $m\geq
k+1$ (remark that, in this case, $r+m-1\geq r+k$). If $i=1$, let $z=x_{1}\otimes c\in J_{1}\otimes J^{m-1}I^{r}M$ such that $0=\partial
_{1,m-1}(z)=x_{1}c$. Then $c\in (0:x_{1})\cap
J^{m-1}I^{r}M=(0:x_{1})\cap I^{r+m-1}M=J_{0}I^{r+m-2}M=0$. Thus $z=0$. If $i=s$, let $z=\sum_{i=1}^{s}x_{i}\otimes c_{i}\in J\otimes
J^{m-1}I^{r}M$ such that $0=\partial _{1,m-1}(z)=\sum
_{i=1}^{s}x_{i}c_{i}$. Then $x_{s}c_{s}=-\sum_{i=1}^{s-1}x_{i}c_{i}$. Thus $c_{s}\in
(J_{s-1}J^{m-1}I^{r}M:x_{s})\cap
J^{m-1}I^{r}M=(J_{s-1}I^{r+m-1}M:x_{s})\cap
I^{r+m-1}M=J_{s-1}I^{r+m-2}M$. Thus $c_{s}=\sum_{i=1}^{s-1}x_{i}\lambda _{i}$, $\lambda _{i}\in
I^{r+m-2}M=J^{m-2}I^{r}M$. Take $u=\sum_{i=1}^{s-1}x_{i}\otimes
(c_{i}+x_{s}\lambda_{i})\in J_{s-1}\otimes J^{m-1}I^{r}M$. Then $\partial _{1,m-1}(u)=0$. By induction hypothesis, there exists $v\in
\Lambda _{2}(J_{s-1})\otimes J^{m-2}I^{r}M$ such that $\partial
_{2,m-2}(v)=u$. Take $w=v+\sum_{i=1}^{s-1}(x_{j}\wedge x_{s})\otimes
\lambda _{i}$ and one has $\partial _{2,m-2}(w)=z$. This proves ${\rm
rt}(J;I^{r}M)\leq k$. The second assertion follows from Proposition \[regu\].
Let $A$ be a noetherian ring, $J\subset I$ two ideals of $A$ and $M$ a finitely generated $A$-module. Let $x_{1},\ldots ,x_{s}$ be a system of generators of $J$. Write, as before, $J_{0}=0$ and $J_{i}=(x_{1},\ldots ,x_{i})$ for $i=1,\ldots ,s$. The sequence $x_{1},\ldots ,x_{s}$ is said to be a [*$d$-sequence with respect to $M$*]{} if any $x_{j}$ is not contained in the ideal generated by the others $x_{i}$ and for all $k\geq i+1$ and all $i\geq 0$, $(J_{i}M:x_{i+1}x_{k})=(J_{i}M:x_{k})$. It is known that this last condition is equivalent to $(J_{i}M:x_{i+1})\cap JM=J_{i}M$ for all $i=0,\ldots ,s-1$. Let ${\mbox{$\mathcal{G}(I;M)$}}$ be the associated graded module of $I$ with respect to $M$ and denote by $x_{1}^{*},\ldots ,x_{s}^{*}$ the images of $x_{1},\ldots,x_{s}$ in $I/I^{2}\subset {\mbox{$\mathcal{G}(I)$}}$. The sequence $x_{1},\ldots ,x_{s}$ is said to be a [*complete $d$-sequence with respect to $I$ and $M$*]{} if $x_{1},\ldots ,x_{s}$ is a $d$-sequence with respect to $M$ and $x_{1}^{*},\ldots ,x_{s-1}^{*}$ is a ${\mbox{$\mathcal{G}(I;M)$}}$-regular sequence (see [@huckaba3] and [@trung2]). If $A$ is local, it can be shown that $x_{1}^{*},\ldots ,x_{s-1}^{*}$ is a ${\mbox{$\mathcal{G}(I;M)$}}$-regular sequence if and only if $x_{1},\ldots ,x_{s-1}$ is an $M$-regular sequence and, for all $n\geq 0$ and all $i=1,\ldots ,s-1$, the $n$-th Valabrega-Valla module $VV_{J_{i}}(I;M)_{n}=J_{i}M\cap
I^{n+1}M/J_{i}I^{n}M$ is equal to zero (see e.g. [@huckaba2], Lemma 2.2, [@cz], Proposition 2.3).
Huckaba proved that if $A$ is noetherian local, if $I$ is an ideal with analytic spread $l(I)$ equal to the height of the ideal ${\rm
ht}(I)$ or ${\rm ht}(I)+1$ and with a minimal reduction $J$ generated by a complete $d$-sequence with respect to $I$, then ${\rm rt}(I)\leq
{\rm rn}_{J}(I)+1$ (see [@huckaba2], Theorem 2.3 and [@huckaba3], Theorem 1.4). Later, Trung proved that, in general, ${\mbox{$\mathrm{rt}$}}(I)\leq {\mbox{$\mathrm{reg}$}}({\mbox{$\mathcal{R}(I)$}})+1$ and that if $I$ has a a reduction $J$ generated by a complete $d$-sequence with respect to $I$, then ${\mbox{$\mathrm{reg}$}}({\mbox{$\mathcal{R}(I)$}})={\mbox{$\mathrm{rn}$}}_{J}(I)$ (see [@trung2], Proposition 2.6 and Theorem 6.4; for more related results on this topic see also [@schenzel] and [@trung1]). From our Propositions \[rtlrtrn\] and \[costacond\], we obtain a generalization of these results. Our proof closely follows ideas of Trung in [@trung2].
\[cds\] Let $A$ be a noetherian ring, let $I$ be an ideal of $A$ and let $M$ be a finitely generated $A$-module. Let $J=(x_{1},\ldots
,x_{s})\subset I$ be a reduction of $I$ with respect to $M$ and with reduction number ${\rm rn}_{J}(I;M)=r$. Suppose that
- $x_{1},\ldots,x_{s}$ is a $d$-sequence with respect to $M$.
- $x_{1},\ldots,x_{s-1}$ is an $M$-regular sequence.
- $(x_{1},\ldots ,x_{i})M\cap I^{r+1}M=(x_{1},\ldots ,
x_{i})I^{r}M$ for all $i=1,\ldots,s-1$.
Then ${\mbox{$\mathrm{rt}$}}(J;I^{r}M)=1$, ${\mbox{$\mathrm{rt}$}}(I;M)\leq {\mbox{$\mathrm{rn}$}}_{J}(I;M)+1$ and ${\mbox{$\mathrm{rn}$}}_{J}(I;M)={\mbox{$\mathrm{reg}$}}({\mbox{$\mathcal{R}(I;M)$}})$.
Write $J_{0}=0$ and $J_{i}=(x_{1},\ldots,x_{i})$ for $i=1,\ldots, s$. Using $(ii)$ and $(iii)$, we obtain $(J_{i-1}M:x_{i})\cap I^{r+1}M=J_{i-1}M\cap I^{r+1}M= J_{i-1}I^{r}M$ for $i=1,\ldots ,s-1$. By $(i)$, $(J_{s-1}M:x_{s})\cap
JM=J_{s-1}M$. Since $I^{r+1}M=JI^{r}M$, then $(J_{s-1}M:x_{s})\cap
I^{r+1}M=J_{s-1}M\cap I^{r+1}M$ which, by $(iii)$, is equal to $J_{s-1}I^{r}M$. Thus, for all $i=1,\ldots ,s$, $$\begin{aligned}
(J_{i-1}M:x_{i})\cap I^{r+1}M=J_{i-1}I^{r}M.\end{aligned}$$ A straighforward generalization to modules of Theorem 4.8, $(i)$ in [@trung2], allows us to assert that for all integers $n\geq r+1$ and for all $i=1,\ldots ,s$, $$\begin{aligned}
(J_{i-1}M:x_{i})\cap I^{n}M=J_{i-1}I^{n-1}M,\end{aligned}$$ which clearly implies for all integers $n\geq r+1$ and for all $i=1,\ldots ,s$, $$\begin{aligned}
(J_{i-1}I^{n}M:x_{i})\cap I^{n}M=J_{i-1}I^{n-1}M.\end{aligned}$$ By Proposition \[rtlrtrn\], ${\mbox{$\mathrm{rt}$}}(I;M)\leq {\mbox{$\mathrm{rn}$}}_{J}(I;M)+{\rm
rt}(J;I^{r}M)$ and, by Proposition \[costacond\], ${\mbox{$\mathrm{rt}$}}(I;J^{r}M)=1$ and ${\mbox{$\mathrm{rn}$}}_{J}(I;M)={\mbox{$\mathrm{reg}$}}({\mbox{$\mathcal{R}(I;M)$}})$.
Integral degree of a ring {#id}
=========================
In this section we introduce the [*integral degree*]{}, an invariant associated to the ring, which later will be used to bound the reduction number. Let $A\subset B$ be a ring extension. Recall that an element $b\in B$ is said to be integral over $A$ if there exist $a_{i}\in A$ and an integral equation of degree $n\geq 1$: $$\begin{aligned}
b^{n}+a_{1}b^{n-1}+a_{2}b^{n-2}+\ldots +a_{n-1}b+a_{n}=0.\end{aligned}$$ If $b\in B$ is integral over $A$, we will call the [*integral degree of*]{} $b$ [*over*]{} $A$ to the integer: $$\begin{aligned}
{\rm id}_{A}(b)={\rm min} \{ n\geq 1\mid b\mbox{ satisfies an integral
equation of degree }n\}.\end{aligned}$$ Let $A\subset C\subset B$, $C$ an $A$-submodule of $B$. Suppose the elements of $C$ are integral over $A$. Then the [*integral degree of*]{} $C$ [*over*]{} $A$ is defined as the integer (possibly infinite): $$\begin{aligned}
{\mbox{$\mathrm{d}$}}_{A}(C)={\rm sup}\, \{ {\rm id}_{A}(c)\mid c\in C\}.\end{aligned}$$ Remark that ${\mbox{$\mathrm{d}$}}_{A}(C)=1$ if and only if $A=C$.
As usual, $\mu _{A}(\cdot )$ stands for the minimal number of generators as an $A$-module.
\[genatiyah\] Let $A\subset B$ be a ring extension, $b\in B$ and $n\geq 1$. Then the following conditions are equivalent:
- $b$ is integral over $A$ and ${\rm id}_{A}(b)\leq n$.
- $A[b]$ is a finitely generated $A$-module and $\mu
_{A}(A[b])\leq n$.
- There exists a ring $C$, $A\subset A[b]\subset C\subset
B$, such that $C$ is a finitely generated $A$-module and $\mu
_{A}(C)\leq n$.
- There exists a faithful $A[b]$-module $M$ such that $M$ is a finitely generated $A$-module and $\mu _{A}(M)\leq n$.
It follows from [@am], Proposition 5.1, just taking into account the definition of ${\rm id}_{A}(b)$.
Next we prove that the integral degree of the sum or product of two integral elements is, in fact, bounded above by the product of their integral degrees.
\[nelements\] Let $A\subset B$ be a ring extension and $b_{1},\ldots ,b_{n}\in B$ integral over $A$. Then $A[b_{1},\ldots ,b_{n}]$ is a finitely generated $A$-module, $A\subset A[b_{1},\ldots ,b_{n}]$ is an integral extension and: $$\begin{aligned}
{\rm max}\{ {\rm id}_{A}(b_{i})\}\leq {\mbox{$\mathrm{d}$}}_{A}(A[b_{1},\ldots
,b_{n}])\leq \mu _{A}(A[b_{1},\ldots ,b_{n}])\leq \prod_{i=1}^{n}{\rm
id}_{A}(b_{i}).\end{aligned}$$ In particular, if $b\in B$ is integral over $A$, then ${\rm
id}_{A}(b)={\mbox{$\mathrm{d}$}}_{A}(A[b])=\mu _{A}(A[b])$.
Let $C=A[b_{1},\ldots ,b_{n}]$ and $m=\prod_{i=1}^{n}{\rm
id}_{A}(b_{i})$. Then it is clear that $\mu _{A}(C)\leq m$. Now take any $b\in C\subset B$. So we have $A\subset A[b]\subset C\subset B$ with $\mu _{A}(C)=r\leq m$. By Proposition \[genatiyah\], $(iii)\Rightarrow (i)$, $b$ is integral over $A$ and ${\rm
id}_{A}(b)\leq r$ and taking the supremum over all $b\in C$, then ${\mbox{$\mathrm{d}$}}_{A}(C)\leq r=\mu_{A}(C)$. As for the second assertion, just take $n=1$.
\[uidlmu\] Let $A\subset B$ be a ring extension. If $B$ is a finitely generated $A$-module, then $A\subset B$ is integral and $$\begin{aligned}
{\mbox{$\mathrm{d}$}}_{A}(B)\leq\mu _{A}(B).\end{aligned}$$
If $b\in B$, take $A\subset A[b]\subset B$, with $B$ a finitely generated $A$-module. By Proposition \[genatiyah\], $(iii)\Rightarrow (i)$, $b$ is integral over $A$ and ${\mbox{$\mathrm{id}$}}_{A}(b)\leq
\mu _{A}(B)$. Taking the supremum, ${\mbox{$\mathrm{d}$}}_{A}(B)\leq \mu _{A}(B)$.
Let $A$ be noetherian domain and let $\overline{A}$ be the integral closure of $A$ in its quotient field. If ${\rm dim}\, A\leq 2$, then $\overline{A}$ is noetherian (see e.g. [@matsumura], 11.7 and [@nagata], 33.12). Nevertheless, $\overline{A}$ may be a non finitely generated $A$-module, as an example of Akizuki shows ([@akizuki] or [@reid], 9.5). Next, we want to prove that the ring $A$ in the example of Akizuki has at least finite integral degree ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})$. Before that, and for easy reference, we state the following lemma.
\[idrtrn\] Let $A$ be a ring and $x,y\in A$, with $x$ regular. The following are equivalent.
- $y/x$ is integral over $A$ and ${\mbox{$\mathrm{id}$}}_{A}(y/x)\leq n$.
- $(x)$ is a reduction of $(x,y)$ and ${\rm
rn}_{(x)}(x,y)\leq n-1$.
- $x(x,y)^{n-1}:y^{n}=A$.
In particular, if $y/x$ is integral over $A$, then ${\mbox{$\mathrm{id}$}}_{A}(y/x)={\rm
rt}(x,y)={\rm rn}_{(x)}(x,y)+1$
Take $y/x\in\overline{A}$ with ${\mbox{$\mathrm{id}$}}_{A}(y/x)\leq n$. Then, there exist $a_{i}\in A$ such that $(y/x)^{n}+a_{1}(y/x)^{n-1}+\ldots
+a_{n}=0$. Multiplying by $x^{n}$, one has $y^{n}\in xI^{n-1}$, where $I=(x,y)$. Thus $I^{n}=xI^{n-1}$, $J=(x)$ is a reduction of $I$ and ${\rm rn}_{J}(I)\leq n-1$. If $J=(x)$ is a reduction of $I=(x,y)$ with ${\rm rn}_{J}(I)\leq n-1$, then $I^{n}=xI^{n-1}$ and $y^{n}\in
xI^{n-1}$. Thus $1\in xI^{n-1}:y^{n}$ and $xI^{n-1}:y^{n}=A$. Finally, if $xI^{n-1}:y^{n}=A$, where $I=(x,y)$, then $y^{n}\in xI^{n-1}$ and $y^{n}=b_{1}xy^{n-1}+\ldots +b_{n}x^{n}$, for some $b_{i}\in
A$. Dividing by $x^{n}$ one obtains an integral equation of $y/x$ over $A$ of degree $n$. In particular, if $y/x\in \overline{A}$ with ${\rm
id}_{A}(y/x)=n\geq 2$, then $xI^{n-2}:y^{n-1}\varsubsetneq
A=xI^{n-1}:y^{n}$, where $I=(x,y)$. By Proposition \[E(x,y)\], $E(I)_{n}\neq 0$ and $E(I)_{n+s}=0$ for all $s\geq 1$. Thus ${\rm
rt}(I)=n={\rm id}_{A}(y/x)$. Moreover, $(i)\Leftrightarrow (ii)$ says that $J=(x)$ is a reduction of $I=(x,y)$ and that ${\rm
rn}_{J}(I)=n-1$.
Now, let us prove that the example of Akizuki has finite integral degree. Denote by $e(A)$ the multiplicity of $A$.
\[akizuki\] Let $A$ be a one-dimensional noetherian local ring. Then $$\begin{aligned}
{\mbox{$\mathrm{d}$}}_{A}(\overline{A})\leq e(A/H^{0}_{\mathfrak{m}}(A))+{\rm
length}(H^{0}_{\mathfrak{m}}(A)).\end{aligned}$$ In particular, if $A$ is a domain (as Akizuki’s example is), then ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})\leq e(A)$.
Take $y/x\in \overline{A}$ with $x,y\in A$, $x$ regular, and $I=(x,y)$ the ideal of $A$ generated by $x,y$. By Lemma \[idrtrn\], ${\mbox{$\mathrm{id}$}}_{A}(y/x)={\rm rt}(I)$. By [@planas], Lemma 6.1, ${\rm
rt}(I)\leq {\rm rt}(IA/J)+{\rm length}(J)$, where $J=H^{0}_{\mathfrak{m}}(A)$. By [@planas], Lemma 6.3, ${\rm
rt}(IA/J)\leq e(A/J)$.
We next see that there exist one-dimensional noetherian domains with infinite integral degree. Remark that the ring in this example must be not local nor excellent so that one can not apply the existence of a uniform bound for the relation type of all ideals (see [@planas], Proposition 6.5 and Theorem 3). The next example is due to Sally and Vasconcelos (see [@sv], Example 1.4, and also [@planas], Remark 7.3).
[ There exists one-dimensional noetherian domains $A$ with ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})$ infinite.]{}
Let $t_{1},t_{2},t_{3},\ldots $ be infinitely many indeterminates over a field $k$. Let $R$ be defined as $R=k[t_{1}^{2},t_{1}^{3},t_{2}^{3},t_{2}^{4}, t_{2}^{5},\ldots
,t_{n}^{n+1},t_{n}^{n+2},\ldots ,t_{n}^{2n+1},\ldots ]$. Take ${\mbox{$\mathfrak{p}$}}_{n}=(t_{n}^{n+1},t_{n}^{n+2},\ldots ,t_{n}^{2n+1})$, which is a prime ideal of height 1. Let $S$ be the multiplicative closed set $R-\cup {\mbox{$\mathfrak{p}$}}_{n}$ and $A=S^{-1}R$. One can prove that $A$ is a one-dimensional noetherian domain and that $t_{n}^{n+2}/t_{n}^{n+1}$ is in $\overline{A}$ and has integral degree $n$. Therefore ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})=\infty$.
We now give two more properties of the integral degree.
Let $A\subset B$ and $B\subset C$ be integral extensions. Then $A\subset C$ is an integral extension and $$\begin{aligned}
{\mbox{$\mathrm{d}$}}_{A}(C)\leq {\mbox{$\mathrm{d}$}}_{A}(B)^{{\rm d}_{B}(C)}\cdot {\mbox{$\mathrm{d}$}}_{B}(C).\end{aligned}$$
If $c\in C$, there exists an equation $c^{n}+b_{1}c^{n-1}+\ldots
+b_{n-1}c+b_{n}=0$, with $b_{i}\in B$ and $n\leq
{\mbox{$\mathrm{d}$}}_{B}(C)$. Take $D=A[b_{1},\ldots ,b_{n}]$. Since $A\subset B$ is an integral extension, all $b_{i}$ are integral over $A$ and, by Corollary \[nelements\], $D$ is a finitely generated $A$-module and $\mu _{A}(D)\leq \prod_{i=1}^{n}{\rm id}_{A}(b_{i})\leq
{\mbox{$\mathrm{d}$}}_{A}(B)^{{\rm d}_{B}(C)}$. On the other hand, $c$ is clearly integral over $D$ and $D[c]$ is a finitely generated $D$-module with $\mu _{D}(D[c])\leq n\leq {\mbox{$\mathrm{d}$}}_{B}(C)$. Since $D$ is a finitely generated $A$-module and $D[c]$ is a finitely generated $D$-module, then $D[c]$ is a finitely generated $A$-module. So we have $A\subset
A[c]\subset D[c]\subset C$ with $D[c]$ a finitely generated $A$-module with $\mu _{A}(D[c])\leq \mu _{A}(D)\mu _{D}(D[c])\leq
{\mbox{$\mathrm{d}$}}_{A}(B)^{{\rm d}_{B}(C)}\cdot {\mbox{$\mathrm{d}$}}_{B}(C)$. Applying Proposition \[genatiyah\], $(iii)\Rightarrow (i)$, we deduce that $c$ is integral over $A$ and ${\mbox{$\mathrm{id}$}}_{A}(c)\leq {\mbox{$\mathrm{d}$}}_{A}(B)^{{\rm d}_{B}(C)}\cdot
{\mbox{$\mathrm{d}$}}_{B}(C)$.
\[uidlocalitzat\] Let $A\subset B$ be an integral extension and $S$ a multiplicatively closed subset of $A$. Then $S^{-1}A\subset S^{-1}B$ is an integral extension and $$\begin{aligned}
{\mbox{$\mathrm{d}$}}_{S^{-1}A}(S^{-1}B)\leq {\mbox{$\mathrm{d}$}}_{A}(B).\end{aligned}$$ In particular, if $A$ is reduced, $\mathfrak{p}$ is a prime ideal of $A$ and $\overline{A}$ and $\overline{A_{{\mbox{$\mathfrak{p}$}}}}$ are the integral closures of $A$ and $A_{{\mbox{$\mathfrak{p}$}}}$ in their total quotient rings, then $$\begin{aligned}
{\mbox{$\mathrm{d}$}}_{A_{{\mbox{$\mathfrak{p}$}}}}(\overline{A_{{\mbox{$\mathfrak{p}$}}}})\leq {\mbox{$\mathrm{d}$}}_{A}(\overline{A}).\end{aligned}$$
Let $b/s\in S^{-1}B$, $b\in B$, $s\in S$. Then $b^{n}+a_{1}b^{n-1}+\ldots +a_{n-1}b+a_{n}=0$, say, and multiplying by $s^{-n}$ we get $(b/s)^{n}+(a_{1}/s)(b/s)^{n-1}+\ldots
+(a_{n-1}/s^{n-1})(b/s)+a_{n}/s^{n}=0$. So $b/s$ is integral over $S^{-1}A$ and ${\rm id}_{S^{-1}A}(b/s)\leq {\rm id}_{A}(b)\leq
{\mbox{$\mathrm{d}$}}_{A}(B)$. If $A$ is reduced, then $S^{-1}\overline{A}=\overline{S^{-1}A}$ (see e.g. [@hw], Lemma 2.1). Therefore, if $S=A-{\mbox{$\mathfrak{p}$}}$, $\overline{(A_{{\mbox{$\mathfrak{p}$}}})}=\overline{S^{-1}A}=S^{-1}\overline{A}$ and ${\mbox{$\mathrm{d}$}}_{A_{{\mbox{$\mathfrak{p}$}}}}(\overline{A_{{\mbox{$\mathfrak{p}$}}}})={\mbox{$\mathrm{d}$}}_{S^{-1}A}(S^{-1}\overline{A})\leq
{\mbox{$\mathrm{d}$}}_{A}(\overline{A})$.
If $A$ is not reduced, Proposition \[uidlocalitzat\] may fail. The next example is taken from [@hw].
[ Let $k$ be a field and $A=k\lbrack\!\lbrack x,y,z\rbrack\!\rbrack
/(x^{3}-y^{2})(x,y,z)$. Since the maximal ideal annihilates the non-zero element $x^{3}-y^{2}$, $A$ is integrally closed and ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})=1$. Set $S$ the multiplicatively closed set $\{z^{n},n\geq 0\}$. Then $S^{-1}A= K\lbrack\!\lbrack
x,y\rbrack\!\rbrack /(x^{3}-y^{2})$, where $K$ is the quotient field of $k\lbrack\!\lbrack z\rbrack\!\rbrack$, and one can prove that ${\mbox{$\mathrm{d}$}}_{S^{-1}A}(\overline{S^{-1}A})=2$.]{}
Integral degree and reduction number {#idrn}
====================================
We now prove the main result of the paper. Recall that if $I$ is a regular ideal having principal reductions $J_{1}$ and $J_{2}$ with ${\rm rn}_{J_{1}}(I)=n$ and ${\rm rn}_{J_{2}}(I)=m$, then Huckaba proved that $n=m$ (see [@huckaba1], where the local assumption is not needed; it could also be deduced from Theorem \[cds\]). We will denote ${\rm rn}(I)$ to the $J$-reduction number of $I$ for any principal reduction $J$ of $I$.
\[id=rn\] Let $A$ be a noetherian ring, $A\supset{\mbox{$\mathbb{Q}$}}$. Then $$\begin{aligned}
{\mbox{$\mathrm{d}$}}_{A}(\overline{A})={\rm sup}\, \{ {\rm rn}(I)\mid I\mbox{ regular
ideal of }A\mbox{ having a principal reduction}\} +1.\end{aligned}$$
Set $\sigma={\rm sup}\, \{\, {\mbox{$\mathrm{rn}$}}(I)\mid I \mbox{ regular ideal
of }A\mbox{ having a principal reduction}\}+1$ and $d={\mbox{$\mathrm{d}$}}_{A}(\overline{A})$. Take $I$ any regular ideal of $A$ having a principal reduction $J=(x)$, which is also regular. Then $I^{n+1}=xI^{n}$ for some $n\geq 0$. Set $H=x^{-1}I$. Then $H$ is a fractional ideal of $A$ with $H^{n+1}=H^{n}$. If $y\in I$, $(y/x)H^{n}\subset H^{n+1}=H^{n}$. Thus $H^{n}$ is a faithful $A[y/x]$-module. By Proposition \[genatiyah\], $y/x$ is integral over $A$. Thus ${\rm id}_{A}(y/x)\leq d$. By Lemma \[idrtrn\], $x(x,y)^{d-1}:y^{d}=A$ and $y^{d}\in x(x,y)^{d-1}\subseteq xI^{d-1}$. Therefore $I^{[d]}\subset xI^{d-1}$, where $I^{[d]}$ stands for the ideal generated by the $d$-th powers of all elements of $I$. If $A\supset {\mbox{$\mathbb{Q}$}}$, then $I^{[d]}=I^{d}$ (see e.g. [@bourbaki], A1, § 8, n$^{\circ}$ 2, page 95). Thus ${\mbox{$\mathrm{rn}$}}(I)\leq d-1$ and $\sigma\leq
d$. Now take $x,y\in A$, with $x$ regular, such that $y/x$ is integral over $A$. By Lemma \[idrtrn\], ${\mbox{$\mathrm{id}$}}_{A}(y/x)={\mbox{$\mathrm{rn}$}}(x,y)+1\leq
\sigma$. Therefore $d\leq \sigma$.
[Let $A$ be a noetherian ring, $A\supset{\mbox{$\mathbb{Q}$}}$. If $I$ is a regular ideal of $A$ having a principal reduction, by Theorem \[cds\], ${\mbox{$\mathrm{rt}$}}(I)\leq{\mbox{$\mathrm{rn}$}}(I)+1$ and ${\mbox{$\mathrm{reg}$}}({\mbox{$\mathcal{R}(I)$}})={\mbox{$\mathrm{rn}$}}(I)$. Moreover, by Lemma \[idrtrn\], ${\mbox{$\mathrm{id}$}}_{A}(y/x)={\mbox{$\mathrm{rt}$}}(x,y)$ for any $x,y\in A$, with $x$ regular and such that $y/x$ is integral over $A$. In other words, ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})$ is less than or equal to the supremum of the relation type of two-generated regular ideals of $A$ having principal reductions. Therefore, in Theorem \[id=rn\], one can replace ${\mbox{$\mathrm{rn}$}}(I)+1$ by else ${\mbox{$\mathrm{reg}$}}({\mbox{$\mathcal{R}(I)$}})+1$ or else ${\mbox{$\mathrm{rt}$}}(I)$. In addition, one can take the supremum just over the set of two-generated regular ideals having principal reductions. ]{}
We state a particular version of Theorem \[id=rn\] which will be used later. Note that here we do not need the hypothesis $A\supset{\mbox{$\mathbb{Q}$}}$.
\[rtm\] Let $(A,{\mbox{$\mathfrak{m}$}})$ be a noetherian local ring with infinite residue field. Then $$\begin{aligned}
{\mbox{$\mathrm{d}$}}_{A}(\overline{A})={\rm sup}\, \{ {\rm rt}_{\mathfrak{m}}(I)\mid
I\mbox{ two-generated regular ideal of $A$}\}.\end{aligned}$$
Set $\sigma={\rm sup}\, \{\, {\mbox{$\mathrm{rt}$}}_{\mathfrak{m}}(I)\mid I \mbox{
two-generated regular ideal of }A \}$ and $d={\mbox{$\mathrm{d}$}}_{A}(\overline{A})$. Take $I$ a two-generated regular ideal of $A$. Since $A$ is noetherian local with infinite residue field, $I$ has a minimal reduction $J$ generated by as many elements as the analytic spread $l(I)$ of $I$ (see [@nr] or [@hs]). If $l(I)=1$, by Remark \[coef-mod\] and Theorems \[cds\] and \[id=rn\], ${\mbox{$\mathrm{rt}$}}_{\mathfrak{m}}(I)\leq
{\mbox{$\mathrm{rt}$}}(I)\leq {\mbox{$\mathrm{rn}$}}(I)+1\leq d$. If $l(I)=2$, then $I$ is generated by two analytically independent elements $x,y$ and the fiber cone of $I$, ${\mbox{$\mathcal{R}_{{\mbox{$\mathfrak{m}$}}}(I)$}}={\mbox{$\mathcal{F}_{\mathfrak{m}}(I)$}}=\oplus _{n\geq 0}I^{n}/{\mbox{$\mathfrak{m}$}}I^{n}$ is isomorphic to a polynomial ring $(A/{\mbox{$\mathfrak{m}$}})[X,Y]$. Thus ${\rm
rt}_{\mathfrak{m}}(I)={\rm rt}({\mbox{$\mathcal{F}_{\mathfrak{m}}(I)$}})=1\leq d$ and $\sigma\leq
d$. Now take $x,y\in A$, with $x$ regular, such that $y/x$ is integral over $A$. Set ${\mbox{$\mathrm{id}$}}_{A}(y/x)=n$. By Lemma \[idrtrn\], $xI^{n-2}:y^{n-1}\varsubsetneq xI^{n-1}:y^{n}=A$. By Proposition \[E(x,y)\], $E_{{\mbox{$\mathfrak{m}$}}}(I)_{n}=A/{\mbox{$\mathfrak{m}$}}$ and $E_{{\mbox{$\mathfrak{m}$}}}(I)_{n+s}=0$ for all $s\geq 1$. Thus ${\rm
rt}_{{\mbox{$\mathfrak{m}$}}}(I)=n$. Therefore ${\rm id}_{A}(y/x)=n={\rm rt}_{{\mbox{$\mathfrak{m}$}}}(I)\leq
\sigma$. Thus $d\leq \sigma$.
\[necprin\] [ Clearly, Theorem \[id=rn\] is no longer true for ideals having reductions generated by regular sequences of length $l\geq 2$. For instance, in the power series ring $A=k\lbrack\!\lbrack
x,y\rbrack\!\rbrack$ over a field $k$, the ideals $I_{n}=(x^{n},y^{n},
x^{n-1}y)$ have reductions $(x^{n},y^{n})$ with reduction number $n-1$ (see [@huneke], Remark 5.8). However, $(x^{n},y^{n})$ does not verify condition $(iii)$ of Theorem \[cds\]. ]{}
Uniform Artin-Rees numbers {#uarp}
==========================
We now can prove all the results related to Artin-Rees properties.
\[modules\] Let $A$ be a noetherian ring with finite integral degree ${\mbox{$\mathrm{d}$}}_{A}(\overline{A})=d$. Suppose that $A\supset{\mbox{$\mathbb{Q}$}}$. Let $N\subset M$ be two finitely generated $A$-modules. Let $I$ be a regular ideal of $A$ having a principal reduction generated by a $d$-sequence with respect to $M/N$. Then, for every integer $n\geq d$, $$\begin{aligned}
I^{n}M\cap N=I^{n-d}(I^{d}M\cap N).\end{aligned}$$
Since $I$ is regular and has a principal reduction, by Theorem \[id=rn\], ${\mbox{$\mathrm{rn}$}}(I)\leq d-1$. It is enough to prove that $s(N,M;I)\leq {\mbox{$\mathrm{rn}$}}(I)+1$. Let $J=(x)$ be a principal reduction of $I$, set $r={\mbox{$\mathrm{rn}$}}(I)$ and take $k\geq 1$. Then $I^{r+k}M\cap
N=x^{k}I^{r}M\cap N$. Since $x$ is a $d$-sequence with respect to $M/N$, $x^{k}I^{r}M\cap N=x^{k-1}(xI^{r}M\cap N)\subseteq
I^{k-1}(I^{r+1}M\cap N)$. Thus $s(N,M;I)\leq r+1$.
[ By Proposition \[slrt\], $s(N,M;I)\leq {\rm rt}(I;M/N)$. If $I$ has a principal reduction $J$ generated by a $d$-sequence with respect to $M/N$, then $J$ is also a principal reduction of $I$ with respect to $M/N$ and, by Theorem \[cds\], ${\rm rt}(I;M/N)\leq {\rm
rn}_{J}(I;M/N)+1\leq {\rm rn}_{J}(I)+1$. Since $I$ is a regular ideal having a principal reduction, by Theorem \[id=rn\], ${\rm rn}(I)\leq
d-1$. Therefore, $s(N,M;I)\leq d$, which also proves Theorem \[modules\]. ]{}
Our ideal-theoretic version of Theorem \[modules\] is the following.
\[ideals\] Let $A$ be a noetherian ring, $A\supset{\mbox{$\mathbb{Q}$}}$. Let ${\mbox{$\mathfrak{a}$}}$ be an ideal of $A$ such that $A/{\mbox{$\mathfrak{a}$}}$ has finite integral degree ${\mbox{$\mathrm{d}$}}_{A/{\mbox{$\mathfrak{a}$}}}(\overline{A/{\mbox{$\mathfrak{a}$}}})=d$. Let $I$ be an ideal of $A$ such that $IA/{\mbox{$\mathfrak{a}$}}$ has an $A/{\mbox{$\mathfrak{a}$}}$-regular principal reduction. Then, for every integer $n\geq d$, $$\begin{aligned}
I^{n}\cap {\mbox{$\mathfrak{a}$}}=I^{n-d}(I^{d}\cap {\mbox{$\mathfrak{a}$}}).\end{aligned}$$
By Proposition \[slrt\], $s({\mbox{$\mathfrak{a}$}},A;I)\leq {\rm
rt}(I;A/{\mbox{$\mathfrak{a}$}})$. By Remark \[coef-mod\], ${\rm rt}(I;A/{\mbox{$\mathfrak{a}$}})={\rm
rt}(IA/{\mbox{$\mathfrak{a}$}})={\rm rt}(IA/{\mbox{$\mathfrak{a}$}};A/{\mbox{$\mathfrak{a}$}})$. Since $IA/{\mbox{$\mathfrak{a}$}}$ is $A/{\mbox{$\mathfrak{a}$}}$-regular and has principal reduction $JA/{\mbox{$\mathfrak{a}$}}$, by Theorem \[cds\], ${\rm
rt}(IA/{\mbox{$\mathfrak{a}$}};A/{\mbox{$\mathfrak{a}$}})\leq {\rm rn}_{JA/{\mbox{$\mathfrak{a}$}}}(IA/{\mbox{$\mathfrak{a}$}};A/{\mbox{$\mathfrak{a}$}})+1={\rm
rn}_{JA/{\mbox{$\mathfrak{a}$}}}(IA/{\mbox{$\mathfrak{a}$}})+1$. By Theorem \[id=rn\], ${\rm
rn}_{JA/{\mbox{$\mathfrak{a}$}}}(IA/{\mbox{$\mathfrak{a}$}})\leq d-1$. So $s({\mbox{$\mathfrak{a}$}},A;I)\leq d$.
As a corollary of Theorem \[ideals\] we obtain a particular version of the main result in [@planas].
\[cor1dim\] Let $(A,{\mbox{$\mathfrak{m}$}})$ be a noetherian local ring, $A\supset{\mbox{$\mathbb{Q}$}}$. Let ${\mbox{$\mathfrak{a}$}}$ be an ideal of $A$ such that $A/{\mbox{$\mathfrak{a}$}}$ has finite integral degree ${\mbox{$\mathrm{d}$}}_{A/{\mbox{$\mathfrak{a}$}}}(\overline{A/{\mbox{$\mathfrak{a}$}}})=d$. Suppose that ${\rm dim}(A/{\mbox{$\mathfrak{a}$}})\leq
1$. Then, for every integer $n\geq d$ and for every ideal $I$ of $A$ such that $IA/{\mbox{$\mathfrak{a}$}}$ is $A/{\mbox{$\mathfrak{a}$}}$-regular, $$\begin{aligned}
I^{n}\cap {\mbox{$\mathfrak{a}$}}=I^{n-d}(I^{d}\cap {\mbox{$\mathfrak{a}$}}).\end{aligned}$$
Since ${\rm dim}(A/{\mbox{$\mathfrak{a}$}})\leq 1$, every ideal $I$ of $A$ is such that $IA/{\mbox{$\mathfrak{a}$}}$ has a principal reduction. Then apply Theorem \[ideals\].
Remark that by a result of Krull, if $(R,{\mbox{$\mathfrak{n}$}})$ is a noetherian local non-reduced ring such that ${\mbox{$\mathfrak{n}$}}$ contains a regular element, then the integral closure $\overline{R}$ is not a finite $R$-module (see e.g. [@matsumura], §33). In particular, in Theorem \[ideals\] and in Corollary \[cor1dim\] (as well as in Theorem \[ideals2\]), setting $R=A/{\mbox{$\mathfrak{a}$}}$, if $A/{\mbox{$\mathfrak{a}$}}$ has a finite integral closure and ${\mbox{$\mathfrak{m}$}}/{\mbox{$\mathfrak{a}$}}$ has a regular element, one deduces that ${\mbox{$\mathfrak{a}$}}$ is forced to be a radical ideal.
The next example, taken from Eisenbud and Hochster in [@eh], shows that if the integral degree is not finite, then the conclusion of Theorem \[ideals\] may be false.
\[ehe\][ There exist $A$, a two-dimensional noetherian domain, ${\mbox{$\mathfrak{a}$}}$, a prime ideal of $A$, and $\{I_{n}\}_{n}$, a family of two-generated ideals of $A$ such that $I_{n}A/{\mbox{$\mathfrak{a}$}}$ has an $A/{\mbox{$\mathfrak{a}$}}$-regular principal reduction, but, for every integer $n\geq 1$, $$\begin{aligned}
I_{n}^{n}\cap {\mbox{$\mathfrak{a}$}}\varsupsetneq I_{n}(I_{n}^{n-1}\cap {\mbox{$\mathfrak{a}$}}) .\end{aligned}$$]{}
Let $k$ be an algebraically closed field and $\{ X_{n}\}$, $\{
Y_{n}\}$ two countable families of indeterminates. Set $f_{n}=X^{n}_{n}-Y^{n+1}_{n}$ and $I_{n}$ the ideal in $T_{n}=k[X_{1},Y_{1},\ldots ,X_{n},Y_{n}]$ generated by $f_{2}-f_{1},\ldots ,f_{n}-f_{1}$. Set $S_{n}=T_{n}/I_{n}$ and $U_{n}=S_{n}-\cup _{i=1}^{n}(X_{i},Y_{i})S_{n}$. $U_{n}$ is a multiplicatively closed subset of $S_{n}$. Set $A_{n}=U_{n}^{-1}S_{n}$, $A=\botrel{\lim}{\rightarrow}A_{n}$ and $x_{n},y_{n}$ and $f$ the images of $X_{n},Y_{n}$ and $f_{n}$ in $A$. Then $A$ is a two-dimensional noetherian regular factorial ring whose maximal ideals $I_{n}=(x_{n},y_{n})$ form a countable set. Their intersection $\cap _{n}I_{n}$ is a prime principal ideal ${\mbox{$\mathfrak{a}$}}=(f)$ whose generator $f$ is in $I_{n}^{n}$. Then $I_{n}A/{\mbox{$\mathfrak{a}$}}$ is $A/{\mbox{$\mathfrak{a}$}}$-regular and $y_{n}A/{\mbox{$\mathfrak{a}$}}$ is a principal reduction of $I_{n}A/{\mbox{$\mathfrak{a}$}}$. Moreover, $$\begin{aligned}
I_{n}(I_{n}^{n-1}\cap {\mbox{$\mathfrak{a}$}})=I_{n}{\mbox{$\mathfrak{a}$}}\varsubsetneq {\mbox{$\mathfrak{a}$}}=I_{n}^{n}\cap {\mbox{$\mathfrak{a}$}}.\end{aligned}$$ In particular, by Theorem \[ideals\], ${\mbox{$\mathrm{d}$}}_{A/{\mbox{$\mathfrak{a}$}}}(\overline{A/{\mbox{$\mathfrak{a}$}}})=\infty$.
We now prove that there exists a uniform Artin-Rees modulo ${\mbox{$\mathfrak{m}$}}$ number for the set of two-generated regular ideals. Here, we do not need $A\supset{\mbox{$\mathbb{Q}$}}$.
\[ideals2\] Let $(A,{\mbox{$\mathfrak{m}$}})$ be a noetherian local ring with infinite residue field. Let ${\mbox{$\mathfrak{a}$}}$ be an ideal of $A$ such that $A/{\mbox{$\mathfrak{a}$}}$ has finite integral degree ${\mbox{$\mathrm{d}$}}_{A/{\mbox{$\mathfrak{a}$}}}(\overline{A/{\mbox{$\mathfrak{a}$}}})=d$. Let $I$ be a two-generated ideal of $A$ such that $IA/{\mbox{$\mathfrak{a}$}}$ is $A/{\mbox{$\mathfrak{a}$}}$-regular. Then, for every $n\geq d$, $$\begin{aligned}
I^{n}\cap {\mbox{$\mathfrak{a}$}}=I^{n-d}(I^{d}\cap {\mbox{$\mathfrak{a}$}}) +{\mbox{$\mathfrak{m}$}}I^{n}\cap {\mbox{$\mathfrak{a}$}}.\end{aligned}$$
Let $I$ be a two-generated ideal of $A$ such that $IA/{\mbox{$\mathfrak{a}$}}$ is $A/{\mbox{$\mathfrak{a}$}}$-regular. By Proposition \[slrt\], $s_{\mathfrak{m}}({\mbox{$\mathfrak{a}$}},A;I)\leq {\rm rt}_{\mathfrak{m}}(I;A/{\mbox{$\mathfrak{a}$}})$. By Remark \[coef-mod\], ${\rm rt}_{\mathfrak{m}}(I;A/{\mbox{$\mathfrak{a}$}})={\rm
rt}_{\mathfrak{m}/\mathfrak{a}}(IA/{\mbox{$\mathfrak{a}$}})$, which is $d$ or less by Proposition \[rtm\].
The next example, taken from Wang in [@wang], shows that even this weaker uniform Artin-Rees property of Theorem \[ideals2\] is not true anymore for the set of three-generated ideals. It also shows that if in Theorem \[ideals\] one changes the set of ideals having principal reductions for the set of ideals having reductions generated by regular sequences of length two, then there may not exist a uniform Artin-Rees (modulo ${\mbox{$\mathfrak{m}$}}$) number.
\[wange\] [There exist $(A,{\mbox{$\mathfrak{m}$}})$, a three-dimensional noetherian local ring with infinite residue field, ${\mbox{$\mathfrak{a}$}}$, a prime ideal of $A$ such that $A/{\mbox{$\mathfrak{a}$}}$ has finite integral closure $\overline{A/{\mbox{$\mathfrak{a}$}}}$, and $\{I_{n}\}_{n}$, a family of three-generated ideals of $A$ such that $I_{n}A/{\mbox{$\mathfrak{a}$}}$ is $A/{\mbox{$\mathfrak{a}$}}$-regular, but, for every $n\geq 1$, $$\begin{aligned}
I_{n}^{n}\cap {\mbox{$\mathfrak{a}$}}\varsupsetneq I_{n}(I_{n}^{n-1}\cap {\mbox{$\mathfrak{a}$}}) +{\mbox{$\mathfrak{m}$}}I_{n}^{n}\cap {\mbox{$\mathfrak{a}$}}. \end{aligned}$$]{}
Take $(A,{\mbox{$\mathfrak{m}$}})$, a three-dimensional regular local ring with infinite residue field, ${\mbox{$\mathfrak{m}$}}=(x,y,z)$, the maximal ideal generated by a regular system of parameters $x,y,z$, and ${\mbox{$\mathfrak{a}$}}=(z)$. Let $I_{n}=(x^{n},y^{n},x^{n-1}y+z^{n})$. Since $x^{n},y^{n},x^{n-1}y+z^{n}$ is a regular sequence of $A$, the relation type of $I_{n}$ is ${\rm rt}(I_{n})=1$. It is not difficult to prove that the relation type of $I_{n}A/{\mbox{$\mathfrak{a}$}}$ and that the relation type of its fiber cone are given by ${\rm rt}(I;A/{\mbox{$\mathfrak{a}$}})={\rm
rt}_{\mathfrak{m}}(I;A/{\mbox{$\mathfrak{a}$}})=n$. Then, by Proposition \[slrt\], $s_{\mathfrak{m}}({\mbox{$\mathfrak{a}$}},A;I_{n})={\rm
rt}_{\mathfrak{m}}(I_{n};A/{\mbox{$\mathfrak{a}$}})=n$. Remark that $(x^{n},y^{n})A/{\mbox{$\mathfrak{a}$}}$ is a reduction of $I_{n}A/{\mbox{$\mathfrak{a}$}}$ generated by a regular sequence of length two.
[*Acknowledgement.*]{} The second author is partially supported by the MTM2004-01850 spanish grant.
[cc]{}
|
---
abstract: 'We construct a model atom for Ti– using more than 3600 measured and predicted energy levels of Ti and 1800 energy levels of Ti, and quantum mechanical photoionisation cross-sections. Non-local thermodynamical equilibrium (NLTE) line formation for Ti and Ti is treated through a wide range of spectral types from A to K, including metal-poor stars with \[Fe/H\] down to $-2.6$ dex. NLTE leads to weakened Ti lines and positive abundance corrections. The magnitude of NLTE corrections is smaller compared to the literature data for FGK atmospheres. NLTE leads to strengthened Ti lines and negative NLTE abundance corrections. For the first time, we performed the NLTE calculations for Ti– in the 6500 K$\leq$ [$T_{\rm eff}$]{} $\leq$ 13000 K range. For four A type stars we derived in LTE an abundance discrepancy of up to 0.22 dex was obtained between Ti and Ti and it vanishes in NLTE. For other four A-B stars, with only Ti lines observed, NLTE leads to decrease of line-to-line scatter. An efficiency of inelastic Ti + H collisions was estimated from analysis of Ti and Ti lines in 17 cool stars with $-2.6 \leq$ \[Fe/H\] $\leq$ 0.0. Consistent NLTE abundances from Ti and Ti were obtained applying classical Drawinian rates for the stars with $\ge$ 4.1, and neglecting inelastic collisions with H for the VMP giant HD 122563. For the VMP turn-off stars (\[Fe/H\] $\leq -2$ and $\leq$ 4.1), we obtained the positive abundance difference Ti– already in LTE and it increases in NLTE. The accurate collisional data for Ti and Ti are desired to find a clue to this problem.'
author:
- |
T. M. Sitnova$^{1,2}$[^1]; L. I. Mashonkina$^{1}$, T. A. Ryabchikova$^{1}$\
$^{1}$Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya 48, 119017, Moscow, Russia\
$^{2}$Sternberg Astronomical Institute; Faculty of Physics, Moscow State University, Universitetsky pr., 13, 119991, Moscow, Russia\
title: 'A NLTE line formation for neutral and singly-ionised titanium in model atmospheres of the reference A-K stars'
---
\[firstpage\]
line: formation – stars: atmospheres – stars: fundamental parameters – stars: abundances.
Introduction
============
Titanium is observed in lines of two ionisation stages, Ti and Ti, in a wide range of spectral types from A to K. Experimental oscillator strengths ($f_{ij}$) for Ti and Ti were measured using a common method [@Lawler2013_ti1; @Wood2013_ti2 respectively], which permits to use Ti and Ti lines for determination of accurate titanium abundances and stellar atmosphere parameters. @Bergemann2011 and Bergemann et al. (2012) investigated the non-local thermodynamic equilibrium (NLTE) line-formation for Ti- in the atmospheres of cool stars. The first paper presents the NLTE calculations for the Sun and four metal-poor stars with [$T_{\rm eff}$]{} $\le$ 6350 K while the second one for red supergiants with 3400 K $\le$ [$T_{\rm eff}$]{} $\le$ 4400 K, $-0.5 \le$ $\le$ 1.0, and $-0.5 \le$ \[Fe/H\] $\le$ 0.5. @Bergemann2011 found that the deviations from LTE are small in the solar atmosphere, with the abundance difference between NLTE and LTE (the NLTE abundance correction, $\Delta_{\rm NLTE}$) not exceeding 0.11 dex for Ti lines. For the Sun @Bergemann2011 derived consistent within 0.04 dex NLTE abundances from Ti and Ti lines. However, she failed to achieve the Ti/Ti ionisation equilibrium for cool metal-poor (MP, $-2.5 \leq$ \[Fe/H\] $\leq -1.3$) dwarfs with well-determined atmospheric parameters. @Bergemann2011 suggested that this can be caused by: (i) neglecting high-excitation levels of Ti in the used model atom; (ii) using hydrogenic photoionisation cross-sections; (iii) using a rough theoretical approximation [@Drawin1968; @Drawin1969] for inelastic collisions with hydrogen atoms. We eliminate the first two points in this study. We still rely on the Drawinian approximation because accurate laboratory measurements or quantum mechanical calculations for inelastic Ti $+$ H collisions are not available. Poorly-known collisions with H atoms is the main source of the uncertainties in the NLTE results for stars with [$T_{\rm eff}$]{} $\le$ 7000 K.
For the atmospheres hotter than [$T_{\rm eff}$]{} $\ge$ 6500 K the NLTE calculations for Ti– were not yet performed, although the observations indicate a discrepancy in LTE abundances between Ti and Ti. For example, @Bikmaev2002 derived under the LTE assumption the abundance difference Ti–Ti[^2] = $-0.17$ dex and $-0.20$ dex for the A-type stars HD 32115 and HD 37954, respectively. @Becker1998 performed the NLTE calculations for Ti in A-type stars (Vega, supergiants $\eta$ Leo and 41-3712 from M31) and found that NLTE leads to weakened Ti lines, with the NLTE abundance corrections being larger for weak lines compared with those calculated for strong lines. Using model atom from @Becker1998, @Przybilla2006 and @Schiller2008 derived the NLTE abundances from lines of Ti in BA-type supergiants and concluded that proper NLTE calculations reduce the line-to-line scatter.
We aim to construct a comprehensive model atom of Ti– and to treat a reliable method of abundance determination from different lines of Ti and Ti in a wide range of stellar spectral types from late B to K, including metal-poor stars. First, we test the new model atom employing the stars with [$T_{\rm eff}$]{} $\ge$ 7100 K, where inelastic collisions with hydrogen atoms do not affect the statistical equilibrium (SE). Then, we empirically constrain an efficiency of collisions with H from analysis of Ti and Ti lines in spectra of cool metal-poor stars. In total, we analyse titanium lines in 25 well-studied stars.
We present the constructed model atom and the NLTE mechanism for Ti and Ti in Section \[method\]. Section \[obspar\] describes observations and stellar parameters of our stellar sample. The obtained results for hot and cool stars are considered in Sections \[hots\] and \[cools\], respectively. Our conclusions and recommendations are given in Sect. \[con\].
Method of NLTE calculations for Ti– {#method}
===================================
In this section we describe the model atom of titanium, the programs used for computing the level populations and spectral line profiles, and mechanisms of departures from LTE for Ti and Ti.
The model atom
--------------
Titanium is almost completely ionised throughout the atmosphere of stars with effective temperatures above 4500 K. For example, the ratio ${ \rm N_{Ti~II}/N_{Ti~I} \simeq 10^2}$ throughout the solar atmosphere. Such minority species as Ti are particularly sensitive to NLTE effects because any small deviation in the intensity of ionising radiation from the Plank function strongly changes their population. For accurate calculations of the SE we include in our model atom high-excitation levels of Ti and Ti, which establish collisional coupling of Ti and Ti levels near the continuum to the ground states of Ti and Ti, respectively. @mash_fe included high-excitation levels of Fe in their Fe– model atom, and found that the SE of iron changed substantially by achieving close collisional coupling of the Fe levels near the continuum to the ground state of Fe. Our model atom of titanium (Fig. \[ti1\], \[ti2\]) is constructed using not only all the known energy levels from NIST [@NIST08], but also the predicted levels from atomic structure calculation of R. Kurucz ($http://kurucz.harvard.edu/atoms.html$). The measured levels of Ti with the excitation energy [$E_{\rm exc}$]{} $\le$ 6 eV belong to 175 terms. Neglecting their fine structure, except for the ground state of Ti, we obtain 177 levels in the model atom. The predicted and measured levels below the threshold, in total 3500 ones with [$E_{\rm exc}$]{} $\ge$ 6 eV, with common parity and close energies were combined whenever the energy separation is smaller than $\Delta E$ = 0.1 eV. This makes up 17 super-levels.
For Ti we use the experimental energy levels belonging to 89 terms with [$E_{\rm exc}$]{}up to 10.5 eV. The fine structure is neglected, except for the ground state of Ti. The 1800 high excitation levels with $10.5 \le$ [$E_{\rm exc}$]{} $\le 13.6$ eV are used to make up 28 super-levels. The ground state of Ti completes the system of levels in the model atom.
In total, 7929 and 3104 allowed transitions of Ti and Ti, respectively, occur in our final model atom. Their average f-values are calculated using the data from R. Kurucz database. We compared predicted gf-values with accurate laboratory data for about 900 transitions of Ti [@Lawler2013_ti1] and found a systematic shift to be minor, with an average difference of log gf$_{lab}$ – log gf$_{Kurucz}$ = $-0.05 \pm$ 0.28. An advantage of the Kurucz’s predicted gf-values is their completeness that is of extreme importance for the statistical equilibrium calculations. For the transitions involving the superlevels the total gf-value was calculated as a sum of gf of individual transitions $g f_{tot} = \sum_{i, j}^{} (g_i f_{i, j})$, i = 1,..., N$_l$, j = 1,..., N$_u$, where N$_l$ and N$_u$ are numbers of individual levels, which form a lower and upper superlevel, respectively. Radiative rates were computed using the Voigt profiles for transitions with $f_{ij} \ge 0.10$ and 1800 Å $ \le \lambda \le$ 4000 Å and the Doppler profiles for the remaining ones. The transitions with $f_{ij} \le 10^{-8}$ were treated as forbidden ones. For 115 terms of Ti with [$E_{\rm exc}$]{} $\le$ 5.5 eV we use photoionisation cross-sections from calculations of @Nahar2015, based on the close-coupling R-matrix method, and for 78 terms of Ti with [$E_{\rm exc}$]{} $\le$ 10.0 eV we use the data from quantum-mechanical calculations of Keith Butler (private communication). For the remaining high-excitation levels we assume a hydrogenic approximation with using an effective principle quantum number. We compare the quantum-mechanical photoionisation cross-sections with the hydrogenic ones for selected levels of Ti and Ti in Fig. \[pic\_ti1\] and \[pic\_ti2\], respectively. For each level the hydrogenic cross-sections fit, on average, the quantum-mechanical ones near the ionisation threshold. The difference at frequencies higher than 3.29 $10^{15}$ Hz ($\lambda \le$ 912 Å) weakly affects the photoionisation rate because of small flux in this spectral range in the investigated stellar atmospheres.
All levels in our model atom are coupled via collisional excitation and ionisation by electrons and by neutral hydrogen atoms. Our calculations of collisional rates rely on the theoretical approximations because no accurate experimental or theoretical data are available. For electron-impact excitation we use the formula of @Reg1962 for the allowed transitions and the formula from @WA1948_cbb with a collision strength of 1.0 for the radiatively forbidden transitions. Ionisation by electronic collisions is calculated from the [@Seaton1962] approximation using the threshold photoionisation cross-section.
For collisions with H atoms, we employ the formula of @SteenbockHolweger1984 based on theory of @Drawin1968 [@Drawin1969] for allowed b-b and b-f transitions and, following @Takeda1994, a simple relation between hydrogen and electron collisional rates, C$_H = C_e \sqrt{(m_e/m_H)}N_H/N_e$, for forbidden transitions. Due to the Drawin formula provides order-of-magnitude estimates, we perform the NLTE calculations using a scaling factor [$S_{\rm H}$]{}=0.1, 0.5 and 1, and constrain its magnitude empirically from analysis of metal-poor stars.
The nearly resonance charge exchange reaction (CER) H$^+$ + Ti $\leftrightarrow$ H + Ti takes place because the ionisation thresholds for Ti and H are 13.57 eV and 13.60 eV, respectively. There are no literature data on cross-sections for this process. In order to inspect an influence of CER on the statistical equilibrium of titanium, we assumed that the analytic fit deduced by @Arnaud1985 for O can also be applied to Ti, because the ionisation threshold for O is close to that for Ti and amounts to 13.62 eV. Test calculations for A-type stars showed that the CER makes the populations of the ground states of Ti and Ti to be in thermodynamic equilibrium, nevertheless, no change in the NLTE abundances from lines of Ti was found. For stars with [$T_{\rm eff}$]{} $\le$ 9000 K the CER weakly affects the SE because of small fraction of Ti.
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Programs and model atmospheres
------------------------------
The coupled radiative transfer and SE equations were solved with a revised version of the <span style="font-variant:small-caps;">detail</span> code by @detail. The opacity package of the <span style="font-variant:small-caps;">detail</span> code was updated as described by @2011JPhCS.328a2015P and @mash_fe, by including the quasi-molecular Ly$_{\alpha}$ satellites following the implementation by @CastelliKurucz2001 of the @Allard1998 theory and using the Opacity Project [see @Seaton1994 for a general review] photoionisation cross-sections for the calculations of b-f absorption of C, N, O, Mg, Si, Al, Ca, and Fe. In addition to the continuous background opacity, the line opacity introduced by H and metal lines was taken into account by explicitly including it in solving the radiation transfer. The metal line list was extracted from the @Kurucz1994 compilation and the VALD database [@vald]. The pre-calculated departure coefficients were then used by <span style="font-variant:small-caps;">synthV\_NLTE</span> code updated in @ryab2015, and based on @Tsymbal1996 to compute the theoretical synthetic spectra. The integration of the <span style="font-variant:small-caps;">synthV\_NLTE</span> code in the <span style="font-variant:small-caps;">idl binmag3</span> code by O. Kochukhov[^3] allows us to obtain the best fit to the observed line profiles with the NLTE effects taken into account.
Throughout this study, the element abundance is determined from line profile fitting. For late type stars we used classical plane-parallel model atmospheres from the <span style="font-variant:small-caps;">marcs</span> model grid [@MARCS], which were interpolated for given [$T_{\rm eff}$]{}, , and \[Fe/H\] using a FORTRAN-based routine written by Thomas Masseron[^4]. For A-B type stars the model atmospheres were calculated under the LTE assumption with the code <span style="font-variant:small-caps;">LLmodels</span> [@LLmodels].
For each star the line list includes unblended lines of various strength ($EW\le$ 150 mÅ, where $EW$ is the line equivalent width) and excitation energies. The full list of the lines is presented in Table \[atomic\] along with the transition information, gf-value, excitation energy, and damping constants (log $\gamma_{rad}$, log $\gamma_4/N_e$, log $\gamma_6/N_H$ at 10000 K). The line list was extracted from the VALD database [@vald; @vald2015]. The adopted oscillator strengths for most lines of both ions were measured by a common method [@Lawler2013_ti1; @Wood2013_ti2 – Wisconsin data], and, hence, represent homogeneous set of gf-values.
\[atomic\]
[|l|c|c|c|c|c|c|]{} **$\lambda$, Å** & & & transition & & &\
\
4008.927 & 0.021 & -1.000 & 3a3F – y3F & 8.000 & -6.080 & -7.750\
4060.262 & 1.052 & -0.690 & a3P – x3P & 8.050 & -6.050 & -7.646\
4287.403 & 0.836 & -0.370 & a5F – x5D & 8.230 & -6.010 & -7.570\
4449.143 & 1.886 & 0.470 & a3G – v3G & 8.120 & -5.560 & -7.579\
4453.699 & 1.872 & 0.100 & a3G – v3G & 8.110 & -4.970 & -7.582\
4512.733 & 0.836 & -0.400 & a5F – y5F & 8.130 & -5.120 & -7.593\
4533.240 & 0.848 & 0.540 & a5F – y5F & 8.130 & -5.120 & -7.593\
4534.776 & 0.836 & 0.350 & a5F – y5F & 8.130 & -5.280 & -7.596\
4548.763 & 0.826 & -0.280 & a5F – y5F & 8.130 & -5.410 & -7.598\
4555.484 & 0.848 & -0.400 & a5F – y5F & 8.130 & -5.280 & -7.596\
\
Statistical equilibrium of Ti-
------------------------------
In this section, we consider the NLTE effects for Ti– in various model atmospheres. The deviations from LTE in level populations are characterized by the departure coefficients b$_{i}$ = n$^{NLTE}_i$/n$^{LTE}_i$, where n$^{NLTE}_i$ and n$^{LTE}_i$ are the statistical equilibrium and thermal (Saha-Boltzmann) number densities, respectively. The departure coefficients for the selected levels of Ti, Ti and the ground state of Ti in the model atmospheres 5777/4.44/0, 6350/4.09/$-2.15$, 9700/4.1/0.4 and 12800/3.75/0 are presented in Fig. \[depart\]. All the levels retain their LTE populations in deep atmospheric layers below log$\tau_{5000}$ = 0. In the higher atmospheric layers a total number density of Ti is lower compared with the TE value. The overionisation is caused by superthermal radiation of non-local origin below the thresholds of the low excitation levels of Ti. In the atmospheres, where Ti is the majority species, collisional recombinations to the Ti high-excitation levels followed by cascades of spontaneous transitions tend to compensate a depopulation of the lower levels of Ti. However, this process can not prevent the overionisation. High superlevels of Ti are collisionally coupled to the ground state of Ti. NLTE leads to weakened lines of Ti compared to their LTE strengths.
High levels of Ti are overpopulated via radiative pumping transitions from the low excitation levels. The NLTE effects for Ti are small in cool atmospheres. In the models 5780/4.44/0.0 and 6350/4.09/$-2.1$ a behaviour of the departure coefficients is qualitatively similar. However a magnitude of the NLTE effects grows towards higher [$T_{\rm eff}$]{} and lower and \[Fe/H\]. In the models representing atmospheres of A-type stars high levels of Ti retain their LTE populations inward log$\tau_{5000}$ = $-1.5$, and become underpopulated in the higher atmospheric layers. This results in strengthening the Ti line cores formed in the uppermost layers compared with LTE. In the hottest model atmosphere 12800/3.75/0.0 Ti becomes the majority species, while the levels of Ti are underpopulated beginning at log $\tau \simeq$ 0.5. Overionisation of Ti results in weakened Ti lines.
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Observations and stellar atmosphere parameters {#obspar}
==============================================
Our sample includes the Sun and 24 well-studied stars. They are listed in Table \[obs\]. Atmospheric parameters ([$T_{\rm eff}$]{}, , \[Fe/H\], ) were either determined in our earlier studies or taken from the literature. These parameters were derived by several independent methods, which gave consistent results. Our hot stellar sample consists of A and late B stars, which do not reveal pulsation activity, chemical stratification and magnetic field. For Sirius, $\pi$ Cet, 21 Peg, HD 32115, HD 37594, HD 73666, HD 145788 atmospheric parameters were derived by common method, based on multicolour photometry, analysis of hydrogen Balmer lines and metal lines in high resolution spectra and comparison of spectrophotometric data with theoretical flux (see Table \[obs\] for the references). For HD 72660 the parameters 9700/4.10/0.45/1.8 were derived by fitting the 4400-5200 Å and 6400-6700 Å spectral regions with <span style="font-variant:small-caps;">sme</span> (Spectroscopy Made Easy) program package [@1996AAS..118..595V]. We used medium-resolution spectrum of HD 72660 extracted from ELODIE archive. <span style="font-variant:small-caps;">sme</span> was tested for Sirius, $\pi$ Cet, 21 Peg, and HD 32115 by @2015ASPC..494..308R where the authors derived practically the same parameters as adopted in the present paper. Atmospheric parameters of HD 72660 agree with the results of @Lemke1989, who derived [$T_{\rm eff}$]{}/ = 9770/4.0 from photometry and H$_{\beta}$, and , who derived 9650/4.05. Each cool star of the sample has photometric [$T_{\rm eff}$]{} and based on the Hipparcos parallax. We checked in advance whether an ionisation equilibrium between Fe and Fe is fulfilled in NLTE when using non-spectroscopic parameters. The iron abundances obtained from the lines of Fe and Fe in dwarfs agree within 0.05 dex in NLTE, when using [$S_{\rm H}$]{} = 0.5 [@lick]. To confirm the adopted parameters, we checked them with evolutionary tracks and derived reasonable masses and ages. Our sample also includes the most metal-poor giant, HD 122563 (\[Fe/H\] = $-2.56$), with the accurate Hipparcos parallax available. The effective temperature of HD 122563 was determined by @creevey2012 based on angular diameter measurements.
\[obs\]
[|l|c|c|c|c|c|c|c|c|]{} Star & [$T_{\rm eff}$]{}, & $\log g$ & \[Fe/H\] & , & Ref. &$\lambda/\Delta\lambda$, & $S/N >$ & source\
& K & & & [kms$^{-1}$]{}& & 10$^3$ & &\
Sun & 5777 & 4.44 & 0.0 & 0.9 & – & 300& 300& KPNO84\
HD 24289 & 5980 & 3.71 & –1.94 & 1.1 & S15 & 60 & 110 & S15\
HD 64090 & 5400 & 4.70 & –1.73 & 0.7 & S15 & 60 & 280 & S15\
HD 74000 & 6225 & 4.13 & –1.97 & 1.3 & S15 & 60 & 140 & S15\
HD 84937 & 6350 & 4.09 & –2.16 & 1.7 & S15 & 80 & 200 & UVESPOP$^1$\
HD 94028 & 5970 & 4.33 & –1.47 & 1.3 & S15 & 60 & 120 & S15\
HD 103095 & 5130 & 4.66 & –1.26 & 0.9 & S15 & 60 & 200 & FOCES$^2$\
HD 108177 & 6100 & 4.22 & –1.67 & 1.1 & S15 & 60 & 60 & S15\
HD 140283 & 5780 & 3.70 & –2.46 & 1.6 & S15 & 80 & 200 & UVESPOP\
BD–4$^\circ$ 3208 & 6390 & 4.08 & –2.20 & 1.4 & S15 & 80 & 200 & UVESPOP\
BD–13$^\circ$ 3442 & 6400 & 3.95 & –2.62 & 1.4 & S15 & 60 & 100 & S15\
BD+7$^\circ$ 4841 & 6130 & 4.15 & –1.46 & 1.3 & S15 & 120 & 150 & S15\
BD+9$^\circ$ 0352 & 6150 & 4.25 & –2.09 & 1.3 & S15 & 120 & 160 & S15\
BD+24$^\circ$ 1676 & 6210 & 3.90 & –2.44 & 1.5 & S15 & 60 & 90 & S15\
BD+29$^\circ$ 2091 & 5860 & 4.67 & –1.91 & 0.8 & S15 & 60 & 80 & S15\
BD+66$^\circ$ 0268 & 5300 & 4.72 & –2.06 & 0.6 & S15 & 60 & 110 & S15\
G 090–003 & 6010 & 3.90 & –2.04 & 1.3 & S15 & 60 & 100 & S15\
HD 122563 & 4600 & 1.60 & –2.60 & 2.0 & M11 & 80 & 200 & UVESPOP\
HD 32115 & 7250 & 4.20 & 0.0 & 2.3 & F11 & 60 & 490 & F11\
HD 37594 & 7150 & 4.20 & -0.30 & 2.5 & F11 & 60 & 535 & F11\
HD 72660 & 9700 & 4.10 & 0.45 & 1.8 & this study & 30 & 150 & STIS$^3$, L98\
HD 73666 & 9380 & 3.78 & 0.10 & 1.8 & F07, F10 & 65 & 660 & F07\
HD 145788 & 9750 & 3.70 & 0.0 & 1.3 & F09 & 115 & 200 & F09\
HD 209459 & 10400 & 3.55 & 0.0 & 0.5 & F09 & 120 & 700 & F09\
(21 Peg) & & & &&&&&\
HD 48915 & 9850 & 4.30 & 0.4 & 1.8$^4$ & H93 & 70 & 500 & F95\
(Sirius) & & & &&&&&\
HD 17081 & 12800 & 3.75 & 0.0 & 1.0 & F09 & 65 & 200 & F09\
($\pi$ Cet) & & & &&&&&\
\
Analysis of Ti and Ti lines in A-B-type stars {#hots}
=============================================
A-B type stars are suitable for testing the treated model atom because the deviations from LTE are large for both Ti and Ti and poorly known inelastic collisions with hydrogen atoms do not or weakly affect the SE. For example, in the model 7170/4.20/$-0.30$ the use of [$S_{\rm H}$]{} = 0 and 0.5 leads to a maximal abundance difference of 0.02 dex and 0.01 dex for individual lines of Ti and Ti, respectively.
The lines of two ionisation stages are observed in HD 32115, HD 37594, HD 73666, and HD 72660. In spectra of Sirius, 21 Peg, $\pi$ Ceti, and HD 145788 only the lines of Ti can be detected. For each star at least 6 lines were used to derive the titanium abundance. The LTE and NLTE abundances are given in Table \[abund\]. In NLTE, the abundance from Ti lines increases by 0.05 dex to 0.14 dex for different stars. In contrast, NLTE leads to up to 0.12 dex lower abundance from the lines of Ti. An exception is the late B star $\pi$ Cet, where NLTE leads to line weakening and to higher titanium abundance compared with LTE. From the eleven lines of Ti we derived log A(Ti) = $-7.41 \pm$0.09 dex and log A(Ti) = $-7.14 \pm$0.08 dex in LTE and NLTE, respectively. Hereafter, the statistical abundance error is the dispersion in the single line measurements: $\sigma = \sqrt{\Sigma (x - x_i )^2 /(N - 1)}$, where N is the total number of lines used, x is their mean abundance, $x_i$ is the abundance of each individual line. In LTE for four our stars the abundance difference Ti–Tiranges between $-0.22$ dex and $-0.09$ dex, while in NLTE Ti–Tidecreases in absolute value and does not exceed 0.07 dex for each of the four stars.
For the A-type stars the LTE abundances from strong lines of Ti are higher than those from the weak lines (see Fig. \[hd145788\] for HD 145788). Such a behavior can be wrongly interpreted as an underestimation of a microturbulent velocity. For example, to derive consistent LTE abundances from different lines of Ti in HD 145788 one needs to adopt a microturbulent velocity of = 1.8 [kms$^{-1}$]{}, while = 1.3 [kms$^{-1}$]{} was found by @fossati09 from lines of Fe. We show that a discrepancy between strong and weak lines vanishes in NLTE. This is because the strong lines are more affected by NLTE compared with the weak lines. For example, in HD 145788, the cores of the Ti lines with $EW\sim$ 100 mÅ form at the optical depth log$\tau_{5000} \simeq$ –2.5, and their NLTE abundance corrections reach –0.24 dex. For the Ti lines with $EW\le$ 70 mÅ the NLTE abundance corrections do not exceed few hundredths in absolute value. We do not recommend to apply the Ti lines with $EW\ge$ 70 mÅ for abundance determination under the LTE assumption. For A-B type stars NLTE leads to significant decrease of line-to-line scatter compared to LTE (Table \[abund\]).
We checked effects of the use of accurate photoionisation cross-sections by @Nahar2015 and K. Butler instead of the hydrogenic approximation. Using quantum-mechanical cross-sections for Ti leads to increasing the photoionisation rates and the deviations from LTE. For example, the NLTE abundance corrections for Ti lines increase by 0.01–0.02 dex in the model 9700/4.10/0.4/1.8. In the atmospheres with [$T_{\rm eff}$]{} $\leq$ 10500 K the NLTE abundances derived from the Ti lines do not change significantly, when using either accurate or hydrogenic cross-sections. This is due to the fact that mechanism of deviations from LTE for Ti is not ruled by the bound-free transitions. For the hottest star of our sample, HD 17081 (B7 IV), where Ti is affected by overionisation, we found that using the accurate cross-sections leads to weakened NLTE effects for Ti and 0.06 dex smaller NLTE abundance compared with that calculated with the hydrogenic cross-sections. Since we adopt the theoretical approximations to calculate electron collision rates, we perform the test calculations. Test calculations with the model atmosphere 7250/4.20/0.0 show that a hundredfold decrease in electron collision rates results in a 0.05 dex increase in the NLTE abundance from Ti, and up to 0.06 dex decrease in NLTE abundance from the strongest lines of Ti with EW of 150 mÅ.
Thus, analysis of the titanium lines in the hot stars gives an evidence for that our NLTE method gives reliable results.
For the 22 lines of Ti and 82 lines of Ti we calculated the NLTE abundance corrections in a grid of model atmospheres with [$T_{\rm eff}$]{} from 6500 K to 13000 K with a step of 250 K, = 4, \[Fe/H\] = 0 and = 2 [kms$^{-1}$]{}. For lines of Ti the NLTE abundance corrections are positive and vary between 0.0 dex to 0.20 dex (Fig. \[corrections\_fig\]). For Ti the NLTE abundance corrections are negative for [$T_{\rm eff}$]{} $\leq$ 10000 K and can be up to $-0.17$ dex. In the atmospheres with [$T_{\rm eff}$]{} $\geq$ 10000 K the lines of neutral titanium can not be detected, and the NLTE abundance corrections for lines of Ti are positive and reach 0.37 dex. The data are available as on-line material (Table \[corrections\]).
\[corrections\]
[|r|r|r|r|r|]{} [$T_{\rm eff}$]{}$_1$, K & [$T_{\rm eff}$]{}$_2$ & ... & [$T_{\rm eff}$]{}$_{26}$ & [$T_{\rm eff}$]{}$_{27}$\
EW$_1$, mÅ& EW$_2$ & ... & EW$_{26}$ & EW$_{27}$\
[$\rm \Delta_{NLTE}$]{}$_1$ & [$\rm \Delta_{NLTE}$]{}$_2$ & ... & [$\rm \Delta_{NLTE}$]{}$_{26}$ & [$\rm \Delta_{NLTE}$]{}$_{27}$\
6500 & 6750 & ... & 12750 & 13000\
\
\
54 & 40 & ... & –1 & –1\
0.17 & 0.17 & ... & –1.00 & –1.00\
\
\
176 & 169 & ... & 15 & 12\
–0.09 & –0.10 & ... & 0.25 & 0.26\
![NLTE (filled circles) and LTE (open circles) abundances from the lines of Ti in HD 145788 as a function of equivalent width.[]{data-label="hd145788"}](hd145788_ti2_lawler_only.ps){width="80mm"}
![NLTE abundance corrections for the selected lines of Ti (circles) and Ti (triangles) shown by different colours. A size of symbol represents an equivalent width of the corresponding line.[]{data-label="corrections_fig"}](hot_corr_sel.ps){width="80mm"}
Analysis of Ti and Ti lines in the reference late-type stars {#cools}
============================================================
Ti and Ti lines in the solar spectrum
-------------------------------------
We used 27 Ti and 12 Ti lines in the solar flux spectrum [@kurucz84] to determine the LTE and NLTE abundances. Under the LTE assumption we derived log A$_{\rm TiI} = -7.11 \pm$ 0.06 dex and log A$_{\rm TiII} = -7.06 \pm$ 0.04 dex from the lines of Ti and Ti, respectively. We calculated the NLTE abundances for [$S_{\rm H}$]{} = 0, 0.1, 0.5 and 1.0. Consistent within 0.03 dex abundances from Ti and Ti were found in NLTE, independent of adopted [$S_{\rm H}$]{} value (Table \[lines5\]). This means that the solar analysis does not help to constrain [$S_{\rm H}$]{}. Solar titanium abundance averaged over Ti and Ti lines, log A = $-7.09 \pm$0.06 (NLTE, [$S_{\rm H}$]{} = 1), agrees with the meteoritic value, log A = $-7.11 \pm$0.03 dex [@Lodders2009].
The treated NLTE method was applied before publication to check the Ti/Ti ionisation equilibrium of 11 stars with 5050 $\le$ [$T_{\rm eff}$]{} $\le$ 6600 K, 3.76 $\le$ $\le$ 4.47 and $-0.48 \le$ \[Fe/H\] $\le 0.24$ [@ryab2015]. For HD 49933 (6600/4.0/$-0.48$), the star with the largest deviations from LTE in the sample, the NLTE calculations provide consistent within the error bars the Ti and Ti based abundances independent of using either [$S_{\rm H}$]{} = 0.5 or 1. For the studied stars the NLTE abundance difference Ti–Ti nowhere exceeds $-0.06$ dex.
Ti and Ti lines in the metal-poor stars
---------------------------------------
Metal-poor stars suit better for a calibration of [$S_{\rm H}$]{} parameter than the solar-metallicity stars. This is due to the deviations from LTE grow with decreasing \[Fe/H\] because of increasing the ultraviolet (UV) flux and decreasing electronic number density. Our sample of cool MP dwarfs includes 15 stars with $-2.6 \leq$ \[Fe/H\] $\leq -1.3$. For all the stars we determined the titanium abundance under the LTE assumption and in NLTE with [$S_{\rm H}$]{} = 1.0, and also with [$S_{\rm H}$]{} = 0.5 and 0.1 for few stars. The abundance differences Ti–Ti are listed in Table \[lines5\] for various line formation scenarios and shown in Fig. \[ti12cool\] for LTE and NLTE with [$S_{\rm H}$]{} = 1.0. For the seven stars the NLTE calculations result in consistent within the error bars abundances from Ti and Ti. For example, in HD 94028 Ti–Ti = $-0.11$ dex in LTE, and reduces to $-0.05$ dex in NLTE ([$S_{\rm H}$]{} = 1). For the other eight stars, on the contrary, an agreement between Ti and Ti is better in LTE compared to that in NLTE. Moreover, for these stars Ti–Ti $\geq$ 0 is obtained already in LTE, and the difference increases in NLTE. For example, in BD $-13^\circ3442$ we derived the largest discrepancy of 0.23 dex when using NLTE with [$S_{\rm H}$]{} = 1, while in LTE Ti–Ti = 0.09 dex. All these stars, except HD 103095, are either turn-off (TO) stars with 6200 $\le$ [$T_{\rm eff}$]{} $\le$ 6400 K, 3.9 $\le$ $\le$ 4.1, $-2.6 \le$\[Fe/H\]$\le -1.9$, or VMP subgiants (SG) with [$T_{\rm eff}$]{} $\ge$ 5780 K. Due to lower [$S_{\rm H}$]{} leads to larger NLTE effects, we do not perform calculations with [$S_{\rm H}$]{} $\le$ 1 for these stars, except HD 84937. For the eight dwarfs with negative LTE abundance difference Ti–Ti we performed NLTE calculations with [$S_{\rm H}$]{} = 0.5. The minimal difference Ti–Tifor maximal number of stars is achieved, when using [$S_{\rm H}$]{} = 1.
. In LTE we derived an abundance difference of Ti–Ti$ = -0.36$ dex, and in NLTE it decreases in absolute value and amounts to Ti–Ti$ = -0.18$ dex, $-0.13$ dex, and $-0.06$ dex, when using [$S_{\rm H}$]{} = 1.0, 0.5, and 0.1, respectively. To achieve an agreement between Ti and Ti, the lower [$S_{\rm H}$]{} is required, compared with that for the dwarfs. It is worth noting that similar conclusion was drawn by @mash_fe from a relative to the Sun line-by-line differential analysis of iron lines in HD 122563. @mash_fe derived an abundance difference of Fe–Fe$ = -0.21$ dex in LTE and Fe–Fe$ = -0.18$ dex, $-0.05$ dex, and 0.03 dex in NLTE, when using [$S_{\rm H}$]{} = 1.0, 0.1, and 0.0, respectively. While to achieve the Fe/Fe balance for MP TO-star HD 84937 [$S_{\rm H}$]{} = 1 is required. In HD 122563, for both Fe and Fe and Ti and Ti NLTE leads to smaller abundance difference between the two ionisation stages compared to LTE.
Comparison with other studies
-----------------------------
We have the four stars in common with @Bergemann2011, namely, the Sun, HD 84937, HD140283, and HD 122563. For the common lines of Ti and Ti used in the solar analyses we recalculated abundances derived by @Bergemann2011 using gf-values adopted in this study. For the majority lines the LTE abundance difference between @Bergemann2011 and our data does not exceed 0.03 dex and nowhere exceeds 0.05 dex. We also compared the NLTE abundance corrections for Ti. @Bergemann2011 adopted [$S_{\rm H}$]{} = 3 in the NLTE calculations, while we use [$S_{\rm H}$]{} = 1. However, for the majority lines she computed larger NLTE abundance corrections, by up to 0.03 dex (for Ti 4981Å). @Bergemann2011 derived with [$S_{\rm H}$]{} = 3 the average abundance difference Ti$_{\rm NLTE}$–Ti$_{\rm LTE}$ = 0.05 dex, while we obtain the same value, when using [$S_{\rm H}$]{} = 0.5. The smaller NLTE effects in this study compared with @Bergemann2011 are due to using a comprehensive model atom that includes predicted high-excitation levels of Ti.
The difference between our and @Bergemann2011 NLTE results grows, when moving to the MP stars. We compare the abundance differences Ti$_{\rm NLTE}$–Ti$_{\rm LTE}$ and Ti-Ti. For HD 84937, @Bergemann2011 derived Ti$_{\rm NLTE}$–Ti$_{ \rm LTE}$ = 0.14 dex using [$S_{\rm H}$]{} = 3 and MAFAGS-OS model atmosphere [@Grupp2009]. Using the same stellar parameters for this star, [$S_{\rm H}$]{} = 3, and MARCS model atmosphere [@MARCS] we derived Ti$_{\rm NLTE}$–Ti$_{ \rm LTE}$ = 0.09 dex. We checked, whether this abundance discrepancy can be attributed to different codes for model atmosphere calculation. We calculated Ti and Ti abundances with MARCS and MAFAGS-OS models, and found that the abundance difference does not exceed 0.02 dex for any line. For HD 84937 @Bergemann2011 derived in LTE Ti–Ti = 0.11 dex, while we found Ti–Ti$_{ \rm LTE} = 0.03$ dex. For HD 140283 she presents abundances calculated only with the MAFAGS-ODF model structure, Ti–Ti = 0.02 dex in LTE and 0.16 dex in NLTE ([$S_{\rm H}$]{} = 3). The corresponding values in our calculations are $-0.05$ dex (LTE) and 0.09 dex (NLTE, [$S_{\rm H}$]{} = 1). Similar situation in our studies was found for HD 122563. We derived discrepancies of Ti–Ti$ = -0.36$ dex and $-0.18$ dex in LTE and NLTE ([$S_{\rm H}$]{} = 1), respectively. The corresponding LTE and NLTE ([$S_{\rm H}$]{} = 3) values from @Bergemann2011 are $-0.40$ dex and $-0.10$ dex. This abundance comparison indicates that our model atom leads to smaller deviations from LTE compared with those computed by @Bergemann2011.
The star HD 84937 is used like a reference star in many studies, since its atmospheric parameters are well-determined by different independent methods. @Sneden2016 investigated the titanium lines under the LTE assumption adopting [$T_{\rm eff}$]{} = 6300 K, log g = 4.0, \[Fe/H\]$ = -2.15$, = 1.5 [kms$^{-1}$]{} and the interpolated model from @Kurucz2011 model grid. In LTE they found consistent abundances from Ti and Ti. Using adopted in their study atmospheric parameters and our linelist we derived in LTE Ti$-$Ti = 0.02 dex. A very similar abundance difference of Ti–Ti = 0.03 dex was found, with our parameters 6350/4.09/$-2.16$/1.7. This is due to higher [$T_{\rm eff}$]{} and higher lead to decrease in abundance from Ti and Ti, respectively, keeping the ionisation balance safe. HD 84937 is one of the stars, where NLTE leads to positive Ti–Tiabundance difference, as discussed above.
What is a source of discrepancy between Ti and Ti in VMP TO-stars?
------------------------------------------------------------------
The main NLTE mechanism for Ti is the UV overionisation and there is no process, which can result in strengthened lines of Ti and negative NLTE abundance corrections. Inelastic collisions with H atoms serve as an additional source of thermalisation that reduces, but does not cancel the overionisation. It is worth noting that in the atmospheres of our VMP (\[Fe/H\] $\leq -2$) TO-stars the lines of Ti are weak (EW $\leq$ 20 mÅ) and form inwards log$\tau_{5000}$ = $-1$. The NLTE abundance corrections for Ti lines are positive in the model 6350/4.09/$-2.15$, $\Delta_{\rm NLTE} \leq$ 0.01 dex when [$S_{\rm H}$]{} = 1, and $\Delta_{\rm NLTE}$ can be up to 0.08 dex, when neglecting collisions with H atoms. To what extent inelastic collisions with H atoms can help to solve the problem of Ti–Ti in the MP TO stars remains unclear until accurate collisional data will be computed for both Ti + H and Ti + H.
The lines of Ti are more sensitive to [$T_{\rm eff}$]{} variation compared with Fe lines, because of lower ionisation threshold for Ti compared to Fe. For example, we found an abundance shift of 0.09 dex for Ti lines, and only 0.05 dex for Fe, when adopting 70 K lower [$T_{\rm eff}$]{} for HD 103095 (5130/4.66/$-1.26$). For this star, a downward revision of [$T_{\rm eff}$]{} by 70 K results in consistent abundances from Ti and Ti, and does not destroy Fe/Fe ionisation equilibrium. However, a different situation was found for the VMP TO stars. For example, we obtained similar abundance shifts of 0.08 dex and 0.06 dex for Ti and Fe, respectively, when adopting 100 K lower [$T_{\rm eff}$]{} for HD 84937 (6350/4.09/$-2.16$). For this star, the [$T_{\rm eff}$]{} decrease results in the ionisation equilibrium for titanium, but not for iron.
The solution of the NLTE problem with such a comprehensive model atom as treated in this study is only possible, at present, with classical plane-parallel (1D) model atmospheres. Neglecting atmospheric inhomogeneities (3D effects) can lead to errors in our results. From hydrodynamical modelling of stellar atmospheres @Collet2007 and @Dobrovolskas2013 predict negative abundance corrections $\Delta_{\rm 3D}$ = log A$_{\rm 3D}$–log A$_{\rm 1D}$ for lines of neutral species in red giant stars. In the models of TO (5900/4.0) stars with \[Fe/H\]$ = -2$, $\Delta_{\rm 3D}$ increases in absolute value with decreasing the excitation energy of the lower level, and reaches $-0.84$ dex and $-0.20$ dex for the $\lambda$ = 4000 Å lines with [$E_{\rm exc}$]{} = 0 and 2 eV, respectively [@Dobrovolskas_phd]. All the lines of Ti used for our MP TO stars have [$E_{\rm exc}$]{} $\le $1.75 eV. The 3D abundance corrections can be either positive or negative, and do not exceed 0.07 dex in absolute value for the lines of Ti. Negative 3D corrections for Ti could help to achieve an agreement between Ti and Ti. We selected two lines of Ti, at 4617 Å ([$E_{\rm exc}$]{} = 1.75 eV) and 4681 Å ([$E_{\rm exc}$]{} = 0.05 eV), and Ti 5336 Å ([$E_{\rm exc}$]{} = 1.58 eV), which give consistent within 0.02 dex LTE abundances and calculated the abundance differences Ti–Tifor different line formation scenarios, taking 3D abundance corrections from @Dobrovolskas_phd. Abundances from individual lines are shown in Fig. \[hd84937\_3d1d\]. In LTE we derived Ti–Ti = 0.05 dex and $-0.49$ dex in 1D and 3D, respectively. In NLTE+3D we derived $-0.27$ dex ([$S_{\rm H}$]{} = 1) and $-0.16$ dex ([$S_{\rm H}$]{} = 0), while Ti–Ti= 0.17 dex in NLTE([$S_{\rm H}$]{} = 1)+1D, which is our standard scenario. The predicted 3D effects are too strong for low-excitation lines of Ti and produce a large discrepancy between Ti lines with different [$E_{\rm exc}$]{}, which reaches 0.66 dex in LTE+3D. We suppose that for MP stars simple co-adding the NLTE(1D) and 3D(LTE) corrections is too rough procedure, because both NLTE and 3D effects are equally significant.
One more source can be connected with a star’s chromosphere that heats the line formation layers. An inspiring insight into this problem was presented by @Dupree2016. Further efforts should be invested to evaluate a possible influence of the star’s chromosphere on the formation of titanium lines.
![Titanium abundances in HD 84937 from individual lines: Ti 4617 Å (triangle), Ti 4681 Å (square), Ti 5336 Å (circle) in different line formation scenarios, namely, 1 = LTE+1D, 2 = LTE+3D, 3 = NLTE([$S_{\rm H}$]{} = 0)+3D, 4 = NLTE([$S_{\rm H}$]{} = 1)+3D, 5 = NLTE([$S_{\rm H}$]{} = 1)+1D.[]{data-label="hd84937_3d1d"}](hd84937_3d1d.ps){width="80mm"}
\[lines5\]
-------------------- ------- ------- ------- -------
Star LTE 1 0.5 0.1
Sun –0.05 –0.03 –0.02 0.00
HD 64090 –0.04 –0.01 0.00 0.06
HD 84937 0.03 0.15 0.19 0.24
HD 94028 –0.11 –0.05 –0.03 0.05
HD 122563 –0.36 –0.18 –0.13 –0.06
BD+07$^\circ$ 4841 –0.06 0.02 0.06
BD+09$^\circ$ 0352 –0.06 0.04 0.08
HD 140283 –0.05 0.09 0.14
BD+29$^\circ$ 2091 –0.14 –0.09 –0.07
G 090-003 –0.07 0.05 0.09
HD 24289 0.00 0.14
HD 74000 0.01 0.12
HD 103095 0.06 0.07
HD 108177 –0.07 0.02
BD–13$^\circ$ 3442 0.09 0.23
BD–04$^\circ$ 3208 0.02 0.15
BD+24$^\circ$ 1676 0.07 0.20
-------------------- ------- ------- ------- -------
: The abundance difference Ti-Ti for cool stars of the sample in different line formation scenarios.
{width="80mm"} {width="80mm"}
\[abund\]
[|l|c|c|c|c|c|c|]{} Star & N$_{\rm Ti I}$ & log A(Ti)$_{\rm LTE}$ & log A(Ti)$_{\rm NLTE}$ & N$_{\rm Ti II}$ & log A(Ti)$_{\rm LTE}$ & log A(Ti)$_{\rm NLTE}$\
HD 37594 & 8 & –7.23$\pm$0.13 & –7.11$\pm$0.11 & 27 & –7.01$\pm$0.15 & –7.04$\pm$0.11\
HD 32115 & 6 & –7.45$\pm$0.05 & –7.31$\pm$0.05 & 9 & –7.23$\pm$0.07 & –7.26$\pm$0.05\
HD 72660 & 5 & –6.63$\pm$0.05 & –6.57$\pm$0.08 & 36 & –6.54$\pm$0.12 & –6.59$\pm$0.08\
HD 73666 & 2 & –6.94$\pm$0.02 & –6.89$\pm$0.09 & 6 & –6.72$\pm$0.20 & –6.84$\pm$0.09\
HD 145788 & & & & 32 & –6.76$\pm$0.15 & –6.81$\pm$0.07\
Sirius & & & & 6 & –6.84$\pm$0.06 & –6.89$\pm$0.04\
21 Peg & & & & 46 & –7.24$\pm$0.05 & –7.24$\pm$0.04\
$\pi$ Cet & & & & 11 & –7.41$\pm$0.09 & –7.14$\pm$0.08\
Sun & 27 & –7.11$\pm$0.05 & –7.09$\pm$0.05 & 12 & –7.06$\pm$0.04 & –7.06$\pm$0.04\
BD–13$^\circ$ 3442 & 3 & -9.25$\pm$0.04 & -9.09$\pm$0.04 & 15 & -9.34$\pm$0.06 & -9.32$\pm$0.06\
BD–04$^\circ$ 3208 & 9 & -8.90$\pm$0.05 & -8.77$\pm$0.05 & 17 & -8.92$\pm$0.06 & -8.92$\pm$0.05\
BD+7$^\circ$ 4841 & 26 & –8.24$\pm$0.05 & –8.17$\pm$0.05 & 34 & –8.17$\pm$0.06 & –8.19$\pm$0.05\
BD+9$^\circ$ 0352 & 9 & –8.87$\pm$0.05 & –8.78$\pm$0.05 & 22 & –8.81$\pm$0.05 & –8.82$\pm$0.04\
BD+24$^\circ$ 1676 & 7 & -9.12$\pm$0.06 & -8.98$\pm$0.06 & 16 & -9.19$\pm$0.06 & -9.18$\pm$0.06\
BD+29$^\circ$ 2091 & 20 & –8.76$\pm$0.06 & –8.72$\pm$0.06 & 24 & –8.62$\pm$0.08 & –8.63$\pm$0.07\
HD 24289 & 16 & –8.79$\pm$0.10 & –8.67$\pm$0.10 & 27 & –8.79$\pm$0.08 & –8.81$\pm$0.09\
HD 64090 & 35 & –8.73$\pm$0.07 & –8.71$\pm$0.07 & 30 & –8.69$\pm$0.06 & –8.70$\pm$0.05\
HD 74000 & 15 & -8.78$\pm$0.07 & -8.68$\pm$0.07 & 26 & -8.79$\pm$0.08 & -8.80$\pm$0.08\
HD 84937 & 12 & –8.84$\pm$0.04 & –8.71$\pm$0.04 & 15 & –8.87$\pm$0.08 & –8.86$\pm$0.08\
HD 94028 & 26 & –8.34$\pm$0.06 & –8.30$\pm$0.07 & 26 & –8.23$\pm$0.04 & –8.24$\pm$0.05\
HD 103095 & 37 & –8.06$\pm$0.09 & –8.05$\pm$0.09 & 29 & –8.12$\pm$0.07 & –8.12$\pm$0.07\
HD 108177 & 14 & –8.50$\pm$0.06 & –8.43$\pm$0.07 & 12 & –8.43$\pm$0.07 & –8.45$\pm$0.06\
HD 122563 & 22 & –9.82$\pm$0.07 & –9.64$\pm$0.08 & 36 & –9.46$\pm$0.06 & –9.46$\pm$0.07\
HD 140283 & 19 & –9.36$\pm$0.07 & –9.21$\pm$0.07 & 25 & –9.31$\pm$0.05 & –9.30$\pm$0.05\
G 090–03 & 18 & –8.85$\pm$0.07 & –8.75$\pm$0.07 & 30 & –8.78$\pm$0.07 & –8.79$\pm$0.06\
\
Conclusions {#con}
===========
We construct a comprehensive model atom for Ti– using the energy levels from laboratory measurements and theoretical predictions and quantum mechanical photoionisation cross-sections. NLTE line formation for Ti and Ti lines was considered in 1D-LTE model atmospheres of the 25 reference stars with reliable stellar parameters, which cover a broad range of effective temperatures 4600 $\le$ [$T_{\rm eff}$]{} $\le$ 12800 K, surface gravities 1.60 $\le$ $\le$ 4.70, and metallicities $-2.5~\le$ \[Fe/H\] $\le$ +0.4.
The NLTE calculations for Ti– in A-type stars were performed for the first time. The NLTE titanium abundances were determined for the eight stars. For the four stars with both Ti and Ti lines observed, NLTE analysis provides consistent within 0.07 dex abundances from Ti and Ti lines, while the corresponding LTE abundance difference can be up to 0.22 dex in absolute value. For each species, NLTE leads to smaller line-to-line scatter compared with LTE. For stars with [$T_{\rm eff}$]{}$\ge$ 7000 K lines of Ti and Ti can be used for atmospheric parameter determination, when taking into account deviations from LTE. For the 22 lines of Ti and 82 lines of Ti we calculated the NLTE abundance corrections in a grid of model atmospheres with [$T_{\rm eff}$]{} from 6500 K to 13000 K, = 4, \[Fe/H\] = 0, and =2 [kms$^{-1}$]{}.
We made progress in determination of NLTE abundance of titanium for cool stars compared with data from the literature. Taking into account a bulk of the predicted high-excitation levels of Ti in the model atom established close collisional coupling of the Ti levels near the continuum to the ground state of Ti resulting in smaller NLTE effects in cool model atmospheres compered with the @Bergemann2011 data. Because no accurate calculations of inelastic collisions of titanium with neutral hydrogen atoms are available, we use the Drawinian formalism with the scaling factor, which was estimated as [$S_{\rm H}$]{} = 1 from abundance comparison between Ti and Ti in the sample of cool main sequence stars over wide metallicity range, $-2.6 \leq$ \[Fe/H\] $\leq$ 0.0. For the VMP TO-stars NLTE fails to achieve agreement between Ti and Ti. Moreover, for these stars we derived positive abundance difference Ti–Tiin LTE, and it increases in NLTE. To clarify this matter, accurate collisional data for Ti and Ti would be extremely helpful.
Appendix {#apend}
========
In this section we present Table \[atomic\] and Table \[corrections\] in theirs entirety.
This is a full version of Table \[atomic\]. The list of Ti and Ti lines with the adopted atomic data.\
$\lambda$ (Å), [$E_{\rm exc}$]{}(eV), log gf, transition, log $\gamma_{rad}$, log $\gamma_4/N_e$, log $\gamma_6/N_H $\
Ti I\
4008.927 0.021 -1.000 3a3F – y3F 8.000 -6.080 -7.750\
4060.262 1.052 -0.690 a3P – x3P 8.050 -6.050 -7.646\
4060.262 1.052 -0.690 a3P – x3P 8.050 -6.050 -7.646\
4287.403 0.836 -0.370 a5F – x5D 8.230 -6.010 -7.570\
4449.143 1.886 0.470 a3G – v3G 8.120 -5.560 -7.579\
4453.699 1.872 0.100 a3G – v3G 8.110 -4.970 -7.582\
4512.733 0.836 -0.400 a5F – y5F 8.130 -5.120 -7.593\
4533.240 0.848 0.540 a5F – y5F 8.130 -5.120 -7.593\
4534.776 0.836 0.350 a5F – y5F 8.130 -5.280 -7.596\
4548.763 0.826 -0.280 a5F – y5F 8.130 -5.410 -7.598\
4555.484 0.848 -0.400 a5F – y5F 8.130 -5.280 -7.596\
4617.268 1.748 0.440 a5P – w5D 8.080 -5.860 -7.626\
4623.097 1.739 0.160 a5P – w5D 8.070 -5.850 -7.627\
4639.361 1.739 -0.050 a5P – w5D 8.070 -5.840 -7.740\
4639.661 1.748 -0.140 a5P – w5D 8.070 -5.850 -7.740\
4639.940 1.733 -0.160 a5P – w5D 8.070 -5.840 -7.630\
4656.468 0.000 -1.290$^2$ 2a3F – z3G 6.380 -6.110 -7.706\
4681.909 0.050 -1.030$^2$ 4a3F – z3G 6.460 -6.110 -7.702\
4758.118 2.248 0.510 a3H – x3H 8.080 -6.040 -7.621\
4759.269 2.255 0.590 a3H – x3H 8.080 -6.040 -7.620\
4820.410 1.502 -0.380 a1G – y1F 8.210 -5.940 -7.625\
4840.874 0.899 -0.430 a1D – y1D 7.530 -6.120 -7.697\
4913.615 1.872 0.220 a3G – y3H 7.850 -5.890 -7.619\
4981.731 0.848 0.570 a5F – y5G 7.950 -6.050 -7.626\
4991.067 0.836 0.450 a5F – y5G 7.940 -6.050 -7.629\
4997.093 0.000 -2.070 2a3F – z3D 6.900 -6.110 -7.722\
4999.502 0.826 0.320 a5F – y5G 7.940 -6.050 -7.632\
5009.645 0.021 -2.200 3a3F – z3D 6.870 -6.110 -7.720\
5016.161 0.848 -0.480 a5F – y5G 7.940 -6.050 -7.629\
5020.025 0.836 -0.330 a5F – y5G 7.940 -6.050 -7.630\
5024.844 0.818 -0.530 a5F – y5G 7.940 -6.050 -7.635\
5025.570 2.041 0.250$^1$ z5G – e5F 7.960 -5.310 -7.550\
5036.464 1.443 0.140 b3F – w3G 8.160 -5.700 -7.539\
5039.955 0.021 -1.080 3a3F – z3D 6.900 -6.110 -7.720\
5064.652 0.048 -0.940 4a3F – z3D 6.870 -6.110 -7.719\
5147.477 0.000 -1.940 2a3F – z3F 6.820 -6.110 -7.727\
5173.740 0.000 -1.060 2a3F – z3F 6.820 -6.110 -7.729\
5192.969 0.021 -0.950 3a3F – z3F 6.820 -6.110 -7.727\
5210.384 0.048 -0.820 4a3F – z3F 6.810 -6.110 -7.724\
5512.524 1.460 -0.400 b3F – w3D 8.190 -6.100 -7.700\
5514.343 1.429 -0.660 b3F – w3D 8.140 -5.990 -7.710\
5514.532 1.443 -0.500 b3F – w3D 8.150 -6.070 -7.710\
5866.449 1.066 -0.790 a3P – y3D 8.000 -6.070 -7.724\
6258.099 1.443 -0.390 b3F – y3G 8.250 -5.990 -7.582\
6261.096 1.429 -0.530 b3F – y3G 8.260 -5.980 -7.585\
8426.506 0.826 -1.200$^2$ a5F – z5D 6.370 -6.090 -7.711\
Ti II\
2827.114 3.687 -0.020$^4$ z4G – e4G 8.850 -5.850 -7.720\
2828.077 3.749 0.870$^3$ z4G – e4H 8.860 -5.820 -7.720\
2834.011 3.716 0.000$^4$ z4G – e4G 8.850 -5.850 -7.720\
2841.935 0.607 -0.590 a2F – y2F 8.420 -6.390 -7.830\
2851.101 1.221 -0.730 a2P – x2D 8.320 -6.470 -7.820\
2853.931 0.607 -1.550 a2F – y2F 8.350 -6.390 -7.840\
2868.741 0.574 -1.380 a2F – y2D 8.260 -6.390 -7.850\
4012.385 0.574 -1.780 a2F – z4G 8.220 -6.390 -7.860\
4028.343 1.891 -0.920 b2G – y2F 8.420 -6.410 -7.830\
4053.820 1.892 -1.070 b2G – y2F 8.350 -6.410 -7.840\
4161.530 1.084 -2.090 a2D – z4D 8.410 -6.430 -7.840\
4163.640 2.589 -0.130 b2F – x2D 8.320 -6.470 -7.820\
4174.070 2.598 -1.260$^4$ b2F – x2D 8.320 -6.470 -7.820\
4188.987 5.423 -0.600$^4$ y2G – e2G 8.900 -5.690 -7.690\
4190.233 1.084 -3.122$^1$ a2D – z4D 8.410 -6.430 -7.840\
4287.870 1.080 -1.790$^4$ a2D – z2D 8.170 -6.430 -7.850\
4290.215 1.164 -0.870 a4P – z4D 8.410 -6.500 -7.840\
4300.049 1.180 -0.460 a4P – z4D 8.410 -6.490 -7.840\
4301.920 1.160 -1.210 a4P – z4D 8.410 -6.490 -7.840\
4316.794 2.047 -1.620 b2P – z2P 8.420 -6.460 -7.840\
4337.915 1.080 -0.960$^4$ a2D – z2D 8.160 -6.430 -7.850\
4374.820 2.060 -1.570 b2P – y2D 8.260 -6.460 -7.850\
4386.844 2.598 -0.960$^4$ b2F – y2G 8.450 -6.540 -7.830\
4391.020 1.231 -2.300 b4P – z4D 8.410 -6.410 -7.840\
4394.059 1.221 -1.770 a2P – z4D 8.410 -6.490 -7.840\
4395.031 1.084 -0.540 a2D – z2F 8.160 -6.430 -7.850\
4395.839 1.242 -1.930 b4P – z4D 8.410 -6.410 -7.840\
4399.772 1.236 -1.200 a2P – z4D 8.410 -6.500 -7.840\
4409.235 1.242 -2.780 b4P – z4D 8.410 -6.410 -7.840\
4409.520 1.231 -2.530 b4P – z4D 8.410 -6.410 -7.840\
4411.070 3.093 -0.650 c2D – x2F 8.290 -6.330 -7.840\
4411.925 1.224 -2.620 b4P – z4D 8.420 -6.410 -7.840\
4417.713 1.165 -1.190$^4$ a4P – z2D 8.170 -6.580 -7.850\
4418.331 1.236 -1.990 a2P – z4D 8.410 -6.490 -7.840\
4421.938 2.060 -1.640 b2P – z2P 8.350 -6.460 -7.840\
4423.239 1.231 -3.066$^1$ b4P – z4D 8.420 -6.410 -7.840\
4432.109 1.236 -3.080 a2P – z4D 8.420 -6.490 -7.840\
4441.730 1.180 -2.330$^4$ a4P – z2D 8.170 -6.580 -7.850\
4443.801 1.080 -0.710 a2D – z2F 8.150 -6.430 -7.850\
4444.554 1.115 -2.200 a2G – z2F 8.160 -6.590 -7.850\
4450.482 1.084 -1.520 a2D – z2F 8.150 -6.430 -7.850\
4464.449 1.161 -1.810$^4$ a4P – z2D 8.160 -6.600 -7.850\
4468.500 1.130 -0.630 a2G – z2F 8.860 -5.710 -7.690\
4468.510 1.130 -0.630 a2G – z2F 8.860 -5.710 -7.690\
4469.151 1.084 -2.550 a2D – z4F 8.380 -6.430 -7.840\
4470.853 1.165 -2.020$^4$ a4P – z2D 8.160 -6.600 -7.850\
4488.324 3.122 -0.500 c2D – x2F 8.280 -6.330 -7.840\
4501.270 1.115 -0.770 a2G – z2F 8.150 -6.590 -7.850\
4518.330 1.080 -2.560 a2D – z4F 8.380 -6.430 -7.840\
4529.474 1.571 -1.750 a2H – z2G 8.310 -6.490 -7.820\
4533.960 1.237 -0.530$^4$ a2P – z2D 8.170 -6.540 -7.850\
4544.020 1.243 -2.580$^4$ a2G – z4F 8.170 -6.410 -7.850\
4549.620 1.583 -0.220 a2H – z2G 8.310 -6.490 -7.820\
4563.757 1.221 -0.795$^1$ a2P – z2D 8.160 -6.550 -7.850\
4568.314 1.224 -3.030$^5$ b4P – z2D 8.160 -6.410 -7.850\
4571.971 1.571 -0.310 a2H – z2G 8.310 -6.490 -7.820\
4583.410 1.164 -2.840 a4P – z2F 8.150 -6.590 -7.850\
4589.958 1.237 -1.620$^5$ a2P – z2D 8.160 -6.540 -7.850\
4636.320 1.165 -3.024$^1$ a4P – z4F 8.380 -6.500 -7.840\
4657.201 1.242 -2.290 b4P – z2F 8.160 -6.410 -7.850\
4708.663 1.236 -2.350 a2P – z2F 8.150 -6.540 -7.850\
4719.515 1.242 -3.320 b4P – z2F 8.150 -6.410 -7.850\
4763.880 1.221 -2.400 a2P – z4F 8.380 -6.510 -7.840\
4764.525 1.236 -2.690 a2P – z4F 8.380 -6.500 -7.840\
4779.985 2.048 -1.260$^5$ b2P – z2S 8.230 -6.460 -7.860\
4798.530 1.080 -2.660 a2D – z4G 8.220 -6.430 -7.860\
4805.085 2.061 -0.960$^5$ b2P – z2S 8.230 -6.460 -7.860\
4865.612 1.115 -2.700 a2G – z4G 8.220 -6.490 -7.860\
4911.190 3.122 -0.640 c2D – y2P 8.260 -6.330 -7.830\
4996.367 1.582 -3.290$^6$ b2D2 – z4D 8.410 -6.490 -7.840\
5005.157 1.565 -2.730 b2D2 – z4D 8.410 -6.490 -7.840\
5010.210 3.093 -1.350 c2D – x2D 8.320 -6.330 -7.820\
5013.330 3.095 -2.028$^1$ c2D – x2D 8.320 -6.330 -7.820\
5013.686 1.581 -2.140 b2D2 – z4D 8.410 -6.500 -7.840\
5072.290 3.122 -1.020 c2D – x2D 8.320 -6.330 -7.820\
5129.160 1.891 -1.340 b2G – z2G 8.310 -6.410 -7.820\
5154.070 1.566 -1.750$^4$ b2D2 – z2D 8.170 -6.580 -7.850\
5185.913 1.892 -1.410 b2G – z2G 8.310 -6.410 -7.820\
5188.680 1.582 -1.050$^4$ b2D2 – z2D 8.170 -6.580 -7.850\
5211.536 2.589 -1.410 b2F – y2F 8.420 -6.480 -7.830\
5226.550 1.570 -1.260$^4$ b2D2 – z2D 8.160 -6.590 -7.850\
5262.140 1.582 -2.250$^4$ b2D2 – z2D 8.160 -6.590 -7.850\
5268.610 2.597 -1.610 b2F – y2F 8.350 -6.480 -7.840\
5336.786 1.581 -1.600 b2D2 – z2F 8.160 -6.590 -7.850\
5381.022 1.565 -1.970 b2D2 – z2F 8.150 -6.590 -7.850\
5418.768 1.581 -2.130 b2D2 – z2F 8.150 -6.590 -7.850\
5490.690 1.566 -2.663$^1$ b2D2 – z4F 8.380 -6.510 -7.840\
6491.566 2.061 -1.942$^1$ b2P – z2D 8.170 -6.460 -7.850\
6606.950 2.060 -2.790$^3$ b2P – z2D 8.160 -6.460 -7.850\
6680.133 3.093 -1.890 c2D – y2F 8.350 -6.330 -7.840\
6998.905 3.122 -1.280 c2D – y2D 8.260 -6.330 -7.850\
sources of gf-values\
1 - Kurucz,\
2 - BLNP, Blackwell-Whitehead, R. J. and Lundberg, H. and Nave, G. and Pickering, J. C. and Jones, H. R. A. and Lyubchik, Y. and Pavlenko, Y. V. and Viti, S., Monthly Notices Roy. Astron. Soc., 373, 1603-1609 (2006);\
3 - MFW, Martin, G.A. and Fuhr, J.R. and Wiese, W.L., J. Phys. Chem. Ref. Data Suppl., 17, 3 (1988);\
4 - PTP, Pickering, J. C. and Thorne, A. P. and Perez, R., Astrophys. J. Suppl. Ser., 132, 403-409 (2001);\
5 - RHL, Ryabchikova, T. A. and Hill, G. M. and Landstreet, J. D. and Piskunov, N. and Sigut, T. A. A., Monthly Notices Roy. Astron. Soc., 267, 697 (1994);\
6 - BHN, Bizzarri, A. and Huber, M. C. E. and Noels, A. and Grevesse, N. and Bergeson, S. D. and Tsekeris, P. and Lawler, J. E., Astronomy and Astrophysics, 273, 707 (1993);\
gf-values taken from Wisconsin [@Lawler2013_ti1; @Wood2013_ti2] if not prescribed.\
This is a full version of Table \[corrections\]. NLTE abundance corrections and equivalent widths for the lines of Ti and Ti depending on [$T_{\rm eff}$]{} in the models with = 4, \[Fe/H\] = 0, and = 2 [kms$^{-1}$]{}. A portion is shown here for guidance regarding its form and content. If EW = $-1$ and [$\rm \Delta_{NLTE}$]{} = $-1$, this means that EW $<$ 5 mÅ in a given model atmosphere.
Calculations are performed for the 27 following effective temperatures (K):\
6500 6750 7000 7250 7500 7750 8000 8250 8500 8750 9000 9250 9500 9750 10000 10250 10500 10750 11000 11250 11500 11750 12000 12250 12500 12750 13000\
The file is constructed as following:\
wavelength, A; Ti species; excitation energy, eV; gf-value\
equivalent width$_1$; ...; equivalent width$_{27}$\
NLTE abundance correction$_1$; ...; NLTE abundance correction$_{27}$\
4287.4028 A Ti1 Eexc = 0.836 log gf = -0.370\
38 29 21 15 11 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.11 0.10 0.09 0.08 0.08 0.08 0.08 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4453.6992 A Ti1 Eexc = 1.872 log gf = 0.100\
18 13 10 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.14 0.13 0.12 0.12 0.12 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4512.7329 A Ti1 Eexc = 0.836 log gf = -0.400\
38 28 21 15 10 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.10 0.09 0.08 0.08 0.08 0.08 0.09 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4533.2402 A Ti1 Eexc = 0.848 log gf = 0.540\
94 82 72 60 49 37 27 18 12 7 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.04 0.05 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.11 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4534.7759 A Ti1 Eexc = 0.836 log gf = 0.350\
86 74 63 52 41 30 21 14 9 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.06 0.06 0.06 0.07 0.07 0.08 0.09 0.10 0.11 0.11 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4548.7632 A Ti1 Eexc = 0.826 log gf = -0.280\
46 35 26 19 13 9 6 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.10 0.09 0.08 0.08 0.08 0.08 0.09 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4617.2681 A Ti1 Eexc = 1.748 log gf = 0.440\
40 31 24 18 13 9 7 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.13 0.13 0.13 0.13 0.12 0.13 0.12 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4656.4678 A Ti1 Eexc = 0.000 log gf = -1.290\
25 17 12 8 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.20 0.19 0.17 0.16 0.15 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4759.2690 A Ti1 Eexc = 2.255 log gf = 0.590\
26 20 16 12 9 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.12 0.11 0.10 0.10 0.10 0.09 0.09 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4913.6152 A Ti1 Eexc = 1.872 log gf = 0.220\
25 18 14 10 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.11 0.10 0.10 0.10 0.09 0.09 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4981.7310 A Ti1 Eexc = 0.848 log gf = 0.570\
106 94 82 70 58 46 34 23 15 9 6 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.02 -0.01 -0.00 0.02 0.03 0.05 0.07 0.08 0.09 0.09 0.09 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4999.5020 A Ti1 Eexc = 0.826 log gf = 0.320\
89 77 65 53 42 31 22 14 9 6 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.09 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5016.1611 A Ti1 Eexc = 0.848 log gf = -0.480\
36 27 19 14 9 6 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.09 0.08 0.07 0.06 0.06 0.07 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5025.5698 A Ti1 Eexc = 2.041 log gf = 0.250\
20 15 12 9 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.16 0.14 0.12 0.11 0.09 0.08 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5036.4639 A Ti1 Eexc = 1.443 log gf = 0.140\
41 31 24 17 13 9 6 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.10 0.09 0.09 0.09 0.08 0.08 0.08 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5173.7402 A Ti1 Eexc = 0.000 log gf = -1.060\
41 29 20 13 9 6 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.18 0.17 0.16 0.16 0.15 0.14 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5192.9692 A Ti1 Eexc = 0.021 log gf = -0.950\
46 33 23 16 11 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.19 0.19 0.17 0.16 0.15 0.14 0.13 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5210.3838 A Ti1 Eexc = 0.048 log gf = -0.820\
54 40 29 20 14 9 6 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.17 0.17 0.17 0.16 0.15 0.14 0.14 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5866.4492 A Ti1 Eexc = 1.066 log gf = -0.790\
14 10 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.15 0.14 0.13 0.13 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
6258.0991 A Ti1 Eexc = 1.443 log gf = -0.390\
19 14 10 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.08 0.06 0.05 0.04 0.04 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
6261.0962 A Ti1 Eexc = 1.429 log gf = -0.530\
14 10 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.08 0.06 0.05 0.04 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
8426.5059 A Ti1 Eexc = 0.826 log gf = -1.200\
16 11 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.03 0.03 0.02 0.02 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
2827.1140 A Ti2 Eexc = 3.687 log gf = -0.020\
47 44 41 38 35 31 28 24 20 17 14 12 10 8 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.01 0.01 0.01 0.00 0.00 -0.00 -0.00 -0.00 0.00 0.00 0.01 0.01 0.02 0.03 0.04 0.05 0.07 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
2828.0769 A Ti2 Eexc = 3.749 log gf = 0.870\
80 78 75 72 69 66 62 58 54 49 45 40 37 33 30 27 24 22 19 16 14 12 9 8 6 5 -1\
0.04 0.03 0.02 0.01 -0.01 -0.02 -0.03 -0.03 -0.03 -0.03 -0.02 -0.01 0.01 0.02 0.04 0.06 0.08 0.09 0.12 0.16 0.20 0.24 0.28 0.32 0.35 0.38 -1.00\
2834.0110 A Ti2 Eexc = 3.716 log gf = 0.000\
48 45 42 39 36 32 29 25 21 18 15 12 10 8 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.01 0.01 0.01 0.00 0.00 -0.00 -0.00 -0.00 -0.00 0.00 0.01 0.01 0.02 0.03 0.04 0.05 0.07 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
2841.9351 A Ti2 Eexc = 0.607 log gf = -0.590\
130 122 115 109 104 98 93 87 82 76 70 65 60 55 51 47 42 37 33 28 23 18 14 11 9 7 5\
-0.01 -0.03 -0.04 -0.05 -0.07 -0.09 -0.10 -0.12 -0.13 -0.13 -0.14 -0.14 -0.13 -0.12 -0.10 -0.08 -0.06 -0.04 -0.01 0.02 0.07 0.11 0.16 0.20 0.24 0.28 0.30\
2851.1011 A Ti2 Eexc = 1.221 log gf = -0.730\
100 96 92 87 82 78 73 67 61 55 49 43 38 33 29 25 21 18 14 12 9 7 5 -1 -1 -1 -1\
-0.03 -0.04 -0.05 -0.06 -0.08 -0.09 -0.09 -0.09 -0.09 -0.09 -0.08 -0.07 -0.05 -0.04 -0.03 -0.01 0.00 0.02 0.05 0.08 0.12 0.17 0.21 -1.00 -1.00 -1.00 -1.00\
2853.9309 A Ti2 Eexc = 0.607 log gf = -1.550\
91 87 82 77 71 66 60 54 47 40 33 28 23 19 15 12 10 8 6 5 -1 -1 -1 -1 -1 -1 -1\
-0.02 -0.03 -0.04 -0.04 -0.05 -0.05 -0.05 -0.05 -0.04 -0.04 -0.04 -0.04 -0.03 -0.03 -0.03 -0.02 -0.01 0.00 0.03 0.05 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
2868.7410 A Ti2 Eexc = 0.574 log gf = -1.380\
97 93 88 83 78 72 66 60 54 47 41 35 29 25 20 17 14 11 9 7 5 -1 -1 -1 -1 -1 -1\
-0.02 -0.03 -0.04 -0.05 -0.06 -0.06 -0.07 -0.07 -0.07 -0.06 -0.06 -0.06 -0.05 -0.05 -0.04 -0.03 -0.02 -0.01 0.01 0.04 0.08 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4012.3850 A Ti2 Eexc = 0.574 log gf = -1.780\
117 113 108 104 99 95 90 84 77 68 59 51 43 36 29 24 19 15 12 9 7 5 -1 -1 -1 -1 -1\
-0.03 -0.03 -0.03 -0.03 -0.03 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 0.01 0.02 0.04 0.07 0.10 0.13 -1.00 -1.00 -1.00 -1.00 -1.00\
4028.3430 A Ti2 Eexc = 1.891 log gf = -0.920\
105 103 100 97 94 90 87 82 76 69 62 54 47 41 35 30 25 21 17 14 11 8 6 5 -1 -1 -1\
-0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.00 0.00 0.01 0.02 0.04 0.06 0.09 0.12 0.15 0.17 -1.00 -1.00 -1.00\
4053.8201 A Ti2 Eexc = 1.892 log gf = -1.070\
98 95 92 89 86 82 78 73 67 60 52 45 38 32 27 23 19 16 12 10 8 6 5 -1 -1 -1 -1\
-0.02 -0.03 -0.03 -0.03 -0.03 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 0.00 0.01 0.02 0.04 0.06 0.09 0.12 0.15 -1.00 -1.00 -1.00 -1.00\
4161.5298 A Ti2 Eexc = 1.084 log gf = -2.090\
84 80 75 70 65 59 52 45 38 31 24 19 15 12 10 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 0.00 0.01 0.01 0.03 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4163.6401 A Ti2 Eexc = 2.589 log gf = -0.130\
116 114 112 110 108 106 103 99 95 89 83 77 70 64 57 51 45 39 33 28 23 19 15 12 10 8 7\
-0.04 -0.04 -0.05 -0.05 -0.06 -0.05 -0.05 -0.05 -0.04 -0.04 -0.03 -0.02 -0.01 -0.01 0.00 0.02 0.03 0.05 0.07 0.09 0.13 0.16 0.19 0.22 0.24 0.26 0.28\
4174.0698 A Ti2 Eexc = 2.598 log gf = -1.260\
54 53 50 47 44 40 37 32 28 23 19 15 13 10 8 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 0.00 0.00 0.01 0.01 0.02 0.03 0.05 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4188.9868 A Ti2 Eexc = 5.423 log gf = -0.600\
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4190.2329 A Ti2 Eexc = 1.084 log gf = -3.122\
24 21 18 15 12 10 8 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4287.8701 A Ti2 Eexc = 1.080 log gf = -1.790\
103 99 95 90 85 80 74 67 59 50 41 34 27 22 18 14 11 9 7 5 -1 -1 -1 -1 -1 -1 -1\
-0.07 -0.07 -0.06 -0.05 -0.05 -0.04 -0.03 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.00 0.00 0.01 0.03 0.05 0.07 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4290.2148 A Ti2 Eexc = 1.164 log gf = -0.870\
148 143 138 133 129 125 120 115 109 102 95 87 80 73 65 58 50 43 36 30 24 19 14 11 9 7 5\
-0.07 -0.08 -0.09 -0.09 -0.09 -0.08 -0.08 -0.07 -0.06 -0.06 -0.05 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.04 0.06 0.10 0.13 0.17 0.20 0.23 0.26 0.27\
4300.0488 A Ti2 Eexc = 1.180 log gf = -0.460\
176 169 162 156 151 146 141 136 130 123 116 110 103 97 90 83 76 69 61 54 45 37 30 24 19 16 13\
-0.06 -0.07 -0.08 -0.09 -0.10 -0.10 -0.10 -0.10 -0.10 -0.09 -0.09 -0.09 -0.08 -0.07 -0.06 -0.05 -0.03 -0.01 0.02 0.05 0.09 0.12 0.16 0.20 0.23 0.25 0.27\
4301.9199 A Ti2 Eexc = 1.160 log gf = -1.210\
129 125 120 116 112 107 102 97 91 83 75 66 58 51 43 37 30 25 20 16 12 9 7 5 -1 -1 -1\
-0.07 -0.07 -0.07 -0.07 -0.06 -0.06 -0.05 -0.04 -0.04 -0.03 -0.03 -0.02 -0.02 -0.01 -0.01 0.00 0.01 0.02 0.05 0.07 0.10 0.14 0.17 0.20 -1.00 -1.00 -1.00\
4374.8198 A Ti2 Eexc = 2.060 log gf = -1.570\
65 62 59 55 51 47 42 37 31 25 21 17 13 11 9 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 -0.00 -0.01 -0.01 -0.00 -0.00 -0.00 0.00 0.01 0.02 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4386.8442 A Ti2 Eexc = 2.598 log gf = -0.960\
75 73 71 68 65 61 57 52 46 39 33 28 23 19 16 13 11 9 7 6 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.02 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 0.00 0.00 0.01 0.02 0.03 0.04 0.07 0.09 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4391.0200 A Ti2 Eexc = 1.231 log gf = -2.300\
66 61 56 51 45 39 34 28 23 18 14 11 8 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4394.0591 A Ti2 Eexc = 1.221 log gf = -1.770\
97 93 89 84 79 74 68 61 53 45 37 30 24 19 16 12 10 8 6 5 -1 -1 -1 -1 -1 -1 -1\
-0.04 -0.04 -0.04 -0.03 -0.03 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 0.00 0.01 0.02 0.03 0.05 0.08 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4395.0308 A Ti2 Eexc = 1.084 log gf = -0.540\
176 169 162 156 151 146 142 137 131 124 117 110 103 97 90 83 76 68 60 52 43 36 28 23 18 15 12\
-0.09 -0.10 -0.12 -0.13 -0.13 -0.13 -0.13 -0.12 -0.12 -0.12 -0.11 -0.11 -0.10 -0.09 -0.07 -0.06 -0.04 -0.02 0.01 0.05 0.09 0.12 0.16 0.20 0.22 0.25 0.26\
4395.8389 A Ti2 Eexc = 1.242 log gf = -1.930\
88 84 79 74 69 63 57 50 43 35 28 22 18 14 11 9 7 6 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.03 -0.03 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 0.00 0.01 0.02 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4399.7720 A Ti2 Eexc = 1.236 log gf = -1.200\
128 124 120 116 111 107 102 96 90 82 74 65 57 49 42 35 29 24 19 15 12 9 7 5 -1 -1 -1\
-0.07 -0.08 -0.08 -0.07 -0.07 -0.06 -0.05 -0.05 -0.04 -0.03 -0.03 -0.02 -0.02 -0.01 -0.01 0.00 0.01 0.03 0.05 0.07 0.11 0.14 0.18 0.21 -1.00 -1.00 -1.00\
4409.2349 A Ti2 Eexc = 1.242 log gf = -2.780\
35 32 28 24 20 17 14 11 9 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4409.5200 A Ti2 Eexc = 1.231 log gf = -2.530\
51 46 41 36 32 27 23 18 15 11 9 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4411.0698 A Ti2 Eexc = 3.093 log gf = -0.650\
67 66 64 62 60 56 53 48 43 38 32 27 23 20 17 14 12 10 8 7 5 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 0.00 0.00 -0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.03 0.05 0.07 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4411.9248 A Ti2 Eexc = 1.224 log gf = -2.620\
45 41 36 31 27 23 19 15 12 9 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4417.7129 A Ti2 Eexc = 1.165 log gf = -1.190\
133 129 125 120 116 111 106 101 94 86 77 69 61 53 45 38 32 26 21 17 13 10 7 6 -1 -1 -1\
-0.11 -0.12 -0.12 -0.11 -0.10 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.02 -0.01 -0.00 0.01 0.02 0.05 0.07 0.10 0.14 0.17 0.20 -1.00 -1.00 -1.00\
4418.3311 A Ti2 Eexc = 1.236 log gf = -1.990\
85 81 76 71 66 60 53 46 39 32 25 20 16 13 10 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.03 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 0.00 0.01 0.02 0.03 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4421.9380 A Ti2 Eexc = 2.060 log gf = -1.640\
61 59 55 51 47 43 38 33 28 23 18 15 12 9 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 0.00 0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4423.2388 A Ti2 Eexc = 1.231 log gf = -3.066\
22 19 16 14 11 9 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4432.1089 A Ti2 Eexc = 1.236 log gf = -3.080\
21 18 16 13 11 9 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4441.7300 A Ti2 Eexc = 1.180 log gf = -2.330\
68 64 58 53 47 41 35 29 24 19 14 11 9 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.03 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4443.8008 A Ti2 Eexc = 1.080 log gf = -0.710\
164 158 152 147 142 138 133 128 122 115 108 101 94 87 80 72 65 57 49 41 34 27 21 17 13 10 8\
-0.09 -0.11 -0.12 -0.12 -0.12 -0.12 -0.12 -0.11 -0.10 -0.10 -0.09 -0.09 -0.08 -0.06 -0.05 -0.04 -0.02 -0.00 0.03 0.06 0.09 0.13 0.17 0.20 0.23 0.25 0.26\
4450.4819 A Ti2 Eexc = 1.084 log gf = -1.520\
118 114 110 105 101 96 91 84 77 68 59 50 43 36 30 24 19 16 12 9 7 5 -1 -1 -1 -1 -1\
-0.08 -0.08 -0.07 -0.07 -0.06 -0.05 -0.04 -0.04 -0.03 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.00 0.01 0.02 0.05 0.07 0.10 0.14 -1.00 -1.00 -1.00 -1.00 -1.00\
4464.4492 A Ti2 Eexc = 1.161 log gf = -1.810\
100 96 91 86 81 75 69 62 54 45 37 30 24 19 16 12 10 8 6 5 -1 -1 -1 -1 -1 -1 -1\
-0.06 -0.06 -0.06 -0.05 -0.04 -0.03 -0.03 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 0.00 0.01 0.03 0.05 0.08 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4468.5098 A Ti2 Eexc = 1.130 log gf = -0.630\
197 187 177 169 162 154 148 141 133 125 117 109 101 94 87 79 71 64 55 47 39 31 25 20 16 12 10\
-0.06 -0.07 -0.08 -0.09 -0.09 -0.09 -0.09 -0.09 -0.09 -0.09 -0.09 -0.09 -0.08 -0.07 -0.06 -0.04 -0.03 -0.01 0.02 0.05 0.09 0.13 0.17 0.20 0.23 0.25 0.27\
4469.1509 A Ti2 Eexc = 1.084 log gf = -2.550\
58 53 48 43 37 32 27 22 18 13 10 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.01 -0.01 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4470.8530 A Ti2 Eexc = 1.165 log gf = -2.020\
88 83 79 73 68 61 55 48 40 33 26 21 16 13 10 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.05 -0.04 -0.04 -0.03 -0.03 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 0.01 0.02 0.03 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4488.3242 A Ti2 Eexc = 3.122 log gf = -0.500\
76 75 73 71 69 66 63 58 53 47 41 35 30 26 22 19 16 14 11 9 7 6 5 -1 -1 -1 -1\
-0.02 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 0.00 0.00 0.01 0.01 0.03 0.05 0.07 0.09 0.12 -1.00 -1.00 -1.00 -1.00\
4501.2700 A Ti2 Eexc = 1.115 log gf = -0.770\
159 154 148 144 139 134 129 124 118 111 104 96 89 82 75 67 59 52 44 37 29 23 18 14 11 9 7\
-0.10 -0.11 -0.12 -0.12 -0.12 -0.12 -0.11 -0.10 -0.10 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.01 0.00 0.03 0.06 0.10 0.13 0.17 0.20 0.23 0.25 0.27\
4518.3301 A Ti2 Eexc = 1.080 log gf = -2.560\
58 53 47 42 37 31 26 22 17 13 10 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.01 -0.01 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4529.4741 A Ti2 Eexc = 1.571 log gf = -1.750\
83 79 75 70 65 60 54 47 40 33 27 21 17 14 11 9 7 6 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.04 -0.04 -0.03 -0.03 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 0.01 0.02 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4533.9600 A Ti2 Eexc = 1.237 log gf = -0.530\
172 166 160 155 150 145 140 135 129 121 114 107 100 92 85 78 70 63 54 47 39 32 25 20 16 13 10\
-0.12 -0.14 -0.15 -0.16 -0.16 -0.16 -0.15 -0.14 -0.14 -0.12 -0.12 -0.10 -0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.04 0.06 0.10 0.14 0.18 0.21 0.24 0.26 0.28\
4544.0200 A Ti2 Eexc = 1.243 log gf = -2.580\
54 49 44 39 34 29 24 20 16 12 9 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4549.6201 A Ti2 Eexc = 1.583 log gf = -0.220\
176 170 164 158 153 149 144 139 133 126 119 113 106 100 93 86 79 72 64 56 47 40 32 26 21 17 14\
-0.11 -0.13 -0.14 -0.15 -0.15 -0.15 -0.15 -0.14 -0.14 -0.14 -0.13 -0.13 -0.12 -0.10 -0.09 -0.07 -0.05 -0.03 -0.00 0.03 0.08 0.12 0.16 0.20 0.23 0.26 0.28\
4563.7568 A Ti2 Eexc = 1.221 log gf = -0.795\
161 156 151 146 141 137 132 126 120 113 105 98 91 83 75 68 60 53 45 38 30 24 19 15 12 9 7\
-0.12 -0.14 -0.15 -0.15 -0.15 -0.15 -0.14 -0.13 -0.12 -0.11 -0.10 -0.08 -0.07 -0.05 -0.04 -0.02 -0.00 0.01 0.04 0.07 0.11 0.14 0.18 0.21 0.24 0.26 0.28\
4568.3140 A Ti2 Eexc = 1.224 log gf = -3.030\
24 21 18 15 13 10 9 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4571.9712 A Ti2 Eexc = 1.571 log gf = -0.310\
169 163 158 153 148 144 139 134 128 122 115 108 101 94 88 81 74 66 58 50 42 35 28 22 18 14 12\
-0.12 -0.13 -0.14 -0.14 -0.15 -0.14 -0.14 -0.14 -0.13 -0.13 -0.12 -0.11 -0.10 -0.09 -0.08 -0.06 -0.04 -0.02 0.01 0.04 0.08 0.12 0.17 0.20 0.23 0.26 0.28\
4583.4102 A Ti2 Eexc = 1.164 log gf = -2.840\
36 32 28 24 20 17 14 11 9 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 -0.00 -0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4589.9580 A Ti2 Eexc = 1.237 log gf = -1.620\
109 105 101 96 91 85 80 73 65 56 47 39 32 26 21 17 14 11 8 6 5 -1 -1 -1 -1 -1 -1\
-0.08 -0.08 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.03 -0.02 -0.02 -0.01 -0.01 -0.01 -0.00 0.00 0.02 0.03 0.05 0.08 0.11 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4636.3198 A Ti2 Eexc = 1.165 log gf = -3.024\
26 23 20 17 14 12 9 8 6 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4657.2012 A Ti2 Eexc = 1.242 log gf = -2.290\
69 64 59 53 47 42 36 30 24 19 15 11 9 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4708.6631 A Ti2 Eexc = 1.236 log gf = -2.350\
65 60 55 49 44 38 33 27 22 17 13 10 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4719.5151 A Ti2 Eexc = 1.242 log gf = -3.320\
14 12 10 8 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4763.8799 A Ti2 Eexc = 1.221 log gf = -2.400\
61 56 51 46 40 35 30 25 20 15 12 9 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4764.5249 A Ti2 Eexc = 1.236 log gf = -2.690\
42 38 34 29 25 21 17 14 11 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4779.9849 A Ti2 Eexc = 2.048 log gf = -1.260\
89 86 83 79 75 70 65 59 52 44 37 31 25 21 17 14 11 9 7 6 -1 -1 -1 -1 -1 -1 -1\
-0.04 -0.05 -0.05 -0.04 -0.04 -0.03 -0.03 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.00 0.00 0.01 0.03 0.05 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4798.5298 A Ti2 Eexc = 1.080 log gf = -2.660\
53 48 43 37 32 27 23 18 15 11 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.00 -0.00 -0.00 -0.00 -0.01 -0.01 -0.01 -0.01 -0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
4805.0850 A Ti2 Eexc = 2.061 log gf = -0.960\
107 104 101 98 94 90 85 79 73 64 56 48 42 35 30 25 21 17 14 11 8 7 5 -1 -1 -1 -1\
-0.07 -0.07 -0.08 -0.07 -0.07 -0.06 -0.05 -0.04 -0.04 -0.03 -0.03 -0.02 -0.02 -0.02 -0.01 -0.01 0.00 0.01 0.03 0.05 0.08 0.11 0.15 -1.00 -1.00 -1.00 -1.00\
4911.1899 A Ti2 Eexc = 3.122 log gf = -0.640\
68 67 66 64 61 58 55 50 45 39 33 28 24 21 17 15 12 10 9 7 6 5 -1 -1 -1 -1 -1\
0.01 0.01 0.00 0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.01 -0.01 -0.01 -0.01 -0.00 0.00 0.01 0.03 0.05 0.08 -1.00 -1.00 -1.00 -1.00 -1.00\
4996.3672 A Ti2 Eexc = 1.582 log gf = -3.290\
8 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.01 -0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5005.1572 A Ti2 Eexc = 1.565 log gf = -2.730\
25 23 20 17 15 12 10 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5010.2100 A Ti2 Eexc = 3.093 log gf = -1.350\
28 27 26 24 22 20 18 16 13 11 9 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.01 -0.01 -0.01 -0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5013.3301 A Ti2 Eexc = 3.095 log gf = -2.028\
7 7 7 6 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5013.6860 A Ti2 Eexc = 1.581 log gf = -2.140\
61 56 52 47 42 37 32 26 22 17 13 10 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5072.2900 A Ti2 Eexc = 3.122 log gf = -1.020\
46 45 43 41 39 36 33 29 25 21 17 14 12 10 8 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.00 0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5129.1602 A Ti2 Eexc = 1.891 log gf = -1.340\
97 93 89 85 80 75 70 63 55 47 39 33 27 22 18 15 12 10 8 6 5 -1 -1 -1 -1 -1 -1\
-0.07 -0.06 -0.06 -0.05 -0.05 -0.04 -0.03 -0.03 -0.03 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.01 -0.00 0.00 0.02 0.05 0.07 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5154.0698 A Ti2 Eexc = 1.566 log gf = -1.750\
90 86 81 76 70 64 58 50 43 35 28 23 18 14 11 9 7 6 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.06 -0.06 -0.06 -0.05 -0.05 -0.04 -0.03 -0.03 -0.02 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.00 0.01 0.02 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5185.9131 A Ti2 Eexc = 1.892 log gf = -1.410\
92 89 85 80 76 70 65 58 51 43 35 29 24 19 16 13 10 8 7 5 -1 -1 -1 -1 -1 -1 -1\
-0.06 -0.06 -0.05 -0.05 -0.04 -0.03 -0.03 -0.03 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.00 0.01 0.02 0.05 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5188.6802 A Ti2 Eexc = 1.582 log gf = -1.050\
134 130 126 121 117 111 106 99 92 83 74 64 56 48 41 34 28 23 19 15 11 9 7 5 -1 -1 -1\
-0.15 -0.15 -0.15 -0.14 -0.14 -0.12 -0.11 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.02 0.04 0.07 0.10 0.14 0.17 0.21 -1.00 -1.00 -1.00\
5211.5361 A Ti2 Eexc = 2.589 log gf = -1.410\
50 48 46 43 39 36 32 27 23 19 15 12 10 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.00 -0.00 -0.01 -0.01 -0.01 -0.01 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 0.00 0.01 0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5262.1401 A Ti2 Eexc = 1.582 log gf = -2.250\
55 51 46 41 37 32 27 22 18 14 11 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.03 -0.03 -0.03 -0.02 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5268.6099 A Ti2 Eexc = 2.597 log gf = -1.610\
38 36 34 31 28 25 22 19 16 13 10 8 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5336.7861 A Ti2 Eexc = 1.581 log gf = -1.600\
100 96 91 86 81 75 68 61 53 44 37 30 24 19 16 13 10 8 6 5 -1 -1 -1 -1 -1 -1 -1\
-0.07 -0.07 -0.06 -0.06 -0.05 -0.04 -0.04 -0.03 -0.03 -0.02 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 0.00 0.02 0.04 0.06 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5381.0220 A Ti2 Eexc = 1.565 log gf = -1.970\
75 71 66 60 55 49 43 36 30 24 19 15 12 9 7 6 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.04 -0.04 -0.03 -0.03 -0.03 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.02 -0.01 -0.01 -0.01 -0.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
5418.7681 A Ti2 Eexc = 1.581 log gf = -2.130\
64 59 55 49 44 39 33 28 23 18 14 11 8 7 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.03 -0.03 -0.02 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
6680.1328 A Ti2 Eexc = 3.093 log gf = -1.890\
11 11 10 9 8 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.00 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
6998.9048 A Ti2 Eexc = 3.122 log gf = -1.280\
36 34 33 30 28 25 22 19 16 13 10 8 7 6 5 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1\
-0.01 -0.01 -0.01 -0.01 -0.01 -0.02 -0.02 -0.02 -0.02 -0.03 -0.03 -0.04 -0.05 -0.05 -0.06 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00\
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Keith Butler for computations of the photoionisation cross-sections for Ti. This research was supported by the Russian Foundation for Basic Research (grants 16-32-00695 and 15-02-06046). TS and LM are grateful to the Swiss National Science Foundation (the SCOPES project IZ73Z0–152485). TS and LM are indebted to the International Space Science Institute (ISSI), Bern, Switzerland, for supporting and funding the international team “First stars in dwarf galaxies” and “The Formation and Evolution of the Galactic Halo”. We made use of the NIST, SIMBAD, and VALD databases.
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[^1]: E-mail: sitnova@inasan.ru
[^2]: Here, log A(X)=log($N_X/N_{tot}$), where N$_{tot}$ is a total number density; X–X means difference in abundance derived from lines of X and X, log A(X)–log A(X).
[^3]: http://www.astro.uu.se/$\sim$oleg/download.html
[^4]: [http://marcs.astro.uu.se/software.php]{}
|
---
abstract: 'We have investigated the motion of a single optically trapped colloidal particle close to a limiting wall at time scales where the inertia of the surrounding fluid plays a significant role. The velocity autocorrelation function exhibits a complex interplay due to the momentum relaxation of the particle, the vortex diffusion in the fluid, the obstruction of flow close to the interface, and the harmonic restoring forces due to the optical trap. We show that already a weak trapping force has a significant impact on the velocity autocorrelation function $C(t)=\langle v(t)v(0)\rangle$ at times where the hydrodynamic memory leads to an algebraic decay. The long-time behavior for the motion parallel and perpendicular to the wall is derived analytically and compared to numerical results. Then, we discuss the power spectral densities of the displacement and provide simple interpolation formulas. The theoretical predictions are finally compared to recent experimental observations.'
author:
- 'Thomas Franosch${}^*$'
- Sylvia Jeney
title: Persistent correlation of constrained colloidal motion
---
Introduction
============
Understanding and controlling the motion of small colloidal particles suspended in fluids and confined to small volumes is essential in many applications in microfluidics [@Squires:2005] and biophysics [@Cicuta:2007]. Thermal fluctuations of the surrounding fluid agitate the colloids giving rise to Brownian motion. Positioning the particle close to an interface or membrane and the direct optical observation of its trajectories allows in principle to use the bead’s motion as a sensor for the chemical and physical properties of these two-dimensional surfaces. For example, the presence of surfactants [@Blawzdziewicz:1999], surface tension and elasticity [@Felderhof:2006b; @Bickel:2007], or an adsorption layer [@Felderhof:2006c] modifies the frequency-dependent mobility close to the interface. To establish Brownian motion as a local reporter of frequency-dependent surface properties a detailed understanding of the complex interplay of the hydrodynamic flow and the colloidal thermal fluctuations is a prerequisite. The simplest effect known as surface confinement, namely the anisotropic reduction of the diffusion coefficient close to a bounding wall [@Happel:LowReynolds; @Lorentz:1907], has been directly observed experimentally only very recently [@carbajal-tinoco:2007; @Schaffer:2007]. To gain insight beyond the transport coefficients, one should record the time-dependent information, e.g. the mean-square displacement or the velocity autocorrelation, at times scales before the regime of simple diffusion is attained. Using weak optical trapping the quasi-free Brownian particle’s trajectory can be monitored interferometrically [@Gittes:1998] with a time resolution of a few microseconds and a simultaneous positional sensitivity in the sub-nanometer regime [@Lukic:2005; @Lukic:2007].
In the diffusive regime the momenta of the colloid are equilibrated with the surrounding fluid and are irrelevant for the Brownian motion. At very short times the particle’s motion is ballistic as expected by Newton’s laws. Hence, only after the colloid’s initial momentum is transferred to the fluid the particle undergoes a random walk characterized by diffusion coefficients. The time scale where this transition occurs may be naïvely estimated by balancing the particle’s inertia with the Stokes drag, $\tau_\text{p} = m_\text{p}/6 \pi \eta a$, where $m_\text{p}$ denotes the mass of the colloid, $a$ its radius, and $\eta$ the shear viscosity of the fluid. Since momentum is conserved it can only be transported by vortex diffusion giving rise to another characteristic time scale $\tau_\text{f} = a^2 \rho_\text{f}/\eta$, where $\rho_\text{f}$ is the density of the fluid. The time a vortex emitted from the colloid to reach the wall separated at a distance $h$ may be estimated by the characteristic time $\tau_\text{w} = h^2 \rho_\text{f}/\eta$. The dynamical information on how the diffusive regime is reached, is encoded in the velocity autocorrelation function (VACF) $C_\parallel(t) = \langle v_\parallel(t) v_\parallel(0)\rangle$ for the motion parallel to the surface and similarly perpendicular to the wall. In bulk the VACF exhibits a power-law decay $t^{-3/2}$ at long times due to slow vortex diffusion. This long-time tail is also expected in the presence of a wall for times up to $\tau_\text{w}$. For longer times this leading non-analytic behavior is canceled and a more rapid decay is expected, since the wall can carry away the momentum much faster. An algebraic decay $t^{-5/2}$ for motion parallel and a $t^{-7/2}$ long-time tail for motion perpendicular to the wall has been predicted by Gotoh and Kaneda [@Gotoh:1982] extending an earlier work of Wakiya [@Wakiya:1964]. However, the result for the perpendicular motion is erroneous, as pointed out by Felderhof [@Felderhof:2005]. He succeeded also in calculating the frequency-dependent mobility for all frequencies in both directions and thus giving an analytic solution for the motion close to the wall up to Fourier transform. Later, he generalized this work to the case of a compressible fluid [@Felderhof:2005b] and to a second bounding wall [@Felderhof:2006]. Computer simulations for a colloidal particle confined between two walls demonstrate that the long-time behavior of the VACF is strongly affected by the confinement [@Hagen:1997; @Pagonabarraga:1998; @Pagonabarraga:1999; @Frydel:2006; @Frydel:2007].
Recently, we have reported direct measurements of the velocity autocorrelation function of a single colloid immersed in water close to a bounding wall [@Jeney:2008]. We have observed that the time-dependent VACF becomes anisotropic and exhibits the non-algebraic tails that had been anticipated much earlier [@Wakiya:1964; @Gotoh:1982; @Felderhof:2005]. To observe the particle for sufficiently long times close to a wall we used a trap constraining the Brownian motion even more. The optical trap introduces a harmonic restoring force $-K {\mathbf}{x}$, where ${\mathbf}{x}$ denotes the displacement from the trap center and $K$ the spring constant. Ignoring the wall a new time scale can be constructed $\tau_\text{k} = 6\pi \eta a/K$ characteristic of the positional equilibration in the trap.
Here we supplement our experimental results with a derivation of the theoretical description of the Brownian motion confined by a harmonic potential and a bounding wall. In particular, we discuss the complex interplay of the weak trap with hydrodynamic memory originating from the obstructed vortex motion. We show that even if the trapping time $\tau_\text{k}$ exceeds the characteristic time scales of particle momentum relaxation $\tau_\text{p}$, fluid momentum diffusion $\tau_\text{f}$, and the wall-vortex reflection $\tau_\text{w}$, by several orders of magnitude it still has a significant influence on the velocity autocorrelation function. Section \[Sec:Felderhof\] provides a brief introduction into the theoretical framework of Felderhof, which is extended by a harmonic restoring force in Sec. \[Sec:trap\]. An analytical discussion of the emerging long-time tails is presented in Sec. \[Sec:analytical\] followed by a numerical study of the VACF and the power spectral density of the displacement in Sec. \[Sec:numerical\]. A comparison to our experimental results is shown in Sec. \[Sec:experimental\].
Felderhof’s framework {#Sec:Felderhof}
=====================
In this Section we review the theoretical basis elaborated by Felderhof [@Felderhof:2005] for the motion of a single colloid in the vicinity of a wall. The fluid is treated as a continuum described by the Navier-Stokes equations, the colloid is modeled as an impenetrable sphere embedded in a viscous incompressible fluid. At the particle’s surface the usual no-slip boundary conditions are imposed. By the fluctuation-dissipation theorem the velocity autocorrelation function is related to the frequency-dependent admittance [@Zwanzig:1970], hence it suffices to calculate the deterministic velocity response of the colloid to an external force.
The external force ${\mathbf}{R}(t)$ causes the motion of both the particle and surrounding fluid, which reacts by exerting a time-dependent drag force. Newton’s second law for the acceleration of the sphere reads after a temporal Fourier transform $-\text{i} \omega (m_{\text{p}}-m_{\text{f}}) {\mathbf}{U}_\omega = -{\mathbf}{F}_\omega+ {\mathbf}{R}_\omega$, where ${\mathbf}{U}_\omega$ is the velocity of the colloidal particle, $m_{\text{p}}$ its mass, and $m_{\text{f}}$ denotes the mass of the displaced fluid. The total *induced force* ${\mathbf}{F}_\omega$ corresponds to the frequency-dependent drag force on the particle up to the acceleration force $-\text{i} \omega m_{\text{f}} {\mathbf}{U}_\omega$ of a rigid sphere of fluid of equal radius. We shall employ the *point particle limit* where inhomogeneities of the flow on the scale of the colloid are ignored.
To calculate the total induced force one maps the problem in the presence of a bounding surface to a corresponding one in infinite space and employs the generalized Faxén theorem of Mazur and Bedeaux [@Mazur:1974]: For an unbounded fluid the frequency-dependent induced force ${\mathbf}{F}_\omega$ is $$\begin{aligned}
{\mathbf}{F}_\omega &= {\left [}\zeta(\omega) - \frac{3}{2}\text{i}\omega m_\text{f} {\right ]}({\mathbf}{U}_\omega-{\mathbf}{v}'_\omega) \, , \label{eq: mazur}\end{aligned}$$ with $\zeta(\omega)= 6 \pi \eta a (1+ \sqrt{-\text{i} \omega \tau_\text{f}}) $ and ${\mathbf}{v}'_\omega$ denoting the unperturbed flow evaluated at the position of the particle ${\mathbf}{r}_0$. The branch cut is chosen such that $\sqrt{-\text{i} \omega \tau_\text{f}} = (1-\text{i}) \sqrt{\omega \tau_\text{f}/2}$. Then consider as acting flow $${\mathbf}{v}'_\omega({\mathbf}{r})= {\mathbf}{v}_\omega({\mathbf}{r}) -{\mathbf}{v}_{0 \omega}({\mathbf}{r}) \, ,$$ where ${\mathbf}{v}_\omega({\mathbf}{r}), {\mathbf}{v}_{0 \omega }({\mathbf}{r})$ denote the solutions of the Navier Stokes equation in response to a point force ${\mathbf}{R}_\omega$ acting at ${\mathbf}{r}_0$ in the presence of a bounding surface and in the infinite space. Clearly, ${\mathbf}{v}'_\omega({\mathbf}{r})$ satisfies the homogenous Stokes equations. The key idea is that ${\mathbf}{v}_\omega({\mathbf}{r})$ can be interpreted as a flow in infinite space resulting from the force ${\mathbf}{R}_\omega$ with ${\mathbf}{v}'_\omega({\mathbf}{r})$ as an externally acting flow. Yet, for flows in infinite space the generalized Faxén theorem applies and the induced force can be calculated easily.
By linearity of the Navier-Stokes equation, the fluid response to a point force ${\mathbf}{R}_\omega$ acting at ${\mathbf}{r}_0$ is obtained by $${\mathbf}{v}_\omega({\mathbf}{r})= G({\mathbf}{r},{\mathbf}{r}_0) \cdot {\mathbf}{R}_\omega \, , \qquad
{\mathbf}{v}_{0 \omega}({\mathbf}{r})= G_0({\mathbf}{r}-{\mathbf}{r}_0) \cdot {\mathbf}{R}_\omega\, ,$$ where $G({\mathbf}{r},{\mathbf}{r}_0)$ and $G_0({\mathbf}{r}-{\mathbf}{r}_0)$ are the corresponding (tensor) Green’s functions. Then the acting flow at the position of the particle reads ${\mathbf}{v}'_\omega=F({\mathbf}{r}_0,\omega) {\mathbf}{R}_\omega$, where $$\label{eq:rft_def}
F({\mathbf}{r}_0,\omega)= \lim_{{\mathbf}{r} \to {\mathbf}{r}_0} {\left (}G({\mathbf}{r},{\mathbf}{r}_0) - G_0({\mathbf}{r},{\mathbf}{r}_0) {\right )}\, ,$$ defines the reaction field tensor. Its frequency dependence only enters via the ratio $h/\delta = \sqrt{\omega \tau_\text{f}/2}$, where $h$ is the distance to the wall and $\delta=\sqrt{2 \eta/\rho_\text{f}\omega }$ denotes the skin penetration depth. Combining Newton’s second law, Faxén’s theorem, and the reaction field approach one finds for the force balance $$\begin{aligned}
\label{eq:force_balance}
\lefteqn{- \text{i} \omega (m_\text{p}-m_\text{f}) {\mathbf}{U}_\omega =} \nonumber \\
&= & - {\left [}\zeta(\omega) - \frac{3}{2}\text{i}\omega m_\text{f} {\right ]}{\left [}{\mathbf}{U}_\omega - F({\mathbf}{r}_0,\omega) {\mathbf}{R}_\omega) {\right ]}+ {\mathbf}{R}_\omega \, .\end{aligned}$$ Solving this equation for the particle velocity yields ${\mathbf}{U}_\omega= {\cal Y}(\omega) {\mathbf}{R}_\omega$, where the response function ${\cal Y}(\omega)$ is called the *admittance tensor* and corresponds to a frequency-dependent mobility.
Consider first the case where the wall is infinitely far away and the reaction field tensor vanishes. Then one finds for the admittance for infinite space $$\label{eq:Y0}
{\cal Y}_0(\omega) = {\left [}-\text{i} \omega m^* + \zeta(\omega) {\right ]}^{-1} \, .$$ Here $m^* = m_\text{p} + m_\text{f}/2$ can be interpreted as the *effective mass* of the particle, since for a particle moving in an ideal fluid with constant velocity ${\mathbf}{U}$ the total kinetic energy including the dragged fluid is given by $m^* {\mathbf}{U}^2/2$ [@Landau:1987].
The general result in presence of a bounding wall can then be expressed as $$\begin{aligned}
\lefteqn{ {\cal Y}({\mathbf}{r}_0,\omega) = } \nonumber \\ &=& {\cal Y}_0(\omega) {\left [}1 + 6 \pi \eta a {\left (}1 + \sqrt{-\text{i}\, \omega \tau_\text{f}} + \frac{-\text{i}\, \omega \tau_\text{f}}{3} {\right )}F({\mathbf}{r}_0,\omega) {\right ]}\, . \nonumber \\
\label{eq:Ywall}\end{aligned}$$ Thus, all modifications due to the bounding wall are described by the reaction field tensor. By symmetry only the motion parallel and perpendicular to the wall ${\cal Y}_{||}({\mathbf}{r}_0,\omega)$, ${\cal Y}_{\perp}({\mathbf}{r}_0,\omega)$ have non-zero components. Felderhof succeeded in providing a full analytical result for the frequency dependence of the corresponding reaction field tensors. We have verified the sophisticated results for the parallel motion, Eq. (3.9) in [@Felderhof:2005] and the perpendicular motion [@Felderhof:2006erratum]. For further analysis, in particular, to evaluate the long-time tails in the velocity autocorrelation function, it is sufficient to know their corresponding low-frequency expansions $$\label{eq:Fxxlow}
F_{\parallel}(h, \omega)
= \frac{1}{6 \pi \eta h} \left( -\frac{9}{16} + v -\frac{9}{8} v^2 \nonumber + {\cal O}(v^3) \right) \, ,$$ and $$F_{\perp}(h,\omega) = \frac{1}{6\pi \eta h} \left(-\frac{9}{8}+ v -
\frac{3}{8} v^2 + {\cal O}(v^4)\right) \, . \label{eq:Fzzlow}$$ where $v= (-\text{i} \omega \tau_\text{w})^{1/2}$. Specializing to the stationary case, $\omega=0$, and inserting into Eq. (\[eq:Ywall\]), one easily obtains the zero-frequency admittance ${\cal Y}_{||,\perp}(h,0)$ recovering Lorentz’s result [@Lorentz:1907] for the mobility close to a wall $$\label{eq:lorentz}
\mu_{\parallel} = \mu_0 {\left [}1 - \frac{9}{16} \frac{a}{h} {\right ]}\, , \, \,
\mu_{\perp} = \mu_0 {\left [}1 - \frac{9}{8} \frac{a}{h} {\right ]}\, ,$$ where $\mu_0 = 1/6 \pi \eta a$ denotes the mobility in bulk.
Influence of a harmonic trap {#Sec:trap}
============================
In the framework presented above it is implicitly assumed that the distance $h$ between the particle and its bounding wall is time-independent. To realize this situation experimentally and to prevent the particle from leaving the detection region a trap is needed, which should be included to complete the model. Already in bulk, the motion of a harmonically bound Brownian colloid differs drastically from a free particle, since at long enough times the restoring forces limit the particle’s mean-square displacement to a finite value. The very first attempt by Uhlenbeck and Ornstein [@Uhlenbeck:1930] to model the interplay of inertia of a particle, Stokes friction and harmonic restoring forces dates back to 1930 but ignores hydrodynamic memory effects. Only much later, Clercx and Schram [@Clercx:1992] incorporated the fluid inertia generalizing the VACF for the free motion. In particular, they found that the long-time anomaly changes from a $t^{-3/2}$ behavior for a free particle to a much more rapid decay according to $t^{-7/2}$. Here, we combine the ideas of Felderhof for the free motion close to a confining wall with the externally acting harmonic restoring force.
Following the chain of arguments of Sec. \[Sec:Felderhof\] we include the harmonic restoring force in Newton’s second law $$- \text{i} \omega (m_\text{p}-m_\text{f}) {\mathbf}{U}_\omega = - {\mathbf}{F}_\omega - K {\mathbf}{x}_\omega + {\mathbf}{R}_\omega \,.$$ Here $K$ denotes the spring constant of the optical trap, which we assume to be isotropic. The displacement of the particle relative to the trap center at ${\mathbf}{r}_0$ will be eliminated in favor of the particle’s velocity via ${\mathbf}{U}_\omega =-
\text{i} \omega {\mathbf}{x}_\omega$. Since Faxén’s law and the reaction field tensor are unaffected by the presence of the trap, one can again eliminate the total induced force ${\mathbf}{F}_\omega$ and merely supplement the force balance, Eq. (\[eq:force\_balance\]), by the harmonic force $$\begin{aligned}
\lefteqn{- \text{i} \omega (m_\text{p}-m_\text{f}) {\mathbf}{U}_\omega -\frac{K}{\text{i} \omega} {\mathbf}{U}_\omega = }\nonumber \\
&= & - {\left [}\zeta(\omega) - \frac{3}{2}\text{i}\,\omega m_\text{f} {\right ]}{\left [}{\mathbf}{U}_\omega - F({\mathbf}{r}_0,\omega) {\mathbf}{R}_\omega) {\right ]}\nonumber + {\mathbf}{R}_\omega \, .\end{aligned}$$ In terms of the admittance tensor the previous relation may be written as $${\mathbf}{U}_\omega = {\cal Y}({\mathbf}{r}_0,\omega) \left[\frac{K}{\text{i}\, \omega} {\mathbf}{U}_\omega + {\mathbf}{R}_\omega \right]\, ,$$ with the following interpretation: The particle’s velocity is still determined by the admittance tensor without trap, provided now the net force is considered as driving the system. Solving for the velocity ${\mathbf}{U}_\omega = {\cal Y}^{(k)}({\mathbf}{r}_0,\omega) {\mathbf}{R}_\omega$, one finds that the harmonic potential modifies the admittance according to the simple rule $$\label{eq:Ytrap}
{\cal Y}^{(k)}({\mathbf}{r}_0,\omega) = \left[ {\cal Y}^{-1}({\mathbf}{r}_0,\omega) + K/-\text{i}\,\omega \right]^{-1} \, .$$ This result includes Felderhof’s result for zero trapping, $K=0$, as well as Clercx and Schram [@Clercx:1992] if the wall is infinitely far away, ${\cal Y}({\mathbf}{r}_0,\omega) \to
{\cal Y}_0(\omega)$. Again by symmetry, the matrix ${\cal Y}^{(k)}({\mathbf}{r}_0,\omega)$ is diagonal with elements ${\cal Y}^{(k)}_{||}({\mathbf}{r}_0,\omega)$ and ${\cal Y}^{(k)}_{\perp}({\mathbf}{r}_0,\omega)$ for the force ${\mathbf}{R}_\omega$ parallel or perpendicular to the wall.
The trap introduces a new characteristic time scale $\tau_\text{k}$ into the problem: balancing the harmonic restoring force $K x$ with the zero-frequency Stokes drag $6\pi \eta a v$, one obtains $\tau_\text{k} = 6\pi \eta a/K = 1/\mu_0 K$. The reference drag force has been chosen as the one acting in bulk, although it is clear that the presence of a boundary suppresses the hydrodynamic friction. The physical interpretation of the dependences is that a stronger trap will lead to a faster relaxation to equilibrium resulting in a reduction of $\tau_\text{k}$, whereas an increase of the Stokes drag induced by a larger viscosity leads to slowing down of the equilibration process.
In the derivation we have assumed that the trapping is isotropic, i.e. the spring constants are identical for all directions of the displacement. In case the trapping potential becomes anisotropic, the restoring force is still given by $K {\mathbf}{x}_\omega$ where the spring constant $K$ has to be interpreted as a symmetric matrix. Then Eq. (\[eq:Ytrap\]) still holds, and if the principal axes of the trap includes the direction perpendicular to the wall, the matrix inversion is achieved by inverting the diagonal elements.
Let us compare Eq. (\[eq:Ytrap\]) to the well known high- and low-frequency limits. For high frequencies the trap becomes increasingly irrelevant and the response is dominated by the inertia of the particle and the displaced fluid. In the low-frequency regime the harmonic potential suppresses the admittance reflecting the fact that no static external force can induce a stationary motion of the particle in confinement.
Analytical discussion of the Velocity autocorrelation function {#Sec:analytical}
==============================================================
Here we focus on the velocity autocorrelation function (VACF) $C_\parallel(t) = \langle v_\parallel(t) v_\parallel(0) \rangle$ for the motion parallel and perpendicular to the wall, $C_\perp(t) = \langle v_\perp(t) v_\perp(0) \rangle$. The VACF are connected to the admittances of the previous Section via the fluctuation-dissipation theorem $$\label{eq:fluctuation-dissipation}
{\cal Y}(\omega) = \frac{1}{k_B T} \int_0^\infty\! {\mathrm{d}}t \, \text{e}^{\text{i} \omega t} C(t) \, ,$$ where $k_B T$ denotes the thermal energy. This relation holds for all cases, i.e. parallel and perpendicular to the wall, with and without trap. To simplify notation we have suppressed the dependence on the position ${\mathbf}{r}_0$.
Due to trapping the particle samples only a region close to the center of the trap and according to a stationary ensemble given by the Gibbs-Boltzmann measure. In particular, time-dependent position correlation functions like $\langle {\mathbf}{x}(t) \cdot {\mathbf}{x}(0) \rangle$ are well defined. In the derivation for the admittances, the position of the particle was assumed to be fixed at ${\mathbf}{r}_0$. However, since the particle fluctuates and takes excursions from the center of the trap of typical magnitude $\sqrt{k_BT/K }$, the spring constant of the trap has to be strong enough in order to render these fluctuations negligible compared to the distance to the wall $h$.
The short-time evolution of the velocity autocorrelation is inferred from the high-frequency behavior of the corresponding admittances. Obviously, the harmonic restoring forces do not affect the initial value and they are still given by $$\begin{aligned}
C_{\parallel}^{(k)}(t=0) &= \frac{k_B T}{m^*} \left( 1 - \frac{a^3}{16 h^3} \right)\, , \nonumber \\
C_{\perp}^{(k)}(t=0) &= \frac{k_B T}{m^*} \left( 1 - \frac{a^3}{8 h^3} \right) \, .\end{aligned}$$ Without a bounding surface $C^{(k)}(t=0) = k_B T /m^*$ [@Clercx:1992] is recovered. The equipartition theorem suggests that the initial value should read $k_B T / m_{\text{p}}$ for all cases. However, as discussed already by Zwanzig and Bixon [@Zwanzig:1970], there is an additional rapid decrease of the VACF if the finite compressibility of the fluid is taken into account.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$6 \pi \eta a {\cal Y}(\omega) $ leading nonanalytic term
---------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
0 $ -(-\text{i} \omega \tau_\text{f})^{1/2}$
$\perp$ $(-\text{i} \omega \tau_{\text{f}})^{3/2} \left[ - \frac{5}{9}+ \frac{\tau_{\text{p}}}{\tau_{\text{f}}} + \frac{1}{4} \sqrt{\frac{\tau_{\text{f}}}{\tau_{\text{w}}}} \left( 1 - \frac{9 \tau_{\text{p}}}{2 \tau_{\text{f}}} \right) \right] $
$\parallel$ $(-\text{i} \omega \tau_{\text{f}})^{3/2} \left[ \frac{\tau_{\text{w}}}{\tau_{\text{f}}} - \frac{5}{9} + \frac{\tau_{\text{p}}}{\tau_{\text{f}}} + \frac{1}{8} \sqrt{\frac{\tau_{\text{f}}}{\tau_{\text{w}}}} \left( 1 - \frac{9 \tau_{\text{p}}}{2 \tau_{\text{f}}} \right) \right] $
$k$ $- (-\text{i} \omega \tau_{\text{f}})^{5/2} \tau_\text{k}^2/\tau_\text{f}^2 $
$k\perp$ $ - \frac{16}{9 (9-8 \sqrt{\tau_{\text{w}}/\tau_{\text{f}}})^2} (- \text{i} \omega \tau_\text{f})^{7/2} \frac{\tau_\text{k}^2}{\tau_\text{f}^2} \times $
$ \times
\left[ 20 \frac{\tau_\text{w}}{\tau_\text{f}} \left( 1 - \frac{9\tau_\text{p}}{5\tau_\text{f}}\right) - 9 \sqrt{ \frac{\tau_\text{w}}{\tau_\text{f}}}
\left( 1 - \frac{9\tau_\text{p}}{2\tau_\text{f}}\right)
\right]
$
$k\parallel$ $ - \frac{32}{9 (9-16 \sqrt{\tau_{\text{w}}/\tau_{\text{f}}})^2} (- \text{i} \omega \tau_\text{f})^{7/2} \frac{\tau_\text{k}^2}{\tau_\text{f}^2} \times $
$ \times
\left[ -72 \frac{\tau_\text{w}^2}{\tau_\text{f}^2} + 40 \frac{\tau_\text{w}}{\tau_\text{f}} \left( 1 - \frac{9\tau_\text{p}}{5\tau_\text{f}}\right) - 9 \sqrt{ \frac{\tau_\text{w}}{\tau_\text{f}}}
\left( 1 - \frac{9\tau_\text{p}}{2\tau_\text{f}}\right)
\right]
$
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Leading non-analytic term in the low-frequency expansion of the dimensionless admittances $6\pi \eta a {\cal Y}(\omega)$ for the free case (0), perpendicular ($\perp$) and parallel ($\parallel$) and again in the presence of the trap (k, k$\perp$, k$\parallel$)[]{data-label="tab:admittances"}
The low-frequency expansion of the admittance exhibits non-integer powers, which by tauberian theorems [@Feller:Probability] correspond to algebraic long-time decays in the VACF. The leading long-time behavior of the normalized velocity autocorrelation, $C(t\to\infty)/C(0) \simeq (\tau/t)^{\alpha}$, can be inferred from the leading non-analytic contribution in the admittance $Y(\omega \to 0) \simeq C(0) \tau (-\text{i} \omega \tau )^{\alpha-1} \Gamma(1-\alpha)/k_B T$. The physical origin of the long-living correlations lies in the long-range fluid flow that is generated by momentum conservation in the Navier-Stokes equation. At low Reynolds number, transverse momentum can only be transported away by vortex diffusion, hence there is a time-dependent growing length scale $R(t) \sim (\eta t/ \rho_\text{f} )^{1/2}$ characterizing how far momentum has penetrated into the fluid. In the simple case of unconfined free motion, the particle’s initial momentum is then shared with a fluid volume $\sim R(t)^3$ leading to a long-time behavior of $C(t)/C(0) \sim t^{-3/2}$. For the free motion close to a bounding surface, the leading term $ \sim t^{-3/2}$ is canceled and a more rapid algebraic decay $t^{-5/2}$[@Felderhof:2005] is expected. Interestingly, the same algebraic decay occurs also in the disordered Lorentz gas [@vanLeeuwen:1967; @Ernst:1971b; @Hoefling:2007], a simple model for transport in porous media. There the long-time memory arises since the particle remembers the presence of an obstructing wall for arbitrarily long time. Confining the motion of a colloidal particle in bulk by a harmonic potential leads to a more rapid decorrelation according to $t^{-7/2}$ [@Clercx:1992]. In the case of a trapped particle close to a wall, a series expansion ${\cal Y}_\parallel^{(k)}(\omega), {\cal Y}_\perp^{(k)}(\omega)$ in powers of the frequency reveals a leading non-analytic term $\omega^{7/2}$ resulting in a $\sim t^{-9/2}$ tail. To obtain this result it is sufficient to know the reaction field tensors $F_\parallel({\mathbf}{r}_0,\omega), F_\perp({\mathbf}{r}_0,\omega)$ including ${\cal O}(\omega)$ terms. Table \[tab:admittances\] summarizes the leading non-analytic behavior of the admittances including all prefactors. The corresponding long-time tails in the normalized velocity autocorrelation function are displayed in Table \[tab:tails\], where for the walls only the leading term in $\tau_\text{w}/\tau_\text{f}$ is shown.
$C(t)/C(0) $ leading algebraic decay
-------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
0 $ B \left(\frac{t}{ \tau_\text{f}} \right)^{-3/2}$
$\perp$ $ \frac{ 3}{2} B \left( \frac{\tau_{\text{p}}}{ \tau_\text{f}} -\frac{5}{9} \right)\left(\frac{t}{ \tau_\text{f}} \right)^{-5/2}$
$\parallel$ $\frac{3}{2} B \left(\frac{\tau_{\text{w}}}{ \tau_\text{f}} \right)\left(\frac{t}{ \tau_\text{f}} \right)^{-5/2}$
$k$ $ \frac{15}{4} B \left(\frac{\tau_\text{k}}{ \tau_\text{f}} \right)^2\left(\frac{t}{ \tau_\text{f}} \right)^{-7/2}$
$k\perp$ $\frac{105}{8} B \frac{\tau_\text{k}^2}{\tau_\text{f}^2} \left( \frac{\tau_\text{p}}{\tau_\text{f}} - \frac{5}{9} \right) \left(\frac{t}{ \tau_\text{f}} \right)^{-9/2} $
$k\parallel$ $\frac{105}{8} B \frac{\tau_\text{w} \tau_\text{k}^2}{\tau_\text{f}^3} \left(\frac{t}{ \tau_\text{f}} \right)^{-9/2} $
: The leading long-time behavior of the normalized velocity autocorrelation function $C(t\to \infty)/C(0)$ for the free case (0), perpendicular ($\perp$) and parallel ($\parallel$) and again in the presence of the trap (k, k$\perp$, k$\parallel$). For the cases of a bounding wall, only the leading contribution in $\tau_\text{w}/\tau_\text{f}$ is displayed. The mass ratio is encoded in the constant $ B = (9 \tau_\text{p}/\tau_\text{f} +1)/18\sqrt{\pi}$. []{data-label="tab:tails"}
For the motion perpendicular to the wall, the true long-time behavior is masked by the next-to-leading term. This has been pointed out by Felderhof [@Felderhof:2005] for the trap-free motion and here we supplement the analysis also for the trapped dynamics. In order to derive the next-to-leading term in the low-frequency expansion the reaction field tensor $F_\perp(\omega,{\mathbf}{r}_0)$ has to be evaluated including the order $(-\text{i} \omega \tau_\text{f})^{5/2}$. Then the two leading nonanalytic terms in the far-field expansion $\tau_\text{w}/\tau_\text{f} \gg 1$ read $$\begin{aligned}
\lefteqn{ 6 \pi \eta a {\cal Y}_\perp(\omega) = } \nonumber \\
&= & (-\text{i} \omega \tau_\text{f})^{3/2} \left[ \left( \frac{\tau_\text{p}}{\tau_\text{f}} - \frac{5}{9} \right)
-\frac{1}{10}\frac{\tau_\text{w}^2}{\tau_\text{f}^2} (-\text{i} \omega \tau_\text{f}) \right] \, ,\end{aligned}$$ without harmonic restoring force, and $$\begin{aligned}
\lefteqn{6 \pi \eta a {\cal Y}_\perp^{(k)}(\omega) = } \nonumber \\
&=& (-\text{i} \omega \tau_\text{f})^{7/2} \frac{\tau_\text{k}^2}{\tau_\text{f}^2} \left[ \left( \frac{\tau_\text{p}}{\tau_\text{f}} - \frac{5}{9} \right)
-\frac{1}{10}\frac{\tau_\text{w}^2}{\tau_\text{f}^2} (-\text{i} \omega \tau_\text{f}) \right] \, ,\end{aligned}$$ including the trap. The second term becomes negligible with respect to the first one only at frequencies below $\omega \ll \tau_\text{f}/\tau_\text{w}^2$. For the corresponding time correlation functions this implies a long-time behavior according to $$\begin{aligned}
\label{eq:Cperp_t}
\frac{C_\perp(t\to\infty)}{C_\perp(0)} &=& \frac{3}{2} B \Big[ \left( \frac{\tau_\text{p}}{\tau_\text{f}}
- \frac{5}{9} \right) \left(\frac{t}{ \tau_\text{f}} \right)^{-5/2} \nonumber \\
& &+ \frac{\tau_\text{w}^2}{4 \tau_\text{f}^2} \left(\frac{t}{ \tau_\text{f}} \right)^{-7/2} \Big] \, ,\end{aligned}$$
$$\begin{aligned}
\frac{C^{(k)}_\perp(t\to \infty)}{C^{(k)}_\perp(0)} &=& \frac{105}{8} B \frac{\tau_\text{k}^2}{\tau_\text{f}^2}
\Big[ \left( \frac{\tau_\text{p}}{\tau_\text{f}} - \frac{5}{9} \right) \left(\frac{t}{ \tau_\text{f}} \right)^{-9/2} \nonumber \\
& &+
\frac{9\tau_\text{w}^2}{20\tau_\text{f}^2} \left(\frac{t}{ \tau_\text{f}} \right)^{-11/2}
\Big] \, .\end{aligned}$$
The amplitude of the leading term depends on the ratio of the mass of the particle to the displaced mass of the fluid or equivalently on the ratio $\tau_\text{p}/\tau_\text{f} = 2 m_\text{p}/9 m_\text{f}$. For typical experimental conditions, e.g. a silica sphere in water, the amplitude is negative and one should expect the correlation function to approach zero from below. The presence of the factor $\tau_\text{w}/\tau_\text{f}$ renders the amplitude of the subleading term to a large quantity. Balancing both terms yields an estimate $\tau_\text{w}^2/\tau_\text{f}$ for the time one has to wait in order to observe the true long-time behavior. Hence the perpendicular motion without trap exhibits a zero crossing on a time scale much longer than the na[ï]{}ve guess $\tau_\text{w}$. We shall see in the next Section that even a weak trap has strong implications for this intermediate time behavior.
Numerical Results {#Sec:numerical}
=================
Velocity autocorrelation function
---------------------------------
The inverse Fourier transformation cannot be calculated analytically, since the admittance tensors are not simple elementary functions. A numerical Fourier transformation can easily be performed, and since the time-correlation functions are real and even in time, a Fourier-Cosine transformation is sufficient $$\begin{aligned}
C(t) &=& \frac{2 k_B T}{\pi} \int_0^\infty {\mathrm{d}}\omega \, \cos(\omega t) \text{Re }[ {\cal Y}(\omega)]\, .\end{aligned}$$ Since the VACF exhibits long-time power-law behavior the admittance has to be sampled over many decades in frequency. A conventional Fast Fourier Transform (FFT) is rather ill-suited to achieve this and would also calculate $C(t)$ on an equidistant time grid not convenient for our purposes. Hence, we apply a simple modified Filon algorithm [@Tuck:1967] with typically $10^3$ frequencies per decade covering 30 decades to calculate $C(t)$ on a logarithmically equidistant grid containing 100 data points per decade. We employed Mathematica to evaluate the various special functions appearing in the reaction field tensors and spliced them together with a high-order series expansion at low and high frequencies. We have checked that our numerical data reproduce the predicted tails over several orders of magnitude, with the exception of $C_\perp^{(k)}(t)$, where the next-to-leading tail dominates the figures.
![(Color online) Double-logarithmic representation of the normalized VACF for the motion parallel (a) and perpendicular (b) to the wall. The parameter $\tau_\text{p}/\tau_\text{f} = 0.5$ corresponds approximately to a silica bead in water. Full lines correspond to weakly trapped particles $\tau_\text{f}/\tau_\text{k} = 10^{-5}$, whereas the dotted lines are without trapping ($\tau_\text{f}/\tau_\text{k} = 0$). The wall is gradually approached, $\tau_\text{w}/\tau_\text{f}= 16,8,4,2$, corresponding to $h/a = 4, 2.83,2, 1.41$, and the initial decay shifts to the left. The straight lines are guides to the eye for the $t^{-5/2}$ power law. []{data-label="fig:Cwall"}](Cxx_pdf "fig:"){width="45.00000%"} ![(Color online) Double-logarithmic representation of the normalized VACF for the motion parallel (a) and perpendicular (b) to the wall. The parameter $\tau_\text{p}/\tau_\text{f} = 0.5$ corresponds approximately to a silica bead in water. Full lines correspond to weakly trapped particles $\tau_\text{f}/\tau_\text{k} = 10^{-5}$, whereas the dotted lines are without trapping ($\tau_\text{f}/\tau_\text{k} = 0$). The wall is gradually approached, $\tau_\text{w}/\tau_\text{f}= 16,8,4,2$, corresponding to $h/a = 4, 2.83,2, 1.41$, and the initial decay shifts to the left. The straight lines are guides to the eye for the $t^{-5/2}$ power law. []{data-label="fig:Cwall"}](Czz_pdf "fig:"){width="45.00000%"}
The velocity autocorrelation for the motion parallel to a wall is displayed in Fig. \[fig:Cwall\]a for a colloid gradually approaching the bounding surface. Without trap the correlation functions remain positive for all times. The long-time behavior exhibits the $t^{-5/2}$ decay with an amplitude that depends sensitively on the distance to the wall. For the weak harmonic restoring force under consideration here, the VACF follows the free behavior up to intermediate times; at longer times the VACF is strongly influenced by the trap. In particular, the VACF exhibits two zeros as is the case also without the bounding wall. Note that the first zero is by orders of magnitude earlier than the characteristic time scale of the trap $\tau_k$, however much later than the Langevin momentum relaxation time $\tau_p$. Between the two zeros a negative flat plateau emerges that extends to time scales longer than $\tau_\text{k}$. The very late decay is governed by the $t^{-9/2}$ tail of Table \[tab:tails\], however its onset exceeds the range of the figure.
The VACF in the direction perpendicular to the wall is exhibited in Fig. \[fig:Cwall\]b. For a typical experimental particle with momentum relaxation time $\tau_\text{p} = 0.5 \tau_\text{f}$ and without confining potential the curves exhibit a single zero. In this case the amplitude of the leading long-time behavior $t^{-5/2}$ becomes negative and the next-to-leading order $t^{-7/2}$ dominates at an intermediate time interval, see Eq. (\[eq:Cperp\_t\]). The correlation functions without trapping thus approach zero from the negative side by an algebraic decay that is fairly insensitive to the distance from the wall. The zero on the other hand shifts rapidly to shorter times as the wall is approached. From Eq. (\[eq:Cperp\_t\]) one infers an asymptotic scaling behavior $\sim\tau_\text{w}^2/\tau_\text{f}$ for the zero far away from the wall. For the two curves corresponding to the distant particle, $\tau_\text{w}/\tau_\text{f} = 8, 16$, the $t^{-7/2}$ decay is visible for intermediate times close to the zero, as has been reported earlier [@Felderhof:2005]. For a weak trap $\tau_\text{f}/\tau_\text{k} =10^{-5}$ the curves follow the ones of the unconstrained motion down to a signal of $10^{-5}$ and start to deviate strongly at later times. Again a flat negative plateau characteristic of the weak restoring force is attained. The decay from this intermediate plateau occurs at times large compared to $\tau_\text{k}$ followed by a rapid algebraic decay $t^{-9/2}$ without passing through another zero (not shown). For different parameters $\tau_\text{k}$, $\tau_\text{w}$ one finds also a scenario with three zero crossing.
To study the hydrodynamic memory effects for the colloidal motion close to the wall, we choose a trapping potential that is as weak as possible. Then the time scale $\tau_\text{k} = 6\pi \eta a/k$ is much larger than the remaining time scales of the problem $\tau_\text{p}, \tau_\text{f}$, and $\tau_\text{w}$ and one is tempted to argue that the trap is irrelevant in the regime of interest. However, as is discussed in Fig. \[fig:Cwall\] the trap has significant impact on the velocity correlation functions even if the characteristic time scales differ by orders of magnitude. The reason for such a behavior is twofold: First, the hydrodynamic memory leads to a scale-free power-law long-time decay and the parameters $\tau_\text{p}, \tau_\text{f}$, and $\tau_\text{w}$ determine merely the amplitude of the algebraic behavior rather than a characteristic decay time. Second, for the point particle limit to apply accurately, $h/a \gtrsim 3$, and the VACFs become small at times $t\gtrsim \tau_\text{w}$ where the tail $t^{-5/2}$ is expected to set in. Thus already a weak trap has a strong influence on the signal where the interesting hydrodynamic memory effect dominates the VACFs.
Power spectral density
----------------------
An alternative way to investigate the interplay of fluid inertia and a trap is to focus on the power spectral density, an approach that has been pursued by Berg-Sørensen *et al.* [@Berg-Sorensen:2004; @Berg-Sorensen:2005]. There the fluctuating position $x(t)$ of the bead in a finite time interval $[-\text{T}/2,\text{T}/2]$ is decomposed into Fourier modes $x_{\text{T}}(\omega) = \int_{-\text{T}/2}^{\text{T}/2} x(t) \exp(\text{i} \omega t){\mathrm{d}}t$ where the angular frequencies $\omega$ are integer multiples of $2\pi/\text{T}$. For long observation times, $\text{T}\to \infty$ the power spectrum $S(\omega)=\left\langle |x_\text{T}(\omega)|^2 \right\rangle/\text{T} $ becomes a quasi-continuous function for frequencies $\omega \gg 2\pi/T$. Since the corresponding velocities fulfill $v_\text{T}(\omega) = -\text{i} \omega x_\text{T}(\omega)$ up to irrelevant boundary terms, the power spectrum can be also expressed as $\omega^2 S(\omega) = \left\langle |v_\text{T}(\omega)|^2 \right\rangle/\text{T} $. Then with the help of the Wiener-Khinchin theorem $ \omega^2 S(\omega) = \int \langle v(t) v(0) \rangle \exp( \text{i} \omega t){\mathrm{d}}t$ and the fluctuation-dissipation theorem Eq. (\[eq:fluctuation-dissipation\]), the power spectral density can be obtained from the admittance $$\label{eq:Srel}
S(\omega)= \frac{2 k_B T \, \mbox{\sf Re} {\left [}{\cal Y}^{(k)}(\omega) {\right ]}}{\omega^2} \, .$$ Since the trap modifies the admittance according to Eq. (\[eq:Ytrap\]), we find for the power spectrum $$S(\omega)= \frac{2 k_B T\mbox{\sf Re} {\left [}{\cal Y}(\omega)^{-1} {\right ]}}{ {\left (}\omega \, \mbox{\sf Re} {\left [}{\cal Y}(\omega)^{-1} {\right ]}{\right )}^2+ {\left (}\omega \, \mbox{\sf Im} {\left [}{\cal Y}(\omega)^{-1} {\right ]}+ K {\right )}^2} \, .$$ This result is again valid for all cases, i.e. in bulk, parallel and perpendicular to the bounding wall. For the zero-frequency limit in the case of a limiting wall we find $$S_{||}(0) = \frac{2k_B T}{\mu_{||} K^2} \, , \qquad
S_{\perp}(0) = \frac{2k_B T}{\mu_{\perp} K^2} \, , \label{eq:Stw}$$ using the zero-frequency limit of the admittance at the wall. Without trap, the power spectrum $S(\omega)$ would diverge for $\omega \to 0$ as can be seen from Eq. (\[eq:Srel\]), reflecting the fact that the particle can take excursions without bounds. The $1/K^2$ dependence on the trap strength arises from two aspects. First, the motion of the particle is harmonically confined leading to equilibrium fluctuations $\langle x^2 \rangle = k_B T/ K$. Second, the time scale to reach equilibrium may be estimated by balancing the restoring force yielding $1/\mu K$. For the bulk motion this is identified with $\tau_k = 6\pi \eta a/K=1/\mu_0 K$, whereas close to the wall the reduction of the mobilities $\mu_\parallel, \mu_\perp$ due to the obstruction of the hydrodynamic flow has to be taken into account.
![(Color online) Semi-logarithmic representation of the power spectral density for a weakly trapped Brownian particle $\tau_\text{f}/\tau_\text{k}= 10^{-5} $ close to a limiting wall for the ratio $\tau_\text{p}/\tau_\text{f}=0.5$ ((a) lateral direction and (b) vertical direction). The distances to the wall are the same as in Fig. \[fig:Cwall\]. The normalization is chosen as the zero-frequency limit of the power spectrum in bulk $S_0(\omega=0) = 2 k_B T/\mu_0 K^2$. The increase of the initial value is due the wall according to Eq. (\[eq:Stw\]). []{data-label="fig:Swall_tauk"}](psd_xx_pdf "fig:"){width="45.00000%"} ![(Color online) Semi-logarithmic representation of the power spectral density for a weakly trapped Brownian particle $\tau_\text{f}/\tau_\text{k}= 10^{-5} $ close to a limiting wall for the ratio $\tau_\text{p}/\tau_\text{f}=0.5$ ((a) lateral direction and (b) vertical direction). The distances to the wall are the same as in Fig. \[fig:Cwall\]. The normalization is chosen as the zero-frequency limit of the power spectrum in bulk $S_0(\omega=0) = 2 k_B T/\mu_0 K^2$. The increase of the initial value is due the wall according to Eq. (\[eq:Stw\]). []{data-label="fig:Swall_tauk"}](psd_zz_pdf "fig:"){width="45.00000%"}
At the scale of the trap relaxation rate $1/\tau_\text{k}$, the main feature of the power spectral density as displayed in Fig \[fig:Swall\_tauk\] is the saturation at a height given by Eq. (\[eq:Stw\]) at low frequencies. A rapid decrease of the power spectrum is manifest at higher frequencies. Ignoring inertial effects of the fluid and the particle, the power spectra assume a Lorentzian shape $$\begin{aligned}
\label{eq:lorentzian}
S^{(L)}(\omega) = \frac{2 k_B T}{\mu K^2} \frac{1}{1+ (\omega/\mu K)^2} \, ,\end{aligned}$$ where $\mu = \mu_0, \mu_\parallel, \mu_\perp$ is the corresponding mobility. For the weak traps employed here, this gives an accurate representation of the power spectral densities for dimensionless frequencies $\omega\tau_\text{k}$ of order unity.
![(Color online) Power spectral density with respect to a Lorentzian, Eq. (\[eq:lorentzian\]), in the frequency regime where fluid inertia plays a role for the parallel (a) and perpendicular motion (b). The parameters correspond to the ones of Fig. \[fig:Swall\_tauk\]. Also included is the harmonic oscillator prediction according to Ornstein and Uhlenbeck. []{data-label="fig:Swall_tauw"}](psd_plot_Sxx_pdf "fig:"){width="45.00000%"} ![(Color online) Power spectral density with respect to a Lorentzian, Eq. (\[eq:lorentzian\]), in the frequency regime where fluid inertia plays a role for the parallel (a) and perpendicular motion (b). The parameters correspond to the ones of Fig. \[fig:Swall\_tauk\]. Also included is the harmonic oscillator prediction according to Ornstein and Uhlenbeck. []{data-label="fig:Swall_tauw"}](psd_plot_Szz_pdf "fig:"){width="45.00000%"}
To highlight the deviations from the simple Lorentzian, we display the ratio $S(\omega)/S^{(L)}(\omega)$ for the parallel and perpendicular motion in Fig. \[fig:Swall\_tauw\]. The differences arise at the frequency scale $1/\tau_\text{f}$, where the motion of the hydrodynamic vortex sets in. At higher frequencies the full power spectrum decays more rapidly than a Lorentzian due the inertia of the particle and the fluid. Interestingly, the deviations are strongest in bulk and they fade out only slowly as the frequency is decreased. The wall suppresses these inertial effects consistent with the notion that the rigid interface carries away part of the particle’s initial momentum. Hence the obstruction of the vortex pattern, manifest in the time-dependent VACF as reduction of a $t^{-3/2}$ to $t^{-5/2}$-tail, is also visible in the power spectral density by a faster approach to the Lorentzian shape for low frequencies $\omega\tau_\text{w} \lesssim 1$. Note that for the perpendicular motion the power spectral density becomes larger than a Lorentzian very close to the wall. This phenomenon is related to the change of sign of the long-time anomaly of the $t^{-5/2}$ in Eq. (\[eq:Cperp\_t\]). The overshoot disappears if denser colloidal particles are used, i.e. $\tau_\text{p}/\tau_\text{f}$ is increased. However, the vanishing of the overshoot does not coincide with the sign change of the power-law tail at $\tau_\text{p}/\tau_\text{f} = 5/9$ but occurs at larger ratios of approximately 1.5 for the parameters used here. For comparison we have also included in both panels the harmonic oscillator prediction for the bulk motion following Ornstein-Uhlenbeck $$S^{(OU)}(\omega) = \frac{2 k_B T}{\mu_0 K^2} \frac{1}{(1-\omega^2 \tau_\text{p} \tau_\text{k})^2 + (\omega \tau_\text{k})^2} \, ,$$ neglecting both, the added mass due to the displaced fluid, as well as the vortex motion. As can be inferred from Fig. \[fig:Swall\_tauw\], the wall suppresses the hydrodynamic memory, and for the parallel motion the power spectral densities are typically in between the bulk behavior where hydrodynamics is included and the Ornstein-Uhlenbeck shape.
For the weak trapping regime under consideration, the time scale $\tau_\text{k}$ is much larger than the other ones. Then one may assume that the processes of the relaxation of the momentum to the surrounding fluid and the relaxation of the position in the trap are decoupled, which suggests to approximate the power spectral density as $$\label{eq:lorentz_approx}
S(\omega) \approx S^{(L)}(\omega) \frac{ \mbox{\sf Re} {\left [}{\cal Y}(\omega) {\right ]}}{\mu} \, ,$$ where again the appropriate mobilities $\mu = \mu_0, \mu_\parallel, \mu_\perp$ and similarly for the admittances ${\cal Y}(\omega)$ have to be inserted for the different cases. We have checked that for the parameters used here this constitutes an excellent approximation for all frequencies. Consequently, Fig. \[fig:Swall\_tauw\] essentially displays the real part of the admittances for different distances to the wall. The reason that the curves superimpose is that the zero-frequency limit serves as a background that overlaps the interesting non-analytic behavior discussed in Sec. \[Sec:analytical\]. Consequently one can not use Eq. (\[eq:lorentz\_approx\]) as input in the numerical Fourier transform since it does not result in the correct long-time behavior. Concluding, although the power spectral density allows for simple approximations on different frequency scales, their deviations encode subtle correlations that are manifest and more directly accessible in the time-dependent VACF.
Comparison to Experiments {#Sec:experimental}
=========================
![\[fig:scheme\] (Color online) (a) 3D lateral view of the experiment; spheres are drawn to scale. A silica particle of radius $a=$1.5m trapped by the laser focus is placed next to the surface of a significantly larger silica sphere. This 100m sphere is immobilized between the two coverglass surfaces of the sample chamber. (b) Optical image of the probing particle’s position relative to the wall created by the big sphere. The 3m probing particle was placed at a distance $h=$11.5m away from the 100m sphere’s surface and gradually approached. The velocity correlation functions, as well as the diffusion coefficients for the motion parallel and perpendicular to the wall are measured. ](Figure_artistic){width="45.00000%"}
Recently, we have performed experimental tests for a small silica sphere trapped by a laser focus in the vicinity of a surface [@Jeney:2008]. The trajectory of the particle is measured interferometrically [@Gittes:1998] with a spatial resolution in the subnanometer range [@Lukic:2005]. We employ an infrared ($\lambda=1064$nm) diode-pumped Nd:YAG laser (IRCL-500-1064-S, CrystaLaser, USA) with a cw output power of 500 mW. The beam is first expanded 20$\times$ and then focused using a 63$\times$ water-immersion objective lens of a numerical aperture NA=$1.2$ resulting in a stable optical trap for the colloid. An InGaAs quadrant photodiode (G6849, Hamamatsu Photonics, Japan) is placed in the back focal plane of the condenser lens recording the modulation of the optical power due to the displacement of the particle near the beam focus. The photodiode signal is amplified and digitized using a data acquisition card with a dynamic range of 12 bits. The detected positions contain $N =10^7$ points separated by 2s, which corresponds to a sampling rate of $f_s = 500$kHz and a recording time of $t_s =20$s.
![\[fig:LTT-wall\] (Color online) Log-log plot of both normalized VACF, $C_\parallel(t)/C_\parallel(0)$ and $C_\perp(t)/C_\perp(0)$ for a sphere ($a=$1.5m, $\tau_\text{p} =1$s, $\tau_\text{f} = 2.25$s) trapped in a weak optical potential ($k$ $\approx$2N/m, $\tau_\text{k}=14$ms). The increasingly anisotropic VACF is measured at three distances from the wall ($h$= 9.8, 6.8 and 4.8m, corresponding to $\tau_\text{w}$ = 96, 46, 23s, respectively). Positive correlations are represented by full symbols, negative ones by open symbols. The characteristic power-laws are represented by thick lines as guide to the eye. ](exp_wall1_pdf "fig:"){width="45.00000%"} ![\[fig:LTT-wall\] (Color online) Log-log plot of both normalized VACF, $C_\parallel(t)/C_\parallel(0)$ and $C_\perp(t)/C_\perp(0)$ for a sphere ($a=$1.5m, $\tau_\text{p} =1$s, $\tau_\text{f} = 2.25$s) trapped in a weak optical potential ($k$ $\approx$2N/m, $\tau_\text{k}=14$ms). The increasingly anisotropic VACF is measured at three distances from the wall ($h$= 9.8, 6.8 and 4.8m, corresponding to $\tau_\text{w}$ = 96, 46, 23s, respectively). Positive correlations are represented by full symbols, negative ones by open symbols. The characteristic power-laws are represented by thick lines as guide to the eye. ](exp_wall2_pdf "fig:"){width="45.00000%"} ![\[fig:LTT-wall\] (Color online) Log-log plot of both normalized VACF, $C_\parallel(t)/C_\parallel(0)$ and $C_\perp(t)/C_\perp(0)$ for a sphere ($a=$1.5m, $\tau_\text{p} =1$s, $\tau_\text{f} = 2.25$s) trapped in a weak optical potential ($k$ $\approx$2N/m, $\tau_\text{k}=14$ms). The increasingly anisotropic VACF is measured at three distances from the wall ($h$= 9.8, 6.8 and 4.8m, corresponding to $\tau_\text{w}$ = 96, 46, 23s, respectively). Positive correlations are represented by full symbols, negative ones by open symbols. The characteristic power-laws are represented by thick lines as guide to the eye. ](exp_wall3_pdf "fig:"){width="45.00000%"}
The Brownian particle is a silica sphere of radius $a=1.5$m and mass density $\rho_{\text{p}}=1.96$g/cm$^3$ immersed in water ($\rho_{\text{f}}=1$g/cm$^3$, $\eta=10^{-3}$Pa$\cdot$s) at ambient temperature. The characteristic time scale corresponding to the particle’s inertia $\tau_\text{p} = m_\text{p}/ 6 \pi \eta a = 2 a^2\rho_\text{p} /9\eta = 1$s is about half the one of the fluid inertia $\tau_\text{f} = a^2 \rho_\text{f}/\eta =2.25$s. Using optical tweezers, we approach the particle incrementally towards a sphere of diameter 100m, much larger than the size of our particle. Then sufficiently close to the large sphere the curvature can be ignored, and the particle undergoes Brownian motion close to a planar wall around the trap’s center, see Fig. \[fig:scheme\]. The large sphere is immobilized since it is in close contact with the two coverslides of our fluid chamber (size $\approx 2$cm $\times$0.5cm and thickness $\approx100$m). In this set-up we have equal sensitivity for the motion parallel and perpendicular to the surface under consideration, since both directions are perpendicular to the axis of the laser beam. The sample is mounted onto a piezo-stage, and the 100m sphere can be positioned at a distance $h$ relative to the trapped particle by moving the piezo-stage in all three dimensions with a precision of $\approx1$nm.
To study the hydrodynamic memory effects induced by the presence of a wall rather than the motion due to the trap confinement, we have applied the weakest trapping force possible. Yet, the optical trap should still be strong enough not to loose the particle from the laser focus during the experiment. Hence, we have optimized the trapping strength in order to suppress the effects of the trap on the VACF without loosing too much sensitivity at the detector. A series of experiments revealed that $k$ $\approx$ 2N/m fulfills all these requirements. Note that the corresponding time scale $\tau_k=14$ ms exceeds the parameters of the bulk motion $\tau_\text{p},
\tau_\text{f}$ by four orders of magnitude and the one of the wall $\tau_\text{w}$ by more than two orders of magnitude even for the farthest distance studied. However, this weak trapping force has still significant influence on the VACF at the time scale where the hydrodynamic memory becomes apparent.
We have recorded the particle’s position $r_\parallel(t_n), r_\perp(t_n)$ at equidistant instants of time $t_n := n\Delta t$, $\Delta t := 1/f_s$, $n\in \mathbb{N}_0$, and derive coarse-grained velocities as $v_\parallel(t_n) := [r_\parallel(t_{n}+ \Delta t)- r_\parallel(t_n)]/\Delta t$. The velocity autocorrelation functions are then evaluated as time-moving averages $C_\parallel(t) = N^{-1} \sum_{n=1}^N \Delta v_\parallel(t_n+t) \Delta v_\parallel(t_n)$, and similarly for the motion perpendicular to the wall. The normalized VACFs $C_\parallel(t)/C_\parallel(0)$ and $C_\perp(t)/C_\perp(0)$ are displayed in Fig. \[fig:LTT-wall\] for different distances $h$ from the wall. For large separation $h=9.8$m both VACFs coincide within our error bars down to where the signal reaches the level of 1%. This regime is clearly dominated by bulk behavior and, in particular, the well-known algebraic decay $t^{-3/2}$ characteristic for the unconstrained vortex diffusion is recovered. At the time scale $\tau_\text{w} = \rho_\text{f} h^2/\eta$ the vortex generated by the thermal fluctuations of the Brownian particle reaches the wall and the correlation functions split as the motion becomes anisotropic. The wall leads to a more rapid decay and for the parallel motion a power law $t^{-5/2}$ enters the observation window . The signal of the perpendicular motion is anticorrelated at these time scales. Later times lead to only weak signals that we cannot resolve within our noise level of $10^{-4}$. For the VACFs the trap manifests itself in two zeros in the parallel motion, the second of which is outside of our observation window. The perpendicular motion exhibits a first zero which is shifted to earlier times by the presence of the optical trap — a second zero induced by the harmonic restoring forces remains unobservable. Since for silica in water $\tau_\text{p}/\tau_\text{f} =0.44 < 5/9$, the theory ignoring the trap also predicts an anticorrelated signal in the long-time behavior for the perpendicular motion. However, the negative signal we observe is dominated by the restoring force of the optical trap. Approaching the wall to $h=6.8$m and 4.8m, the characteristic time $\tau_\text{w} =46$s, respectively 23s, decreases and the splitting into parallel and perpendicular motion shifts to earlier times. For the parameters chosen, the bulk behavior is dominated by the $t^{-3/2}$ tail, and the splitting sets in at higher values of the correlation functions rendering it easier to observe. The crossover in the parallel VACF from the bulk dominated behavior $t^{-3/2}$ to the wall dominated algebraic decay $t^{-5/2}$ becomes more and more pronounced. The zero due to the trap is dragged to shorter times since the surface constrains the colloid’s motion increasingly. The perpendicular component $C_\perp(t)$ decreases more rapidly and practically enters our noise floor at the expected zero induced by the trap. The asymptotic expansion for the long-time behavior of $C_\perp(t)$ suggests that a power law of $t^{-7/2}$ with positive amplitude should be present at intermediate times. Such a behavior may be inferred from the data for a narrow regime of times, but the presence of the trap suppresses its amplitude by a factor of two.
Next, we define the time-dependent diffusion coefficients $$\begin{aligned}
D_\parallel(t) = \int_0^t C_\parallel(t'){\mathrm{d}}t'\, ,\end{aligned}$$ and similarly for the motion perpendicular to the wall. From the recorded time series, $D_\parallel(t) = N^{-1} \sum_{n=1}^N [ r_\parallel(t_n+t)
-r_\parallel(t_n)] v_\parallel(t_n)$ is directly evaluated. We have checked that this gives the same result as integrating the VACF. The diffusion coefficients $D_\parallel(t), D_\perp(t)$ corresponding to the distances $h =9.8$, 6.8 and 4.8m of Fig. \[fig:LTT-wall\] as well as the motion in bulk ($h=37.8$m) are displayed in Fig. \[fig:Diffusion\]. The most prominent feature is a plateau extending to the time $\tau_\text{k}$ where the trapping becomes effective. The height of each plateau approaches the diffusion coefficients parallel and perpendicular to the wall that are obtained by the Einstein-Smoluchowski relation from the mobilities, $D_\parallel = k_B T \mu_\parallel$, $D_\perp = k_B T \mu_\perp$. The reduction of the zero-frequency mobilities due to the wall in Eq. (\[eq:lorentz\]) ignores higher order correction terms in $a/h$ which can be calculated exactly [@Happel:LowReynolds]. For the distances studied here $h/a>3$ the point particle limit is accurate within 2% and the correction terms can be safely ignored. Since the VACF exhibits long-time tails, the approach to the plateau is slow. Ignoring the trap, we expect asymptotically for $t\to \infty$ in leading order in $a/h$ $$\begin{aligned}
D_\parallel(t) = D \left( 1 - \frac{9a}{8h} \right) - D \frac{ \tau_\text{w}}{\tau_\text{f} \sqrt{4\pi}} \left( \frac{t}{\tau_\text{f}} \right)^{-3/2} \, ,\end{aligned}$$ where $D= k_B T /6 \pi \eta a$ denotes the bulk diffusion constant. Even for the weak trapping used in the experiment the maximum in the time-dependent diffusion coefficient deviates from the asymptotic value by a few percent. The maximum corresponds to the zero crossing in the VACF, which for the parallel motion is entirely due to the trap. At longer times the diffusion coefficient $D_\parallel(t)$ decreases and eventually reaches zero. The diffusion constant is given by the Green-Kubo relation $D_\parallel = \int_0^\infty C_\parallel(t') {\mathrm{d}}t'$ or the long-time limit of the time-dependent diffusion coefficient. Hence, at long times no diffusion occurs, which is consistent with the observation that the harmonic restoring forces localize the particle at sufficiently long times. Then the data at long times are sensitive to the precise value of the trap time $\tau_\text{k}$, and we have used this observation to optimize the fit for the trap stiffness. Since the laser power is held at a constant value for all distances, the trapping time should be identical for all data. Figure \[fig:LTT-wall\] shows that the terminal decay to zero can indeed be fitted by a single $\tau_\text{k}$, however for the closest distance $h=4.8$m the trap appears to be slightly weaker. We attribute this observation to deformations of the laser field at such close distances to the large sphere. Nevertheless, the overall agreement is excellent and validates the theoretical approach on a quantitative level of a few percent. In particular, we conclude that even at the closest distance the point-particle limit is accurate not only for the stationary diffusion coefficient but also for the time-dependent motion.
![\[fig:Diffusion\] (Color online) Normalized time-dependent diffusion coefficients, $D_\parallel(t)/D$ and $D_\perp(t)/D$, with the bulk diffusion constant $D=k_B T/6 \pi \eta a$ for the direction parallel and perpendicular to the wall at the same distances $h$ as in Fig. \[fig:LTT-wall\]. The experimental data are represented by symbols and the full lines correspond to the theoretical fits. ](diffusion_pdf){width="45.00000%"}
In the experimental set-up the planar wall has been realized by inserting a much larger spherical particle in the fluid chamber. Although theoretical models have not addressed the Brownian motion close to a curved interface, some back-on-the envelope calculations may be made to confirm the validity of the approximation. First, the zero-frequency mobility or steady diffusion constant will be modified and the suppression of mobility predicted by Lorentz, Eq. (\[eq:lorentz\]), should include also terms $h/R$, where $R$ denotes the radius of the large immobilized sphere. For the experiments far away from the wall, $h=9.8$m, the suppression is less than 20%, and $h/R \sim 1/10$ suggests that the curvature effect adds another 2%. Close to the wall, $h =4.8$m, the suppression is already $35\%$, yet for the curvature an additional 1% should be anticipated. One may also ask at what time scale $\tau_\text{curv}$ the effects of curvature should be manifest in the correlation functions. Assuming that curvature is relevant if the fictitious flat wall is separated from the immobilized sphere by more than $h$, simple geometric considerations lead to $\tau_\text{curv} = 2 \rho_\text{f} R h/\eta$. In our experiments these times are at least a factor of 20 larger than the corresponding $\tau_\text{w}$ and the signal is indistinguishable from noise.
Conclusion
==========
Optical trapping interferometry allows monitoring the motion of a single colloidal particle on time scales where its momentum plays an important role. Due to the hydrodynamic memory of the fluid friction, the velocity autocorrelation function (VACF) exhibits an algebraic long-time decay rather than an exponential relaxation. If the colloid is placed in close vicinity to a bounding wall the vortex diffusion is significantly hindered leading to a more rapid decay of the VACF. Although weak optical trapping is employed, where the trap relaxation time scale exceeds the ones of the fluid by orders of magnitude, the influence of the confining harmonic potential becomes significant. At times where the algebraic decay due to the wall should be visible the trap introduces additional features in the signal which have to be disentangled carefully. To analyze experimental data the full frequency dependence of the admittance including the trap has to be used to obtain a consistent interpretation.
Our study should be useful to analyze experiments of colloidal particles close to an interface and/or in visco-elastic fluids in a straightforward way. Employing the time-domain rather than the frequency-domain should make the interpretation of data on visco-elastic solutions, e.g. conducted by Atakhorrami *et al* [@Atakhorrami:2006], more transparent and significant, and improve the trapping force calibration method based on the power spectral density suggested by Fischer and Berg-Sørensen [@Fischer:2007]. In visco-elastic media the viscosity becomes itself frequency-dependent which is readily incorporated in the admittances. The velocity autocorrelation functions can be calculated numerically once the frequency dependence is known. Provided the experimental data exhibit little noise one may determine the frequency-dependent elastic moduli by adjusting the numerically generated curves to the experiment. Similarly, the admittances change as the properties of the interface change by adding surfactants, by capillary fluctuations, surface viscosity, etc. If the admittances for each case are known one may use the fluctuating bead as a probe for the local environment and determine material properties on scales ranging from nano- to micrometers.
We thank F. H[ö]{}fling for discussions and help to implement the Filon algorithm. SJ thanks Ecole Polytechnique Fédérale de Lausanne (EPFL) for funding the experimental equipment. This work is supported by the Swiss National Foundation under grant no. 200021-113529. TF gratefully acknowledges support by the Nanosystems Initiative Munich (NIM).
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---
abstract: 'Recent developments on the four dimensional (4d) lattice studies of the finite temperature electroweak phase transition (EWPT) are summarized. The phase diagram is given in the continuum limit. The finite temperature SU(2)-Higgs phase transition is of first order for Higgs-boson masses $m_H<66.5 \pm 1.4$ GeV. Above this endpoint only a rapid cross-over can be seen. The full 4d result agrees completely with that of the dimensional reduction approximation. The Higgs-boson endpoint mass in the Standard Model (SM) would be $72.1 \pm 1.4$ GeV. Taking into account the LEP Higgs-boson mass lower bound excludes any EWPT in the SM. A one-loop calculation of the static potential in the SU(2)-Higgs model enables a precise comparison between lattice simulations and perturbative results. The most popular extension of the SM, the Minimal Supersymmetric SM (MSSM) is also studied on 4d lattices.'
address: |
Institute for Theoretical Physics, Eötvös University,\
H-1088 Budapest, HUNGARY
author:
- 'Z. Fodor'
title: Electroweak Phase Transitions
---
INTRODUCTION
============
The visible Universe is made of matter. This fact is based on observations of the cosmic diffuse $\gamma$-ray background, which could be larger than the present limits, if boundaries between “worlds” and “antiworlds” had existed [@cohen98]. The observed baryon asymmetry of the universe was eventually determined at the EWPT [@KRS85]. On the one hand this phase transition was the last instance during which baryon asymmetry could have been generated around $T \approx 100$ GeV, on the other hand at these temperatures any B+L asymmetry could have been washed out. The possibility of baryogenesis at the EWPT is a particularly attractive one, since the underlying physics can be –and already largely has been– tested at collider experiments. Thus, the detailed understanding of this phase transition is very important.
A succesfull baryogenesis scenario consists three ingredients, the Sakharov’s conditions.\
1. Baryon number violating processes\
2. C and CP violation\
3. Departure from equilibrium.\
All of the three conditions has non-perturbative features and are studied on the lattice (e.g. at this conference baryon number violating sphalerons have been discussed by [@moore99], spontaneous CP violation by [@laine99], whereas this contribution mostly studies the out of equilibrium condition).
It is rather easy to see the necessity of the first two conditions. Without baryon number violation no net baryon asymmetry can be generated. C and CP violation are needed to give a direction to the processes. The standard picture concerning the third condition is the turn-off of the baryon number violating rate after the phase transition, which means a smaller sphaleron rate than the Hubble rate. Inspecting the formula for the sphaleron rate one needs a strong enough phase transition, thus $v/T_c {\mathop{\gsi}}1$. This ratio is of particular interest and both perturbative and lattice studies have the main goal to determine it.
The first-order nature of the EWPT for light Higgs bosons can be shown within perturbation theory. However, perturbation theory breaks down for Higgs boson masses ($m_H$) larger than about 60 GeV due to bad infrared behavior of the gauge-Higgs part of the electroweak theory [@perturb]. Solutions of gap-equations even suggest an end-point scenario for the first order EWPT [@gap]. Numerical simulations are needed to analyze the nature of the transition for realistic Higgs bosons.
One very succesfull possibility is to construct an effective three dimensional (3d) theory by using dimensional reduction, which is a perturbative step. The non-perturbative study is carried out in this effective 3d model [@3d-sim]. The end-point of the phase transition is determined and its universality class is studied [@3d-end].
Another approach is to use 4d simulations. The complete lattice analysis of the SM is not feasible due to the presence of chiral fermions, however, the infrared problems are connected only with the bosonic sector. These are the reasons why the problem is usually studied by simulating the SU(2)-Higgs model on 4d lattices, and perturbative steps are used to include the U(1) gauge group and the fermions. Finite temperature simulations are carried out on lattices with volumes $L_t \cdot L_s^3$, where $L_t \ll L_s$ are the temporal and spatial extensions of the lattice, respectively. Systematic studies were carried out for $m_H \approx$ 20 GeV, 35 GeV, 50 GeV and 75 GeV [@4d]. The lattice spacing is basically fixed by the number of the lattice points in the temporal direction ($T_c=1/(L_t a)$, where $T_c$ is the critical temperature in physical units); therefore huge lattices are needed to study the soft modes. This problem is particularly severe for Higgs boson masses around the W mass, for which the phase transition is weak and typical correlation lengths are much larger than the lattice spacing. In this case asymmetric lattice spacings are used [@4d-asym].
END-POINT IN FOUR DIMENSIONS
============================
The 4-d SU(2)-Higgs model is studied on both symmetric and asymmetric [@4d-asym] lattices, i.e. lattices with equal or different spacings in temporal ($a_t$) and spatial ($a_s$) directions. The asymmetry of the lattice spacings is given by the asymmetry factor $\xi=a_s/a_t$. The different lattice spacings can be ensured by different coupling strengths in the action for time-like and space-like directions. The action reads in standard notation [@4d] $$\begin{aligned}
&& S[U,\varphi]= \nonumber \\
&& \beta_s \sum_{sp}
\left( 1 - {1 \over 2} {\rm Tr\,} U_{pl} \right)
+\beta_t \sum_{tp}
\left( 1 - {1 \over 2} {\rm Tr\,} U_{pl} \right)
\nonumber \\
&&+ \sum_x \left\{ {1 \over 2}{\rm Tr\,}(\varphi_x^+\varphi_x)+
\lambda \left[ {1 \over 2}{\rm Tr\,}(\varphi_x^+\varphi_x) - 1 \right]^2
\right. \nonumber \\
&&\left.
-\kappa_s\sum_{\mu=1}^3
{\rm Tr\,}(\varphi^+_{x+\hat{\mu}}U_{x,\mu}\,\varphi_x) \right. \nonumber \\
&& \left. -\kappa_t {\rm Tr\,}(\varphi^+_{x+\hat{4}}U_{x,4}\,\varphi_x)\right\},\end{aligned}$$ We introduce $\kappa^2=\kappa_s\kappa_t$ and $\beta^2=\beta_s\beta_t$. The anisotropies $\gamma_\beta^2=\beta_t/\beta_s$ and $\gamma_\kappa^2=\kappa_t/\kappa_s$ are functions of $\xi$. We use $\xi=4.052$, which corresponds to $\gamma_\kappa=4$ and $\gamma_\beta=3.919$.
=0.9
The determination of the end-point of the finite temperature EWPT is done by the use of the Lee-Yang zeros of the partition function ${\cal Z}$. Near the first order phase transition point the partition function reads ${\cal Z}={\cal Z}_s + {\cal Z}_b \propto \exp (-V f_s)+ \exp (-V f_b) \ ,$ where the indices s(b) refer to the symmetric (broken) phase and $f$ stands for the free-energy densities. We also have $f_b = f_s + \alpha (\kappa - \kappa _c ) \ ,$ , since the free-energy density is continuous. It follows that ${\cal Z} \propto \exp [ -V ( f_s +f_b )/2 ]
\cosh [ -V \alpha (\kappa -\kappa_c )] \ ,$ which shows that for complex $\kappa$ ${\cal Z}$ vanishes at ${\rm Im} (\kappa )=2 \pi \cdot (n-1/2) / (V\alpha )$ for integer $n$. In case a first order phase transition is present, these Lee-Yang zeros move to the real axis as the volume goes to infinity. In case a phase transition is absent the Lee-Yang zeros stay away from the real $\kappa $ axis. Denoting $\kappa_0$ the lowest zero of ${\cal Z}$, i.e. the position of the zero closest to the real axis, one expects in the vicinity of the end-point the scaling law ${\rm Im}(\kappa_0)=C(L_t,\lambda)V^{-\nu}+\kappa_0^c(L_t,\lambda)$. In order to pin down the end-point we are looking for a $\lambda$ value for which $\kappa_0^c$ vanishes. In practice we analytically continue ${\cal Z}$ to complex values of $\kappa $ by reweighting. Small changes in $\lambda$ were taken into account by reweighting. The dependence of $\kappa_0^c$ on $\lambda$ [@Aoki99] is shown in fig. 1. To determine the critical value of $\lambda$ i.e. the largest value, where $\kappa_0^c=0$, we have performed fits linear in $\lambda$ to the non-negative $\kappa_0^c$ values.
=0.85
In the isotropic case [@Aoki99], we have used $L_t =2$. The Lee-Yang analysis gave $\lambda_c =0.00116(16)$ for the end-point. Performing $T=0$ simulations with the same parameters this can be converted to $m_{H,c}=73.3 \pm 6.4$GeV. In the anisotropic lattice simulation case [@Csikor99] we also performed a continuum extrapolation for $L_t=2,3,4,5$ (fig. 2), moving along the lines of constant physics (LCP), and obtained $66.5 \pm 1.4$ GeV, which is our final result for the end-point in the SU(2)-Higgs model.
Based on previous 4d simulation results one can determine the phase diagram of the finite temperature EWPT and compare it with the 3d analysis (fig. 3.) as it has been done in ref. [@Laine99]. The phase transition lines $T_c(m_H)$, are in perfect agreement for $m_H{\mathop{\gsi}}25$ GeV. For strong first order phase transition close to the Coleman-Weinberg limit the 3d approach seems to be less accurate. The error bars on the endpoints are on the few percent level, thus uncertainty of the dimensional reduction around the end-point is also in this range. This indicates that the analogous perturbative inclusion of the fermions results also in few percent error on $m_H$.
=1.10
One can determine what is the endpoint value in the full SM. As it was shown previously the perturbative integration of the heavy modes is correct within our error bars. Therefore we use perturbation theory to transform the SU(2)-Higgs model endpoint value to the full SM. We obtain $72.1 \pm 1.4$ GeV, where the dominant error comes from the measured error of $R_{HW,cont.}$. The error on $g_R^2$ is eliminated by calculating the relationship between the coupling definitions used in perturbation theory (${\overline {\rm {MS}}}$) and lattice simulations (from static potential) [@Laine99; @Csikor99a]. The calculation of this relationship and a comparison of the perturbative and lattice results on the EWPT will be shortly discussed in the next section.
The full SM result needs some explanation. Based on vacuum stability the measured top mass ($m_{top} \approx 175$ GeV) results in a lower bound for the Higgs boson mass (approx. 130 GeV). This value is higher than the previously mentioned $72.1 \pm 1.4$ GeV. For the pure SU(2)-Higgs model the endpoint Higgs mass is $66.5 \pm 1.4$ GeV. The inclusion of the fermions, especially the top increases the endpoint slightly. For a hypothetical top quark mass less than approximately 150 GeV the lower bound is less than $\approx 70$ GeV, thus it is below the endpoint and it gives a reliable theory. Increasing the top quark mass the lower bound gets larger than the endpoint. This means that independently of the direct experimental bounds on the Higgs boson mass no EWPT exists in the SM.
RELATIONSHIP BETWEEN GAUGE COUPLINGS
====================================
Despite the fact that the perturbative and lattice approaches are systematic and well-defined, it is not easy to compare their predictions. The reason is that in lattice simulations the gauge coupling constant is determined from the static potential, whereas in perturbation theory the ${\overline {\rm {MS}}}$ scheme is used. One can calculate the static potential on the one-loop level in the SU(2)-Higgs model [@Laine99; @Csikor99a]. As expected the numerical difference between the two conventions is not that large, it is within a few percent, for details see [@Laine99; @Csikor99a]. With this connection we could perform a precise comparison between the predictions of perturbative and lattice approaches (fig. 4).
In [@Csikor99a] the existing lattice data was reanalyzed and a continuum limit extrapolation was performed whenever it was possible. The only quantity which is measured so precisely that the definition of the gauge coupling constant is essential is the ratio of the critical temperature to the Higgs boson mass. As it has been observed already for $M_H \approx 35$ GeV the perturbative value of $T_c$ is larger than in lattice simulations. This sort of discrepancy disappears for larger Higgs boson masses. A plausible reason for this fact is the convergence of the high temperature expansion used in the perturbative approach.
The most dramatic differences appear clearly as we get closer to the end point. The perturbative approach gives non-vanishing jump of the order parameter, non-vanishing latent heat and interface tension, while the lattice results suggest rapid decrease of these quantities and no phase transition beyond the end-point.
=1.15
PHASE TRANSITION IN THE MSSM
============================
As it was demonstrated in the previous sections the SM is not suitable for baryogenesis, not even for a first order EWPT. Several extended models were studied in order to obtain a stronger first order phase transition and a reliable baryon asymmetry. The most popular model is the MSSM, which perturbatively shows a much stronger phase transition than the SM [@mssm] (even an intermediate colour breaking phase transition is possible in these scenarios) . Lattice studies in a 3d reduced model (with one Higgs doublet) basically confirmed the perturbative results [@mssm3d].
We performed a 4d lattice study with the bosonic sector of the MSSM [@mssm4d]. The lattice action is too long to be presented here, thus only the fields involved are listed. Both of the Higgs doublets, the stop, sbottom scalars and SU(2), SU(3) gauge fields were included. It is of particular importance to keep both of the Higgs doublets, since according to the standard scenario the generated baryon asymmetry is directly proportional to the change of the ratio of their expectation values $n_B \propto \langle v^2 \rangle \Delta \beta(T_c)$. Here the length squared of the Higgs field ($v^2=v_1^2+v_2^2$) is integrated over the bubble wall. The ratio of the expectation values of the two Higgs fields is $\tan \beta=v_1/v_2$, and the difference between the $\beta$ values are taken in the “symmetric” and in the “broken” phases.
We had simulations at $L_t=2,3,4,5$ and moved along the line of constant physics. Our simulation point corresponds to $\tan\beta(T=0) \approx 6$, and the mass of the lightest Higgs bosons is approx. 35 GeV (in the bosonic theory).
Two values of $\alpha_s$ were taken (the physical and a smaller one). The physical $\alpha_s$ resulted in $v/T_c \approx 1.5$, whereas the smaller value of $\alpha_s$ gave a stronger phase transition $v/T_c \approx 2$. Perturbation theory predicts just the opposite behaviour (stop-gluon setting sun graphs are proportional to the strong coupling and they are responsible for the strengthening of the phase transition). The reason can be the difference between the renormalization effects in the stop sector.
We measured the $\beta$ parameter in both phases at the phase transition. One obtains $\tan^2 \beta(sym) = 38.13(15)$, $\tan^2
\beta(brok) = 36.04(15)$, which gives $\Delta \beta = 0.0045(7)$. This result is far below the perturbative prediction $\Delta \beta (pert.)=0.017$.
CONCLUSIONS
===========
The endpoint of hot EWPT with the technique of Lee-Yang zeros from simulations in 4d SU(2)-Higgs model was determined. The phase transition is first order for Higgs masses less than $66.5 \pm 1.4$ GeV, while for larger Higgs masses only a rapid cross-over is expected. The phase diagram of the model was given.
It was shown non-perturbatively that for the bosonic sector of the SM the dimensional reduction procedure works within a few percent. This indicates that the analogous perturbative inclusion of the fermionic sector results also in few percent error. In the full SM we get $72.1 \pm 1.4$ GeV for the end-point, which is below the lower experimental bound. This fact is a clear sign for physics beyond the SM.
Based on a one-loop calculation on the static potential of the SU(2)-Higgs model a direct comparison between the perturbative and lattice results was performed.
The MSSM is more promising for a succesfull baryogenesis. Some 4d results were shown, indicating a strong first order phase transition.
[**Acknowledgments:**]{} This work was partially supported by Hung. Grants No. OTKA-T22929-29803-M28413-FKFP-0128/1997.
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---
abstract: 'We discuss phenomenology of light scalar sgoldstino in context of CERN electron beam dump experiment NA64. We calculate sgoldstino production rate for this experiment taking into account sgoldstino mixing with the Higgs boson and find a region in the model parameter space which can be tested in NA64.'
author:
- 'K.O.Astapov'
- 'D.V.Kirpichnikov'
title: Prospects of models with light sgoldstino in electron beam dump experiment at CERN SPS
---
Motivation {#sec:Intro}
==========
Supersymmetry is a promising extention of the Standard Model [@Haber:1984rc; @Martin:1997ns]. However according to experimentally observed absence of superpartners at low energies, SUSY models imply supersymmetry to be spontaneously broken at some scale. The breaking mechanism is provided by underlying microscopic theory. The breaking can happen when a hidden sector dynamics results in a nonzero vacuum expectation value $F$ to an auxiliary component of the superfield [@Brignole:2003cm] and $\sqrt{F}$ is SUSY breaking scale. According to supersymmetric analog of the Goldstone theorem [@Volkov:1972jx] there should exist a massless fermionic degree of freedom, goldstino.
In the simplest case this fermion belongs to a chiral multiplet, dubbed goldstino supermultiplet. Apart from goldstino, it contains scalar - sgoldstino and an auxiliary field acquiring nonzero vacuum expectation value $F$, which trigger spontaneous SUSY breaking. The quantity $\sqrt{F}$ is supersymmetry breaking scale. Couplings of sgoldstino to SM fields are suppressed by $\frac{1}{F}$ and expected to be quite small. Being included into supergravity framework goldstino becomes longitudinal component of gravitino with mass related to the scale of supersymmetry breaking $\sqrt{F}$ as follows $m_{3/2} = \frac{F}{\sqrt{3}M_{pl}},$ where $M_{pl}$ is the Planck mass [@Cremmer:1978iv]. In the present work we consider the mass of sgoldstino to be light (less than $1$ GeV) as phenomenologically interesting case; therefore a high intensity beam is required to test the model via production of sgoldstino. Beam dump experiment can perform the task. A preliminary estimate of the search for sgoldstino at beam dump experiment SHiP for sgoldstino production and decay mechanisms can be found in Ref.[@Astapov:2015otc]; here we complete that study by considering production of sgoldstino in electron-proton collisions.
The purpose of the present paper is to estimate the signal rate of sgoldstino decays expected to detect at the NA64 electron beam-dump experiment in context of present model (see Refs.[@Andreas:2013lya; @Banerjee:2016tad; @Gninenko:2016kpg] for description of the experiment in context of the search for Dark photon). We consider the case when mostly the sgoldstino decays into electron-positron pairs as searching signature.
The paper is organized as follows. In section \[sec:Lagrangian\] we present effective interaction Lagranginan of sgoldstino to SM particles and set of chosen values of parameters of MSSM. Section \[sec:3\] contains calculation of sgoldstino production cross section. In section \[sec:4\] we discuss sgoldstino decay channels and calculate its lifetime for given model parameters. In Secs. \[sec:5\] we calculate NA64 sensitivity to the SUSY breaking scale and put new limits on the model parameters. We conclude in Sec.\[conclude\] by summarizing the results obtained.
Lagrangian and parameters {#sec:Lagrangian}
=========================
To the leading order in $1/F$, sgoldstino couplings to SM gauge fields - photons $F_{\mu\nu}$, gluons $G_{\mu\nu}$ and matter fields - leptons $l_a$, up and down quarks $u_a$ and $d_a$, where index $a$ runs over three gererations at the mass scale above $\Lambda_{QCD}$ but below electroweak symmetry breaking reads as [@Astapov:2014mea; @Gorbunov:2000th]
$$\begin{gathered}
{\cal L}^{s}_{eff}=-\frac{M_{\gamma\gamma}}{2\sqrt{2}F}sF^{\mu\nu}F_{\mu\nu}-\frac{M_2}{\sqrt{2}F}sW^{\mu\nu}W_{\mu\nu}-\frac{M_{ZZ}}{2\sqrt{2}F}sZ^{\mu\nu}Z_{\mu\nu}- \frac{M_3}{2\sqrt{2}F}s{}{{\rm Tr}}{}G^{\mu\nu}G_{\mu\nu}-\frac{A^{U}_{ab}v}{\sqrt{2}F}su_{a}u_{b}-\frac{A^{D}_{ab}v}{\sqrt{2}F}sd_{a}d_{b}-\\
-\frac{A_{ab}^{L}v}{\sqrt{2}F}sl_{a}l_{b}
\label{eef}\end{gathered}$$
Here $M_3$ is the gluino mass, $M_{\gamma\gamma}=M_1\sin^2\theta_W+M_2\cos^2\theta_W$ and $M_{ZZ}=M_1\cos^2\theta_W+M_2\sin^2\theta_W$ with $M_1$ and $M_2$ being $U(1)_Y$- and $SU(2)_W$-gaugino masses and $\theta_W$ the weak mixing angle, and $A^{U}_{ab}$, $A^{D}_{ab}$ and $A_{ab}^{L}$ are soft trilinear coupling constants. Lagrangian includes only single-sgoldstino interaction terms; considered in Refs.[@Perazzi:2000id; @Perazzi:2000ty; @Gorbunov:2000th; @Demidov:2011rd], double-sgoldstino terms are suppressed by $1/F^2$ and are not probable for testing at the NA64 experiment.
In general sgoldstino also mixes with neutral Higgs bosons as discussed in Refs.[@Dudas:2012fa; @Bellazzini:2012mh; @Astapov:2014mea; @Sobolev:2016gmr]: the scalar sgoldstino $S$ mixes with neutral light $h$ and heavy $H$ Higgs bosons, while pseudoscalar $P$ mixes with their axial partner $A$. We account for the mixing with $h$ only, since the other two do not change light scalar sgoldstino phenomenology at NA64 for considered set of parameters of the model. Mixing of the scalar sgoldstino and the lightest MSSM Higgs boson (SM-like Higgs) $h$ can be written as[@Astapov:2014mea] $$\label{L-mixing}
{\cal L}_{mixing}=\frac{X}{F}\,Sh\,,$$ where the mixing parameter $X$ is $$\label{X_mixing}
X = 2\mu^3v\sin{2\beta} +
\frac{1}{2}v^3(g_1^2M_1+g_2^2M_2)\cos^2{2\beta}\,,$$ here $\mu$ is Higgsino mixing mass parameter, $v=174$GeV is the Higgs vacuum expectation value (vev), $\tan\beta$ is describing the Higgs vev ratio, and $g_2$ and $g_1$ are $SU(2)_W$ and $U(1)_Y$ gauge coupling constants.
Since we are considering sgoldstino $S$ mass to be less than $1$ GeV (much lighter than the SM-like Higgs boson of mass $m_h\approx125$GeV) all the Higgs-like couplings of scalar resonanse are suppressed by the mixing angle $$\label{mix_angle}
\theta =-\frac{X}{Fm_{h}^2}\,.$$
In Table\[MSSMpoint\] we set numerical values for parameters of the MSSM so that $h$ asquire its experimentally observed value of $125$GeV by loop corrections from squark masses and trilinear couplings; and $H$ along with $A$ fields acquire heavy masses over $1$TeV to not to contribute into mixing with scalar and pseudoscalar sgoldstinos. In this arbitrary choice we suppose that all the model parameters take experimentally allowed values.
$M_1,$ GeV $M_2,$ GeV $M_3,$ GeV $\mu,$ GeV $\tan\beta$
------------ ------------ ------------ ------------ -------------
100 250 1500 1000 6
$m_A,$ GeV $A_l,$ GeV $m_l,$ GeV $A_Q,$ GeV $m_Q,$ GeV
1000 2800 1000 2800 1000
: \[MSSMpoint\]MSSM benchmark point.
In the table we denoted $A^U_{aa}$, $A^D_{aa}$ as $A_Q$ and $A^L_{aa}$ as $A_l$, all the off-diagonal $A^{U,D,L}_{ab}$ are set to zero.
Production mechanism {#sec:3}
====================
In this section we describe scalar sgoldstino production in electron-proton collisions as $100$ GeV electron beam hitting heavy nuclei lead ($Z = 82$) target. We take into account interactions of sgoldstino with nuclei and electrons. Feynman diagrams of the process are presented on Fig.\[feynD\]. We denote the four-momenta of the initial beam and scattered electrons by $k_e=(E_e, \vec{k_e})$ and $k_e^{\prime} = (E_e^{\prime}, \vec{k_e^{\prime}})$; the four-momenta of the initial and final target state by $k_N^{\prime} = (E_N^{\prime}, \vec{k_N})$ and $k_N^{\prime} = (E_N^{\prime}, \vec{k_N^{\prime}})$; for outgoing sgoldstino particle $k=(E, \vec{k})$. Expressions for corresponding diagrams read as:
$$\begin{gathered}
i\mathcal{M}^{(b)}=\frac{M_{\gamma\gamma}}{2\sqrt{2}F}(-ie)^2\frac{i}{(k_p-k_p^{\prime})^2}\frac{i}{(k_e-k_e^{\prime})^2}\times\\
\times\Big[-2(k_e-k_e^{\prime},k_p-k_p^{\prime})g^{\alpha\beta}+\\
(k_e-k_e^{\prime})^{\alpha}(k_p-k_p^{\prime})^{\beta}+(k_e-k_e^{\prime})^{\beta}(k_p-k_p^{\prime})^{\alpha}\Big]j^e_{\alpha}j^N_\beta,\end{gathered}$$
$$i\mathcal{M}^{(a)}+i\mathcal{M}^{(c)}=\frac{A_Lv}{\sqrt{2}F}\times(-ie)^2\frac{i}{(k_p-k_p^{\prime})^2}L_{\alpha}j^N_{\alpha},$$
where leptonic and hadronic currents read as $$j^e_{\alpha} = \bar{u}_e(k_e)\gamma_{\alpha}u(k_e)$$ and
$$j^N_{\beta} = Z F(Q_t)(k_p+k_p^{\prime})_{\alpha}$$ correspondingly. Here $F(Q)$ is the nuclear charge form factor [@Beranek:2013nqa]. Note, that we did not consider diagrams similar to (a), (b) and (c) but with $Z^0$-boson exchange, since they are suppressed by $Z^0$ mass.
Leptonic tensor: $$L_{\alpha}=\bar{u}_e(k_e)\Bigg(\gamma_{\alpha}\frac{-(\slashed{k}_e-\slashed{k})+m_e}{(k_e-k)^2-m_e^2}+\frac{-(\slashed{k}+\slashed{k}_e^{\prime})+m_e}{(k+k_e^{\prime})^2-m_e^2}\gamma_{\alpha}\Bigg)u(k_e),$$
Therefore full amplitude of the process reads as $$i\mathcal{M}=i\mathcal{M}^a+i\mathcal{M}^b+i\mathcal{M}^c$$
The differential cross section of $2\to3$ process for $m_S=100$ MeV is presented on Fig.\[CS\]
![Differential production cross section of sgoldstino as a function of its mass. Mass of the sgoldstino in taken 100 MeV. \[CS\]](CS_NA64){width="50.00000%"}
![Feynman diagrams describing $2\to3$ sgoldstino production process. \[feynD\]](feyn_new){width="50.00000%"}
Decay channels {#sec:4}
==============
For (sub-)GeV mass-range sgoldstino decay channels into pairs of SM particles, if kinematically allowed are: $\gamma\gamma$, $e^+e^-,$ $\mu^+\mu^-$, $\pi^0\pi^0$, $\pi^+\pi^-$. Decay width of sgoldstino into photons: $$\label{Sgammagamma}
\Gamma(S\to\gamma\gamma)=\left(\frac{\alpha(m_S)
\beta(\alpha(M_{\gamma\gamma}))}{\beta(\alpha(m_S))\alpha(M_{\gamma\gamma})}
\right)^2\frac{m_{S}^3M_{\gamma\gamma}^2}{32\pi F^2}.$$ Here the dimensionless multiplicative factor accounts for the renormalization group evolution of the photonic operator at different mass scales. Lepton channels are: $$\label{Sll}
\Gamma(S\to {}l^+l^-)={m_S^3A_l^2\over 16\pi F^2}{m_{l}^2\over m_S^2}\l
1-{4m_{l}^2\over m_S^2}\r^{3/2}.$$
Decay into light mesons is provided with gluonic operator at a low energy scale.
$$\begin{gathered}
\Gamma(S\to\pi^0\pi^0)={\alpha^2_s(M_3)\over\beta^2(\alpha_s(M_3))}
{\pi m_S\over
4}{m_S^2M_3^2\over F^2}\\ \left( \!\!1\!
-\!{\beta(\alpha_s(M_3))\over\alpha_s(M_3)}
{9\over 4\pi}{B_0\over
m_S}{m_u+m_d\over m_S}{A_Q\over M_3}\!\right)^{\!\!2}\!
\sqrt{1\!-\!{4m_{\pi^0}^2\over m_S^2}},
\label{StoPiPi}\end{gathered}$$
$$\Gamma(S\to\pi^0\pi^0)\approx{\alpha^2_s(M_3)\over\beta^2(\alpha_s(M_3))}
{\pi m_S^3M_3^2\over 4F^2}\sqrt{1-{4m_{\pi^0}^2\over m_S^2}},$$
$$\Gamma(S\to\pi^+\pi^-)=2\Gamma(S\to\pi^0\pi^0)\,.$$
See Ref.[@Astapov:2015otc] for notations for above formulas.
Sgoldstino decay branching ratios for the values of MSSM parameters given in Table\[MSSMpoint\] are shown in Fig.\[SBranching\].
![Branching ratios of a scalar sgoldstino. \[SBranching\]](BR_new){width="50.00000%"}
Hadronic channel $\pi\pi$ dominate when it is kinematically allowed, while $\gamma\gamma$ and $\mu^+\mu^-$ give small but noticeable contributions.
The sgoldstino lifetime for given $\sqrt{F} = 10$ TeV is presented in Fig.\[SLifetime\].
![Lifetime of a scalar sgoldstino as a function of its mass. \[SLifetime\]](LT){width="50.00000%"}
Results {#sec:5}
=======
Here we estimate the number of $e^{-}N\to{}S\to{}e^{+}e^{-}$ events inside the fiducial volume of the NA64 experimental setup. Experimental setup is designed to search for rare decays with charged particles in the final state. Detailed setup scheme is outlined in the Ref.[@Andreas:2013lya]. The experiment utilize clean high energy $e^-$ beam with less then $10^{-2}$ level of impurities and momenta of $100$GeV. Electron beam is produced by primary 400 GeV proton beam from SPS hitting the primary beryllium target. Electron beam strikes on the electron calorimeter target and produces sgoldstinos directly through the processes described in the previous sections. Target calorimeter thickness is $l_{sh}=0.15$m. The vacuum vessel length is about $l_{det}=15$m. It forms a cylinder along the beam axis with an circle base of 30 cm in diameter. At the back end of the vacuum vessel another electronic calorimeter serves for count of electromagnetic shower produced by subsequent sgoldstino decays into charged particles.
The number of signal events reads as
$$\begin{gathered}
N_{\text{signal}}=N_{\text{EOT}}\frac{N_0X_0}{A}\int^{E_0-m_e}_{m_S}dE_S\int^{E_0}_{E_S+m_e}dE_e\\
\times\int^{T}_{0}dt\Bigg[I_e(E_0,E_e,t)\frac{1}{E_e}\frac{d\sigma}{dx_e}\Bigg]w_{det}\text{BR}_{det},\end{gathered}$$
where the expected number of electons on the target is $N_{\text{EOT}}=10^{9}$, $N_0$ is Avagadro’s number, $X_0$ is the unit radiation length of the target material, A is atomic mass number, $E_s$ is sgoldstino energy, $E_0$ and $E_e$ are beam and initial electron energies correspondingly, $x_e=\frac{E_s}{E_e}$ and $w_{det}$ denotes the probability for the sgoldstino to decay inside the fiducial volume of the detector, $$\begin{gathered}
w_{det}(E_{S(P)}, m_{S(P)}, \sqrt{F})=\exp(-l_{sh}/\gamma{}c\tau_{S(P)})\times\\
\times\left[1-\exp(-l_{det}/\gamma{}c\tau_{S(P)})\right],\end{gathered}$$ with the sgoldstino gamma factor $\gamma{}=E_{S(P)}/m_{S(P)}$.
Since electron beam with energy $E_0$ becomes degraded as electrons pass trough and interact with its nucleus. Energy distribution of electrons (see Ref.[@Andreas:2012mt]) after passing through material by $t$ radiation length is given by: $$I_e(E_0,E_e,t) = \frac{1}{E_0}\frac{\Big[{\mathop{\rm ln}\nolimits}(\frac{E_0}{E_e})\Big]^{bt-1}}{\Gamma(bt)},$$ where $\Gamma$ is Gamma function, $b=4/3$, $E_0$ is initial beam energy at $t=0$.
In Fig.\[Scalar\]
![The shaded region will be probed at the NA64 experiment. \[Scalar\]](NA64){width="45.00000%"}
we indicate the region in the model parameter space $(m_{S},1/\sqrt{F})$, where the number of sgoldstino decay events inside the fiducial volume exceeds 3, $N_{\text{signal}}>3$. That is, if no events were observed the region is excluded at the confidence level of 95%, in accordance with the Poisson statistics. The lower boundary in Fig.\[Scalar\] is the region where the couplings are so small that sgoldstinos escape from the detector without decay. The upper boundary corresponds to case when couplings are so large that sgoldstinos decay before the detector. The scalings of the signal events imply that models with a higher (as compared to that presented in Fig.\[Scalar\]) scale of supersymmetry breaking can be tested if MSSM parameters $\mu$, $M_{\gamma\gamma}$ are appropriately larger (as compared to those presented in Table\[MSSMpoint\]).
Conclusions {#conclude}
===========
We have estimated sensitivity of the NA64 experiment to supersymmetric extensions of the SM where sgoldstinos are light. The experiment will be able to probe the supersymmetry breaking scale $\sqrt{F}$ up to $10^4$TeV. We have obtained exclusion regions of the scalar sgoldstino parameter space ($m_S$ vs. $1/\sqrt{F}$).
#### Acknowledgments {#acknowledgments .unnumbered}
We thank D. Gorbunov, S. Demidov, S. Gninenko, M. Kirsanov, N. Krasnikov and S. Kulagin for valuable discussions. The work was supported by the RSF Grant No. 14-12-01430.
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|
---
abstract: 'We analyze galaxies in 300 nearby groups and clusters identified in the Sloan Digital Sky Survey using a photometric gas mass indicator that is useful for estimating the degree to which the interstellar medium of a cluster galaxy has been depleted. We study the radial dependence of inferred gas mass fractions for galaxies of different stellar masses and stellar surface densities. At fixed clustercentric distance and at fixed stellar mass, lower density galaxies are more strongly depleted of their gas than higher density galaxies. An analysis of depletion trends in the two-dimensional plane of stellar mass $M_*$ and stellar mass surface density $\mu_*$ reveals that gas depletion at fixed clustercentric radius is much more sensitive to the density of a galaxy than to its mass. We suggest that low density galaxies are more easily depleted of their gas, because they are more easily affected by ram-pressure and/or tidal forces. We also look at the dependence of our gas fraction/radius relations on the velocity dispersion of the cluster, finding no clear systematic trend.'
author:
- |
Wei Zhang$^{1}$[^1], Cheng Li$^{2}$, Guinevere Kauffmann$^{3}$, Ting Xiao$^{2}
\thanks{LAMOST fellow}$\
${^1}$ National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, China\
${^2}$ Partner Group of Max Planck Institut for Astrophysics and Key Laboratory for Research in Galaxies and Cosmology\
of Chinese Academy of Sciences, Shanghai Astronomical Observatory, Nandan Road 80, Shanghai 200030, China\
${^3}$ Max Planck Institut für Astrophysik, Karl-Schwarzschild-Strasse 1, 85748 Garching, Germany
bibliography:
- 'ref.bib'
- 'addon.bib'
date: 'Accepted ........ Received ........; in original form ........'
title: Gas depletion in cluster galaxies depends strongly on their internal structure
---
\[firstpage\]
galaxies: clusters: general – galaxies: distances and redshifts.
Introduction
============
The observed variation in galaxy colours, star formation rates, cold gas fractions as a function of distance from the centers of groups and clusters places important constraints on the physical processes that affect the gas in galaxies as they evolve within a hierarchy of merging dark matter halos [e.g. @Diaferio-01; @Okamoto-Nagashima-03].
Semi-analytic models of galaxy formation developed in the early 1990’s [e.g. @Kauffmann-White-Guiderdoni-93; @Cole-94] made the simplistic assumption that after a galaxy was accreted by a larger dark matter halo and became a satellite, its reservoir of hot gas would be stripped instantaneously and would form part of the hot atmosphere bound to the common halo. It was also assumed that the cold gas in a galaxy was not affected by stripping processes and the timescale for the satellite galaxy to redden was set by the rate at which the cold gas was used up by star formation. Comparison of these models with group and cluster galaxies from the Sloan Digital Sky Survey revealed that the predicted fraction of blue satellite galaxies was too low [@Weinmann-06]. This discovery was then followed by considerable effort to fix the models by relaxing the assumption that the hot gas reservoir around a satellite is stripped instantaneously following accretion [e.g. @Font-08; @Weinmann-10; @Guo-11]. If satellite galaxies are able to retain a significant fraction of their hot gas for several gigayears following accretion, their colours are found to be in much better agreement with data.
One important constraint from the observations that has not been considered in much detail up to now, is the fact that the increase in the fraction of red galaxies from the outskirts of rich groups and clusters to their centers, is strongly dependent on galaxy mass. As shown in Figure 7 of @vonderLinden-10 , the fraction of red galaxies with stellar masses in the range $3 \times 10^9 M_{\odot}$ to $10^{10} M_{\odot}$ increases by a factor 4 from 0.2 at the virial radius of the cluster to 0.8 at the cluster center. In contrast, for galaxies with stellar masses greater than $5 \times 10^{10} M_{\odot}$, the red fraction only increases by 30% from 0.6 at the virial radius to 0.8 at the cluster center. Because cold gas consumption times in low mass field galaxies are long, one might ask whether stripping of an external reservoir of ionized or hot gas can lead to such strong effects with cluster-centric radius. In a recent paper, @Guo-11 implemented a model in which the hot gas mass around satellites is reduced in direct proportion to the mass of its surrounding dark matter subhalo, which loses mass continuously due to tidal stripping. In addition, Guo et al computed the radius at which ram-pressure forces due to the satellite’s motion through the intracluster medium would remove the hot gas. The mimimum of the tidal radius and the ram-pressure stripping radius define the radius beyond which gas is removed from the subhalo. As shown in their Figure 2, this model does not reproduce the stellar mass dependence of cluster galaxy red fractions at distances less than $\sim 0.5 R_{vir}$ from the cluster center.
In this paper, we analyze 300 clusters and groups from the samples of @Berlind-06 and @vonderLinden-07, looking for clues as to why environmental effects on low mass galaxies are so dramatic. In section 2, we describe the cluster and galaxy samples and introduce a photometric gas mass indicator that is useful for estimating the degree to which the interstellar medium of a cluster galaxy has been depleted with respect to a similar galaxy in the field. In section 3, we study the radial dependence galaxy colours/gas fractions in the two-dimensional plane of stellar mass $M_*$ and stellar mass surface density $\mu_*$. At fixed cluster-centric radius, we find that decrease in inferred cold gas mass fraction is a stronger function of stellar surface mass density than stellar mass. In section 4, we look at the dependence of stripping effects on the cluster/group velocity dispersion and in section 5, we summarize and discuss the implications of our results. Throughout this paper we assume a spatially flat concordance cosmology with $\Omega_m=0.3$, $\Omega_\Lambda=0.7$ and $H_0=100h$kms$^{-1}$Mpc$^{-1}$, where $h=0.7$.
Data
====
The cluster catalogue
---------------------
The cluster catalogue used in this paper includes 300 unique galaxy clusters, of which 181 were identified by @vonderLinden-07 [hereafter vdL07] from the SDSS data release 4 (DR4) and 119 were identified by @Berlind-06 [hereafter B06] from an earlier SDSS data release (DR3). Our cluster sample is restricted to have redshift $z<0.06$ (see next section), to have richness $N_{member}\ge10$, velocity dispersion $\sigma_V \ge 150$kms$^{-1}$ and area completeness $f_{cover} \ge 0.5$. The area completeness is defined by $$f_{cover}=\frac{A_{survey}(<2r_{200})}{A_{full}(<2r_{200})},$$ where $A_{full}(<2r_{200})$ is a circular sky area centered on the brightest cluster galaxy (BCG) with radius twice the virial radius of the cluster ($r_{200}$), and $A_{survey}$ is the area inside the same circle that is covered by the SDSS survey. The area completeness is unity for clusters far enough from the survey edges, but can be very low for clusters near the edges. Following [@Finn-05], we estimate a virial radius for each of our clusters, $r_{200}$ as $$\rm r_{200}=1.73\frac{\sigma_v}{1000km~s^{-1}}
\frac{1}{\sqrt{\Omega_\Lambda+\Omega_m(1+z)^3}}h^{-1}Mpc.$$
In Figure \[fig:sigmaV\] we show a histogram of $\sigma_V$ for the cluster sample. We note that the vdL07 sample is actually a subset of clusters in the C4 catalogue of @Miller-05, who identified clusters based on the presence of a “red sequence” of galaxies with similar positions and redshifts. vdL07 implemented an improved method for identifying the brightest cluster galaxy (BCG) and recalculated the velocity dispersion for each cluster accordingly. The cluster-finding algorithm of B06 was based on a redshift-space friends-of-friends method with no requirement that there be a clearly defined red sequence. As can be seen from the figure, the clusters from vdL07 have higher velocity dispersions than the clusters from B06. The combination of the two gives a sample with a wide coverage of velocity dispersion, ranging from $\sim150$km/s to $\sim800$km/s. This corresponds to a wide range in dark matter halo mass, from $M_h\sim2\times 10^{12}M_\odot$ for the Milky Way-type halos up to $M_h\sim5\times 10^{14}M_\odot$ for the most massive halos in the local Universe [see e.g. @Li-12c].
In this paper, we show results only for the combined sample of clusters. We have tested that all conclusions presented in this paper hold when we analyze the vdL07 and B06 cluster samples separately. The quantitative relations between H[i]{} gas mass fraction and cluster-centric radius in bins of stellar mass and stellar surface density agree for the two samples within the statistical errors. Combining the samples allows us to test whether there are any clear differences in galaxy properties at fixed $r/r_{200}$ for clusters with low and high velocity dispersions. As we will show in Section 4, these differences are very small.
Sample Stellar mass Redshift $N_{gal}$
-------- ---------------------------------- ---------- -----------
M1 $9.60<\log(M_\ast/M_\odot)<10.0$ $z<0.04$ 4169
M2 $10.0<\log(M_\ast/M_\odot)<10.3$ $z<0.06$ 9004
M3 $10.3<\log(M_\ast/M_\odot)<10.5$ $z<0.06$ 5577
M4 $10.5<\log(M_\ast/M_\odot)<10.7$ $z<0.06$ 4763
: Volume-limited samples selected by stellar mass from the SDSS/DR7 galaxy sample
\[tbl:samples\]
The galaxy sample
-----------------
We begin with the parent galaxy sample constructed from the New York University Value Added Catalogue (NYU-VAGC) [sample dr72]{} [@Blanton-05a], which consists of about half a million galaxies with $r<17.6$, $-24<M_{^{0.1}r}<-16$ and redshifts in the range $0.01<z<0.5$. Here, $r$ is the $r$-band Petrosian apparent magnitude, corrected for Galactic extinction, and $M_{^{0.1}r}$ is the $r$-band Petrosian absolute magnitude, corrected for evolution and $K$-corrected to its value at $z=0.1$. We use stellar masses from the MPA/JHU SDSS/DR7 database [^2], which are estimated from fits to the SDSS [*ugriz*]{} photometry following the philosophy of [@Kauffmann-03a] and [@Salim-07], assuming a universal stellar initial mass function of @Kroupa-01.
From the parent sample, we select four volume-limited samples of galaxies according to their stellar mass and redshift. The stellar mass range, the redshift range and the number of galaxies of the samples are listed in Table \[tbl:samples\]. The four volume-limited samples include a total of 23,513 galaxies. In Figure \[fig:mstar\_z\] we indicate the selection criteria of our samples in the stellar mass versus redshift plane. The parent sample is plotted in the background, grey-coded by the number of galaxies. The samples we choose are the same as in , except that we impose an upper redshift cut at $z=0.06$.
H[i]{} mass fraction and H[i]{} deficiency
------------------------------------------
When discussing the effect of the cluster environment on galaxies, it is much more physically intuitive to carry out analyses using [*cold gas mass fractions*]{} rather than colours or star formation rates. This is because we are trying to understand the impact of processes such as ram-pressure or tidal stripping on the interstellar medium of galaxies [@Kormendy-Bender-12].
In recent years, ‘pseudo’ H[i]{} gas mass estimates have been introduced that rely on the fact that the H[i]{} gas mass fraction is strongly correlated with properties such optical and optical/IR colours [e.g. @Kannappan-04]. Subsequent studies established that a combination of colour and stellar surface mass density provides a more accurate estimation of the H[i]{} mass fraction [e.g. @Zhang-09; @Catinella-10; @Li-12b].
The most recent version of such estimators was proposed by and utilizes a combination of four galaxy parameters: $$\begin{aligned}
\label{eqn:hiplane}
\log(M_{\mbox{H{\sc i}}}/M_\ast) & = & -0.325\log\mu_\ast-0.237(NUV-r) \nonumber \\
& & -0.354\log M_\ast-0.513\Delta_{g-i}+6.504,\end{aligned}$$ $M_{*}$ is the stellar mass; $\mu_\ast$ is the surface stellar mass density given by $\log\mu_\ast=\log M_\ast-\log(2\pi R_{50}^2)$ ($R_{50}$ is the radius enclosing half the total $z$-band Petrosian flux and is in units of kpc). $NUV-r$ is the global near-ultraviolet (NUV) to $r$-band colour. The $NUV$ magnitude is provided by the GALEX pipeline and the $NUV-r$ colour is corrected for Galactic extinction following [@Wyder-07] with $A_{NUV-r} = 1.9807A_r$ , where $A_r$ is the extinction in r-band derived from the dust maps of [@Schlegel-Finkbeiner-Davis-98]. $\Delta_{g-i}$ is the colour gradient defined as the difference in $g-i$ colour between the outer and inner regions of the galaxy. The inner region is defined to be the region within $R_{50}$ and the outer region is the region between $R_{50}$ and $R_{90}$. The estimator has been calibrated using samples of nearby galaxies ($0.025<z<0.05$) with H[i]{} line detections from the GALEX Arecibo SDSS Survey [GASS; @Catinella-10], and is demonstrated to provide unbiased H[i]{}-to-stellar mass ratio estimates, $M_{\mbox{H\sc i}}/M_\ast$, even for H[i]{}-rich galaxies.
As well as the H[i]{} mass fraction, we will also work with an ‘H[i]{} deficiency parameter’, $H_{def}$ which we define as the deviation in $\log(M_{HI}/M_\ast)$ from the value predicted from the mean relation between $\log(M_{HI}/M_\ast)$ and galaxy mass $M_\ast$ and stellar surface mass density $\mu_\ast$ [see @Li-12b]: $$\label{eqn:hidef}
H_{def}=\log(M_{\mbox{H{\sc i}}}/M_\ast)-\log(M_{\mbox{H{\sc
i}}}/M_\ast)|(M_\ast,\mu_*)$$ where $\log(M_{\mbox{H{\sc
i}}}/M_\ast)$ is estimated by equation (\[eqn:hiplane\]) and $$\label{eqn:hiplane2} \log(M_{\mbox{H{\sc
i}}}/M_\ast)|(M_\ast,\mu_*)= -0.227\log M_\ast - 0.646 \log \mu_*+7.166.$$ The relation between the stellar mass of a galaxy and its structural parameters such as stellar surface mass density and concentration index have been found to depend extremely weakly on environment [e.g. @Kauffmann-04; @Weinmann-09]. The H[i]{} deficiency parameter is thus the most direct measure of the degree to which gas has been depleted in a given cluster galaxy.
Because the photometric estimator in equation (\[eqn:hiplane\]) has been calibrated using the GASS sample, one might question whether it is still valid for galaxies in cluster environments. @Cortese-11 used a sample of $\sim300$ nearby galaxies to investigate the effect of the environment on H[i]{} scaling relations and found that Virgo cluster galaxies still lie on the same ‘plane’ relating H[i]{} gas mass fraction to stellar surface mass density and colour, even though they are significantly offset towards lower gas content compared to field galaxies.
In Figure \[fig:residual\] we use $\sim8000$ galaxies from the Arecibo Legacy Fast ALFA survey[ALFALFA; @Giovanelli-05] to test whether L12 estimator exhibits any significant environmental dependence. We plot the residual in the estimated H[i]{} mass fraction, defined as the difference between the estimated $\log({M_{HI}/M_\ast)}$ value and the observed value, as a function of overdensity, $\ln(1+\delta)$. The overdensity parameter, $\delta$ has been estimated by @Jasche-10 through reconstruction of the 3D density field from the SDSS DR7 data. In the right-hand panel of the same figure we plot the histograms of the residual for subsets of galaxies selected by $\ln(1+\delta)$. We find that the residual in the estimated H[i]{} mass fraction exhibits a weak, but systematic trend with overdensity, in the sense that the gas fractions of galaxies in high-density regions are slightly underestimated. Cluster galaxies detected by ALFALFA that are included in the B06 group/cluster catalogue are plotted as coloured symbols in Figure \[fig:residual\] and the distribution of residuals is shown as a solid black line in the right hand panel. As can be seen, the H[i]{} content of cluster galaxies is underestimated by 0.1 dex on average, but the galaxies that cause this shift are the [*gas-rich*]{} rather than the gas-poor group/cluster members. This particular paper is focused on the physical processes that cause galaxies to become gas-deficient, so we will leave aside this curious phenomenon for the moment.
Results
=======
We begin by analyzing trends in the $NUV-r$ colours of galaxies as a function of cluster-centric radius. This is a directly observed (as opposed to inferred) quantity that has been found to correlate strongly with the H[i]{} gas mass fraction of nearby galaxies [@Catinella-10]. In Figure \[fig:nuv\_radius\_mustar\], we plot the [*difference*]{} in the median $NUV-r$ colour compared to galaxies of the same stellar mass and stellar surface density in the field, where the value of the “field galaxies” is given by the median value of the subsample of the NYU-VAGC [sample dr72]{} in the same coverage of stellar mass and stellar surface density. Results are plotted as a function of distance from the BCG, scaled to the virial radius of the cluster. The four panels show results for galaxies in different stellar mass intervals. [^3] Black lines show results for all galaxies in a given mass interval, while red and blue curves show results for high and low values of $\mu_\ast$, respectively. Errors are estimated using a standard bootstrap resampling technique.
In all stellar mass intervals, the difference in $NUV-r$ color increases with decreasing cluster-centric distance. The strength of the effect depends strongly on stellar mass. For galaxies with stellar masses less than $10^{10}
M_{\odot}$, the $NUV-r$ colour reddens by more than 2 magnitudes from the virial radius to the center of the cluster. For galaxies with stellar masses greater than $3 \times 10^{10} M_{\odot}$, the reddening is only $\sim 0.5$ magnitudes. The very large differences between the blue and red curves in Figure \[fig:nuv\_radius\_mustar\] show that the effect at fixed mass is mainly driven by the dependence of the reddening on surface mass density. We will come back to this point in more detail later.
We now turn to trends in “inferred” H[i]{} mass fraction with cluster-centric radius. In the left panel of Figure \[fig:fg\_radius\] we plot the difference in the median H[i]{} mass fraction with respect to field galaxies of the same stellar mass, as a function of scaled distance from the brightest cluster galaxy (BCG). In the right panel of Figure \[fig:fg\_radius\] we show the same plot, except that the H[i]{} mass fractions are normalized to field galaxies of the same stellar mass and stellar surface mass density, i.e. this is a plot of the H[i]{} deficiency parameter defined in the previous section. Figure \[fig:fg\_radius\] shows that H[i]{} mass fractions drop by a factor of 2-3 at the centers of clusters with respect to “similar” galaxies in the field. The effects are stronger for less massive galaxies. At all masses, the strongest decrease in H[i]{} gas fraction occurs just at the virial radius. Interestingly, the decrease in gas fraction with radius is quite shallow in the central region of the cluster. In order to understand whether this is caused by possible offsets between the BCG and the true center of the potential well, we have repeated our analysis, using the stellar mass-weighted center of the clusters instead of the BCG as their center, and found very similar results. We also note that recent studies of X-ray selected clusters have shown that the offset between the BCG and the peak in X-ray maps is typically $\sim20$ kpc [e.g. @vonderLinden-12], much smaller than the scale ($\sim200$ kpc) where our gas fraction profiles become shallow. Therefore the offset from the ture center is unlikely the reason for the shallow profiles seen from our figure.
We now divide the galaxies in each stellar mass interval into two subsets with higher and lower surface mass densities, $\log\mu_\ast$, and repeat the analysis. The results are shown in Figure \[fig:fg\_radius\_mustar\]. The decrease in H[i]{} gas content with cluster-centric distance is always stronger for galaxies with low densities. For galaxies with stellar surface densities greater than $10^{9} M_{\odot}$ kpc$^{-1}$, there is essentially no change in inferred H[i]{} mass fraction with radius.
Figure \[fig:gas\_bin\] allows the reader to compare the dependence of H[i]{} gas depletion on stellar surface density and on stellar mass at a given clustercentric distance. We plot the mean difference in H[i]{} mass fraction with respect to field galaxies of the same stellar mass and stellar surface mass density in the plane of $\mu_\ast$ versus $M_\ast$. The four panels show results for galaxies located at different cluster-centric distances. We used a two-dimensional adaptive binning technique to adjust the cell size so that each cell includes a fixed number of galaxies (20, 40, 80 and 80, respectively, for the four cluster-centric distances). It is clear that the decline in H[i]{} gas content depends more strongly on stellar surface density than on stellar mass. This is true at all radii within the virial radius of the cluster, but is especially pronounced in the first panel of Figure \[fig:gas\_bin\], which shows results for galaxies in the inner cores of the clusters.
Dependence on halo mass
-----------------------
There are a number of possible reasons why galaxies with low stellar surface densities may lose their gas more efficiently in clusters.
The intracluster medium may exert a drag force on the gas in galactic disks. If the density of the gas in the disk scales with the density of its stars, then one might expect ram-pressure to act more effectively on galaxies with low stellar surface densities. [^4] Ram-pressure scales with the square of the velocity of the galaxy through the surrounding galaxy, so if this is the main process at work, one would also expect to see it operate more efficiently in clusters with higher velocity dispersions.
Alternatively, encounters between galaxies in groups and clusters may result in stars and other material being pulled out of galaxies. Tidal stripping is most effective when galaxies encounter each other with relative velocities comparable to the internal velocity dispersion of their stars. Tidal stripping is thus not thought to be effective in rich cluster environments, where galaxies are moving with average velocities in excess of 1000 km/s.
In Figure \[fig:fg\_radius\_mustar\_sigmav\], we divide the galaxies in each stellar mass and stellar surface density subsample into two further subsets according to the velocity dispersion of their associated clusters. Our main conclusion is that we do not find any dependence of the gas-radius relation on cluster velocity dispersion, even for low stellar surface mass density galaxies. Considering the relatively large scatter in the velocity dispersion-halo mass relation [e.g. @Weinmann-10], we would like to point out that our result does not neccessarily imply that there is no trend in the gas-radius relation with dark matter halo mass.
Summary
=======
In this paper, we have analyzed galaxies in 300 nearby groups and clusters identified in the Sloan Digital Sky Survey using a photometric gas mass indicator that is useful for estimating the degree to which the interstellar medium of a cluster galaxy has been depleted. We study the radial dependence of inferred gas mass fractions for galaxies of different stellar masses and stellar surface densities. Our main results may be summarized as follows.
- The H[i]{} mass fraction of galaxies decrease by a factor of $2-3$ from the outskirts of clusters to their centres. The decrease in gas fraction is most pronounced around the virial radius and is strongest for low mass galaxies.
- At fixed stellar mass and at fixed clustercentric distance, the depletion of gas in cluster galaxies clearly depends on the stellar surface mass density of the galaxy, in the sense that low density galaxies are more H[i]{} deficient.
- An analysis of depletion trends in the two-dimensional plane of stellar mass $M_*$ 1and stellar mass surface density $\mu_*$ reveals that gas depletion at fixed clustercentric radius is more sensitive to the density of a galaxy than to its mass.
- The gas mass fraction-radius relations exhibit little dependence on the velocity dispersion of the cluster.
We suggest that low density galaxies are more easily depleted of their gas, because they are more easily affected by ram-pressure and/or tidal forces.
In recent work , @Fabello-12 compared the environmental density dependence of the atomic gas mass fractions of nearby galaxies with that of their central and global specific star formation rates. For galaxies less massive than $10^{10.5}M_\odot$, the authors found both the H[i]{} mass fraction and the sSFR to decrease with increasing density, with the H[i]{} mass fraction exhibiting stronger trends than the sSFR. This was interpreted as evidence for ram-pressure stripping of atomic gas from the outer disks of low-mass galaxies. The authors also compared their results with predictions from the semi-analytic model (SAM) of @Guo-11. They found the opposite trend in the models: the decline in H[i]{} mass fraction with density was weaker than the decline in sSFR. This indicated that the recipe of gas stripping assumed in the current SAMs, in which only the diffuse cold gas surrounding satellite galaxies is stripped (often called “strangulation”), is insufficient, and ram-pressure stripping of the cold interstellar medium is likely to play a significant role.
[@Li-12b] studied the bias in the clustering of H[i]{}-rich and H[i]{}-poor galaxies with respect to galaxies with normal H[i]{} content on scales between 100 kpc and $\sim 5$ Mpc, and compared the results with predictions from the current semi-analytic models (SAMs) of galaxy formation of @Fu-10 and @Guo-11. They found that, for the H[i]{}-deficient population, the strongest bias effects arise when the H[i]{} deficiency is defined in comparison to galaxies of the same stellar mass and size. This is not produced by the SAMs, where the quenching of star formation does not depend on the internal structure of galaxies. The authors proposed that the disagreement between the observations and the models might be resolved, if processes such as ram-pressure stripping, which depend on the density of the interstellar medium (ISM), are included in the models.
The results of this paper point in much the same direction. The finding that is somewhat puzzling is that the gas mass fraction-radius relation does not depend significantly on the velocity dispersion of the cluster, as might be expected if ram-pressure stripping is the main process at work in groups and clusters. One possible solution is that [*both*]{} tidal and ram-pressure stripping are at work and they act on the galaxy population in such a way as to cancel out any significant velocity-dispersion dependent effects. There have been recent detailed studies of ram-pressure stripping and tidal interactions in individual galaxy groups with extensive coverage of the X-ray emitting hot gas from $Chandra$ observations and detailed H[i]{} maps from Very Large Array (VLA) mosaic observations that have indicated that both ram-pressure and tidal stripping can be at work at the same time in a single group [@Rasmussen-12]. Next generation X-ray observations from the eROSITA satellite and wide-field H[i]{} surveys planned at the Westerbork telescope [APERTIF; @Verheijen-08] and at the Australian SKA Pathfinder telescope (ASKAP) will clarify the relative importance of these processes as a function of dark matter halo mass and location of galaxy within their halos.
Acknowledgments {#acknowledgments .unnumbered}
===============
WZ and CL thank the Max-Planck Institute for Astrophysics (MPA) for warm hospitality while this work was being completed. It is a pleasure to thank the anonymous referee for helpful comments. This work is supported by the Chinese National Natural Science Foundation grants 10903011 and 11173045, and the CAS/SAFEA International Partnership Program for Creative Research Teams (KJCX2-YW-T23). CL acknowledges the support of the 100 Talents Program of Chinese Academy of Sciences (CAS), Shanghai Pujiang Program (no. 11PJ1411600) and the exchange program between Max Planck Society and CAS. GK thank the Aspen Center for Physics and the NSF Grant \#1066293 for hospitality during the writing of this paper. TX acknowledges the support of the LAMOST postdoctoral fellowship.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
\[lastpage\]
[^1]: E-mail:xtwfn@bao.ac.cn
[^2]: http://www.mpa-garching.mpg.de/SDSS/DR7/
[^3]: For a given stellar mass subsample we use the clusters below the same redshift limit of the galaxy subsample. As a result, for the lowest mass range, $9.6<\log_{10}(M_\ast/M_\odot)<10.0$, only the clusters with $z<0.04$ is used, while for the other three mass ranges all the clusters with $z<0.06$ are used.
[^4]: We note that in practice, low density galaxies have larger H[i]{} gas mass fractions [@Catinella-10], so this simple assumption is unlikely to hold in detail.
|
---
abstract: 'Many complex networks exhibit vulnerability to spreading of epidemics, and such vulnerability relates to the viral strain as well as to the network characteristics. For instance, the structure of the network plays an important role in spreading of epidemics. Additionally, properties of previous epidemic models require prior knowledge of the complex network structure, which means the models are limited to only well-known network structures. In this paper, we propose a new epidemiological SIR model based on the continuous time Markov chain, which is generalized to any type of network. The new model is capable of evaluating the states of every individual in the network. Through mathematical analysis, we prove an epidemic threshold exists below which an epidemic does not propagate in the network. We also show that the new epidemic threshold is inversely proportional to the spectral radius of the network. In particular, we employ the new epidemic model as a novel measure to assess the vulnerability of networks to the spread of epidemics. The new measure considers all possible effective infection rates that an epidemic might possess. Next, we apply the measure to correlated networks to evaluate the vulnerability of disassortative and assortative scale-free networks. Ultimately, we verify the accuracy of the theoretical epidemic threshold through extensive numerical simulations. Within the set of tested networks, the numerical results show that disassortative scale-free networks are more vulnerable to spreading of epidemics than assortative scale-free networks.'
author:
- Mina Youssef
- Caterina Scoglio
title: 'Generalized individual-based epidemic model for vulnerability assessment of correlated scale-free complex networks'
---
I. Introduction
===============
Complex networks, such as social networks [@WS:98; @ASBS:00; @LEASA:01; @GN:02], food-webs [@S:01], biological networks [@JTAOB:00; @JMBO:01; @FW:00; @WM:00; @MS:02], the world-wide-web (WWW), and the Internet, largely represent many systems from a topological structure point of view. Inherently, a complex network has many dynamics that describe the state of the system and its functionality. Among these dynamics, the spread of epidemics process has attracted the attention of many researchers in multidisciplinary fields. Essentially, epidemics, like human contagious, are represented by the standard compartmental epidemiological model so-called susceptible/infected/removed (SIR), which emulates the spread process.
The SIR model analytically reveals how an individual’state is changed among the three SIR states in the complex networks. To clarify, during the spread of an epidemic, an individual is in one of the three SIR states. First, a susceptible individual can receive the infection from an infectious neighbor and become infected. Accordingly, the infected individual becomes infectious with infection rate $\beta$. Also, an infected individual can cure itself with a cure rate $\delta$. The curing process represents either the death (removal) or the complete recovery of the individual after the infection. Additionally, the ratio between $\beta$ and $\delta$ is called the effective infection rate. An epidemic threshold $\tau$ is a specific value of the effective infection rate above which an epidemic outbreak takes place. Moreover, it is a function of the network characteristics. Different SIR models are applied to some classes of complex networks [@BPV:03; @N:02a; @MSV:02; @BBSV:05; @VM:09; @FDP:09; @KR:07; @ML:01; @YWRBSWZ:07; @BMNP:09; @TTV:98] depending on the network characteristics. Early SIR models are homogeneous, i.e., all individuals have a similar probability of being infected and infectious. On the other hand, SIR models are also applied on structured networks considering the local connectivity of the network’s individuals. For example, scale-free (SF) networks, which are networks owning power-law node degree distribution $P(k) \sim k^{-2-\nu}$ with $0<\nu\leq1$, show a high level of vulnerability to the spreading of epidemics due to highly heterogeneous node degrees distribution when the minimum node degree is greater than two [@BPV:03]. In addition, the spread of epidemics was studied on correlated networks and uncorrelated networks separately. Thus, we concluded that properties (e.g. epidemic threshold) of previous models are not generalized, and therefore the properties depend on the network structure (e.g. scale-free, small-world, correlated, uncorrelated, regular, exponential, ... etc). Moreover, given the network topology and a suitable SIR model, assessing the vulnerability of the network with respect to the spread of epidemics is difficult. In fact, the epidemic threshold is not always a complete vulnerability measure since it is a binary indication of the epidemic outbreak, and it does not account for the number of infected individuals. Moreover, it has been proven that an epidemic prevails on any SF network regardless of its node degree correlation due to the absence of the epidemic threshold in the limit of a very large number of individuals residing in the network [@BPV:03]. However, within the class of SF networks, correlated and uncorrelated topologies exist, and they behave differently with respect to spread of epidemics. Therefore, the epidemic threshold is not an adequate measure, and consequently, vulnerability assessment becomes a tough task. In this paper, we propose a novel network vulnerability assessment method. The new method considers all possible effective infection strengths that can harm the network. We focus our study on SF correlated networks to evaluate the vulnerability of disassortative and assortative SF networks. Next, we present a novel individual-based SIR model, which is inspired by the Markov chain approach. We separately study the state of each individual during the infection process, revealing the role of the individual’s local connectivity in spreading the infection across the network. Although the exact SIR model, based on the Markov chain stochastic process, describes the global change in the state probabilities of the network, it is limited to small networks due to the exponential divergence in the number of possible network states $3^{N}$ with the growth of network size $N$. Instead, our new model aims to reduce the complexity of the problem and to offer insights into the epidemic spreading mechanism. Through the new SIR model, we study the spread of epidemics on any type of network regardless of its topological structure. Finally, we analytically derive the epidemic threshold for the new model. We find that the new epidemic threshold is inversely proportional to the spectral radius $\lambda_{max}$ (the supremum eigenvalue within the eigenvalue spectrum) of the network. We perform extensive simulations to validate the new SIR model and the new epidemic threshold. Quantitatively, we show that disassortative SF networks are more vulnerable to the spread of epidemics than are assortative SF networks given the same number of individuals $N$ and the same number of connections $L$. The paper is organized as follows. In Sec. II, we shed some light on the homogeneous and heterogeneous mixing hypothesis models that exist in the literature. In Sec. III, we introduce the new individual-based SIR model, and we show how it is inspired by the Markov chain model. We also derive the new epidemic threshold, we determine the condition of the existence of a maximum value of number of infected individuals, and we study the role of the network eigenvalue spectrum on the spread of epidemics. In Sec. IV we introduce the new vulnerability assessment of any complex networks, and validate and discuss our analytical findings through extensive simulations. Finally, in Sec. V we conclude our work.
II. SIR Model {#sec:model}
=============
The science of the spread of epidemics is based on compartmental models that assume individuals are classified into non-intersecting sets [@AM:92; @M:93]. Thus, the classical susceptible/infected/removed SIR model characterizes diseases that lead to either immunization or death of individuals. The infected individuals are in the infected set, the healthy ones are in the susceptible set, and the cured or removed ones are in the removed set. Initially, a small number of infected individuals exist that try to infect their susceptible (healthy) neighbors. After receiving the infection, susceptible individuals become infected, and later they try to infect their susceptible neighbors. In this case, infected individuals are infectious. Subsequently, every infected individual either is cured due to immunization or removed due to death. This process was early described by the homogeneous mixing SIR model [@AM:92], which evaluates the change in the susceptible $s(t)$, such that infected $i(t)$ and removed $r(t)$ population densities with time, while preserving the overall density at any time $t$, $s(t)+i(t)+r(t)=1$. In the homogeneous mixing model, the rates of changes in densities are governed by the following continuous time differential equations:
$$\begin{aligned}
\frac{ds(t)}{dt} & = & -<k> \beta i(t) s(t), \\
\frac{di(t)}{dt} & = & -\delta i(t) + <k> \beta i(t) s(t), \\
\label{eq:hrt}
\frac{dr(t)}{dt} & = & \delta i(t).\end{aligned}$$
These differential equations interpret the infection and cure processes. Initially, the spreading process starts with a small infected density $i(0)\simeq0$, the susceptible density is almost one $s(0)\simeq1$, and the removed density is zero $r(0)=0$. Every infected individual infects on average $<k>$ susceptible neighbors, each with an infection rate $\beta$, where $<k>=\sum_{d}dp(d)$ is the average node degree (average number of contacts), and $p(d)$ is the probability of having an individual with degree $d$. Following the differential Eq. (\[eq:hrt\]), an infected individual is removed at a rate $\delta$. The removed density increases with time until it reaches a certain density level depending on the strength of the epidemic. A non-zero epidemic threshold exist and it is equal to $<k>^{-1}$. If the effective infection rate $\frac{\beta}{\delta}$ is above the threshold, the epidemic prevails in the network. On the other hand, if the effective infection rate is below the threshold, the infected density is very small in the thermodynamic limit. Since on average every infected individual infects a constant number of neighbors, the homogeneous model does not count the heterogeneity in the node degrees of individuals in the network. Another model in the literature is the heterogeneous mixing SIR model [@MSV:02; @BPV:03], which was proposed to overcome the shortcomings of the homogeneous model. In this model, individuals are classified according to their node degrees. Thus, for a given node degree $d$, the states’ densities $s_{d}(t)$, $i_{d}(t)$and $r_{d}(t)$ evolve with time $t$, and their sum is constant, such that $s_{d}(t)+i_{d}(t)+r_{d}(t)=1$. The rates of changes in the three states for a given node degree $d$ are governed by the following set of differential equations:
$$\begin{aligned}
\frac{ds_{d}(t)}{dt} & = & -d \beta s_{d}(t) \theta(t),\\
\frac{di_{d}(t)}{dt} & = & -\delta i_{d}(t) + d \beta s_{d}(t) \theta(t),\\
\frac{dr_{d}(t)}{dt} & = & \delta i_{d}(t).\end{aligned}$$
The probability that a link is pointing to an infected individual is given by the factor $\theta(t)$, where $\theta(t)$ is found to be $\frac{\sum_{d}d p(d)i_{d}(t)}{<k>}$. This model was applied to both uncorrelated and correlated complex networks, leading to further analysis of the epidemic threshold. For uncorrelated networks, the epidemic threshold is $\tau^{ucr}=\frac{<k>}{<k^{2}>-<k>}$, where $<k^{2}>$ is the second moment of the node degrees. On the other hand, the epidemic threshold for correlated networks is $\tau^{cr}=\frac{1}{\overline{\Lambda}_{m}}$, where $\overline{\Lambda}_{m}$ is the maximum eigenvalue of the connectivity matrix $\overline{C}_{dd'}=\frac{d(d'-1)}{d'}p(d'\mid d)$. Although this model considers the heterogeneous connectivity in the networks, it does not reveal the state of each individual in the network. It only reflects the evolution of the densities over time for a given node degree, while neglecting the states of individuals within the same node degree.
III. Individual-based SIR model {#sec:individualbased}
===============================
In this paper, we present a new individual-based SIR model in which each node can be either susceptible $S$, infected $I$ or recovered $R$ with a given probability for each state. The new model is inspired by the continuous time Markov chain SIR model. However, instead of considering the combinatorial states of the individuals in the network, we study each individual deliberately [@MOK:09], by decomposing the infinitesimal $Q_{3^{N} \times 3^{N}}$ matrix to $N$ infinitesimal matrices, each with three states as follows:
$$\begin{aligned}
q_{k}(t)=
\left[
\begin{array}{ccc}
-\beta \sum_j a_{k,j} 1_{[i_{j}(t)=1]} & \beta \sum_j a_{k,j} 1_{[i_{j}(t)=1]} & 0 \\
0 & -\delta & \delta \\
0 & 0 & 0
\end{array}\right]
\nonumber\end{aligned}$$
where $a_{k,j}$ is the binary entry in the network adjacency matrix, representing the existence of a contact between individual $k$ and individual $j$, and the indicator function $1_{[i_{j}(t)=1]}=1$ represents the event that individual $j$ is infected and zero otherwise. In this model, we replace the actual event with its effective probability, and therefore the event $i_{j}(t)=1$ is replaced by $I_{j}(t)=p(i_{j}(t)=1)$. For every individual $k$, we derive the system of differential equations as follows:
$$\frac{dState_{k}(t)}{dt}=q_{k}^{T}(t) State_{k}(t)$$
where $q_{k}^{T}(t)$ is the transpose of $q_{k}(t)$. The obtained differential equations are
$$\begin{aligned}
\label{eq:Seq}
\frac{dS_{k}(t)}{dt} & = & -S_{k}(t) \beta \sum_{j} a_{k,j} I_{k}(t), \\
\label{eq:Ieq}
\frac{dI_{k}(t)}{dt} & = & S_{k}(t) \beta \sum_{j} a_{k,j} I_{k}(t) - \delta I_{k}(t), \\
\label{eq:Req}
\frac{dR_{k}(t)}{dt} & = & \delta I_{k}(t).\end{aligned}$$
At any time $t$, each individual will be in any of the states with total probability of 1, $S_{k}(t)+I_{k}(t)+R_{k}(t)=1$. In addition, the sum of rates of changes in the state probabilities is zero $\frac{dS_{k}(t)}{dt}+\frac{dI_{k}(t)}{dt}+\frac{dR_{k}(t)}{dt}=0$. Therefore, we only solve $2N$ simultaneous differential equations instead of $3N$. Figure \[fig:timePlot\] shows the time evolution of new infected individuals in assortative and disassortative SF networks with different $<k>$=4, 8, 12, 16 and 20 given $\beta=0.1$ and $\delta=0.2$.
![\[fig:timePlot\] Normalized new infected individuals as a function of time for $\beta=0.1$ and $\delta=0.2$ given correlated networks with $N=10^{4}$ and different average node degree $<k>$. Two different types of correlated networks are simulated (a) assortative SF networks, and (b) disassortative SF networks. The peak of the new infected individuals in disassortative networks leads to the corresponding peak in asssortative networks.](timePlot){width="9.5cm"}
Steady-state population
-----------------------
To evaluate the behavior of the system of differential equations at the steady state, we equate the differential Eqs. in (\[eq:Seq\] - \[eq:Req\]) to zero. The steady-state probability of infection $I_{\infty}$ is always zero, while the steady-state probability of recovery $R_{\infty}$ always have a positive value, which is $\delta \int_{0}^{t_{I_{new}}=0} u^{T}I(z) dz$ where $t_{I_{new}}=0$, the time at which there are no more new infected individuals in the network, and $u^{T}$ is the transpose of a vector of 1’s. On the other hand, the steady-state probability of being susceptible $S_{\infty}$ is zero if, and only if, $R_{\infty}=1$, otherwise, it is a positive value.
Epidemic threshold
------------------
The epidemic threshold is the condition that the epidemic prevails in the network. To compute the threshold, we follow the analysis presented in [@BPV:03]. We assume that the initial fraction of infected individuals is very small and therefore $S_{k}(0) \backsimeq 1$. The differential Eq. (\[eq:Ieq\]) is written as follows:
$$\frac{dI_{k}(t)}{dt} \backsimeq \sum_{j} \tilde{L_{k,j}} I_{j}(t)
\label{eq:approxdifeq}$$
where the element $\tilde{L}_{k,j}=\beta a_{k,j} - \delta\delta_{k,j}$ is the entry of the Jacobian matrix $\tilde{L}=\{\tilde{L}_{k,j}\}=\beta A - \delta I_{N \times N}$, and $\delta_{k,j}$ is the Kronecker delta function and equals 1 forall $k=j$. Since any element $a_{k,j}$ of the symmetric adjacency matrix $A$ is either 0 or 1, and according to Frobenius theorem, the maximum eigenvalue $\lambda_{max,A}$ of $A$ is positive and real, the eigenvalues of the matrix $\tilde{L}$ have the form of $\beta \lambda_{i,A} - \delta$, and the eigenvectors are the same as those for the adjacency matrix $A$. Thus, the stability condition of the solution $I=0$ of the differential Eq. (\[eq:approxdifeq\]) is $-\delta +\beta \lambda_{max,A}<0$, and the SIR threshold for any undirected network becomes:
$$\frac{\beta}{\delta} < \frac{1}{\lambda_{max,A}} = \tau
\label{eq:threshold}$$
The threshold states that whenever $\frac{1}{\lambda_{max,A}}$ is greater than the effective infection rate $\frac{\beta}{\delta}$, an epidemic does not prevail in the network.
The existence of a maximum number of infected individuals
---------------------------------------------------------
The number of infected individuals increases in time following a certain profile [@BBV:08] depending on the infection strain. Below, we derive the condition for which a maximum number of infected individuals occurs, and how the condition is related to the epidemic threshold. Let $u^{T}I(t)=\sum_{k}I_{k}(t)$ be the total number of infected individuals in the network. The existence of a maximum value for $I(t)$ is determined through $\frac{du^{T}I(t)}{dt}=\sum_{k}\frac{dI_{k}(t)}{dt} = 0$, and we obtain:
$$\label{eq:maxI}
\sum_{k} \left[ S_{k}(t) \beta \sum_{j} a_{k,j} I_{j}(t) - \delta I_{k}(t)\right] = 0$$
By rewriting Eq. (\[eq:maxI\]) in the matrix form, we obtain the following equation:
$$\label{eq:maxzero}
\left[\beta S^{T}(t) A - \delta u^{T} \right]I(t) = 0$$
Eq. (\[eq:maxzero\]) suggests the possible solutions for $I(t)$ are either $I(t)$ equals zero, which happens at the steady state, or $\beta S^{T}(t) A - \delta u^{T}$ equals zero. The second solution derives a condition for the existence of a positive maximum value of $I(t)$. Consequently, the second solution $A S(t) = \frac{\delta}{\beta} u$ is on the form of $Wx=\rho x$, where $x$ and $\rho$ are an eigenvector and an eigenvalue of the matrix $W$, respectively. The vector $S(t)$ is equal to the vector $u$ only if $\frac{\delta}{\beta}$ is equal to the maximum eigenvalue $\lambda_{max,A}$ of $A$, which follows Frobenius theorem and takes place for $t \to 0$ and $S(0) \to 1$. Moreover, this solution proves the existence of the epidemic threshold shown in inequality (\[eq:threshold\]) whenever $\frac{\delta}{\beta} < \lambda_{max,A}$, and therefore the epidemic spreads in the network, and $S_{k}(t)\leq 1$ forall $k$.
The effect of the network spectrum
----------------------------------
To address the effect of the spectrum of the adjacency matrix $A$, we write the rate of change as a total fraction of infected individuals $u^{T}I(t)$ as follows:
$$\frac{du^{T}I(t)}{dt} = \beta (u^{T}-I^{T}(t)-R^{T}(t)) A I(t) - \delta u^{T}(t) I(t).
\label{eq:spectrum1}$$
Denote the vector of node degrees $D=u^{T} A$, and the eigenvalue decomposition of the adjacency matrix $A=U\Lambda U^{T}$. We rewrite the differential Eq. (\[eq:spectrum1\]) as follows:
$$\begin{aligned}
\frac{du^{T}I(t)}{dt} = && (\beta D - \delta u)^{T} I(t) - \beta (U^{T}I(t))^{T} \Lambda (U^{T}I(t)) \nonumber \\
&& - \beta (U^{T}R(t))^{T} \Lambda (U^{T}I(t))
\label{eq:spectrum2}\end{aligned}$$
Let $x_{j}$ be the $j^{th}$ element in the vector $U^{T}I(t)$, and let $y_{j}$ be the $j^{th}$ element in the vector $U^{T}R(t)$. We rewrite the differential equation as follows:
$$\frac{du^{T}I(t)}{dt} = (\beta D - \delta u)^{T} I(t) - \beta \sum_{j=1}^{N} \lambda_{j} x_{j}^{2} - \beta \sum_{j=1}^{N} \lambda_{j} x_{j} y_{j}
\label{eq:spectrum3}$$
To relate $I_{max}$ with the spectrum $\lambda_{j}$ and the eigenvectors $U$, let $\frac{du^{T}I(t)}{dt}$ equal zero, and therefore we obtain the following equation:
$$\sum_{k=1}^{N} (d_{k} - \frac{\delta}{\beta}) I_{k_{max}}=\sum_{j=1}^{N} \lambda_{j} x_{j}^{2} - \sum_{j=1}^{N} \lambda_{j} x_{j} y_{j}
\label{eq:spectrum4}$$
Since the matrix $A$ is symmetric, we can see that $\lambda_{max}$ is a positive eigenvalue and therefore the dominant eigenvalue within the spectrum, and elements of the corresponding eigenvector are positive as well. Eq. (\[eq:spectrum4\]) states that as $\delta$ decreases, the LHS increases, and so $I_{max}$ increases with the eigenvectors corresponding to $\lambda_{max}$, while on the other hand, the corresponding $R$ decreases.
IV. Vulnerability measure {#sec:vulnerability}
=========================
We employ the individual-based SIR model to assess the vulnerability of a complex network such that the total number of new infected individuals reflects the vulnerability of the network to the spread of epidemics given any infection strength. In this section, we introduce a new vulnerability assessment measure $\Psi$ with respect to the spread of epidemics, and we define it as the ability of an epidemic to prevail in a complex network given all possible effective infection rates. Mathematically, we define the assessment measure $\Psi$ by fixing $\beta=\frac{1}{\lambda_{max,A}}$ and for a given cure rate $\delta$, the total number of new infected individuals is $\int_{0}^{t_{I_{new}=0}}\sum_{k} S_{k}(t) \beta \sum_{j} a_{k,j} I_{k}(t,\delta) dt$. By integrating over the defined range of cure rate $0 \leq \delta \leq 1$, we obtain $\Psi$ as follows:
$$\label{eq:psi}
\Psi = \int_{0}^{1} \int_{0}^{t_{I_{new}=0}}\sum_{k} S_{k}(t) \beta \sum_{j} a_{k,j} I_{k}(t,\delta) dt d\delta$$
Figures \[fig:assortEpsy\] and \[fig:disassortEpsy\] show the numerical simulations of the spread of an epidemic for $0\leq\frac{\delta}{\beta}\leq \lambda_{max,A}$ on assortative and disassortative SF networks given different average node degrees $<k>$, where $\frac{\delta}{\beta}$ is the inverse of the effective infection rate.
![\[fig:assortEpsy\] Normalized total of new infected individuals as a function of the inverse of effective infection rate for assortative SF networks given $N=10^{4}$ and different average node degree $<k>$. The curve starts from the point where $\frac{\delta}{\beta}=0$, and the normalized total new infected cases is 1, and then it decreases until it reaches the value zero when the value of $\frac{\delta}{\beta}$ equals the spectral radius of the network.](assortEpsy){width="9.0cm"}
![\[fig:disassortEpsy\] Normalized total of new infected individuals as a function of the inverse of effective infection rate for disassortative SF networks given $N=10^{4}$ and different average node degree $<k>$. The curve starts from the point where $\frac{\delta}{\beta}=0$, and the normalized total new infected cases is 1, and then it decreases until it reaches the value zero when the value of $\frac{\delta}{\beta}$ equals the spectral radius of the network.](disassortEpsy){width="9.0cm"}
Previous work [@KSSY:09] introduced a measure that takes into account the number of infected individuals at steady state for the susceptible/infected/susceptible compartmental model. We use the new measure $\Psi$ to evaluate the vulnerability of correlated networks, these in which node degree correlation is observed. They are also classified as assortative and disassortative networks. For example, social networks are classified as assortative networks, while technological and biological networks are classified as disassortative networks [@N:02b]. In assortative networks, individuals of small node degree are connected with other individuals of small node degree, while individuals with large node degree are connected with other individuals with large node degree. On the other hand, the opposite is true for disassortative networks. Pearson assortativity coefficient [@N:02b; @SVV:03] was proposed to characterize the node degree correlation numerically. However, it does not give an accurate measure for networks with complicated degree correlation functions. To accurately describe the degree correlations, we evaluate the average connectivity of the neighbors of an individual $k$ by following the technique presented in [@SVV:01; @VSV:02; @BBV:08]:
$$d_{n,n,k}=\frac{1}{d_{k}}\sum_{j\in neighbors(k)} d_{j}$$
The average connectivity of neighbors of an individual is averaged overall of all individuals for a given node degree $d$, $$d_{n,n}(d)=\frac{1}{N_{d}}\sum_{k/d_{k}=d}d_{n,n,k}$$ where $N_{d}$ is the number of individuals of degree $d$. Figures \[fig:assort8knnkPlot\] and \[fig:disassort8knnkPlot\] show two examples for correlated networks, one for an assortative network and the other for a disassortative network, respectively.
![\[fig:assort8knnkPlot\] Node degree as a function of average neighbors connectivity $d_{n,n}(d)$ of individuals with the same node degree for a sample of an assortative SF network with $N=10^{4}$ and $<k>=8$. The node degree correlation is an increasing function for an assortative network.](assort8knnkPlot){width="9.0cm"}
![\[fig:disassort8knnkPlot\] Node degree as a function of average neighbors connectivity $d_{n,n}(d)$ of individuals with the same node degree for a sample of a disassortative SF network with $N=10^{4}$ and $<k>=8$. The node degree correlation is a decreasing function for a disassortative network.](disassort8knnkPlot){width="9.0cm"}
We focus on the vulnerability assessment of correlated SF networks. We generate assortative and disassortative SF networks using the algorithm in [@GZLBWZ:06]. The algorithm starts with a connected graph with $m_{0} \ll N$ individuals. Every new individual is connected to the already existing individuals through two stages: In the first stage, a new individual is connected to an existing individual $k$ with probability $\pi_{k}=\frac{d_{k}}{\sum_{j}d_{j}}$; in the second stage, a new link between the new individual and one of the neighbors $s$ of the chosen individual $k$ in the first stage is added with probability $p_{s}=\frac{d_{s}^{\alpha}}{\sum_{v\in \Gamma_{k}} d_{v}^{\alpha}}$, where $\alpha$ is an assortative tunning coefficient, and $\Gamma_{k}$ is the set of neighbors of individual $k$ chosen in the first stage. To simplify the evaluation of numerical results, both the constructed assortative and disassortative networks have the same number of individuals $N$ and links $L$ with average node degrees $<k>$=4, 8, 12, 16 and 20. Next, we apply the new measure $\Psi$ in Eq. (\[eq:psi\]) to quantitatively assess the vulnerability of both assortative and disassortative networks. All the simulations are averaged over 10 runs. Table \[tab:psimeasure\] summarizes the values of $\Psi$ for both types of networks for different average node degrees. We notice that the disassortative networks have higher values of vulnerability measure $\Psi$ than those of assortative networks regardless of the average node degree value. In addition, the $\Psi$ value increases with increases in $L$ (i.e.$<k>$) due to the increase in the effective spreading rate of any infected individual for its susceptible neighbors. Moreover, in Fig. \[fig:timePlot\], we observe that the peaks of normalized new infected individuals in disassortative networks are greater than the peaks in assortative networks; meanwhile, the peaks in disassortative networks lead the corresponding peaks in assortative networks. In other words, an epidemic widely spreads in disassortative networks, and it spread faster than in assortative networks. Fig. \[fig:timePlot\] also reveals insights about any future immunization strategy that could be applied to both networks. For example, we can assume that immunization strategies on assortative and disassortative networks are different. Therefore, in assortative SF networks, mitigtion strategies are going to be more effective than in disassortative SF networks.
[|c|c|c|]{} $<k>$ & Assortative networks & Disassortative networks 4& 3.32 & 6.54 8& 6.54 & 12.58 12& 9.76 & 17.65 16& 12.98 & 23.47 20& 16.22 & 28.68
V. Conclusions {#sec:conclusions}
==============
In this paper, we have reviewed the well-known homogeneous and heterogeneous SIR models, and we have shown how both models do not evaluate the state of every individual in the complex networks. To account for this, we have presented a new individual-based SIR model that is derived from the continuous time Markov chain model. The new model evaluates the probability of infection of every individual separately considering the probability of infection of the individual’s neighbors. Unlike previous models in the literature whose their properties require a priori knowledge of the topological structure of the network under study, the new individual-based model can be applied to any type of network regardless its structure. We have also derived the epidemic threshold above which an epidemic prevails in the network. We found that the reciprocal of the spectral radius of the complex network is the epidemic threshold showing the role of the network characteristics in the spread of epidemics. In addition, we have shown the condition for the existence of a maximum number of new infected individuals, and how it is related to the epidemic threshold. Moreover, we have shown that the spectral radius and its corresponding eigenvector of the complex network and the effective infection rate determine the maximum number of the new infected individuals. Furthermore, we have presented a new technique $\Psi$ to quantitatively measure the vulnerability of any type of network structures. We have applied the new measure on assortative and disassortative SF networks, and through numerical simulations we have shown that disassortative scale-free networks are more vulnerable than assortative scale-free networks. The new SIR model and its properties could have implications for many systems that are viewed as complex networks, and the new measure could rank different networks based on their vulnerability with respect to spread of epidemics.
This work was partially supported by National Agricultural Biosecurity Center NABC at Kansas State University.
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---
abstract: 'We investigate the possibility that the observed behavior of test particles outside galaxies, which is usually explained by assuming the presence of dark matter, is the result of the dynamical evolution of particles in higher dimensional space-times. Hence, dark matter may be a direct consequence of the presence of an extra force, generated by the presence of extra-dimensions, which modifies the dynamic law of motion, but does not change the intrinsic properties of the particles, like, for example, the mass (inertia). We discuss in some detail several possible particular forms for the extra force, and the acceleration law of the particles is derived. Therefore, the constancy of the galactic rotation curves may be considered as an empirical evidence for the existence of the extra dimensions.'
author:
- 'M. E. Kahil'
- 'T. Harko'
title: 'Is dark matter an extra-dimensional effect?'
---
Introduction
============
There is a large amount of observational evidence showing that the standard gravitational theories cannot describe correctly the large scale dynamics of massive astrophysical systems. The observed rotational curves of spiral galaxies cannot be explained by applying Newtonian or general relativistic mechanics to the visible matter in galaxies and clusters. Neutral hydrogen clouds are observed at large distances from the galactic center, much beyond the extent of the luminous matter. A large number of independent observations have shown that the rotational velocities $v_{tg}(r)$ of these clouds tend toward constant and slightly rising values of the order of $%
v_{tg\infty }\sim 200-300$ km/s as a function of the distance $r$ from the center of the galaxy [@Bi87; @PeSaSt96; @BoSa01]. This is in sharp contrast to the Newtonian inverse square force law, which implies a decline in velocity. For clouds moving in circular orbits with velocity $v_{tg}(r)$, the balance between the centrifugal acceleration $v_{tg}^{2}/r$ and the gravitational attraction force $GM(r)/r^{2}$ allows to express the mass-distance relation $M(r)$ in the form $M(r)=rv_{tg}^{2}/G$. In the constant rotation velocity region this leads to a mass profile $M(r)=rv_{tg\infty }^{2}/G$. Consequently, the mass within a distance $r$ from the center of the galaxy increases linearly with $%
r$, even at large distances where very little luminous matter can be detected.
This behavior of the galactic rotation curves is usually explained by postulating the existence of some dark (invisible) matter, distributed in a spherical halo around the galaxies. The dark matter is assumed to be a cold, pressureless medium. There are many possible candidates for dark matter, the most popular ones being the weakly interacting massive particles (WIMP) (for a review of the particle physics aspects of dark matter see [@OvWe04]). Their interaction cross section with normal baryonic matter, while extremely small, are expected to be non-zero and we may expect to detect them directly. It has also been suggested that the dark matter in the Universe might be composed of superheavy particles, with mass $\geq 10^{10}$ GeV. But observational results show the dark matter can be composed of superheavy particles only if these interact weakly with normal matter, or if their mass is above $10^{15}$ GeV [@AlBa03].
Another interesting candidate for the dark matter is the dilatonic dark matter - the fundamental scalar field which exists in all existing unified field theories. The cosmological implications of the dilatonic dark matter have been explored in [@Cho90], where a higher-dimensional generalization of the standard big-bang cosmology has been proposed. It was shown that the missing-mass problem as well as the horizon problem, and the flatness problem of the standard model can be resolved within the context of this unified cosmology. The possibility that the dilaton plays the role of the dark matter of the universe was investigated in [@Cho3]. The condition for the dilaton to be the dark matter strongly restricts its mass to be around $0.5$ keV or $270$ MeV. For the other mass ranges, the dilaton contradicts the cosmological observations. The 0.5 keV dilaton has a free-streaming distance of about 1.4 Mpc and is an excellent candidate for warm dark matter, while the 270 MeV one has a free-streaming distance of about 7.4 pc and is a candidate for cold dark matter. An experiment to detect the relic dilaton using the electromagnetic resonant cavity, based on the dilaton-photon conversion in strong electromagnetic background was proposed in [@Cho07]. The density of the relic dilaton, as well as an estimate of the dilaton mass for which the dilaton becomes the dark matter of the universe were calculated. The dilaton detection power in the resonant cavity were also obtained, and they were compared with the axion detection power in similar resonant cavity experiment. Based on the fact that the scalar curvature of the internal space determines the mass of the dilaton in higher-dimensional unified theories, the dilaton mass can explain the origin of the mass, and resolve the hierarchy problem [@Cho07a]. Moreover, cosmological observations put a strong constraint on the dilaton mass, and requires that the scale of the internal space to be larger than $10^{-9}$ m. Scalar fields or other long range coherent fields coupled to gravity have also intensively been used to model galactic dark matter [@scal1; @scal2; @scal3; @scal4; @scal5; @scal6; @scal7; @Dah; @Let1; @Let2].
However, despite more than 20 years of intense experimental and observational effort, up to now no *non-gravitational* evidence for dark matter has ever been found: no direct evidence of it and no annihilation radiation from it. Moreover, accelerator and reactor experiments do not support the physics (beyond the standard model) on which the dark matter hypothesis is based.
Therefore, it seems that the possibility that Einstein’s (and the Newtonian) gravity breaks down at the scale of galaxies cannot be excluded *a priori*. Several theoretical models, based on a modification of Newton’s law or of general relativity, have been proposed to explain the behavior of the galactic rotation curves. A modified gravitational potential of the form $%
\phi =-GM\left[ 1+\alpha \exp \left( -r/r_{0}\right) \right] /\left(
1+\alpha \right) r,$with $\alpha =-0.9$ and $r_{0}\approx 30$ kpc can explain flat rotational curves for most of the galaxies [@Sa84; @Sa86].
In an other model, called MOND, and proposed by Milgrom [@Mi1; @Mi2; @Mi3; @Miin], the Poisson equation for the gravitational potential $\nabla
^{2}\phi =4\pi G\rho $ is replaced by an equation of the form $\nabla \left[ \mu \left( x\right) \left( \left| \nabla \phi
\right| /a_{0}\right) \right] =4\pi G\rho $, where $a_{0}$ is a fixed constant and $\mu \left( x\right) $ a function satisfying the conditions $\mu \left( x\right) =x$ for $x<<1$ and $\mu
\left( x\right) =1$ for $x>>1$. The force law, giving the acceleration $\vec{%
a}$ of a test particle, becomes $\mu \left( a/a_{0}\right)
\vec{a}=\vec{a}_{N}$, where $\vec{a}_{N}$ is the usual Newtonian acceleration. $a=a_{N}$ for $%
a_{N}>>a_{0}$ and $a=\sqrt{a_{N}a_{0}}$ for $a_{N}<<a_{0}$. The rotation curves of the galaxies are predicted to be flat, and they can be calculated once the distribution of the baryonic matter is known. The value of the constant $a_{0}$ is given by $a_{0}\approx 1.2\times 10^{-8}\mathrm{cm}/%
\mathrm{s}^{-2}\approx cH_{0}/6\approx c\left( \Lambda /3\right) ^{1/2}/6$, where $H_{0}$ is Hubble’s constant and $\Lambda $ is the cosmological constant. MOND is a purely phenomenological theory, but still it can explain most of the galaxy rotation curves without introducing dark matter. But despite its achievements, MOND has many problems of its own, like, for example, the lack of conserved quantities, like energy, and a theoretical justifications to the MOND phenomenology. A relativistic gravitation theory for MOND was proposed by Bekenstein [@Be04]. In this model gravitation is mediated by the tensor field $g_{\alpha \beta }$, a scalar field $\phi $ and a vector field $U_{\alpha }$, all three dynamical. For a simple choice of its free function, the theory has a Newtonian limit for non-relativistic dynamics with significant acceleration, but a MOND limit when accelerations are small. A tensor-vector-scalar theory that reconciles the galaxy scale success of modified Newtonian dynamics with the cosmological scale evidence for cold dark matter (CDM) has been proposed by Sanders [@Sa05]. The theory provides a cosmological basis for MOND by showing that the predicted phenomenology only arises in a cosmological background.
Alternative theoretical models to explain the galactic rotation curves have been elaborated recently by Mannheim [@Ma931; @Ma932], Moffat and Sokolov [@Mo96], Brownstein and Moffat [@Br061; @Br062] and Roberts [@Ro04]. The constancy of the tangential velocity of test particles orbiting around galaxies can be also explained in the brane world models [@RS99a; @RS99b], where the effects of the projection of the Weyl tensor from the bulk plays the role of the dark matter [@Ma041; @Ma042; @Ma043; @Ma044; @Ma045; @Pal], in the $f(R)$ modified gravity models [@fR1; @fR2; @fR3; @fR4], and by assuming that dark matter is in the form of a Bose-Einstein condensate [@Bo1; @Bo2; @Bo3; @Bo4; @Bo5; @Bo6; @Bo7], or of an Einstein cluster [@Ein1; @Ein2].
It is the purpose of the present paper to show that there is an alternative general physical interpretation of the ”dark matter” paradigm. More exactly, the constancy of the galactic rotation curves can be obtained from the assumption that the motion of the test particles in circular orbits around galaxies is non-geodesic. The galactic dynamics of test particles is a direct consequence of the presence of an extra force $f^{\mu } $, which modifies the dynamic law of motion. Such a scenario, in which the galactic rotation curves are explained by the presence of an extra force may be called EFDOD (extra force dominated orbital dynamics).
One of the most interesting possibilities is that the extra force is due to the presence of the extra dimensions. In such a model, which we may call multidimensional EFDOD, the motion of the particles takes place on geodesics in higher dimensions. A comprehensive geometric treatment of Kaluza - Klein type unifications of non-Abelian gauge theories with gravitation was first introduced in [@Cho1], where the appearance of a cosmological constant was also noted, and further developed in [@Cho1a; @Cho1b]. The possible modifications of Einstein’s theory of gravitation due to the fifth force generated by the Kaluza - Klein dilaton were discussed in [@Cho2], including the effects on the gravitational redshift, the deflection of light, the precession of perihelia, and the time-delay of radar echo around a spherically symmetric black hole in multidimensional space times. The long-range effect of the higher-dimensional fifth force is characterized by the dilatonic charge carried by the black hole even when it is neutral. In [@Cho92] it was emphasized that in the Brans-Dicke theory it is the Pauli metric, not the Jordan metric, which describes the massless spin-two graviton. Similarly, in the Jordan-Brans-Dicke theory, based on Kaluza-Klein unification, only the Pauli metric can correctly describe Einstein¡¦s theory of gravitation. This necessitates a completely new reinterpretation of the Kaluza-Klein cosmology, as well as of the Brans-Dicke theory. More significantly, this analysis shows that the Kaluza-Klein dilaton must generate a fifth force, which could violate the equivalence principle. Recent torsion-balance experiments [@tors] have tested the gravitational inverse-square law at separations between 9.53 mm and 55 $\mu $m, thus probing distances less than the dark-energy length scale $d=85~\mu $m. It has been found with 95% confidence that the inverse-square law holds down to a length scale of around $56~\mu $m, and that an extra dimension must have a size $\leq 44~\mu $m.
However, it is known for some time that the effects of extra-dimensions on the trajectory of test particles as observed in four dimensions can be modeled in terms of an extra force $f^{\mu }$, for both compactified and non-compactified spaces [@Cho3; @Po01; @Po04]. The presence of such a force may explain the phenomenology and behavior of the galactic rotation curves. We investigate in detail the dynamics of the test particles in extra-dimensional models, and we find the conditions which must be satisfied by the five-dimensional metric tensor in order to explain the observed rotation curves. As a physical test of our model we suggest that the lensing effects could be able to find evidences for the multi-dimensional geometry.
We also investigate the acceleration law in EFDOD, and we find that it has a striking similarity with the acceleration law in MOND. This leads us to the conclusion that the MOND theory, which can be considered, from a physical point of view, as describing the non-geodesic motion of a test particle in a gravitational field under the action of an extra force generated by the supplementary vector and scalar fields introduced in the model, is a particular case of the EFDOD models.
The present paper is organized as follows. Physical models determining a non-geodesic motion of particles are considered in Section II. The behavior of test particles in stable circular orbits in multidimensional models is considered in Section III. In Section IV we consider the possibility that dark matter is an extra-dimensional effect, and we obtain the general metric tensor in the flat rotation curve region and we check the consistency of the model. The acceleration law and the relation of our model with MOND is discussed in Section V. We conclude and discuss our results in Section VI. Throughout the paper we use the Landau-Lifshitz conventions [@LaLi] for signature and metric.
Scalar field generated extra force models
=========================================
There are several physical situations in which an extra force may be present, determining a non-geodesic motion of the particles, like, for example, the case of a real scalar field minimally coupled to gravity and interacting with matter, the case of the extra-force generated by the non-trivial coupling between matter and geometry in the $f(R)$ modified gravity models, and the cases of the extra forces generated by the presence of the compactified and non-compactified higher dimensions of the space-time, respectively. In the present Section we will review briefly the first case.
In order to give a systematic treatment of the extra forces in the presence of a scalar field we will use the Bazanski approach for obtaining the geodesic equation [@Ba89]. According to this approach, the equation of motion in any dimensions can be obtained by applying the action principle to the Lagrangian [@Ka06] $$L=m(s)g_{AB}u^{A}\frac{D\Psi ^{B}}{Ds}+f_{A}\Psi
^{A},A,B=0,1,...,D,$$ where $\Psi ^{B}$ is the deviation vector, $u^{A}$ is the tangent vector to the geodesic and $f_{A}$ and $m(s)$ are functions which depend on the specific physical models. The covariant derivative $D\Psi ^{B}/Ds$ is defined as $D\Psi ^{B}/Ds=d\Psi ^{B}/ds+\Gamma
_{CD}^{B}\Psi ^{C}u^{D}$.
The equation describing a real scalar field $\psi $ minimally coupled to gravity and interacting with matter can be given as [@Mb04] $$\nabla _{\alpha }\nabla ^{\alpha }\psi =-J-\frac{\partial J}{\partial \psi }-%
\frac{\partial U}{\partial \psi },$$ where $U=U\left( \psi \right) $ is the self-interaction potential and $%
J=J\left( x^{\beta }\right) $ is the source term of the scalar field. In the following we neglect the effect of $U$ (assumed to be of a breaking symmetry type). As for the source term $J$ we assume that it is of the general form $%
J=4\pi Gg(\psi )T_{\mu }^{\mu }/c^{2}$, where $T_{\mu }^{\mu }$ is the trace of the energy momentum tensor of the matter and $g$ is a coupling function satisfying the conditions $g\left( \psi
_{\infty }\right) =0$ and $\partial g\left( \psi _{\infty }\right)
/\partial \psi \neq 0$, respectively, where $\psi _{\infty }$ is the value of the scalar field at the minimum of the potential. In four dimensions the equation of motion of a test particle in the presence of a scalar field can be derived from the Lagrangian $$L=m(s)g_{\mu \nu }u^{\mu }\frac{D\Psi ^{\nu }}{Ds}+m(s)_{,\mu
}\Psi ^{\mu },$$ and is given by $$\frac{du^{\mu }}{ds}+\Gamma _{\alpha \beta }^{\mu }u^{\alpha }u^{\beta }=%
\frac{1}{m(s)}\left( g^{\mu \sigma }m_{,\sigma
}-\frac{dm}{ds}u^{\mu }\right) .$$
By assuming that the effective mass is of the form $m\sim \exp \left( -g\left( \psi \right) \psi \right) $ we obtain [@Mb04] $$\frac{du^{\mu }}{ds}+\Gamma _{\alpha \beta }^{\mu }u^{\alpha }u^{\beta }=%
\frac{d\left( g\left( \psi \right) \psi \right) }{ds}u^{\mu
}-\partial ^{\mu }\left( g\left( \psi \right) \psi \right) .$$
This equation of motion can also be derived from the variational principle $%
\delta \int mc\sqrt{g_{\mu \nu }u^{\mu }u^{\nu }}ds=0$. The force $\vec{f}$ has two components, one proportional to the velocity $%
\vec{v}$ of the particles, and which, being perpendicular to the acceleration, does not give any contribution in Eq. (\[eq4\]) and therefore can be neglected, and a second component given by $\vec{f}%
_{\parallel }=-\nabla \left[ g\left( \psi \right) \psi \right] $. By assuming that the scalar field has a spherical symmetry, $\psi
=\psi \left( r\right) $, evaluating the force $\vec{f}_{\parallel
}$ around $\psi =\psi _{\infty }$ gives $f_{\parallel }\approx
-\left[ \partial g\left( \psi
_{\infty }\right) /\partial \psi \right] \psi ^{\prime }\psi $, where $%
^{\prime }$denotes the derivative with respect to $r$. In order to obtain concordance with MOND, which means a constant $a_{0}$, it is necessary that $f_{\parallel }\sim -1/r$, which implies that $\partial g\left( \psi _{\infty }\right) /\partial \psi =g^{\prime
}\left( \psi _{\infty }\right) >0$ and $\psi \left( r\right) =\psi _{0}%
\sqrt{\ln \left( r/R_{0}\right) }$, with $\psi _{0},R_{0}=$ constant.
As for the scalar field we assume that is satisfies the equation [@Mb04] $$\Delta \psi =\frac{4\pi G}{c^{2}} g^{\prime }\left( \psi _{\infty
}\right) \psi _{\infty }\rho ,$$ where $\rho $ is the mass density of the matter fields other than $\psi $. With the obtained form of the scalar field it follows that the density of the matter interacting with the scalar field has a density profile given by $$\label{dens}
\rho \left( r\right) =\frac{c^{2}}{16\pi G}\frac{\psi
_{0}}{g^{\prime
}\left( \psi _{\infty }\right) \psi _{\infty }} \frac{1}{r^{2}}%
\frac{8\ln ^{2}\left( r/R_{0}\right) +6\ln \left( r/R_{0}\right)
-1}{\ln ^{3/2}\left( r/R_{0}\right) }.$$
Therefore EFDOD may be due to a scalar field interacting with a matter distribution, whose density varies according to Eq. (\[dens\]). The presence of such a field could explain the constancy of the galactic rotation curves and the corresponding MOND phenomenology.
Motion of test particles in stable circular orbits in multidimensional space-times
==================================================================================
The above approach can naturally be implemented in Kaluza-Klein theory [@Cho90; @Cho2]. In the following we will consider the case of the extra forces generated by the presence of the compactified and non-compactified higher dimensions of the space-time.
Let the coordinates of the five-dimensional manifold, with metric tensor $%
\gamma _{AB}$, be $x^{A}$ $\left( A=0,1,2,3,4\right) $. The $5D$ interval is given by $dS^{2}=\gamma _{AB}dx^{A}dx^{B}$. Usually it is assumed that the first four coordinates $x^{\mu }$ are the coordinates of the space-time $%
x^{\mu }$ $\left( \mu =0,1,2,3\right) $, while $x^{4}=\xi $ is the extra-dimension. Setting $\gamma _{\mu 4}=\gamma _{44}A_{\mu }$ and $\gamma
_{44}=\varepsilon \Phi ^{2}$, where $A_{\mu }$ and $\Phi $ are the vector and scalar potentials, respectively, and $\varepsilon =\pm 1$, we may write the line element without any loss of generality as [@Po01; @Po04] $$dS^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }+\varepsilon \Phi ^{2}\left( d\xi
+A_{\mu }dx^{\mu }\right) ^{2},$$ where $g_{\mu \nu }=\gamma _{\mu \nu }-\varepsilon \Phi ^{2}A_{\mu }A_{\nu }$.
In the non-compact Kaluza-Klein theories, like, for example, the brane world models [@RS99a], all test particles travel on five dimensional geodesics, but the observers, bounded to the usual four-dimensional space-time, have access only to the $4D$ part of the trajectory. Mathematically, this means that the equations governing the motion in $4D$ are projections of the $5D$ equations on the $4D$-hypersurfaces orthogonal to some vector field $\psi ^{A}$. Generally, the background metric in $5D$ can be written as [@Po01; @Po04; @You00] $$dS^{2}=\gamma _{\mu \nu }\left( x^{\alpha },\xi \right) dx^{\mu }dx^{\nu
}+\epsilon \Phi ^{2}\left( x^{\alpha },\xi \right) d\xi ^{2},$$ where $\gamma _{\mu \nu }$ is the induced metric in $4D$. In brane world theory the physical space time four dimensional metric $g_{\mu \nu }$ is generally identified with $\gamma _{\mu \nu }$. However, in some approaches, the physical metric in $4D$ is assumed to be conformally related to the induced one, $$dS^{2}=\Omega \left( \xi \right) g_{\mu \nu }\left( x^{\alpha },\xi \right)
dx^{\mu }dx^{\nu }+\epsilon \Phi ^{2}\left( x^{\alpha },\xi \right) d\xi
^{2}=\Omega \left( \xi \right) ds^{2}+\epsilon \Phi ^{2}\left( x^{\alpha
},\xi \right) d\xi ^{2}, \label{metr}$$ where $\Omega \left( \xi \right) >0$ is called the warp factor [@Po04].
In both compact and non-compact Kaluza-Klein theories, the motion of test particles takes place in higher dimensions (usually the number of dimensions of the space-time is assumed to be five), along the geodesics lines, and with the equation of motion given by $$\frac{du^{A}}{dS}+\hat{\Gamma}_{BC}^{A}u^{B}u^{C}=0.$$ where $u^{A}=\left( dx^{\mu }/dS,d\xi /dS\right) $ is the five-velocity and $%
\hat{\Gamma}_{BC}^{A}$ are the Christoffel symbols formed with the $5D$ metric [@Cho2; @Po01; @Po04; @Ka06].
In order to obtain results which are relevant to the galactic dynamics, in the following we will restrict our study to the static spherically-symmetric five-dimensional metric given by $$dS^{2}=e^{\nu \left( r,\xi \right) }c^{2}dt^{2}-e^{\lambda \left( r,\xi
\right) }dr^{2}-r^{2}\left( d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}\right)
+\epsilon \Phi ^{2}\left( r,\xi \right) d\xi ^{2}=ds^{2}+\epsilon \Phi
^{2}\left( r,\xi \right) d\xi ^{2}, \label{metr1}$$ where the coordinates have been chosen so that $x^{A}=\left( ct,r,\theta
,\phi ,\xi \right) $. The components of the five-velocity $U^{A}$ are given by $U^{A}=dx^{A}/dS$. In particular, $U^{4}=d\xi /dS$. In the following we also denote $u^{A}=dx^{A}/ds$, which represent the components of the five-velocity with respect to the four-dimensional space-time with interval $%
ds$. The four-dimensional interval $ds$ is related to the five-dimensional interval $dS$ by the relations $dS=ds/\sqrt{1-\epsilon \Phi ^{2}\left( r,\xi
\right) \left( U^{4}\right) ^{2}}$ or, equivalently, $dS=ds\sqrt{1+\epsilon
\Phi ^{2}\left( r,\xi \right) \left( u^{4}\right) ^{2}}$. The velocity $%
u^{A} $ is given as a function of $U^{A}$ by $u^{A}=U^{A}/\sqrt{1-\epsilon
\Phi ^{2}\left( r,\xi \right) \left( U^{4}\right) ^{2}}$.
The Lagrangian $L$ of a massive test particle traveling in the five-dimensional space-time with metric given by Eq. (\[metr1\]) is $$2L=e^{\nu \left( r,\xi \right) }\left( \frac{cdt}{dS}\right)
^{2}-e^{\lambda \left( r,\xi \right) }\left( \frac{dr}{dS}\right)
^{2}-r^{2}\left[ \left( \frac{d\theta }{dS}\right) ^{2}+\sin ^2
\theta \left( \frac{d\phi }{dS}\right) ^{2}\right] +\epsilon \Phi
^{2}\left( r,\xi \right) \left( \frac{d\xi }{dS}\right) ^{2}.
\label{lag}$$
Since the metric tensor coefficients do not explicitly depend on $ct$, $%
\theta $ and $\phi $, the Lagrangian (\[lag\]) gives the following conserved quantities (generalized momenta) in five dimensions: $$e^{\nu \left( r,\xi \right) }\frac{cdt}{dS}=E={\rm const.},r^{2}\frac{%
d\theta }{dS}=L_{\theta }={\rm const.},r^{2}\sin ^{2}\theta \frac{d\phi }{dS%
}=L_{\phi }={\rm const.}, \label{cons}$$ where $E$ is the total energy of the particle (in five-dimensions) and $%
L_{\theta }$ and $L_{\phi }$ are the components of the angular moment, respectively. With the use of conserved quantities we obtain from Eq. (\[metr1\]) the geodesic equation for material particles as $$e^{\nu +\lambda }\left( \frac{ds}{dS}\right) ^{2}\left( \frac{dr}{ds}\right)
^{2}+e^{\nu }\left[ 1+\frac{L_{T}^{2}}{r^{2}}-\epsilon \Phi ^{2}\left(
U^{4}\right) ^{2}\right] =E^{2}, \label{geod1}$$ where we have denoted $L_{T}^{2}=L_{\theta }^{2}+L_{\phi }^{2}/\sin
^{2}\theta $. Eq. (\[geod1\]) can be written as $$e^{\nu +\lambda }\left( \frac{ds}{dS}\right) ^{2}\left( \frac{dr}{ds}\right)
^{2}+V_{eff}\left( r,\xi \right) =E^{2},$$ where $$V_{eff}\left( r,\xi \right) =e^{\nu }\left[ 1+\frac{L_{T}^{2}}{r^{2}}%
-\epsilon \Phi ^{2}\left( U^{4}\right) ^{2}\right] , \label{pot}$$ is the effective potential of the motion, which also contains the effects of the presence of the extra-dimension. If $\Phi \equiv 0$, we obtain the well-known four-dimensional expression.
For the case of the motion of particles in circular and stable orbits the generalized potential must satisfy the following conditions: a) $dr/ds=0$ (circular motion) b) $\partial V_{eff}/\partial r$ $=0$ (extreme motion) and c) $\partial ^{2}V_{eff}/\partial r$ $^{2}!_{extr}>0$ (stable orbit), respectively. Conditions a) and b) immediately give the conserved quantities as $$E^{2}=e^{\nu }\left[ 1+\frac{L_{T}^{2}}{r^{2}}-\epsilon \Phi ^{2}\left(
U^{4}\right) ^{2}\right] , \label{cons1}$$ and $$\frac{L_{T}^{2}}{r^{2}}=E^{2}\frac{r\nu ^{\prime }e^{-\nu }}{2}-\frac{r}{2}%
\frac{\partial }{\partial r}\left[ \epsilon \Phi ^{2}\left( U^{4}\right) ^{2}%
\right] , \label{cons2}$$ respectively. Eqs. (\[cons1\]) and (\[cons2\]) allow us to express the constants of the motion in the equivalent form $$E^{2}=\frac{e^{\nu }}{1-\frac{r\nu ^{\prime }}{2}}\left\{ 1-\frac{1}{2r}%
\frac{\partial }{\partial r}\left[ r^{2}\epsilon \Phi ^{2}\left(
U^{4}\right) ^{2}\right] \right\} , \label{cons3}$$ and $$L_{T}^{2}=\frac{r^{3}\nu ^{\prime }}{2}\frac{1}{1-\frac{r\nu ^{\prime }}{2}}%
\left\{ 1-\frac{e^{-\nu }}{\nu ^{\prime }}\frac{\partial }{\partial r}\left[
e^{\nu }\epsilon \Phi ^{2}\left( U^{4}\right) ^{2}\right] \right\} ,
\label{cons4}$$ respectively.
We define the tangential velocity $v_{tg}$ of a test particle in four dimensions, measured in terms of the proper time, that is, by an observer located at the given point, as [@LaLi] $$v_{tg}^{2}=e^{-\nu }r^{2}c^{2}\left[ \left( \frac{d\theta }{cdt}\right)
^{2}+\sin ^{2}\theta \left( \frac{d\phi }{cdt}\right) ^{2}\right] =e^{-\nu
}r^{2}c^{2}\left[ \left( \frac{d\theta }{dS}\right) ^{2}+\sin ^{2}\theta
\left( \frac{d\phi }{dS}\right) ^{2}\right] \left( \frac{dS}{cdt}\right)
^{2}.$$
By using the constants of motion from Eqs. (\[cons\]) we immediately obtain $$\frac{v_{tg}^{2}}{c^{2}}=\frac{L_{T}^{2}}{E^{2}}\frac{e^{\nu }}{r^{2}}.
\label{vtg}$$
By eliminating the constant quantity $L_{T}^{2}/E^{2}$ between Eqs. (\[cons3\]) and (\[cons4\]) gives $$\frac{v_{tg}^{2}}{c^{2}}=\frac{\nu ^{\prime }r}{2}\frac{1-\frac{e^{-\nu }}{%
\nu ^{\prime }}\frac{\partial }{\partial r}\left[ e^{\nu }\epsilon \Phi
^{2}\left( U^{4}\right) ^{2}\right] }{1-\frac{1}{2r}\frac{\partial }{%
\partial r}\left[ r^{2}\epsilon \Phi ^{2}\left( U^{4}\right) ^{2}\right] }.
\label{vtg1}$$
An alternative expression for the tangential velocity can be obtained directly from the line element Eq. (\[metr1\]), by using the constants of motion. The result is $$\frac{v_{tg}^{2}}{c^{2}}=\frac{e^{-\nu }E^{2}-1+\epsilon \Phi ^{2}\left(
U^{4}\right) ^{2}}{e^{-\nu }E^{2}}.$$
With the use of Eq. (\[cons1\]) we can express the tangential velocity as $$\frac{v_{tg}^{2}}{c^{2}}=\frac{L_{T}^{2}}{L_{T}^{2}+r^{2}\left[ 1-\epsilon
\Phi ^{2}\left( U^{4}\right) ^{2}\right] }. \label{vtg2}$$
In order to completely solve the problem of the stable circular motion of the test particles in extra-dimensional models we need an equation determining $U^{4}$. This can be taken as the fifth component of the geodesic equation, and is given by $$\frac{d}{dS}\left( \epsilon \Phi ^{2}U^{4}\right) =\frac{1}{2}\frac{\partial
g_{\alpha \beta }}{\partial \xi }U^{\alpha }U^{\beta }.$$
Taking into account that $U^{1}\equiv 0$, and that the $g_{22}$ and $g_{33}$ metric tensor components do not depend on $\xi $, we have $$\frac{d}{dS}\left( \epsilon \Phi ^{2}U^{4}\right) =\frac{1}{2}\frac{\partial
g_{00}}{\partial \xi }U^{0}U^{0}=-\frac{1}{2}E^{2}\frac{\partial }{\partial
\xi }e^{-\nu }, \label{geod5}$$ where we have again used the conservation law for the energy.
Dark matter as an extra-dimensional effect
==========================================
The galactic rotation curves provide the most direct method of analyzing the gravitational field inside a spiral galaxy. The rotation curves have been determined for a great number of spiral galaxies. They are obtained by measuring the frequency shifts $z$ of the light emitted from stars and from the 21-cm radiation emission from the neutral gas clouds. Usually the astronomers report the resulting $z$ in terms of a velocity field $v_{tg}$. The observations show that at distances large enough from the galactic center $$v_{tg}\approx 200-300\text{ km/s}=\text{\textrm{constant.}}$$
This behavior has been observed for a large number of galaxies [@Bi87].
For a test particle traveling on a stable circular orbit in a multi-dimensional space-time, the tangential velocity is given by Eq. (\[vtg2\]). In a purely four-dimensional space-time, $\Phi ^{2}\equiv 0$, and the tangential velocity is given by $v_{tg}^{2}/c^{2}=L_{T}^{2}/\left(
L_{T}^{2}+r^{2}\right) $. In the limit of large $r$, $r\rightarrow \infty $, and taking into account that $L_{T}^{2}$ is a finite (conserved) quantity, we obtain $%
\lim_{r\rightarrow \infty }v_{tg}=0$. However, the situation is quite different in the multi-dimensional models. If the condition $$1-\epsilon \Phi ^{2}\left( U^{4}\right) ^{2}=\frac{C^{2}\left( \xi \right)L_T^2}{%
r^{2}}, \label{cond1}$$ holds true for large $r$, where $C^{2}\geq 0$ is an arbitrary function of the fifth coordinate, then the tangential velocity of a test particle in circular stable motion around the galactic center is given by $$\frac{v_{tg}^{2}}{c^{2}}=\frac{1 }{1 +C^{2}\left( \xi \right) }%
.$$
If $C^{2}\left( \xi \right) $ is a true (galaxy-dependent) constant, then the tangential velocity is an absolute constant, too. Therefore, the constancy of the galactic rotation curves can be explained naturally in the multi-dimensional physical models, without the necessity of introducing the ad hoc hypothesis of the dark matter. Of course, there are several other choices in Eq. (\[vtg2\]) which may lead to constant or slightly increasing rotation velocity curves.
Since the tangential velocity is a constant or fifth-dimension dependent quantity, one can solve Eq. (\[vtg1\]) and find the value of the metric tensor component $\exp \left( \nu \right) $ in the constant rotational curves region. After some simple transformations Eq. (\[vtg1\]) can be written in the form $$\nu ^{\prime }=2\frac{v_{tg}^{2}}{c^{2}}\frac{1}{r}+\left( 1-\frac{v_{tg}^{2}%
}{c^{2}}\right) \frac{1}{1-\epsilon \Phi ^{2}\left( U^{4}\right) ^{2}}\frac{%
\partial }{\partial r}\left[ \epsilon \Phi ^{2}\left( U^{4}\right) ^{2}%
\right] ,$$ giving $$e^{\nu }=D\left( \xi \right) r^{2v_{tg}^{2}/c^{2}}\left| \epsilon
\Phi ^{2}\left( U^{4}\right) ^{2}-1\right| ^{-\left(
1-v_{tg}^{2}/c^{2}\right) }, \label{enu}$$ where $D\left( \xi \right) $ is an arbitrary integration function. In the four-dimensional limit $\Phi ^{2}\equiv 0$ we obtain the well-known result $%
\exp \left( \nu \right) =Dr^{2v_{tg}^{2}/c^{2}}$, which has been extensively used to discuss the properties of dark matter [@scal1; @scal2; @scal3; @scal4; @scal5; @scal6], [@Ma041; @Ma042; @Ma043; @Ma044]. Hence, the expression of the metric tensor component $g_{00}$ in the constant rotation curves region can be obtained in an exact form.
An important particular case corresponds to the situation in which $e^{\nu }$ is independent of $\xi $. Then, the geodesic equation Eq. (\[geod5\]) can be immediately integrated to give $$\epsilon \Phi ^{2}U^{4}=B^2={\rm constant}.$$
Together with Eq. (\[cond1\]), the above first integral of the equations of motion allows the determination of the functional form of the metric tensor component $\Phi ^{2}$ in the constant rotation curves region as $$\epsilon \Phi ^{2}=\frac{B^{2}}{1-C^2L_T^2/r^{2}}.$$
In this case $B$, $C$ and $D$ are true constants, and they are independent of the fifth dimension $\xi $. This model corresponds to a compactified Kaluza-Klein type theory, in which all the metric tensor components are independent of the fifth coordinate.
The multi-dimensional effects are dominant in the vacuum at large distances from the galaxy. In the presence of matter, that it, inside or at the vacuum boundary of the galaxy, these effects are very small, as compared to the gravitational effect of the normal four-dimensional matter. Hence we may assume that at the boundary of a galaxy with baryonic mass $M_{B}$ and radius $R_{B}$ the four-dimensional geometry is approximately the Schwarzschild geometry, and therefore $$e^{\nu }|_{r=R_{B}}\approx 1-\frac{2GM_{B}}{c^{2}R_{B}}.$$
This matching condition (approximately) determines the constant $D$ in Eq. (\[enu\]).
The acceleration law in EFDOD
=============================
We start by assuming that the motion of a test particle in a $4D$ space-time with metric $g_{\mu \nu }$ is given by $$\frac{D^{(4)}u^{\mu }}{ds}\equiv \frac{du^{\mu }}{ds}+\Gamma _{\alpha \beta
}^{\mu }u^{\alpha }u^{\beta }=f^{\mu }, \label{eq1}$$ where $u^{\mu }=dx^{\mu }/ds$ is the usual four-dimensional velocity of the particle and $\Gamma _{\alpha \beta }^{\mu }$ are the Christoffel symbols constructed by using the $4D$ metric. The presence of the extra force $%
f^{\mu }$ makes the motion of the particle non-geodesic. For $f^{\mu }\equiv
0$ we recover the geodesic equation of motion. All the usual gravitational effects, due to the presence of an arbitrary mass distribution, are assumed to be contained in the term $a_{N}^{\mu }=\Gamma _{\alpha \beta }^{\mu
}u^{\alpha }u^{\beta }$. In three dimensions and in the Newtonian limit, Eq. (\[eq1\]) can be formally represented as a three-vector equation of the form $$\vec{a}=\vec{a}_{N}+\vec{a}_f, \label{eq2}$$ where $\vec{a}$ is the total acceleration of the particle, $\vec{a}_{N}$ is the gravitational acceleration and $\vec{a}_f$ is the acceleration (per unit mass) due to the presence of the extra force. If $\vec{a}_f=0$, the equation of motion is the usual Newtonian one, $\vec{a}=\vec{a}_{N}$, or, equivalently, $\vec{a}=-GM\vec{r}/r^{3}$.
In the following we denote by $v^{2}=\vec{v}\cdot \vec{v}=\left| \vec{v}%
\right| ^{2}$ the magnitude of a vector $\vec{v}$. Taking the square of Eq. (\[eq2\]) gives $$\vec{a}_f\cdot \vec{a}_{N}=\frac{1}{2}\left( a^{2}-a_{N}^{2}-a_f^{2}\right) ,
\label{eq3}$$ where $\cdot $ represents the three-dimensional scalar product. Eq. (\[eq3\]) can be interpreted as a general relation which gives the unknown vector $\vec{a}_{N}$ as a function of the total acceleration $\vec{a}$, of the acceleration $\vec{a}_f$ due to the extra force, and of the magnitudes $a^{2}$, $a_{N}^{2}$ and $%
a_f^{2}$, respectively. From Eq. (\[eq3\]) one can express the vector $\vec{a%
}_{N}$, as one can easily check, in the form $$\vec{a}_{N}=\frac{1}{2}\left( a^{2}-a_{N}^{2}-a_f^{2}\right) \frac{\vec{a}}{%
\vec{a}_f\cdot \vec{a}}+\vec{C}\times \vec{a}_f, \label{eq4}$$ where $\vec{C}$ is an arbitrary vector perpendicular to the vector $\vec{a}_f$. In the following, for simplicity, we assume $\vec{C}\equiv 0$.
The mathematical consistency of Eq. (\[eq4\]) requires $\vec{a}_f\cdot \vec{a%
}\neq 0$, that is, the vectors $\vec{a}_f$ and $\vec{a}$ cannot be perpendiculars. Generally, $\vec{a}_f\cdot \vec{a}=a_fa\cos \alpha $, where $%
\alpha $ is the angle between $\vec{a}_f$ and $\vec{a}$. Again, for simplicity, we take $\alpha =0$, that is, we assume that the vectors $\vec{a}_f
$ and $\vec{a}$ are parallels. For $\vec{a}_f=0$, Eq. (\[eq4\]) gives $%
a_{N}^{2}=a^{2}$, as required.
Therefore, we can represent the gravitational acceleration of a test particle in the presence of an extra force as $$\vec{a}_{N}=\frac{1}{2}\left( a^{2}-a_{N}^{2}-a_f^{2}\right) \frac{\vec{a}}{a_fa}%
. \label{eq5}$$
In the limit of very small gravitational accelerations $a_{N}<<a$, we obtain the relation $$\vec{a}_{N}\approx \frac{1}{2}a\left( 1-\frac{a_f^{2}}{a^{2}}\right) \frac{1}{a_f%
}\vec{a}. \label{eq6}$$
By denoting $(1/2a_f)\left( 1-a_f^{2}/a^{2}\right) =1/a_{E}$, Eq. (\[eq6\]) immediately gives $$\vec{a}_{N}\approx \frac{a}{a_{E}}\vec{a},$$ which is similar to the equation proposed phenomenologically by Milgrom [@Mi1; @Mi2; @Mi3]. From this equation we obtain $a\approx \sqrt{a_{E}a_{N}}$, and since $%
a_{N}=GM/r^{2}$, we have $a\approx \sqrt{a_{E}GM}/r=v_{tg}^{2}/r$, where $%
v_{tg}$ is the rotational velocity of the test particle. Therefore, it follows that $v_{tg}^{2}\rightarrow v_{\infty }^{2}=\sqrt{a_{E}GM}$, giving the Tully-Fisher relation $v_{\infty }^{4}=a_{E}GM\sim L$, where $L$ is the luminosity, assumed to be proportional to the mass [@Mi1; @Mi2; @Mi3].
However, in the framework of EFDOD, $a_{E}$ is not generally a universal constant, but it may be a position, acceleration or galaxy characteristics dependent quantity.
Generally, from the given definition of $a_{E}$, we can formally represent the extra acceleration as a function of $a$ and $a_{E}$ as $%
a_f/a_{E}=-a^{2}/a_{E}^{2}\pm \left( a/a_{E}\right) \sqrt{1+a^{2}/a_{E}^{2}}$. Then, by means of some simple calculations, Eq. (\[eq5\]) can be represented as $$\vec{a}_{N}=\frac{a}{a_{E}}\left[ F\left( \frac{a}{a_{E}}\right) \left(
\frac{a_{N}}{a}\right) ^{2}+1\right] \vec{a},$$ where $$F\left( \frac{a}{a_{E}}\right) =\frac{1}{2}\left( \frac{a}{a_{E}}\right)
^{-1}\left( \frac{a}{a_{E}}\mp \sqrt{1+\frac{a^{2}}{a_{E}^{2}}}\right) ^{-1}.$$
With the use of the equation $a^{2}=v_{\infty
}^{4}/r^{2}=a_{E}GM/r^{2} $ we obtain the following expression for $a_{E}$: $$a_{E}\approx \frac{a_f^{2}r^{2}}{GM}+2a_f.$$
If $a_f\sim GMa_0 /r$, where $a_0$ is a constant, then in the large $r$ limit, when $a_f\rightarrow 0$, $a_{E}\approx a_0 ^{2}$ is a constant, whose numerical value is determined by the physical properties of the extra force. If the extra-force is universal in its nature, than $a_{E}$ is a universal constant.
Therefore the MOND paradigm (which also postulates the existence of a universal constant $a_{0}$) is equivalent, and can be derived, from the assumption of the non-geodesic motion of the test particles around the galactic centers under the action of a specific force.
Discussions and final remarks
=============================
In the present paper we have shown that the dynamics of the test particles in circular orbits around galaxies can be attributed to the presence of an extra force, generated by the presence of the extra dimensions, which modifies the standard (Newtonian or general relativistic) motion, by giving a supplementary contribution to the acceleration. The total acceleration has a general form, which is formally identical to the one proposed on a phenomenological-empirical basis in MOND. In an equivalent formulation, MOND is the result of the non-geodesic motion of particles under the influence of a specific force. On the other hand, depending on the physical nature of the extra force, more general physical models than MOND can be obtained. We have considered several possibilities for the extra force. The extra force may be generated, for example, by a scalar field coupled with matter. However, in such a scenario, some extra (dark?) matter is required, and, in order to obtain a constant $a_0$, a specific density profile for the matter is necessary. On the other hand, for the scalar field to influence the galactic rotational velocity, it has to be massless. But generally massless scalar fields could not exist in nature.
However, one could relate extra dimensions to dark matter through the dilatonic dark matter [@Cho90; @Cho92]. As an essential part of higher dimensional metric the dilaton plays a crucial role to determine the higher dimensional geodesic. But this higher dimensional geodesic equation, expressed in the lower dimension, contains a non-geodesic force, created by the dilaton, which requires a modification of general relativity. Generally, in $4D$ the motion can be described as taking place under the effect of a tensor (metric) field and of a vector and scalar field, respectively. Interestingly enough, the relativistic version of MOND [@Be04] requires exactly such a modification of general relativity, but with the extra-fields introduced by hand. However, for this model to be true, it is necessary that the extra- force from the extra dimension has to be long ranged (in a galactic scale). On the other hand the fifth force obtained in the framework of the compactified Kaluza-Klein theories cannot be long ranged [@Cho3]. Some possibilities of overcoming these difficulties may be by assuming that the extra-dimensions are large, as is the case in the brane-world models [@RS99a; @Po01; @Po04].
In order to explore in more detail the connections between EFDOD, MOND and dark matter, some explicit physical models are necessary to be built. This will be done in some forthcoming papers.
We would like to thank to the anonymous referee for comments and suggestions that helped us to significantly improve the manuscript. The work of T. H. was supported by the GRF grant No. 7018/08P of the government of the Hong Kong SAR.
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---
abstract: 'The Kac-Ward formula allows to compute the Ising partition function on any finite graph $G$ from the determinant of $2^{2g}$ matrices, where $g$ is the genus of a surface in which $G$ embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph $G$, exactly when the genus $g$ is zero or one. Finally, they share several formal properties with the Ray-Singer $\overline\partial$-torsions of the Riemann surface in which $G$ embeds.'
address: 'Section de mathématiques, 2-4 rue du Lièvre, 1211 Genève 4, Switzerland'
author:
- David Cimasoni
bibliography:
- 'Ising.bib'
title: 'The critical Ising model via Kac-Ward matrices'
---
Introduction {#sec:intro}
============
Most of the exact results for the two-dimensional Ising model rely on the so-called [*Pfaffian method*]{}. The idea, due independently to Hurst-Green [@H-G], Kasteleyn [@Ka1] and Fisher [@Fi1; @Fi2], is to associate to the given graph $G$ an auxiliary graph ${\Gamma}_G$ such that the dimer partition function on ${\Gamma}_G$ is equal to the Ising partition function on $G$. The dimer model technology can then be applied to solve the Ising model on $G$. In particular, if $G$ can be embedded in an orientable surface of genus $g$, then so can ${\Gamma}_G$, and its dimer partition function can be computed as an alternated sum of the Pfaffians of $2^{2g}$ well-chosen skew-adjacency matrices [@G-L; @Tes; @D96; @C-RI]. Even though this method is very standard at least for planar graphs (see for example the classical book [@MCW]), it is not a very natural one. First of all, this so-called [*Fisher correspondence*]{} $G\mapsto{\Gamma}_G$ is by no means unique: for example, each of the articles [@Ka1; @Fi2; @D96; @BdT1; @M-L; @Cim2] contains a different version of it. Secondly, whatever the chosen correspondence, virtually all the geometric and combinatorial properties of $G$ will get lost when passing to ${\Gamma}_G$ – the only obvious exception being the genus. This will not be a problem if one is studying a topological class of graphs, such as all planar graphs, or all finite graphs of a given genus. However, if one is interested in a geometric class – as we will be – the Pfaffian method leads to unnecessary complications.
There is another combinatorial method to solve the two-dimensional Ising model which, although naturally related to the Pfaffian method (see [@Cim2 Subsection 4.3]), is in our opinion much more natural. It is due to Kac-Ward [@KW] (even though a rigorous proof awaited many years [@DZMSS]), was originally formulated for planar graphs, and recently extended to any finite graph [@Loe; @Cim2]. With this method, no auxiliary graph is needed: the Ising partition function on a finite graph $G$ is computed as an alternated sum of the square roots of the determinants of $2^{2g}$ [*Kac-Ward matrices*]{} naturally associated to the graph $G$.
In this paper, we initiate the study of the critical Ising model on graphs of [*arbitrary*]{} genus, and we do so using these Kac-Ward matrices. What we mean by “critical Ising model" will be formally defined and thoroughly motivated in Section \[sec:def\], but for now, we will content ourselves with an informal description. Consider a finite number of planar rhombi $\{R_e\}_{e\in E}$ of equal side length, each rhombus $R_e$ having a fixed diagonal $e$ and corresponding half-rhombus angle $\theta_e\in(0,\pi/2)$, as illustrated below.
2.5pt at 113 68 at 45 77
Paste these rhombi together along their sides so that extremities of diagonals are glued to extremities of diagonals. The result is a graph $G$ with edge set $E(G)=E$ embedded in a flat surface ${\Sigma}$ with so-called [*cone-type singularities*]{} in the vertex set $V(G)$ of $G$, and in the middle of the faces of $G\subset{\Sigma}$. We shall say that $G$ is [*isoradially embedded*]{} in the flat surface ${\Sigma}$ (see Definition \[def:crit\]). Using the high-temperature representation, we define the partition function for the [*critical Ising model*]{} on such a graph $G$ by $$Z(G,\nu)=\sum_{\gamma\in\mathcal{E}(G)}\prod_{e\in\gamma}\nu_e,$$ where $\mathcal{E}(G)$ denotes the set of even subgraphs of $G$, and the [*critical weights*]{} $\nu_e$ are given by $\nu_e=\tan(\theta_e/2)$. This is a natural generalization of the critical Z-invariant Ising model [@Bax2], which corresponds to the special case where ${\Sigma}$ is a domain in the (flat) plane. Note that only a specific class of planar graphs admit a Z-invariant Ising model (see [@K-S; @CS]). On the other hand, any finite graph can be isoradially embedded in a flat surface as explained above, and therefore admits a critical Ising model (Proposition \[prop:real\]).
Since its introduction by Baxter, the critical Z-invariant Ising model has been extensively studied, as well as its analog on the flat torus (see for example the papers [@BdT1; @BdT2], where Boutillier and de Tilière make use of the Pfaffian method). On the other hand, very little is known about the critical Ising model on graphs in higher genus. It is widely believed that such models should be discrete analogs of some conformal field theory [@AMV], but in genus $g\ge 2$, such statements are only supported by numerical experiments on very specific examples [@CSM1; @CSM2]. It is our belief that, in order to try to tackle such outstanding conjectures, the Kac-Ward method will prove useful.
Before doing so, we need to settle some fundamental questions about the critical Ising model on graphs of arbitrary genus and the associated Kac-Ward matrices, and this is exactly what the present paper is about.
In the general case of an arbitrary graph embedded in a topological surface, the (generalized) Kac-Ward matrices can be quite complicated (see Definition 1 in [@Cim2]). The first nice surprise is that for a weighted graph $(G,x)$ embedded in a flat surface, the corresponding Kac-Ward matrices take a remarkably simple form. For each homomorphism $\varphi$ from the fundamental group $\pi_1({\Sigma})$ of ${\Sigma}$ to the group $S^1$, we get a [*$\varphi$-twisted Kac-Ward matrix*]{} of order $2|E(G)|$ whose determinant we denote by $\tau^\varphi(G,x)$ (see Definition \[def:KW\]). For a specific type of $\varphi$’s – the ones that belong to the set ${\mathcal S}$ of [*discrete spin structures*]{} on $G\subset{\Sigma}$ – $\tau^\varphi(G,x)$ turns out to be the square of a polynomial in the variables $\{x_e\}_e$. The main theorem of [@Cim2] then easily implies the following result (see Theorem \[thm:Arf\] for the complete statement).
If all cone angles are odd multiple of $2\pi$, then the Ising partition function on the weighted graph $(G,x)$ is given by $$Z(G,x)=\frac{1}{2^g}\sum_{\lambda\in{\mathcal S}}(-1)^{{\mathrm{Arf}}(\lambda)}\tau^\lambda(G,x)^{1/2},$$ where $g$ is the genus of ${\Sigma}$ and ${\mathrm{Arf}}(\lambda)\in{\mathbb Z}_2$ the Arf invariant of the discrete spin structure $\lambda$.
Let us now turn to our main results. In a nutshell, we show that when the weight system is critical the corresponding Kac-Ward determinants $\tau^\varphi(G,\nu)$ exhibit several remarkable properties.
Firstly, these determinants turn out to admit a relatively simple combinatorial interpretation, as described in Proposition \[prop:tech\]. Furthermore, they satisfy the following duality property.
Let $G$ be a graph isoradially embedded in a flat surface ${\Sigma}$, and let $\nu$ be the critical weight system on $G$. The dual graph $G^*$ is also isoradially embedded in ${\Sigma}$, and therefore admits a critical weight system $\nu^*$. If all cone angles are odd multiples of $2\pi$, then for any $\varphi$, $$2^{|V(G^*)|}\hskip-2pt\prod_{e^*\in E(G^*)}(1+\cos(\theta_{e^*}))\,\tau^\varphi(G^*,\nu^*)=2^{|V(G)|}\hskip-2pt\prod_{e\in E(G)}(1+\cos(\theta_{e}))\,\tau^\varphi(G,\nu).$$
This can be interpreted as a generalization of the celebrated Kramers-Wannier duality [@K-W] from the case of planar graphs to the case of graphs of arbitrary genus.
Our final result relates the Kac-Ward matrices to some [*a priori*]{} totally different operator. Given a weighted graph $(G,x)$ and a homomorphism $\varphi\colon\pi_1(G)\to S^1$, the associated [*discrete Laplacian*]{} on $G$ is the operator $\Delta^\varphi=\Delta^\varphi(G,x)$ acting on $f\in{\mathbb C}^{V(G)}$ by $$(\Delta^\varphi f)(v)=\sum_{e=(v,w)}x_e\,\left(f(v)-f(w)\varphi(e)\right),$$ the sum being over all oriented edges $e$ of the form $(v,w)$. Note that if $G$ is a (planar) isoradial graph, then the corresponding critical weights are given by $c_e=\tan(\theta_e)$ (see [@Ken]). We prove:
Let $G\subset\Sigma$ be an isoradially embedded graph in a flat surface, and let us assume that all cone angles $\vartheta_v$ of singularities $v\in V(G)$ are odd multiples of $2\pi$. If the genus of ${\Sigma}$ is $0$ or $1$, then for any $\varphi\colon\pi_1({\Sigma})\to S^1$, $$\tau^\varphi(G,\nu)=(-1)^N\,2^{-\chi(G)}\prod_{e\in E(G)}\frac{\cos(\theta_e)}{1+\cos(\theta_{e})}\,\det\Delta^\varphi(G,c),$$ where $N$ is the number of vertices $v\in V(G)$ such that $\vartheta_v/2\pi$ is congruent to $3$ modulo $4$, and $\chi(G)=|V(G)|-|E(G)|$. On the other hand, the functions $\tau^\varphi(G,\nu)$ and $\det\Delta^\varphi(G,c)$ are never proportional if the genus of ${\Sigma}$ is greater or equal to two.
Via the method developed in [@Cim2 Subsection 4.3], this theorem can be interpreted as a wide-reaching generalization of the main result of [@BdT1; @BdT2] which was obtained via the Pfaffian method. In our opinion, this is a good example of how simpler and more natural a proof can get, when the Kac-Ward method is used instead of the Pfaffian one. (See Remark \[rem:BdT\] below for a more detailed comparison.) In the case of the flat torus, the theorem above implies a relation between the free energy of the critical Z-invariant Ising model on $G$ and the normalized determinant of the critical discrete Laplacian on $G$. The former quantity was computed by Baxter [@Bax2], the latter by Kenyon [@Ken], and our equality allows to obtain any of these two results as a corollary of the other one.
The paper is organized as follows. In Section \[sec:def\], we define our model: the Ising model on graphs isoradially embedded in flat surfaces (Definition \[def:crit\]), with critical weights (Definition \[def:nu\]). In Section \[sec:KW\], we introduce the $\varphi$-twisted Kac-Ward matrices for graphs in flat surfaces (Definition \[def:KW\]), and we show how they can be used to compute the Ising partition function (Theorem \[thm:Arf\]). Section \[sec:crit\] deals with the case of isoradially embedded graphs with critical weights, and contains our main results. We start with the combinatorial interpretation for the Kac-Ward determinants with critical weights (Proposition \[prop:tech\]). Then, we prove the equality relating the Kac-Ward determinants of dual isoradially embedded graphs (Theorem \[thm:duality\]). Also, we relate the Kac-Ward determinants with the determinant of the critical discrete Laplacian on $G$ (Theorem \[thm:Delta\]). In a last paragraph, we explain how the Kac-Ward determinant can be understood as a discrete version of the $\overline\partial$-torsion of the underlying Riemann surface (Subsection \[sub:RS\]).
Acknowledgments {#acknowledgments .unnumbered}
---------------
This research was supported by the European Research Council AG CONFRA and by the Swiss NSF. Part of this paper was done at the ETH in Zurich, and it is a pleasure to thank the Mathematics Department of the ETHZ for providing such an excellent working environment. The author also wishes to thank Cédric Boutillier and Hugo Duminil-Copin for comments on an earlier version of the manuscript, and Martin Loebl for valuable discussions.
The critical Ising model on isoradial graphs {#sec:def}
============================================
The aim of this section is to explain the setup of the model that we will be studying in this paper: the critical Ising model on graphs isoradially embedded in a flat surface (Definitions \[def:crit\] and \[def:nu\]). To motivate this definition, we start by recalling what is meant by high and low-temperature expansions for the Ising model, leading to the Kramers-Wannier duality argument (Subsection \[sub:KW\]). The models on which such an argument can be applied are called Z-invariant Ising models (Subsection \[sub:Z\]). They are all defined on planar (or toric) graphs, but a generalization of these models to surfaces of arbitrary genus then naturally leads to our definition (Subsection \[sub:flat\]).
Kramers-Wannier duality {#sub:KW}
-----------------------
Let $G$ be a finite graph with vertex set $V(G)$ and edge set $E(G)$. A [*spin configuration*]{} on $G$ is a map $\sigma\colon V(G)\to\{-1,+1\}$. Any positive edge weight system $J=(J_e)_{e\in E(G)}$ on $G$ determines a probability measure on the set $\Omega(G)$ of such spin configurations by $$P(\sigma)=\frac{1}{Z^J(G)}\exp\Big(\sum_{e=(u,v)\in E(G)}J_e\sigma_u\sigma_v\Big),$$ where $$Z^J(G)=\sum_{\sigma\in\Omega(G)}\exp\Big(\sum_{e=(u,v)\in E(G)}J_e\sigma_u\sigma_v\Big)$$ is the [*partition function*]{} of the [*Ising model on $G$ with coupling constants $J$*]{}.
As observed by van der Waerden [@vdW], the identity $$\exp(J_e\sigma_u\sigma_v)=\cosh(J_e)(1+\tanh(J_e)\sigma_u\sigma_v)$$ allows to express this partition function as $$\begin{aligned}
Z^J(G)&=\Big(\prod_{e\in E(G)}\cosh(J_e)\Big)\sum_{\sigma\in\Omega(G)}\prod_{e=(u,v)\in E(G)}(1+\tanh(J_e)\sigma_u\sigma_v)\\
&=\Big(\prod_{e\in E(G)}\cosh(J_e)\Big)2^{|V(G)|}\sum_{\gamma\in\mathcal{E}(G)}\prod_{e\in\gamma}\tanh(J_e),\\\end{aligned}$$ where $\mathcal{E}(G)$ denotes the set of even subgraphs of $G$, that is, the set of subgraphs $\gamma$ of $G$ such that every vertex of $G$ is adjacent to an even number of edges of $\gamma$. This is called the [*high-temperature expansion*]{} of the partition function.
Let us now assume that the graph $G$ is planar, and let $G^*$ denote its dual graph. For any spin configuration $\sigma\in\Omega(G)$, consider the subgraph of $G^*$ given by all edges $e^*\in E(G^*)$ dual to $e=(u,v)$ with $\sigma_u\neq\sigma_v$. Clearly, this is an even subgraph of $G^*$. Furthermore, since $G$ is planar, this defines a surjective map $\Omega(G)\to\mathcal{E}(G^*)$ such that $\sigma$ and $\sigma'$ have same image if and only if $\sigma'=-\sigma$. This leads to the following [*low-temperature expansion*]{} of the Ising partition function: $$\begin{aligned}
Z^J(G)&=2\sum_{\gamma^*\in\mathcal{E}(G^*)}\prod_{e\in E(G)}\exp(J_e)\prod_{e*\in\gamma^*}\exp(-2J_e)\\
&=2\Big(\prod_{e\in E(G)}\exp(J_e)\Big)\sum_{\gamma^*\in\mathcal{E}(G^*)}\prod_{e*\in\gamma^*}\exp(-2J_e).\\\end{aligned}$$ Thus, if we assign weights $J$ to $E(G)$ and $J^*$ to $E(G^*)$ in such a way that $\tanh(J_{e^*})=\exp(-2J_e)$, or more symmetrically, $$\sinh(2J_e)\sinh(2J_{e^*})=1,$$ we obtain that the partition functions $Z^J(G)$ and $Z^{J^*}(G^*)$ are proportional to each other.
In the case of the square lattice with constant weight system $J$, this is enough to determine the critical value $J_c$ of the coupling constant. Indeed, assuming that the free energy of the model is analytic everywhere except at a single point, this point must be equal to $J_c$ [*and*]{} to $J_c^*$, since the square lattice is self-dual. The equality $J_c=J^*_c$ then leads to the explicit value $J_c=\log\sqrt{1+\sqrt{2}}$.
2.5pt at 145 100
The Z-invariant Ising models {#sub:Z}
----------------------------
This beautifully simple argument, the celebrated [*Kramers-Wannier duality*]{} [@K-W], is not sufficient to determine the critical value of the coupling constant on a graph that is not self-dual: it only relates this critical value to the one for the dual graph. However, for some planar graphs, this can be obtained “with little additional labor" [@Wan].
Let us start with the example of the hexagonal lattice $H$, and let $H'$ denote the graph obtained from $H$ by a [*star-triangle transformation*]{} at a vertex $v$ as illustrated in Figure \[fig:star\]. Assume one can assign coupling constants $J$ to the edges of $H$ and $J'$ to the newly created edges of $H'$ in such a way that $Z^{J'}(H')=R_v\,Z^J(H)$ for some function $R_v$ of the coupling constants of the edges around $v$. Since the triangular lattice $T$ can be obtained from $H$ by such transformations, it would follow that $Z^K(T)=R\,Z^J(H)$ for some controlled $R$ and well-chosen coupling constants $K,J$. This, together with the Kramers-Wannier duality, would lead to an equality of the form $Z^K(T)=k\,Z^{K^*}(T)$, with $K\mapsto K^*$ some involution and $k$ an explicit function of $K$ and $K^*$. Arguing as above, the critical points should be self-dual under this involution, leading to the equality $k=1$ and an exact description of these critical points for the triangular and hexagonal lattices.
This strategy of using invariance under the star-triangle transformation (or, [*Z-invariance*]{}) can be applied not only to the hexagonal-triangular lattices, but to a wide class of planar graphs [@Bax; @Bax2]. It turns out that this class of graphs on which a Z-invariant Ising model can be defined coincides with the graphs that admit an [*isoradial embedding*]{} in the plane [@K-S; @CS]: this is an embedding such that each face is inscribed in a circle of radius one, with the circumcenter in the closure of the face. Furthermore, the corresponding critical coupling constants admit a very simple geometric description: they are given by $$J_e=\frac{1}{2}\log\left(\frac{1+\sin\theta_e}{\cos\theta_e}\right),$$ where $\theta_e\in(0,\pi/2)$ is the half-rhombus angle associated to the edge $e$, as illustrated in Figure \[fig:theta\]. For example, the square lattice is isoradially embedded with all half-rhombus angles equal to $\theta=\pi/4$, leading to the critical coupling constant $J_c=\log\sqrt{1+\sqrt{2}}$ as above. On the other hand, the triangular and hexagonal lattices are isoradially embedded with angles $\theta=\pi/6$ (resp. $\pi/3$), so the corresponding critical values are equal to $J_c=\log\sqrt{\sqrt{3}}$ (resp. $\log\sqrt{2+\sqrt{3}}$).
This geometric description of the critical coupling constants becomes even nicer when using the high-temperature expansion, as we obtain the [*critical weights*]{} $$\nu_e=\tanh(J_e)=\tanh\left(\frac{1}{2}\log\left(\frac{1+\sin\theta_e}{\cos\theta_e}\right)\right)=\frac{1-\cos\theta_e}{\sin\theta_e}=\tan(\theta_e/2).$$
2.5pt at 180 135 at 115 140
Isoradial graphs in flat surfaces {#sub:flat}
---------------------------------
In the present paper, we will study more general models where the graph $G$ is not assumed to be planar. The proper generalization of planar isoradiality is obtained by considering so-called flat surfaces with cone-type singularities. Let us quickly recall their definition and main properties, referring to [@Tro] for further details.
Given a positive real number $\vartheta$, the space $$C_\vartheta=\{(r,t)\,:\,\hbox{$r\ge 0$, $t\in{\mathbb R}/\vartheta{\mathbb Z}$}\}/(0,t)\sim(0,t')$$ endowed with the metric $\mathit{ds}^2=\mathit{dr}^2+r^2\mathit{dt}^2$ is called the [*standard cone of angle $\vartheta$*]{}. Note that the cone without its tip is locally isometric to the Euclidean plane. Let $\Sigma$ be a surface with a discrete subset $S$. A [*flat metric on $\Sigma$ with cone-type singularities*]{} of angles $\{\vartheta_x\}_{x\in S}$ supported at $S$ is an atlas $\{\phi_x\colon U_x\to U'_x\subset C_{\vartheta_x}\}_{x\in S}$, where $U_x$ is an open neighborhood of $x\in S$, $\phi_x$ maps $x$ to the tip of the cone $C_{\vartheta_x}$, and the transition maps are Euclidean isometries.
This seemingly technical definition should not hide the fact that these objects are extremely simple and natural: any such flat surface can be obtained by gluing polygons embedded in ${\mathbb R}^2$ along pairs of sides of equal length. For example, a rectangle with opposite sides identified will define a flat torus with no singularity. On the other hand, a regular $4g$-gon with opposite sides identified gives a flat surface of genus $g$ with a single singularity of angle $2\pi(2g-1)$. In general, the topology of the surface is related to the cone angles by the following Gauss-Bonnet Formula: if ${\Sigma}$ is a closed flat surface with cone angles $\{\vartheta_x\}_{x\in S}$, then $$\sum_{x\in S}(2\pi-\vartheta_x)=2\pi\chi(\Sigma),$$ where $\chi({\Sigma})$ is the Euler characteristic of ${\Sigma}$.
\[def:crit\] A graph $G$ is [*isoradially embedded in a flat surface $\Sigma$*]{} if the following conditions are satisfied:
[$\Sigma$ is a compact orientable flat surface with cone-type singularities;]{}
[each edge of $G$ is a straight line in ${\Sigma}$;]{}
[each closed face $f$ of $G\subset\Sigma$ contains an element $x_f$ at distance $1$ from all vertices of $\partial f$;]{}
[a singularity of $\Sigma$ is either a vertex of $G$ or a vertex $x_f$ of the dual graph $G^*$, that is, the singular set is contained in $V(G)\cup V(G^*)$.]{}
Given an isoradially embedded graph $G\subset\Sigma$, each edge $e\in E(G)$ has an associated rhombus as illustrated in Figure \[fig:theta\]. Therefore, the metric space $\Sigma$ should simply be understood as rhombi pasted together along their boundary edges. This observation also leads to the following fact.
\[prop:real\] Any finite graph $G$ can be isoradially embedded in a flat surface.
Fix an arbitrary angle $\theta\in(0,\pi/2)$ and associate to each half-edge $\tilde{e}$ of $G$ the isosceles triangle illustrated below.
2.5pt at 175 60 at 320 110 at 80 110 at 175 110 at 220 110
For each vertex $v\in V(G)$, choose a cyclic ordering of the $d_v$ half-edges of $G$ coming out of $v$ and build the associated star $\mathit{St}(v)$ as follows: glue together the corresponding $d_v$ triangles following the chosen cyclic ordering around $v$. Note that $\mathit{St}(v)$ is a flat surface with one singularity of cone angle $2\theta d_v$, and that the cyclic ordering endows this surface with an orientation. For each edge $e=(v,w)\in E(G)$, glue together the stars $\mathit{St}(v)$ and $\mathit{St}(w)$ along their boundary by pasting together the two triangles associated to $e$ in the unique way that is consistent with the orientation of the stars. (If $e$ is a loop at $v$, just glue together the corresponding sides of the star $\mathit{St}(v)$.) By construction, $G$ is isoradially embedded in the resulting metric space ${\Sigma}$, which is a compact oriented flat surface.
Following the discussion of the previous subsections, we shall adopt the following terminology.
\[def:nu\] Let $G$ be a graph isoradially embedded in a flat surface ${\Sigma}$. The [*critical weight*]{} associated to the edge $e\in E(G)$ is defined by $$\nu_e=\tan(\theta_e/2),$$ where $\theta_e\in(0,\pi/2)$ is the half-rhombus angle associated to the edge $e$. The partition function for the [*critical Ising model*]{} on $G$ is given by $$Z(G,\nu)=\sum_{\gamma\in\mathcal{E}(G)}\prod_{e\in\gamma}\nu_e,$$ where $\mathcal{E}(G)$ denotes the set of even subgraphs of $G$.
The Kac-Ward formula for graphs in flat surfaces {#sec:KW}
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In [@Cim2], we gave a generalized Kac-Ward formula for the Ising partition function on any finite weighted graph $(G,x)$. The aim of this section is to show that when the graph $G$ is embedded in a flat surface, the generalized Kac-Ward matrices take a particularly simple form – whatever the weight system $x$ on $G$ is. In the next section, we will consider the case of isoradial graphs with critical weights.
Kac-Ward matrices for graphs in flat surfaces
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Let us start with some general terminology and notation. Given a weighted graph $(G,x)$, let ${{\mathbb E}}={{\mathbb E}}(G)$ be the set of oriented edges of $G$. Following [@Ser], we shall denote by $o(e)$ the origin of an oriented edge $e\in{\mathbb E}$, by $t(e)$ its terminus, and by $\bar{e}$ the same edge with the opposite orientation. By abuse of notation, we shall write $x_e=x_{\bar{e}}$ for the weight associated to the unoriented edge corresponding to $e$ and $\bar{e}$.
Now, assume that $G$ is embedded in an orientable flat surface ${\Sigma}$ so that each edge of $G$ is a straight line, ${\Sigma}\setminus G$ consists of topological discs, and the set $S$ of cone-type singularities is contained in $V(G)\cup V(G^*)$. As above, let $\vartheta_x$ denote the cone angle of the singularity $x\in S$. Fix a unitary character $\varphi$ of the fundamental group of ${\Sigma}$, that is, an element of $${\mathrm{Hom}}(\pi_1({\Sigma}),U(1))={\mathrm{Hom}}(\pi_1({\Sigma}),S^1)={\mathrm{Hom}}(H_1({\Sigma}),S^1)=H^1(\Sigma;S^1).$$ As $G\subset{\Sigma}$ induces a cellular decomposition of ${\Sigma}$, one can represent such a cohomology class by a cellular 1-cocycle, that we shall also denote by $\varphi$. This is nothing but a map from the set ${{\mathbb E}}$ of oriented edges of $G$ into $S^1$, such that $\varphi(\bar{e})=\overline{\varphi(e)}$ and $\varphi(\partial f)=\prod_{e\in\partial f}\varphi(e)=1$ for each face $f$ of $G\subset{\Sigma}$.
\[def:KW\] Let $T^\varphi$ denote the $|{{\mathbb E}}|\times|{{\mathbb E}}|$ matrix defined by $$T^\varphi_{e,e'}=
\begin{cases}
\varphi(e)\,i\exp\left(-\frac{i}{2}\beta(e',\bar{e})\right)\,x_e& \text{if $t(e)=o(e')$ but $e'\neq \bar{e}$;} \\
0 & \text{otherwise,}
\end{cases}$$ where $\beta(e',\bar{e})\in (0,\vartheta_v)$ denotes the angle from $e'$ to $\bar{e}$, as illustrated in the left part of Figure \[fig:alpha\]. We shall call the matrix $I-T^\varphi$ the [*$\varphi$-twisted Kac-Ward matrix*]{} associated to the weighted graph $(G,x)$, and denote its determinant by $\tau^\varphi(G,x)=\det(I-T^\varphi)$.
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\[rem:alpha\] If $t(e)=o(e')=v$ is not a singularity (that is, if the cone angle $\vartheta_v$ is equal to $2\pi$), then the complex number $i\exp\left(-\frac{i}{2}\beta(e',\bar{e})\right)$ is equal to $\exp\left(\frac{i}{2}\alpha(e,e')\right)$, with $\alpha(e,e')\in (-\pi,\pi)$ the angle from $e$ to $e'$ illustrated in the right part of Figure \[fig:alpha\]. If $v$ does belong to $S$, then the dotted line drawn there does not make sense anymore, hence the necessity to adopt this slightly less intuitive definition.
Discrete spin structures on flat surfaces
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The Kac-Ward matrices are particularly useful when $\varphi$ is a special type of 1-cocycle, namely a discrete spin structure. In this paragraph, we shall recall the definition and main properties of these objects, slightly generalizing Section 3.1 of [@Cim1].
Loosely speaking, a [*spin structure*]{} on an oriented surface ${\Sigma}$ is a way to count parity of rotation numbers for closed curves in ${\Sigma}$. In the plane, there is a unique way to do so, and therefore a unique spin structure. On a closed orientable surface of genus $g$ however, there are exactly $2^{2g}$ distinct spin structures. More precisely, one can identify the set ${\mathcal S}({\Sigma})$ of spin structures on ${\Sigma}$ with the set of [*quadratic forms*]{} on ${\Sigma}$ [@Joh]: these are the maps $q\colon H_1({\Sigma};{\mathbb Z}_2)\to{\mathbb Z}_2$ such that $q(x+y)=q(x)+q(y)+x\cdot y$ for all $x,y\in H_1({\Sigma};{\mathbb Z}_2)$, where $x\cdot y$ denotes the intersection number of $x$ and $y$. Note that the difference of two quadratic forms is a linear form. Therefore, this set admits a freely transitive action of the abelian group $H^1({\Sigma};{\mathbb Z}_2)$; in other words, it is an affine $H^1({\Sigma};{\mathbb Z}_2)$-space. The [*Arf invariant*]{} of a spin structure is defined as the Arf invariant of the associated quadratic form, that is, the number ${\mathrm{Arf}}(q)\in{\mathbb Z}_2$ satisfying $$(-1)^{{\mathrm{Arf}}(q)}=\frac{1}{2^g}\sum_{\alpha\in H_1({\Sigma};{\mathbb Z}_2)}(-1)^{q(\alpha)}.$$
Coming back to flat surfaces, let us assume that $G$ is a graph embedded in a flat surface ${\Sigma}$ so that ${\Sigma}\setminus G$ consists of topological discs, and let $X$ denote the induced cellular decomposition. We shall now explain how, in such a situation, it is possible to encode spin structures on ${\Sigma}$ by some cocycles $\lambda\in Z^1(X;S^1)$. Let us assume that all cone angles $\vartheta_x$ are positive multiples of $2\pi$, i.e. that ${\Sigma}$ has trivial local holonomy. Then, the holonomy defines an element $\mathrm{Hol}$ of $\mathrm{Hom}(\pi_1(\Sigma),S^1)=H^1(\Sigma;S^1)=H^1(X;S^1)$. We shall call a cocycle $\kappa\in Z^1(X;S^1)$ such that $[\kappa^{-1}]=\mathrm{Hol}$ a [*discrete canonical bundle*]{} over $\Sigma$. Note that such a cocycle $\kappa$ is very easy to determine. Indeed, it is always possible to represent $\Sigma$ as planar polygons $P$ with boundary identifications. Furthermore, these polygons can be chosen so that $G$ intersects $\partial P$ transversally, except at possible singularities in $S\cap V(G)$. Define $\kappa$ by $$\kappa(e)=
\begin{cases}
1 & \text{if $e$ is contained in the interior of $P$;} \\
\exp(-i\theta) & \text{if $e$ meets $\partial P$ transversally,}
\end{cases}$$ where $\theta$ denotes the angle between the sides of $\partial P\subset{\mathbb C}$ met by the edge $e$. If $S\cap V(G)$ is empty, this defines completely a natural choice of discrete canonical bundle $\kappa$. Otherwise, the partially defined $\kappa$ above can be extended to a cocycle yielding a discrete canonical bundle.
Mimicking the continuous case (in the version developed by Atiyah [@Ati]), let us define a [*discrete spin structure*]{} on $\Sigma$ as any cellular 1-cocycle $\lambda\in Z^1(X;S^1)$ such that $\lambda^2=\kappa$. Two discrete spin structures will be called [*equivalent*]{} if they are cohomologous. The set ${\mathcal S}(X)$ of equivalent classes of discrete spin structures on $\Sigma$ is then given by $${\mathcal S}(X)=\{[\lambda]\in H^1(X;S^1)\,|\,[\lambda]^2=[\kappa]\}.$$ Note that if the flat surface $\Sigma$ has trivial holonomy, then $[\kappa]$ is trivial, so the set ${\mathcal S}(X)$ is equal to the $2g$-dimensional vector space $H^1(\Sigma;{\mathbb Z}_2)$. In general, $G\subset\Sigma$ can be described via planar polygons as explained earlier. In such a case, and assuming that the singular set $S\cap V(G)$ is empty, a discrete spin structure is given by $$\lambda(e)=
\begin{cases}
1 & \text{if $e$ is contained in the interior of $P$;} \\
\exp(-i\theta/2) & \text{if $e$ meets $\partial P$,}
\end{cases}$$ where $\exp(-i\theta/2)$ denotes one of the square roots of the angle between the sides of $\partial P\subset{\mathbb C}$ met by the edge $e$.
One easily checks that the set ${\mathcal S}(X)$ is an affine $H^1(\Sigma;{\mathbb Z}_2)$-space. Furthermore:
\[prop:spin\] If all cone angles of ${\Sigma}$ are odd multiples of $2\pi$, then there exists a canonical $H^1(\Sigma;{\mathbb Z}_2)$-equivariant bijection ${\mathcal S}(X)\to{\mathcal S}(\Sigma)$.
Let $\kappa\in Z^1(X;S^1)$ be a fixed discrete canonical bundle over $\Sigma$. For each $\lambda\in Z^1(X;S^1)$ such that $\lambda^2=\kappa$, we shall now construct a vector field $V_\lambda$ on ${\Sigma}$ with zeroes of even index. Such a vector field is well-known to define a spin structure, or equivalently – by Johnson’s theorem [@Joh] – a quadratic form $q_\lambda$ on $H_1({\Sigma};{\mathbb Z}_2)$. The proof will be completed with the verification that two equivalent $\lambda$’s induce identical quadratic forms, and that the assignment $[\lambda]\mapsto q_\lambda$ is $H^1({\Sigma};{\mathbb Z}_2)$-equivariant.
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First replace the cellular decomposition $X$ of ${\Sigma}$ by $X'$, where each singularity $v\in V(G)\cap S$ is removed as illustrated above. Obviously, $\lambda$ induces $\lambda'\in Z^1(X';S^1)$ by setting $\lambda'(e)=1$ for each newly created edge $e$. Fix an arbitrary orientation $\omega$ of the edges of $X'$. This allows to represent $\lambda'\in Z^1(X';S^1)$ as follows: write $\lambda'(e)=\exp(i\beta_{\lambda'}(e))$ with $0\le\beta_{\lambda'}(e)<2\pi$ if $e$ is an edge oriented by $\omega$, and set $\beta_{\lambda'}(\bar{e})=-\beta_{\lambda'}(e)$ for the reverse edge. Fix an arbitrary tangent vector $V_{\lambda}(v)$ at some arbitrary vertex $v$ of $X'$, and extend it to the 1-skeleton $G'$ of $X'$ as follows: running along an edge $e$, rotate the tangent vector by an angle of $2\beta_{\lambda'}(e)$ in the negative direction. Since $\lambda$ is a cocycle such that $\lambda^2=\kappa$, and since each cone angle is a multiple of $2\pi$, this gives a well-defined vector field along $G'$. Extend it to the whole surface ${\Sigma}$ by the cone construction, creating one zero in each face of $X$ and at each element of $V(G)\cap S$. Obviously, the resulting vector field $V_\lambda$ depends on the choice of $\omega$, but not in a crucial way. Indeed, reversing the orientation $\omega$ on a given edge $e$ either does nothing (if $\lambda'(e)=1$), or corresponds to adding two full twists to the vector field along $e$. Therefore, the parity of winding numbers with respect $V_\lambda$ are independent of $\omega$. In particular, one easily checks that a zero of $V_\lambda$ is of even index if and only if the corresponding cone angle is an odd multiple of $2\pi$, which we assumed.
As explained in [@Joh], the quadratic form $q_\lambda\colon H_1({\Sigma};{\mathbb Z}_2)\to{\mathbb Z}_2$ corresponding to $V_\lambda$ is determined as follows: for any regular oriented simple closed curve $C\subset\Sigma\setminus S$, the number $q_\lambda([C])+1$ is equal to the mod 2 winding number of the tangential vector field along $C$ with respect to the vector field $V_\lambda$. For an oriented simple closed curve $C\subset G$, we obtain the following equality modulo 2: $$q_\lambda([C])=1+\frac{1}{\pi}\Big(\sum_{e\subset C} \beta_\lambda(e)+\frac{1}{2}\sum_{v\in C}\alpha_v(C)\Big),$$ where the first sum is over all oriented edges in the oriented curve $C$, and $\alpha_v(C)$ is the angle illustrated below. (This angle should be interpreted as explained in Remark \[rem:alpha\].)
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Obviously, equivalent $\lambda$’s induce the same quadratic form $q_\lambda$. Finally, given two discrete spin structures $\lambda_1,\lambda_2$, the cohomology class of the 1-cocycle $\lambda_1/\lambda_2\in Z^1(X;\{\pm 1\})$ is determined by its value on oriented simple closed curves in $G$. For such a curve $C$, we have $$(\lambda_1/\lambda_2)(C)=\exp\Big(i\sum_{e\subset C}(\beta_{\lambda_1}(e)-\beta_{\lambda_2}(e))\Big)=\exp\big(i\pi(q_{\lambda_1}-q_{\lambda_2})([C])\big).$$ Therefore, the assignment $[\lambda]\mapsto q_\lambda$ is $H^1({\Sigma};{\mathbb Z}_2)$-equivariant, which concludes the proof.
The Kac-Ward formula for flat surfaces
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We are finally ready to prove the main result of this section, motivating the introduction of twisted Kac-Ward matrices and discrete spin structures. As before, let $(G,x)$ be a weighted graph embedded in a closed orientable flat surface $\Sigma$ so that each edge of $G$ is a straight line, $\Sigma\setminus G$ consists of topological discs, and the set $S$ of cone-type singularities is contained in $V(G)\cup V(G^*)$.
\[thm:Arf\] If all cone angles are odd multiple of $2\pi$, then the Kac-Ward determinant $\tau^\varphi(G,x)$ is the square of a polynomial in the $x_e$’s whenever $\varphi$ is a discrete spin structure on $G\subset{\Sigma}$. Furthermore, if $\tau^\varphi(G,x)^{1/2}$ denotes the square root with constant coefficient equal to $+1$, then the Ising partition function on $G$ is given by $$Z(G,x)=\frac{1}{2^g}\sum_{\lambda\in{\mathcal S}(X)}(-1)^{{\mathrm{Arf}}(\lambda)}\tau^\lambda(G,x)^{1/2},$$ where $g$ is the genus of ${\Sigma}$ and ${\mathrm{Arf}}(\lambda)\in{\mathbb Z}_2$ the Arf invariant of the spin structure corresponding to $\lambda$.
The demonstration given below is by no means self-contained: it only consists in recasting the flat surface case in the more general (and more complicated) topological setting discussed by the author in [@Cim2]. We refer to this article for further details.
By Bass’ Theorem [@Bas], $\tau^\varphi(G,x)$ is given by $$\tau^\varphi(G,x)=\det(I-T^\varphi)=\prod_{\gamma\in\mathcal{P}(G)}\Big(1-\prod_{(e,e')\in\gamma}T^\varphi_{e,e'}\Big),$$ where $\mathcal{P}(G)$ is the (infinite) set of prime reduced oriented closed paths in $G$, and the second product is over all pairs of consecutive oriented edges in the oriented path $\gamma$. By definition of $T^\varphi$, $$\prod_{(e,e')\in\gamma}T^\varphi_{e,e'}=\varphi(\gamma)\,\exp\left(\textstyle{\frac{i}{2}}\alpha(\gamma)\right)\,x(\gamma),$$ where $x(\gamma)=\prod_{e\in\gamma}x_e$ and $\alpha(\gamma)$ is the sum of the angles $\alpha(e,e')$ along $\gamma$ (interpreted as in Remark \[rem:alpha\]). This equality already shows that $\tau^\varphi(G,x)$ does not depend on the choice of the 1-cocycle representing the cohomology class $\varphi$. Furthermore, if $\varphi\in Z^1(X;S^1)$ is a discrete spin structure, then $$\left(\varphi(\gamma)\,\exp\left(\textstyle{\frac{i}{2}}\alpha(\gamma)\right)\right)^2=\varphi(\gamma)^2\,\exp(i\alpha(\gamma))=\varphi(\gamma)^2\,\kappa^{-1}(\gamma)=1$$ for all oriented closed path $\gamma$ in $G$. This implies that $\varphi(\gamma)\,\exp\left(\frac{i}{2}\alpha(\gamma)\right)$ always belongs to $\{\pm 1\}$. Therefore, for such a $\varphi=\lambda\in{\mathcal S}(X)$, $$\tau^\lambda(G,x)=\prod_{[\gamma]\in\mathcal{P}(G)/\sim}\left(1-(-1)^{w_\lambda({\gamma})}x(\gamma)\right)^2,$$ for some $w_\lambda(\gamma)\in{\mathbb Z}_2$, where the equivalence relation on $\mathcal{P}(G)$ is given by $\gamma\sim-\gamma$. Using the notations of the proof of Proposition \[prop:spin\], the element $w_\lambda(\gamma)$ satisfies $$(-1)^{w_\lambda(\gamma)}=\lambda(\gamma)\,\exp\left(\textstyle{\frac{i}{2}}\alpha(\gamma)\right)=
\exp\Big(i\sum_{e\subset\gamma}\beta_\lambda(e)+{\frac{i}{2}}\sum_{v\in\gamma}\alpha_v(\gamma)\Big).$$ Therefore, $w_\lambda(\gamma)$ is nothing but the mod 2 winding number of the tangent vector field along $\gamma$ with respect to the vector field $V_\lambda$ associated to the discrete spin structure $\lambda$. The formula now follows from Proposition \[prop:spin\] and [@Cim2 Corollary 2.2].
The whole setting can be extended to encompass graphs embedded in flat surfaces with boundary. If there is exactly one boundary component, then Theorem \[thm:Arf\] extends verbatim. (In particular, it applies to domains in the plane, where this theorem is exactly the original Kac-Ward formula [@K-W].) If the flat surface has several boundary components, then the formula is slightly more complicated.
The Kac-Ward matrices with critical weights {#sec:crit}
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As proved in the previous section, the $\varphi$-twisted Kac-Ward matrices can be used to compute the Ising partition function for any weighted graph embedded in a flat surface. We shall now assume the graph to be isoradial and the weights to be critical (recall Definitions \[def:crit\] and \[def:nu\]). We will start with a combinatorial interpretation for $\tau^\varphi(G,\nu)$ (Proposition \[prop:tech\]), that we then use for two of our main results. First, we prove a duality theorem relating $\tau^\varphi(G,\nu)$ and $\tau^\varphi(G^*,\nu^*)$ (Theorem \[thm:duality\]). Then, we show that $\tau^\varphi(G,\nu)$ coincides up to a multiplicative constant with the determinant of the critical discrete Laplacian on $G$ if and only if the genus of $\Sigma$ is zero or one (Theorem \[thm:Delta\]). In a last subsection, we explain how $\tau^\varphi(G,\nu)$ can be understood as a discrete version of the $\overline\partial$-torsion of the underlying Riemann surface.
A combinatorial interpretation for $\tau^\varphi(G,\nu)$ {#sub:comb}
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Let us start with some notations.
Given a graph $G$, let ${\mathcal F}(G)$ denote the set of subgraphs $F\subset G$ such that $F$ spans all vertices of $G$, and no connected component of $F$ is a tree. Also, for a graph $F$ embedded in an oriented surface ${\Sigma}$, we shall denote by $N(F)$ a small tubular neighborhood of $F$ in ${\Sigma}$. (This is simply $F$ “thickened", as illustrated in the middle of Figure \[fig:N\].) Since ${\Sigma}$ is oriented, so is $N(F)$, and this induces an orientation on the boundary $\partial N(F)$ of $N(F)$. Therefore, $\partial N(F)$ consists of a disjoint union of oriented simple closed curves $\gamma$ on ${\Sigma}$. This is illustrated in Figure \[fig:N\].
\[prop:tech\] Let $G$ be a graph isoradially embedded in a flat surface ${\Sigma}$, and let us assume that all cone angles $\vartheta_v$ of singularities $v\in V(G)$ are odd multiple of $2\pi$. Let $\nu_e=\tan(\theta_e/2)$ denote the critical weight system on $G\subset\Sigma$, and set $\mu_e=i\tan(\theta_e)$. Then for any 1-cocycle $\varphi$, $$\tau^\varphi(G,\nu)=C\,\sum_{F\in{\mathcal F}(G)}\prod_{\gamma\subset\partial N(F)}(1-\varphi(\gamma))\,\mu(F),$$ where the product is over all connected components $\gamma$ of the (clockwise oriented) boundary of a tubular neighborhood $N(F)$ of $F$ in $\Sigma$, $\mu(F)=\prod_{e\in F}\mu_e$, and the constant $C$ is equal to $$C=(-1)^{|V(G)|}\,2^{-\chi(G)}\prod_{v\in V(G)}\exp(i\vartheta_v/4)\prod_{e\in E(G)}\frac{\cos(\theta_e)}{1+\cos(\theta_e)},$$ with $\chi(G)=|V(G)|-|E(G)|$ the Euler characteristic of $G$.
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The proof being quite substantial, we will split it into two lemmas. Let us begin with some notation. We shall write $V=V(G)$ and $E=E(G)$ for the sets of vertices and edges of the graph $G$, ${{\mathbb E}}={{\mathbb E}}(G)$ for the set of oriented edges of $G$ and ${\mathcal L}({{\mathbb E}})$ for the complex vector space spanned by ${{\mathbb E}}$. Obviously, ${{\mathbb E}}$ can be partitioned into ${{\mathbb E}}=\bigsqcup_{v\in V}E_v$, where $E_v$ contains all oriented edges $e$ with origin $o(e)=v$. Now, let us cyclically order the elements of $E_v$ by turning counterclockwise around $v$. (As ${\Sigma}$ is orientable, this can be done in a consistent way.) Given $e\in E_v$, let $R(e)$ denote the next edge with respect to this cyclic order, as illustrated in Figure \[fig:R\]. This induces an endomorphism $R$ of ${\mathcal L}({{\mathbb E}})$. Also, let $J$ denote the endomorphism of ${\mathcal L}({{\mathbb E}})$ given by $J(e)=\bar{e}$. Finally, we shall write $\mu$ for the endomorphism of ${\mathcal L}({\mathbb E})$ given by $\mu(e)=\mu_e\,e$, and similarly for any weight system and for $\varphi$.
The aim of the first lemma is to relate $\tau^\varphi(G,\nu)$ to the determinant of a more tractable matrix.
\[lemma:KW\] For $G\subset{\Sigma}$ as in Proposition \[prop:tech\], and for any 1-cocycle $\varphi$, $$\tau^\varphi(G,\nu)=2^{-\chi(G)}\prod_{e\in E}\frac{\cos^2(\theta_e)}{1+\cos(\theta_e)}\det M,$$ with $M=I+J\varphi\mu-R(\mu+1)\in\mathit{End}({\mathcal L}({{\mathbb E}}))$ and $\chi(G)=|V(G)|-|E(G)|$.
Let $\mathit{Succ}\in\mathit{End}({\mathcal L}({{\mathbb E}}))$ be defined as follows: if $e$ is an oriented edge with terminus $t(e)=v$, then $$\mathit{Succ}(e)=i\varphi(e)\nu_e\sum_{e'\in E_v}\omega(e',\bar{e})\,e',$$ where $\omega(e',\bar{e})=\exp(-\frac{i}{2}\beta(e',\bar{e}))$ for $e'\neq \bar{e}\in E_v$ with $\beta(e',\bar{e})$ as in Figure \[fig:alpha\], and $\omega(\bar{e},\bar{e})=-1$. Also, let $S\in\mathit{End}({\mathcal L}({{\mathbb E}}))$ be the endomorphism given by $S=\mathit{Succ}+iJ\varphi\nu$. By definition, the $\varphi$-twisted Kac-Ward matrix is the transposed of $I-S$. Now, consider the matrix $$A=(I-S)(I+iJ\varphi\nu)=I-\mathit{Succ}+\mathit{Com},$$ where $$\mathit{Com}(e)=-i\varphi(e)\nu_eS(\bar{e})=\nu_e^2\sum_{\genfrac{}{}{0pt}{}{e'\in E_v}{e'\neq e}}\omega(e',e)\,e'$$ if $e$ has origin $o(e)=v$. Since $$\det(I+iJ\varphi\nu)=\prod_{e\in E}\det\left(\begin{array}{cc}1&i\varphi(\bar{e})\nu_e\cr i\varphi(e)\nu_e&1\cr\end{array}\right)=\prod_{e\in E}(1+\nu_e^2),$$ we get the equality $$\label{equ:A}
\tau^\varphi(G,\nu)=\prod_{e\in E}(1+\nu_e^2)^{-1}\det A.$$ (The computation above is a variation on a trick due to Foata and Zeilberger, see [@F-Z Section 8].)
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Let $Q\in\mathit{End}({\mathcal L}({{\mathbb E}}))$ be defined by $$Q(e)=\exp\left({\textstyle\frac{i}{2}}\beta(e,R(e))\right)e,$$ and set $P=I-RQ$. Obviously, this endomorphism decomposes into $P=\bigoplus_{v\in V}P_v$ with $P_v\in\mathit{End}({\mathcal L}({E}_v))$, and one easily computes $$\det P_v=1-\prod_{e\in E_v}\exp\left({\textstyle\frac{i}{2}}\beta(e,R(e))\right)=1-\exp\left({\textstyle\frac{i}{2}}\vartheta_v\right)=2,$$ as the cone angle $\vartheta_v$ is an odd multiple of $2\pi$. Hence, the determinant of $P$ is $2^{|V|}$. The point of introducing this $P$ is that it can be used to greatly simplify the matrix $A$. Indeed, let us compute the composition $PA$. If $e$ has terminus $v$, then $$\begin{aligned}
P\mathit{Succ}(e)&=(I-RQ)i\varphi(e)\nu_e\sum_{e'\in E_v}\omega(e',\bar{e})\,e'\cr
&=i\varphi(e)\nu_e\sum_{e'\in E_v}\left(\omega(e',\bar{e})-\omega(R^{-1}(e'),\bar{e})\exp\left({\textstyle\frac{i}{2}}\beta(R^{-1}(e'),e')\right)\right)e'\cr
&=-2i\varphi(e)\nu_e\,(\bar{e}).\end{aligned}$$ (To check that the coefficient of $e'=R(\bar{e})$ vanishes, we use once again the fact that $\vartheta_v$ is an odd multiple of $2\pi$.) Therefore, we have the equality $P\mathit{Succ}=-2iJ\varphi\nu$. Similarly, given $e$ with $o(e)=v$, $$\begin{aligned}
P\mathit{Com}(e)=&(I-RQ)\nu_e^2\sum_{e'\in E_v\setminus\{e\}}\omega(e',e)\,e'\cr
=&\nu_e^2\hskip-.2cm\sum_{e'\in E_v\setminus\{e,R(e)\}}\hskip-.2cm\left(\omega(e',e)-\omega(R^{-1}(e'),e)\exp\left({\textstyle\frac{i}{2}}\beta(R^{-1}(e'),e')\right)\right)e'\cr
&+\nu_e^2\left(\omega(R(e),e)\,R(e)-\omega(R^{-1}(e),e)\exp\left({\textstyle\frac{i}{2}}\beta(R^{-1}(e),e)\right)\,e\right)\cr
=&-\nu^2_e(I+RQ)(e).\end{aligned}$$ These two equalities lead to $$\begin{aligned}
PA&=P(I-\mathit{Succ}+\mathit{Com})\cr
&=(I-RQ)+2iJ\varphi\nu-(I+RQ)\nu^2\cr
&=(1-\nu^2)I+2iJ\varphi\nu-RQ(1+\nu^2).\end{aligned}$$ We finally get $$\label{equ:A'}
\det A=2^{-|V|}\prod_{e\in E}(1-\nu_e^2)^2\det A',$$ where $A'$ is given by $$A'=I+iJ\varphi\,\frac{2\nu}{1-\nu^2}-RQ\,\frac{1+\nu^2}{1-\nu^2}.$$
It is now time to use the fact that the weights $\nu_e$ are not any weights, but the critical ones given by $\nu_e=\tan(\theta_e/2)$. First observe that $$i\frac{2\nu_e}{1-\nu_e^2}=i\tan(\theta_e)=\mu_e\qquad\text{and}\qquad\frac{1+\nu_e^2}{1-\nu_e^2}=\frac{1}{\cos(\theta_e)}.$$ Next, note the equality $\beta(e,R(e))=\theta_e+\theta_{R(e)}$ illustrated to the right of Figure \[fig:R\]. This implies that $RQ(e)=\exp\left({\textstyle\frac{i}{2}}\theta_e\right)\exp\left({\textstyle\frac{i}{2}}\theta_{R(e)}\right)R(e)$. Multiplying each column of $A'$ (corresponding to $e$) by $\exp\left({\textstyle\frac{i}{2}}\theta_e\right)$ and each line (corresponding to $e'$) by $\exp\left(-{\textstyle\frac{i}{2}}\theta_{e'}\right)$, we obtain a new matrix $M$ with $$\label{equ:M}
\det A'=\det M\qquad\text{and}\qquad M=I+J\varphi\mu-R(1+\mu),$$ since $\theta_{\bar{e}}=\theta_e$ and $\frac{\exp(i\theta_e)}{\cos(\theta_e)}=1+\mu_e$. Equations (\[equ:A\]), (\[equ:A’\]) and (\[equ:M\]) give the statement of the lemma.
In a second lemma, we now give a combinatorial interpretation of the determinant of this matrix $M$.
\[lemma:M\] The endomorphism $M=I+J\varphi\mu-R(1+\mu)$ of ${\mathcal L}({{\mathbb E}})$ satisfies $$\det M=(-1)^{|V|}\prod_{e\in E}(1+\mu_e)\sum_{\genfrac{}{}{0pt}{}{F\subset G}{V(F)=V}}\prod_{\gamma\subset\partial N(F)}(1-\varphi(\gamma))\,\mu(F),$$ the sum being on all subgraphs $F$ of $G$ spanning all vertices of $G$.
By definition, the coefficients of $M$ are given by $$M_{e,e'}=
\begin{cases}
1& \text{if $e=e'$;} \\
\varphi(e)\mu_e& \text{if $e'=\bar{e}$;} \\
-(1+\mu_e)&\text{if $e'=R(e)$,}
\end{cases}$$ and vanish otherwise. Let us compute directly the determinant of $M$ as $$\det M=\sum_{\sigma\in S({{\mathbb E}})}(-1)^{\text{sgn}(\sigma)}\prod_{e\in{{\mathbb E}}}M_{e,\sigma(e)}.$$ Each permutation $\sigma\in S({{\mathbb E}})$ decomposes into disjoint cycles, inducing a partition ${{\mathbb E}}=\bigsqcup_j E_j(\sigma)$ into orbits of length $\ell_j(\sigma)$. The corresponding contribution to the determinant is $$(-1)^{\text{sgn}(\sigma)}\prod_{e\in E}M_{e,\sigma(e)}=\prod_j(-1)^{\ell_j(\sigma)+1}\prod_{e\in E_j(\sigma)}M_{e,\sigma(e)}.$$ Since $M_{e,e}$ is equal to $1$, the oriented edges that are fixed by $\sigma$ contribute a trivial factor $1$ to this product and the corresponding orbits can be removed. By definition of $M$, a permutation $\sigma\in S({{\mathbb E}})$ will have a non-zero contribution only if each of the $n(\sigma)$ remaining orbits forms a cycle of oriented edges of $G$ such that each oriented edge $e$ is either followed by $J(e)=\bar{e}$ or by $R(e)$, and such that these cycles pass through each edge of $G$ at most twice, and if so, in opposite directions. If ${\Gamma}(G)$ denotes the set of such union of cycles, we have $$\det M=\sum_{\gamma\in{\Gamma}(G)}(-1)^{n(\gamma)}\prod_j\prod_{e\in\gamma_j}-M_{e,\gamma_j(e)},$$ where each $\gamma\in{\Gamma}(G)$ is written as a union of cycles $\gamma=\bigcup_{j=1}^{n(\gamma)}\gamma_j$. Now, for any fixed $\gamma$, a given edge $e\in E$ will fall in one of the following five categories:
[$e$ is covered by $\gamma$ in both directions, as part of a cycle of the form $(\dots,R^{-1}(e),e,\bar{e},R(\bar{e}),\dots)$; the corresponding contribution to the determinant of $M$ is $-\varphi(e)\mu_e(1+\mu_e)$.]{}
[$e$ is covered in both directions by the cycle $(e,\bar{e})$, so the contribution is $-\mu_e^2$. (The minus sign comes from the contribution of this cycle to $n(\gamma)$.)]{}
[$e$ is covered by $\gamma$ in both directions, as part of cycles of the form $(\dots,R^{-1}(e),e,R(e),\dots,R^{-1}(\bar{e}),\bar{e},R(\bar{e}),\dots)$; in this case, the contribution is $(1+\mu_e)^2$.]{}
[$e$ is only covered in one direction, so the contribution is $(1+\mu_e)$.]{}
[$e$ is not covered at all, and the contribution is $1$.]{}
For any given element $\gamma\in\Gamma(G)$, any edge of type ${\mathit{(ii)}}$ in $\gamma$ can be removed (i.e. replaced by an edge of type $\mathit{(v)}$) and the resulting union of cycles will still belong to $\Gamma(G)$. The converse also holds: any type $\mathit{(v)}$ edge in an element of $\Gamma(G)$ can be replaced by a type ${\mathit{(ii)}}$ edge, the result will be in $\Gamma(G)$. Therefore, each time an edge appears as an edge of type ${\mathit{(ii)}}$ of some element of $\Gamma(G)$, it also appears as an edge of type $\mathit{(v)}$ of some other element of $\Gamma(G)$ and vice versa. Using the equality $(1-\mu_e^2)=(1+\mu_e)(1-\mu_e)$, we therefore can factor out a term $1+\mu_e$ for each $e\in E$, leading to $$\det M=\prod_{e\in E}(1+\mu_e)\sum_{\gamma\in{\Gamma}(G)}(-1)^{n(\gamma)}\prod_{e\in\gamma{\mathit{(i)}}}\varphi(e)(-\mu_e)\prod_{e\in\gamma{\mathit{(ii)}}}\mu_e\prod_{e\in\gamma{\mathit{(iii)}}}(1+\mu_e),$$ where $\gamma{\mathit{(i)}}$ (resp. $\gamma{\mathit{(ii)}},\gamma{\mathit{(iii)}}$) denotes the set edges of type ${\mathit{(i)}}$ (resp. ${\mathit{(ii)}},{\mathit{(iii)}}$) of $\gamma$. Next, we wish to expand the last product above as $$\prod_{e\in\gamma{\mathit{(iii)}}}(1+\mu_e)=\sum_{\gamma'\subset\gamma{\mathit{(iii)}}}\prod_{e\in\gamma'}\mu_e.$$ Using the notation $$\widetilde{{\Gamma}}(G)=\{(\gamma,\gamma')\,|\,\gamma\in{\Gamma}(G),\;\gamma'\subset\gamma{\mathit{(iii)}}\},$$ we get the equality $$\det M=\prod_{e\in E}(1+\mu_e)\sum_{\widetilde\gamma\in\widetilde{\Gamma}(G)}(-1)^{n(\widetilde\gamma)}\varphi(\widetilde{\gamma})\prod_{e\in\mathit{Supp}(\widetilde{\gamma})}\mu_e,$$ where $n(\widetilde{\gamma})=n(\gamma)+|\gamma{\mathit{(i)}}|$, the support of $\widetilde{\gamma}$ is $\mathit{Supp}(\widetilde{\gamma})=\gamma{\mathit{(i)}}\cup\gamma{\mathit{(ii)}}\cup\gamma'$ and $\varphi(\widetilde{\gamma})=\prod_{e\in\gamma{\mathit{(i)}}}\varphi(e)$ for any $\widetilde{\gamma}\in\widetilde{{\Gamma}}(G)$. In other words, $$\label{equ:tilde}
\frac{\det M}{\prod_{e\in E}(1+\mu_e)}=\sum_{F\subset E}c_F^\varphi\,\mu(F),\quad\text{where }
c_F^\varphi=\sum_{\genfrac{}{}{0pt}{}{\widetilde{\gamma}\in\widetilde{{\Gamma}}(G)}{\mathit{Supp}(\widetilde{\gamma})=F}}(-1)^{n(\widetilde\gamma)}\varphi(\widetilde{\gamma}).$$
We shall now check that $c^\varphi_F$ vanishes whenever $F$ does not span all vertices of $G$. Let us assume that the vertex $v$ is not spanned by $F$. Let $\gamma_v$ be the cycle $(e,R(e),R^2(e),\dots,R^{d-1}(e))$ with $e\in E_v$ and $d=|E_v|$. The set of $\widetilde{\gamma}\in\widetilde{{\Gamma}}(G)$ with $\mathit{Supp}(\widetilde{\gamma})=F$ is equal to the disjoint union of ${\Gamma}_0$ and ${\Gamma}_1$, where ${\Gamma}_0$ (resp. ${\Gamma}_1$) denotes the set of such $\widetilde{\gamma}=(\gamma,\gamma')$ with $\gamma$ containing (resp. not containing) $\gamma_v$. As $v$ does not belong to $\mathit{Supp}(\widetilde{\gamma})$, the mapping $(\gamma,\gamma')\mapsto(\gamma\cup\gamma_v,\gamma')$ gives a well-defined bijection $f\colon{\Gamma}_0\to{\Gamma}_1$. Since $n(f(\widetilde{\gamma}))=n(\widetilde{\gamma})+1$ and $\varphi(f(\widetilde{\gamma}))=\varphi(\widetilde{\gamma})$, we get $$c_F^\varphi=\sum_{\widetilde{\gamma}\in{\Gamma}_0\sqcup{\Gamma}_1}(-1)^{n(\widetilde\gamma)}\varphi(\widetilde{\gamma})=
\sum_{\widetilde{\gamma}\in{\Gamma}_0}((-1)^{n(\widetilde{\gamma})}+(-1)^{n(f(\widetilde{\gamma}))})\varphi(\widetilde{\gamma})=0.$$
So, let us assume that $F\subset E$ spans all vertices of $G$, and let $\{T_k\}_k$ denote the connected components of $F$ viewed as a subgraph of $G$. Given $\gamma\in{\Gamma}(G)$, let $\gamma|_{T_k}$ denote the restriction of the cycles composing $\gamma$ to the oriented edges of $T_k$. One easily checks that if $(\gamma,\gamma')\in\widetilde{{\Gamma}}(G)$ has support equal to $F$, then $\gamma|_{T_k}$ belongs to ${\Gamma}(T_k)$ and $\gamma|_{T_k}(\bullet)=\gamma(\bullet)\cap T_k$ for $\bullet\in\{\mathit{i,ii,iii}\}$. Hence, sending $(\gamma,\gamma')$ to $(\gamma|_{T_k},\gamma'\cap T_k)$ gives well-defined maps $$\pi_k\colon\{\widetilde{\gamma}\in\widetilde{{\Gamma}}(G)\,|\,\mathit{Supp}(\widetilde{\gamma})=F\}\to\{\widetilde{\gamma}_k\in\widetilde{{\Gamma}}(T_k)\,|\,\mathit{Supp}(\widetilde{\gamma}_k)=T_k\},$$ which in turn induce $$\{\widetilde{\gamma}\in\widetilde{{\Gamma}}(G)\,|\,\mathit{Supp}(\widetilde{\gamma})=F\}\to\prod_k\{\widetilde{\gamma}_k\in\widetilde{{\Gamma}}(T_k)\,|\,\mathit{Supp}(\widetilde{\gamma}_k)=T_k\}.$$ Using the fact that $F$ spans all vertices of $G$, one can check that this map is a bijection. Since $n(\widetilde{\gamma})=\sum_kn(\pi_k(\widetilde{\gamma}))$ and $\varphi(\widetilde{\gamma})=\prod_k\varphi(\pi_k(\widetilde{\gamma}))$ for any $\widetilde{\gamma}$ with support equal to $F$, it follows that $$\label{equ:c}
c^\varphi_F=\prod_k c^\varphi_{T_k}\quad\text{where }
c_T^\varphi=\sum_{\genfrac{}{}{0pt}{}{\widetilde{\gamma}\in\widetilde{{\Gamma}}(T)}{\mathit{Supp}(\widetilde{\gamma})=T}}(-1)^{n(\widetilde\gamma)}\varphi(\widetilde{\gamma}).$$ Note that the value of $c^\varphi_T$ does not depend on $G$ anymore. Furthermore, since $\mathit{Supp}(\widetilde{\gamma})$ must be equal to the whole of $T$, $\widetilde{\gamma}=(\gamma,\gamma')$ must satisfy $\gamma'=\gamma{\mathit{(iii)}}$. Therefore, $$c_T^\varphi=\sum_{\genfrac{}{}{0pt}{}{\gamma\in{\Gamma}(T)}{\mathit{Supp}(\gamma)=T}}(-1)^{n(\gamma)+|\gamma{\mathit{(i)}}|}\varphi(\gamma),$$ where $\mathit{Supp}(\gamma)=\gamma{\mathit{(i)}}\cup\gamma{\mathit{(ii)}}\cup\gamma{\mathit{(iii)}}$ and $\varphi(\gamma)=\prod_{e\in\gamma{\mathit{(i)}}}\varphi(e)$.
Let $N^\varphi_T\in\mathit{End}({\mathcal L}({{\mathbb E}}(T))$ be the endomorphism given by $N^\varphi_T=R-J\varphi$. Developing the determinant of $N^\varphi_T$ explicitely, we get $$\det N^\varphi_T=\sum_{\genfrac{}{}{0pt}{}{\gamma\in{\Gamma}(T)}{\mathit{Supp}(\gamma)=T}}(-1)^{n(\gamma)+|\gamma{\mathit{(i)}}|}\varphi(\gamma)=c^\varphi_T.$$ As before, the set ${{\mathbb E}}(T)$ is equal to $\bigsqcup_{v\in V(T)}E_v(T)$, and the endomorphism $R$ splits into $R=\bigoplus_{v\in V(T)}R_v$. Since $\det R_v=(-1)^{|E_v(T)|+1}$, we get $$c_T^\varphi=\det(R)\det(I-R^{-1}J\varphi)=(-1)^{|V(T)|}\det(I-R^{-1}J\varphi).$$ Now, the orbits $E_j$ of the action of $R^{-1}J$ on the set ${{\mathbb E}}(T)$ correspond exactly to the connected components of the oriented boundary of a tubular neighborhood $N(T)$ of $T$ in ${\Sigma}$, where $N(T)$ is endowed with the clockwise orientation. Hence $$\label{equ:N}
c_T^\varphi=(-1)^{|V(T)|}\prod_j\Big(1-\prod_{e\in E_j}\varphi(e)\Big)=(-1)^{|V(T)|}\prod_{\gamma\subset\partial N(T)}(1-\varphi(\gamma)).$$ Since $|V|=|V(F)|=\sum_k|V(T_k)|$ and $N(F)=\bigsqcup_k N(T_k)$, Equations (\[equ:tilde\]), (\[equ:c\]) and (\[equ:N\]) give the statement of the lemma.
Lemmas \[lemma:KW\] and \[lemma:M\] give the equality $$\tau^\varphi(G,\nu)=C\,\sum_{\genfrac{}{}{0pt}{}{F\subset G}{V(F)=V}}\prod_{\gamma\subset\partial N(F)}(1-\varphi(\gamma))\,\mu(F),$$ where the constant $C$ is equal to $$\begin{aligned}
C&=(-1)^{|V|}2^{-\chi(G)}\prod_{e\in E}\frac{\cos^2(\theta_e)}{1+\cos(\theta_e)}(1+i\tan(\theta_e))\cr
&=(-1)^{|V|}2^{-\chi(G)}\prod_{e\in E}\frac{\cos(\theta_e)}{1+\cos(\theta_e)}\exp(i\theta_e).\end{aligned}$$ This is equal to the value given in the statement of the proposition, since $$\prod_{e\in E}\exp(i\theta_e)=\prod_{v\in V}\prod_{e\in E_v}\exp(i\theta_e/2)=\prod_{v\in V}\exp(i\vartheta_v/4).$$ Finally, note that if a component of $F$ is a tree, then $\partial N(F)$ will contain a trivial cycle and the corresponding coefficient will vanish. Therefore, we can sum over all elements of the set ${\mathcal F}(G)$ defined at the beginning of the section. This concludes the proof of the proposition.
The duality theorem {#sub:duality}
-------------------
We are now ready to state and prove one of our main results.
\[thm:duality\] Let $G$ be a graph isoradially embedded in a flat surface ${\Sigma}$, and let $\nu$ be the critical weight system on $G$. The dual graph $G^*$ is also isoradially embedded in ${\Sigma}$, and therefore admits a critical weight system $\nu^*$. If all cone angles are odd multiples of $2\pi$, then for any $\varphi\in H^1({\Sigma};S^1)$, $$2^{|V(G^*)|}\hskip-2pt\prod_{e^*\in E(G^*)}(1+\cos(\theta_{e^*}))\,\tau^\varphi(G^*,\nu^*)=2^{|V(G)|}\hskip-2pt\prod_{e\in E(G)}(1+\cos(\theta_{e}))\,\tau^\varphi(G,\nu).$$
Given a subgraph $F$ of $G$, let $\psi(F)$ denote the subgraph of $G^*$ given by $$\psi(F)=\{e^*\in E(G^*)\,|\,e\notin F\}.$$ Obviously, $\psi$ defines a bijection from the set of subgraphs of $G$ onto the set of subgraphs of $G^*$, with inverse $\psi^{-1}(F^*)=\{e\in E(G)\,|\,e^*\notin F^*\}$. Let $\widetilde{{\mathcal F}}(G)$ denote the set of subgraphs of $G$ spanning all vertices of $G$, and containing no cycle that is the boundary of a face. These two conditions being dual to each other, the map $\psi$ defines a bijection $\psi\colon\widetilde{{\mathcal F}}(G)\to\widetilde{{\mathcal F}}(G^*)$. Now, consider the sum $$D^\varphi(G,\mu)=\sum_{F\in\widetilde{{\mathcal F}}(G)}c_F^\varphi\,\mu(F),\quad\text{with } c_F^\varphi=\prod_{\gamma\subset\partial N(F)}(1-\varphi(\gamma)).$$ Since dual edges have rhombus half-angles related by $\theta_{e^*}=\frac{\pi}{2}-\theta_e$, the weight $\mu_e=i\tan(\theta_e)$ satisfies $\mu_{e^*}=-\mu_e^{-1}$. Therefore, $$\prod_{e\in E(G)}\mu_e^{-1} D^\varphi(G,\mu)=\sum_{F\in\widetilde{{\mathcal F}}(G)}c_F^\varphi\,\mu^{-1}(\psi(F))=\hskip-2.2pt\sum_{F^*\in\widetilde{{\mathcal F}}(G^*)}(-1)^{|E(F^*)|}c_F^\varphi\,\mu^*(F^*).$$ Furthermore, for any $F\in\widetilde{{\mathcal F}}(G)$, one can decompose the surface ${\Sigma}$ as the union of two tubular neighborhoods $N(F)$ and $N(\psi(F))$ of $F$ and $\psi(F)$, pasted along their common boundary. (This is illutrated in Figure \[fig:dec\].)
In particular, the oriented boundaries satisfy $\partial N(F)=-\partial N(\psi(F))$. Therefore, since $\varphi$ is a 1-cocycle, $$\begin{aligned}
c_{F^*}^\varphi=&\prod_{\gamma^*\subset\partial N(F^*)}(1-\varphi(\gamma^*))=\prod_{\gamma\subset\partial N(F)}(1-\varphi^{-1}(\gamma))\cr
=&(-1)^{|\partial N(F)|}\prod_{\gamma\subset\partial N(F)}\varphi^{-1}(\gamma)\,c_F^\varphi=(-1)^{|\partial N(F)|}\varphi^{-1}(\partial N(F))\,c_F^\varphi\cr
=&(-1)^{|\partial N(F)|}c_F^\varphi,\end{aligned}$$ where $|\partial N(F)|$ denotes the number of boundary components of $N(F)$. Note also that, since $F$ spans all vertices of $G$, adding or removing an edge to $F$ changes the parity of $|E(F^*)|$ and of $|\partial N(F)|$. Therefore, the parity of their sum does not depend on $F$, and it is easily seen to be equal to the parity of $|V(G^*)|$, the number of faces of $G\subset{\Sigma}$. We have proved: $$\prod_{e\in E(G)}\mu_e^{-1} D^\varphi(G,\mu)=(-1)^{|V(G^*)|}D^\varphi(G^*,\mu^*).$$ Since $c_F^\varphi$ vanishes whenever $F\in\widetilde{{\mathcal F}}(G)$ does not belong to ${\mathcal F}(G)$, Proposition \[prop:tech\] gives the equalities $$\begin{aligned}
\tau=&(-1)^{|V(G)|}2^{-\chi(G)}\prod_{v\in V(G)}\exp(i\vartheta_v/4)\prod_{e\in E(G)}\frac{\cos(\theta_e)}{1+\cos(\theta_e)}D^\varphi(G,\mu)\cr
\tau^*=&(-1)^{|V(G^*)|}2^{-\chi(G^*)}\prod_{v\in V(G^*)}\exp(i\vartheta_{v^*}/4)\prod_{e^*\in E(G^*)}\frac{\cos(\theta_{e^*})}{1+\cos(\theta_{e^*})}D^\varphi(G^*,\mu^*),\end{aligned}$$ where $\tau$ and $\tau^*$ stand for $\tau^\varphi(G,\nu)$ and $\tau^\varphi(G^*,\nu^*)$, respectively. The three equations above yield the equality $$2^{|V(G^*)|}\prod_{e^*\in E(G^*)}(1+\cos(\theta_{e^*}))\,\tau^*=A\cdot2^{|V(G)|}\prod_{e\in E(G)}(1+\cos(\theta_{e}))\,\tau,$$ where the constant $A$ is equal to $$\begin{aligned}
A=& (-1)^{|V(G)|}\frac{\prod_{v}\exp(i\vartheta_v/4)}{\prod_{v^*}\exp(i\vartheta_{v^*}/4)}\prod_{e}\frac{\cos(\theta_e)}{\cos(\theta_{e^*})}i\tan(\theta_e)\cr
=&(-1)^{|V(G)|+|V(G^*)|}\prod_{v}\exp(i\vartheta_v/4)\prod_{v^*}\exp(i\vartheta_{v^*}/4)(-1)^{|E(G)|}i^{-|E(G)|} \cr
=& i^{\chi({\Sigma})}(-1)^{N+N^*},\end{aligned}$$ with $N$ the number of $v\in V(G)$ such that $\vartheta_v/2\pi$ is congruent to $3$ modulo $4$ (and similarly for $N^*$). By the discrete Gauss-Bonnet formula (recall Subsection \[sub:flat\]), $$2\pi\chi({\Sigma})=\sum_v(2\pi-\vartheta_v)+\sum_{v^*}(2\pi-\vartheta_{v^*}).$$ This means exactly that $N+N^*$ and $\frac{1}{2}\chi({\Sigma})$ have the same parity, so $A$ is equal to $1$ and the theorem is proved.
Kac-Ward matrices versus discrete Laplacians
--------------------------------------------
Quite surprisingly, if the surface ${\Sigma}$ has genus zero or one, then our Kac-Ward determinants with critical weights turn out to be proportional to the determinants of discrete critical Laplacians. Let us briefly recall the definition and main properties of these objects before stating the precise result.
As first observed by Eckmann [@Eck] (in a much more general context), the Laplace operator on the space of complex valued smooth functions on a Riemann surface admits a beautifully simple discretization. It is the operator $\Delta$ on $C^0(G;{\mathbb C})={\mathbb C}^{V(G)}$ given by $$(\Delta f)(v)=\sum_{e=(v,w)}x_e\,(f(v)-f(w)),$$ for any $f\in{\mathbb C}^{V(G)}$, where the sum is over all oriented edges $e$ of the form $(v,w)$. Its codimension-one minors are very useful, as they count the number of (weighted) spanning trees in $G$: this is Kirchhoff’s celebrated matrix tree theorem [@Kir]. On the other hand, the determinant of $\Delta$ always vanishes.
This construction admits a straightforward generalization:
\[def:Delta\] Let $(G,x)$ be any weighted graph, and $\varphi\colon\pi_1(G)\to S^1$ any representation. The associated [*discrete Laplacian*]{} is the operator $\Delta^\varphi=\Delta^\varphi(G,x)$ on ${\mathbb C}^{V(G)}$ defined by $$(\Delta^\varphi f)(v)=\sum_{e=(v,w)}x_e\,\left(f(v)-f(w)\varphi(e)\right),$$ for any $f\in{\mathbb C}^{V(G)}$, where the sum is over all oriented edges $e=(v,w)$.
The determinant of this discrete Laplacian has a nice combinatorial interpretation, which goes back at least to Forman [@For]. We include a proof here for the sake of completeness.
Given a graph $G$, let ${\mathcal F}_1(G)$ denote the set of subgraphs $F\subset G$ such that $F$ spans all the vertices of $G$, and each connected component $T$ of $F$ has a unique cycle (i.e: $|V(T)|=|E(T)|$).
\[prop:For\] For any weighted graph $(G,x)$ and any $\varphi\colon\pi_1(G)\to S^1$, $$\det\Delta^\varphi(G,x)=\sum_{F\in{\mathcal F}_1(G)}\prod_{T\subset F}(2-\varphi(T)-\varphi(T)^{-1})\,x(F),$$ where the product is over all connected components $T$ of $F$, $x(F)=\prod_{e\in F} x_e$, and $\varphi(T)=\prod_{e\in C_T}\varphi(e)$ with $C_T$ the unique cycle in $T$ endowed with an arbitrary orientation.
Evaluating directly the determinant of $\Delta^\varphi(G,x)$ as a sum over permutations of $V=V(G)$, we get $$\det\Delta^\varphi(G,x)=\sum_{\sigma\in S(V)}(-1)^{\text{sgn}(\sigma)}\prod_{v\in V}\Delta_{v,\sigma(v)}.$$ Each permutation $\sigma\in S(V)$ decomposes into disjoint cycles, inducing a partition $V=\bigsqcup_j V_j(\sigma)$. A permutation will have a non-zero contribution to the determinant only if these orbits correspond to the disjoint union of oriented simple closed curves $\gamma=\bigsqcup_{j=1}^{|\gamma|}\gamma_j$ in $G$. If $\Lambda(G)$ denotes the set of such union of curves, we get $$\det\Delta^\varphi(G,x)=\sum_{\gamma\in\Lambda(G)}(-1)^{|\gamma|}\varphi(\gamma)x(\gamma)\prod_{v\notin\gamma}\sum_{e\ni v}x_e,$$ with $x(\gamma)=\prod_{e\in\gamma}x_e$ and $\varphi(\gamma)=\prod_{e\in\gamma}\varphi(e)$.
Given any subset $W\subset V$, let $\mathcal{X}(W)$ denote the set of maps $X$ assigning to each $v\in W$ an oriented edge $X(v)\in{{\mathbb E}}$ with origin $v$, and set $x(X)=\prod_{v\in W}x_{W(v)}$. For any “vector field" $X\in\mathcal{X}(V)$, let $\Lambda(X)$ denote the set of disjoint unions of “trajectories" of $X$: these are oriented simple closed curves $C$ in $G$ such that any oriented edge $e\in C$ with origin $o(e)$ satisfies $X(o(e))=e$. We have $$\begin{aligned}
\det\Delta^\varphi(G,x)&=\sum_{\gamma\in\Lambda(G)}(-1)^{|\gamma|}\varphi(\gamma)x(\gamma)\sum_{\widetilde{X}\in\mathcal{X}(V\setminus(V\cap\gamma))}x(\widetilde{X})\cr
&=\sum_{X\in\mathcal{X}(V)}\sum_{\gamma\in\Lambda(X)}(-1)^{|\gamma|}\varphi(\gamma)\,x(X)\cr
&=\sum_{X\in\mathcal{X}(V)}\prod_C(1-\varphi(C))\,x(X),\end{aligned}$$ the product being over all trajectories $C$ of $X$. (This is the original formula of Forman [@For].) The map assigning to each $X\in\mathcal{X}(V)$ the subgraph $F=\{X(v)\}_{v\in V}$ is surjective onto the set ${\mathcal F}_1(G)$ defined above, and each $F\in{\mathcal F}_1(G)$ has $2^{|F|}$ preimages given by the possible choices of orientations of the cycle in each of the $|F|$ connected components of $F$. The proposition now follows from the equality $(1-\varphi(C))(1-\varphi(-C))=2-\varphi(C)-\varphi(C)^{-1}$.
Let us mention one more fact: if $G$ is a planar isoradial graph, then the corresponding critical weights for the discrete Laplacian are given by $c_e=\tan(\theta_e)$, where $\theta_e$ denotes the half-rhombus angle [@Ken]. Therefore, we shall refer to $\Delta^\varphi(G,c)$ as the [*critical discrete Laplacian*]{} on $G$. Finally, we shall make the usual abuse of notation and denote by the same letter $\varphi$ a homomorphism $\pi_1(\Sigma)\to S^1$ and the induced homomorphism on $\pi_1(G)$ for $G\subset{\Sigma}$.
\[thm:Delta\] Let $G$ be a graph isoradially embedded in a flat surface ${\Sigma}$, and let us assume that all cone angles $\vartheta_v$ of singularities $v\in V(G)$ are odd multiples of $2\pi$. Let $\nu_e=\tan(\theta_e/2)$ denote the critical weight system on $G\subset\Sigma$, and set $c_e=\tan(\theta_e)$. If the genus of ${\Sigma}$ is zero or one, then for any $\varphi\colon\pi_1(\Sigma)\to S^1$, $$\tau^\varphi(G,\nu)=(-1)^N\,2^{-\chi(G)}\prod_{e\in E(G)}\frac{\cos(\theta_e)}{1+\cos(\theta_{e})}\,\det\Delta^\varphi(G,c),$$ where $N$ is the number of vertices $v\in V(G)$ such that $\vartheta_v/2\pi$ is congruent to $3$ modulo $4$ and $\chi(G)=|V(G)|-|E(G)|$. On the other hand, the functions $\tau^\varphi(G,\nu)$ and $\det\Delta^\varphi(G,c)$ are never proportional if the genus of ${\Sigma}$ is greater or equal to two.
If $\varphi$ is trivial, then both sides of the equality vanish by Propositions \[prop:tech\] and \[prop:For\]. In particular, the equality holds in the genus zero case, so it can be assumed that the genus $g$ of ${\Sigma}$ is positive. Let $F$ be an element of the set ${\mathcal F}(G)$, that is, a spanning subgraph of $G$ such that no connected component of $F$ is a tree. For any connected component $T$ of $F$, let $c^\varphi_T$ denote the corresponding coefficient $c^\varphi_T=\prod_{\gamma\subset\partial N(F)}(1-\varphi(\gamma))$. First note that $c_T^\varphi$ only depends on the homotopy type of $T$ in ${\Sigma}$. (Basically, one can deform $T$ continuously in ${\Sigma}$ without changing $c_T^\varphi$.) Therefore, it can be assumed that $T$ is a wedge of $n$ circles, with $n=1-\chi(T)=1+|E(T)|-|V(T)|\ge 1$. If $n$ is greater then $2g$, then these $n$ cycles are linearly dependant in $H_1({\Sigma};{\mathbb Z})$. Via a homotopy of $({\Sigma},T)$, one can therefore assume that one of these cycles is null-homologous, leading to $c_T^\varphi=0$. If $n$ is equal to $2g$, then either these cycles are linearly dependant in $H_1({\Sigma};{\mathbb Z})$ and $c_T^\varphi$ vanishes as above, or these cycles are independent in homology. In this case, $T$ induces a cellular decomposition of ${\Sigma}$, so $\partial N(T)$ is the boundary of $f$ faces with $$2-2g=\chi({\Sigma})=|V(T)|-|E(T)|+f=1-2g+f.$$ Therefore, $f$ is equal to $1$, so $\partial N(T)$ is connected, hence null-homologous, and $c_T^\varphi$ vanishes in this case as well. We have proved that $c_F^\varphi$ vanishes unless each connected component $T$ of $F$ satisfies $0\le |E(T)|-|V(T)| \le 2g-2$.
In the case of genus $1$, this shows that $c_F^\varphi$ vanishes unless $F$ belongs to the set ${\mathcal F}_1(G)$. For such an element $F$, the contribution of each connected component $T$ can be easily computed: $\partial N(T)$ consists of two connected components homologous to $T$ and $-T$, so $$c_T^\varphi=(1-\varphi(T))(1-\varphi(-T))=2-\varphi(T)-\varphi(T)^{-1}.$$ By Propositions \[prop:tech\] and \[prop:For\], we now have $$\begin{aligned}
\tau^\varphi(G,\nu)&=C\sum_{F\in{\mathcal F}(G)}\prod_{\gamma\subset\partial N(F)}(1-\varphi(\gamma))\,\mu(F)\cr
&=C\sum_{F\in{\mathcal F}_1(G)}\prod_{T\subset F}(2-\varphi(T)-\varphi(T)^{-1})\,\mu(F)\cr
&=C\,\det\Delta^\varphi(G,\mu)\cr
&=C\,i^{|V|}\det\Delta^\varphi(G,c),\end{aligned}$$ since $\mu_e=ic_e$. The equality now easily follows from the explicit value for the constant $C$ given in Proposition \[prop:tech\].
Let us finally assume that the genus of ${\Sigma}$ is greater or equal to two. By the argument above, it is enough to show that there is some element $F\in{\mathcal F}(G)\setminus{\mathcal F}_1(G)$ with non-zero coefficient $c^\varphi_F$. In particular, we just need to find a spanning connected subgraph $T\subset G$ with $|E(T)|-|V(T)|=1$ and such that none of the boundary components of $N(T)$ is trivial in $H_1({\Sigma};{\mathbb Z})$. Such a $T$ is obtained as follows: choose a spanning tree $T_0\subset G$ and add two edges of $G$ so that the resulting two cycles are linearly independent in $H_1({\Sigma};{\mathbb Z})$ but have zero intersection number. (This is possible since ${\Sigma}$ has genus $g>1$ and $G$ induces a cellular decomposition of ${\Sigma}$.) The resulting graph $T$ satisfies the conditions listed above, and the proof is completed.
\[rem:BdT\] Boutillier and de Tilière obtained a similar result in [@BdT1; @BdT2] (see also [@dT]): given any graph $G$ isoradially embedded in the flat torus, they relate $\det\Delta^\varphi(G,c)$ with the determinant of some $\varphi$-twisted Kasteleyn matrix of $({\Gamma}_G,\nu)$, where ${\Gamma}_G$ is the graph associated to $G$ via some variation of the Fisher correspondence [@Fi2]. Actually, using the methods developed in [@Cim2 Subsection 4.3], one can show that Corollary 12 of [@BdT2] is equivalent to our Theorem \[thm:Delta\] in the flat toric case (that is, when $g=1$ and $S$ is empty). In our opinion, our approach has several advantages. It is more general, as we allow singularities and understand the higher genus while Boutillier-de Tilière only deal with the flat toric case; it is more natural, as we work on the same graph $G$ throughout without using one of many possible auxiliary graphs ${\Gamma}_G$; and it is simpler, as our whole proof relies solely on Propositions \[prop:tech\] and \[prop:For\]. The demonstration of Boutillier-de Tilière, on the other hand, is quite substantial and relies on highly non-trivial results of Kenyon [@Ken] and Kenyon-Okounkov [@K-O].
We conclude this section with one last remark. Let us assume that $G$ is isoradially embedded in the flat torus ${\Sigma}={\mathbb C}/\Lambda$, with $\Lambda$ some lattice in ${\mathbb C}$, and let $\widetilde{G}$ be the corresponding $\Lambda$-periodic planar graph. For $n\ge 1$, set $G_n=\widetilde{G}/n\Lambda$. The [*free energy per fundamental domain*]{} of the critical Z-invariant Ising model on $\widetilde{G}$ is defined by $$f^I=-\lim_{n\to\infty}\frac{1}{n^2}\log Z^J(G_n),$$ with $J_e=\frac{1}{2}\log\left(\frac{1+\sin\theta_e}{\cos\theta_e}\right)$ as explained in Subsection \[sub:Z\]. The equality $$Z^J(G_n)=\Big(\prod_{e\in E(G)}\cosh(J_e)\Big)^{n^2}2^{n^2|V(G)|}Z(G_n,\nu)$$ together with Theorems \[thm:Arf\] and \[thm:Delta\] imply $$f^I=-|V(G)|\frac{\log(2)}{2}-\frac{1}{2}\log{\textstyle\det_1}\Delta(G,c),$$ where $\log\det_1\Delta(G,c)=\lim_{n\to\infty}\frac{1}{n^2}\log\det\Delta^\lambda(G_n,c)$ and $\lambda$ is one of the three non-trivial spin structures on the torus. (On the torus, spin structures are canonically identified with $H^1({\Sigma};{\mathbb Z}_2)$, the trivial spin structure has Arf invariant $1$ and the three others have Arf invariant $0$. The trivial spin structure does not contribute to the partition function by Proposition \[prop:tech\], and it is a fact that the three other spin structures will have the same contribution to the free energy of the model.) The free energy $f^I$ was computed by Baxter in [@Bax2], and the normalized determinant by Kenyon in [@Ken] – even though the existence of the limit was not proved there. One can check that these two results are related in the way displayed above. This gives a reality check to our computations, and allows to obtain any of these two results as a corollary of the other one.
The Kac-Ward determinants as discrete $\overline\partial$-torsions {#sub:RS}
------------------------------------------------------------------
In this last subsection, we wish to relate (in an informal way) the critical Kac-Ward determinants with the $\overline\partial$-torsions of the underlying Riemann surface. Let us start by briefly recalling the definition of this invariant, in the special case relevant to us.
To any closed N-dimensional complex manifold ${\Sigma}$ endowed with a unitary representation $\varphi\colon\pi_1({\Sigma})\to U(n)$ and a Hermitian metric, Ray and Singer [@RS2] associate a sequence of numbers $T_p({\Sigma},\varphi)$ with $p=0,1,\dots,N$. In the case of a Riemann surface ($N=1$), the numbers $T_0({\Sigma},\varphi)$ and $T_1({\Sigma},\varphi)$ coincide, leading to a single invariant $T({\Sigma},\varphi)$. If $n=1$, then $\varphi$ is a unitary character (that is, an element of $H^1({\Sigma};S^1)$), and it induces a complex line bundle $L(\varphi)$ over ${\Sigma}$. Since ${\Sigma}$ is endowed with a Hermitian metric, one can consider the associated Laplacian $\Delta$ on the space of smooth sections of $L(\varphi)$. The [*$\overline\partial$-torsion*]{} $T({\Sigma},\varphi)$ is defined as the square root of the zeta-regularized determinant of this Laplacian, that is, $$T({\Sigma},\varphi)=\exp\Big(-{\textstyle\frac{1}{2}}\,\zeta^\prime(0)\Big),$$ where $\zeta(s)$ is the zeta function of the Laplacian $\Delta$. Of course, this depends on the choice of Hermitian metric. However, for any two non-trivial characters $\varphi,\varphi'$, the ratio $T(\Sigma,\varphi)/T(\Sigma,\varphi')$ is independent of this choice [@RS2 Theorem 2.1].
Let us come back to the discrete setting. As before, let $G$ be a graph isoradially embedded in a flat surface ${\Sigma}$ with all cone angles at singularities in $V(G)$ being odd multiples of $2\pi$. Recall that the flat metric defines a conformal structure on the underlying surface, so that ${\Sigma}$ is now a Riemann surface. Consider the following statement.
[*As a function of $\varphi\in H^1({\Sigma};S^1)$, the Kac-Ward determinant $\tau^\varphi(G,\nu)$ behaves – up to a multiplicative constant – as a discrete version of $T({\Sigma},\varphi)^2$, the square of the corresponding $\overline\partial$-torsion.*]{}
We shall not give any proof of this vague statement, not even formulate a precise conjecture (although this is very tempting). Instead, we shall simply give a list of evidences towards such a statement.
First of all, recall that given any two non-trivial characters $\varphi,\varphi'$, the ratio $T(\Sigma,\varphi)/T(\Sigma,\varphi')$ only depends on the Riemann surface $\Sigma$ and on $\varphi,\varphi'$, but not on the Hermitian metric. Therefore, one expects the corresponding ratios $\tau^\varphi(G,\nu)/\tau^{\varphi'}(G,\nu)$ to exhibit some independence of the choice of the graph $G$ embedded in the surface $\Sigma$. As mentioned in the introduction, the papers [@CSM1; @CSM2] provide numerical evidences towards such a claim at the scaling limit (in the special case where $\varphi$ and $\varphi'$ are discrete spin structures). Now, our duality result can be understood as a further step in that direction. Indeed, it is a direct consequence of Theorem \[thm:duality\] that the ratios $\tau^\varphi(G,\nu)/\tau^{\varphi'}(G,\nu)$ and $\tau^\varphi(G^*,\nu^*)/\tau^{\varphi'}(G^*,\nu^*)$ actually [*coincide*]{}, without taking any scaling limit, for any two non-trivial characters $\varphi,\varphi'$.
In the remainder of the discussion, we shall distinguish between the genus zero, the genus one, and the higher genus cases.
[**The genus zero case.**]{} Whenever the character $\varphi$ is trivial, $\tau^\varphi(G,\nu)$ vanishes by Proposition \[prop:tech\] (actually, Lemma \[lemma:KW\] is enough). Since the kernel of the continuous Laplacian has dimension one, $T({\Sigma},\varphi)$ is also equal to zero. In particular, $\tau^\varphi(G,\nu)$ and $T({\Sigma},\varphi)^2$ trivially coincide in the case of genus zero.
[**The genus one case.**]{} In the toric case, Theorem \[thm:Delta\] shows that $\tau^\varphi(G,\nu)$ is proportional to the most ‘natural’ discretization of $T({\Sigma},\varphi)^2$, that is, the determinant of the critical discrete Laplacian.
[**The case of genus $\mathbf{g>1}$.**]{} Let us now consider the case of a Riemann surface of genus $g>1$. It can be written ${\Sigma}=H/\Gamma$, where $H$ is the Poincaré upper half plane and $\Gamma$ a discrete subgroup of $\mathit{PSL}(2,{\mathbb R})$. Since $\Sigma$ is compact, each element $\gamma\in\pi_1(\Sigma)=\Gamma$ acts on $H$ via $\gamma(z)=\exp(2\rho_\gamma)\frac{z-z_0}{z-z_1}$ for some real fixed points $z_0,z_1$ and $\rho_\gamma>0$. Given any unitary character $\varphi$ of $\pi_1({\Sigma})=\Gamma$, the corresponding $\overline\partial$-torsion satisfies $$T({\Sigma},\varphi)^2=C\cdot Z_\Gamma(1,\varphi),$$ with $C$ a constant depending on the Hermitian metric and $Z_\Gamma(\sigma,\varphi)$ the Selberg zeta function defined by $$Z_\Gamma(\sigma,\varphi)=\prod_{\{\gamma\}}\prod_{k\ge 0}(1-\varphi(\gamma)\exp(-2\rho_\gamma(\sigma+k))).$$ Here, the product is over all conjugacy classes of primitive elements of $\Gamma$, that is, elements that are not powers in $\Gamma$ [@RS2 Theorem 4.6]. As explained in [@Sar], there is a more geometric way to understand this zeta function: conjugacy classes of primitive elements of $\Gamma$ correspond to primitive closed geodesics on $\Sigma$, and $2\rho_\gamma$ is the length $\ell_\gamma$ of the corresponding geodesic. Summing up (and neglecting the problems of convergence), we have the equality $$T({\Sigma},\varphi)^2=C\prod_{\gamma\in\mathcal{P}({\Sigma})}\prod_{k\ge 1}(1-\varphi(\gamma)\exp(-k\ell_\gamma)),\eqno{(\star)}$$ where $\mathcal{P}({\Sigma})$ denotes the set of primitive closed geodesics in ${\Sigma}$.
Let us make the corresponding study on the discrete side. Given an oriented closed path $\gamma$ on a graph $G$, we will say that $\gamma$ is [*reduced*]{} if it never backtracks, that is, if no oriented edge $e$ is immediately followed by the oriented edge $\bar{e}$. The oriented closed path $\gamma$ will be called [*prime*]{} if, when viewed as a cyclic word, it cannot be expressed as the product $\delta^r$ of a given closed path $\delta$ for any $r\ge 2$. Finally, we shall denote by $\mathcal{P}(G)$ the set of prime reduced oriented closed paths in a graph $G$. Let us now assume that $G$ is isoradially embedded in a flat surface ${\Sigma}$. As explained in the proof of Theorem \[thm:Arf\], Bass’ Theorem [@Bas] implies the equality $$\tau^\varphi(G,\nu)=\prod_{\gamma\in\mathcal{P}(G)}\Big(1-\varphi(\gamma)\exp\left(\textstyle{\frac{i}{2}}\alpha_\gamma\right)\nu(\gamma)\Big),\eqno{(\star\star)}$$ where $\nu(\gamma)=\prod_{e\in\gamma}\nu_e$ and $\alpha_\gamma$ is the sum of the angles $\alpha(e,e')$ along $\gamma$ (recall Figure \[fig:alpha\]).
The expression $(\star\star)$ can be understood as a discrete version of $(\star)$ in the following sense. Primitive closed geodesics on ${\Sigma}$ are replaced by homotopically non-trivial reduced prime closed paths in $G\subset{\Sigma}$. The factor $\exp\left(\textstyle{\frac{i}{2}}\alpha_\gamma\right)$ measures to which extend $\gamma$ differs from a straight line (i.e. a geodesic on ${\Sigma}$ with respect to the flat metric). And finally, the critical weight $\nu_e=\tan(\theta_e/2)$ plays the role of $\exp(-\ell_e)$, with $\ell_e$ the length of the edge $e$.
|
---
abstract: 'Stuttering is a speech impediment affecting tens of millions of people on an everyday basis. Even with its commonality, there is minimal data and research on the identification and classification of stuttered speech. This paper tackles the problem of detection and classification of different forms of stutter. As opposed to most existing works that identify stutters with language models, our work proposes a model that relies solely on acoustic features, allowing for identification of several variations of stutter disfluencies without the need for speech recognition. Our model uses a deep residual network and bidirectional long short-term memory layers to classify different types of stutters and achieves an average miss rate of 10.03%, outperforming the state-of-the-art by almost 27%.'
address: |
Department of Electrical and Computer Engineering\
Queen’s University, Kingston, Ontario, Canada\
bibliography:
- 'refs.bib'
title: 'Detecting Multiple Speech Disfluencies using a Deep Residual Network with Bidirectional Long Short-Term Memory'
---
Speech, stuttering, disfluency, deep learning, residual network, LSTM.
Introduction {#sec:intro}
============
Speech disfluencies are inconsistencies and interruptions in the flow of otherwise normal speech. Of these speech impediments, stuttering is one of the most prominent, affecting over 70 million people, about one percent of the global population [@stutteringfoundation]. 5-10% of children stutter at some point in their childhood, with a quarter of these children maintaining their stutters throughout their entire lives [@nidcd]. Common therapy methods often involve helping the patient monitor and maintain awareness of their speaking patterns in order to correct them [@stutteringbook]. Moreover, therapeutic success rates have been reported to be over 80%, especially when detected and dealt with in early stages [@Saltuklaroglu2004]. Accordingly, with the recent advances in machine learning, deep learning, and language/speech processing techniques, developing smart and interactive tools for detection and therapy is now a real possibility.
In addition to interactive therapy purposes, other applications can be realized for automated stutter recognition. Fluent speech is crucial and influential in presentations such as talks and business communications [@morreale2000]. There are currently a number of applications available to assist speakers in monitoring and improving their presentation skills. For example, monitoring of features like volume, rate of speech, and intonation, among others have been explored in this context [@ROBOCOP] [@Kotz2003]. However, detection and quantification of stutters has not yet been fully explored for such applications.
Despite the many potential applications for automated stutter detection, little research has been done in this area. This is partially due to the fact that the notion of detecting and classifying the type and location of stutters can be a difficult problem, especially when factoring in variables such as gender, speech rate, accent, and phone-realization [@speechfactors]. Existing works in the area mostly rely on automatic speech recognition (ASR) to first convert audio signals to text, and then utilize language models to detect and identify the stutters [@Heeman2016] [@interspeech2017] [@interspeech2018]. While this approach has proven effective and achieved promising results, the reliance on ASR can both be a potential source for error, as well as an unnecessary additional computational step.
In this paper, we propose a model that directly utilizes audio speech signals to detect and classify stutters, skipping the ASR step and the need for language models. Our method uses spectrogram features to train a deep neural network with residual layers followed by bidirectional long short-term memory (Bi-LSTM) units to learn to locate and identify different types of stutters. The overview of our method is presented in Figure \[fig:intro\_diagram\]. Our experiments show the effectiveness of our approach in generalizing across multiple classes of stutters while maintaining a high accuracy and strong consistency between classes.
![Proposed stutter detection system diagram.[]{data-label="fig:intro_diagram"}](Intro_Diagram_V3.png){width="0.8\columnwidth"}
Related Work {#sec:format}
============
Early studies on the topic focused on the feasibility of stutter differentiation, with training and testing often being performed on a small set of specific stuttered words. For example, a hidden Markov model (HMM) was used to create a stutter recognition assistance tool [@tan2007]. Testing results averaged to 96% and 90% accuracy on human and artificially generated stuttered speech samples respectively for a single pre-determined word [@tan2007].
A number of assumptions are often made in order to simplify the problem of disfluency detection. For example, as different disfluencies vary heavily by nature, proposed solutions often tackle one single type of stutter (such as interjections, prolongations, or repetitions) at a time. In [@ravikumar2009], for instance, sound repetition stutters were accurately detected on a small set of trained words. Another common assumption used for simplification has been to remove under-represented subject classes (for example based on gender or age) [@ravikumar2009], [@Chee2009], [@interspeech2018].
As ASR and natural language processing (NLP) has evolved greatly in recent years, such methods have become increasing popular for the problem of stutter classification and recognition. One such method incorporated annotations from speech language pathologist to a word lattice model, improving the baseline method by a relative 7.5% [@Heeman2016]. Another model using Bi-LSTMs with condition random fields (CRFs) to get an average F-score of 85.9% across all stutter types [@zayats2016]. The current state-of-the-art stutter classification method uses task-oriented finite state transducer (FST) lattices to detect repetition stutters with an average 37% miss rate across 4 different types of [@interspeech2018].
Proposed Network
================
Our proposed method first generates spectrogram feature vectors from the audio clips. The spectrograms are then passed through a deep residual neural network, mapping the spectrogram matrices to a linear vector. These are then be passed through a bidirectional LSTM to learn the extracted feature embeddings for different types of stutters. Following the different steps of our proposed pipeline are described.
Feature extraction
------------------
Spectrograms are commonly used features in speech analysis in different applications ranging from speech recognition to noise cancellation [@Kingsbury1998] [@Szabelska2013]. We use spectrograms as the sole feature for our model. These features are generated every 10 *ms* on a 25 *ms* window for each 4-second audio clip.
Feature embedding layers
------------------------
We utilize a residual network [@resnet] in our model in order to effectively learn the stutter-specific features while avoiding issues such as the vanishing gradient problem. The use of this type of network also allows a deep architecture (a depth of 18 convolution layers) without overfitting, especially when considering the relatively small size of the dataset. Moreover, architectures with residual components have recently shown considerable promise in speech analysis [@xiong2018microsoft] [@hajavi2019]. In our proposed solution, each group of 3 convolutional layers is referred to as a convolutional block. Figure \[fig:res\] presents the convolutional blocks and the stacked blocks in our mode. We used batch normalization and ReLu activation functions in the model. Table \[table:hyperparameters\] presents the hyperparameters of our network. The detection task for each stutter is formulated as a binary problem, with the same architecture mentioned being used for every disfluency type.
{width="0.9\linewidth"}
Recurrent layers
----------------
The learned feature embeddings are provided to 2 recurrent layers, each consisting of 512 bidirectional LSTM units [@BiLSTM]. We utilized LSTM layers as they have been proven to be effective in classification when dealing with short sequential data, and are a popular approach in speech and NLP [@LSTM]. In the context of the problem at hand, most stutters tend to be quick and last only a fraction of the 4-second audio clip that they are contained in. Therefore the LSTM layers don’t suffer from memory issues [@dropout]. Lastly, the use of *bidirectional* LSTMs allow the model to learn both past and future embeddings, providing further context for our problem. Dropout rates of 0.2 and 0.4 are utilized after each recurrent layer.
Experiment Setup and Results {#sec:pagestyle}
============================
Data and annotation
-------------------
Speech samples were collected from the University College London’s Archive of Stuttered Speech (UCLASS) Release One [@UCLASS] dataset, created by the Division of Psychology and Language Sciences within the university. The dataset contains samples of monologues from 139 participants, ranging between 8 and 18 years of age, with known stuttered speech impediments of different severity. Of these recordings, 25 unique participants were used due to the availability of their orthographic transcriptions of the monologues.
Forced time-alignment was used on the audio and transcriptions to generate a timestamp for each word and stutter spoken [@timealign]. The stutter annotation approach is similar to previously used methods [@yairiambrose; @justeandrade]. We then manually annotated each recording for one of 7 stutter disfluencies [@Yaruss1997]: sound repetition, word repetition, phrase repetition, revision, interjection, or prolongation. A description of each type of stutter can be found in Table \[table: stutters\]. We leave out part-word repetition disfluencies as the dataset contained only few samples of such stutters, preventing our deep learning approach from properly learning the classification task. Each monologue recording was segmented into 4-second samples, totalling to 800 labeled audio clips.
Label Stutter Disfluency Description Example
------- -------------------- ---------------------------------------------------- ------------------------------
S Sound Repetition Repetion of any phenome th-th-this
W Word Repetition Repetition of any word why why
PH Phrase Repetition Repetition of multiple successive words I know I know that
R Revision Repetition of thought, rephrased mid sentence I think that- I believe that
I Interjection Fabricated word or sounds, added to stall for time um, uh
PR Prolongation Prolonged sounds whoooooo is it
Implementation details
----------------------
The model was built using TensorFlow’s Keras API [@chollet2015keras]. It was trained with a learning rate of $10^{-4}$ over 30 epochs, with minimal improvement in results seen in following epochs. A root means square propagation (RMSProp) optimizer was used, as well as the softmax loss function. An Nvidia 1080 Ti GPU was used to perform the training.
----------------------------------- -------------------- ----------- ----------- ------- ------- ---------- ----------- ---------- ----------- ----------- ----------- ------- -------
Paper Method MR Acc MR Acc MR Acc MR Acc MR Acc MR Acc
Alharbi et al. [@interspeech2018] Word Lat. 60 – **0** – 25 – **0** –
Ours (baseline) ResNet+LSTM 20.13 83.20 3.40 95.60 4.93 95.07 3.00 96.99 25.31 80.80 6.12 93.88
**Ours (proposed)** **ResNet+Bi-LSTM** **18.10** **84.10** 3.20 96.60 **4.46** **95.54** **2.86** **97.14** **25.12** **81.40** 5.92 94.08
----------------------------------- -------------------- ----------- ----------- ------- ------- ---------- ----------- ---------- ----------- ----------- ----------- ------- -------
Paper Method Ave. MR Ave. Acc
----------------------------------- -------------------- ------------ ------------
Alharbi et al. [@interspeech2018] Word Lat. 37% –
Ours (baseline) ResNet+LSTM 10.45% 90.96%
**Ours (proposed)** **ResNet+Bi-LSTM** **10.03%** **91.15%**
: Average accuracy and miss-rate of stutter classification models.[]{data-label="table: results_ave"}
Validation
----------
To rigorously test our proposed model, leave-one-subject-out (LOSO) cross validation was used: the model was trained on the speech of 24 of the UCLASS participants, while the last subject’s audio was used for testing. The process was repeated 25 times, testing the model on a different subject every time. For this dataset, accuracy (Acc) and miss rate (MR) values have been reported in prior work. Lastly, it should be noted that we target 6 categories of stutter disfluencies, as opposed to most prior work where fewer classes are considered.
Performance and Comparison
--------------------------
The results of our experiments for the UCLASS dataset is summarized in Table \[table: results\], where we compare our method to [@interspeech2018]. Additionally, to evaluate the need for *bidirectional* LSTM as opposed to a unidirectional LSTM, we compare our results to a baseline model where a ResNet with LSTM is used instead of our proposed model. The table shows that our method outperforms the state-of-the-art in detection of sound repetition and revisions by considerable margins (an improvements of 41.90% and 22.14% respectively).
The statistical language models and task-oriented word lattices used in other methods rely heavily on generating a strong orthographic transcriptions for each speaker. As a result, while these methods struggle with sub-word stutters such as sound repetition or revision, they perform well for word repetition or prolongation. This can be observed in Table \[table: results\] as [@interspeech2018] performs better than our method by a small margin (3.2%) for word repetition. Additionally, [@interspeech2018] performs with a lower miss rate than ours for detection of prolongation (5.92%). Since our method relies on spectrogram features as opposed to a language model, some longer utterances can exceed the four-second windows or suffer from alignment issues, causing those stutters to be misclassified. Hence our approach produces slightly more false negatives in classification of prolongation.
As shown in Table \[table: results\], our classifier is able to identify interjections with an accuracy of 81.4%. Many other works on stutter classification tend to avoid interjection disfluencies as a class, since interjection stutters tend to be more diverse and lack the consistency of repetition and prolongation stutters, making them more difficult to classify. While other works such as [@Mahesha2016; @Mahesha2017] were able to robustly detect interjections with mel-frequency cepstral coefficients (MFCC), small subsets of the UCLASS dataset were used, preventing us from performing a fair comparison to our model.
The comparison between our proposed method and the baseline approach (using LSTM instead of bidirectional LSTM) fares similarly, with the bidirectional LSTM having slightly better or similar results for every class. This lack of significant difference between the two LSTM variations is most likely due to the fact that the feature embeddings learned using the ResNet portion of our pipeline are quite robust, accurately capturing the information required to represent different stutters. While the difference in bidirectional and unidirectional LSTMs is marginal, we opt to use the Bi-LSTM approach as the additional computational cost for Bi-LSTM is not significant.
Table \[table: results\_ave\] presents the average performance of our model compared to [@interspeech2018] and the baseline approach. It can be observed that our model achieves an improvement of 26.97% lower miss rate on the UCLASS dataset over the previous state-of-the-art. Moreover, as previously shown, our method slightly outperforms the unidirectional LSTM baseline when averaged across all stutter types. Lastly, Figure \[fig:acc\] shows the performance of our method for different stutter types against different training epochs. It can be seen that after approximately 20 epochs, our model reaches a steady-state, indicating stable learning of disfluency-related features throughout the learning phase.
![Average training accuracy for the considered types of stutter.[]{data-label="fig:acc"}](Acc_Curve_Pyplot_PDF.pdf){width="0.9\columnwidth"}
Conclusion and Future Work
==========================
We present a method for detection and classification of different types of stutter disfluencies. Our model utilizes a residual network and bidirectional LSTM units trained using input spectrogram features calculated from labeled audio segments of stuttered speech. Six classes of stutter were considered in this paper: sound repetition, word repetition, phrase repetition, revision, interjection, and prolongation. Investigations show that our method performs robustly across all classes and performs with very high average accuracy and low average miss rate, achieving state-of-the-art with a significant improvement over previous the previous state-of-the-art for stutter detection.
In future work, building upon the current model, we will conduct research on multi-class learning of different stutter disfluencies. As multiple stutter types may occur at once (e.g. *’I went to uh to to uh to’*), this approach may result in more robust classification of stutters.
Acknowledgements
================
The authors would like to thank Prof. Jim Hamilton for his support and valuable feedback throughout the course of this of this work.
|
---
abstract: 'We study the influence of the nonlinearity in the Schrödinger equation on the motion of quantum particles in a harmonic trap. In order to obtain exact analytic solutions, we have chosen the logarithmic nonlinearity. The unexpected result of our study is the existence in the presence of nonlinearity of two or even three coexisting Gaussian solutions.'
author:
- 'Iwo Bia[ł]{}ynicki-Birula'
- 'Tomasz Sowi[ń]{}ski'
title: |
Solutions of the Logarithmic Schrödinger Equation\
in a Rotating Harmonic Trap
---
Introduction
============
The nonlinear Schrödinger equation with the logarithmic nonlinearity (we use the units $\hbar = 1$ and $m = 1$) $$\begin{aligned}
\label{lse}
i\partial_t\psi({\bf r}, t) = \left(-\frac{1}{2}
\Delta + V({\bf r}, t) - b\log(\vert\psi({\bf r}, t)\vert^2/a^3)\right)
\psi({\bf r}, t)\end{aligned}$$ was introduced [@bbm1] long time ago to seek possible departures of quantum mechanics from the linear regime. The parameter $b$ measures the strength of the nonlinear interaction (positive $b$ means attraction) and $a$ is needed to make the argument of the logarithm dimensionless — it plays no significant role since the change of $a$ results only in an additive constant to the potential. In what follows, we shall absorb the parameter $a$ into the wave function that amounts effectively to putting $a = 1$.
It has been proven in beautiful experiments with neutron beams [@shimony; @shull; @gkz] that the nonlinear effects in quantum mechanics, if they exist at all, are extremely small. The upper limit for the constant $b$ was determined to be $3.3\;10^{-15}$ eV. Thus, the applicability of the logarithmic Schrödinger equation to the time evolution of wave functions seems to have been ruled out. Nevertheless this equation, owing to its unique mathematical properties, has been used in many branches of physics to model the nonlinear behavior of various phenomena. It has been applied in the study of dissipative systems [@hr], in nuclear physics [@hefter], in optics [@keb; @b4sc], and even in geophysics [@dfgl]. In contrast to the properties of other nonlinear equations, the logarithmic Schrödinger equation in any number of dimensions possesses analytic solutions, called Gaussons in [@bbm2]. Gaussons represent localized nonspreading solutions of the Gaussian shape. The internal structure of the Gaussons may also change in time. The existence of these analytic solutions enables one to study in detail the influence of nonlinearities. In this paper we focus our attention on the behavior of the solutions of the logarithmic Schrödinger equation in a rotating harmonic trap. The aim of our study was to see to what extent the nonlinear interaction may change the dynamics and affect the stability of solutions. Perhaps, our results will help to better understand the behavior of the Bose-Einstein condensate in a rotating trap. Previous studies of these problems (for example, [@rzs] and [@csbrd]) were often based on the hydrodynamic equations and we plan in the future to express our results in terms of the hydrodynamic variables.
Formulation of the problem
==========================
The logarithmic Schrödinger equation in a rotating trap has the form $$\label{nonleq}
i\partial_t\psi({\mathbf r}, t)
= \left(-\frac{1}{2}\Delta +
\frac{1}{2}{\mathbf r}\!\cdot\!{\hat V}(t)\!\cdot\!{\mathbf r} -
b\log(\vert\psi({\mathbf r}, t)\vert^2)\right)\psi({\mathbf r}, t),$$ where the symmetric $3\times3$ matrix ${\hat V}(t)$ depends on time due to rotation. In order to simplify the analysis of stability, we assume that the trap is subjected to a uniform rotation and we shall use the coordinate system co-rotating with the trap. In this manner the potential becomes time-independent but due to rotation there appears an additional term in the equation. $$\label{nonleq1}
i\partial_t\psi({\mathbf r}, t)
= \left(-\frac{1}{2}\Delta +
\frac{1}{2}{\mathbf r}\!\cdot\!{\hat V}\!\cdot\!{\mathbf r} -
b\log(\vert\psi({\mathbf r}, t)\vert^2)
- {\mathbf\Omega}\!\cdot\!{\bf M}\right)\psi({\mathbf r}, t),$$ where ${\mathbf\Omega}$ is the vector of angular velocity and ${\bf M} = {\bf
r}\times{\bf p}$ is the operator of angular momentum. We shall seek the solutions of Eq. (\[nonleq1\]) in the Gaussian form $$\begin{aligned}
\label{gausson}
\psi({\mathbf r}, t)
= N(t)e^{if(t)}\exp\left(-\frac{1}{2}
{\tilde{\mathbf r}}\!\cdot\!({\hat A}(t)+i{\hat B}(t))\!\cdot\!
{\tilde{\mathbf r}}(t) + i{\boldsymbol\pi}(t)\!\cdot\!{\bf r}\right),\end{aligned}$$ where ${\tilde{\mathbf r}}={\mathbf r}-{\boldsymbol\xi}(t)$. The time-dependent vectors ${\boldsymbol\xi}(t)$ and ${\boldsymbol\pi}(t)$ specify the position and momentum of the center of mass of the Gaussian wave packet and the time-dependent real symmetric matrices ${\hat A}(t)$ and ${\hat B}(t)$ specify the shape and the internal motion of the wave packet, respectively. The two real functions $N(t)$ and $f(t)$ define the normalization and the overall phase of the Gausson. Substituting this Ansatz into Eq. (\[nonleq1\]), we arrive at the following set of [*ordinary*]{} differential equations for all the functions entering our formula (\[gausson\]) $$\begin{aligned}
\label{ode}
\frac{d{\hat A}(t)}{dt} &=& {\hat B}(t){\hat A}(t) + {\hat A}(t){\hat B}(t)
-\left[{\hat\Omega},{\hat A}(t)\right] ,\\
\frac{d{\hat B}(t)}{dt} &=& {\hat B}(t)^2 - {\hat A}(t)^2 + {\hat V}
+ 2b{\hat A}(t) - \left[{\hat\Omega},{\hat B}(t)\right],\\
\frac{d{\boldsymbol\xi}(t)}{dt} &=& {\boldsymbol\pi}(t) -
{\boldsymbol\Omega}\times{\boldsymbol\xi}(t),\\
\frac{d{\boldsymbol\pi}(t)}{dt} &=& - {\hat V}\!\cdot\!{\boldsymbol\xi}(t) -
{\boldsymbol\Omega}\times{\boldsymbol\pi}(t),\\
\frac{dN(t)}{dt} &=& \frac{1}{2}{\rm Tr}\{{\hat B}(t)\}N(t),\\
\frac{df(t)}{dt} &=& -\frac{1}{2}\left({\rm Tr}\{{\hat A}(t)\}
+ {\boldsymbol\pi}(t)\!\cdot\!{\boldsymbol\pi}(t)
- {\boldsymbol\xi}(t)\!\cdot\!{\hat V}\!\cdot\!{\boldsymbol\xi}(t)\right),\end{aligned}$$ where the antisymmetric matrix ${\hat\Omega}$ and the components of the angular velocity vector ${\boldsymbol\Omega}$ are related through the formula $\Omega_{ij} = \epsilon_{ijk}\Omega^k$. Note, that the internal motion (described by ${\hat A}(t)$ and ${\hat B}(t)$) completely decouples from the motion of the center of mass (described by ${\boldsymbol\xi}(t)$ and ${\boldsymbol\pi}(t)$). In turn, the equations for the normalization factor and the phase can be integrated after the internal and the center of mass motion has been determined. This decoupling follows from the general theorem [@gpv] and [@bb1] stating that from every solution of a nonlinear Schrödinger equation in a harmonic potential (including time-dependent potential) one may obtain a solution displaced by a classical trajectory fully preserving the shape of the wave function.
Solutions and their stability
=============================
In what follows, for simplicity, we shall assume that the trap rotates along one of its principal axis. In this case the motion in the direction perpendicular to the rotation plane decouples and we are left with a two-dimensional problem. In the stationary state of our system the center of mass motion must be absent (${\boldsymbol\xi}(t)=0,{\boldsymbol\pi}(t)=0$) The stationary state of the system is described by the wave function characterized by the solution of the following two time-independent equations for two $2\times 2$ matrices $A$ and $B$ $$\begin{aligned}
0 &=& {\hat B}{\hat A} + {\hat A}{\hat B} - \left[{\hat\Omega},{\hat
A}\right],\label{stat1}\\
0 &=& {\hat B}^2 - {\hat A}^2 + {\hat V}
+ 2 b{\hat A} - \left[{\hat\Omega},{\hat B}\right].\label{stat2}\end{aligned}$$ We shall seek the solutions of these equations in the coordinate frame in which the matrix ${\hat V}$ is diagonal, ${\hat V} = {\rm
Diag}\{\omega_1^2,\omega_2^2\}$. We assume, for definitness, that $\omega_1 <
\omega_2$. It follows from Eqs. (\[stat1\]–\[stat2\]) that in this frame the matrix ${\hat A}$ is also diagonal and the matrix ${\hat B}$ is off-diagonal. Finally, we are left with three equations for two matrix elements $\alpha_1$, $\alpha_2$ of ${\hat A}$ and one matrix element $\beta$ of ${\hat
B}$ $$\begin{aligned}
(\alpha_1+\alpha_2)\beta -(\alpha_1-\alpha_2)\Omega &=& 0,\label{eq1}\\
\beta^2 - \alpha_1^2 + \omega_1^2 + 2 b\alpha_1 + 2\beta\Omega &=& 0,\label{eq2}\\
\beta^2 - \alpha_2^2 + \omega_2^2 + 2 b\alpha_2 - 2\beta\Omega &=& 0.\label{eq3}\end{aligned}$$ It follows from Eq. (\[eq1\]) that in the absence of rotation $\beta$ must vanish and we obtain immediately two physically acceptable solutions of the decoupled quadratic equations for the parameters $\alpha$ $$\begin{aligned}
\alpha_1 &=& (\omega_1\sqrt{1 + b^2/\omega_1^2} + b),\label{sols01}\\
\alpha_2 &=& (\omega_2\sqrt{1 + b^2/\omega_2^2} + b),\label{sols02}\\
\beta &=& 0.\label{sols03}\end{aligned}$$ The two remaining solutions yield negative values of the $\alpha$’s and must be rejected. Thus, in the absence of rotation the nonlinearity modifies only the size of the Gaussian wave function without introducing any significant changes. Even for negative values of $b$ (nonlinear repulsion), stable solutions described by (\[sols01\]) and (\[sols02\]) always exist, no matter how strong is the repulsion.
Simple analytic formulas can also be obtained in the presence of rotation but without nonlinearity. The formulas for the Gausson parameters read in this case $$\begin{aligned}
\alpha_1 &=& \frac{\sqrt{\omega_1^2+\omega_2^2+
2\Omega^2 \pm 2\sqrt{(\omega_1^2-\Omega^2)(\omega_2^2-\Omega^2)}}}
{1+\sqrt{(\omega_2^2-\Omega^2)/(\omega_1^2-\Omega^2)}},\label{sols11}\\
\alpha_2 &=& \frac{\sqrt{\omega_1^2+\omega_2^2+
2\Omega^2 \pm 2\sqrt{(\omega_1^2-\Omega^2)(\omega_2^2-\Omega^2)}}}
{1+\sqrt{(\omega_1^2-\Omega^2)/(\omega_2^2-\Omega^2)}},\label{sols12}\\
\beta &=& \Omega\frac{1-\sqrt{(\omega_2^2-\Omega^2)/(\omega_1^2-\Omega^2)}}
{1+\sqrt{(\omega_2^2-\Omega^2)/(\omega_1^2-\Omega^2)}}.\label{sols13}\end{aligned}$$ The values of $\alpha$ are real in the two regions of stability when $\Omega <
\omega_1$ (region 1) and $\Omega > \omega_2$ (region 2). The same regions of stability were obtained in the analysis of the characteristic frequencies in classical or quantum-mechanical center-of-mass motion [@bb1]. In the formulas (\[sols11\]) and (\[sols12\]) the + and – sign is to be chosen for the region 1 and the region 2, respectively.
In the presence of both rotation and nonlinearity the properties of solutions change significantly. The most striking difference is the appearance of additional stationary Gaussian solutions. This is an unexpected result because in the linear theory a purely Gaussian shape always is found for [*only one*]{} fundamental state of the system — all other states have polynomial prefactors. We have not been able to find closed expressions for the parameters $\alpha$ and $\beta$, so we had to resort to numerical analysis of the solutions of Eqs. (\[eq1\]–\[eq3\]). We present our results in three plots showing the calculated values of the parameters $\alpha_1$ and $\alpha_2$ that determine the shape of the Gaussian wave function. These values are plotted as functions of the angular velocity $\Omega$. In all plots we have fixed the trap parameters to be $\omega_1 = \sqrt{2/3}, \omega_2 = \sqrt{4/3}$. We have chosen three values of $b$ to describe the following characteristic cases. In Fig. \[fig:g1\] we plot the values of $\alpha$’s without the nonlinear interaction ($b = 0$). In Fig. \[fig:g2\] we added the attractive nonlinear interaction ($b = 1$) and in Fig. \[fig:g3\] the repulsive nonlinear interaction ($b = -1$).
Conclusions
===========
Knowing the exact analytic form of the solutions of our nonlinear Schrödinger equation we were able determine the influence of rotation and nonlinearity on the stability of solutions. The unexpected result of our analysis is that the repulsive interaction [*expands*]{} the region of stability. We have to admit, however, that this may be true only for the special form of the nonlinearity: the logarithmic nonlinearity.
![This plot shows the values of $\alpha_1$ and $\alpha_2$ in the absence of the nonlinearity. For each value of $\Omega$ in the stability regions there is just one Gaussian wave function whose shape is described by the values of $\alpha$’s.[]{data-label="fig:g1"}](g1.eps)
![This plot shows the values of $\alpha_1$ and $\alpha_2$ in the case of attractive nonlinear interaction. For sufficiently large values of $b$, as in this case, there are no regions of instability. For small and for large values of $\Omega$ there is just one Gaussian wave function but for intermediate values there are two or even three solutions. Matching pairs of $\alpha$’s are distinguished by the lines of the same style: solid, dashed, and dotted.[]{data-label="fig:g2"}](g2.eps)
\[fig3\]
![This plot shows the values of $\alpha_1$ and $\alpha_2$ in the case of repulsive nonlinear interaction. The upper region of stability is extended now downwards as compared to the case without the nonlinear term. Moreover, there are two solutions (solid and dashed lines) that coexist in the newly established region of stability.[]{data-label="fig:g3"}](g3.eps)
[15]{} I. Bialynicki-Birula and J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Cl. III [**23**]{}, 461 (1975). I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. of Phys. (N.Y.), [**100**]{}, 62 (1976). I. Bialynicki-Birula and J. Mycielski, Gaussons: Solitons of the logarithmic Schrödinger equation, Physica Scripta, [**20**]{}, 539 (1978). I. Bialynicki-Birula and Z. Bialynicka-Birula, Center of mass motion in many-body theory of Bose-Einstein condensates, Phys. Rev. A [**65**]{}, 063606 (2002). H. Buljan, A. [Š]{}iber, M. Solja[č]{}ic, T. Schwartz, M. Segev, and D. N. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E [**68**]{}, 036607 (2003). M. Cozzini, S. Stringari, V. Bretin, P. Rosenbusch, and J. Dalibard, Scissors mode of a rotating Bose-Einstein condensate, Phys. Rev. A [**67**]{}, 021602(R) (2003). S. De Martino, M. Falanga, C. Godano, and G. Lauro, Logarithmic Schrödinger-like equation as a model for magma transport, Europhys. Lett. [**63**]{}, 472 (2003). J. J. Garc[í]{}a-Ripoll, V. M. P[é]{}rez-Garc[í]{}a, and V. Vekslerchik, Construction of exact solutions by spatial translations in inhomogeneous nonlinear Schrödinger equations, Phys. Rev. E [**64**]{}, 056602 (2001). R. Gähler, A. G. Klein, and A. Zeilinger, Neutron optical tests of nonlinear wave mechanics, Phys. Rev. A [**23**]{}, 1611 (1981). E. F. Hefter, Application of the nolinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev. [**A 32**]{}, 1201 (1985). E. S. Hernandez and B. Remaud, General properties of Gausson-conserving descriptions of quantal damped motion, Physica [**105A**]{}, 130 (1980). W. Krolikowski, D. Edmundson, and O. Bang, Unified model for partially coherent solitons in logaritmically nonlinear media, Phys. Rev. E [**61**]{}, 3122 (2000). A. Recati, F. Zambelli, and S. Stringari, Overcritical rotation of a trapped Bose-Einstein condensate, Phys. Rev. Lett. [**86**]{}, 377 (2001). A. Shimony, Proposed neutron interferomenter test of some nonlinear variants of wave mechanics, Phys. Rev. A [**20**]{}, 394 (1979). C. G. Shull, D. K. Atwood, J. Arthur, and M. A. Horne, Search for a nonlinear variant of the Schrödinger equation by neutron interferometry, Phys. Rev. Lett. [**44**]{}, 765 (1980).
|
---
author:
- 'E. Koulouridis, I. Georgantopoulos, G. Loukaidou, A. Corral, A. Akylas, L. Koutoulidis, E. F. Jiménez-Andrade, J. León Tavares, P. Ranalli'
title: The XMM spectral catalog of SDSS optically selected Seyfert 2 galaxies
---
Introduction
============
Nearly thirty years ago, the first discovery by Miller & Antonucci (1983) of broad permitted emission lines and a clearly non-stellar continuum in the polarized spectrum of the archetypal Seyfert 2 (Sy2), NGC 1068, was just the beginning of numerous similar observations in a wide variety of galaxies. Ten years later, the unification model of active galactic nuclei (AGN) was formulated upon these observations (Antonucci 1993). According to the unification model, all AGN are intrinsically identical, while the only cause of their different observational features is the orientation of an obscuring torus with respect to our line of sight. In more detail, the AGN type depends on the obscuration of the broad line region (BLR), a small area at close proximity to the SMBH where the broad permitted lines are produced. If the torus happens to be between the observer and the BLR, the optical emission and even the soft X-rays are absorbed. Optical spectropolarimetric observations can reveal the hidden broad line region (HBLR) by highlighting its scattered emission. The observed narrow permitted emission lines are produced at far larger distances from the core, where the torus is irrelevant. As a prediction of this model, the presence and strength of the broad optical emission lines, hence the derived optical spectral type (from type 1 AGN/Sy1s to type 2 AGN/Sy2s, and the intermediate types), should correlate with the amount of intervening material as measured in X-rays.
X-ray observations can reveal the exact density of the obscuring torus, even for mildly obscured sources. X-ray surveys with [*Ginga*]{} (Smith and Done 1996) and [*ASCA*]{} (Turner et al. 1997) measured column densities between $10^{22}$ and up to a few times $10^{24}$ $\rm cm^{-2}$ in type 2 AGN samples. More recently, Akylas & Georgantopoulos (2009) and Brightman & Nandra (2011), using [*XMM-Newton*]{}, and Jia et al. (2013, JJ13 hereafter), using [*Chandra,*]{} also studied the obscuration of type 2 X-ray sources in detail (see also Brandt & Alexander (2015) for a recent review). However, even in the hard X-ray band, the X-ray surveys may be missing a fraction of highly obscured sources. These sources are called Compton-thick AGN (see reviews by Comastri 2004 and Georgantopoulos 2013), and they present very high obscuring column densities ($>10^{24} cm^{-2}$ , corresponding to an optical reddening of $A_V>$100). Even though Compton-thick AGN are abundant in the optically selected samples of nearby Seyferts (e.g., Risaliti et al. 1999), only a few tens of Compton-thick sources have been identified from X-ray data. Moreover, Krumpe et al. (2008) found no Compton-thick QSO in their high redshift ($z>0.5$), X-ray selected sample, implying a possible redshift evolution, though this may be due to selection.
Although the population of Compton-thick sources remains elusive, there is concrete evidence of its presence. The X-ray background synthesis models can explain the peak of the X-ray background at 30-40 keV, where most of its energy density lies, (Frontera et al. 2007; Churazov et al. 2007) only by invoking a large number of Compton-thick AGN (Gilli et al. 2007). We note, however, that other models (e.g., Treister, Urry & Virani 2009; Akylas et al. 2012) succeed in explaining the X-ray background (XRB) spectrum assuming a lower fraction of CT sources. Additional evidence of a Compton-thick population comes from the directly measured space density of black holes in the local Universe. It is found that this space density could be up to a factor of two higher than predicted from the X-ray luminosity function (Marconi et al. 2004). This immediately suggests that the X-ray luminosity function is missing an appreciable number of obscured AGN.
On the other hand, although widely accepted today, the unification model cannot explain a series of observations. For example, Tran et al. (2001) noticed the absence of a HBLR in polarized light in many Sy2 galaxies (non-HBLR Sy2 galaxies), suggesting that there is a class of true Sy2 galaxies that intrinsically lack the broad-line region (see Ho et al. 2008 for a review). Theoretical models attributed the absence of a BLR to either a low Eddington ratio (Nicastro 2000) or to low luminosity (Elitzur & Shlossman 2006). Many studies propose an evolutionary model where a fraction of Sy2 represents the first or the last phase in the life of an AGN (Hunt & Malkan 1999; Dultzin-Hacyan, 1999; Krongold et al. 2002; Levenson et al. 2001; Koulouridis et al. 2006a,b, 2013; Koulouridis 2014, Elitzur, Ho & Trump 2014). This was supported by studies of the local environment of Seyfert galaxies, which showed that Sy2s reside in richer environments compared to Sy1s (e.g., Villaroel & Korn 2014). Unobscured low-luminosity Sy2s were detected via investigation of their X-ray properties (e.g., Pappa et al. 2000, Panessa & Bassani 2002; Akylas & Georgantopoulos 2009). Models of galaxy formation also support this scenario: for example, Hopkins et al. (2008) assert that the AGN is heavily obscured during its birth. During the build-up of its black hole mass, it blows away its cocoon, becoming an unobscured AGN.
In this paper, we compile a sample of bona fide optically selected Sy2 galaxies using the SDSS spectra from the data release 10 (DR10). We cross-correlate our sample with the 3XMM/XMMFITCAT spectral catalog (Corral et al. 2015), which contains good quality spectra (at least 50 net counts per XMM detector). We identify a sample of 31 Sy2 galaxies with available X-ray spectra in the redshift range z=0.05-0.3. Our study is complemented by X-ray, mid-IR, and \[OIII\] luminosity ratio diagnostics (Georgantopoulos et al. 2013, Trouille & Barger 2010). This study provides an extension of previous X-ray studies in the local Universe (e.g., Akylas & Georgantopoulos 2009) but also of similar studies at higher redshifts (e.g., JJ13) because of the high S/N X-ray spectra used.
We describe our sample selection in §2, the X-ray analysis in \$3, while our results and conclusions are presented in §4 and §5, respectively. Throughout this paper we use $H_0=72$ km/s/Mpc, $\Omega_m=0.27$, and $\Omega_{\Lambda}=0.73$.
Sample selection
================
Our sample is composed of Seyfert 2 galaxies with available X-ray spectra within the XMM-Newton Serendipitous Source catalog (Watson et al. 2009, Rosen et al. 2015) and optical spectra within the SDSS-DR10. The names of the sources are taken from the SDSS database. Also, a sequence number is given to each source in the current paper (see Table 1). In the diagrams, interesting sources are followed by their sequence numbers. In the text, the names are followed by the sequence number in parenthesis to make it easier for the reader to trace the sources in the tables and the diagrams.
3 pt
----------- --------------------- ----------- ----------- ----------- ----------- --------------------------- ------------------- --------------
N name obsid ra dec z $\rm N_H (\times10^{22}$) exposure time counts
[*(1)*]{} [*(2)*]{} [*(3)*]{} [*(4)*]{} [*(5)*]{} [*(6)*]{} [*(7)*]{} [*(8)*]{} [*(9)*]{}
1 J080429.14+235444.1 504102101 121.1219 23.9127 0.07432 3.18 18300/–/– 86/0/0
2 J080535.00+240950.3 203280201 121.3961 24.1645 0.05971 3.08 5598/8425/– 125/83/0
3 J083139.08+524205.6 92800201 127.9131 52.7016 0.05855 1.16 60280/70920/71790 334/127/166
4 J084002.36+294902.6 504120101 130.0095 29.8175 0.06481 1.83 17870/22630/22640 1072/424/428
5 J085331.05+175339.0 305480301 133.3791 17.8942 0.18659 2.75 34560/–/– 173/0/0
6 J091636.53+301749.3 150620301 139.1524 30.2969 0.12339 1.25 9049/9392/– 431/151/0
7 J100129.41+013633.8 302351001 150.3724 1.6095 0.10423 3.07 31650/42310/42540 310/98/127
8 J101830.79+000504.9 402781401 154.6286 0.0845 0.06233 3.00 15700/20540/20600 753/398/397
9 J103408.58+600152.1 306050701 158.5360 60.0307 0.05101 1.51 8311/–/11420 465/0/133
10 J103456.37+393941.0 506440101 158.7349 39.6614 0.15081 1.96 68400/83070/83850 422/147/120
11 J103515.64+393909.5 506440101 158.8154 39.6527 0.10710 2.03 –/83170/83870 0/112/ 92
12 J104426.70+063753.8 405240901 161.1109 6.6317 0.20991 3.02 24960/–/– 92/0/0
13 J112026.64+431518.4 107860201 170.1109 43.2554 0.14591 1.32 13870/–/– 182/0/0
14 J113549.08+565708.2 504101001 173.9555 56.9522 0.05112 1.07 17490/21310/21320 448/130/127
15 J114826.24+530417.1 204260101 177.1089 53.0717 0.09826 1.23 1701/3632/– 112/94/0
16 J121839.40+470627.6 400560301 184.6649 47.1077 0.09390 1.00 –/37830/37570 0/86/137
17 J123056.11+155212.2 112552101 187.4978 13.5183 0.09816 2.31 8394/–/– 80/0/0
18 J122959.45+133105.7 106061001 187.7338 15.87 0.18768 2.00 4660/–/8979 96/0/98
19 J124214.47+141147.0 504240101 190.5607 14.196 0.15710 2.22 59590/–/80240 1465/0/665
20 J125743.06+273628.2 124710201 194.4296 27.608 0.06839 1.52 30010/–/– 119/0/0
21 J130920.52+212642.7 163560101 197.3359 21.4453 0.27858 1.57 –/28390/28700 0/191/215
22 J131104.66+272807.2 21740201 197.7694 27.469 0.23975 2.06 35000/43000/43100 267/73/75
23 J132525.63+073607.5 200730201 201.3567 7.6022 0.12402 5.02 26900/–/– 174/0/0
24 J134245.85+403913.6 70340701 205.6908 40.6537 0.08926 1.57 26010/35340/35100 393/220/216
25 J135436.29+051524.5 404240101 208.6515 5.2564 0.08152 7.65 11020/–/15780 168/0/101
26 J141602.13+360923.2 14862010 214.0089 36.1567 0.17100 2.56 10910/15780/16140 271/146/145
27 J145720.44–011103.6 502780601 224.3353 –1.1844 0.08735 11.4 7942/–/– 71/0/0
28 J150719.93+002905.0 305750801 226.8330 0.4847 0.18219 10.5 9931/–/– 227/0/0
29 J150754.38+010816.8 402781001 226.9764 1.1381 0.06099 9.81 14330/17900/17890 222/81/81
30 J215649.51–074532.4 654440101 329.2059 –7.7589 0.05541 5.22 42310/73600/75600 134/63/56
31 J224323.18–093105.8 503490201 340.8464 –9.5185 0.14509 2.72 –/113700/114500 0/246/247
----------- --------------------- ----------- ----------- ----------- ----------- --------------------------- ------------------- --------------
X-ray selection
---------------
The XMM-Newton catalog is the largest catalog of X-ray sources ever built. Its current version, 3XMM-DR4 (http://xmmssc-www.star.le.ac.uk/Catalogue/3XMM-DR4/), contains photometric information for half a million source detections, and in addition, spectral and timing data for $\sim$ 120000 of them. The count limit adopted by the 3XMM-DR4 pipeline to derive spectral products is of 100 EPIC net (background subtracted) counts, in order to allow reliable X-ray spectral extraction and analysis.
The starting sample was extracted from the XMM-Newton/SDSS-DR7 cross-correlation presented in Georgakakis & Nandra (2011), including more than 40000 X-ray sources. We first selected the sources detected in the X-ray hard band (2-8 keV), a band less affected by obscuration than is the soft one (0.5-2 keV). A total of 1275 sources were found to have available optical spectra within SDSS-DR7. Out of these, 1018 sources had available 3XMM-DR4 spectral data. The corresponding SDSS optical spectra of these 1018 sources were manually examined in order to identify Seyfert 2 galaxies, resulting in our final sample of Sy2s (see next section). It is worth noting that two of these sources have more than one XMM-Newton observation with spectra within 3XMM-DR4, from which we used the longest one.
3 pt
-------------- ------------------------------ ----------------------- -------------------------- --------------------------- -------------------- ------------------- ------------- -------------- ------------------- ------------------- ------------------- ---------------
N $\rm N_H $ $\Gamma_{soft}$ $\Gamma_{hard}$ EW flux $L_X$ p1/p2 cstat/dof $L_{[OIII]}$ $L_{12}$ $L_{bol}$ log($M_{BH}$)
$(\times10^{22}$) $(\times10^{-14})$ $(\times10^{43})$ $(\times10^{42})$ $(\times10^{43})$ $(\times10^{43})$
[*(1)*]{} [*(2)*]{} [*(3)*]{} [*(4)*]{} [*(5)*]{} [*(6)*]{} [*(7)*]{} [*(8)*]{} [*(9)*]{} [*(10)*]{} [*(11)*]{} [*(12)*]{} [*(13)*]{}
1 $18.84_{-5.56}^{+7.68}$ $ 7.6_{-4.6}^{+7.6}$ 1.8$\ddagger$ $ <0.33 $ 22.2 0.27 0.001 105.2/96 0.18 0.4 1.1 6.9
2 $45.83_{-13.94}^{+21.30}$ $ 2.4_{-0.4}^{+0.4}$ 1.8$\ddagger$ $ 0.29_{-0.24}^{+0.71}$ 25.0 0.20 0.015 165.25/210 0.60 0.2 0.3 6.8
3 $23.56_{-3.79}^{+4.00} $ $2.1_{-0.4}^{+0.4}$ $2.1_{-0.4}^{+0.4}$ $ 0.26_{-0.13}^{+0.20}$ 9.3 0.07 0.012 537.63/618 0.17 0.7 1.0 6.3
4 $54.24_{-5.82}^{+5.32} $ $2.8_{-0.1}^{+0.2}$ $ 1.4_{-0.7}^{+0.7}$ $ 0.30_{-0.80}^{+0.80}$ 69.7 0.63 0.024 1088.68/1403 1.11 8.8 19.9 7.6
5 $26.38_{-20.54}^{+27.12} $ $ 2.5_{-0.7}^{+0.7}$ 1.8$\ddagger$ $ <0.17 $ 17.8 1.20 0.293 147.19/189 5.77 11.4 25.9 7.9
6 $<0.06 $ $ <0.74 $ 22.4 0.87 425.68/522 3.47 1.8 2.8 8.2
7 $4.75_{-1.40}^{+1.28} $ $ 1.4_{-0.4}^{+0.4}$ $1.4_{-0.4}^{+0.4}$ $ 0.29_{-0.17}^{+0.20}$ 15.6 0.39 0.018 465.96/529 0.07 1.4 1.9 7.2
8 $2.34_{-0.32}^{+0.38} $ $ <0.10 $ 60.1 0.54 825.83/1078 0.04 0.3 0.4 6.2
9$^\dagger$ $30.32_{-11.62}^{+19.24} $ $ 2.9_{-0.2}^{+0.2} $ 1.8$\ddagger$ $ 1.30_{-0.48}^{+0.80}$ 17.8 0.07 0.293 430.95/474 4.98 7.4 10.3 8.2
10 $64.58_{-23.33}^{+29.99} $ $ 3.0_{-0.2}^{+0.2}$ 1.8$\ddagger$ $ 0.52_{-0.25}^{+0.26}$ 4.1 0.20 0.102 505.46/546 5.37 4.4 6.2 8.2
11 $14.35_{-5.85}^{+6.61}$ $ 6.3_{-1.5}^{+2.1}$ $ 2.0_{-1.1}^{+1.2}$ $ <1.21 $ 9.6 0.25 0.007 189.43/210 0.22 7.4 16.7 7.6
12 $90.26_{-48.60}^{+31.61} $ $ 1.5_{-0.9}^{+0.9}$ $1.5_{-0.9}^{+0.9}$ $ <13.68 $ 11.6 0.92 0.004 82.28/98 1.90 87.9 118.0 8.7
13 $5.49_{-2.51}^{+2.68} $ $ 1.3_{-0.7}^{+0.6}$ $1.3_{-0.7}^{+0.6}$ $ 0.32_{-0.28}^{+0.85}$ 19.1 0.95 0.054 140.57/186 0.23 2.8 6.3 6.0
14 $130.46_{-56.34}^{+69.32}$ $ 2.9_{-0.2}^{+0.2}$ 1.8$\ddagger$ $ <0.65 $ 7.6 0.04 0.006 434.45/462 6.62 22.2 50.8 7.6
15 $1.92_{-0.69}^{+0.88} $ $ <0.48 $ 144 3.37 169.96/204 0.64 0.9 1.2 7.3
16$^\dagger$ $16.07_{-10.34}^{+39.22} $ $ 2.9_{-0.4}^{+0.4}$ 1.8$\ddagger$ $ 0.85_{-0.66}^{+0.75}$ 7.9 0.16 0.173 145.04/155 5.47 9.4 13.1 7.3
17 $4.83_{-2.48}^{+2.80} $ $ 2.3_{-1.1}^{+1.0}$ $2.3_{-1.1}^{+1.0}$ $\star$ 12.6 0.30 0.015 82.11/85 2.06 4.8 6.3 7.4
18 $1.81_{-0.87}^{+1.08} $ $ <0.35 $ 45.2 3.81 183.1/184 0.59 9.3 15.0 6.5
19 $<0.04 $ $ 0.38_{-0.21}^{+0.21}$ 9.9 0.65 748.87/885 1.45 1.4 4.0 8.0
20 $12.62_{-7.20}^{+6.96} $ $1.7_{-1.1}^{+1.0}$ $1.7_{-1.1}^{+1.0}$ $ 0.35_{-0.35}^{+0.51}$ 9.7 0.10 0.052 140.82/167 0.04 0.3 0.4 5.9
21 $<0.08 $ $\star$ 4.7 1.21 214.46/247 0.10 0.5 0.5 6.9
22$^\dagger$ $243.07_{-115.26}^{+303.45}$ $ 2.6_{-0.2}^{+0.2}$ 1.8$\ddagger$ $ 0.65_{-0.60}^{+0.81}$ 4.7 0.37 0.003 296.79/365 3.07 11.7 23.3 7.8
23 $0.39_{-0.22}^{+0.39} $ $ <0.88$ 16.5 0.62 172.56/170 0.13 0.4 1.1 7.4
24 $6.47_{-1.07}^{+1.31} $ $ 2.0_{-0.3}^{+0.4}$ $2.0_{-0.3}^{+0.4}$ $ 0.17_{-0.13}^{+0.17}$ 41.1 0.77 0.009 659.27/707 0.26 2.3 3.4 7.1
25$^\dagger$ $<0.07 $ $<0.70$ 22.7 0.34 274.4/258 0.16 0.9 2.5 6.3
26 $1.98_{-0.47}^{+0.50} $ $ <0.37 $ 46.5 3.47 383.72/507 1.48 42.3 97.8 8.1
27 $5.29_{-3.53}^{+4.69} $ $<0.60$ 13.1 0.23 97.74/80 0.22 1.4 1.9 7.1
28 $28.78_{-12.22}^{+15.82} $ $ 1.7_{-0.7}^{+0.5}$ $1.7_{-0.7}^{+0.5}$ $ <0.46 $ 34.6 2.56 0.045 246.1/310 10.66 2.4 32.3 8.8
29$^\dagger$ $32.18_{-12.68}^{+19.90}$ $ 3.4_{-0.3}^{+0.4}$ 1.8$\ddagger$ $ 1.22_{-0.73}^{+1.89}$ 10.6 0.09 0.155 339.5/354 1.71 1.5 2.1 7.4
30$^\dagger$ $14.95_{-7.28}^{+11.92} $ $3.6_{-0.4}^{+0.5} $ 1.8$\ddagger$ $ 2.08_{-1.24}^{+3.05}$ 3.8 0.03 0.280 266.06/300 0.96 2.7 4.9 7.4
31 $2.61_{-0.68}^{+0.89} $ $ <0.25 $ 10.3 0.51 355.49/417 0.45 0.2 0.4 6.2
-------------- ------------------------------ ----------------------- -------------------------- --------------------------- -------------------- ------------------- ------------- -------------- ------------------- ------------------- ------------------- ---------------
Optical selection
-----------------
We built the final Sy2 sample based on the emission line properties of their SDSS optical spectra. Initially, we selected only emission line galaxies with redshifts between $z$=0.05 and $z$=0.35. The lower redshift limit excludes all already extensively studied and well-known Seyferts (e.g., Akylas & Georgantopoulos 2009), while the upper limit ensures that the ${\mathrm{H}\alpha}$ and \[NII\] emission lines are within the SDSS spectral range. Furthermore, we excluded all objects where the velocity dispersion of the $\rm H\alpha$ line is greater than 500 km/s, since these objects are certainly broadline AGN. The rest of the objects were placed on a BPT diagram (Baldwin, Phillips, and Terlevich, 1981) and star-forming galaxies, composite galaxies, and LINERS were removed according to the criteria of Kewley et al. (2001) and Schawinski et al. (2007).
We used the MPA-JHU emission line fluxes published in DR8 (Brinchmann et al. 2004; Tremonti et al. 2004), although DR10 also contains data from the recent spectroscopic analysis of the Portsmouth Group (Thomas et al. 2013). However, the latter includes only those galaxies from the first two years of observations of the SDSS-III/Baryonic Oscillation Spectroscopic Survey (BOSS) collaboration. We note that a comparison between the two databases by Thomas et al. (2013) has shown that the discrepancy between the calculated emission line fluxes is small. However, the comparison was made after rescaling the Portsmouth values with a factor provided by the “spectofiber” keyword in the MPA-JHU database. This rescaling was originally applied to the MPA-JHU data so that the synthetic r-band magnitude computed from the spectrum matches the r-band fiber magnitude measured by the photometric pipeline. The use of either database does not significantly affect the BPT diagram, since we only need the emission line ratios.
We note that in some cases the broadening of the Balmer lines cannot be automatically detected (Seyfert 1.5, 1.8, and especially 1.9), since it only affects the lower part of the lines. As a result, the automated modeling of the line by a single Gaussian may result in lower velocity dispersion values than what is expected from a broad line profile, and the source may be misclassified as a narrow-line AGN. However, since we sought a broad-line-free sample, the spectra of all remaining AGN were eye-inspected with the “interactive spectrum” tool of the SDSS, and all evident intermediate-type Seyferts were removed. After the above filtering, the catalog of Sy2s included 40 objects.
Despite the above selection, a number of sources in our sample still have discrepant classifications in the literature; i.e., eight of the sources are listed as Sy1s in Veron-Cetty & Veron (2010, V&V10 hereafter) catalog, plus another one in the NED (NASA extragalactic database). Although none of these objects can actually be a Sy1, we proceeded with our own optical spectrum analysis to determine whether there is any broadening of the permitted emission lines.
Optical spectrum analysis
-------------------------
The spectra have been retrieved from the SDSS-DR10 and corrected for Galactic extinction using the maps of Schlegel (1998). We use the stellar population synthesis code [^1] to obtain the best fit to an observed spectrum $O_{\lambda}$, taking the corresponding flux error into account. The best fit is a combination of single stellar populations (SSP) from the evolutionary synthesis models of (Bruzual 2003) and a set of power laws to represent the AGN continuum emission. Following the latter approach, several studies have been successful at disentangling the host galaxy and AGN emission components in SDSS spectra (Cid-Fernandes 2011; Tavares 2011).
We use a base of 150 SSPs plus six power laws in the form F($\lambda$) = 10$^{20}$($\lambda$ / 4020)$^\beta$, where $\beta$= -0.5, -1, -1.5, -2, -2.5, -3. Each SSP spans six metallicities, Z = 0.005, 0.02, 0.2, 0.4, 1, and 2.5, $Z_{\odot}$, with 25 different ages between 1 Myr and 18 Gyr. Extinction in the galaxy is taken into account in the synthesis, assuming that it arises from a foreground screen with the extinction law of (Cardelli 1989). The code finds the minimum $\chi^{2}$, $$\chi^{2} = \sum_{\lambda} \left( \frac{O_{\lambda}- M_{\lambda}} {\sigma_{obs}}\right)
,$$ where $M_{\lambda}$ is the model spectrum (SSP and power laws), obtaining the corresponding physical parameters of the modeled spectrum: star formation history, $x_{j}$, as a function of a base of SSP models normalized at $\lambda_{0}$, $b_{j,\lambda}$, extinction coefficient of predefined extinction laws, $r_{\lambda}$, and velocity dispersion $\sigma_{\star}$, which obeys the relation
$$M_{\lambda}= M_{\lambda 0} \left ( \sum_{j=1}^{N_{SSP}} x_{j}, b_{j,
\lambda} r_{\lambda} \right) \otimes G(v_{\star}, \sigma_{\star})
.$$
A detailed description of the code can be found in the publications of the SEAGal collaboration (Cid-Fernandes 2005 ; Cid-Fernandes 2007; Mateus 2006, Asari 2007). In Fig. 1 we present two examples of the spectral decomposition results.
After subtracting the stellar background, we use the commercial software [PEAKFIT]{}, by [*Systat Software Inc.*]{}, to model the emission lines. We analyze separately the red ($\rm H\alpha$, N\[II\] and S\[II\] emission lines) and the blue ($\rm H\beta$ and \[OIII\] emission lines) parts of the spectrum. We initially model the emission lines in the blue part, since we are mostly interested in the profile of the \[OIII\]${\lambda5007}$ narrow emission line, with which we also try to fit the lines in the blue part and especially the $\rm H\alpha$. We model the \[OIII\] line with a mixed Gaussian and Lorentzian profile. The contribution of each profile to the fit is a free parameter. If the same profile can also be applied to the red part of the spectrum, we consider this source as a narrow line AGN and keep it in our sample. If there is still a need for an extra broad component to model the $\rm H\alpha$ the source is discarded. In any case, the \[NII\]${\lambda6583}$/\[NII\]${\lambda6548}$ flux ratio should be $\sim3$. We find that seven out of the 40 sources present a broad $\rm H\alpha$ component. Most of these sources belong to the list of ambiguous-type Seyferts that we described in the previous section.
Finally, we plot the BPT diagram anew, this time with the line ratios calculated by the above spectral analysis. Although the differences are small, we find that a source that was already close to the AGN-LINER separating line, falls in the LINER region and is therefore excluded. The BPT diagram is plotted in Fig. 2.
X-ray spectral fitting
======================
The X-ray data have been obtained with the EPIC (European Photon Imaging Cameras, Strüder et al. 2001; Turner et al. 2001) onboard XMM-Newton. X-ray photons are collected by three detectors (PN, MOS1, and MOS2). All available instrument spectra are modeled simultaneously by using XSPEC, the standard package for X-ray spectral analysis (Arnaud 1996). We used Cash statistics (C-statistics), implemented as cstat in XSPEC to obtain reliable spectral-fitting results even for the lowest quality spectra in our sample. Many of our sources were detected in only one or two of the three detectors (see Table 1).
The X-ray spectra of type 2 AGN are usually complicated and consist of multiple components: power-law, thermal, scattering, reflection, and emission lines (see Turner et al. 1997; Risaliti 2002; Ptak et al. 2006; LaMassa et al. 2009). Therefore, no single model could successfully fit the spectra in all cases. We initially tried to model all spectra with a single absorbed power law, but if the fit was not acceptable we added a second power law. Since a strong line is expected in obscured sources, we fit a Gaussian line for the FeK$\alpha$ emission line in both cases. In more detail, this includes:
- [Single absorbed power law plus Gaussian FeK$\alpha$ line.]{}
We assumed a standard power-law model with two absorption components (wabs\*zwabs\*pow in XSPEC notation) to fit the source continuum emission. The first component models the Galactic absorption. Its fixed values are obtained from Dickey & Lockman (1990) and are listed in Table 1. The second component represents the AGN intrinsic absorption and is left as a free parameter during the modeling procedure. A Gaussian component has also been included to describe the FeK$\alpha$ emission line. We fix the line energy at 6.4 keV in the source rest frame (except in the case of J090036.85+205340.3 (N6) where the line was found at 6.7 keV and implies ionized Fe) and the line width $\sigma$ at 0.01 keV ($\sim$10% of the instrumental line resolution of XMM-Newton). In 12 cases the fitting procedure gives a rejection probability less than 90 per cent and we can accept the model. However, when this simple parametrization is not sufficient to model the whole spectrum, additional components must be included as described in the next paragraph.\
- [Double power law plus Gaussian FeK$\alpha$ line.]{}
In the remaining 20 cases, an additional power law was necessary to obtain an acceptable fit (wabs\*(pow+zwabs\*pow), in XSPEC notation). The additional power law is only absorbed by the galactic column density. Initially, the photon indices of the soft (scattered/unabsorbed) and hard (intrinsic/absorbed) power-law components were tied together. However, in 13 cases the value of the hard power-law photon index $\Gamma_{hard}$ was too high (the average photon index of the intrinsic power-law measured in AGN is usually $\sim1.8-2$), and we needed to untie it from the soft one to obtain an acceptable fit. In the cases where the data quality was not high enough to constrain $\Gamma_{hard}$, we fixed it to 1.8 (see Table 2).
The X-ray analysis revealed that one of the sources is the brightest galaxy of a contaminating X-ray luminous cluster. We chose to exclude this source from our sample since we cannot provide any reliable X-ray measurements. Our final sample comprises 31 Seyfert 2. In Fig. 3 we present some examples of the X-ray spectra of unobscured ($<10^{22}cm^{-2}$, left panels) and strong FeK$\alpha$-line sources (right panels).
Results
=======
In the next sections we use various criteria and diagnostic diagrams to investigate the possibility that some objects are more obscured than we can infer from their $\rm N_H$ values and that Compton-thick candidate sources are indeed heavily obscured.
Candidate Compton-thick sources
-------------------------------
Only two of the sources have $\rm N_H>10^{24}$ cm$^{-2}$, consistent with the high values that define Compton-thick sources. Also, sources (N10) and (N12) are consistent with being CT within the uncertainties. However, except for the column density as a direct indicator of obscuration, there are other criteria, based not only on the X-ray but also on the optical and the infrared emission, that could point to possible Compton-thick sources within our sample. In more detail, a heavily obscured source can have one or more of the following characteristics:
1. Flat X-ray spectrum ($\Gamma<1$). This implies the presence of a strong reflection component that intrinsically flattens the X-ray spectrum at higher energies (e.g., Matt et al. 2000).
2. High equivalent width of the FeK$\alpha$ line ($\sim$1 keV). In this case a Compton-thick nucleus is evident since the line is measured against a heavily obscured continuum (Leahy & Creighton 1993) or only against the reflected component.
3. Low X-ray to mid-infrared ($\rm L_{12}$) luminosity ratio. All Compton-thick sources should have low $\rm L_{2-10 keV}$ to $\rm L_{12}$ ratios, since the mid-IR luminosity of an AGN should be dominated by very hot dust and the X-ray emission should be suppressed by high amounts of absorption (e.g., Lutz et al. 2004; Maiolino et al. 2007).
4. Low X-ray to optical luminosity ratio. The \[OIII\] line emission originates in the narrow line region and is not affected by the circum-nuclear obscuration. Therefore, the ratio between the observed hard X-ray (2-10 keV) and \[OIII\] line luminosity could be used as an indicator of the obscuration of the hard X-ray emission (Mulchaey et al. 1994; Heckman et al. 2005; Panessa et al. 2006; Lamastra et al. 2009; LaMassa et al. 2009; Trouille & Barger2010).
### Flat X-ray spectrum as an indicator of obscuration
The first criterion of $\Gamma$ $<1$ is satisfied only by J135436.29+051524.5 (N25). However, this source cannot be included in the Compton-thick candidate sources because there is evidence of partial covering. For more detail see the notes on individual objects in the appendix.
### High equivalent width of the FeK$\alpha$ line as an indicator of obscuration
The second criterion of a strong FeK$\alpha$ line is satisfied by four objects (see Table 2). Although the presence of the strong line provides robust evidence of their obscuration, all four exhibit lower $\rm N_H$ values than what is expected by a Compton-thick source. Therefore, we also fit these sources with the model of Brightman & Nandra (2011), which is based on Monte-Carlo simulations. The advantage of this model is that it fits an iron line consistently with the computed $\rm N_H$. Thus, it cannot result in a good fit with a low $\rm N_H$ value and at the same time a high-EW iron line, and vice versa. The fitting confirms that these four sources are indeed Compton-thick. More details can be found in the notes on individual objects in the appendix. Therefore, we do include them in our list of CT sources.
Also, we need to examine the X-ray spectra of the unobscured sources carefully for the FeK$\alpha$ line that could give away the presence of obscuration. However, as we can see in Table 2, the line is actually detected only in one out of the five sources, and the equivalent width (EW) is relatively small ($0.38^{+0.21}_{-0.21}$). We do not detect the line in the spectra of any other unobscured source, and the given value of the EW is just the upper limit. Thus, there is no evidence of obscuration based on the the presence of a FeK$\alpha$ line.
We note that this criterion is not explicit. High equivalent width lines may also appear in the case of anisotropic distribution of the scattering medium (Ghisellini et al. 1991) or in the case of a time lag between the reprocessed and the direct component (e.g., NGC 2992, Weaver et al. 1996). On the other hand, Compton-thick sources with FeK$\alpha$ EW well below 1 keV have been reported (e.g., Awaki et al. 2000, for Mkn1210).
### The $L_X/L_{12}$ ratio as an indicator of obscuration
The detection of a low X-ray to mid-IR luminosity ratio has been widely used as the main instrument for detecting faint Compton-thick AGN, which cannot be easily identified in X-ray wavelengths (e.g., Goulding et al. 2011). This is because the mid-IR luminosity (e.g., 12 $\mu $m or 6 $\mu $m) is a good proxy for the AGN power because it should be dominated by very hot dust that is heated by the AGN (e.g., Lutz et al. 2004; Maiolino et al. 2007). At these wavelengths, the contribution of the stellar light and of colder dust heated by young stars should be small. Gandhi et al. (2009) presented high angular resolution mid-IR (12 $\mu $m) observations of the nuclei of 42 nearby Seyfert galaxies. These observations provide the least contaminated core fluxes of AGN. These authors find a tight correlation between the near-IR fluxes and the intrinsic X-ray luminosity (the Gandhi relation).
Spitzer observations do not have the spatial resolution to resolve the core, and the infrared luminosity of an AGN is probably contaminated by the stellar background and the star-forming activity of the galaxy. To obtain an estimate of the purely nuclear 12 $\mu $m infrared luminosity of our sources, we constructed their spectral energy distributions (SED) and computed the various contributions. To model the spectra we used optical data from the SDSS (five optical bands), photometry in the four WISE bands (3.4, 4.6, 12, and 22 $\mu $m) (Wright et al. 2010), and photometry in the three 2MASS bands (J, H, and K) for all sources. Although WISE does include the 12 $\mu $m band, we are only interested in the AGN contribution, so that the construction of the SED and the decomposition of the AGN and host galaxy component is essential. For more details about the code used, the interested reader should refer to Rovilos et al. (2014, Appendix A).
In Fig. 4 we present the obscured X-ray luminosities against the 12 $\mu$m luminosities. All our unobscured sources seem to follow the Gandhi-relation closely, and none of them shows unusually high infrared luminosity compared to the X-ray. On the other hand, candidate CT sources are found closer to the dashed line that demarcates the purely CT region. The sources located below this line are all candidate CT according to our analysis. Therefore, it is is unlikely that we are missing any CT candidates among the Sy2 sample.
3 pt
----------- --------------------- ------------------- ------------- ------------------ ----------------------
N name $\rm N_H$ FeK$\alpha$ $\rm L_x/L_{12}$ $\rm L_X/L_{[OIII]}$
[*(1)*]{} [*(2)*]{} [*(3)*]{} [*(4)*]{} [*(5)*]{} [*(6)*]{}
9 J103408.58+600152.1 $>10^{24}\dagger$ x x x
10 J103456.37+393941.0 $>5\times10^{23}$
12 J104426.70+063753.8 $>9\times10^{23}$ x
14 J113549.08+565708.2 $>10^{24}$ x x
16 J121839.40+470627.6 $>10^{24}\dagger$ x x x
22 J131104.66+272807.2 $>10^{24}$ x
29 J150754.38+010816.8 $>10^{24}\dagger$ x
30 J215649.51–074532.4 $>10^{24}\dagger$ x x x
----------- --------------------- ------------------- ------------- ------------------ ----------------------
: Candidate Compton-thick criteria
### The $\rm N_H$ vs. $L_x/L[OIII]$ ratio as an indicator of obscuration
In this section we investigate the possibility that some of the sources are more obscured than we can infer from their column density.
In Fig. 5 we plot the column density obtained from the X-ray spectral modeling as a function of the X-ray to optical luminosity ratio. The \[OIII\] luminosities are corrected for reddening using the formula described in Basanni et al. (1999): $\rm L_{[OIII]_{COR}} = L_{[OIII]_{OBS}} [(H\alpha /H\beta)/(H\alpha /H\beta)_o]^{2.94}$, where the intrinsic Balmer decrement $\rm (H\alpha /H\beta )_o$ equals 3. The lower left region in this plot could be possibly occupied by highly obscured or Compton-thick AGN, although their $\rm N_H$ values show the opposite (Akylas & Georgantopoulos 2009). In our case, however, none of the unobscured sources is located in this region, and therefore there is no evidence that their nuclei are heavily obscured.
On the other hand, three sources with $\rm N_H>10^{23}$ cm$^{-2}$ are found marginally outside the 3$\sigma$ limit. This implies that they are probably even more obscured than what we calculated by fitting their X-ray spectra. Interestingly, these are the three out of four sources (J103408.58+600152.1 (N9), J121839.40+470627.6 (N16), J215649.51–074532.4 (N30)) for which a high FeK$\alpha$ EW is reported, and they are also found below the CT line in Fig. 4. Therefore, despite the value of the $\rm N_H$, it is evident that the iron line is a robust indicator of obscuration. Once again we can infer that our classification of unobscured and CT sources is valid.
Discussion and conclusions
==========================
Candidate Compton-thick sources
-------------------------------
X-ray spectroscopy shows that the number of Compton-thick AGN in our sample could be as high as eight. N10 was initially included in the CT candidates because it is consistent with being CT within the uncertainties of the calculated column density. However, we chose to exclude this source since it is not confirmed by any of the diagnostics presented in this study (see also LaMassa et al. 2014). Therefore, we are left with seven CT sources, translating to a percentage of $\sim$23%.
We find that the number of CT AGN found in our survey agrees with those in other X-ray surveys of optically selected Seyfert galaxies. In more detail, Akylas & Georgantopoulos (2009), using XMM-Newton observations, estimate the number of CT sources among the Seyfert galaxies from the Palomar spectroscopic sample of nearby galaxies (Ho, Filippenko & Sargent 1995). They find a percentage of CT sources of 15-20 %. Since their sample consists of nearby ($<$120 Mpc) Sy2 galaxies, the X-ray observations provide excellent spectra, hence accurate column density measurements classifications of all the AGN in their sample. Also, Malizia et al. (2009) reports that $\sim18\%$ of their hard X-ray selected Sy2 sample is Compton-thick. Nevertheless, considering only the low-redshift sources ($z<0.015$) to remove the selection bias that affects their sample against the detection of CT objects, the percentage becomes $\sim35\%$. They argue that this result is in excellent agreement with the percentage of CT AGN in the optically selected sample of Risaliti, Maiolino & Salvati (1999). We note that because of our sample selection, which requires a sufficient number of photons in order to derive X-ray spectra, we may also be biased against heavily obscured sources.
On the other hand, JJ13 in their SDSS optically selected sample of type 2 QSOs, estimate a higher percentage of CT sources that could be as high as 50%, albeit with limited photon statistics. Initially, the percentage they calculate based on the X-ray spectral modeling and the intensity of the FeK$\alpha$ line is significantly lower. However, it reaches 50% after they conclude that at least half of the $\rm N_H$ values of their sources are underestimated, based on their $L_{2-10 keV}/L_{[OIII]}$ ratios.
Nevertheless, four out of the seven CT sources in our study are in common with JJ13. Three of them are also reported as CT in JJ13. N12 is not a CT source in JJ13 despite its high $\rm N_H$ and the detection of the FeK$\alpha$ line in their work. A probable reason is that they report an X-ray luminosity that is one order of magnitude higher than the one we measure in the current study. Therefore the $L_{2-10 keV}/L_{[OIII]}$ ratio is higher than their threshold for a CT source.
We note that four sources in our sample were initially considered heavily obscured because of the high FeK$\alpha$ EW ($>$1 keV), although their column density was only a few times $10^{23}$ $\rm cm^{-2}$. This suggests that these sources may be attenuated by CT absorbers. Indeed, all CT sources in the local Universe appear to present high EW of the FeK$\alpha$ line (e.g., Fukazawa et al. 2011) owing to suppression of their continuum emission. The discrepancy between the estimated column density and the EW could be attributed to a more complex spectral model that involves a double screen absorber with one of them being CT. In all four cases, by fitting the X-ray spectra with the model of Brightman & Nandra (2011), we confirm that they are indeed heavily obscured ($\rm N_H>10^{24} cm^{-2}$). In addition, according to Table 3, most of them satisfy all our CT criteria. Interestingly, three out of the above four high-EW sources, J103408.58+600152.1 (N9), J215649.51–074532.4 (N30), and J121839.40+470627.6 (N16) lie in the CT regime in the $\rm L_x/L_{12}$ diagram, and (Fig. 4) the same three sources have the lowest $\rm L_X/L_{[OIII]}$ ratios (Fig. 5), again supporting their CT nature. Two of these sources are in common with JJ13 (N9 and N16), and present a high EW in both studies.
Unabsorbed Sy2 nuclei
---------------------
The X-ray spectral analysis revealed that four[^2] Sy2 galaxies ($\sim$13%) present very low absorption, below $10^{22}$ cm$^{-2}$, in sharp contrast with the unification model of AGN. The percentage of unobscured Sy2 sources varies in the literature, from a few percent ($\sim 3-4\%$) in Risality, Maiolino & Salvati (1999) and in Malizia et al. (2009), to 40% in Page et al. (2006) and 66% in Garcet et al. (2007). Our value is in better agreement with Panessa & Bassani (2002) and Akylas & Georgantopoulos (2009). However, considering that the number of unobscured Sy2s discovered in any of these studies is less than eight, we argue that we roughly agree with all of them, except perhaps with Garcet et al. (2007). Also, we note that our criteria for selecting narrow line AGN are more stringent than in most of the above studies; for example, Risality, Maiolino & Salvati (1999) include Sy1.9 in their sample, and Garcet et al (2007) allow narrow line AGN up to FWHM$_{{\mathrm{H}\alpha}}$=1500 km s$^{-1}$.
As we have already discussed, none of our unobscured sources present a low X-ray to \[OIII\] or $\rm L_{12}$ luminosity ratio. They also do not present a strong FeK$\alpha$ line, and therefore we cannot associate them with a highly obscured Compton-thick nucleus. In addition, the FWHM of their ${\mathrm{H}\alpha}$ line is less than 500 km s$^{-1}$, which excludes the possibility of a narrow-line Sy1 classification. Although Tran (2001) argues about the presence of this kind of type 2 AGN in his sample of non-HBLR Sy2s fifteen years ago, their existence is still being strongly debated (see discussion in Antonucci 2012). Below, we summarize important observational and theoretical studies in the field, which attempt to approach this problem from various angles.
There is strong evidence that the dusty obscuring torus in low luminosity AGN is absent or is thinner than expected in higher luminosities (e.g., Elitzur & Shlosman 2006; Perlman et al. 2007; van der Wolk et al. 2010). Accordingly, all low luminosity AGN should have been Type 1 sources, which of course is not the case. The only reasonable explanation of this problem is the additional absence of the BLR in such systems. Some authors (e.g., Nicastro 2000; Nicastro, Martocchia & Matt 2003; Bian & Gu. 2007; Marinucci et al. 2012; Elitzur, Ho & Trump 2014) presented arguments that below a specific accretion rate of material into the black hole, and therefore at lower luminosities, the BLR might also be absent.
Using data from nearby bright AGN, Elitzur & Ho (2009) conclude that the BLR disappears at bolometric luminosities that are lower than $5 \times 10^{39} (M_{BH}/10^7 M_{\sun})^{2/3}
\rm erg\; s^{-1}$, where $M_{BH}$ is the mass of the black hole. They also argue that the quenching of the BLR and the disappearance of the torus can occur either simultaneously or in sequence, with decreasing black hole accretion rate and luminosity. Thus, a possible scenario would be that non-HBLR Sy2 AGN are objects lacking the BLR and possibly the torus. Nicastro, Martocchia & Matt (2003) conclude that the BLR probably does not exist below an accretion rate threshold of $\log(L_{bol}/L_{Edd})=-3$, while Marinucci et al. (2012) argue that true Sy2s can be found below the relatively higher limits of bolometric luminosity $\log L_{bol}=43.9$ and Eddington ratio $\log(L_{bol}/L_{Edd})=-1.9$. Marinucci et al. (2012) derived the bolometric luminosity from the X-ray and the \[OIV\] luminosity and conclude that $L_{[\rm OIII]}$ is not as reliable (see also relevant discussion in Elitzur 2012). We note that Elitzur & Ho (2009) thresholds are relatively low, not only compared to other studies but also for the general Sy2 population (see discussion in the recent review by Netzer 2015). However, the idea that the accretion rate is essential in the formation of the BLR seems to be valid, although the exact limits have not yet been defined and probably also depend on other factors (see discussion in Koulouridis 2014).
To evaluate the above limits for our four unobscured sources, we computed their bolometric luminosities from the SED modeling (see §4.2.4). We also calculated their black hole masses using the $M_{BH}-\sigma*$ relation (Tremaine et al. 2002), where $\sigma*$ is the stellar velocity dispersion, calculated from the FWHM of the \[OIII\] emission lines (Greene & Ho 2005). We find that the Elitzur & Ho (2009) limits are very low for our unobscured sources. Nevertheless, all satisfy the bolometric luminosity and Eddington ratio limit of Marrinucci et al. (2012). We note, however, that our Eddington ratios may be overestimated since the Eddington luminosities, derived from the FWHM of the \[OIII\] lines, are probably underestimated (e.g., Bian & Gu 2007).
By conducting a two-sample Student’s t-test between the accretion rates of the unobscured and the obscured sources, we conclude that their mean values are significantly different at the 99.9% confidence level. In Fig. 6 we plot the Eddington ratio versus the bolometric luminosity of our Sy2s, but also the discarded intermediate type Seyferts (crosses). We also plot the lines that apparently separate the unobscured sources from the rest of the Sy2 population. These limits are similar to the respective ones found by Marinucci et al. (dashed lines in Fig.6) for HBLR and non-HBLR sources. All four unobscured sources fall into the area where non-HBLR Sy2s are found and the BLR is predicted to not exist. We note that the limits of previous works were based on the differences between HBLR and non-HBLR Sy2s, while our sample is divided into obscured and unobscured sources. The unobscured Sy2s are non-HBLR Sy2s by definition, whereas the obscured sources are not necessarily HBLR Sy2s. Therefore, the presence of obscured Sy2s in the bottom left quarter of the plot may imply the lack of their BLR as well. Interestingly, a number of Compton-thick sources exhibit low accretion rates. This agrees with the evolutionary scheme of AGN proposed by Koulouridis (2014), where a fraction of Compton-thick sources are predicted to emerge shortly after a galaxy interaction or merging event that causes the inflow of gas and dust toward the central region of the galaxy, enhances circumnuclear star formation and triggers the AGN. During this phase the accretion rate is expected to be low and the BLR absent. However, the failure to detect the BLR in CT sources may as well be due to the heavy obscuration and the large covering factor of the nucleus (see next paragraph). We note that the uncertainties that enter the above calculations are large (see Greene & Ho 2005) and our samples fairly small. However, the general tendency of low accretion type-2 AGN to lack any evidence of a BLR is once more evident.
An alternative scenario that can explain the lack of detectable BLR in many CT sources is that heavy obscuration does not allow the detection of the BLR even in the polarized spectrum. Marinucci et al. (2012) conclude that 64% of their compton-thick non-HBLR Sy2s exhibit higher accretion rates than the threshold clearly separating the two Sy2 classes. They attributed this discrepancy to heavy absorption along our line of sight, preventing the detection of the actual BLR in their nuclei. Evidently, merging systems constitute a class of extragalactic objects where heavy obscuration occurs (e.g., Hopkins et al. 2008). The merging process may also lead to rapid black hole growth, giving birth to a heavily absorbed and possibly Compton-thick AGN. Thus, we could presume that a number of our non-HBLR mergers, if not all of them, might actually be BLR AGN galaxies, where the high concentration of gas and dust prohibits even the indirect detection of the broad line emission (e.g., Shu et al. 2007). However, other studies have concluded that there is no evidence that non-HBLR Sy2s are more obscured than their HBLR peers (Tran 2003; Yu 2005; Wu 2011), while totally unobscured low-luminosity non-HBLR Sy2s were detected via investigation of their X-ray properties (e.g., Panessa & Bassani 2002; Akylas & Georgantopoulos 2009). The total population of non-HBLR Sy2s is probably a mixture of objects with low accretion rate and/or high obscuration.
Koulouridis (2014) argue that both of the above scenarios agree with an AGN evolutionary scheme (Krongold et al. 2002; Koulouridis et al. 2006a, b, 2013), where a low accretion rate is predicted at the beginning and the end of the Seyfert duty cycle, without ruling out the possibility that some HBLR Sy2s could also be created by minor disturbances or even secular processes.
Finally, we note that there is always the possibility that the discrepancy between the optical and the X-ray spectra is due to variability, since they were not obtained simultaneously.\
In a nutshell:
1. We found four unobscured sources ($\sim$13%) at odds with the simplest unification scheme. These sources exhibit low accretion rates that agree with previous studies that predict the lack of the BLR in low-accretion-rate AGN.
2. 64% of the Sy2s are obscured with a median column density value of $\rm N_H\sim1.0\times10^{23}cm^{-2}$.
3. The percentage of CT AGN is at $\sim$23%, although direct comparison with previous studies is difficult because of the different selection methodologies. Their heavy obscuration was confirmed using a variety of criteria and diagnostics.
We thank the anonymous referee for the insightful comments and suggestions that significantly contributed to improving the quality of the publication. EK acknowledges fellowship funding provided by the Greek General Secretariat of Research and Technology in the framework of the program Support of Postdoctoral Researchers, PE-1145. This work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA). Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.
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notes on individual objects
===========================
- Source 9 - J103408.58+600152.1 Because of the large EW of the FeK$\alpha$ line, but the relatively low $\rm N_H$, we fit the spectrum with the model of Brightman & Nandra (2011). The result of the fit is a high column density, $\rm N_H=220^{+\infty}_{-70}$, characteristic of the CT sources. Other useful values: p1/p2=0.008, cstat/dof=51.9/34, $\Gamma_{soft}=3^{+0.2}_{-0.4}$, $\Gamma_{hard}=1.8$ (fixed).\
- Source 16 - J121839.40+470627.6 Because of the large EW of the FeK$\alpha$ line, but the relatively low $\rm N_H$, we fit the spectrum with the model of Brightman & Nandra (2011). The result of the fit is a high column density, $\rm N_H=2009^{+\infty}_{-135}$, characteristic of the CT sources. Other useful values: p1/p2=0.003, cstat/dof=145/155, $\Gamma_{soft}=3.4^{+0.8}_{-0.6}$, $\Gamma_{hard}=1.8$ (fixed).\
- Source 22 - J131104.66+272807.2 Because of the high $\rm N_H$, but the small EW of the FeK$\alpha$ line, we fit the spectrum with the model of Brightman & Nandra (2011). The result of the fit is a column density value of $\rm N_H=114^{+87}_{-29}\times10^{22}$, which is relatively lower than what is reported in the current study, but again above the limit that characterize CT sources. A strong FeK$\alpha$ line is only present in the pn detector. Other useful values: p1/p2=0.003, cstat/dof=300/366, $\Gamma_{soft}=2.6^{+0.2}_{-0.2}$, $\Gamma_{hard}=1.8$ (fixed).\
- Source 25 - J135436.29+051524.5We chose not to include this source in the unabsorbed list because its photon index $\Gamma$ is extremely flat ($\sim$0.8) if left as a free parameter, and in addition there seems to be a strong FeK$\alpha$ line. It may be a reflection -dominated Compton-thick source, but we cannot confirm this because of the relatively low quality X-ray spectrum. Also, even though the EW seems high, it cannot be considered as a Compton-thick candidate because the scattered percentage is too large ($>30$%) implying partial covering instead of scattered emission.\
- Source 29 - J150754.38+010816.8 Because of the large EW of the FeK$\alpha$ line, but the relatively low $\rm N_H$, we fit the spectrum with the model of Brightman & Nandra (2011). The result of the fit is a high column density, $\rm N_H=211^{+\infty}_{-61}$, characteristic of the CT sources. Other useful values: p1/p2=0.003, cstat/dof=320/355, $\Gamma_{soft}=3.2^{+0.4}_{-0.4}$, $\Gamma_{hard}=1.8$ (fixed).\
- Source 30 - J215649.51–074532.4 Because of the large EW of the FeK$\alpha$ line, but the relatively low $\rm N_H$, we fit the spectrum with the model of Brightman & Nandra (2011). The result of the fit is a high column density, $\rm N_H=1500^{+\infty}_{-1200}$, which is characteristic of the CT sources. Other useful values: p1/p2=0.003, cstat/dof=265/3, $\Gamma_{soft}=3.5^{+0.5}_{-0.5}$, $\Gamma_{hard}=1.8$ (fixed).
[^1]: http://www.starlight.ufsc.br/
[^2]: We note that (N25) is a flat X-ray spectrum source with a visible FeK$\alpha$ line, and we have excluded it from the list of unobscured sources (see appendix for more details)
|
---
abstract: 'Motivated by recent surprising experimental results for the noise output of superconducting microfabricated resonators used in quantum computing applications and astronomy, we develop a fully quantum theoretical model to describe quantum dynamics of these circuits. Building on theoretical techniques from quantum optics, we calculate the noise in the output voltage due to two-level system (TLS) defects. The theory predicts squeezing for the noise in the amplitude quadrature with respect to the input noise, which qualitatively reproduces the noise ellipse observed in experiment. We show that noise enhancement along the phase direction persists for pump frequencies away from resonance. Our results also suggest that intrinsic TLS fluctuations must be incorporated in the model in order to describe the experimentally observed dependence of the phase noise on input power.'
author:
- So Takei
- 'Victor M. Galitski'
- 'Kevin D. Osborn'
title: 'Squeezed noise due to two-level system defects in superconducting resonator circuits'
---
Amorphous dielectrics contain weakly-coupled two-level system (TLS) defects [@philippsrev; @book] that are known to modify the complex permittivity at low temperatures, including the imaginary part which is responsible for loss [@schickfus]. In Josephson qubits the purposeful amorphous dielectrics were found to contain TLSs which leads to decoherence [@martinisetal]. Similarly, the loss in superconducting resonators is generally attributed to TLS defects which can arise from the native oxides on the superconducting leads [@wangetal] or deposited amorphous dielectrics [@simmondsetal; @kevins]. Superconducting resonators are used in applications of quantum computing [@qubits] and as single photon detectors for astronomy, where noise can limit the detection performance [@thesis] and may be attributable to TLSs [@gaokumar]. Surprisingly, recent studies have revealed that the amplitude noise quadrature of resonators is limited by the experimental sensitivity near the vacuum noise limit [@gaoarxiv]. In contrast, noise in the phase quadrature is relatively large and decreases as the square root of the measurement power [@gaoetal]. A semi-empirical model was used to describe this phase noise saturation, but a quantitative model of phase-amplitude noise asymmetry is still lacking [@gaoetal2].
Previous related noise studies have considered the effects of TLS noise on superconducting qubits [@yunoise], and noise squeezing in various nonlinear systems including Josephson junction parametric amplifiers [@yurkeetal; @beltranetal], micro-cantilevers [@rugar], and nanomechanical resonators coupled to a Cooper-pair box [@suhetal]. Atoms have also been used for four-wave mixing experiments to squeeze optical cavity states [@slusheretal].
In this letter, we develop a theory for noise due to TLS defects in a superconducting resonator circuit. We compute the noise in the transmitted voltage for the system shown in Fig. \[fig:circuit\](a) assuming that the lone source of stochastic fluctuations are the quadrature-independent white noise in the incident voltages on the transmission lines. The theory predicts that squeezing in the transmitted noise generally occurs even if the dynamics of the internal resonator, including the TLSs, is fully deterministic. The results imply that intrinsic stochastic dynamics of TLSs is not necessary in explaining the observed squeezing in the transmitted voltage noise. Our work shows a similarity between noise squeezing from TLS defects in these devices [@gaoarxiv; @gaoetal] and squeezing of coherent light due to two-level atoms in quantum optics [@lugiatorev]. We also compute noise power for off-resonant pump frequencies and confirm that enhancement in the phase noise and squeezing in the amplitude noise persists away from resonance.
![\[fig:circuit\] (a) A circuit diagram of the considered setup. (b) Actual image (taken from Ref. ) of a notch-type aluminum LC resonator which can be modelled using (a). The components are labelled to clarify its correspondence to circuit in (a).](Fig1.pdf)
We have two identical semi-infinite 1D transmission lines, labeled $-$ and $+$, which are attached to each other end-to-end at $x=0$ (see Fig. \[fig:circuit\](a)). They are coupled to the internal superconducting LC resonator at $x=0$ through the coupling capacitor $C_c$. The capacitor and inductor in the LC circuit have capacitance $C$ and inductance $L$. Here, we take $C_c\ll C$, and consider the case where TLS defects reside only inside the dielectric of capacitor $C$. A microwave source (not shown) sends in a pump signal from $x=-\infty$, and we focus on the transmitted voltage noise on transmission line +.
The Hamiltonian for the full system has three terms, $\hat H=\hat H_0+\hat H_{\rm TLS}+
\hat H_{{\rm TLS}-R}$. $\hat H_0=\sum_{m=\{+,-\}}\hat H_m+\hat H_R$ models the two transmission lines and the internal resonator, $$\begin{aligned}
\centering
\label{li}
\hat H_m&=&\int dx\, \Theta(mx){\left[ \frac{\hat P_m^2(x)}{2\ell}+\frac{{\left( \nabla_x\hat Q_m(x) \right)}^2}{2c} \right]}\\
\label{lr}
\hat H_R&=&\frac{\hat P_L^2}{2L}+\frac{\hat Q_0^2}{2C_c}+\frac{(\hat Q_0-\hat Q_L)^2}{2C},\end{aligned}$$ where $m=\{+,-\}$ labels the left ($-$) and right ($+$) transmission lines, and $\Theta(x)$ is the unit step function. $\hat Q_m(x)$ is the operator for total charge residing to the right of $x$ on transmission line $m$, $\hat Q_L$ is the operator for total charge that has flowed through the inductor, and $\hat Q_0=(\hat Q_--\hat Q_+)|_{x=0}$ denotes the total charge operator on capacitor $C_c$. $\hat P_m$ and $\hat P_L$ are the conjugate momenta for $\hat Q_m$ and $\hat Q_L$, respectively, and they satisfy $[\hat Q_m(x),\hat P_{m'}(x')]=i\hbar\delta_{mm'}\delta(x-x')$ and $[\hat Q_L,\hat P_L]=i\hbar$, where $m,m'=\{+,-\}$. The transmission lines are modeled as conductors with inductance per unit length $\ell$, capacitance to ground per unit length $c$, and characteristic impedance $Z=\sqrt{\ell/c}$ [@yurke; @yurkedenker]. We assume we have $N$ identical and independent TLSs, all with asymmetry energy $\Delta_A$ and tunneling energy $\Delta_0$. The TLS Hamiltonian is then given by $\hat H_{\rm TLS}=\Delta_A\hat S_z+\Delta_0\hat S_x$, where $\hat S_i=\sum_{\alpha=1}^N\hat s_{i\alpha}$ is the collective spin operator which represents the $N$ TLSs, and $\hat s_{i\alpha}
=\sigma_i/2$ with the usual Pauli matrices $\sigma_i$. The components of the collective spin operator obey $[\hat S_i,\hat S_j]=i\epsilon_{ijk}\hat S_k$. After diagonalization $\hat H_{\rm TLS}=E\hat S_z$ where $E=[\Delta_A^2+\Delta_0^2]^{1/2}$. Each TLS interacts with the uniform electric field inside the capacitor through its electric dipole moment $\bf p$. We assume the dipoles fluctuate by making $180^\circ$ flips between parallel and anti-parallel orientations with respect to the field. The field affects the asymmetry energy and here we ignore the relatively small changes to the tunnel barrier, similar to other treatments of TLSs derived from the tunneling model [@philippsrev]. In the diagonalized basis the interaction between the TLSs and the field can then be written as $\hat H_{{\rm TLS}-R}=g\hbar[\hat S_z\cot2\xi+(\hat S^++\hat S^-)/2](\hat Q_0-\hat Q_L)$, where the coupling constant $g=2|{\bf p}|\sin 2\xi/\hbar Cd$, $d$ is the separation between the plates of the capacitor, $\xi$ is defined via $\tan2\xi=\Delta_0/\Delta_A$, and $\hat S^\pm=\hat S_x\pm i\hat S_y$. We use $|{\bf p}|=1$ Debye, which is comparable to the dipole moment sizes observed for TLSs in amorphous SiO$_2$ at microwave frequencies and OH rotors in AlO$_x$ [@musgrave].
For $x\ne 0$, $\hat Q_m(x,t)$ obey the massless scalar Klein-Gordon equation, and the solution can be written as a sum of right- ($R$) and left-propagating ($L$) components, i.e. $\hat Q_m(x,t)=\hat Q_{mR}(x,t)+\hat Q_{mL}(x,t)$, where $$\label{chargefieldexp}
\hat Q_{m\{R,L\}}(x,t)=\int_0^\infty\frac{d\omega}{2\pi}\sqrt{\frac{\Omega_r}{\omega}}
(\hat q_{m\{R,L\}}^\dag(\omega)e^{i\omega(t\mp x/v)}+h.c.),$$ $v=(\ell c)^{-1/2}$ is the velocity of wave propagation and $\Omega_r$ is the resonance frequency of the loaded LC circuit. The charge operators obey $[\hat q_{m\tau}(\omega),\hat q^\dag_{m'\tau'}(\omega')]=(\pi\hbar/\Omega_rZ)\delta_{mm'}\delta_{\tau\tau'}
\delta(\omega-\omega')$, where $\tau,\tau'=\{R,L\}$. We also introduce the ladder operator for the inductor charge via $\hat Q_L=\hat q_L+ \hat q_L^\dag$, where $[\hat q_L,\hat q_L^\dag]=\hbar/2\omega_0L$ and $\omega_0=(LC)^{-1/2}$.
At $x=0$, the quantum Maxwell-Bloch equations for our problem read $$\begin{aligned}
\label{eom1na}
\left.\nabla_x(\hat Q_{-R}+\hat Q_{-L})\right|_{x=0^-}&=&
\left.\nabla_x(\hat Q_{+R}+\hat Q_{+L})\right|_{x=0^+}\\
\label{eom2na}
\left.\frac{\nabla_x(\hat Q_{+R}+\hat Q_{+L})}{c}\right|_{x=0^+}&=&
-{\left( \frac{\hat Q_0}{C_c}+L\ddot {\hat Q}_L \right)}\\
\label{eom3na}
LC\ddot {\hat Q}_L-(\hat Q_0-\hat Q_L)&=&Cg\hbar(\hat S_z\cot2\xi+(\hat S^++\hat S^-)/2)\\
\dot{\hat S}^-+[i(E/\hbar)+\Gamma_2]\hat S^-&=&ig(\hat S_z-\cot2\xi\hat S^-)(\hat Q_0-\hat Q_L)\\
\label{eom5na}
\dot{\hat S}_z+\Gamma_1(\hat S_z-S_z^0)&=&(ig/2)(\hat S^--\hat S^+)(\hat Q_0-\hat Q_L).\end{aligned}$$ Here, we have introduced phenomenological longitudinal and transverse TLS relaxation rates, $\Gamma_1=T_1^{-1}$ and $\Gamma_2=T_2^{-1}$, and $\hat Q_0=[\left.\hat Q_{-R}+\hat Q_{-L}-\hat Q_{+R}-\hat Q_{+L}]\right|_{x=0}$. $S_z^0=-(N/2)\tanh(E/2k_BT)$ is the equilibrium expectation value for $\hat S_z$ when the TLSs are decoupled from the fields. Assuming we are in the low-temperature regime, i.e. $k_BT \ll E$, we take $S_z^0\approx -N/2$. The first three equations above are quantum generalizations of the Kirchhoff voltage law, and the last two equations are quantum Bloch equations describing the coupling of the TLSs to the photon fields.
We now perform a series of standard approximations which maps the above equations to a form often seen in quantum optics literature describing an ensemble of two-level atoms in an optical cavity [@lugiatorev]. Since we have a high-$Q$ resonator we perform the Markov approximation, where we replace $\sqrt{\omega}$ by $\sqrt{\Omega_r}$ in the mode expansion Eq. (\[chargefieldexp\]) and extend the lower limit of the integral to negative infinity. We now go to a frame rotating at the pump frequency $\Omega$, and use Greek letters to define operators in the rotating frame, i.e. $\hat\theta_{m\tau}(t)=\hat q_{m\tau}(t)e^{i\Omega t}$, $\hat\theta_L(t)=
\hat q_L(t)e^{i\Omega t}$, $\hat\Sigma^\pm(t)=\hat S^\pm(t)e^{\mp i\Omega t}$, and $\hat\Sigma_z(t)=\hat S_z(t)$. Performing the slowly-varying envelope approximation on the $\ddot Q_L(t)$ term and with the usual rotating-wave approximation, Eqs. (\[eom1na\])-(\[eom5na\]) can then be rewritten as $$\begin{aligned}
\label{eom1a}
(\partial_t-i\Omega)\hat\theta_1&=&0\\
\label{eom2a}
ZC_c(\partial_t-i\Omega)(\hat\theta_{+R}-\hat\theta_{+L})&=&\hat\theta_{0}
-LC_c{\left( \Omega^2+2i\Omega\partial_t \right)}\hat\theta_L\\
\label{eom3a}
-LC{\left( \Omega^2+2i\Omega\partial_t \right)}\hat\theta_L&=&
\hat\theta_{0}-\hat\theta_L+(Cg\hbar/2)\hat\Sigma^-\\
(\partial_t+i\Delta+\Gamma_2)\hat\Sigma^-&=&ig\hat\Sigma_z(\hat\theta_{0}-\hat\theta_L)\\
\label{eom5a}
2\partial_t\hat\Sigma_z+2\Gamma_1(\hat\Sigma_z-\Sigma_z^0)&=&
ig[\hat\Sigma^-(\hat\theta_0^\dag-\hat\theta_L^\dag)-h.c.],\end{aligned}$$ where $\Delta=E/\hbar-\Omega$ is the TLS detuning, and $\hat\theta_{\{0,1\}}=\hat\theta_{-R}\pm\hat\theta_{-L}-\hat\theta_{+R}\mp\hat\theta_{+L}$. We note that the voltage fields propagating along the transmission lines relate to the charge fields $\hat\theta_{m\tau}$ through $\hat v_{m\tau}(t)=-c^{-1}\nabla_x\hat\theta_{m\tau}(x,t)|_{x=0}$.
We solve Eqs. (\[eom1a\])-(\[eom5a\]) for $N\gg 1$ where the dynamics of the quantum system can be described semi-classically [@altlandetal]. We treat each field $\phi(t)$ as a $c$-number and assume it has a steady-state mean part $\bar\phi$ and a fluctuating part $\delta\phi(t)$ which models the noise arising from the incident voltage on the transmission lines. Assuming small fluctuations, we linearize Eqs. (\[eom1a\])-(\[eom5a\]) with respect to these fluctuations about the steady-state value.
![\[fig:MFres\] (a) Steady-state solution for the transmission with $N=100$, and (i) $\bar v_{-R}=10\mu$V; (ii) $\bar v_{-R}=1\mu$V; and (iii) $\bar v_{-R}=0.05\mu$V. Here, $\Delta\Omega\approx 0.0003\omega_0$. (b) Noise ellipses for $\bar v_{-R}=2\mu$V, $\Omega=\Omega_r$, and $N=100$ (red), $300$ (green), $600$ (blue), and $1000$ (magenta). The dashed line is quadrature-independent noise with no TLSs. (c) Plot of population imbalance, $\bar S_z$, as a function of input power for $N=100$.](Fig2.pdf)
The steady-state solution to the transmission amplitude $\bar v_{+R}/\bar v_{-R}$ is plotted in Fig. \[fig:MFres\](a) for real $\bar v_{-R}$ (input voltage amplitude) near the resonance frequency $\Omega_r$. Blue points correspond to pump frequencies below $\Omega_r$ and red points to those above $\Omega_r$. The solutions are plotted for three different values of $\bar v_{-R}$, and the parameters used are $C_c=0.01$pF, $C=0.3$pF, $Z=50\Omega$, $f_0=\omega_0/2\pi=6$ GHz, $T_1=300$ns, $T_2=30$ns, $d\approx 70$nm and $N=100$. We also assume the TLSs to be in resonance with the LC resonator and fix the TLS energy to $E=\Omega_r$ throughout and take $\Delta_A=\Delta_0$. We then find $\Omega_r\approx 0.9837\omega_0$. In Fig. \[fig:MFres\](c), $\bar S_z$ (TLS population imbalance) is plotted as a function of the input power, $P_{in}$. As $P_{in}$ increases $\bar S_z$ approaches zero signifying TLS saturation. We see that saturation occurs for $P_{in}\gtrsim 10^{-13}$W. Fig. \[fig:MFres\](a) shows that the transmission deviates very little from the defect-free (no TLSs) limit for saturated TLSs. This is expected since only a small fraction of the total energy is stored in the TLSs when they are saturated.
We now move on to obtaining the noise in the transmitted voltage. Solving the linearized equations the transmitted charge fluctuations can be written in terms of the two input fluctuations as $$\centering
\label{tfluc}
{\bf A}_{+R}(\omega)\delta\Theta_{+R}(\omega)={\bf A}_{-R}(\omega)\delta\Theta_{-R}(\omega)
+{\bf A}_{+L}(\omega)\delta\Theta_{+L}(\omega),$$ where $\omega$ is the frequency deviation measured from the pump frequency $\Omega$, $\delta\Theta_{m\tau}(\omega)={\left( \delta\theta_{m\tau}(\omega),\delta\theta_{m\tau}^*(-\omega) \right)}^T$, and the elements of the coefficient matrices ${\bf A}_{m\tau}(\omega)$ are given in the Supplementary Material. To study the quadrature-dependence of the noise in the transmitted voltage we introduce generalized voltage fluctuation variables $\delta v_{m\tau}(\omega,\varphi)=[e^{-i\varphi}\delta v_{m\tau}(\omega)+e^{i\varphi}\delta v_{m\tau}^*(-\omega)]/2$, where $\varphi$ is the quadrature angle measured with respect to the real axis. We focus on the symmetrized correlator of these fluctuations, i.e. $S_{+R}(\omega,\omega',\varphi)={\left\langle \{\delta v_{+R}(\omega,\varphi),\delta v_{+R}^*
(\omega',\varphi)\} \right\rangle}$, where $\{A,B\}=AB+BA$. Assuming that the incoming fluctuations, $\delta v_{-R}(t)$ and $\delta v_{+L}(t)$, are characterized by a quadrature-independent white noise spectrum $s_0$ (for thermalized transmission lines it is $s_0^{\rm th}=2\pi\hbar Z\Omega_r\coth(\hbar\Omega_r/2k_BT)$), we have $$\centering
\label{noisein}
{\left\langle \{\delta v_{\{-R.+L\}}(\omega,\varphi),\delta v_{\{-R,+L\}}^*(\omega',\varphi)\} \right\rangle}=s_0\delta(\omega-\omega').$$ Using Eqs. (\[tfluc\]) and (\[noisein\]) together with $\delta v_{+R}(\omega,\varphi)\approx-2iZ\Omega_r
\delta\theta_{+R}(\omega,\varphi)$, $S_{+R}(\omega,\omega',\varphi)=s_{+R}(\omega,\varphi)\delta(\omega-\omega')$ can be straightforwardly obtained. (see Supplementary Material for details). In the following, we focus on the resulting spectral density of the noise, $s_{+R}(\omega,\varphi)$.
![\[fig:noises2\] Colour intensity plot of the normalized noise power $s_{+R}(\omega,\varphi)/s_0$ on the ($\omega$, $\varphi$)-plane for pump frequencies on and away from resonance, $\bar v_{-R}=2\mu$V and $N=1000$. Plots correspond to noise centred around (a) west; (b) northwest; and (c) north points, where north is defined as the top point on the resonance circle.](Fig3.pdf)
We first focus on the on-resonance case (where $\mbox{Im}\{\bar v_{+R}\}=0$). Fig. \[fig:MFres\](b) plots noise ellipses centred at this resonance point for $\bar v_{-R}=2\mu$V and various $N$. Here, we are plotting the $\omega=0$ component of the noise spectral density normalized by $s_0$. The noise ellipse is defined such that every vector from the origin to a point on the ellipse makes an angle $\varphi$ (quadrature angle) with respect to the positive real axis and has length $s_{+R}(\omega=0,\varphi)/s_0$. In the defect-free limit, the noise ellipse is circular with radius 1 (the dashed line), which shows that in the absence of non-linearity the transmitted noise remains quadrature-independent. In the presence of TLSs, we obtain *squeezing* where the noise along amplitude (phase) quadrature is reduced below (enhanced above) the noise of the incoming fluctuations. Eccentricity of the noise ellipse increases as the number of TLSs is increased.
Fig. \[fig:noises2\] is a colour intensity plot of the normalized noise power $s_{+R}(\omega,\varphi)/s_0$ on the $(\omega,\varphi)$-plane for on- and off-resonance pump frequencies. Here, $\bar v_{-R}=2\mu$V and $N=1000$ are both fixed. We now use the convention where north corresponds to the point at the top of the resonance circle in Fig. \[fig:MFres\](a), and west to the left-most point on the circle and so on. Then, the three plots correspond to the noise centred around (a) west; (b) northwest; and (c) north. The results show that the major axis of the noise ellipse is always along the phase direction. Our results are consistent with experiments, where fluctuations are also primarily observed in the direction tangent to the resonance circle [@gaoetal].
In Fig. \[fig:powerdep\](a) we plot the normalized excess phase noise power, $s^{\rm ph}_{\rm exc}(\omega):=[s_{+R}(\omega,\varphi=\pi/2)-s_0]/s_0$ and negative of the normalized excess amplitude noise, $-s^{\rm amp}_{\rm exc}(\omega):=-[s_{+R}(\omega,\varphi=0)-s_0]/s_0$, as a function of frequency deviation away from resonance, $\omega$. Here we use $\bar v_{-R}=2\mu$V and $N=1000$. Both noises roll off at the resonator bandwidth $\omega_{\rm roll}/\omega_0\approx 10^{-4}$. From the steady-state solution the quality factor of the resonator is estimated to be $Q_r\approx 9800$, which is consistent with the plot. We see that the phase noise stays above the incoming noise $s_0$ while the amplitude noise remains below this value signifying squeezing. They both approach $s_0$ as the frequency deviates sufficiently beyond the resonator bandwidth (see also Fig. \[fig:noises2\]). For $\omega\gg\omega_{\rm roll}$ we find $s^{\rm ph}_{\rm exc}(\omega)\sim(\omega/\omega_0)^{-2}$ and $-s^{\rm amp}_{\rm exc}(\omega)\sim(\omega/\omega_0)^{-2.9}$.
![\[fig:powerdep\] (a) Frequency-dependence of the excess phase noise power and negative of the excess amplitude noise power for $\bar v_{-R}=2\mu$V and $N=1000$. Roll-off at $\approx\omega/\omega_0=10^{-4}$ can be seen. (b) Normalized excess phase noise at $\omega=0$ and inverse of the internal quality factor as a function of input power ($\Omega=\Omega_r$).](Fig4.pdf)
In Fig. \[fig:powerdep\](b) we plot the loss tangent, $Q_{\rm int}^{-1}$ for the pump frequency on resonance and $N=100$. This was found numerically by fitting the transmission amplitude to a circuit model as a function of $[V(1+Q_{\rm ext}/Q_{\rm int})]^2/Z\approx V^2/Z$. Here, $V$ is the voltage across capacitor $C$ (as shown in Fig. \[fig:circuit\]) and $Q_{\rm ext}=2(C+C_c)/(\omega_0ZC_c^2)$ is the external quality factor. The above approximation holds because our resonator is over-coupled. At low amplitude the loss is constant, and at high amplitude the slope is $\approx -1$ as the spins are saturated. This is in agreement with the semi-classical theory for a single TLS type. For the standard tunneling model distribution the density of defects follows $P(\Delta_A,\Delta_0)=P_0/\Delta_0$ and the superposition of the loss from different TLSs give a slope of $-1/2$ in the high power regime [@schickfus], as observed in amorphous films.
In the same figure the excess phase noise $s_{\rm exc}(\omega=0)$ is plotted. In the same high power regime as the loss tangent, the partially saturated TLSs exhibit phase noise with a slope of $-1$, which disagrees with the power dependence observed in experiment [@thesis]. This suggests that a correct description for the phase noise power dependence necessitates the inclusion of intrinsic stochastic fluctuations of the TLSs. Indeed, intrinsic TLS noise causes dielectric constant fluctuations and can lead to observable noise [@gaoetal2]. We reiterate, however, that squeezing phenomenon itself is present regardless of deterministic or stochastic nature of TLS dynamics.
In conclusion, we have developed a theoretical framework applicable to circuits containing transmission lines, lumped circuit elements, and TLS defects, building upon previous work in quantum optics. It serves as a first step toward a quantitative theory for noise in these circuits. We found that quadrature-independent incident noise is generally squeezed once transmitted through a lumped circuit containing TLSs even when the latter evolve deterministically. Extensions of the model could allow for further quantitative results on noise due to TLSs, including the treatment of the standard tunneling model distribution of TLSs and incorporating intrinsic TLS fluctuations.
*Acknowlegments*: S. T. thanks Lev S. Bishop for discussions. S. T. and V. G. were supported by the Intelligence Advanced Research Projects Activity (IARPA) through the US army Research Office award W911NF-09-1-0351.
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---
abstract: 'derive the uniqueness of weak solutions to the Shigesada-Kawasaki-Teramoto (SKT) systems using the adjoint problem argument. Combining with [@PT16] we then derive the well-posedness for the SKT systems in space dimension $d\le 4$.'
author:
- '[[Pham]{}]{}[^1]'
- '[[Temam]{}]{}[^2]'
title: '[[A Result of Uniqueness of Solutions of the Shigesada-Kawasaki-Teramoto Equations]{}]{} '
---
#### Keywords and phrases:
wellposedness; quasi-linear parabolic equations; global existence.
#### 2010 Mathematics Subject Classification:
35K59, 35B40, 92D25.
Introduction\[sec:intro\]
=========================
is an important issue to address when one considers the global well-posedness for a system of differential equations. For systems of partial differential equations like cross diffusion systems, the uniqueness has remained a challenge for solutions with mild regularity since the comparison/maximum principle for the cross diffusion systems like SKT is not available. We also note here that in our recent work [@PT16], we showed a weak maximum principle for non-negativeness of solutions that allowed us to prove the existence of positive weak solutions of SKT systems directly using finite difference approximations, a priori estimates and passage to the limit, which avoid the change of variables (or entropy function) being used in other works as in e.g. [@CJ04; @CDJ16; @Jun15]. Together with our existence result for weak solutions of SKT systems in [@PT16], this article provides the well-posedness for these systems in space dimension $d\le 4$.
The available uniqueness results for the KST systems are rather scarce and require high regularity of the solutions. In [@Yag93 Theorem 3.5], the author proved a uniqueness result for solutions in ${\mathcal{C}}((0,T];H^2(\Omega))\cap{\mathcal{C}}^1((0,T];L^2(\Omega))$ using an abstract theory for parabolic equations in space dimension 2. In [@Ama89; @Ama90], the author proved global existence and uniqueness results for solutions of general systems of parabolic equations with high regularity in space in the semigroup settings, $W^{1,p}(\Omega)$ for $p>n$, which require Hölder a priori estimates when applied to SKT equations.
In this work, we use the argument of adjoint problems to build specific test functions to show the uniqueness for solutions in a more general space setting, in $L^\infty(0,T;H^1(\Omega)^2)$ with time derivatives in $L^\f43(\Omega_T)^2$ for space dimension $d\le 4$, see Remark \[rmk:soln reg\] below. This argument of using adjoint problems has been used to show uniqueness results for scalar partial differential differential equations describing flows of gas or fluid in porous media or the spread of a certain biological population, see e.g. [@Aro85; @ACP82]. It is also systematically used in the context of linear equations in [@LM72].
Throughout our work, we denote by $\Omega$ an open bounded domain in $\mathbb{R}^d$, with $d\le 4$, and we set $\Omega_T=\Omega\times (0,T)$ for any $T>0$. We aim to show the uniqueness result and then combine this result to our previous global existence results of weak solutions in [@PT16] to show the global wellposedness for the following SKT system of diffusion reaction equations, see [@SKT79]: $$\label{SKT vector}
\begin{cases}
&\dps \partial_t {\mathbf{u}}- \Delta {\mathbf{p}}({\mathbf{u}}) + {\mathbf{q}}({\mathbf{u}}) = {\boldsymbol{\ell}}(u)\text{ in } \Omega_T,\\
&\partial_\nu {\mathbf{u}}= {\mathbf{0}}\text{ on } \partial \Omega \times (0,T) \text{ or }{\mathbf{u}}={\mathbf{0}}\text{ on }\partial \Omega\times (0,T), \\
&{\mathbf{u}}(x,0) = {\mathbf{u}}_0(x) \ge {\mathbf{0}}, \text{ in } \Omega,
\end{cases}$$ where ${\mathbf{u}}=(u,v)$ and
\[pi qi li\] $$\begin{aligned}
&{\mathbf{p}}({\mathbf{u}}) = \begin{pmatrix}
p_1(u,v) \\ p_2(u,v)
\end{pmatrix} = \begin{pmatrix}
(d_1 + a_{11} u+ a_{12}v) u \\ (d_2 + a_{21} u+ a_{22}v) v
\end{pmatrix} , \\
&{\mathbf{q}}({\mathbf{u}}) = \begin{pmatrix}
q_1(u,v) \\ q_2(u,v)
\end{pmatrix}
=\begin{pmatrix}
(b_1u+c_1v)u
\\ (b_2u+c_2v)v
\end{pmatrix},
\\& \text{and }
{\boldsymbol{\ell}}({\mathbf{u}}) =\begin{pmatrix}
\ell_1(u)\\ \ell_2(v)
\end{pmatrix}
=\begin{pmatrix}
a_1u\\ a_2v
\end{pmatrix}.
\end{aligned}$$
Here $a_{ij} \ge 0, b_i \ge 0, c_i \ge 0, a_i \ge 0, d_i \ge 0$ are such that $$\label{1.5c}
0<a_{12} a_{21} < 64 a_{11}a_{22}. $$ It can be shown in [@Yag08] that the condition [[[(\[1.5c\])]{}]{}]{} is equivalent to $$\label{coef cond}
0<a_{12}^2 < 8 a_{11}a_{21} \text{ and } 0<a_{21}^2 <8a_{22}a_{12}, $$ as far as existence and uniqueness of solutions are concerned.
One of the difficulties with the SKT equations is that they are not parabolic equations. Whereas Amann [@Ama89; @Ama90] has proven the existence and uniqueness of regular solutions for general parabolic equations, which can be applied to SKT equations using $L^p$ estimates, we proved in [@PT16] the existence of weak solutions (see also [@Jun15]) to the SKT equations. It is important, to validate this concept of weak solutions, to show that the weak solutions are unique. This is precisely what we are doing in this article in dimension $d\le 4$.
Throughout the article, we often use the following alternate form of [[[(\[SKT vector\])]{}]{}]{}: $$\label{SKE alternate vector}
\partial_t {\mathbf{u}}- \grad \cdot \Big( {\mathbf{P}}({\mathbf{u}}) \grad {\mathbf{u}}\Big) + {\mathbf{q}}({\mathbf{u}}) ={\boldsymbol{\ell}}({\mathbf{u}}),$$ where $$\label{P}
{\mathbf{P}}({\mathbf{u}}) = \begin{pmatrix}
p_{11}(u,v) & p_{12}(u,v)\\
p_{21}(u,v) & p_{22}(u,v)
\end{pmatrix}
=\begin{pmatrix} d_1 +2a_{11}u + a_{12}v & a_{12} u \\ a_{21}v & d_2 + a_{21}u +a_{22}v \end{pmatrix}.$$
When the condition [[[(\[coef cond\])]{}]{}]{} is satisfied and $u\ge 0, v\ge 0$, we can prove that the matrix ${\mathbf{P}}({\mathbf{u}})$ is (pointwise) positive definite and that: $$\label{P positive definite}
\left({\mathbf{P}}({\mathbf{u}}){\boldsymbol{\xi}}\right) \cdot {\boldsymbol{\xi}}\ge \alpha(u+v){ \left| {\boldsymbol{\xi}}\right|}^2 + d_0 { \left| {\boldsymbol{\xi}}\right|}^2,\quad \forall {\boldsymbol{\xi}}\in \mathbb{R}^2,$$ where $d_0 = \min (d_1,d_2)$ and $$\label{alpha}
0<\alpha < \min \left(a_{11}, a_{12},a_{21 },a_{22},\delta_0\right) ;$$ Here we refer the readers to a proof of [[[(\[P positive definite\])]{}]{}]{} in our recent article [@PT16].
We consider later on the mappings $$\label{P Q map} {\mathcal{P}}: {\mathbf{u}}=(u,v) \mapsto {\mathbf{p}}=(p_1,p_2), \quad {\mathcal{Q}}: {\mathbf{u}}=(u,v) \mapsto {\mathbf{q}}=(q_1,q_2),$$ and we observe that $$\label{P Q jac} {\mathbf{P}}({\mathbf{u}}) = \f{D{\mathcal{P}}}{D {\mathbf{u}}}({\mathbf{u}}),\quad {\mathbf{Q}}({\mathbf{u}}) = \f{D{\mathcal{Q}}}{D {\mathbf{u}}}({\mathbf{u}}),$$ and $$\label{P grad u} \grad {\mathbf{p}}({\mathbf{u}}) = {\mathbf{P}}({\mathbf{u}}) \grad {\mathbf{u}}.$$ We see that the explicit form of ${\mathbf{P}}({\mathbf{u}})$ is given in [[[(\[P\])]{}]{}]{} and that of ${\mathbf{Q}}({\mathbf{u}})$ is $$\label{Q}{\mathbf{Q}}({\mathbf{u}}) = \begin{pmatrix}
2b_1 u+c_1 v & c_1 u \\ b_2v & b_2u+2c_2v
\end{pmatrix}.$$
Note that [[[(\[P positive definite\])]{}]{}]{} implies that, for $u,v \ge 0$, ${\mathbf{P}}({\mathbf{u}})$ is invertible (as a $2\times 2$ matrix), and that, pointwise (i.e. for a.e. $x\in \Omega$), $$\label{2.5b}
{ \left| {\mathbf{P}}({\mathbf{u}})^{-1} \right|}_{\mathcal{L}(\mathbb{R}^2)} \le \f{1}{d_0+\alpha(u+v)}.$$ Our work is organized as follows. We show our main result in Section \[sec:uniq\], where the uniqueness for weak solutions to the SKT system is derived using solutions of adjoint problems. Since the proof of the uniqueness relies on the existence of solutions to the adjoint problem, we show the existence for these problems in Section \[sec:adj\], together with the apriori estimates in dimension $d\le4$. We finally show in Section \[sec:global\] that the newly derived uniqueness result in Section \[sec:uniq\] together with our existence result in [@PT16] leads to the global well-posedness for the SKT systems in space dimension $d\le 4$.
Uniqueness result for SKT systems\[sec:uniq\]
=============================================
mentioned earlier, our uniqueness result is proven using an argument of an adjoint problem, see e.g. [@ACP82]; see also [@LM72] in the context of linear parabolic problems. The existence of solution of our adjoint problem will be granted if the solution ${\mathbf{u}}$ of [[[(\[SKT vector\])]{}]{}]{} enjoys the following regularity properties $$\label{uniqueness cond}
{\mathbf{u}}\in L^\infty(0,T;H^1(\Omega)^2), \text{ and }\partial_t {\mathbf{u}}\in L^\f43(\Omega_T)^2.$$
Although this was not explicitly stated in [@PT16], the solutions that we constructed in dimension $d\le 4$ belong to $L^\infty(0,T;H^1(\Omega)^2)$ with $\partial_t {\mathbf{u}}\in L^2(0,T;L^2(\Omega)^2)$; see Appendix \[appen: C\].\[rmk:soln reg\]
Introducing a test function ${\boldsymbol \varphi}$ which satisfies [[[(\[phi\])]{}]{}]{} below and the same boundary condition as ${\mathbf{u}}$, we multiply [[[(\[SKT vector\])]{}]{}]{} by ${\boldsymbol \varphi}$, integrate, integrate by parts and obtain the variational weak form of [[[(\[SKT vector\])]{}]{}]{}: $$\label{ui eq}
\begin{cases}
&\dps {\left\langle \partial_t {\mathbf{u}}, {\boldsymbol \varphi}\right\rangle} - {\left\langle {\mathbf{p}}({\mathbf{u}}), \Delta {\boldsymbol \varphi}\right\rangle} + {\left\langle {\mathbf{q}}({\mathbf{u}}), {\boldsymbol \varphi}\right\rangle} = {\left\langle {\boldsymbol{\ell}}({\mathbf{u}}), {\boldsymbol \varphi}\right\rangle},\\
&\partial_\nu {\mathbf{u}}= {\mathbf{0}}\text{ on } \partial \Omega \times (0,T) \text{ or }{\mathbf{u}}={\mathbf{0}}\text{ on }\partial \Omega\times (0,T), \\
&{\mathbf{u}}(x,0) = {\mathbf{u}}_0, \text{ in } \Omega,
\end{cases}$$ for all test functions ${\boldsymbol \varphi}$ such that $$\label{phi}
\begin{cases}
&{\boldsymbol \varphi}\in L^2(0,T;H^2(\Omega)^2) \cap L^\infty(0,T;H^1(\Omega)^2), \text{ and }\partial_t{\boldsymbol \varphi}\in L^\f43(\Omega_T)^2, \\
&\partial_\nu {\boldsymbol \varphi}= {\mathbf{0}}\text{ or } {\boldsymbol \varphi}={\mathbf{0}}\text{ on }\partial \Omega \text{ (${\boldsymbol \varphi}$ satifies the same b.c. as ${\mathbf{u}}$)}.
\end{cases}$$ Note that the boundary terms disappear because ${\mathbf{p}}({\mathbf{u}})$ satisfies the same b.c. as ${\mathbf{u}}$. To show that the solutions of [[[(\[SKT vector\])]{}]{}]{} are unique, we introduce the difference of two solutions ${\mathbf{u}}_1,\,{\mathbf{u}}_2$ of [[[(\[ui eq\])]{}]{}]{}, $\bar{{\mathbf{u}}} = {\mathbf{u}}_1 -{\mathbf{u}}_2$, and we will eventually show that $\bar{{\mathbf{u}}}={\mathbf{0}}$ for a.e. ${\mathbf{x}}\in \Omega$ and $t>0$. We first observe that $\bar{{\mathbf{u}}}$ satisfies $$\label{u diff eq}
\begin{cases}
&\dps {\left\langle \partial_t \bar{{\mathbf{u}}}, {\boldsymbol \varphi}\right\rangle} - {\left\langle {\mathbf{p}}({\mathbf{u}}_1)-{\mathbf{p}}({\mathbf{u}}_2), \Delta {\boldsymbol \varphi}\right\rangle} + {\left\langle {\mathbf{q}}({\mathbf{u}}_1)-{\mathbf{q}}({\mathbf{u}}_2), {\boldsymbol \varphi}\right\rangle} = {\left\langle {\boldsymbol{\ell}}({\mathbf{u}}_1) -{\boldsymbol{\ell}}({\mathbf{u}}_2), {\boldsymbol \varphi}\right\rangle},\\
&\partial_\nu \bar{{\mathbf{u}}} = {\mathbf{0}}\text{ on } \partial \Omega \times (0,T) \text{ or }\bar{{\mathbf{u}}}={\mathbf{0}}\text{ on }\partial \Omega\times (0,T), \\
&\bar{{\mathbf{u}}}(x,0) = {\mathbf{0}}, \text{ in } \Omega,
\end{cases}$$ for any test function ${\boldsymbol \varphi}$ that satisfies [[[(\[phi\])]{}]{}]{}.
Using the notations ${\mathbf{P}}(\star),\,{\mathbf{Q}}(\star)$ introduced earlier in [[[(\[P Q jac\])]{}]{}]{} and the relations [[[(\[p diff\])]{}]{}]{}, [[[(\[q diff\])]{}]{}]{} from Lemma \[lem: u diff\], we find $${\left\langle \bar{{\mathbf{u}}}, {\boldsymbol \varphi}\right\rangle}_t-{\left\langle \bar{{\mathbf{u}}}, {\boldsymbol \varphi}_t \right\rangle}-{\left\langle {\mathbf{P}}(\tilde{{\mathbf{u}}})\bar{{\mathbf{u}}}, \Delta{\boldsymbol \varphi}\right\rangle} +{\left\langle {\mathbf{Q}}(\tilde{{\mathbf{u}}})\bar{{\mathbf{u}}}, {\boldsymbol \varphi}\right\rangle} = {\left\langle {\boldsymbol{\ell}}(\bar{{\mathbf{u}}}), {\boldsymbol \varphi}\right\rangle},$$ where $\tilde{{\mathbf{u}}} = ({\mathbf{u}}_1+{\mathbf{u}}_2)/2$.
Thus $$\label{skt inner}
{\left\langle \bar{{\mathbf{u}}}, {\boldsymbol \varphi}\right\rangle}_t-{\left\langle \bar{{\mathbf{u}}}, {\boldsymbol \varphi}_t \right\rangle}-{\left\langle \bar{{\mathbf{u}}}, {\mathbf{P}}(\tilde{{\mathbf{u}}})^T\Delta{\boldsymbol \varphi}\right\rangle} +{\left\langle \bar{{\mathbf{u}}}, {\mathbf{Q}}(\tilde{{\mathbf{u}}})^T{\boldsymbol \varphi}\right\rangle} = {\left\langle {\boldsymbol{\ell}}(\bar{{\mathbf{u}}}), {\boldsymbol \varphi}\right\rangle}.$$ We notice that $\tilde{\mathbf{u}}\in L^\infty(0,T;H^1(\Omega)^2)$ because ${\mathbf{u}}_1,{\mathbf{u}}_2\in L^\infty(0,T;H^1(\Omega)^2)$.
We now consider the test function $ {\boldsymbol \varphi}$ to be solution of the following backward adjoint problem $$\label{adj eq phi}
\begin{cases}
&-\partial_t{\boldsymbol \varphi}- {\mathbf{P}}(\tilde{{\mathbf{u}}})^T \Delta{\boldsymbol \varphi}+ {\mathbf{Q}}(\tilde{{\mathbf{u}}})^T{\boldsymbol \varphi}= {\boldsymbol \varphi}\text{ in }\Omega_T,
\\& \partial_\nu {\boldsymbol \varphi}= {\mathbf{0}}\text{ or } {\boldsymbol \varphi}= {\mathbf{0}}\text{ on }\partial \Omega\times(0,T),
\\& {\boldsymbol \varphi}(T) ={\boldsymbol{\chi}}({\mathbf{x}})\text{ in }\Omega,
\end{cases}$$ where ${\boldsymbol{\chi}}({\mathbf{x}})= (\chi^u({\mathbf{x}}),\chi^v({\mathbf{x}})) \in H^1(\Omega)^2$.
Before showing the uniqueness result of solutions of [[[(\[SKT vector\])]{}]{}]{} using the test function ${\boldsymbol \varphi}$ as a solution of [[[(\[adj eq phi\])]{}]{}]{}, we first show the existence of ${\boldsymbol \varphi}=(\phi^u,\phi^v)\in L^2(0,T;H^2(\Omega)^2)$ with $\partial_t {\boldsymbol \varphi}\in L^\f43(\Omega_T)$ in the following section.
Existence of solutions for the adjoint systems \[sec:adj\]
----------------------------------------------------------
this section, we continue to assume that $d\le4$ and we show the existence of a solution ${\boldsymbol \varphi}$ of [[[(\[adj eq phi\])]{}]{}]{} satisfying [[[(\[phi\])]{}]{}]{} by building approximate systems where the classical existence theory can be applied to show the existence of approximate solutions. We suppose throughout this section that the functions $\tilde{\mathbf{u}}\ge {\mathbf{0}}$ in [[[(\[adj eq phi\])]{}]{}]{}$_1$ satisfies $$\label{tilde u}
\tilde{\mathbf{u}}\in L^\infty(0,T;H^1(\Omega)^2).$$ We observe that the diffusive matrix ${\mathbf{P}}(\tilde{\mathbf{u}})$ in [[[(\[adj eq phi\])]{}]{}]{} may not be uniformly parabolic[^3] unless $\tilde{\mathbf{u}}\in L^\infty(\Omega_T)^2$. Thus, we can not directly apply the classical results for parabolic equations to show the existence of ${\boldsymbol \varphi}$, see e.g. [@Lad68 Theorem 5.1]. We therefore use an approximation approach as in [@ACP82].
The existence of solution ${\boldsymbol \varphi}$ of [[[(\[adj eq phi\])]{}]{}]{} is obtained in three steps:
- Define approximations ${\boldsymbol \varphi}_\ep$ of ${\boldsymbol \varphi}$, which are solutions of the approximate systems [[[(\[adj eq ep\])]{}]{}]{} below.
- Derive a priori estimates for the functions ${\boldsymbol \varphi}_\ep$.
- Pass to the limit as $\ep \rightarrow 0$ to show the existence of ${\boldsymbol \varphi}$, solution of [[[(\[adj eq phi\])]{}]{}]{}.
We start now with the first step of building approximate solutions ${\boldsymbol \varphi}_\ep$:
### Approximate adjoint systems
We know that $\tilde{\mathbf{u}}= ({\mathbf{u}}_1+{\mathbf{u}}_2)/2 \ge {\mathbf{0}}$ and $\tilde{\mathbf{u}}\in L^\infty(0,T;H^1(\Omega)^2)$ (see Theorem \[thm: existence\]). We build approximations $\tilde{\mathbf{u}}_\ep$ of $\tilde{\mathbf{u}}$, as a sequence in $L^\infty(\Omega_T)^2$, that converges to $\tilde{\mathbf{u}}$ in $L^4(\Omega_T)^2$. We can define such $\tilde{\mathbf{u}}_\ep$ as follows $$\label{u ep}
\tilde{\mathbf{u}}_\ep = {\boldsymbol{\theta}}_\ep(\tilde{\mathbf{u}}),$$ where ${\boldsymbol{\theta}}_\ep $ is a smooth function with derivative bounded by a constant independent of $\ep$, which we assume to be $1$, such that $$\label{theta ep} {\boldsymbol{\theta}}_\ep(\tilde{{\mathbf{u}}} ) =\begin{cases}
& \tilde{{\mathbf{u}}} \text{ for } \tilde{{\mathbf{u}}} \le \f{1}{\ep},
\\ & \f{1}{\ep} \text{ for } \tilde{{\mathbf{u}}} \ge \f{2}{\ep}.
\end{cases}$$ We easily see that ${\mathbf{u}}_\ep \in L^\infty(\Omega_T)^2$. Furthermore, we have $\tilde{\mathbf{u}}_\ep \rightarrow \tilde {\mathbf{u}}$ a.e. and ${ \left| \tilde{\mathbf{u}}_\ep \right|}_{L^4} \le { \left| \tilde{\mathbf{u}}\right|}_{L^4}<\infty$, and we obtain by the Lebesque dominated convergence that $\tilde{\mathbf{u}}_\ep$ converges to $\tilde{\mathbf{u}}$ in $L^4(\Omega_T)^2$. Finally, we easily see the following by straightforward calculations $$\label{u ep bounded by u}
{\lVert\tilde{\mathbf{u}}_\ep\rVert}_{L^\infty(0,T;H^1(\Omega)^2)} \le \kappa {\lVert\tilde{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1(\Omega)^2)},$$ where $\kappa$ depends on the maximum value of $\theta_\ep'$ which is independent of $\ep$.
We then let ${\boldsymbol \varphi}_\ep = (\varphi^\ep_u, \varphi^\ep_v) $ satisfy the following approximate system $$\label{adj eq ep}
\begin{cases}
&-\partial_t{\boldsymbol \varphi}_\ep - {\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep)^T \Delta{\boldsymbol \varphi}_\ep+ {\mathbf{Q}}(\tilde{{\mathbf{u}}}_\ep)^T{\boldsymbol \varphi}_\ep = {\boldsymbol \varphi}_\ep \text{ in }\Omega_T,
\\& \partial_\nu {\boldsymbol \varphi}_\ep = {\mathbf{0}}\text{ or } {\boldsymbol \varphi}_\ep = {\mathbf{0}}\text{ on }\partial \Omega\times(0,T),
\\& {\boldsymbol \varphi}_\ep(T) ={\boldsymbol{\chi}}({\mathbf{x}})\text{ in }\Omega.
\end{cases}$$ We know that $\tilde{\mathbf{u}}_\ep \in L^\infty(\Omega_T)^2$ which yields ${\mathbf{P}}(\tilde{\mathbf{u}}_\ep) \in L^\infty(\Omega_T)^4$, and this in turn implies that $$\left({\mathbf{P}}(\tilde{\mathbf{u}}_\ep){\boldsymbol{\xi}}\right) \cdot {\boldsymbol{\xi}}\le \kappa(\ep) { \left| {\boldsymbol{\xi}}\right|}^2,$$ where $\kappa(\ep)$ is a constant depending on $\ep$. This bound from above of ${\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep)$ and its bound from the below in [[[(\[P positive definite\])]{}]{}]{} give the uniform parabolic condition for the approximate system [[[(\[adj eq phi\])]{}]{}]{}. The existence of a smooth function ${\boldsymbol \varphi}_\ep$ is hence given by the classical theory of the equations of parabolic type, see e.g. [@Lad68 Theorem 5.1]. We now bound the approximate solutions ${\boldsymbol \varphi}_\ep$ independently of $\ep$:
\[lem: phi bound\] Assume that $d\le4$ and $\tilde{\mathbf{u}}\in L^\infty (0,T;H^1(\Omega)^2)$. We then have the following a priori bounds independent of $\ep$ for the solution ${\boldsymbol \varphi}_\ep$ of [[[(\[adj eq ep\])]{}]{}]{}:
$$\begin{aligned}
& \label{phi bound} \sup_{t\in [0,T]} {\lVert{\boldsymbol \varphi}_\ep\rVert}_{H^1(\Omega)^2} \le \kappa {\lVert{\boldsymbol{\chi}}\rVert}_{H^1(\Omega)^2}, \\
& \int_0^T (1+\tilde u_\ep+\tilde v_\ep) { \left| \Delta {\boldsymbol \varphi}_\ep \right|}^2 dt \le \kappa {\lVert{\boldsymbol{\chi}}\rVert}_{H^1(\Omega)^2}, \label{phi bound1}
\end{aligned}$$
and $$\label{time der bound}
{\lVert\partial_t {\boldsymbol \varphi}_\ep\rVert}_{ L^\f43(\Omega_T)} \le \kappa {\lVert{\boldsymbol{\chi}}\rVert}.$$
Here, in this lemma, $\kappa$ depends on ${\lVert{\mathbf{u}}\rVert}_{ L^\infty (0,T;H^1(\Omega)^2)}$ and on the coefficients but is independent of $\ep$.
#### Proof of Lemma \[lem: phi bound\]:
Multiplying [[[(\[adj eq ep\])]{}]{}]{} by $ {\boldsymbol \varphi}_\ep$, we find $$\label{phi 1}
-\f12 \f{d}{dt}{ \left| {\boldsymbol \varphi}_\ep \right|}^2 - {\left\langle {\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep)^T\Delta {\boldsymbol \varphi}_\ep, {\boldsymbol \varphi}_\ep \right\rangle} + {\left\langle {\mathbf{Q}}(\tilde{{\mathbf{u}}}_\ep)^T{\boldsymbol \varphi}_\ep, {\boldsymbol \varphi}_\ep \right\rangle} ={ \left| {\boldsymbol \varphi}_\ep \right|}^2.$$ Multiplying [[[(\[adj eq ep\])]{}]{}]{} by $-\Delta {\boldsymbol \varphi}_\ep$, we also find after integration by parts $$\label{phi 2}
-\f12 \f{d}{dt}{ \left| \grad {\boldsymbol \varphi}_\ep \right|}^2 + {\left\langle {\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep)^T\Delta {\boldsymbol \varphi}_\ep, \Delta {\boldsymbol \varphi}_\ep \right\rangle} - {\left\langle {\mathbf{Q}}(\tilde{{\mathbf{u}}}_\ep)^T{\boldsymbol \varphi}_\ep, \Delta {\boldsymbol \varphi}_\ep \right\rangle} ={ \left| \grad {\boldsymbol \varphi}_\ep \right|}^2$$ Adding equations [[[(\[phi 1\])]{}]{}]{} and [[[(\[phi 2\])]{}]{}]{} and regrouping the terms, we find $$\begin{gathered}
\label{phi 3}
-\f12\f{d}{dt}\left( { \left| {\boldsymbol \varphi}_\ep \right|}^2+{ \left| \grad {\boldsymbol \varphi}_\ep \right|}^2\right) + {\left\langle {\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep)^T\Delta {\boldsymbol \varphi}_\ep, \Delta {\boldsymbol \varphi}_\ep \right\rangle} \\= {\left\langle \Big({\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep) + {\mathbf{Q}}(\tilde{\mathbf{u}}_\ep)^T\Big){\boldsymbol \varphi}_\ep, \Delta {\boldsymbol \varphi}_\ep \right\rangle} - {\left\langle {\mathbf{Q}}(\tilde{{\mathbf{u}}}_\ep)^T{\boldsymbol \varphi}_\ep, {\boldsymbol \varphi}_\ep \right\rangle}+ { \left| {\boldsymbol \varphi}_\ep \right|}^2+{ \left| \grad {\boldsymbol \varphi}_\ep \right|}^2.\end{gathered}$$ We bound the first two terms on the right hand side of [[[(\[phi 3\])]{}]{}]{} as follows:
- We first bound the easier term ${\left\langle {\mathbf{Q}}(\tilde{{\mathbf{u}}}_\ep)^T{\boldsymbol \varphi}_\ep, {\boldsymbol \varphi}_\ep \right\rangle}$ using the Hölder inequality for three functions with powers $(2,4,4)$, the Sobolev embedding from $H^1$ to $L^4$ in dimension $d\le 4$, and [[[(\[u ep bounded by u\])]{}]{}]{}: $$\begin{aligned}
& { \left| {\left\langle {\mathbf{Q}}(\tilde{{\mathbf{u}}}_\ep)^T{\boldsymbol \varphi}_\ep, {\boldsymbol \varphi}_\ep \right\rangle} \right|} \le c_0{ \left| \tilde{\mathbf{u}}_\ep \right|}_{L^2} { \left| {\boldsymbol \varphi}_\ep \right|}_{L^4}^2 \le c_1 {\lVert\tilde{\mathbf{u}}_\ep\rVert}_{H^1} {\lVert {\boldsymbol \varphi}_\ep\rVert}_{H^1}^2 \le \kappa_1\left( {\lVert\tilde{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}\right) {\lVert {\boldsymbol \varphi}_\ep\rVert}_{H^1}^2 .
\end{aligned}$$ Here $ \kappa_1\left( {\lVert\tilde{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}\right) $ is a constant which depends on $ {\lVert\tilde{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)} $ but not on $\ep$.
- We now bound the term ${\left\langle \Big({\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep) + {\mathbf{Q}}(\tilde{\mathbf{u}}_\ep)^T\Big){\boldsymbol \varphi}_\ep, \Delta {\boldsymbol \varphi}_\ep \right\rangle} $ using Hölder’s inequality for three functions with powers $(4,4,2)$, the previously used Sobolev embedding from $H^1$ to $L^4$ which assumes $d\le 4$, the Young inequality, and [[[(\[u ep bounded by u\])]{}]{}]{}: $$\begin{gathered}
{ \left| {\left\langle \Big({\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep) + {\mathbf{Q}}(\tilde{\mathbf{u}}_\ep)^T\Big){\boldsymbol \varphi}_\ep, \Delta {\boldsymbol \varphi}_\ep \right\rangle} \right|} \le c_0 { \left| \tilde{\mathbf{u}}_\ep \right|}_{L^4} { \left| {\boldsymbol \varphi}_\ep \right|}_{L^4} { \left| \Delta{\boldsymbol \varphi}_\ep \right|}_{L^2}
\\ \le c_1 {\lVert\tilde{\mathbf{u}}_\ep\rVert}_{H^1} {\lVert{\boldsymbol \varphi}_\ep\rVert}_{H^1} { \left| \Delta{\boldsymbol \varphi}_\ep \right|}_{L^2} \hspace{4cm}
\\ \le c_1 {\lVert\tilde{\mathbf{u}}_\ep\rVert}_{L^\infty(0,T;H^1)} {\lVert{\boldsymbol \varphi}_\ep\rVert}_{H^1} { \left| \Delta{\boldsymbol \varphi}_\ep \right|}_{L^2}
\\ \le \kappa_2 \left({\lVert\tilde{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}\right){\lVert{\boldsymbol \varphi}_\ep\rVert}_{H^1}^2 + \f{d_0}{2}{ \left| \Delta{\boldsymbol \varphi}_\ep \right|}_{L^2} ^2,
\end{gathered}$$
where $d_0=\min(d_1,d_2)$ and $ \kappa_2\left( {\lVert\tilde{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}\right) $ is independent of $\ep$.
Using these two bounds in [[[(\[phi 3\])]{}]{}]{}, we find $$\begin{gathered}
-\f12\f{d}{dt}\left( { \left| {\boldsymbol \varphi}_\ep \right|}^2+{ \left| \grad {\boldsymbol \varphi}_\ep \right|}^2\right) + {\left\langle {\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep)^T\Delta {\boldsymbol \varphi}_\ep, \Delta {\boldsymbol \varphi}_\ep \right\rangle} \\\le \kappa \left({\lVert\tilde{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}\right) \left( { \left| {\boldsymbol \varphi}_\ep \right|}^2+{ \left| \grad {\boldsymbol \varphi}_\ep \right|}^2\right) + \f{d_0}{2}{ \left| \Delta{\boldsymbol \varphi}_\ep \right|}_{L^2} ^2,\end{gathered}$$ where $ \kappa\left( {\lVert\tilde{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}\right) $ is a constant which depends on $ {\lVert\tilde{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)} $ but is independent of $\ep$. Thanks to the positivity of ${\mathbf{P}}(\star)$ in [[[(\[P\])]{}]{}]{}, we have using [[[(\[P positive definite\])]{}]{}]{} $$\begin{gathered}
\label{phi 4}
-\f12\f{d}{dt}\left( { \left| {\boldsymbol \varphi}_\ep \right|}^2+{ \left| \grad {\boldsymbol \varphi}_\ep \right|}^2\right) + \left(\f{d_0}{2} +\alpha(\tilde{u}_\ep+\tilde v_\ep)\right) { \left| \Delta {\boldsymbol \varphi}_\ep \right|}^2
\\ \le \kappa \left({\lVert\tilde{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}\right) \left( { \left| {\boldsymbol \varphi}_\ep \right|}^2+{ \left| \grad {\boldsymbol \varphi}_\ep \right|}^2\right) .\end{gathered}$$ This implies $$\label{3.10b}
-\f{d}{dt}\left( { \left| {\boldsymbol \varphi}_\ep \right|}^2+{ \left| \grad {\boldsymbol \varphi}_\ep \right|}^2\right) + \alpha(1+\tilde u_\ep +\tilde v_\ep){ \left| \Delta {\boldsymbol \varphi}_\ep \right|}^2 \le \kappa \left( { \left| {\boldsymbol \varphi}_\ep \right|}^2+{ \left| \grad {\boldsymbol \varphi}_\ep \right|}^2 \right),$$ where again $\kappa = \kappa \left({\lVert\tilde{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}\right)$.
Recall that ${\boldsymbol \varphi}(T) = {\boldsymbol{\chi}}\in H^1(\Omega)^2$. By multiplying [[[(\[3.10b\])]{}]{}]{} by $e^{2t}$ and integrating over $[t,T]$ for $t\in [0,T]$, we infer [[[(\[phi bound\])]{}]{}]{} and [[[(\[phi bound1\])]{}]{}]{}.
Now, to derive the bound independent of $\ep$ for $\partial_t{\boldsymbol \varphi}_\ep$, we write using [[[(\[adj eq ep\])]{}]{}]{}$_1$: $$\label{eq}
{\lVert\partial_t{\boldsymbol \varphi}_\ep \rVert}_{L^\f43} = {\lVert{\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep)^T\Delta {\boldsymbol \varphi}_\ep - {\mathbf{Q}}(\tilde{{\mathbf{u}}}_\ep)^T{\boldsymbol \varphi}_\ep +{\boldsymbol \varphi}_\ep \rVert}_{L^\f43}.$$ We bound the most challenging norm term ${\lVert{\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep)^T\Delta {\boldsymbol \varphi}_\ep\rVert}_{L^\f43} $ on the right hand side of [[[(\[eq\])]{}]{}]{}. We consider a function ${\mathbf{z}}\in L^4(\Omega_T)^2$ and write $$\begin{gathered}
\int_{\Omega_T}{\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep)^T\Delta {\boldsymbol \varphi}_\ep {\mathbf{z}}\,dxdt \\
\le \left( \max_{i=1,2}d_i { \left| \Omega \right|}^\f14 + 2\max_{i,j=1,2}a_{ij} \Big({\lVertu_\ep\rVert}_{L^4}+{\lVertv_\ep\rVert}_{L^4}\Big)\right) \left({\lVert\Delta \varphi^u_\ep\rVert}_{L^2} + {\lVert\Delta \varphi^v_\ep\rVert}_{L^2}\right) {\lVert{\mathbf{z}}\rVert}_{L^4}.\end{gathered}$$ Observing that ${\lVertu_\ep\rVert}_{L^4} \le {\lVertu\rVert}_{L^4} \le {\lVertu\rVert}_{L^\infty(0,T;H^1)}$, a similar bound for ${\lVertv_\ep\rVert}_{L^4}$, and using [[[(\[phi bound1\])]{}]{}]{}, we find the following bound for any ${\mathbf{z}}\in L^4(\Omega_T)$ $$\int_{\Omega_T}{\mathbf{P}}(\tilde{{\mathbf{u}}}_\ep)^T\Delta {\boldsymbol \varphi}_\ep {\mathbf{z}}\,dxdt \le \kappa({\lVert{\mathbf{u}}\rVert}_{L^4}) { \left| \Delta {\boldsymbol \varphi}_\ep \right|}{\lVert{\mathbf{z}}\rVert}_{L^4}\le \kappa({\lVert{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}) {\lVert{\boldsymbol{\chi}}\rVert}{\lVert{\mathbf{z}}\rVert}_{L^4},$$ where $\kappa$ depends on ${\lVert{\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}$ and on the coefficients $d_i, a_{ij}$ but is independent of $\ep.$ This gives the a priori bound [[[(\[time der bound\])]{}]{}]{}.
### Passage to the limit for the solutions of the approximate systems
We now pass to the limit as $\ep \rightarrow 0$ in the approximate adjoint system [[[(\[adj eq ep\])]{}]{}]{}. We have from the a priori estimates [[[(\[phi bound\])]{}]{}]{} – [[[(\[time der bound\])]{}]{}]{} that there exist a subsequence of ${\boldsymbol \varphi}_\ep$, still denoted by ${\boldsymbol \varphi}_\ep$, such that as $\ep \rightarrow 0$
\[phi conv\] $$\begin{aligned}
&\text{{{\color{RoyalBlue}\ding{93}}}}\quad{\boldsymbol \varphi}_\ep \rightharpoonup{\boldsymbol \varphi}\text{ in }L^\infty(0,T;H^1(\Omega)^2) \text{ weak-star},\\
\label{phi convb}&\text{{{\color{RoyalBlue}\ding{93}}}}\quad\Delta{\boldsymbol \varphi}_\ep \rightharpoonup \Delta {\boldsymbol \varphi}\text{ in }L^2(\Omega_T)^2 \text{ weakly},\\
\label{phi conv} &\text{{{\color{RoyalBlue}\ding{93}}}}\quad\partial_t {\boldsymbol \varphi}_\ep \rightharpoonup\partial_t{\boldsymbol \varphi}\text{ in }L^\f43(\Omega_T)^2 \text{ weakly}.
\end{aligned}$$
We then pass to the limit term by term in [[[(\[adj eq ep\])]{}]{}]{}, where the most challenging product term ${\mathbf{P}}(\tilde{\mathbf{u}}_\ep)\Delta{\boldsymbol \varphi}_\ep$ is treated as follows: for any ${\mathbf{z}}\in L^4(\Omega_T)^2$, we write $$\begin{gathered}
\int_{\Omega_T}\left({\mathbf{P}}(\tilde{\boldsymbol \varphi}_\ep)^T\Delta{\boldsymbol \varphi}_\ep - {\mathbf{P}}(\tilde{\boldsymbol \varphi})^T\Delta{\boldsymbol \varphi}\right){\mathbf{z}}\,dxdt \\
= {\begingroup
\color{RoyalBlue}
\underbrace{\color{black}\int_{\Omega_T}\Delta{\boldsymbol \varphi}_\ep\left({\mathbf{P}}(\tilde{\mathbf{u}}_\ep) - {\mathbf{P}}(\tilde{\mathbf{u}})\right){\mathbf{z}}\,dxdt}_{T_\ep^1}
\endgroup
} + {\begingroup
\color{RoyalBlue}
\underbrace{\color{black}\int_{\Omega_T}\left(\Delta{\boldsymbol \varphi}_\ep - \Delta{\boldsymbol \varphi}\right){\mathbf{P}}(\tilde{\mathbf{u}}){\mathbf{z}}\,dxdt }_{T_\ep^2}
\endgroup
}.\end{gathered}$$
1. We first deal with the easier term $T_\ep^2$. We easily see that $T_\ep^2 \rightarrow 0$ as $\ep\rightarrow0$ thanks to [[[(\[phi convb\])]{}]{}]{} and the facts that ${\mathbf{P}}(\tilde{\mathbf{u}})\in L^4(\Omega_T)^4,{\mathbf{z}}\in L^4(\Omega_T)^2$ which gives ${\mathbf{P}}(\tilde{\mathbf{u}}){\mathbf{z}}\in L^2(\Omega_T)^2$.
2. For $T_\ep^1$, as $\ep \rightarrow 0$, we know that $\tilde{\mathbf{u}}_\ep \rightarrow \tilde{\mathbf{u}}$ in $L^4(\Omega_T)^2$ strongly which gives ${\mathbf{P}}(\tilde{\mathbf{u}}_\ep) \rightarrow {\mathbf{P}}(\tilde{\mathbf{u}})$ in $L^4(\Omega_T)^4$ strongly. Thus $\left({\mathbf{P}}(\tilde{\mathbf{u}}_\ep) - {\mathbf{P}}(\tilde{\mathbf{u}})\right){\mathbf{z}}$ converges to ${\mathbf{0}}$ in $L^2(\Omega_T)^2$ strongly. Since $\Delta{\boldsymbol \varphi}_\ep$ is bounded in $L^2(\Omega_T)^2$ (thanks to [[[(\[phi bound\])]{}]{}]{}$_2$), we conclude that $T_\ep^1\rightarrow 0$ as $\ep\rightarrow 0$.
We hence have ${\mathbf{P}}(\tilde{\mathbf{u}}_\ep)\Delta{\boldsymbol \varphi}_\ep \rightharpoonup{\mathbf{P}}(\tilde{\mathbf{u}})\Delta{\boldsymbol \varphi}$ weakly in $L^\f43(\Omega_T)^2$ as $\ep\rightarrow 0$. We thus conclude that ${\boldsymbol \varphi}$ is a weak solution of [[[(\[adj eq phi\])]{}]{}]{} as below
\[thm: phi exist\] Under the assumptions that $d\le 4$ and $\tilde{\mathbf{u}}\in L^\infty(0,T;H^1(\Omega)^2)$, the adjoint system [[[(\[adj eq phi\])]{}]{}]{} admits a solution ${\boldsymbol \varphi}$ in $L^\infty(0,T;H^1(\Omega)) \cap L^2(0,T;H^2(\Omega)^2)$ such that $\partial_t {\boldsymbol \varphi}\in L^\f43(\Omega_T)^2$.
We now resume the work of showing the uniqueness of solution ${\mathbf{u}}$ of [[[(\[SKT vector\])]{}]{}]{}:
Uniqueness result for the SKT system
------------------------------------
We recall that ${\mathbf{u}}_1,\,{\mathbf{u}}_2$ are two solutions of [[[(\[ui eq\])]{}]{}]{}, and we have written $\bar{{\mathbf{u}}} = {\mathbf{u}}_1 -{\mathbf{u}}_2$; we will eventually show that $\bar{{\mathbf{u}}}={\mathbf{0}}$ for a.e. ${\mathbf{x}}\in \Omega$ and $t>0$. We also recall that that $\bar{{\mathbf{u}}}$ satisfies [[[(\[u diff eq\])]{}]{}]{}.
Using the existence result of the adjoint problem in Theorem \[thm: phi exist\], we have a solution ${\boldsymbol \varphi}=(\phi^u,\phi^v)\in L^2(0,T;H^2(\Omega)^2)$ of [[[(\[adj eq phi\])]{}]{}]{} with $\partial_t {\boldsymbol \varphi}\in L^\f43(\Omega_T)$.
We henceforth infer from [[[(\[skt inner\])]{}]{}]{} and [[[(\[adj eq phi\])]{}]{}]{} that $$\label{eq u bar phi}
{\left\langle \bar{{\mathbf{u}}}, {\boldsymbol \varphi}\right\rangle}_t+{\left\langle \bar{{\mathbf{u}}}, {\boldsymbol \varphi}\right\rangle} = {\left\langle {\boldsymbol{\ell}}(\bar{{\mathbf{u}}}), {\boldsymbol \varphi}\right\rangle}.$$ We look at the first component in [[[(\[eq u bar phi\])]{}]{}]{}: $${\left\langle u_1(t)-u_2(t), \varphi^u \right\rangle}_t = (a_1-1) {\left\langle u_1(t)-u_2(t)), \varphi^u \right\rangle} .$$ Multiplying the equation by $e^{-(a_1-1)t}$ and integrating over the time interval $[0,T]$, we find $${\left\langle u_1(T)-u_2(T), \chi^u \right\rangle} = 0.$$ This is true for any $\chi^u\in H^1(\Omega)$ and we thus find $u_1(T)=u_2(T)$ for a.e ${\mathbf{x}}\in \Omega$. The argument is also valid for any other time $t<T$ which gives $u_1(t)=u_2(t)$ a.e..
Similarly, we have $v_1(t) = v_2(t)$ a.e..
We have thus shown the following result:
In space dimension $d\le 4$, the SKT system [[[(\[SKT vector\])]{}]{}]{} admits at most one weak solution ${\mathbf{u}}\ge {\mathbf{0}}$ such that \[thm:uniq\] $${\mathbf{u}}\in L^\infty(0,T;H^1(\Omega)^2), \text{ and }\partial_t {\mathbf{u}}\in L^\f43(\Omega_T)^2.$$
Global well-posedness for the SKT system\[sec:global\]
======================================================
this section, we assume that $d\le 4$ and show that our uniqueness result in Section \[sec:uniq\] yields the global well-posedness for solutions of the SKT system [[[(\[SKT vector\])]{}]{}]{} with the following initial datum conditions $$\label{u0 cond}
{\mathbf{u}}_0 \in L^2(\Omega)^2 \text{ and } \grad {\mathbf{p}}({\mathbf{u}}_0) \in L^2(\Omega)^4.$$
Thanks to the existence result in our prior work [@PT16], whose main result is stated as Theorem \[thm: existence\], we see that for all $T>0$, under the assumptions that the space dimension $d\le 4$ and the initial data satisfies [[[(\[u0 cond\])]{}]{}]{}, the SKT system [[[(\[SKT vector\])]{}]{}]{} possesses solutions ${\mathbf{u}}\in L^\infty(0,T;H^1(\Omega)^2)$ with $\partial_t{\mathbf{u}}\in L^\f43(\Omega_T)^\f43$ as consequences of [[[(\[3.38c\])]{}]{}]{} and [[[(\[3.36\])]{}]{}]{}. Theorem \[thm:uniq\] thus applies and gives the uniqueness of such a solution ${\mathbf{u}}$ of [[[(\[SKT vector\])]{}]{}]{}. We then conclude that the solution ${\mathbf{u}}$ exists globally and uniquely.
Our main result in this section is as follows:
Suppose that $d\le 4$, that ${\mathbf{u}}_0$ satisfies [[[(\[u0 cond\])]{}]{}]{}, and that the coefficients satisfy [[[(\[coef cond\])]{}]{}]{}. The system [[[(\[SKT vector\])]{}]{}]{} possesses a unique global solution ${\mathbf{u}}\in L^\infty(0,\infty;H^1(\Omega)^2) $ with $\partial_t {\mathbf{u}}\in L^2(\Omega\times(0,\infty))^2$. Furthermore, the mapping ${\mathbf{u}}_0 \mapsto {\mathbf{u}}$ is continuous from $L^q(\Omega)$ into $L^2(\Omega)$ endowed with the norm ${ \left| \star \right|}_w$ $${ \left| \star \right|}_w = \sup_{v\in H^1} \f{{\left\langle \star, v \right\rangle}}{{\lVertv\rVert}}.$$ Here $q=\max(2d/(6-d),4d/(d+2))$.
To show the continuous dependance on the initial data, we suppose that ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ are two solutions with initial data ${\mathbf{u}}_1(0),{\mathbf{u}}_2(0) $ satisfying [[[(\[u0 cond\])]{}]{}]{}. We proceed as in Section \[sec:uniq\] by denoting $\bar{\mathbf{u}}= {\mathbf{u}}_1 -{\mathbf{u}}_2$, $\tilde {\mathbf{u}}= ({\mathbf{u}}_1+{\mathbf{u}}_2)/2$, and recall from [[[(\[skt inner\])]{}]{}]{} that $$\label{skt inner2}
{\left\langle \bar{{\mathbf{u}}}, {\boldsymbol \varphi}\right\rangle}_t-{\left\langle \bar{{\mathbf{u}}}, {\boldsymbol \varphi}_t \right\rangle}-{\left\langle \bar{{\mathbf{u}}}, {\mathbf{P}}(\tilde{{\mathbf{u}}})^T\Delta{\boldsymbol \varphi}\right\rangle} +{\left\langle \bar{{\mathbf{u}}}, {\mathbf{Q}}(\tilde{{\mathbf{u}}})^T{\boldsymbol \varphi}\right\rangle} = {\left\langle {\boldsymbol{\ell}}(\bar{{\mathbf{u}}}), {\boldsymbol \varphi}\right\rangle},$$ where ${\boldsymbol \varphi}$ solves the following adjoint problem $$\label{adj eq phi2}
\begin{cases}
&-\partial_t{\boldsymbol \varphi}- {\mathbf{P}}(\tilde{{\mathbf{u}}})^T \Delta{\boldsymbol \varphi}+ {\mathbf{Q}}(\tilde{{\mathbf{u}}})^T{\boldsymbol \varphi}= {\boldsymbol{\ell}}({\boldsymbol \varphi}) \text{ in }\Omega_\tau =\Omega \times (0,\tau),
\\& \partial_\nu {\boldsymbol \varphi}= {\mathbf{0}}\text{ or } {\boldsymbol \varphi}= {\mathbf{0}}\text{ on }\partial \Omega\times(0,\tau),
\\& {\boldsymbol \varphi}(\tau) ={\boldsymbol{\chi}}\text{ in }\Omega,
\end{cases}$$ for ${\boldsymbol{\chi}}({\mathbf{x}})= (\chi^u({\mathbf{x}}),\chi^v({\mathbf{x}})) \in H^1(\Omega)^2$ (arbitrary) with appropriate compatible boundary conditions.
The existence of solution that satisfies the following a priori estimates was proven in Lemma \[lem: phi bound\] in Section \[sec:adj\]:
\[lem: phi bound2\] Assume that $d\le4$ and $\tilde{\mathbf{u}}=(\tilde u,\tilde v)\in L^\infty (0,T;H^1(\Omega)^2)$. We then have the following a priori bounds independent of $\tau\in [0,T]$ for the solutions ${\boldsymbol \varphi}$ of [[[(\[adj eq phi2\])]{}]{}]{}:
$$\begin{aligned}
& \label{phi bounda} \sup_{t\in [0,\tau]} {\lVert{\boldsymbol \varphi}\rVert}_{H^1(\Omega)^2} \le \kappa {\lVert{\boldsymbol{\chi}}\rVert}_{H^1(\Omega)^2}, \\
& \int_0^\tau (1+\tilde u+\tilde v) { \left| \Delta {\boldsymbol \varphi}\right|}^2 dt \le \kappa {\lVert{\boldsymbol{\chi}}\rVert}_{H^1(\Omega)^2}, \label{phi boundb}
\end{aligned}$$
Here, $\kappa$ depends on ${\lVert\tilde{\mathbf{u}}\rVert}_{ L^\infty (0,T;H^1(\Omega)^2)},T$ and on the coefficients but is independent of $\tau$.
We now continue to show the continuous dependance of ${\mathbf{u}}$ on the data. We find from equations [[[(\[skt inner2\])]{}]{}]{} and [[[(\[adj eq phi2\])]{}]{}]{} that $$\label{eq u bar phi2}
{\left\langle \bar{{\mathbf{u}}}, {\boldsymbol \varphi}\right\rangle}_t = 0.$$ Thus $${\left\langle \bar{\mathbf{u}}(\tau), {\boldsymbol{\chi}}\right\rangle} = {\left\langle \bar{\mathbf{u}}(0), {\boldsymbol \varphi}(0) \right\rangle},$$ where we have used ${\boldsymbol \varphi}(\tau) ={\boldsymbol{\chi}}$.
We now use [[[(\[adj eq phi2\])]{}]{}]{}$_1$ and find $$\begin{aligned}
\nonumber
{\left\langle \bar{\mathbf{u}}(\tau), {\boldsymbol{\chi}}\right\rangle} & = {\left\langle \bar{\mathbf{u}}(0), {\boldsymbol{\chi}}+ \int_0^\tau \big[ {\mathbf{P}}(\tilde{\mathbf{u}})^T \Delta {\boldsymbol \varphi}-{\mathbf{Q}}(\tilde{\mathbf{u}}){\boldsymbol \varphi}+ {\boldsymbol{\ell}}({\boldsymbol \varphi})\big]dt \right\rangle}
\\& \label{4.5}= {\left\langle \bar{\mathbf{u}}(0), {\boldsymbol{\chi}}\right\rangle} + {\left\langle \bar{\mathbf{u}}(0), \int_0^\tau \big[{\mathbf{P}}(\tilde{\mathbf{u}})^T \Delta {\boldsymbol \varphi}-{\mathbf{Q}}(\tilde{\mathbf{u}}){\boldsymbol \varphi}+ {\boldsymbol{\ell}}({\boldsymbol \varphi})\big] dt \right\rangle}\end{aligned}$$ We next bound the typical terms on the RHS of [[[(\[4.5\])]{}]{}]{}:
1. We first bound a typical term $ {\left\langle \bar u(0), \tilde u \Delta \phi^u \right\rangle}$ in ${\left\langle \bar{\mathbf{u}}(0), {\mathbf{P}}^T(\tilde {\mathbf{u}}) \Delta {\boldsymbol \varphi}\right\rangle}$ as follows: $$\begin{gathered}
{\left\langle \bar u(0), \tilde u \Delta \phi^u \right\rangle} = {\left\langle {\begingroup
\color{RoyalBlue}
\underbrace{\color{black}\bar u(0)}_{\in L^\f{4d}{d+2}}
\endgroup
}, {\begingroup
\color{RoyalBlue}
\underbrace{\color{black}\tilde u^\f12}_{\in L^\f{4d}{d-2}}
\endgroup
} {\begingroup
\color{RoyalBlue}
\underbrace{\color{black}\tilde u^\f12 \Delta \phi^u}_{\in L^2}
\endgroup
} \right\rangle} \le { \left| \bar u(0) \right|}_{L^\f{4d}{d+2}} { \left| \tilde u \right|}_{L^\f{2d}{d-2}}^\f12 { \left| \tilde u^\f12 \Delta \phi^u \right|}_{L^2}
\\ \le c { \left| \bar u(0) \right|}_{L^\f{4d}{d+2}} { \left| \tilde u \right|}_{L^\infty(0,T;H^1)}^\f12 { \left| \tilde u^\f12 \Delta \phi^u \right|}_{L^2}.\end{gathered}$$ Thus, by [[[(\[phi boundb\])]{}]{}]{}, we find $$\begin{gathered}
\label{4.7}
{\left\langle \bar {\mathbf{u}}(0), \int_0^\tau\tilde {\mathbf{u}}\Delta {\boldsymbol \varphi}\,dt \right\rangle} \le c { \left| \bar {\mathbf{u}}(0) \right|}_{L^\f{4d}{d+2}} { \left| \tilde {\mathbf{u}}\right|}_{L^\infty(0,T;H^1)}^\f12 \int_0^\tau { \left| \tilde {\mathbf{u}}^\f12 \Delta {\boldsymbol \varphi}\right|}_{L^2}
\\ \le c T^\f12 { \left| \bar {\mathbf{u}}(0) \right|}_{L^\f{4d}{d+2}} { \left| \tilde {\mathbf{u}}\right|}_{L^\infty(0,T;H^1)}^\f12 {\lVert{\boldsymbol{\chi}}\rVert}_{H^1}^\f12.\end{gathered}$$
2. We now bound the term $\dps{\left\langle \bar {\mathbf{u}}(0), \int_0^\tau {\mathbf{Q}}(\tilde {\mathbf{u}}) {\boldsymbol \varphi}\,dt \right\rangle}$ by bounding its typical term $\dps \int_0^\tau{\left\langle \bar u(0), \tilde u^2 \phi^u \right\rangle}dt $:
$$\begin{gathered}
{\left\langle {\begingroup
\color{RoyalBlue}
\underbrace{\color{black}\bar u(0)}_{\in L^\f{2d}{6-d} }
\endgroup
} , {\begingroup
\color{RoyalBlue}
\underbrace{\color{black}\tilde u^2}_{L^\f{d}{d-2}}
\endgroup
} {\begingroup
\color{RoyalBlue}
\underbrace{\color{black}\phi^u}_{\in L^\f{2d}{d-2}}
\endgroup
} \right\rangle} \le { \left| \bar u(0) \right|}_{L^\f{2d}{6-d}} { \left| \tilde u \right|}_{L^\f{2d}{d-2}}^2 { \left| \phi^u \right|}_{L^\f{2d}{d-2}} \\\le c { \left| \bar u(0) \right|}_{L^\f{2d}{6-d}} {\lVert\tilde u\rVert}_{L^\infty(0,T;H^1)}^2 {\lVert\phi^u\rVert}_{L^\infty(0,T;H^1)}.\end{gathered}$$
Therefore, by [[[(\[phi bounda\])]{}]{}]{}, we find $$\label{4.8}
{\left\langle \bar {\mathbf{u}}(0), \int_0^\tau {\mathbf{Q}}(\tilde {\mathbf{u}}) {\boldsymbol \varphi}\,dt \right\rangle} \le cT { \left| \bar {\mathbf{u}}(0) \right|}_{L^\f{2d}{6-d}} {\lVert\tilde {\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}^2 {\lVert{\boldsymbol{\chi}}\rVert}_{H^1}.$$
3. We finally bound $$\label{4.9}
{\left\langle \bar {\mathbf{u}}(0), \int_0^\tau {\boldsymbol{\ell}}({\boldsymbol \varphi})dt \right\rangle} \le c { \left| \bar {\mathbf{u}}(0) \right|} \int_0^ \tau { \left| {\boldsymbol \varphi}\right|}dt \le c T^\f12 { \left| \bar {\mathbf{u}}(0) \right|} {\lVert{\boldsymbol{\chi}}\rVert}^\f12.$$
We therefore infer from [[[(\[4.5\])]{}]{}]{}–[[[(\[4.9\])]{}]{}]{} that $${\left\langle \bar{\mathbf{u}}(\tau), {\boldsymbol{\chi}}\right\rangle} \le {\left\langle \bar{{\mathbf{u}}}(0), {\boldsymbol{\chi}}\right\rangle} + \kappa\left( \left[{ \left| \tilde {\mathbf{u}}\right|}_{L^\infty(0,T;H^1)}^\f12 +1\right]{\lVert{\boldsymbol{\chi}}\rVert}_{H^1}^\f12+ {\lVert\tilde {\mathbf{u}}\rVert}_{L^\infty(0,T;H^1)}^2 {\lVert{\boldsymbol{\chi}}\rVert}_{H^1} \right) { \left| \bar{{\mathbf{u}}}(0) \right|}_{L^q(\Omega)^2} ,$$ where $\kappa = \kappa(T)$ is independent of $\tau$ and $q=\max(2d/(6-d),4d/(d+2))$.
By taking the supremum over ${\boldsymbol{\chi}}$ with ${\lVert{\boldsymbol{\chi}}\rVert}= {\lVert\chi\rVert}_{H^1(\Omega)^2} \le 1$, we conclude that $$\label{initial dep}
\sup_{{\boldsymbol{\chi}}\in H^1: {\lVert{\boldsymbol{\chi}}\rVert} \le 1}{\left\langle {\mathbf{u}}_1(\tau) -{\mathbf{u}}_2(\tau), {\boldsymbol{\chi}}\right\rangle} \le { \left| {\mathbf{u}}_1(0) -{\mathbf{u}}_2(0) \right|} +\kappa { \left| {\mathbf{u}}_1(0) -{\mathbf{u}}_2(0) \right|}_{L^q(\Omega)^2},$$ where $\kappa =\kappa (T,{\lVert{\mathbf{u}}_1+ {\mathbf{u}}_2\rVert}_{L^\infty(0,T;H^1)})$ .
Appendices {#appendices .unnumbered}
==========
A technical lemma
=================
\[lem: u diff\] Suppose that ${\mathbf{p}},{\mathbf{q}}$ are as in [[[(\[pi qi li\])]{}]{}]{} and ${\mathbf{P}},{\mathbf{Q}}$ are as in [[[(\[P Q jac\])]{}]{}]{}. We then have $$\label{p diff}
{\mathbf{p}}({\mathbf{u}}_1) - {\mathbf{p}}({\mathbf{u}}_2) = {\mathbf{P}}(\tilde{{\mathbf{u}}})\bar{{\mathbf{u}}},$$ and $$\label{q diff}
{\mathbf{q}}({\mathbf{u}}_1) - {\mathbf{q}}({\mathbf{u}}_2) = {\mathbf{Q}}(\tilde{{\mathbf{u}}})\bar{{\mathbf{u}}},$$ where $\tilde{{\mathbf{u}}}={({\mathbf{u}}_1+{\mathbf{u}}_2)}/{2}$ and $\bar{{\mathbf{u}}}={\mathbf{u}}_1-{\mathbf{u}}_2$.
We write $$\begin{aligned}
&{\mathbf{p}}({\mathbf{u}}_1) -{\mathbf{p}}({\mathbf{u}}_2) = {\mathbf{p}}({\mathbf{u}}_2 + \bar{\mathbf{u}}) -{\mathbf{p}}({\mathbf{u}}_2) = \int_0^1 \f{d}{dt} {\mathbf{p}}({\mathbf{u}}_2 + t\bar{{\mathbf{u}}})\;dt
=\int_0^1 \f{D {\mathcal{P}}}{D{\mathbf{u}}}({\mathbf{u}}_2+t\bar{{\mathbf{u}}}) \cdot \bar{{\mathbf{u}}}\;dt
\\& = \int_0^1 \begin{pmatrix}
d_1 +2a_{11}(u_2+t\bar{u}) + a_{12}(v_2+t\bar{v}) & a_{12}(u_2+t\bar{u})
\\ a_{21}(v_2+t\bar{v}) & d_2+a_{21}(u_2+t\bar{u})+2a_{22}(v_2+t\bar{v})
\end{pmatrix}
\cdot \bar{{\mathbf{u}}} \;dt
\\& = \begin{pmatrix}
d_1 +a_{11}(u_1+u_2) + a_{12}{(v_1+v_2)}/{2} & a_{12}{(u_1+u_2)}/{2}
\\ a_{21}{(v_1+v_2)}/{2} & d_2+a_{21}{(u_1+u_2)}/{2}+a_{22}(v_1+v_2)
\end{pmatrix} \cdot \bar{{\mathbf{u}}}
\\& = {\mathbf{P}}(\tilde{\mathbf{u}})\cdot \bar{{\mathbf{u}}}.
\end{aligned}$$
We thus proved [[[(\[p diff\])]{}]{}]{} and we can derive [[[(\[q diff\])]{}]{}]{} in the same fashion.
Existence result for SKT systems
================================
In [@PT16 Theorem 3.1], we proved the following existence result for SKT system [[[(\[SKT vector\])]{}]{}]{}:
\[thm: existence\][s]{}
1. We assume that that $d\le 4$, that the condition [[[(\[coef cond\])]{}]{}]{} hold, and that ${\mathbf{u}}_0$ is given, ${\mathbf{u}}_0\in L^2(\Omega)^2,{\mathbf{u}}_0\ge 0$. Then equation [[[(\[SKT vector\])]{}]{}]{} possesses a solution ${\mathbf{u}}\ge {\mathbf{0}}$ such that, for every $T>0$:
\[3.38\] $$\begin{aligned}
\label{3.36a} & {\mathbf{u}}\in L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega)^2)\\
&(\sqrt{u} +\sqrt{v})({ \left| \grad u \right|}+ { \left| \grad v \right|}) \in L^2(0,T;L^2(\Omega))\\
\label{3.38c}&{\mathbf{u}}\in L^4(0,T;L^4(\Omega)).\end{aligned}$$
with the norms in these spaces bounded by a constant depending on $T$, on the coefficients, and on the norms in $L^2(\Omega)$ of $u_0$ and $v_0$.
2. If, in addition, $\grad {\mathbf{p}}({\mathbf{u}}_0)\in L^2(\Omega)^4$, then the solution ${\mathbf{u}}$ also satisfies
$$\begin{aligned}
\label{3.36}
&\grad {\mathbf{p}}({\mathbf{u}}) \in L^\infty(0,T;L^2(\Omega)^4), \quad (1+{ \left| u \right|}+{ \left| v \right|})^\f12 \left({ \left| \partial_t u \right|}+{ \left| \partial_t v \right|}\right) \in L^2(0,T;L^2(\Omega)),
\\ & \label{3.37}\Delta {\mathbf{p}}({\mathbf{u}}) \in L^2(0,T;L^2(\Omega)^2),\end{aligned}$$
with the norms in these spaces bounded by a constant depending on the norms of ${\mathbf{u}}_0$ and $\grad {\mathbf{p}}({\mathbf{u}}_0)$ in $L^2$ (and on $T$ and the coefficients).
Additional regularity of weak solutions\[appen: C\]
===================================================
Although this was not explicitly stated in [@PT16], the solutions that we constructed in dimension $d\le 4$ belong to $L^\infty_t(H^1)$ with $\partial_t {\mathbf{u}}$ in $L^2_t(L^2)$:
1. From [[[(\[3.36a\])]{}]{}]{} and [[[(\[3.36\])]{}]{}]{}, we have ${\mathbf{u}},\grad {\mathbf{p}}({\mathbf{u}})\in L^\infty(0,T;L^2(\Omega)^2)$. To show that $\grad {\mathbf{u}}\in L^\infty(0,T;L^2(\Omega)^2)$, we note that [[[(\[P positive definite\])]{}]{}]{} implies that, for $u,v \ge 0$, ${\mathbf{P}}({\mathbf{u}})$ is invertible (as a $2\times 2$ matrix), and that, pointwise (i.e. for a.e. $x\in \Omega$), $$\label{2.5b}
{ \left| {\mathbf{P}}({\mathbf{u}})^{-1} \right|}_{\mathcal{L}(\mathbb{R}^2)} \le \f{1}{d_0+\alpha(u+v)}.$$ We thus find $\grad {\mathbf{u}}\in L^\infty(0,T;L^2(\Omega)^2)$ which says that ${\mathbf{u}}\in L^\infty(0,T;H^1(\Omega)^2)$.
2. From [[[(\[3.36\])]{}]{}]{}, we have $\partial_t{\mathbf{u}}\in L^2(\Omega_T)^2$.
#### Acknowledgement.
This work was supported in part by NSF grant DMS151024 and by the Research Fund of Indiana University.
\#1[7 71000017 10000 -17 100007]{}
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[^1]: du.pham@utsa.edu
[^2]: temam@indiana.edu
[^3]: ${\mathbf{P}}(\star)$ is uniformly parabolic if $ \kappa_1 { \left| {\boldsymbol{\xi}}\right|}^2 \le \left({\mathbf{P}}(\star){\boldsymbol{\xi}}\right) \cdot {\boldsymbol{\xi}}\le \kappa_2 { \left| {\boldsymbol{\xi}}\right|}^2,$ for some $\kappa_1,\kappa_2>0$; see the definition in e.g. [@Lad68]
|
---
abstract: 'In the paper we prove that there exists a simultaneous reduction of one-parameter family of $\mathfrak{m}_{n}$-primary ideals in the ring of germs of holomorphic functions. As a corollary we generalize the result of A. Płoski [@ploski] on the semicontinuity of the Łojasiewicz exponent in a multiplicity-constant deformation.'
address: |
University of Łódź\
Faculty of Mathematics and Computer Science\
S. Banacha 22, 90-238 Łódź, Poland
author:
- Tomasz Rodak
bibliography:
- '/home/tomek/documents/tex/rezultaty/bibliografia.bib'
title: Reduction of a family of ideals
---
[^1]
Introduction
============
Let $R$ be a ring and $I$ an ideal. We say that an ideal $J$ is a *reduction of* $I$ if it satisfies the following condition: $$J\subset I,\quad\text{and for some}\quad r>0\quad\text{we have}\quad I^{r+1}=JI^{r}.$$
The notion of reduction is closely related to the notions of *Hilbert-Samuel multiplicity* and *integral closure* of an ideal.
Recall that if $\left(R,\mathfrak{m}\right)$ is a Noetherian local ring of dimension $n$ and $I$ is an $\mathfrak{m}$-primary ideal of $R$, then the *Hilbert-Samuel multiplicity* of $I$ is given by the formula $$e(I)=n!\lim_{k\to\infty}\frac{\mathrm{length}_{R}R/I^{k}}{k^{n}}.$$
For the multiplicity theory in local rings see for example [@matsumura] or [@huneke].
Let $I$ be an ideal in a ring $R$. An element $x\in R$ is said to be *integral over* $I$ if there exists an integer $n$ and elements $a_{k}\in I^{k}$, $k=1,\ldots,n$, such that $$x^{n}+a_{1}x^{n-1}+\cdots+a_{n}=0.$$
The set of all elements of $R$ that are integral over $I$ is called the *integral closure of* $I$, and is denoted $\overline{I}$. If $I=\overline{I}$ then $I$ is called *integrally closed*. It is well known that $\overline{I}$ is an ideal.
The relationship between the above notions is given in the following Theorem due to D. Rees:
\[thm:Rees\]Let $(R,\mathfrak{m})$ be a formally equidimensional Noetherian local ring and let $J\subset I$ be two $\mathfrak{m}$-primary ideals. Then the following conditions are equivalent:
1. $J$ is a reduction of $I$;
2. $e\left(I\right)=e\left(J\right)$;
3. $\overline{I}=\overline{J}$.
It is an important fact that a reduction of an ideal is often generated by a system of parameters. More precisely we have
\[thm:rzutowanie\]Let $(R,\mathfrak{m})$ be a $d$-dimensional Noetherian local ring, and suppose that $k=R/\mathfrak{m}$ is an infinite field; let $I=(u_{1},\ldots,u_{s})$ be an $\mathfrak{m}$-primary ideal. Then there exist a finite number of polynomials $D_{\alpha}\in k[Z_{ij};1\leqslant i\leqslant d,1\leqslant j\leqslant s]$, $1\leqslant\alpha\leqslant\nu$ such that if $y_{i}=\sum a_{ij}u_{j}$, $i=1,\ldots,d$ and at least one of $D_{\alpha}(\overline{a}_{ij};1\leqslant i\leqslant d,1\leqslant j\leqslant s)\ne0$, then the ideal $(y_{1},\ldots,y_{d})R$ is a reduction of $I$ and $\{y_{1},\ldots,y_{d}\}$ is a system of parameters of $R$.
Let $\left(\mathcal{O}_{n},\mathfrak{m}_{n}\right)$ be the ring of germs of holomorphic functions $\left(\mathbb{C}^{n},0\right)\to\mathbb{C}$. The aim of this note is to prove the following:
\[thm:Main\]Let $F=F_{t}(x)=F(x,t)\colon(\cc^{n}\times\cc,0)\to(\cc^{m},0)$ be a holomorphic map. Assume that $(F_{t})\mathcal{O}_{n}$ is an $\mathfrak{m}_{n}$-primary ideal for all $t$. Then there exists a complex linear map $\pi\colon\mathbb{C}^{m}\to\mathbb{C}^{n}$ such that for all $t$ the ideal $\left(\pi\circ F_{t}\right)\mathcal{O}_{n}$ is a reduction of $\left(F_{t}\right)\mathcal{O}_{n}$.
In the next section we get as a corollary that if the above family $\left(F_{t}\right)\mathcal{O}_{n}$ is of constant multiplicity then the Łojasiewicz exponent in this family is a lower semicontinuos function of $t$. A. Płoski proved this result under additional restriction $m=n$ but with space of parameters of arbitrary dimension.
The proof of Theorem \[thm:Main\] is based on some geometric property of Hilbert-Samuel multiplicity, given in section 3.
Semicontinuity of the Łojasiewicz exponent
==========================================
Let $(R,\mathfrak{m})$ be a local ring and let $I$ be an $\mathfrak{m}$-primary ideal. By the *Łojasiewicz exponent* $\mathcal{L}(I)$ of $I$ we define the infimum of $$\left\{ \frac{p}{q}:\mathfrak{m}^{p}\subset\overline{I^{q}}\right\} .$$ It was proved in [@ljt1974] that if $F\colon\left(\cc^{n},0\right)\to\left(\cc^{m},0\right)$ is a holomorphic map with an isolated zero at the origin and $I:=\left(F\right)\mathcal{O}_{n}$, then $\mathcal{L}\left(I\right)$ is an optimal exponent $\nu$ in the inequality $$\left|F\left(x\right)\right|\geqslant C\left|x\right|^{\nu},$$ where $C$ is some positive constant and $x$ runs through sufficiently small neighbourhood of $0\in\cc^{n}$.
\[lem:wykladnik - redukcja\]Let $\left(R,\mathfrak{m}\right)$ be a Noetherian local ring. If $I$ is an $\mathfrak{m}$-primary ideal of $R$ and $J$ is a reduction of $I$ then $\mathcal{L}\left(I\right)=\mathcal{L}\left(J\right)$.
Obviously $\mathcal{L}(I)\leqslant\mathcal{L}(J)$. Assume that $\mathfrak{m}^{p}\subset\overline{I^{q}}$. Since $J$ is a reduction of $I$, then also $J^{q}$ is a reduction of $I^{q}$ [@huneke Prop. 8.1.5]. Thus $\overline{J^{q}}=\overline{I^{q}}$ by Theorem \[thm:Rees\], which gives $\mathfrak{m}^{p}\subset\overline{J^{q}}$. This proves the inequality $\mathcal{L}(J)\leqslant\mathcal{L}(I)$ and ends the proof.
Let $F\colon\left(\cc^{n}\times\cc,0\right)\to\left(\cc^{m},0\right)$ be a holomorphic map. Put $I_{t}:=\left(F_{t}\right)\mathcal{O}_{n}$. If the function $t\mapsto e\left(I_{t}\right)$ is constant and finite then the function $t\mapsto\mathcal{L}\left(I_{t}\right)$ is lower semicontinuos.
By Theorem \[thm:Main\] there exists a linear map $\pi\colon\cc^{m}\to\cc^{n}$ such that $J_{t}:=\left(\pi\circ F_{t}\right)\mathcal{O}_{n}$ is a reduction of $I_{t}$ for all $t$. Thus $\mathcal{L}\left(J_{t}\right)=\mathcal{L}\left(I_{t}\right)$ and $e\left(J_{t}\right)=e\left(I_{t}\right)$ by Theorem \[thm:Rees\] and Lemma \[lem:wykladnik - redukcja\]. Consequently $t\mapsto e\left(J_{t}\right)$ is constant and finite and the assertion follows from the case $m=n$ proved by A. Płoski.
Improper intersection multiplicity
==================================
Let $I$ be an $\mathfrak{m}_{n}$-primary ideal of $\mathcal{O}_{n}$ and let $f_{1},\ldots,f_{m}$ be its generators. We put $f=\left(f_{1},\ldots,f_{m}\right)$. It is well known that if $m=n$ then $$e(I)=\dim_{\cc}\mathcal{O}_{n}/I.$$ On the other hand, if $m>n$ then we may define so-called *improper intersection multiplicity* $i_{0}\left(I\right)$ of $I$ as the improper intersection multiplicity $i(\mathrm{graph}f\cdot(\cc^{n}\times\{0\});(0,0))$ of $\mathrm{graph}f$ and $\cc^{n}\times\{0\}$ at the point $(0,0)\in\cc^{n}\times\cc^{m}$ (see [@atw]).
Let $C_{f}$ be the (Whitney) tangent cone of the germ of the image of $f$ at the origin. The following observation is due to S. Spodzieja.
\[thm:sp\]The number $i_{0}\left(I\right)$ is well defined. Moreover, if $\pi\colon\cc^{m}\to\cc^{l}$ is a linear map such that $\ker\pi\cap C_{f}=\{0\}$, then the ideal $J$ generated by $\pi\circ f$ is $\mathfrak{m}_{n}$-primary and we have $i_{0}(I)=i_{0}(J)$. If additionally $l=n$ then $i_{0}(I)=e(J)$.
\[prop:i0=00003De\]If $I$ is an $\mathfrak{m}_{n}$-primary ideal in $\mathcal{O}_{n}$, then $i_{0}(I)=e(I)$.
Let $I=(f_{1},\ldots,f_{m})\mathcal{O}_{n}$. By Theorems \[thm:rzutowanie\] and \[thm:sp\] there exists linear combinations $g_{i}=\sum a_{ij}f_{j}$, $i=1,\ldots,n$ such that $J=(g_{1},\ldots,g_{n})\mathcal{O}_{n}$ is a reduction of $I$, $\{g_{1},\ldots,g_{n}\}$ is a system of parameters of $\mathcal{O}_{n}$ and $i_{0}(I)=i_{0}(J)=e(J)$. From Theorem \[thm:Rees\] we get $e(I)=e(J)$. This ends the proof.
\[cor:cone-reduction\]If $\pi\colon\cc^{m}\to\cc^{l}$ is a linear map such that $\ker\pi\cap C_{f}=\{0\}$, then the ideal $J$ generated by $\pi\circ f$ is a reduction of $I$.
We have $J\subset I$ and $e\left(J\right)=e\left(I\right)$. This and Theorem \[thm:Rees\] give the assertion.
Elementary blowing-up
=====================
Here we recall the notion of an elementary blowing-up after [@lojasiewicz].
Let $U\subset\cc^{n}$ be an open and connected neighbourhood of $0\in\cc^{n}$; let $f=(f_{0},\ldots,f_{m})\ne0$ be a sequence of holomorphic functions on $U$. Put $S=\{x\in U:f(x)=0\}$ and $$E(f)=\{(x,u)\in U\times\pm:f_{i}(x)u_{j}=f_{j}(x)u_{i},i,j=0,\ldots,m\},$$ where $u=[u_{0}:\cdots:u_{m}]\in\pm$.
Let $Y$ be the closure of $E(f)\setminus S$ in $U\times\pm$. The natural projection $$\pi:Y\to U$$ is called the *(elementary) blowing-up of $U$ by means of $f_{0},\ldots,f_{m}$.* The analytic subset $S$ is called a *centre of the blowing-up* and its inverse image $\pi^{-1}(S)\subset Y$ is called the *exceptional set* of the blowing-up.
\[blow\] Under above notations we have:
1. $Y$ is an analytic subset of $U\times\pm$;\[enu:-is-an\]
2. $\pi$ is proper, its range is $U$ and the restriction $\pi_{|Y\setminus\pi^{-1}(S)}$ is a biholomorphism onto $U\setminus S$;\[enu:-is-proper,\]
3. $Y$ is irreducible;\[enu:-is-irreducible\]
4. The exceptional set $\pi^{-1}(S)$ is analytic in $U\times\pm$ and it is of pure dimension $n-1$.\[enu:The-exceptional-set\]
Although the above proposition is well known, we think that point is worth proving. Let us consider the analytic map $$F\colon U\times\pm\ni(x,u)\mapsto(f(x),u)\in\cc^{m+1}\times\pm.$$ Let $y_{0},\ldots,y_{m}$ be coordinates in $\cc^{m+1}$. If we denote by $\pi_{m+1}\colon\Pi_{m+1}\to\cc^{m+1}$ the blowing-up of $\cc^{m+1}$ by means of $y_{0},\ldots,y_{m}$ then for the restriction $\widetilde{f}=F_{|Y}$ we get the following commutative diagram of analytic maps:
$\begin{CD}Y @>\widetilde{f}>> \Pi_{m+1} \\
@VV\pi V @VV\pi_{m+1} V\\
U @>f>> \cc^{m+1}
\end{CD}$
Take $(x_{0},u_{0})\in\pi^{-1}(0)$. Let $\Omega\subset\Pi_{m+1}$ be a neighbourhood of $(0,u_{0})$, $h\colon\Omega\to\cc$ an analytic function such that $$\pi_{m+1}^{-1}(0)\cap\Omega=\{(y,u)\in\Omega:h(y,u)=0\}.$$
Let $\widetilde{\Omega}\subset Y$ be a neighbourhood of $(x_{0},u_{0})$ such that $\widetilde{f}(\widetilde{\Omega})\subset\Omega$. Since $\widetilde{f}^{-1}(\pi_{m+1}^{-1}(0))=\pi^{-1}(S)$ we get $$\pi^{-1}(S)\cap\widetilde{\Omega}=\{(x,u)\in\widetilde{\Omega}:h\circ\widetilde{f}(x,u)=0\}.$$
Thus there exists a neighbourhood $\Delta\subset U\times\pm$ of $(x_{0},u_{0})$ and an analytic set $V\subset\Delta$ of pure dimension $n+m-1$ such that $$\pi^{-1}(S)\cap\Delta=V\cap Y\cap\Delta.$$
This gives $$\dim_{(x_{0},u_{0})}\pi^{-1}(S)\geqslant\dim_{(x_{0},u_{0})}Y-1=n-1.$$
Since $Y$ is irreducible and $\pi^{-1}(S)\varsubsetneq Y$ we get that $\dim_{p}\pi^{-1}(S)=n-1$ for any $p\in\pi^{-1}(S)$. This ends the proof.
Proof of Theorem \[thm:Main\]
=============================
\[lem:blow lemma\]Let $F\colon(\cc^{n}\times\cc,0)\to(\cc^{m+1},0)$, $m\geqslant n$ be a holomorphic map. Assume that $0$ is an isolated point of $F_{t}^{-1}(0)$ for $|t|<\delta$. Then there exists $\delta>\epsilon>0$ and a complex line $V\subset\cc^{m+1}$, such that $V\cap C_{F_{t}}=\{0\}$ for $|t|<\epsilon$.
Let $F\colon U\to\cc^{m+1}$, where $U\subset\cc^{n}\times\cc$ is a connected neighbourhood of the origin. Put $S=\{(z,t)\in U:F(z,t)=0\}$ and let $\pi\colon U\times\pm\supset Y\to U$ be the elementary blowing-up of $U$ by $F$. By Proposition \[blow\] its exceptional set $E:=\pi^{-1}(S)$ is an analytic set of pure dimension $n$. Let $\mathcal{E}$ be a set of those irreducible components $W$ of $E$ for which origin in $\cc^{n+1}$ is an accumulation point of $\pi(W)\cap(\{0\}\times\cc)$. Then $\mathcal{E}$ is finite. Denote by $\widetilde{C_{F_{t}}}$ the image of the cone $C_{F_{t}}$ in $\pm$. Observe that $$\{(0,t)\}\times\widetilde{C_{F_{t}}}\subset\bigcup\mathcal{E},\quad|t|<\delta.$$ On the other hand for any $W\in\mathcal{E}$ we have $$\dim W\cap(\{0\}\times\pm)\leqslant n-1<m.$$ Thus there exists $\epsilon>0$ and an open set $G\subset\pm$ such that $$(\{(0,t)\}\times G)\cap\bigcup\mathcal{E}=\emptyset,\quad0<|t|<\epsilon$$ As a result if $V$ is a line in $\cc^{m+1}$ corresponding to some point in $G$ then $V\cap C_{F_{t}}=\{0\}$ for $0<|t|<\epsilon$. Since $G$ is not a subset of $C_{F_{0}}$ we get the assertion.
Induction on $m$. In the case $m=n$ there is nothing to prove. Let us assume that the assertion is true for some $m\geqslant n$ and let $F\colon(\cc^{n}\times\cc,0)\to(\cc^{m+1},0)$ be a holomorphic map such that the ideals $\left(F_{t}\right)\mathcal{O}_{n}$ are $\mathfrak{m}_{n}$-primary. By Lemma \[lem:blow lemma\] there exists $\epsilon>0$ and a linear mapping $\pi'\colon\cc^{m+1}\to\cc^{m}$ such that $\ker\pi'\cap C_{F_{t}}=\left\{ 0\right\} $ for $|t|<\epsilon$. Thus, by Corollary \[cor:cone-reduction\] the ideal $\left(\pi'\circ F_{t}\right)\mathcal{O}_{n}$ is a reduction of $\left(F_{t}\right)\mathcal{O}_{n}$. On the other hand, by induction hypothesis, there exists a linear map $\pi''\colon\mathbb{C}^{m}\to\mathbb{C}^{n}$ such that $\left(\pi''\circ\pi'\circ F_{t}\right)\mathcal{O}_{n}$ is a reduction of $\left(\pi'\circ F_{t}\right)\mathcal{O}_{n}$ for small $t$. Thus if we put $\pi:=\pi''\circ\pi'$ we get the assertion.
[^1]: This research was partially supported by the Polish OPUS Grant No 2012/07/B/ST1/03293
|
---
abstract: 'We provide a general approach for the analysis of optical state evolution under conditional measurement schemes, and identify the necessary and sufficient conditions for such schemes to simulate unitary evolution on the freely propagating modes. If such unitary evolution holds, an effective photon nonlinearity can be identified. Our analysis extends to conditional measurement schemes more general than those based solely on linear optics.'
author:
- 'G.G. Lapaire$^1$, Pieter Kok$^2$, Jonathan P. Dowling$^2$, and J.E. Sipe$^1$'
title: 'Conditional linear-optical measurement schemes generate effective photon nonlinearities'
---
Introduction
============
One of the main problems that optical quantum computing has to overcome is the efficient construction of two-photon gates [@kok00]. We can use Kerr nonlinearities to induce a phase shift in one mode that depends on the photon number in the other mode, and this nonlinearity is sufficient to generate a universal set of gates [@kerr]. However, passive Kerr media have typically small nonlinearities (of the order of $10^{-16}\,\text{cm}%
^{2}\,\text{sV}^{-1}$ [@boyd99]). We can also construct large Kerr nonlinearities using slow light, but these techniques are experimentally difficult [@lukin00].
On the other hand, we can employ linear optics with projective measurements. The benefit is that linear optical schemes are experimentally much easier to implement than Kerr-media approaches, but the downside is that the measurement-induced nonlinearities are less versatile and the success rate can be quite low (especially when inefficient detectors are involved). However, Knill, Laflamme and Milburn [@klm] showed that with sufficient ancilla systems, these linear-optical quantum computing (LOQC) devices can be made near-deterministic with only polynomial resources. This makes linear optics a viable candidate for quantum computing. Indeed, many linear optical schemes and approaches have been proposed since [@gottesman; @franson; @simple; @mathis; @kyi; @snmk], and significant experimental progress has already been made [@feedfwd; @franson2].
The general working of a device that implements linear optical processing with projective measurements is shown in Fig. 1. The computational input and the ancilla systems add up to $N$ optical modes that are subjected to a unitary transformation $U$, which is implemented with beam splitters, phase shifters, *etc*. This is called an optical $N$-port device. In order to induce a transformation of interest on the computational input, the output is conditioned on a particular measurement outcome of the ancilla system. For example, one can build a single-photon quantum nondemolition detector with an optical $N$-port device [@kok02]. In general, $N$-port devices have been studied in a variety of applications [@nport].
The class of such devices of interest here is that in which a unitary evolution on the computational input is effected. To date these devices have been proposed and studied on a more-or-less case by case basis. Our approach is to address this class in a more general way, and identify the conditions that such a device must satisfy to implement a unitary evolution on the computational input. Once that unitary evolution is established, an effective photon nonlinearity associated with the device can be identified.
In this paper, we present necessary and sufficient conditions for the unitarity of the optical transformation of the computational input, and we derive the effective nonlinearities that are associated with some of the more common optical gates in LOQC. We begin section \[sec:formalism\] by introducing the formalism. In sections \[sec:consec\]-\[sec:nscon\], we examine the transformation equation under the assumption that it is unitary. We show that there are two necessary and sufficient conditions for the transformation to be unitary and we provide a simple test condition. In section \[sec:post\], we expand the formalism and conditions to include measurement dependent output processing (see Fig. 2), which is used in several schemes. In section \[sec:examples\], we show how the formalism can be applied to quantum computing gates. We choose as examples two quantum gates already proposed, the conditional sign flip of Knill, Laflamme, and Milburn [@klm], and the polarization-encoded CNOT of Pittman *et al*. [@franson]. Our concluding remarks are presented in section \[sec:conclusions\], where we note that our main results extend to devices where the unitary transformation $U$ is more general than those implementable with linear optics alone.
The general formalism {#sec:formalism}
=====================
We consider a class of optical devices that map the computational input state onto an output state, conditioned on a particular measurement outcome of an ancilla state (see Fig. 1). We introduce a factorization of the entire Hilbert space into a space $\mathcal{H}_{C}$ involving the input computing channels (*i.e.*, both “target” and “control” in a typical quantum gate), and a Hilbert space $\mathcal{H}_{A}$ involving the input ancilla channels, $$\mathcal{H}=\mathcal{H}_{C}\otimes \mathcal{H}_{A}\; .$$ We assume that the input computing and ancilla channels are uncorrelated and unentangled, so we can write the full initial density operator as $\rho
\otimes \sigma $, where $\rho $ is the initial density operator for the computing channels, and $\sigma $ the initial density operator for the ancilla channels.
Let $U$ be the unitary operator describing the pre-measurement evolution of the optical multi-port device. At the end of this process we have a full density operator given by $U\left( \rho \otimes \sigma \right) U^{\dagger }$. In anticipation of the projective measurement, it is useful to introduce a new factorization of the full Hilbert space into an output computing space $%
\mathcal{H}_{\bar{C}}$ and a new ancilla space $\mathcal{H}_{\bar{A}}$, $$\mathcal{H=H}_{\bar{C}}\otimes \mathcal{H}_{\bar{A}}\; .$$
The Von Neumann projective measurements of interest are described by projector-valued measures (or PVMs) of the type $\left\{ \bar{P},{I-}\bar{P}%
\right\} $, where ${I}$ is the identity operator for the whole Hilbert space, and the projector $\bar{P}$ is of the form $$\bar{P}={I}_{\bar{C}}\otimes \sum_{\bar{k}}s_{\bar{k}}\left| \bar{k}%
\right\rangle \left\langle \bar{k}\right| \; , \label{projector}$$ where ${I}_{\bar{C}}$ is the identity operator in $\mathcal{H}_{\bar{C}}$, and we use Roman letters with an overbar, *e.g.,* $\left| \bar{k}%
\right\rangle $, to label a set of orthonormal states, $\left\langle \bar{k}|%
\bar{l}\right\rangle =\delta _{\bar{k}\bar{l}}$, spanning the Hilbert space $%
\mathcal{H}_{\bar{A}}$; each $s_{\bar{k}}$ is equal to zero or unity. The number of nonzero $s_{\bar{k}}$ identifies the rank of the projector $\bar{P}
$ in $\mathcal{H}_{\bar{A}}$. “Success” is defined as a measurement outcome associated with the projector $\bar{P}$, and the probability of success is thus $$d(\rho )\equiv \mathrm{Tr}_{\bar{C},\bar{A}}\left( U\left( \rho \otimes
\sigma \right) U^{\dagger }\bar{P}\right)\; . \label{Ddef}$$ Clearly, in general $d(\rho )$ depends on the ancilla density operator $%
\sigma $, the unitary evolution $U$, and the projector $\bar{P}$, as well as on $\rho $. However, we consider the first three of these quantities fixed by the protocol of interest and thus only display the dependence of the success probability on the input density operator $\rho $. In the event of a successful measurement, the output of the channels associated with $\mathcal{%
H}_{\bar{C}}$ is identified as the computational result, and it is described by the reduced density operator $$\bar{\rho}=\frac{\mathrm{Tr}_{\bar{A}}\left( \bar{P}U\left( \rho \otimes
\sigma \right) U^{\dagger }\bar{P}\right) }{\mathrm{Tr}_{\bar{C},\bar{A}%
}\left( U\left( \rho \otimes \sigma \right) U^{\dagger }\bar{P}\right) }.
\label{rhobar}$$ For any $\rho $ with $d(\rho )\neq 0$, this defines a so-called completely positive (CP), trace preserving map $\mathcal{T}$ that takes each $\rho $ to its associated $\bar{\rho}$: $\bar{\rho} = \mathcal{T}(\rho )$, relating density operators in $\mathcal{H}_{C}$ to density operators in $\mathcal{H}_{%
\bar{C}}$. It will be convenient to write $\mathcal{T}(\rho )=\mathcal{V}%
(\rho )/d(\rho )$, where $$\mathcal{V}(\rho )\equiv \mathrm{Tr}_{\bar{A}}\left( \bar{P}U\left( \rho
\otimes \sigma \right) U^{\dagger }\bar{P}\right) \label{Vdef}$$ is a linear (non-trace preserving) CP map of density operators in $\mathcal{H%
}_{C}$ to positive operators in $\mathcal{H}_{\bar{C}}$ that is defined for all density operators $\rho $ in $\mathcal{H}_{C}$. We restrict ourselves to density operators $\rho $ over a subspace $\mathcal{S}_{C}$ of $\mathcal{H}%
_{C}$. This is usually the subspace in which the quantum gate operates.
As an example, consider the gate that turns the computational basis into the Bell basis. In terms of polarization states, the subspace $\mathcal{S}_{C}$ might be spanned by the computational basis $\{|H,H\rangle ,|H,V\rangle
,|V,H\rangle ,|V,V\rangle \}$ (whereas $\mathcal{H}_{C}$ is spanned by the full Fock basis). The Bell basis on $\mathcal{S}_{C}$ is then given by $\{
|\Psi^+\rangle, |\Psi^-\rangle, |\Phi^+\rangle, |\Phi^-\rangle \}$, where $$|\Psi ^{\pm }\rangle =\frac{1}{\sqrt{2}}\left( |H,V\rangle \pm |V,H\rangle
\right) \text{~and~} |\Phi ^{\pm }\rangle =\frac{1}{\sqrt{2}}\left(
|H,H\rangle \pm |V,V\rangle \right) \;.$$ This gate is very important in quantum information theory, because it produces maximal entanglement, and its inverse can be used to perform Bell measurements. Both functions are necessary in, *e.g.,* quantum teleportation [@bennett93]. However, it is well known that such gates cannot be constructed deterministically, and we therefore need to include an ancilla state $\sigma$ and a projective measurement. We consider gates such as these in this paper.
Suppose the subspace $\mathcal{S}_{C}$ is spanned by a set of vectors labeled by Greek letters, *e.g.,* $\left| \alpha \right\rangle $. We can then write $$\rho =\sum_{\alpha ,\beta }\left| \alpha \right\rangle \rho ^{\alpha \beta
}\left\langle \beta \right| , \label{rhodecompose}$$ where $\rho ^{\alpha \beta }\equiv \left\langle \alpha |\rho |\beta
\right\rangle $. We identify a convex decomposition of the ancilla density operator $\sigma $ as $$\sigma =\sum_{i}p_{i}\left| \chi _{i}\right\rangle \left\langle \chi
_{i}\right| ,$$ where the normalized (but not necessarily orthogonal) vectors $\left| \chi
_{i}\right\rangle $ are elements of $\mathcal{H}_{A}$, and the $p_{i}$ are all non-negative and sum to unity, $$\sum_{i}p_{i}=1\; .$$ We can then use (\[rhobar\]) to write down an expression for the matrix elements of $\bar{\rho}$. Note that it is possible to work with the eigenkets of $\sigma $ so that $\left\{ \left| \chi _{i}\right\rangle
\right\} $ is an orthonormal set; however, this does not simplify the analysis so we do not introduce the restriction. Furthermore, dealing with non-orthogonal states in the ancilla convex decomposition may be more convenient, depending on the system of interest. Choosing an orthonormal basis of $\mathcal{H}_{\bar{C}}$ that we label by Greek letters with overbars, *e.g.,* $\left| \bar{\alpha}\right\rangle $, we find $$\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }\sum_{i,\bar{k}%
}\left( W_{\bar{k},i}^{\bar{\alpha}\beta }(\rho )\right) \rho ^{\beta \gamma
}\left( W_{\bar{k},i}^{\bar{\delta}\gamma }(\rho )\right) ^{*},
\label{rhobarwork}$$ where $$W_{\bar{k},i}^{\bar{\alpha}\beta }(\rho )=s_{\bar{k}}\sqrt{\frac{p_{i}}{%
d(\rho )}}\left( \left\langle \bar{k}\right| \left\langle \bar{\alpha}%
\right| \right) U\left( \left| \beta \right\rangle \left| \chi
_{i}\right\rangle \right) .$$ Note that $$\sum_{\bar{\alpha}}\sum_{i,\bar{k}}\left( W_{\bar{k},i}^{\bar{\alpha}\gamma
}(\rho )\right) ^{*}\left( W_{\bar{k},i}^{\bar{\alpha}\beta }(\rho )\right)
=\delta _{\gamma \beta }\;,$$ which is confirmed by $$\mathrm{Tr}_{\bar{C}}(\bar{\rho})=\sum_{\bar{\alpha}}\bar{\rho}^{\bar{\alpha}%
\bar{\alpha}}=\sum_{\beta }\rho ^{\beta \beta }=\mathrm{Tr}_{C}(\rho ),
\label{tracepreserve}$$ This last equation follows immediately from (\[rhobar\]), since $\mathcal{T%
}$ is a trace preserving CP map and $\mathrm{Tr}_{C}(\rho )=1$.
In this paper, we consider a special class of maps that constitute a unitary transformation on the computational subspace $\mathcal{S}_{C}$. In particular, such transformations include the CNOT, the C-SIGN, and the controlled bit flip. These are not the only useful maps in linear optical quantum computing, but they arguably constitute the most important class. Before we continue, we introduce the following definition:
Definition:
: We call a CP map $\rho \rightarrow \bar{\rho}=\mathcal{T}%
(\rho )$ an *operationally unitary transformation* on density operators $\rho $ over a subspace $\mathcal{S}_{C}$ if and only if:
- For each $\rho $ over the subspace $\mathcal{S}_{C}$ we have $d(\rho
)\neq 0$, and
- For each $\rho $ defined by Eq. (\[rhodecompose\]) over the subspace $\mathcal{S}_{C}$, the map $\mathcal{T}(\rho )$ yields a $\bar{\rho}
$ given by $$\bar{\rho}=\sum_{\alpha ,\beta }\left| \bar{\nu}_{\alpha }\right\rangle \rho
^{\alpha \beta }\left\langle \bar{\nu}_{\beta }\right| , \label{rhobareu}$$ where the $\left| \bar{\nu}_{\alpha }\right\rangle $ are fixed vectors in $%
\mathcal{H}_{\bar{C}}$ satisfying $\left\langle \bar{\nu}_{\alpha }|\bar{\nu}%
_{\beta }\right\rangle =\left\langle \alpha |\beta \right\rangle =\delta
_{\alpha \beta }$.
This forms the obvious generalization of usual unitary evolution, since it maintains the inner products of vectors under the transformation. Much of our concern in this paper is in identifying the necessary and sufficient conditions for a general map $\mathcal{T}(\rho )$ of Eqs. (\[rhobar\]) and (\[rhobarwork\]) to constitute an operationally unitary map. We begin in the next section by considering what can be said about such maps.
Consequences of operational unitarity {#sec:consec}
=====================================
In this section we restrict ourselves to CP maps $\mathcal{T}(\rho )$ that are operationally unitary \[see Eqs. (\[rhodecompose\]) and (\[rhobareu\])\] for density operators $\rho $ over a subspace $\mathcal{S}_{C}$ of $%
\mathcal{H}_{C}$. The linearity of such maps implies that the convex sum of two density operators is again a density operator: $$\rho _{c}=x\rho _{a}+(1-x)\rho _{b},$$ with $0\leq x\leq 1$. Applying Eqs. (\[rhodecompose\]) and (\[rhobareu\]) to the three density operators $\rho _{a}$, $\rho _{b}$, and $\rho _{c}$ it follows immediately that $$\bar{\rho}_{c}=x\bar{\rho}_{a}+(1-x)\bar{\rho}_{b}. \label{tranfirst}$$ Now a second expression for $\bar{\rho}_{c}$ can be worked out by using the defining relation (\[rhobar\]) directly, $$\begin{aligned}
\bar{\rho}_{c} &=&\mathcal{T}(\rho _{c})=\frac{\mathcal{V}(\rho _{c})}{%
d(\rho _{c})} \label{transecond} \\
&=&\frac{x\mathcal{V}(\rho _{a})+(1-x)\mathcal{V}(\rho _{b})}{xd(\rho
_{a})+(1-x)d(\rho _{b})} \nonumber \\
&=&\frac{xd(\rho _{a})\bar{\rho}_{a}+(1-x)d(\rho _{b})\bar{\rho}_{b}}{%
xd(\rho _{a})+(1-x)d(\rho _{b})} \nonumber\end{aligned}$$ where in the second line we have used the linearity of $\mathcal{V}(\rho )$ (\[Vdef\]) and $d(\rho )$ (\[Ddef\]), and in the third line we have used the corresponding relations for $\bar{\rho}_{a}$ in terms of $\rho _{a}$, and $\bar{\rho}_{b}$ in terms of $\rho _{b}$. Setting the right-hand-sides of Eqs. (\[tranfirst\]) and (\[transecond\]) equal, we find $$x(1-x)\left[ d(\rho _{b})-d(\rho _{a})\right] (\bar{\rho}_{a}-\bar{\rho}%
_{b})=0. \label{eureka}$$ Since it is easy to see from Eqs. (\[rhodecompose\]) and (\[rhobareu\]) that if $\rho _{a}$ and $\rho _{b}$ are distinct then $\bar{\rho}_{a}$ and $%
\bar{\rho}_{b}$ are as well; choosing $0<x<1$ it is clear that the only way the operator equation (\[eureka\]) can be satisfied is if $d(\rho
_{a})=d(\rho _{b})$. But since this must hold for *any* two density operators acting over $\mathcal{S}_{C}$, we have established that:
- If a map $\mathcal{T}(\rho )$ is operationally unitary for $\rho $ (acting on a subspace $\mathcal{S}_{C}$), then $d(\rho )$ is independent of $%
\rho $: $d(\rho )=d$, for all $\rho $ acting on that subspace.
With this result in hand we can simplify Eq. (\[rhobarwork\]) for a map that is operationally unitary, writing $$\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }\sum_{J}w_{J}^{%
\bar{\alpha}\beta }\rho ^{\beta \gamma }\left( w_{J}^{\bar{\delta}\gamma
}\right) ^{*}, \label{tsimpform}$$ where now $$w_{J}^{\bar{\alpha}\beta }=w_{\bar{k},i}^{\bar{\alpha}\beta }=s_{\bar{k}}%
\sqrt{\frac{p_{i}}{d}}\left( \left\langle \bar{k}\right| \left\langle \bar{%
\alpha}\right| \right) U\left( \left| \beta \right\rangle \left| \chi
_{i}\right\rangle \right)$$ is independent of $\rho $; we have also introduced a single label $J$ to refer to the pair of indices $\bar{k},i$. A further simplification arises because the condition of operational unitarity guarantees that the subspace $%
\mathcal{S}_{\bar{C}}$ of $\mathcal{H}_{\bar{C}}$, over which the range of density operators $\bar{\rho}$ generated by $\mathcal{T}(\rho )$ act as $%
\rho $ ranges over $\mathcal{S}_{C}$, has the same dimension as $\mathcal{S}%
_{C}$. We can thus adopt a set of orthonormal vectors $\left| \bar{\alpha}%
\right\rangle $ that span that subspace $\mathcal{S}_{\bar{C}}$, and the matrices $w_{J}^{\bar{\alpha}\beta }$ are square.
At this point we can formally construct a unitary map on $\mathcal{S}_{C}$: $%
\tilde{\rho}\equiv \mathcal{U}(\rho )$, which is isomorphic in its effect on density operators $\rho $ with our operationally unitary map $\bar{\rho}=%
\mathcal{T}(\rho )$. We do this by associating each $\left| \bar{\alpha}%
\right\rangle $ with the corresponding $\left| \alpha \right\rangle $, introducing a density operator $\tilde{\rho}$ acting over $\mathcal{S}_{C}$, and putting $$\begin{aligned}
\tilde{\rho}^{\alpha \delta } &\equiv &\bar{\rho}^{\bar{\alpha}\bar{\delta}},
\label{correspondence} \\
M_{J}^{\alpha \beta } &\equiv &w_{J}^{\bar{\alpha}\beta }. \nonumber\end{aligned}$$ The unitary map $\tilde{\rho}\equiv \mathcal{U}(\rho )$ is defined by the CP map $$\tilde{\rho}^{\alpha \delta }=\sum_{\beta ,\gamma }\sum_{J}M_{J}^{\alpha
\beta }\rho ^{\beta \gamma }(M_{J}^{\delta \gamma })^{*},$$ or simply $$\tilde{\rho}=\sum_{J}M_{J}\rho M_{J}^{\dagger }. \label{unitransform}$$ This is often what is done implicitly when describing an operationally unitary map, and we will see examples later in section \[sec:examples\]; here we find this strategy useful to simplify our reasoning below.
Since the map $\tilde{\rho}\equiv \mathcal{U}(\rho )$ is unitary it can be implemented by a unitary operator $M$, $$\tilde{\rho}=M\rho M^{\dagger },$$ where $M^{\dagger }=M^{-1}$. Thus $(M_{1},M_{2},....)$ and $(M,0,0,....)$, where we add enough copies of the zero operator so that the two lists have the same number of elements, constitute two sets of Kraus operators that implement the same map $\tilde{\rho}\equiv \mathcal{U}(\rho )$. From Nielsen and Chuang [@NandC] we have the following theorem:
Theorem:
: Suppose $\{E_{1},\ldots ,E_{n}\}$ and $\{F_{1},\ldots
,F_{m}\}$ are Kraus operators giving rise to CP linear maps $\mathcal{E}$ and $\mathcal{F}$ respectively. By appending zero operators to the shorter list of elements we may ensure that $m=n$. Then $\mathcal{E}=\mathcal{F}$ if and only if there exists complex numbers $u_{jk}$ such that $%
E_{j}=\sum_{k}u_{jk}F_{k}$, and $u_{jk}$ is an $m\times m$ unitary matrix.
Hence, $(M_{1},M_{2},....)$ must be related to $(M,0,0,....)$ by a unitary matrix, and each $M_{J}$ is proportional to the single operator $M$. This proof carries over immediately to the operationally unitary map $\mathcal{%
T(\rho )}$ under consideration, and we have
- If a map $\mathcal{T}(\rho )$ is operationally unitary for $\rho $ acting over a subspace $\mathcal{S}_{C}$, then for fixed $\bar{k}$ and $i$ the square matrix defined by $$w_{\bar{k},i}^{\bar{\alpha}\beta }=s_{\bar{k}}\sqrt{\frac{p_{i}}{d}}\left(
\left\langle \bar{k}\right| \left\langle \bar{\alpha}\right| \right) U\left(
\left| \beta \right\rangle \left| \chi _{i}\right\rangle \right) ,$$ with $\bar{\alpha}$ labeling the row and $\beta $ the column, either vanishes or is proportional to all other nonvanishing matrices identified by different $\bar{k}$ and $i$. We can thus define a matrix $w^{\bar{\alpha}%
\beta }$ proportional to all the nonvanishing $w_{\bar{k},i}^{\bar{\alpha}%
\beta }$ such that we can write our map (\[tsimpform\]) as $$\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }w^{\bar{\alpha}%
\beta }\rho ^{\beta \gamma }\left( w^{\bar{\delta}\gamma }\right) ^{*}.
\label{tfinal}$$
It is in fact easy to show that the two *necessary* conditions we have established here for a map $\mathcal{T}(\rho )$ to be an operationally unitary transformation are also *sufficient* conditions to guarantee that it is. We show this in section \[sec:nscon\]. First, however, we establish a simple way of identifying whether or not $d(\rho )$ is independent of $\rho $.
The test condition
==================
In this section we consider a general map $\mathcal{T}(\rho )$ of the form of Eq. (\[rhobar\]), and seek a simple condition equivalent to the independence of $d(\rho )$ on $\rho $ for all $\rho $ acting over $\mathcal{S%
}_{C}$. To do this we write $d(\rho )$ of Eq. (\[Ddef\]) by taking the complete trace over $\mathcal{H}_{C}$ and $\mathcal{H}_{A}$ rather than over $\mathcal{H}_{\bar{C}}$ and $\mathcal{H}_{\bar{A}}$, $$\begin{aligned}
d(\rho ) &=&\mathrm{Tr}_{C,A}\left( U\left( \rho \otimes \sigma \right)
U^{\dagger }\bar{P}\right) \\
&=&\mathrm{Tr}_{C,A}\left( \left( \rho \otimes \sigma \right) U^{\dagger }%
\bar{P}U\right) \\
&=&\mathrm{Tr}_{C}(\rho\, T)\end{aligned}$$ where we have introduced a *test operator* $T$ over the Hilbert space $\mathcal{H}_{C}$ as $$T=\mathrm{Tr}_{A}\left( \sigma\, U^{\dagger }\bar{P}U\right) ,$$ which does not depend on $\rho$. The operator $T$ is clearly Hermitian; it is also a positive operator, since the probability for success $d(\rho )\geq
0$ for all $\rho $. We can now identify a condition for $d(\rho )$ to be independent of $\rho$:
Theorem:
: $d(\rho )$ is independent of $\rho $, for density operators $\rho $ acting over a subspace $\mathcal{S}_{C}$ of $\mathcal{H}_{C}$, if and only if the test operator $T\,$ is proportional to the identity operator ${I}_{\mathcal{S}_{C}}$ over the subspace $\mathcal{S}_{C}.$ We refer to this condition on $T$ as the *test condition.*
Proof:
: The sufficiency of the test condition for a $d(\rho )$ independent of $\rho $ is clear. Necessity is easily established by contradiction: Suppose that $d(\rho )$ were independent of $\rho $ but $T$ not proportional to ${I}_{\mathcal{S}_{C}}$. Then at least two of the eigenkets of $T$ must have different eigenvalues; call those eigenkets $%
\left| \mu _{a}\right\rangle $ and $\left| \mu _{b}\right\rangle $. It follows that $d(\rho _{a})\neq d(\rho _{b})$, where $\rho _{a}=$ $\left| \mu
_{a}\right\rangle \left\langle \mu _{a}\right| $ and $\rho _{b}=\left| \mu
_{b}\right\rangle \left\langle \mu _{b}\right| $, in contradiction with our assumption. $\square $
When the test condition is satisfied we denote the single eigenvalue of $T$ over $\mathcal{S}_{C}$ as $\tau $, i.e., $T=\tau {I}_{\mathcal{S}_{C}}$. Then $d(\rho )=\tau $, and $\tau $ is identified as the probability that the measurement indicated success. For any given protocol the calculation of the operator $T$ gives an easy way to identify whether or not $d(\rho )$ is independent of $\rho $.
Necessary and sufficient conditions {#sec:nscon}
===================================
We can now identify necessary and sufficient conditions for a map $\bar{\rho}%
=\mathcal{T}(\rho )$, to be an operationally unitary map for $\rho $ acting on a subspace $\mathcal{S}_{C}$ of $\mathcal{H}_{C}$. They are:
1. The test condition is satisfied: Namely, the operator $$T=\mathrm{Tr}_{A}\left( \sigma \,U^{\dagger }\bar{P}U\right)$$ is proportional to the identity operator ${I}_{\mathcal{S}_{C}}$ over the subspace $\mathcal{S}_{C}$.
2. Each matrix $$w_{\bar{k},i}^{\bar{\alpha}\beta }=s_{\bar{k}}\sqrt{\frac{p_{i}}{\tau }}%
\left( \left\langle \bar{k}\right| \left\langle \bar{\alpha}\right| \right)
U\left( \left| \beta \right\rangle \left| \chi _{i}\right\rangle \right) ,$$ identified by the indices $\bar{k}$ and $i$, with row and column labels $%
\bar{\alpha}$ and $\beta $ respectively, either vanishes or is proportional to all other such nonvanishing matrices; here $\tau $ is the eigenvalue of $T
$.
The necessity of the first condition follows because it is equivalent to the independence of $d(\rho )$ on $\rho $, which was established above as a necessary condition for the transformation to be operationally unitary, as was the second condition given here. So we need only demonstrate sufficiency, which follows immediately: If the first condition is satisfied then $d(\rho )=\tau $ is independent of $\rho $, and if the second is satisfied then, from Eq. (\[tsimpform\]), we can introduce a single matrix $w^{\bar{\alpha}\beta }$ such that (\[tfinal\]) is satisfied. Then $$\sum_{\bar{\alpha}}\bar{\rho}^{\bar{\alpha}\bar{\alpha}}=\sum_{\beta ,\gamma
}\rho ^{\beta \gamma }\sum_{\bar{\alpha}}\left( w^{\bar{\alpha}\gamma
}\right) ^{*}w^{\bar{\alpha}\beta }.$$ Now the Hermitian matrix $$Y^{\gamma \beta }\equiv \sum_{\bar{\alpha}}\left( w^{\bar{\alpha}\gamma
}\right) ^{*}w^{\bar{\alpha}\beta }$$ must in fact be the unit matrix: $Y^{\gamma \beta }=\delta _{\gamma \beta }$, otherwise we would not have $$\sum_{\bar{\alpha}}\bar{\rho}^{\bar{\alpha}\bar{\alpha}}=\sum_{\beta }\rho
^{\beta \beta }$$ for an arbitrary $\rho $ over $\mathcal{S}_{C}$, and we know our general map $\bar{\rho}=\mathcal{T}(\rho )$ satisfies that condition \[see Eq. (\[tracepreserve\])\]. Thus $w^{\bar{\alpha}\beta }$ is a unitary matrix, and from the form of Eq. (\[tfinal\]) of the map from $\rho $ to $\bar{\rho}$ it follows immediately that the map is operationally unitary \[see Eqs. (\[rhodecompose\]) and (\[rhobareu\])\].
The physics of the two necessary and sufficient conditions given above is intuitively clear, and indeed the results we have derived here could have been guessed beforehand. For if the probability for success $d(\rho )$ of the measurement were dependent of the input density operator $\rho $, by monitoring the success rate in an assembly of experiments all characterized by the same input $\rho $, one could learn something about $\rho $, and we would not expect operationally unitary evolution in the presence of this kind of gain of information. And the independence of the nonvanishing matrices $w_{\bar{k},i}^{\bar{\alpha}\beta }$ on $\bar{k}$ and $i$, except for overall factors, can be understood as preventing the ‘mixedness’ of both the input ancilla state $\sigma $ and the generally high rank projector $%
\bar{P}$, from degrading the operationally unitary transformation and leading to a decrease in purity.
If a map is found to be operationally unitary, we can introduce the formally equivalent unitary operator $M$ on $\mathcal{H}_{C}$, as in Eq. (\[correspondence\]), which can then be written in terms of an effective action operator $Q$, $$M=e^{-iQ/\hbar }\; . \label{action}$$ The operator $Q$ can be determined simply by diagonalizing $M$, and its form reveals the nature of the Hamiltonian evolution simulated by the conditional measurement process. We can define an effective Hamiltonian $H_{eff}$ that characterizes an effective photon nonlinearity acting through a time $%
t_{eff}\,$by putting $H_{eff}\equiv Q/t_{eff}$, where $t_{eff}$ can be taken as the time of operation of the device.
In a special but common case, the input ancilla state is pure and the projector $\bar{P}$ is of unit rank in $\mathcal{H}_{\bar{A}}$. For cases such as this there is only one matrix $w^{\bar{\alpha}\beta }$ in the problem, and thus there is only a single necessary and sufficient condition for the map to be operationally unitary:
- In the special case of a projector $\bar{P}$ of rank 1 in $\mathcal{H}%
_{\bar{A}}$, where $\bar{P}=$ ${I}_{\bar{C}}\otimes \left| \bar{K}%
\right\rangle \left\langle \bar{K}\right| $, and a pure input ancilla state, $\sigma =\left| \chi \right\rangle \left\langle \chi \right| $, then map $%
\bar{\rho}=\mathcal{T}(\rho )$ is operationally unitary for $\rho $ acting on a subspace $\mathcal{S}_{C}$ of $\mathcal{H}_{C}$ if and only if $T$ satisfies the test condition. Here $$T=\left\langle \chi |U^{\dagger }\bar{P}U|\chi \right\rangle ,$$ which is an operator in $\mathcal{H}_{C}$. If it does satisfy this condition, then the transformation is given by $$\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }w^{\bar{\alpha}%
\beta }\rho ^{\beta \gamma }\left( w^{\bar{\delta}\gamma }\right) ^{*},
\label{spectrans}$$ where $$w^{\bar{\alpha}\beta }=\sqrt{\frac{1}{\tau }}\left( \left\langle \bar{K}%
\right| \left\langle \bar{\alpha}\right| \right) U\left( \left| \beta
\right\rangle \left| \chi \right\rangle \right) ,$$ and $\tau $ is the single eigenvalue of $T$ over $\mathcal{S}_{C}$.
Generalization to include feed-forward processing {#sec:post}
=================================================
Suppose that the measurement outcome of the ancilla does not yield the desired result, but that it signals that the output can be transformed by simply applying a (deterministic) unitary mode transformation on the output (see Fig. 2). This is called feed-forward processing and is widely used. For example, in teleportation, Alice sends Bob a classical message which allows him to correct for ‘wrong’ outcomes of Alice’s Bell measurement. Here, we can explicitly take into account feed-forward processing.
Suppose the projective measurement is characterized by a set of projectors, each identifying a different detection signature, $\left\{ \bar{P}_{(1)},%
\bar{P}_{(2)},...\bar{P}_{(N)},\bar{P}_{\perp }\right\} $, where $$\bar{P}_{\perp }={I-}\sum_{L=1}^{N}\bar{P}_{(L)},$$ and $$\bar{P}_{(L)}={I}_{\bar{C}}\otimes \sum_{\bar{k}}s_{L,\bar{k}}\left| \bar{k}%
\right\rangle \left\langle \bar{k}\right| .$$ All the $s_{L,\bar{k}}$ are equal to zero or unity, such that $$\bar{P}_{(L)}\bar{P}_{(L^{\prime })}=\bar{P}_{(L)}\delta _{LL^{\prime }}.$$ Here success arises if the measurement outcome is associated with *any* of the operators $\bar{P}_{(L)}$. And if outcome $L$ is achieved, then the computational output is processed by application of the unitary operator $\bar{V}_{(L)}$ acting over $\mathcal{H}_{\bar{C}}$. The probability of achieving outcome $L$ is $$d_{(L)}(\rho )\equiv \mathrm{Tr}_{\bar{C},\bar{A}}\left( U\left( \rho
\otimes \sigma \right) U^{\dagger }\bar{P}_{(L)}\right)$$ and if outcome $L$ is achieved the feed-forward processed computational output is then $$\bar{\rho}_{(L)}=\frac{\bar{V}_{(L)}\left[ \mathrm{Tr}_{\bar{A}}\left( \bar{P%
}_{(L)}U\left( \rho \otimes \sigma \right) U^{\dagger }\bar{P}_{(L)}\right)
\right] \bar{V}_{(L)}^{\dagger }}{\mathrm{Tr}_{\bar{C},\bar{A}}\left(
U\left( \rho \otimes \sigma \right) U^{\dagger }\bar{P}_{(L)}\right) }.$$ which defines a map $\bar{\rho}_{(L)}=\mathcal{T}_{(L)}(\rho )$ for those $%
\rho $ for which $d_{(L)}(\rho )\neq 0$. In this more general case we define the *set* of maps $\left\{ \mathcal{T}_{(L)}\right\} $ to be operationally unitary for density operators $\rho $ over the subspace $%
\mathcal{S}_{C}$ when:
- For each $\rho $ over the subspace $\mathcal{S}_{C}$ at least one of the $d_{(L)}(\rho )\neq 0$, and
- For each $\rho $ over the subspace $\mathcal{S}_{C}$, for each $L$ for which $d_{(L)}(\rho )\neq 0$ the map $\mathcal{T}_{(L)}(\rho )$ yields a $\bar{\rho}_{(L)}$ of the form of Eq. (\[rhobareu\]), independent of $L$.
The kind of arguments we have presented above can be extended to show that the necessary and sufficient conditions for such a set of maps to be operationally unitary for density operators $\rho $ over the subspace $%
\mathcal{S}_{C}$ are:
1. Test conditions are satisfied: The operators $$T_{(L)}=\mathrm{Tr}_{A}\left( \sigma U^{\dagger }\bar{P}_{(L)}U\right)$$ are each proportional to the identity operator ${I}_{\mathcal{S}C}$ over the subspace $\mathcal{S}_{C}$. The proportionality constants $\tau _{(L)}$ need not be the same for all $L$.
2. Omitting matrices associated with any $L$ for which $\tau _{(L)}=0$, each matrix $$w_{L,\bar{k},i}^{\bar{\alpha}\beta }=s_{L,\bar{k}}\sqrt{\frac{p_{i}}{\tau
_{(L)}}}\sum_{\bar{\lambda}}\bar{V}_{(L)}^{\bar{\alpha}\bar{\lambda}}\left(
\left\langle \bar{k}\right| \left\langle \bar{\lambda}\right| \right)
U\left( \left| \beta \right\rangle \left| \chi _{i}\right\rangle \right) ,$$ identified by the indices $L,\bar{k},$ and $i$, with row and column labels $%
\bar{\alpha}$ and $\beta $ respectively, either vanishes or is proportional to all other such nonvanishing matrices.
The probability of success is $\sum_{L}\tau _{(L)}=\tau .$ This expanded formalism applies to the feed-forward schemes discussed by Pittman *et al*. [@franson2] and the teleportation schemes of Gottesman and Chuang [@gottesman]. In devices such as these, a measurement provides classical information that is used in the subsequent evolution of the output state.
In a common special case, the input ancilla state is pure, $\sigma =\left|
\chi \right\rangle \left\langle \chi \right| $, and each of the projectors $%
\bar{P}_{(L)}$ is of unit rank in $\mathcal{H}_{\bar{A}}$, $\bar{P}_{(L)}={I}%
_{\bar{C}}\otimes \left| \overline{k_{L}}\right\rangle \left\langle
\overline{k_{L}}\right| $. Here the two necessary and sufficient conditions for the set of maps to be operationally unitary for density operators $\rho $ over the subspace $\mathcal{S}_{C}$ simplify to:
1. All the operators $$T_{(L)}=\left\langle \chi |U^{\dagger }\bar{P}_{(L)}U|\chi \right\rangle$$ over $\mathcal{H}_{C}$ satisfy the test condition.
2. Omitting matrices associated with any $L$ for which $\tau _{(L)}=0$, each matrix $$w_{L}^{\bar{\alpha}\beta }=\frac{1}{\sqrt{\tau _{(L)}}}\sum_{\bar{\lambda}}%
\bar{V}_{(L)}^{\bar{\alpha}\bar{\lambda}}\left( \left\langle \overline{k_{L}}%
\right| \left\langle \bar{\lambda}\right| \right) U\left( \left| \beta
\right\rangle \left| \chi \right\rangle \right) ,$$ identified by the indices $L,$ with row and column labels $\bar{\alpha}$ and $\beta $ respectively, either vanishes or is proportional to all other such nonvanishing matrices.
If these conditions are met, then the operationally unitary transformation is given by $$\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }w_{L}^{\bar{\alpha%
}\beta }\rho ^{\beta \gamma }\left( w_{L}^{\bar{\delta}\gamma }\right) ^{*},$$ which is independent of $L$.
Another extension of the standard Von Neumann, or projection, measurements is to the class of measurements described by more general positive operator valued measures, or POVMs. * *These can be used to describe more complicated measurements, often resulting from imperfections in a designed PVM. Our analysis can be generalized to POVMs by expanding the ancilla space, and then describing the POVMs by PVMs in this expanded space. In some instances operationally unitarity might still be possible; in others, the extension would allow us to study of the effect of realistic limitations such as detector loss and the lack of single-photon resolution.
Examples {#sec:examples}
========
In this section we will apply the formalism developed above to two proposed optical quantum gates for LOQC. The straightforward calculation of the effects of these gates presented in the original publications make it clear that they are operationally unitary; our purpose here is merely to illustrate how the approach we have introduced here is applied.
To evaluate the test operators $T_{(L)}$ and matrix elements $w_{L,\bar{k}%
,i}^{\bar{\alpha}\beta }$ it is useful to have expression for quantities such as $Ua_{\Omega }U^{\dagger }$, where we use capital Greek letters as subscripts on the letter $a$ to denote annihilation operators for input (computing and ancilla) channels; similarly, we use $a_{\bar{\Delta}}$ to denote annihilation operators for output (computing and ancilla) channels. We now characterize the unitary transformation $U$ by a set of quantities $_{\Omega \bar{\Delta}}^{*}$ that give the complex amplitude for an output photon in mode $\bar{\Delta}$ given an input photon in mode $%
\Omega $. That is, $$U\left( a_{\Omega }^{\dagger }\left| \text{vac}\right\rangle \right) =\sum_{%
\bar{\Delta}}\mathsf{U}_{\Omega \bar{\Delta}}^{*}\left( a_{\bar{\Delta}%
}^{\dagger }\left| \text{vac}\right\rangle \right) , \label{Uadagger}$$ where $\left| \text{vac}\right\rangle $ is the vacuum of the full Hilbert space $\mathcal{H}$. Since only linear optical elements are involved we have $U^{\dagger }\left| \text{vac}\right\rangle =\left| \text{vac}\right\rangle $, and it further follows from (\[Uadagger\]) that $$Ua_{\Omega }^{\dagger }U^{\dagger }=\sum_{\bar{\Delta}}\mathsf{U}_{\Omega
\bar{\Delta}}^{*}a_{\bar{\Delta}}^{\dagger }, \label{Udaggerevol}$$ or $$Ua_{\Omega }U^{\dagger }=\sum_{\bar{\Delta}}\mathsf{U}_{\Omega \bar{\Delta}%
}a_{\bar{\Delta}}. \label{Ufund}$$ Using the commutation relations satisfied by the creation and annihilation operators, it immediately follows that the matrix $\mathsf{U}_{\Omega \bar{%
\Delta}}$ , which identifies the unitary transformation $U$, is itself a unitary matrix. Certain calculations can be simplified by its diagonalization, but for the kind of analysis of few photon states that we require this is not necessary. We will need to express, in terms of few photon states with respect to the decomposition $\mathcal{H}_{\bar{C}}$ $%
\otimes $ $\mathcal{H}_{\bar{A}}$, the result of acting with $U$ on few photon states of the decomposition $\mathcal{H}_{C}$ $\otimes \mathcal{H}_{A}
$; this follows directly from (\[Udaggerevol\]). For example, denoting by $%
\left| 1_{\Omega _{1}}2_{\Omega _{2}}\right\rangle $ the state with one photon in mode $\Omega _{1}$ and two in mode $\Omega _{2}$, we have $$\begin{aligned}
U\left| 1_{\Omega _{1}}2_{\Omega _{2}}\right\rangle &=&Ua_{\Omega
_{1}}^{\dagger }\frac{\left( a_{\Omega _{2}}^{\dagger }\right) ^{2}}{\sqrt{2}%
}\left| \text{vac}\right\rangle \label{Uuse} \\
&=&\frac{1}{\sqrt{2}}\left( Ua_{\Omega _{1}}^{\dagger }U^{\dagger }\right)
\left( Ua_{\Omega _{2}}^{\dagger }U^{\dagger }\right) \left( Ua_{\Omega
_{2}}^{\dagger }U^{\dagger }\right) \left| \text{vac}\right\rangle
\nonumber \\
&=&\frac{1}{\sqrt{2}}\sum_{\bar{\Delta}_{1},\bar{\Delta}_{2},\bar{\Delta}%
_{3}}\mathsf{U}_{\Omega _{1}\bar{\Delta}_{1}}^{*}\mathsf{U}_{\Omega _{2}\bar{%
\Delta}_{2}}^{*}\mathsf{U}_{\Omega _{2}\bar{\Delta}_{3}}^{*}\left( a_{\bar{%
\Delta}_{1}}^{\dagger }a_{\bar{\Delta}_{2}}^{\dagger }a_{\bar{\Delta}%
_{3}}^{\dagger }\right) \left| \text{vac}\right\rangle , \nonumber\end{aligned}$$ and doing the sums in the last line allow us to indeed accomplish our goal.
KLM conditional sign flip
-------------------------
The first example we consider is the conditional sign flip discussed by Knill, Laflamme, and Milburn [@klm]. Note that in this case the input ancilla state is pure, there is no feed-forward processing, and the projector $\bar{P%
}$ is of unit rank in $\mathcal{H}_{\bar{A}}$. The necessary and sufficient conditions for the transformation to be operationally unitary are those of the special case discussed in section \[sec:nscon\]. The gate consists of one computational input port (labeled 1) and two ancilla input ports (2 and 3). The projective measurement is performed on two output ports (b,c) and the one remaining port is the computational output (a). The subspace $\mathcal{S}%
_{C}$ is spanned by the Fock states $|0\rangle$, $|1\rangle$, and $|2\rangle$ in each optical mode.
The pre-measurement evolution, which is done *via* beam splitters and a phase shifter, is given by the unitary transformation $U$ and characterized by the matrix
$$\mathsf{U}=\mathsf{U}^{*}=\left[
\begin{array}{lll}
1-\sqrt{2} & 2^{-1/4} & (3/\sqrt{2}-2)^{1/2} \\
2^{-1/4} & 1/2 & 1/2-1/\sqrt{2} \\
(3/\sqrt{2}-2)^{1/2} & 1/2-1/\sqrt{2} & \sqrt{2}-1/2
\end{array}
\right] .$$
The ancilla input state is $$\left| \chi \right\rangle =a_{2}^{\dagger }\left| \text{vac}%
_{A}\right\rangle , \label{ancilla input}$$ denoting a single photon in the 2 mode, where $\left| \text{vac}%
_{A}\right\rangle $ denotes the vacuum of $\mathcal{H}_{A}$. The projective measurement operator is given by
$$\bar{P}={I}_{\bar{C}}\otimes \left| \bar{K}\right\rangle \left\langle \bar{K}%
\right| ={I}_{\bar{C}}\otimes a_{b}^{\dagger }\left| \text{vac}_{\bar{A}%
}\right\rangle \left\langle \text{vac}_{\bar{A}}\right| a_{b}$$ which corresponds to the detection of one and only one photon in mode b, and zero photons in mode c. The basis states that define the subspace $\mathcal{S%
}_{C}$ are $$\left| 0\right\rangle =\left| \text{vac}_{C}\right\rangle , \quad \left|
1\right\rangle =a_{1}^{\dagger }\left| \text{vac}_{C}\right\rangle , \quad
\left| 2\right\rangle =\frac{\left( a_{1}^{\dagger }\right) ^{2}}{\sqrt{2}}%
\left| \text{vac}_{C}\right\rangle ,$$ and the basis states of $\mathcal{H}_{\bar{C}}$ are
$$\left| \overline{0}\right\rangle =\left| \text{vac}_{\bar{C}}\right\rangle ,
\quad \left| \overline{1}\right\rangle =a_{a}^{\dagger }\left| \text{vac}_{%
\bar{C}}\right\rangle , \quad \left| \overline{2}\right\rangle =\frac{\left(
a_{a}^{\dagger }\right) ^{2}}{\sqrt{2}}\left| \text{vac}_{\bar{C}%
}\right\rangle ,$$ In order to evaluate the test function, we first write $$U^{\dagger }\bar{P}U=\sum_{\overline{\alpha }}U^{\dagger }\left(
a_{b}^{\dagger }\left| \text{vac}_{\bar{A}}\right\rangle \otimes \left|
\overline{\alpha }\right\rangle \right) \left( \left\langle \overline{\alpha
}\right| \otimes \left\langle \text{vac}_{\bar{A}}\right| a_{b}\right) U$$ and look at the matrix elements
$$\begin{aligned}
&&\left( \left\langle \alpha \right| \otimes \left\langle \chi\right|
\right) U^{\dagger }\bar{P}U\left( \left| \chi\right\rangle \otimes \left|
\beta \right\rangle \right) \label{alphasum} \\
&=&\sum_{\overline{\alpha }}\left( \left\langle \alpha \right| \otimes
\left\langle \text{vac}_{A}\right| a_{2}\right) U^{\dagger }\left(
a_{b}^{\dagger }\left| \text{vac}_{\bar{A}}\right\rangle \otimes \left|
\overline{\alpha }\right\rangle \right) \left( \left\langle \overline{\alpha
}\right| \otimes \left\langle \text{vac}_{\bar{A}}\right| a_{b}\right)
U\left( a_{2}^{\dagger }\left| \text{vac}_{A}\right\rangle \otimes \left|
\beta \right\rangle \right) \nonumber\end{aligned}$$
over the computational subspace, $\mathcal{S}_{C}$. The calculation is straightforward. Applying the operator $U$ on each of the states $%
a_{2}^{\dagger }\left| \text{vac}_{A}\right\rangle \otimes \left| \beta
\right\rangle $ gives the following states in the $\mathcal{H}_{\bar{C}}$ $%
\otimes $ $\mathcal{H}_{\bar{A}}$ decomposition
$$\begin{aligned}
U\left( a_{2}^{\dagger }\left| \text{vac}_{A}\right\rangle \otimes \left|
0\right\rangle \right) &=&\left( 2^{-1/4}a_{a}^{\dagger }+\frac{1}{2}%
a_{b}^{\dagger }+\left[ \frac{1}{2}-\frac{1}{\sqrt{2}}\right] a_{c}^{\dagger
}\right) \left| \text{vac}\right\rangle \\
U\left( a_{2}^{\dagger }\left| \text{vac}_{A}\right\rangle \otimes \left|
1\right\rangle \right) &=&\left( 2^{-1/4}a_{a}^{\dagger }+\frac{1}{2}%
a_{b}^{\dagger }+\left[ \frac{1}{2}-\frac{1}{\sqrt{2}}\right] a_{c}^{\dagger
}\right) \\
&&\times \left( \left[ 1-\sqrt{2}\right] a_{a}^{\dagger
}+2^{-1/4}a_{b}^{\dagger }+\left[ \frac{3}{\sqrt{2}}-2\right]
^{1/2}a_{c}^{\dagger }\right) \left| \text{vac}\right\rangle \\
U\left( a_{2}^{\dagger }\left| \text{vac}_{A}\right\rangle \otimes \left|
2\right\rangle \right) &=&\frac{1}{\sqrt{2}}\left( 2^{-1/4}a_{a}^{\dagger }+%
\frac{1}{2}a_{b}^{\dagger }+\left[ \frac{1}{2}-\frac{1}{\sqrt{2}}\right]
a_{c}^{\dagger }\right) \\
&&\times \left( \left[ 1-\sqrt{2}\right] a_{a}^{\dagger
}+2^{-1/4}a_{b}^{\dagger }+\left[ \frac{3}{\sqrt{2}}-2\right]
^{1/2}a_{c}^{\dagger }\right) ^{2}\left| \text{vac}\right\rangle .\end{aligned}$$
and we can then separately evaluate the terms in the sum (\[alphasum\]), noting that the non-zero elements are $$\begin{aligned}
\left| \left( \left\langle \overline{0}\right| \otimes \left\langle \text{vac%
}_{\bar{A}}\right| a_{b}\right) U\left( a_{2}^{\dagger }\left| \text{vac}%
_{A}\right\rangle \otimes \left| 0\right\rangle \right) \right| ^{2} &=&%
\frac{1}{4} \\
\left| \left( \left\langle \overline{1}\right| \otimes \left\langle \text{vac%
}_{\bar{A}}\right| a_{b}\right) U\left( a_{2}^{\dagger }\left| \text{vac}%
_{A}\right\rangle \otimes \left| 1\right\rangle \right) \right| ^{2} &=&%
\frac{1}{4} \\
\left| \left( \left\langle \overline{2}\right| \otimes \left\langle \text{vac%
}_{\bar{A}}\right| a_{b}\right) U\left( a_{2}^{\dagger }\left| \text{vac}%
_{A}\right\rangle \otimes \left| 2\right\rangle \right) \right| ^{2} &=&%
\frac{1}{4}\end{aligned}$$ The test operator $T$ is then
$$\begin{aligned}
T &=&\frac{1}{4}\left[ \left| \text{vac}_{C}\right\rangle \left\langle \text{%
vac}_{C}\right| +a_{1}^{\dagger }\left| \text{vac}_{C}\right\rangle
\left\langle \text{vac}_{C}\right| a_{1}+\frac{\left( a_{1}^{\dagger
}\right) ^{2}}{\sqrt{2}}\left| \text{vac}_{C}\right\rangle \left\langle
\text{vac}_{C}\right| \frac{\left( a_{1}\right) ^{2}}{\sqrt{2}}\right] \\
&=&\frac{1}{4}{I}_{\mathcal{S}C},\end{aligned}$$
and is indeed a multiple of the unit operator in the computational input space. The probability of a success-indicating measurement is 1/4, independent of the computational input state. Since this test condition is satisfied, the transformation (\[spectrans\]) is operationally unitary. The terms of the transformation matrix $w^{\bar{\alpha}\beta }$ can be calculated noting that the non-zero $\left\langle \bar{K}\right|
\left\langle \bar{\alpha}\right| U\left| \beta \right\rangle \left|
\chi\right\rangle $ terms are
$$\begin{aligned}
\left\langle \bar{K}\right| \left\langle \overline{0}\right| U\left|
0\right\rangle \left| \chi\right\rangle &=&\frac{1}{2}, \\
\left\langle \bar{K}\right| \left\langle \overline{1}\right| U\left|
1\right\rangle \left| \chi\right\rangle &=&\frac{1}{2}, \\
\left\langle \bar{K}\right| \left\langle \overline{2}\right| U\left|
2\right\rangle \left| \chi\right\rangle &=&-\frac{1}{2},\end{aligned}$$
and since $\tau =1/4$ the non-zero elements of the transformation matrix are $$\begin{aligned}
w^{\overline{0}0} &=&1, \\
w^{\overline{1}1} &=&1, \\
w^{\overline{2}2} &=&-1,\end{aligned}$$ which corresponds to the conditional sign flip, since with probability 1/4 the gate takes the input state $\left| \psi \right\rangle =\alpha _{0}\left|
0\right\rangle +\alpha _{1}\left| 1\right\rangle +\alpha _{2}\left|
2\right\rangle $ and produces the state $\left| \bar{\psi}\right\rangle
=\alpha _{0}\left| \overline{0}\right\rangle +\alpha _{1}\left| \overline{1}%
\right\rangle -\alpha _{2}\left| \overline{2}\right\rangle $.
This map can be seen to exhibit an effective nonlinear interaction between the photons, since the formally equivalent unitary map (see section \[sec:consec\]) is characterized by the unitary operator $M$ (\[correspondence\]), $$\left| \tilde{\psi}\right\rangle =\alpha _{0}\left| 0\right\rangle +\alpha
_{1}\left| 1\right\rangle -\alpha _{2}\left| 2\right\rangle =M\left( \alpha
_{0}\left| 0\right\rangle +\alpha _{1}\left| 1\right\rangle +\alpha
_{2}\left| 2\right\rangle \right) ,$$ which can be written in terms of an effective action operator $Q$ (\[action\]), where we can take $$Q=\frac{\pi \hbar }{2}\left( 5\widehat{n}-\widehat{n}^{2}\right) ,$$ with $\hat{n}$ the photon number operator. But such an effective action operator exists only if we restrict ourselves to the three-dimensional subspace $\mathcal{S}_{C}$, spanned by the kets $\left| 0\right\rangle $, $%
\left| 1\right\rangle $, and $\left| 2\right\rangle $. For consider an attempt to expand this subspace to that spanned by the kets $\left( \left|
0\right\rangle ,\left| 1\right\rangle ,\left| 2\right\rangle ,\left|
3\right\rangle \right) $. The device guarantees that a computational input of three photons can only produce a computational three-photon output, since a successful measurement requires the detection of one and only one photon in the ancilla space. The test operator is therefore still diagonal in the photon number basis. However, we find $$\left| \left( \left\langle \overline{3}\right| \otimes \left\langle \text{vac%
}_{\bar{A}}\right| a_{b}\right) U\left( a_{2}^{\dagger }\left| \text{vac}%
_{A}\right\rangle \otimes \left| 3\right\rangle \right) \right| ^{2}=\left( 2%
\sqrt{2}-\frac{5}{2}\right) ^{2},$$ and thus the test operator $T$ is no longer a multiple of the unit operator in this enlarged subspace. In this larger space the probability of a success-indicating measurement is dependent on the input, and the map is not operationally unitary.
Polarization encoded CNOT
-------------------------
The second example is the polarization-encoded Gottesman-Chuang protocol discussed by Pittman *et al*. [@franson]. In this case the input ancilla state is pure, there is feed-forward processing, and there are several projectors $\bar{P}_{(L)}$ of unit rank in $\mathcal{H}_{\bar{A}}$. The necessary and sufficient conditions for the transformation to be operationally unitary are therefore those of the special case discussed in section \[sec:post\]. The device has two computational input ports (labeled $a$ and $b$) and four ancilla input ports (1-4). A projective measurement is made on four output ports ($p$,$q$,$n$,$m$) while the two remaining ports are the computational output (5 and 6). A photon of horizontal polarization represents a logical 0, and a vertically polarized photon represents a logical 1. We use the same notation as Pittman *et al.* [@franson]. For example, $\left| H(V)_{a}\right\rangle $ represents a horizontally(vertically) polarized photon in port ‘a’ and the Hadamard transformed modes are $\left| F(S)_{a}\right\rangle =\frac{1}{2}\left[
\left| H_{a}\right\rangle \pm \left| V_{a}\right\rangle \right] $. The four basis states of the computational input are $\left| 00\right\rangle =$ $%
\left| H_{a}\right\rangle \left| H_{b}\right\rangle ,$ $\left|
01\right\rangle =$ $\left| H_{a}\right\rangle \left| V_{b}\right\rangle ,$ $%
\left| 10\right\rangle =$ $\left| V_{a}\right\rangle \left|
H_{b}\right\rangle ,$ $\left| 11\right\rangle =$ $\left| V_{a}\right\rangle
\left| V_{b}\right\rangle $ and the output states are labeled as $\left|
\overline{00}\right\rangle =\left| H_{5}\right\rangle \left|
H_{6}\right\rangle ,\left| \overline{01}\right\rangle =\left|
H_{5}\right\rangle \left| V_{6}\right\rangle ,\left| \overline{10}%
\right\rangle =\left| V_{5}\right\rangle \left| H_{6}\right\rangle ,\left|
\overline{11}\right\rangle =\left| V_{5}\right\rangle \left|
V_{6}\right\rangle .$ The input ancilla state is $$\begin{aligned}
\left| \chi \right\rangle &=&\frac{1}{2}\left( \left| H_{1}\right\rangle
\left| H_{4}\right\rangle \left| H_{2}\right\rangle \left|
H_{3}\right\rangle +\left| H_{1}\right\rangle \left| V_{4}\right\rangle
\left| H_{2}\right\rangle \left| V_{3}\right\rangle \right) \\
&&+\frac{1}{2}\left( \left| V_{1}\right\rangle \left| H_{4}\right\rangle
\left| V_{2}\right\rangle \left| V_{3}\right\rangle +\left|
V_{1}\right\rangle \left| V_{4}\right\rangle \left| V_{2}\right\rangle
\left| H_{3}\right\rangle \right) ,\end{aligned}$$ and the measurement projectors, $\bar{P}_{(L)}={I}_{\bar{C}}\otimes \left|
\overline{k_{L}}\right\rangle \left\langle \overline{k_{L}}\right| ,$ represent the 16 possible success outcomes: $$\begin{aligned}
\left| \overline{k_{1}}\right\rangle &=&\left| F_{p}\right\rangle \left|
F_{q}\right\rangle \left| F_{n}\right\rangle \left| F_{m}\right\rangle \\
&=&\frac{1}{4}\left( \left| H_{p}\right\rangle +\left| V_{p}\right\rangle
\right) \left( \left| H_{q}\right\rangle +\left| V_{q}\right\rangle \right)
\left( \left| H_{n}\right\rangle +\left| V_{n}\right\rangle \right) \left(
\left| H_{m}\right\rangle +\left| V_{m}\right\rangle \right) \\
\left| \overline{k_{2}}\right\rangle &=&\left| F_{p}\right\rangle \left|
F_{q}\right\rangle \left| F_{n}\right\rangle \left| S_{m}\right\rangle \\
&=&\frac{1}{4}\left( \left| H_{p}\right\rangle +\left| V_{p}\right\rangle
\right) \left( \left| H_{q}\right\rangle +\left| V_{q}\right\rangle \right)
\left( \left| H_{n}\right\rangle +\left| V_{n}\right\rangle \right) \left(
\left| H_{m}\right\rangle -\left| V_{m}\right\rangle \right) \\
&& \\
&&\vdots \\
&& \\
\left| \overline{k_{15}}\right\rangle &=&\left| S_{p}\right\rangle \left|
S_{q}\right\rangle \left| S_{n}\right\rangle \left| F_{m}\right\rangle \\
&=&\frac{1}{4}\left( \left| H_{p}\right\rangle -\left| V_{p}\right\rangle
\right) \left( \left| H_{q}\right\rangle -\left| V_{q}\right\rangle \right)
\left( \left| H_{n}\right\rangle -\left| V_{n}\right\rangle \right) \left(
\left| H_{m}\right\rangle +\left| V_{m}\right\rangle \right) \\
\left| \overline{k_{16}}\right\rangle &=&\left| S_{p}\right\rangle \left|
S_{q}\right\rangle \left| S_{n}\right\rangle \left| S_{m}\right\rangle \\
&=&\frac{1}{4}\left( \left| H_{p}\right\rangle -\left| V_{p}\right\rangle
\right) \left( \left| H_{q}\right\rangle -\left| V_{q}\right\rangle \right)
\left( \left| H_{n}\right\rangle -\left| V_{n}\right\rangle \right) \left(
\left| H_{m}\right\rangle -\left| V_{m}\right\rangle \right) \end{aligned}$$ The polarizing beam splitters perform a unitary evolution on the input ports, characterized by the set of quantities $_{\Omega \bar{\Delta%
}}^{*}$. One can summarize the evolution of modes in $\mathcal{H}_{C}$ $%
\otimes$ $\mathcal{H}_{A}$ to modes in $\mathcal{H}_{\bar{C}}$ $\otimes $ $%
\mathcal{H}_{\bar{A}}$ with the following linear map $$\begin{aligned}
\left| H_{1}\right\rangle &\rightarrow &\left| H_{p}\right\rangle ,\left|
V_{1}\right\rangle \rightarrow -i\left| V_{q}\right\rangle , \\
\left| H_{2}\right\rangle &\rightarrow &\left| H_{5}\right\rangle ,\left|
V_{2}\right\rangle \rightarrow \left| V_{5}\right\rangle , \\
\left| H_{3}\right\rangle &\rightarrow &\left| H_{6}\right\rangle ,\left|
V_{3}\right\rangle \rightarrow \left| V_{6}\right\rangle , \\
\left| H_{4}\right\rangle &\rightarrow &\left| H_{m}\right\rangle ,\left|
V_{4}\right\rangle \rightarrow -i\left| V_{n}\right\rangle , \\
\left| H_{a}\right\rangle &\rightarrow &\left| H_{q}\right\rangle ,\left|
V_{a}\right\rangle \rightarrow -i\left| V_{p}\right\rangle , \\
\left| H_{b}\right\rangle &\rightarrow &\left| H_{n}\right\rangle ,\left|
V_{b}\right\rangle \rightarrow -i\left| V_{m}\right\rangle ,\end{aligned}$$ since $_{H_{1}H_{p}}^{*}=1,$ $_{V_{1}V_{q}}^{*}=-i,$ *etc*. As in the previous example, to evaluate the test operators, we first look at the terms $$\begin{aligned}
U\left( \left| 00\right\rangle \left| \chi \right\rangle \right) &=&\frac{%
\left| H_{q}\right\rangle \left| H_{n}\right\rangle }{2}\left[
\begin{array}{c}
\left| H_{p}\right\rangle \left| H_{m}\right\rangle \left|
H_{5}\right\rangle \left| H_{6}\right\rangle -i\left| H_{p}\right\rangle
\left| V_{n}\right\rangle \left| H_{5}\right\rangle \left|
V_{6}\right\rangle \\
-i\left| V_{q}\right\rangle \left| H_{m}\right\rangle \left|
V_{5}\right\rangle \left| V_{6}\right\rangle -\left| V_{q}\right\rangle
\left| V_{n}\right\rangle \left| V_{5}\right\rangle \left|
H_{6}\right\rangle
\end{array}
\right] \\
U\left( \left| 01\right\rangle \left| \chi \right\rangle \right) &=&\frac{%
-i\left| H_{q}\right\rangle \left| V_{m}\right\rangle }{2}\left[
\begin{array}{c}
\left| H_{p}\right\rangle \left| H_{m}\right\rangle \left|
H_{5}\right\rangle \left| H_{6}\right\rangle -i\left| H_{p}\right\rangle
\left| V_{n}\right\rangle \left| H_{5}\right\rangle \left|
V_{6}\right\rangle \\
-i\left| V_{q}\right\rangle \left| H_{m}\right\rangle \left|
V_{5}\right\rangle \left| V_{6}\right\rangle -\left| V_{q}\right\rangle
\left| V_{n}\right\rangle \left| V_{5}\right\rangle \left|
H_{6}\right\rangle
\end{array}
\right] \\
U\left( \left| 10\right\rangle \left| \chi \right\rangle \right) &=&\frac{%
-i\left| V_{p}\right\rangle \left| H_{n}\right\rangle }{2}\left[
\begin{array}{c}
\left| H_{p}\right\rangle \left| H_{m}\right\rangle \left|
H_{5}\right\rangle \left| H_{6}\right\rangle -i\left| H_{p}\right\rangle
\left| V_{n}\right\rangle \left| H_{5}\right\rangle \left|
V_{6}\right\rangle \\
-i\left| V_{q}\right\rangle \left| H_{m}\right\rangle \left|
V_{5}\right\rangle \left| V_{6}\right\rangle -\left| V_{q}\right\rangle
\left| V_{n}\right\rangle \left| V_{5}\right\rangle \left|
H_{6}\right\rangle
\end{array}
\right] \\
U\left( \left| 11\right\rangle \left| \chi \right\rangle \right) &=&\frac{%
-\left| V_{p}\right\rangle \left| V_{m}\right\rangle }{2}\left[
\begin{array}{c}
\left| H_{p}\right\rangle \left| H_{m}\right\rangle \left|
H_{5}\right\rangle \left| H_{6}\right\rangle -i\left| H_{p}\right\rangle
\left| V_{n}\right\rangle \left| H_{5}\right\rangle \left|
V_{6}\right\rangle \\
-i\left| V_{q}\right\rangle \left| H_{m}\right\rangle \left|
V_{5}\right\rangle \left| V_{6}\right\rangle -\left| V_{q}\right\rangle
\left| V_{n}\right\rangle \left| V_{5}\right\rangle \left|
H_{6}\right\rangle
\end{array}
\right] \end{aligned}$$ The matrix elements of interest are now
$$\begin{aligned}
&&\left( \left\langle \alpha \right| \otimes \left\langle \chi \right|
\right) U^{\dagger }\bar{P}_{(L)}U\left( \left| \chi \right\rangle \otimes
\left| \beta \right\rangle \right) \label{alphasum2} \\
&=&\sum_{\overline{\alpha }}\left( \left\langle \alpha \right| \otimes
\left\langle \chi \right| \right) U^{\dagger }\left| \overline{k_{L}}%
\right\rangle \left| \overline{\alpha }\right\rangle \left\langle \overline{%
\alpha }\right| \left\langle \overline{k_{L}}\right| U\left( \left| \chi
\right\rangle \otimes \left| \beta \right\rangle \right) \nonumber\end{aligned}$$
and the non-zero terms of the sum in (\[alphasum2\]) are $$\begin{aligned}
\left| \left\langle \overline{00}\right| \left\langle \overline{k_{L}}%
\right| U\left( \left| \chi \right\rangle \otimes \left| 00\right\rangle
\right) \right| ^{2} &=&\frac{1}{16} \\
\left| \left\langle \overline{01}\right| \left\langle \overline{k_{L}}%
\right| U\left( \left| \chi \right\rangle \otimes \left| \overline{01}%
\right\rangle \right) \right| ^{2} &=&\frac{1}{16} \\
\left| \left\langle \overline{11}\right| \left\langle \overline{k_{L}}%
\right| U\left( \left| \chi \right\rangle \otimes \left| \overline{10}%
\right\rangle \right) \right| ^{2} &=&\frac{1}{16} \\
\left| \left\langle \overline{10}\right| \left\langle \overline{k_{L}}%
\right| U\left( \left| \chi \right\rangle \otimes \left| \overline{11}%
\right\rangle \right) \right| ^{2} &=&\frac{1}{16}\end{aligned}$$ for all $L$. The test functions, $\left\{ T_{(L)}\right\} $ , are then
$$\begin{aligned}
T_{(L)} &=&\frac{1}{64}\left[
\begin{array}{c}
\left| H_{a}\right\rangle \left| H_{b}\right\rangle \left\langle
H_{b}\right| \left\langle H_{a}\right| +\left| H_{a}\right\rangle \left|
V_{b}\right\rangle \left\langle V_{b}\right| \left\langle H_{a}\right| \\
+\left| V_{a}\right\rangle \left| H_{b}\right\rangle \left\langle
H_{b}\right| \left\langle V_{a}\right| +\left| V_{a}\right\rangle \left|
V_{b}\right\rangle \left\langle V_{b}\right| \left\langle V_{a}\right|
\end{array}
\right] \\
&=&\frac{1}{64}\;{I}_{\mathcal{S}C}\end{aligned}$$
and are indeed multiples of the unit operator in the computational input space. In this scheme $\tau _{(L)}=1/64$, and the probability of success is the sum of the individual probabilities of the 16 detection outcomes, $%
\sum_{L}\tau _{(L)}=1/4.$ The terms of the transformation matrices $w_{L}^{%
\bar{\alpha}\beta }$ can be calculated noting that the non-zero $%
\left\langle \overline{k_{L}}\right| \left\langle \bar{\lambda}\right|
U\left| \beta \right\rangle \left| \chi \right\rangle $ terms are $$\begin{aligned}
\left\langle \overline{k_{L}}\right| \left\langle \overline{00}\right|
U\left| 00\right\rangle \left| \chi \right\rangle &=&e^{i\phi _{L,0}}/8, \\
\left\langle \overline{k_{L}}\right| \left\langle \overline{01}\right|
U\left| 01\right\rangle \left| \chi \right\rangle &=&e^{i\phi _{L,1}}/8, \\
\left\langle \overline{k_{L}}\right| \left\langle \overline{11}\right|
U\left| 10\right\rangle \left| \chi \right\rangle &=&e^{i\phi _{L,2}}/8, \\
\left\langle \overline{k_{L}}\right| \left\langle \overline{10}\right|
U\left| 11\right\rangle \left| \chi \right\rangle &=&e^{i\phi _{L,3}}/8,\end{aligned}$$ where $e^{i\phi _{L,0}}=1,e^{i\phi _{1,1}}=-1,e^{i\phi _{2,1}}=1,\ldots
,e^{i\phi _{16,3}}=1$ are phase factors of $\pm $1$.$ For this transformation to be operationally unitary, the $w_{L}^{\bar{\alpha}\beta }$ matrices must all be proportional to each other. In certain outcomes, single-qubit operations ($\pi $-phase shifts) are required to correct the phase factors so that the transformation is operationally unitary and the desired output is produced. The feed-forward processing matrices, $\bar{V}_{(L)}^{%
\bar{\alpha}\bar{\lambda}}$, represent these single qubit operations. Setting $$\bar{V}_{(L)}^{\overline{00},\overline{00}}=e^{i\phi _{L,0}},\quad \bar{V}%
_{(L)}^{\overline{01},\overline{01}}=e^{i\phi _{L,1}},\quad \bar{V}_{(L)}^{%
\overline{11},\overline{11}}=e^{i\phi _{L,2}},\quad \bar{V}_{(L)}^{\overline{%
10},\overline{10}}=e^{i\phi _{L,3}}\;,$$ with all other elements equal to zero gives the appropriate corrections. The non-zero transformation matrix elements, are then $$\begin{aligned}
w_{L}^{\overline{00},00} &=&1 \\
w_{L}^{\overline{01},01} &=&1 \\
w_{L}^{\overline{11},10} &=&1 \\
w_{L}^{\overline{10},11} &=&1\end{aligned}$$ for all $L$. Since the 16 evolution matrices are identical, the proportionality condition is satisfied. The transformation is then $$\bar{\rho}^{\bar{\alpha}\bar{\delta}}=\sum_{\beta ,\gamma }w_{L}^{\bar{\alpha%
}\beta }\rho ^{\beta \gamma }\left( w_{L}^{\bar{\delta}\gamma }\right) ^{*}$$ which is the CNOT operation. This gate takes the input state $\left| \psi
\right\rangle =\alpha _{0}\left| 00\right\rangle +\alpha _{1}\left|
01\right\rangle +\alpha _{2}\left| 10\right\rangle +\alpha _{3}\left|
11\right\rangle $ and produces the state $\alpha _{0}\left| \overline{00}%
\right\rangle +\alpha _{1}\left| \overline{01}\right\rangle +\alpha
_{2}\left| \overline{11}\right\rangle +\alpha _{3}\left| \overline{10}%
\right\rangle $ with probability 1/4. Again, this map exhibits an effective nonlinear interaction between the photons since the formally equivalent unitary map is characterized by a nonlinear effective action operator $Q$ (\[action\]). In this case one could choose $$Q=\frac{\pi \hbar }{2}(3+a_{b}^{\dagger }(1-\hat{n}_{b})+(1-\hat{n}%
_{b})a_{b})\hat{n}_{a}.$$ Again, however, the operational unitarity is restricted to the subspace. Suppose we expand the computational subspace to include an extra photon in one of the input modes. As an example, consider the special state $\left|
S\right\rangle =$ $\left| H_{a}\right\rangle \left| H_{b}\right\rangle
\left| H_{b}\right\rangle $. The form of the projectors indicates that the detection events involve one and only one photon in the appropriate modes. Evaluating the corresponding test operator elements we find $$\left| \left\langle \overline{\alpha }\right| \left\langle \overline{k_{L}}%
\right| U\left( \left| \chi \right\rangle \otimes \left| S\right\rangle
\right) \right| ^{2}=0\;,$$ since the extra photon inhibits a success-indicating measurement result. The evolution cannot be operationally unitary in this expanded subspace because the test operator is no longer proportional to the unit operator.
Conclusion {#sec:conclusions}
==========
In this paper we introduced a general approach to the investigation of conditional measurement devices. We considered an important class of optical $N$-port devices, including those employing projectors of rank greater than unity, mixed input ancilla states, multiple success outcomes, and feed-forward processing. We also sketched how more general POVMs, rather than PVMs, could be included. The necessary and sufficient conditions for these devices to simulate unitary evolution have been derived. They are not surprising, and indeed from a physical point of view are fairly obvious. But to our knowledge they have not been discussed in this general way before. One of the conditions is that the probability of each successful outcome must be independent of the input density operator. Whether or not this holds can be checked by evaluating a set of test operators over the input computational Hilbert space, which is easily done for any proposed device. In the special case of only one successful outcome there is only one test operator to be computed; furthermore, if the ancilla state is pure and the success projector of rank one, then the passing of a test condition by that single test operator guarantees that the map is operationally unitary. In the case of more than one successful outcome it is a necessary consequence of operational unitarity that each of the test operators pass the test condition. This is not sufficient to imply operational unitarity in the multiple projector case unless the proportionality condition is also satisfied. The proportionality condition can often be satisfied by introducing feed-forward processing.
Besides application in the analysis of particular proposed devices, we believe the general framework presented here will be useful in exploring the different types of pre-measurement evolution and measurements that might be useful in the design, optimization, and characterization of such devices. In particular, the conditional sign flip and polarization-encoded CNOT devices we considered functioned as operationally unitary maps only over the input computational subspaces for which they were originally proposed. So while effective photon nonlinearities could be introduced, the degree to which they are physically meaningful is somewhat limited. An outstanding issue, perhaps even of interest more from the general perspective of nonlinear optics than from that of quantum computer design, is the study of potential devices that provide effective photon nonlinearities over much larger input computational subspaces. The question remains: to what extent are such devices possible in theory and feasible in practice?
Finally, we note that only in section \[sec:examples\] did we assume that the pre-measurement unitary evolution $U$ is associated with linear elements in an optical system. The more general framework of the earlier sections may find application in describing other proposed devices for quantum information processing that involve conditional measurement schemes in the presence of more complicated interactions [@dfs].
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the Natural Sciences and Engineering Research Council of Canada and the Walter C. Sumner Foundation. Part of this work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. In addition, P.K. acknowledges the United States National Research Council. Support was also received from the Advanced Research and Development Activity, the National Security Agency, the Defense Advanced Research Projects Agency, and the Office of Naval Research. We would like to thank Alexei Gilchrist, James Franson, and Gerard Milburn for stimulating discussions.
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(400,150)(0,0) (150,0)[(60,120)[U]{}]{} (75,90)[(1,0)[75]{}]{} (75,30)[(1,0)[75]{}]{} (210,90)[(1,0)[75]{}]{} (210,30)[(1,0)[25]{}]{} (235,10)[(70,40)]{} (30,75)[Computational Input]{} (220,75)[Computational Output]{} (40,15)[Ancilla Input]{} (240,33)[Measurement]{} (260,18)[on $\mathcal{H}_{\bar{A}}$]{} (290,90)[$\mathcal{H}_{\bar{C}}$]{} (55,30)[$\mathcal{H}_{A}$]{} (55,90)[$\mathcal{H}_{C}$]{}
(400,200) (120,0)[(60,120)[U]{}]{} (75,90)[(1,0)[45]{}]{} (75,30)[(1,0)[45]{}]{} (275,90)[(1,0)[20]{}]{} (180,30)[(1,0)[25]{}]{} (180,90)[(1,0)[25]{}]{} (205,70)[(70,40)]{} (205,10)[(70,40)]{} (210,33)[Measurement]{} (229,18)[on $\mathcal{H}_{\bar{A}}$]{} (212,93)[Feed-forward]{} (212,78)[processing]{} (300,90)[$\mathcal{H}_{\bar{C}}$]{} (55,30)[$\mathcal{H}_{A}$]{} (55,90)[$\mathcal{H}_{C}$]{} (242,50)[(0,1)[20]{}]{} (238,50)[(0,1)[20]{}]{}
|
---
abstract: 'In this paper we investigate the following question: under what conditions can a second-order homogeneous ordinary differential equation (spray) be the geodesic equation of a Finsler space. We show that the Euler-Lagrange partial differential system on the energy function can be reduced to a first order system on this same function. In this way we are able to give effective necessary and sufficient conditions for the local existence of a such Finsler metric in terms of the holonomy algebra generated by horizontal vector-fields. We also consider the Landsberg metrizability problem and prove similar results. This reduction is a significant step in solving the problem whether or not there exists a non-Berwald Landsberg space.'
address: 'Institute of Mathematics, University of Debrecen, Debrecen, H-4010, PBox 12, Hungary'
author:
- Zoltán Muzsnay
title: 'The Euler-Lagrange PDE and Finsler metrizability'
---
Introduction {#sec:introduction}
============
A Finsler structure on an $n$-manifold $M$ is a nonnegative function $F:TM \to {\mbox{$\Bbb R$}}$ that is smooth and positive away from the zero section of $TM$, positively homogeneous of degree 1, and strictly convex on each tangent space. The energy function $E:TM \to {\mbox{$\Bbb R$}}$ associated to a Finsler structure $F$ is defined as $E:=\frac{1}{2}F^2$. This is a direct generalization of a Riemannian structure. The fundamental tensor $g_{_E}$ associated to $E$ is formally analogous to the metric tensor in Riemannian geometry. It is defined by $$\label{eq:g}
(g_{_E})_{ij} := \frac{\partial^2 E}{ \partial {y^i} \partial
{y^j}},$$ in an induced standard coordinate system $(x,y)$ on $TM$.
As in Riemannian geometry, a canonical connection $\Gamma$ can be defined for a Finsler space [@Gri].
However, since the energy function is not necessarily quadratic and only homogeneous, the connection is in general non-linear. We mention two special types of Finsler spaces: *Berwald spaces*, where the connection $\Gamma$ is linear, and *Landsberg spaces*, where the connection $\Gamma$ is metric, i.e. the parallel transport preserves the norm defined by $g_{\scriptscriptstyle E}$.
Suppose that $M$ is an $n$-manifold endowed with a Finsler structure. The geodesics are the extremals of the variational problem in which the Lagrangian is the energy function. Since $g_{\scriptscriptstyle
E}$ is non-degenerate, the parametrization of the extremals is fixed. The geodesic equation associated to a Finsler structure is described by the Euler-Lagrange equations $$\label{eq:E-L_loc}
\hphantom{\qquad i=1,...,n}
\frac{d}{dt}\frac{\partial E}{\partial y^i }- \frac{\partial E}{\partial
x^i} = 0, \qquad i=1,...,n.$$ Recently several papers were devoted to the problem of characterizing second-order differential equations coming from a Finsler, a special Finsler, or a generalized Finsler structure (see for example [@Bao_Shen], [@bryant], [@Kozma_1], [@ok], [@CST], [@shen], [@szil_vat]). In this paper we offer a contribution to the solution of this problem. Now we formulate the problem from our point of view.
A second-order differential equation on $M$, locally given by $$\label{eq:sode}
\hphantom{\qquad i=1,...,n} \ddot{x}^i=f^i(x, \dot{x}), \qquad
i=1,...,n,$$ where the functions $f^i$ are positive homogeneous of degree 2 in the $\dot{x}$ variable, is called Finsler metrizable, if there exists a Finsler structure whose geodesics are described by (\[eq:sode\]). Moreover, (\[eq:sode\]) is Landsberg metrizable, if it is Finsler metrizable, and in addition we also have $$\label{eq:E_h_metric_loc}
\frac{\partial g_{jk}}{\partial x^i } - \Gamma_i^l \frac{\partial
g_{jk}}{\partial y^l} - \Gamma_{ik}^l g_{l j} - \Gamma_{ij}^{l}
g_{lk}= 0,$$ where $\Gamma^i_j:= -\frac{1}{2} \frac{\partial f^i}{\partial y^j}$ are the components of the connection $\Gamma$ associated to (\[eq:sode\]), $\Gamma^i_{jk}:= \frac{\partial
\Gamma^i_j}{\partial y^k}$, and $g_{ij}=(g_{_E})_{ij}$.
It follows that a second-order system (\[eq:sode\]) is Finsler metrizable if and only if there exists a function $E:TM \to {\mbox{$\Bbb R$}}$ (energy function), so that
1. $E$ is homogeneous of degree 2,
2. $E$ is a solution of the Euler-Lagrange system (\[eq:E-L\_loc\]) considered as a second-order partial differential equation with respect to $E$,
3. the quadratic form $g_{\scriptscriptstyle E}$ defined by (\[eq:g\]) is positive definite.
Euler’s theorem for homogeneous functions implies that the homogeneity condition on $E$ can be described by the equation $$\label{eq:E_hom_loc}
y^i \frac{\partial E}{\partial y^i}-2E = 0.$$ The Euler-Lagrange partial differential equation associated to (\[eq:sode\]) is $$\label{eq:E_EL_loc}
\hphantom{\qquad j=1,..,n} y^j \frac{\partial^2 E}{\partial x^j
\partial y^i}+f^j \frac{\partial^2 E}{\partial y^j \partial y^i}-
\frac{\partial E}{\partial x^i } = 0, \qquad i=1,..,n,$$ in induced local coordinates $(x,y)$ on $TM$. We arrive at the reformulation of the metrizability property in terms of a partial differential system:
\[theo:reformulation\] A second-order differential equation (\[eq:sode\]) is
1. *Finsler metrizable*, if and only if, there exists a solution $E:TM \to {\mbox{$\Bbb R$}}$ to the second-order PDE system formed by the equations (\[eq:E\_hom\_loc\]) and (\[eq:E\_EL\_loc\]) so that the quadratic form $g_E$ defined in (\[eq:g\]) is positive definite;
2. *Landsberg metrizable*, if and only if there exists a solution $E:TM \to {\mbox{$\Bbb R$}}$ to the third-order[^1] PDE system (\[eq:E\_h\_metric\_loc\]), (\[eq:E\_hom\_loc\]) and (\[eq:E\_EL\_loc\]) so that the quadratic form $g_E$ is positive definite.
The main results of this paper can be found in Sections \[cha:finsler\] and \[cha:landsberg\].
In Section \[cha:finsler\] we consider the problem of Finsler metrizability. Using the integrability conditions of the corresponding PDE, we show that the system is equivalent to a first order PDE on the same unknown function (Theorem \[theo:finsler\]). We formulate a necessary and sufficient condition for the local metrizability in terms of a distribution $\H$ associated to the spray (Theorem \[theo:C\_D\] and \[theo:finsler\_2\]). $\H$ is called *holonomy distribution* or *holonomy algebra* [@Kle], and it is generated by the horizontal vector fields and their successive Lie-brackets.
In Section \[cha:landsberg\] we consider the problem of Landsberg metrizability. We show that the corresponding third-order system can be reduced to a first order PDE on the same energy function (Theorem \[theo:landsberg\]). As in the previous case, we are able to formulate a necessary and sufficient condition for the metrizability in terms of a distribution $\L$ (Theorem \[theo:landsberg\_2\]). The distribution $\L$ is generated by the holonomy algebra and the image of the Berwald curvature.
In Sections \[sec:remarks\] and \[sec:exist-non-berw\] we illustrate some consequences of the results on Finsler and Landsberg metrizability. We also discuss the famous problem of whether there exists a non-Berwald Landsberg space. As we show through several examples, Theorem \[theo:landsberg\_2\] offers a promising alternative approach to solve this problem.
Preliminaries
=============
Notations, conventions
----------------------
Throughout this paper $M$ will denote an $n$-dimensional smooth manifold. $C^\infty(M)$ denotes the ring of real-valued smooth functions, $\mathfrak X(M)$ is the $C^\infty(M)$-module of vector fields on $M$, $\pi:TM \to M$ is the tangent bundle of $M$, $\TM=TM \setminus 0$ is the slit tangent space. We will essentially work on the manifold $TM$ and on its tangent space $TTM$. When there is no danger of confusion, $TTM$ and $T^*TM$ will simply be denoted by $T$ and $T^*$, respectively. $T^v=\mathrm{Ker}\,\pi_*$ will be the vertical sub-bundle of $T$.
The exterior differential, the Lie differential (with respect to $X
\in \mathfrak X (M)$) and the interior product (induced by $X$) are denoted by $d$, ${\mathcal L}_X$ and $i_X$, respectively.
We denote by $\Lambda^k(M)$ and $S^k(M)$ the $C^\infty (M)$-modules of the skew-symmetric and symmetric $k$-forms. The Frölicher-Nijenhuis theory provides a complete description of the derivation of $\Lambda(M)$ with the help of vector-valued differential forms, for details we refer to [@FN]. The $i_*$ and the $d_*$ type derivation associated to a vector valued $l$-form $L$ will be denoted by $i_L$ and $d_L$. They can be defined in the following way:
1. if ${\rm deg}\,L = 0$, i.e. $L \in \mathfrak{X}(M)$, then $i_L
\omega : = \omega(L)$, and $d_L\omega : = {\mathcal L}_L\omega$;
2. if ${\rm deg} \ L =l>1$, then
$ i_L\omega(X_1,...,X_l) : = \omega(L(X_1,...,X_l)),
$ for $\omega \in \Lambda^1(M)$;
$ d_Lf(X_1,\cdots,X_l) : = df(L(X_1,\cdots,X_l)),
$ for $f \in C^\infty(M)$.
Geometry associated to a spray
------------------------------
Let $J$ be the canonical vertical endomorphism of $T~(=TTM)$ and $C \in
\mathfrak X(TM)$ the canonical vertical vector field. In an induced local coordinate system $(x^i,y^i)$ on $TM$ we have $$J = dx^i \otimes \frac {\partial}{\partial y^i },
\qquad C = y^i \frac {\partial}{\partial y^i }.$$
\[theo:Pc\] Using the canonical vector-field, equation (\[eq:E\_hom\_loc\]) can be written in the form $\Pc E = 0$, where $ \Pc : C^\infty(TM) \to C^\infty(TM)
$ is a first-order differential operator defined on a function $E:TM
\to {\mbox{$\Bbb R$}}$ by $$\label{eq:E_hom}
\Pc E:= {\mathcal L}_C E - 2E.$$
A *spray* is a vector field $S \in \mathfrak X(TM)$ on $TM$ satisfying the relations $JS = C$ and $[C,S]=S$. The coordinate representation of a spray $S$ takes the form $$\label{eq:S}
S = y ^i \frac {\partial}{\partial x^i } +f^i (x,y)
\frac {\partial}{\partial y^i },$$ where $f^i(x,y)$ is positive-homogeneous of degree 2 in $y=(y^j)$. The integral curves of a spray are curves $\gamma : I \to M$ so that $S
\circ \dot \gamma = \ddot \gamma$. They are the solutions of the equations $ \ddot{x}^i = f^i(x,\dot{x}).$
To every spray $S$ a *connection* $\Gamma: = [J,S]$ can be associated [@Gri]. We have $\Gamma ^2 = \textrm{id}_T$, and the eigenspace of $\Gamma$ corresponding to the eigenvalue $-1$ is the vertical space $T^v$. We denote the eigenspace belonging to the eigenvalue +1 of $\Gamma$ by $T^h$ and we call it the *horizontal space*. Then $$T = T^h \oplus T^v.$$ The horizontal and the vertical projector belonging to $\Gamma$ are $ h : = \frac{1}{2}(\textrm{I} + \Gamma ),$ and $v := \mathrm{id}_T-h$. The almost complex structure associated to $\Gamma$ is the vector valued 1-form $F$ on $TM$ such that $FJ = h$ and $Fh = -J$. The *curvature* of the connection $\Gamma$ is the vector-valued 2-form $$\label{def:R}
R : = -\frac{1}{2}[h,h].$$ A linear connection on $TM$, called the [*Berwald connection*]{}, can also be associated to $S$. It is defined by: $$\n \Gamma = 0, \qquad \n_{hX} JY = [ h, JY]X, \qquad \n_{JX} JY = [ J,
JY]X;$$ $X, Y \in \mathfrak X (TM)$. In an induced coordinate system $(x,y)$ we have $$\label{berwald}
\left\{ \quad
\begin{aligned}{}
\n_{_{\p{y^i}}} {\scriptstyle \p{y^j}}&=0,
\\
\n_{_{ \p{x^i}}} {\scriptstyle \p{y^j}}&= \n_{_{\p {y^j}}}
{\scriptstyle \p{x^i}} = \Gamma_{i j}^k \, {\scriptstyle \p
{y^k}},
\\
\n_{_{\p{x^i}}} {\scriptstyle \p{x^j}} &= \Gamma_{i j}^k
{\scriptstyle \p {x^k}}
+ \Bigl( \tfrac{\partial \Gamma_j^l} {\partial x^i}
+ \Gamma_j^k \Gamma_{i k}^l
- \Gamma_k^l \Gamma_{i j}^k \Bigl)\, {\scriptstyle \p {y^l}}.
\end{aligned}
\right.$$ where $\Gamma^k_i:= -\frac{1}{2}\frac{\partial f^k}{\partial y^i}$ and $\Gamma^k_{ij}:= \frac{\partial \Gamma^k_i}{\partial y^j}$. Considering the $(h,v,v)$ components of the classical curvature of the Berwald connection we obtain a tensor-field $$\label{eq:P}
\P (X,Y,Z)=\n_{hX}\n_{JY}JZ - \n_{JY}\n_{hX}JZ - \n_{[hX, JY]}JZ$$ called the *Berwald curvature* in Shen’s monograph [@shen].
Using the coordinate expressions (\[berwald\]), it is easy to see that locally we have $$\P = -\frac{1}{2} \frac{\partial^3 f^l}{\partial y^i \partial y^j
\partial y^k} \, dx^i \otimes dx^j \otimes dx^k \otimes \p
{y^l}.$$ Therefore the connection $\Gamma$ is linear, and the corresponding Finsler space is of Berwald type, if and only if, $\P=0$.
\[theo:Pg\] Using the Berwald connection, we can introduce a third-order differential operator $ \Pg : C^\infty(TM) \longrightarrow Sec \, (T^* \otimes S^2 T^*),
$ given by $$\label{eq:E_h_metric_P}
(\Pg E)(X,Y,Z):=\n_{hX} g_{_E}(JY, JZ),$$ for $X,Y,Z \in \mathfrak X (TM)$. Then (\[eq:E\_h\_metric\_loc\]) takes the form $ \label{eq:E_h_metric}
\Pg E = 0.
$
Lagrangian and spray
--------------------
A Lagrangian $E: TM \to {\mbox{$\Bbb R$}}$ is called *regular*, if the 2-form $$\Omega_E : = dd_JE$$ is symplectic. This holds if and only if $ \det \Bigl( \frac {\partial^2E}{\partial y^\alpha \partial y^\beta }
\Bigl) \neq 0.$ Let $S \in \mathfrak X (TM)$ be a spray. We introduce a second-order differential operator $ \Pe : C^\infty(TM) \to Sec~T^*, $ given by $$\label{eq:E_EL_P}
\Pe E := i_S \Omega_E +d{\mathcal L}_C E -dE.$$ It is not difficult to see that $\Pe E$ is a semi-basic 1-form for all $E \in C^\infty(TM)$, and its coordinate representation takes the form $\Pe E = \omega_i \, dx^i$ where the coefficients $\omega_i$ are the functions appearing in the left-hand side of the Euler-Lagrange equation (\[eq:E\_EL\_loc\]). Therefore $S$ corresponds to the geodesic equation of $E$ if and only if the equation $\Pe E = 0$ is valid. So we have the
\[theo:Pe\] If $S$ is a spray, then $\Pe E=0$ is the coordinate-free expression of the Euler-Lagrange partial differential equation (\[eq:E\_EL\_loc\]) associated to $S$.
Formal integrability
--------------------
In order to solve the metrizability problems formulated above, we have to deal with partial differential systems. We shall use Spencer’s technique of formal integrability in the form explained in [@grif_muzs_2]; for a detailed account see [@BCGGG]. We recall here only some basic notions in order to fix the terminology.
Let $B$ be a vector bundle over $M$. If $s$ is a section of $B$, then $j_{k,p}s=(j_k s)_p$ will denote the $k$th order jet of $s$ at the point $p \in M$. The bundle of $k$th order jets of the sections of $B$ is denoted by $J_kB$. In particular $J_k({\mbox{$\Bbb R$}}_M)$ will denote the $k$th order jet of the sections of the trivial line bundle, i.e. the real valued functions. If $B_1$ and $B_2$ are two vector bundles over the same manifold $M$ and $$P:Sec\,(B_1) \to Sec \,(B_2)$$ is a linear differential operator of order $k$, then the morphism $p_{k+l}(P):\, J_{k+l}(B_1) \to J_l(B_2)$ defined by $$\hphantom{\qquad l=0,1,2,...} p_{k+l}(P) \,
\bigl(j_{k+l,p}(s)\bigl):=j_{l,p}(Ps), \qquad l=0,1,2,...$$ is called the $l$th order prolongation of $P$. $R_{k+l,
p}(P):=\mathrm{Ker}\, p_{k+l}(P)_p$ will denote the bundle of the formal solutions of order $k+l$ at $p$. A differential operator $P$ is called *formally integrable* at $p\in M$, if $R_{k+l}(P)$ is a vector bundle for all $l \geq 0$, and $\overline{\pi}_{k+l,p} :
R_{{k+l},p}(P)\rightarrow R_{k+l-1,p}(P)$ is onto for every $l\geq 1$. In analytical terms, formal integrability implies for arbitrary initial data the existence of solutions (see. [@BCGGG], p. 397).
$\sigma_{k}(P):S^{k}T^*M \otimes B_1 \to B_2$ is the symbol of $P$, defined as the highest order terms of the operator, and $\sigma_{k+l}(P):S^{k+l}T^*M \otimes B_1 \to S^lT^*M \otimes B_2$ is the symbol of the $l$-th order prolongation of $P$. We write $$\begin{aligned}
{2}
& g_{k,p} (P) & & = \mathrm{Ker} \, \sigma_{k,p}(P),
\\
& g_{k,p} (P)_{e_1...e_j} & & = \bigl\{A\in g_{k,p}(P) \mid i_{e_1}A
= .... = i_{e_j}A = 0 \bigl\}, \quad j=1,...,n,\end{aligned}$$ where $\{e_1,...,e_n\}$ is a basis of $T_p M$. A basis $\{e_i\}_{i=1}^n$ of $T_pM$ is called *quasi-regular* if $$\textrm{dim} \, g_{k+1,p}(P) = \textrm{dim} \, g_{k,p}(P) + {\sum
_{j=1}^{n}} \textrm{dim} \, g_{k,p}(P)_{e_1...e_j} .$$ A symbol is called *involutive*[^2] at $p$, if there exists a quasi-regular basis at $p$. The notion of involutivity allows us to check the formal integrability in quite a simple way:
**Theorem** \[theo:cartan\_kahler\] (Cartan-Kähler). *Let $P$ be a linear partial differential operator. Suppose that $g_{k+1}(P)$ is regular, i.e. $R_{k+1}(P)$ is a vector bundle on $R_k(P)$. If the map $\overline {\pi }_{k} : R_{k+1}(P)\longrightarrow R_k(P)$ is onto and the symbol is involutive, then $P$ is formally integrable.*
Finsler metrics with prescribed geodesics {#cha:finsler}
=========================================
In this paragraph we are going to investigate the following problem: *under which conditions can a second order differential equation (\[eq:sode\]) be the geodesic equation of a Finsler metric*. As we explained in Section \[sec:introduction\] (Proposition \[theo:reformulation\]) we have to look for a solution of the PDE comprised of (\[eq:E\_hom\_loc\]) and (\[eq:E\_EL\_loc\]). Therefore we have to deal with the second-order system $$\label{eq:e_L_hom}
\Pf := (\Pc , \, \Pe )$$ where $\Pc$ and $\Pe$ are defined in (\[eq:E\_hom\]) and (\[eq:E\_EL\_P\]). We will prove the following theorems:
\[theo:finsler\](Reduction of $\Pf$.) A Lagrangian $E: TM \to {\mbox{$\Bbb R$}}$ is a solution of the second order operator $\Pf$, if and only if, it is a solution of the first order system $$\label{eq:syst_2_1}
\left\{\quad
\begin{aligned}
{\mathcal L}_CE-2E&=0,
\\
d_\h E & =0,
\end{aligned}
\right.$$ where ${\mathcal H} \subset T (=TTM)$ is the holonomy algebra generated by the horizontal vector fields and their successive Lie-brackets, and $\h:T\to \H$ is an arbitrary projection on $\H$.
For $X \in \X {TM}$ we have $ d_\h E (X) = \h X (E) = {\mathcal L}_{\h X} E,
$ so the second equation of (\[eq:syst\_2\_1\]) means simply that the Lie-derivative of $E$ with respect to vector-fields in the holonomy distribution $\H= \mathrm{Im}\, \h$ is zero. This property is independent of the projection $\h$ of $\H$ chosen.
*Proof of Theorem \[theo:finsler\].* Let us suppose that $E:TM \to {\mbox{$\Bbb R$}}$ is a solution of (\[eq:syst\_2\_1\]). Since $T^h \subset \H$, we have $\h \circ h = h
$. Therefore $$d_hE = d_{\h \circ h}E = i_h d_\h E - d_\h i_h E + i_{[h, \h]} E
= i_h d_\h E = 0$$ since the action of an $i_*$-type derivation is trivial on functions. Moreover as $S$ is homogeneous, $hS=S$ and $${\mathcal L}_{S}E= {\mathcal L}_{hS}E=d_h E (S) =0.$$ Writing the Euler-Lagrange operator in the form $$\Pe E = i_Sdd_JE + d{\mathcal L}_CE - dE = d_J{\mathcal L}_SE -i_{[J,S]}dE =
d_J{\mathcal L}_SE -2d_hE$$ we obtain that $\Pe E =0$ and $E$ is a solution of (\[eq:e\_L\_hom\]).
Let us suppose now that $E:TM \to {\mbox{$\Bbb R$}}$ is a solution of (\[eq:e\_L\_hom\]). We have $$\label{eq:el}
i_S \Omega_E = d(E - {\mathcal L}_CE) = -dE.$$ Since $[J,J] = 0$, we have $ d^2_J = d_J \circ d_J = d_{[J,J]}=0,$ and $ i_J \Omega_E = 0,$ so $$\label{eq:ic}
i_C \Omega_E=i_{JS} \Omega_E = i_Si_J\Omega_E - i_J i_S \Omega_E =
i_J dE.$$ On the other hand, for every $X \in \X{TM}$ we have $$i_S \Omega_E(X) = \Omega_E(S, X) = - \Omega_E(C, FX) = - i_F i_C
\Omega_E (X),$$ i.e. $$\label{eq:is}
i_S \Omega_E = i_F i_C \Omega_E .$$ Putting (\[eq:ic\]) into (\[eq:is\]) we obtain $$\label{eq:elh}
i_S \Omega_E = - i_Fi_C \Omega_E = -i_F i_J dE = -d_v E = -d E + d_h
E.$$ Comparing (\[eq:elh\]) with (\[eq:el\]) we obtain that $ d_h E=0.$ It follows that $hX(E)=0$, i.e. $E$ is constant with respect to horizontal vector fields. Therefore it must be constant on the distribution generated by the horizontal sub-bundle taking the recursive Lie-bracket operations, i.e. on $\H$. This means that we have $d_\h E =0$ and $E$ is a solution of (\[eq:syst\_2\_1\]).
\[rem:num\_1\] $E$ is a solution of (\[eq:syst\_2\_1\]) if and only if it is a solution of $$\tag{\ref{eq:syst_2_1}'}
\label{eq:syst_2_1_1}
\left\{\quad
\begin{aligned}
{\mathcal L}_CE-2E&=0,
\\
d_h E & =0,
\end{aligned}
\right.$$ where $h$ is simply the horizontal projection associated to $\Gamma$, so (\[eq:syst\_2\_1\]) and (\[eq:syst\_2\_1\_1\]) are equivalent. However, as we will see in Proposition \[theo:D\_int\], under regularity assumption the system (\[eq:syst\_2\_1\]) is integrable while (\[eq:syst\_2\_1\_1\]) is not, unless the curvature is zero. Indeed, we have $$d_R E= - \tfrac{1}{2} d_{[h,h]}E = - \tfrac{1}{2} d_h d_h E,$$ therefore $d_R E=0$ is a compatibility condition for (\[eq:syst\_2\_1\_1\]).
\[rem:num\_2\] Let us introduce the first order differential operator $\PH
:C^\infty(TM) \longrightarrow Sec\,(T^*)$ by the rule $$\label{eq:d_p}
\PH E\, (X):=\pr X(E),$$ $E \in C^\infty(TM)$, $X\in \mathfrak{X}(TM)$, and the differential operator $$\label{eq:D_3}
\Pfff:=(\Pc, \, \PH)$$ corresponding to the system (\[eq:syst\_2\_1\]). Theorem \[theo:finsler\] shows that a Lagrangian is a solution of $\Pf$ if and only if it is a solution of $\Pfff$.
\[theo:C\_D\] Let $S$ be a spray over the manifold $M$. If $C \in \H$, then there is no Finsler metric whose geodesics are given by $S$.
Let $S$ be a spray and $E:TM \to {\mbox{$\Bbb R$}}$ a Lagrangian. From Proposition \[theo:reformulation\] we know that if $E$ is an energy function associated to $S$, then it is a solution of $\Pf=(\Pc, \, \Pe)$, and by Theorem \[theo:finsler\] we obtain that $E$ satisfies the equations $\mathcal{L}_CE -2E = 0$ and $d_\h E =0$. If $C\in \H$, we have also $$0=\PH E (C)=(\h C) E = CE ={\mathcal L}_CE,$$ therefore $E = 0$. Since $E$ has to be a regular Lagrangian, this is impossible and the proposition is proved.
Let us consider the case when $C \not \in \H$. We have the following
\[theo:finsler\_2\] Let $S$ be an analytical spray over the analytical manifold $M$. If $C \not \in \H$ and $\H$ has constant rank in a neighbourhood of $v\in \TM$, then there exists an analytical Finsler metric in a neighbourhood of $v$ such that the geodesics are given by $S$ if and only if the kernel of the first prolongation of (\[eq:syst\_2\_1\]) at $v$ contains positive definite initial data.
Let $(x^i)$ be a local coordinate system on $M$, $(x^i,y^i)$ the associated coordinate system on $TM$ in the neighborhood of $v$. If $p:=j_k(E)_v \in J_{2}({\mbox{$\Bbb R$}}_{TM})$ is a $k$th order jet of a real valued function $E$ on $TM$ we set $$\label{eq:ch5_1}
s_{i_1...i_a \underline{i_{a+1}..i_l}}(p):= \frac{\partial^l E}{\partial
x^{i_1} ... \, \partial x^{i_a} \partial y^{i_{a+1}} ... \, \partial
y^{i_l}}(v), \qquad 1 \leq l \leq k.$$ Then $ (s,s_j,s_{\underline{j}}, s_{jk}, s_{j\underline{k}},
s_{\underline{jk}})
$ gives a coordinate system on $J_{2,v}({\mbox{$\Bbb R$}}_{TM})$. Using the notation of (\[eq:D\_3\]) introduced in Remark \[rem:num\_2\], positive definite initial data for the first prolongation of (\[eq:syst\_2\_1\]) at $v$ is simply an element $s_{2,v} \in J_{2,v}({\mbox{$\Bbb R$}}_{TM})$ represented as $ s_{2,v}=(s,s_i,s_{\underline{i}}, s_{ij}, s_{i\underline{j}},
s_{\underline{ij}}) \in \R^{1+ (n+n)+ \frac{n(n+1)}{2}+
n^2+\frac{n(n+1)}{2}}
$ such that ${(s_{\underline{ij}})}_{1 \leq i,j\leq n}$ determines a positive definite quadratic form, and $s_{2,v}$ is a second order solution of $\Pfff$ at $v$. This last condition gives *linear algebraic* equations on the coordinates of $s_{2,v}$.
*Proof of Theorem \[theo:finsler\_2\].* The proof is based on Theorem \[theo:finsler\] and on Proposition \[theo:D\_int\] proved below. Indeed, if $\Pfff$ is formally integrable (see Proposition \[theo:D\_int\]), then for every initial condition we have an infinite order formal solution of $\Pfff$. In the analytic case, this formal solution gives an analytical solution in an open neighborhood of $\TM$. Theorem \[theo:finsler\] shows that this solution is also a solution of the operator $\Pf$. In this way we obtain an analytical solution of $\Pf$, i.e. a homogeneous function which satisfies the Euler-Lagrange equation associated to $S$. Therefore $S$ is locally Finsler metrizable.
\[theo:D\_int\] Let $S$ be a spray over $M$ so that $C \not \in \H$ and the rank of $\H$ is locally constant. Then the differential operator $\Pfff=(\Pc, \, \PH)$ is formally integrable.
First of all remark that $\Pfff$ is a regular differential operator because, by the hypothesis, rank of $\H=\mathrm{Im}\, \h$ is locally constant. Moreover, using Lemma \[lemma\_1\], Lemma \[lemma\_2\] and the Cartan-Kähler theorem on formal integrability (see page ), we obtain the proposition.
\[lemma\_1\] Every first order solution of $\Pfff$ can be lifted to a second order solution.
It is easy to see from their local description that the symbol of $\Pc$ and $\PH$ can be interpreted as a map $$\begin{aligned}
{2}
& \sigma_1(\Pc) : T^* \to {\mbox{$\Bbb R$}}, & & \sigma_1(\Pc)B_1 = B_1(C)
\\
& \sigma_1(\PH) : T^* \to T^* \qquad & & (\sigma_1(\PH)B_1) (X) =
B_1(\pr X)\end{aligned}$$ for all $B_1 \in T^*$, $X \in T$. The symbol of the first prolongations are defined by $$\begin{aligned}
{2}
& \sigma_2(\Pc) : S^2T^* \to T^*, & & (\sigma_2(\Pc)B_2) (X) =
B_2(X, C)
\\
& \sigma_2(\PH) : S^2T^* \to T^* \otimes T^* \qquad & &
(\sigma_2(\PH)B_2) (X,Y) = B_2(X, \pr Y)\end{aligned}$$ for all $B_2 \in S^2T^*$, $X,Y \in T$. Comparing the first prolongation of the symbols, we can easily find that for every $B_2
\in S^2 T^*$ and $X\in T$ we have $$(\sigma_2 (\Pc)B_2)(\pr X) - (\sigma_2 (\PH)B_2)(C,X) = B_2(\pr X,
C ) - B_2(C, \pr X) = 0,$$ and there is no more relation between the two symbols. That is, if we consider the map $ \tau: T^* \oplus (T^* \otimes T^*) \longrightarrow T^*
$ defined for $B_1 \in T^*$, $B_2 \in T^* \otimes T^*$ and for $X \in
T$ as $$\bigl(\tau (B_1, B_2)\bigl) (X):= B_1 (\pr X) - B_2 (C, X),$$ then we find the commutative diagram:
$$\begin{CD}
& 0 && 0 && &0
\\
& \downarrow && \downarrow & & & \downarrow
\\
0 \longrightarrow & \ g_2(\Pfff) @>i>> S^2 T^*
@>\sigma_2(\Pfff)>> & T^* \oplus (T^* \otimes T^*) @>\tau>> T^*
& \longrightarrow 0
\\
& @VVV @VVV & @VVV
\\
0 \longrightarrow & R_2 @>i >> J_2({\mbox{$\Bbb R$}}_{TM}) @>p_2(\Pfff)>> &
J_1({\mbox{$\Bbb R$}}_{TM} \oplus T^*)
\\
& @VV\overline{\pi}V @VV\pi V & @VV\pi V
\\
0 \longrightarrow & R_1 @>i >> J_1({\mbox{$\Bbb R$}}_{TM}) @>p_1(\Pfff)>> & {\mbox{$\Bbb R$}}\oplus T^*
\\
& && @VVV & @VVV
\\
& && 0 & & & 0
\end{CD}$$
where the successive arrows represent exact sequences. ($R_1$ and $R_2$ denote the spaces of the first and second order formal solutions of $\Pfff$.)
Every first order solution of $\Pfff$ can be lifted into a second-order formal solution if and only if the map $\overline{\pi}:
{\mathcal R}_2 \to {\mathcal R}_1$ is onto. We know by a lemma of homological algebra that there exists a map $ \varphi : R_1 \longrightarrow T^* \, (= \mathrm{Im}\,\tau)
$ such that $$\label{eq:ker_im}
\mathrm{Im}\, \overline{\pi} = \mathrm{Ker} \,\varphi.$$ This map can be constructed for a first order formal solution $j_{1,v}(E)\in R_1$ of $\Pfff$ at $v \in \TM$ as follows: $$\label{varphi_pi}
\varphi_v(E) : = \tau (\n \Pfff E)_v.$$ Let us compute how this map acts. If $E:TM \to {\mbox{$\Bbb R$}}$ is a function such that $j_{1,v}(E) \in R_{1,v}$, then $(\Pfff E)_v=0$, that is $(\PH E)_v=0$ and $(\Pc E)_v=0$. Evaluating $\varphi_v (E)$ on an arbitrary vector $X\in T$ we find that $$\label{varphi}
\begin{aligned}
&\varphi_v (E)(X) = \tau (\n \Pfff E)_v(X) = (\n \Pc E)_v(\h X)-
(\n \PH E)_v (C, X)
\\
& = \bigl({\mathcal L}_{\pr X}({\mathcal L}_CE-2E)-{\mathcal
L}_C({\mathcal L}_{\pr X} E)\bigl)_v = \bigl({\mathcal L}_{\pr
X}({\mathcal L}_{C}E) -{\mathcal L}_{C}({\mathcal L}_{\pr X}E)
-2 {\mathcal L}_{\pr X}E\bigl)_v
\\
& = \bigl({\mathcal L}_{[\pr X,C]}E\bigl)_v -2 (\PH E)_v(X) =
\bigl({\mathcal L}_{[\pr X,C]}E\bigl)_v .
\end{aligned}$$ Now we can remark, that if $X \in \X {TM}$, then $[\h X, C] \in \H
$. Indeed $\H$ can be generated by the successive brackets of the horizontal basis $ \{h_1, ... h_n\},
$ where $ h_i := h \bigl(\p {x^i}\bigl)= \p {x^i} - \Gamma_i^\alpha \p
{y^\alpha}.
$ Since $S$ is homogeneous, we have $[h_i, C]=0$. By the Jacobi identity, this is also true for the successive brackets of the $h_i$’s. If we consider an arbitrary $Y \in \H$, then it can be written as a linear combination of the elements $Y=g^\alpha
Y_\alpha$, where $Y_\alpha$ can be obtained by successive brackets of the $h_i$’s. Thus we have $$[Y,C]=[g^\alpha Y_\alpha, C]=-(Cg^\alpha) Y_\alpha +
g^\alpha[Y_\alpha,C] = -(Cg^\alpha)Y_\alpha,$$ which shows that $[Y,C] \in \H$.
Continuing the above computation of $\varphi_v (E)$ we find that $$\varphi_v (E)(X) = \bigl({\mathcal L}_{[\pr X,C]}E\bigl)_v =
\bigl({\mathcal L}_{\h [\pr X,C]}E\bigl)_v = (\PH E )_v([\pr
X,C])=0,$$ since $(\PH E)_v$ vanishes on $\H_v$. It follows that $\varphi_v$ is identically zero, by (\[eq:ker\_im\]) we conclude that $\mathrm{Ker} \, \varphi_v = R_1$, and $\overline{\pi}$ is onto. Hence every first order solution of $\Pfff$ can be lifted into a second order solution.
\[lemma\_2\] The symbol of $\Pfff$ is involutive.
Let $k$ be the co-dimension of $\H$. Since $\mathrm{dim} \, \H \geq
n$ and $C \not \in \H$, we have $n \leq \mathrm{dim} \, \H \leq 2n-1
$ and $1 \leq k \leq n$. Let us consider the basis $$\label{quasi_reg}
\{e_1, ..., e_{2n}\}:=\{ v_1, ..., v_n, h_1, ..., h_n\}$$ of $T$ at $v \in TM$ where $v_1, ..., v_n$ are vertical, $h_1, ...,
h_n$ are horizontal, the last $2n-k$ vectors generate $\H$ and $v_k:=C$. Then we have $$\begin{aligned}
{1}
g_1(\Pfff)&:=\mathrm{Ker}\, (\sigma_1(\Pfff))= \{B_1\in T^* \ | \
B_1(e_i)=0, \ i=k,...,2n \},
\\
g_2(\Pfff)&:=\mathrm{Ker}\, (\sigma_2(\Pfff))= \{B_2\in S^2T^* \ | \
B_2(e_i, e_j)=0, \ i=1,...,2n, \ j=k,...,2n \}
\end{aligned}$$ and for $1 \leq m < k$, $$\begin{aligned}
{1}
g_{1,e_1, ..., e_m}(\Pfff) :&= \mathrm{Ker}\, (\sigma_1(\Pfff))
\cap \{B_1\in T^* \ | \ B_1(e_i)=0, \ i=1,..,m \}
\\
& = \bigl\{B_1\in T^* \ | \ B_1(e_i)=0, \ i\in \{1,..,m\} \cup
\{k, ..., 2n\} \bigl\}.
\end{aligned}$$ The dimension of these spaces are $$\begin{aligned}
{1}
&\mathrm{dim}\,\bigl(g_1(\Pfff)\bigl)= k-1,
\\
&\mathrm{dim}\,\bigl(g_2(\Pfff)\bigl)= \frac{k(k-1)}{2},
\\
&\mathrm{dim}\,\bigl( g_{1,e_1, ..., e_m}(\Pfff)\bigl) = \left\{
\begin{aligned}
k-&1-m, & & \quad \mathrm{for} \quad m=1,...,k-1,
\\
&0, & & \quad \mathrm{for} \quad m=k,...,2n,
\end{aligned}
\right.
\end{aligned}$$ therefore $$\begin{aligned}
{1}
\mathrm{dim}\,\bigl(g_1(\Pfff)\bigl) & + \sum_{m=1}^{2n}
\mathrm{dim}\,\bigl( g_{1,e_1, ..., e_m}(\Pfff)\bigl) = (k-1) +
\sum_{m=1}^{k-1} (k-1-m) = \frac{(k-1)k}{2}
\\
& = \mathrm{dim}\,\bigl(g_2(\Pfff)\bigl).
\end{aligned}$$ This shows that (\[quasi\_reg\]) is a quasi-regular basis for $\Pfff$. The existence of such a basis proves Lemma \[lemma\_2\].
Landsberg metrizability {#cha:landsberg}
=======================
In this paragraph we will investigate the following problem: *under what conditions can a given second order differential equation (\[eq:sode\]) be the geodesic equation of a Finsler metric of Landsberg type?* As we explained in Proposition \[theo:reformulation\], to answer this question we have to look for a solution of the PDE system consisting of (\[eq:E\_hom\_loc\]), (\[eq:E\_EL\_loc\]) and (\[eq:E\_h\_metric\_loc\]). Let us consider the third order system $$\label{eq:e_L_hom_g}
\Pl = (\Pc , \, \Pe, \, \Pg)$$ where $\Pc$, $\Pg$ and $\Pe$ are defined by (\[eq:E\_hom\]), (\[eq:E\_h\_metric\_P\]) and (\[eq:E\_EL\_P\]). We will prove the following theorems:
\[theo:landsberg\] (Reduction of $\Pl$) The third-order partial differential system $\Pl E=0$ is equivalent to the first order system $$\label{eq:L_syst_2_1}
\left\{\quad
\begin{aligned}
{\mathcal L}_CE-2E&=0,
\\
d_{\l} E & =0,
\end{aligned}
\right.$$ where $\mathfrak L$ is the distribution generated by the horizontal vector fields, the image of the Berwald curvature and their successive Lie-brackets and $\l : TTM \to \L$ is an arbitrary projection of $TTM$ onto $\L$.
The second equation of (\[eq:L\_syst\_2\_1\]) means simply that the Lie-derivative of $E$ with respect to vector-fields in the distribution $\L= \mathrm{Im}\, \l$ is zero. This property is independent of the projection $\l$ of $\L$ chosen.
$E$ is a solution of (\[eq:L\_syst\_2\_1\]) if and only if it is a solution of $$\tag{\ref{eq:L_syst_2_1}'}
\label{eq:L_syst_2_1_1}
\left\{\quad
\begin{aligned}
{\mathcal L}_CE-2E&=0,
\\
d_h E & =0,
\\
d_{\mathcal R} E & =0,
\end{aligned}
\right.$$ where $h$ is simply the horizontal projection associated to $\Gamma$. However, under the assumption of regularity, the system (\[eq:L\_syst\_2\_1\]) is integrable but (\[eq:L\_syst\_2\_1\_1\]) in general is not, because it is not containing its compatibility conditions.
\[theo:landsberg\_2\] Let $S$ and $M$ be analytical, and suppose that rank of $\L$ constant in a neighborhood of $v \in \TM$. Then there exists a Finsler metric of Landsberg type in a neighborhood of $v$ whose geodesics are given by $S$, if and only if, $C \not \in \L$, and the kernel of the first prolongation of (\[eq:L\_syst\_2\_1\]) at $v$ contains a positive definite initial condition.
In order to prove the above theorems, we need the following
\[theo:lemma\_3\] Let us consider the differential operator $d_\P: C^\infty(TM) \to
{\mathcal S}ec\,(S^3T^*)$, where $\P$ is the Berwald curvature. For all $X,Y,Z \in T$ we have $$\label{eq:3}
\Pg E \, (X,Y,Z) = \n^2 \PH E \, (JY, JZ,hX) + d_\P E \, (X, Y, Z),$$ where $\nabla$ is the Berwald connection and $\PH$ is introduced in (\[eq:d\_p\]).
*Proof.* The three terms in (\[eq:3\]) are all semi-basic in $X$, $Y$ and $Z$. Putting $X = \p{x^i}$, $Y = \p{x^j}$ and $Z = \p {x^k}$, we have $$\begin{aligned}
{1}
& (\Pg E) \Bigl( \p {x^i}, \p {x^j}, \p {x^k}\Bigl)- (\n^2 \PH E)
\Bigl(\p {y^j}, \p {y^k}, h \Bigl(\p {x^k}\Bigl)\Bigl)
= \frac{\partial ^3 E}{\partial x^i \partial y^j \partial y^k}
\\
& \quad \phantom{=} - \Gamma_i^l \frac{\partial ^3 E}{\partial
y^l \partial y^j \partial y^k} - \Gamma_{ij}^l
\frac{\partial^2 E}{\partial y^l \partial y^k} -
\Gamma_{ik}^l \frac{\partial^2 E}{\partial y^l \partial
y^j} - \p {y^j} \p {y^k} \left( \frac{\partial E}{\partial x^i } -
\Gamma_i^l \frac{\partial E}{\partial y^l} \right)
\\
& \quad = \frac{\partial ^2 \Gamma^l_{i}}{\partial y^j \partial
y^k} \frac{\partial E}{\partial y^l}
= -\frac{1}{2}\frac{\partial^3 f^l}{\partial y^i \partial y^j
\partial y^k} \frac{\partial E}{\partial y^l}
= d_\P E \, \Bigl(\p {x^i}, \p {x^j}, \p {x^k}\Bigl).\end{aligned}$$
Since $\h \circ h = h$, we have $\PH(hX) = d_hE(hX)$, and we have the relation $$\tag{\ref{eq:3}'} \Pg E \, (X,Y,Z) = \n^2 d_hE \, (JY, JZ,hX) +
d_\P E \, (X, Y, Z).$$ expressed in terms of the horizontal projection $h$.
*Proof of Theorem \[theo:landsberg\]*.
1\) If $E:TM \to {\mbox{$\Bbb R$}}$ is a solution of $\Pl=(\Pc, \, \Pe, \, \Pg)$, then by Theorem \[theo:finsler\] we obtain that $\PH E =0$. In particular, $E$ is constant on the horizontal distribution. Moreover, we can find from (\[eq:3\]) that $d_\P E=0$, i.e. $E$ is constant on the image of the Berwald curvature. Consequently $E$ has to be constant on the distribution $\L$ generated by the horizontal vector-field and the image of Berwald curvature.
2\) Conversely, let $E:TM \to {\mbox{$\Bbb R$}}$ be a solution of (\[eq:L\_syst\_2\_1\]). By the construction $\H \subset \L$, we obtain that $E$ is a solution of $\PH$. By Theorem \[theo:finsler\], $E$ is also a solution of $\Pc$ and $\Pe$. Moreover, $\mathrm{Im} \, \P
\subset \L$ implies $d_\P E=0$ and by Lemma \[theo:lemma\_3\] we have $\Pg E=0$. Therefore $E$ is a solution of the system $\Pl=(\Pc, \,
\Pe, \, \Pg)$.
By 1) and 2) we conclude that the system $\Pl=(\Pc, \, \Pe, \, \Pg)$ is equivalent to the $1^{\mathrm{st}}$ order system (\[eq:L\_syst\_2\_1\]) which proves Theorem \[theo:landsberg\].
*Proof of Theorem \[theo:landsberg\_2\].*
The reasoning is completely analogous to the proof of Theorem \[theo:finsler\_2\]. Indeed, $E:TM \to {\mbox{$\Bbb R$}}$ is a Landsberg-type Finsler metric associated to $S$, if and only if, $g_E$ is positive definite and $E$ is a solution of the system $\Pl =(\Pc, \, \Pe , \,
\Pg )$.
If $C \in \L$ and $E:TM \to {\mbox{$\Bbb R$}}$ is a solution of (\[eq:L\_syst\_2\_1\]), then $dE=0$. So $E$ is not a regular Lagrangian, and $S$ cannot be variational.
Suppose that $C \not \in \L$ and that $\L$ has constant rank in a neighbourhood of $v\in \TM$. By Theorem \[theo:landsberg\] we know that $E$ is a solution of $\Pl$ if and only if it is a solution of (\[eq:L\_syst\_2\_1\]). Therefore, it is sufficient to consider this first order PDE and show that it has a solution.
By the hypotheses, $\L$ is of constant rank in a neighbourhood of $v\in TM$, the system (\[eq:L\_syst\_2\_1\]) is regular.
A computation, completely analogous to that of made in the proof of Proposition \[theo:D\_int\], shows that (\[eq:L\_syst\_2\_1\]) is formally integrable. Consequently, for every initial condition, there exists an analytical solution to (\[eq:L\_syst\_2\_1\]) in a neighbourhood of $v \in \TM$. Using Theorem \[theo:landsberg\], this function will be a solution of the system $\Pl=(\Pc, \, \Pe , \, \Pg
)$, and therefore it will be a Landsberg type Finsler metric in a neighborhood of $v$ with geodesics determined by $S$.
Remarks and examples of Finsler and Landsberg metrizability {#sec:remarks}
===========================================================
Theorems \[theo:finsler\], \[theo:C\_D\], \[theo:finsler\_2\], \[theo:landsberg\] and \[theo:landsberg\_2\] give us a powerful method to test the metrizability of a second order ordinary differential system. We mention here only some direct consequences.
\[cor:remarks\] A quadratic second order differential equation is Landsberg metrizable if and only if it is Finsler metrizable.
Indeed, in the quadratic case, the functions $f^i(x, \dot{x})$ are quadratic in the $\dot{x}$ variable and the Berwald curvature $\P$ vanishes identically. Therefore the distribution $\L$ coincides with $\H$.
\[theo:L=T\] If $\mathrm{rank} \, \L=2n$ (resp. $\mathrm{rank}\,\H=2n$), then the spray is not Landsberg (resp. Finsler) metrizable.
Indeed, in this cases $\L=T$ (resp. $\H=T$). If $E:TM \to {\mbox{$\Bbb R$}}$ is a solution of (\[eq:L\_syst\_2\_1\]) (resp. (\[eq:syst\_2\_1\])), then $dE=0$, and $E$ cannot be a regular Lagrangian.
*Examples*
1. For a generic spray, the image of the curvature $R$ and the image of $\P$ generate the whole vertical space. In this case $\L=T$, and therefore there is no a regular solution to (\[eq:L\_syst\_2\_1\]).
2. In some cases, even if the image of the curvature $R$ and the image of $\P$ do not generate the whole vertical space, nevertheless $\L=T$. For example let $f(t):=a \sqrt{t^2+bt+c}$ with $a$, $b$, $c$ nonzero reals, and consider the system $$\label{ex:3}
\ddot{x}_1={\dot x}_1^2 \, f \left( \frac{{\dot x}_2}{{\dot x}_1}
\right) , \quad \ddot{x}_2={\dot x}_1{\dot x}_2 \, f \left(
\frac{{\dot x}_2}{{\dot x}_1} \right).$$ In this case $ \mathrm{Im} \, \P = \mathrm{Im} \, R
$ is a 1-dimensional distribution of $T$. However, by computing the Lie-brackets of horizontal vector fields with the generator of $\textrm{Im}\, \P$ we find that $$\Bigl[h \p{x^i}, \P \Bigl(\p {x^1}, \p {x^1}, \p
{x^1}\Bigl)\Bigl]_{i=1,2} \in \textrm{Im} \, \P \quad
\Leftrightarrow \quad b=c=0.$$ Therefore we have $\mathrm{dim}\, \L =4$, so $\L =T$ and (\[ex:3\]) is not Landsberg metrizable.
\[theo:constant\] If $\H$ (resp. $ \L$) contains the vertical lift of a non-zero vector field on $M$, then the spray is not Finsler (resp. Landsberg) metrizable.
Let $X \in \H$ (resp. $X \in \L$) be a vertical lift, namely $X=Z^v$, $Z \in \mathfrak{X}(M)$. Then, locally, $X = (X^\alpha
\circ \pi) \frac{\partial}{\partial y^\alpha}$, where the functions $X^\alpha$ are defined on a domain of $M$. If $S$ is Finsler (resp. Landsberg) metrizable, then the corresponding energy function $E:TM \to {\mbox{$\Bbb R$}}$ is a regular Lagrangian, and it is a solution of $\Pf$ (resp. $\Pl$). Using Theorem \[theo:finsler\] (resp. Theorem \[theo:landsberg\]) we get that $E$ is a solution of (\[eq:syst\_2\_1\]) (resp. (\[eq:L\_syst\_2\_1\])), and in particular $E$ is constant on every vector fields of $\H$ (resp. $\L$).
Since $X \in \H$ (resp. $X \in \L$) we have ${\mathcal L}_XE=0$. Taking the derivatives with respect to the vertical directions and using the special form of $X$ we obtain that $$0=\frac{\partial {\mathcal L}_X E}{\partial {y^i}}
= \p {y^i} \left( (X^j \circ \pi) \frac{\partial E}{\partial y^j }
\right)
= (X^j \circ \pi) \frac{\partial^2 E}{\partial y^j \partial
{y^i}},$$ so $E$ cannot be a regular Lagrangian. This contradicts the hypothesis. Therefore the spray is not Finsler (resp. Landsberg) metrizable.
*Example.* Let us consider the system $$\label{eq:ex_1}
\left\{
\begin{aligned}
\ddot{x}^1:= \lambda_1(x) \, f(x, \dot{x}),
\\
\ddot{x}^2:= \lambda_2 (x)\, f(x, \dot{x}),
\end{aligned}
\right.$$ where $f(x, y)$ is an arbitrary second order homogeneous but non-quadratic function in $y=(y^1, y^2)$ and $\lambda_1$, $\lambda_2$ arbitrary functions of $x=(x^1, x^2)$. In this case the image of the Berwald curvature is generated by the vertically lifted vector field $X= \lambda_1 \p {y^1} + \lambda_2 \p {y^2}$. Thus, by Theorem \[theo:constant\], the system is not Landsberg metrizable.
On the existence of non-Berwald type Landsberg spaces {#sec:exist-non-berw}
=====================================================
A Landsberg metric is said to be of Berwald type if the connection $\Gamma$ is linear, that is in its geodesic equations $\ddot{x}^i =
f^i(x, \dot{x})$ the functions are quadratic in $\dot{x}$. These types of spaces can be characterized in terms of the Berwald curvature: a Landsberg space is of Berwald type if and only if the Berwald curvature introduced in (\[eq:P\]) vanishes. One of the most exciting questions in Finsler geometry is the following:
*Are there any non-Berwald Landsberg metrics on a manifold?*
To answer this question a promising strategy is to investigate the solvability of the system $\Pl=0$, which is a third order differential system. Theorems \[theo:landsberg\] and \[theo:landsberg\_2\] can be useful for this purpose, because they provide a reduction of $\Pl$ to a much simpler first order differential system. Far from exploring fully the possibilities offered by the above theorems, we shall be content to make the following observations.
There is no a nontrivial analytic function $f$ such that the equations $$\begin{aligned}
\ddot{x}_1 & = {\dot x}_1^2 \, f({\dot x}_2/{\dot x}_1),
\\
\ddot{x}_2 & = {\dot x}_1{\dot x}_2 \, f({\dot x}_2/{\dot x}_1)
\end{aligned}$$ constitute the geodesic system of a non-Berwald type Landsberg metric.
If $f \not \equiv 0$, then $\P \neq 0$. Unless $f$ satisfies the equation $ 3 f'' f' + f f''' =0,
$ we have $ \mathrm{Im} \, \P \neq \mathrm{Im} \, R.
$ In this case $\L$ is the entire second tangent bundle $TTM$, and consequently there is no corresponding Landsberg metric.
If $f$ satisfies the above equation, then it has the form $ f(t) = a \sqrt{t^2+bt+c}
$ with $a,b,c \in {\mbox{$\Bbb R$}}$. Computing the Lie brackets $[h (\p {x^i}),
\P(\p {x^1}, \p {x^1}, \p {x^1})]$ we find that they are in the subspace generated by $\mathrm{Im} \, \P$ if and only if $b=c=0$. But in this case $\P=0$ which contradicts our hypotheses.
The system $$\label{ex:4}
\begin{aligned}
\phantom{\qquad a \in {\mbox{$\Bbb R$}}, \ t \in \Bbb N,} \ddot{x}_1&=a \,
{\dot x}_1^{2-t}\, {\dot x}_2^t, \qquad a \in {\mbox{$\Bbb R$}}, \ t \in \Bbb N,
\\
\phantom{\qquad b \in {\mbox{$\Bbb R$}}, \ s \in \Bbb N,} \ddot{x}_2&=b \,
{\dot x}_1^{2-s} \, {\dot x}_2^s , \qquad b \in {\mbox{$\Bbb R$}}, \ s \in \Bbb N,
\end{aligned}$$ cannot be the geodesic system of a non-Berwald type Landsberg metric.
Let us consider the spray $S$ corresponding to (\[ex:4\]): $$S = y^1 \p {x^1}+ y^2 \p {x^2} + a \, y_1^{2-t}\, y_2^t \p {y^1}+b
\, y_1^{2-s}\, y_2^s \p {y^2}.$$ If $s \in \{0,1,2\}$ or $t \in \{0,1,2\}$, then $S$ is not the geodesic equation of a non-Berwald type Landsberg metric.
Indeed, in case of $s, t \in \{0,1,2\}$, then $\P=0$ and therefore the Berwald connection is linear. If the system is Finsler-metrizable, then it is also Landsberg metrizable (Corollary \[cor:remarks\]), and the corresponding Finsler space is of Berwald type. If $s \in
\{0,1,2\}$ or $t \in \{0,1,2\}$, then $\mathrm{Im} \, \P$ generated by $\p {y^1}$ or $\p {y^2}$. As we explained in Theorem \[theo:constant\], in these cases there is no regular Lagrangian associated to the system.
If $s, t \not \in \{0,1,2\}$, then the image of $\P$ is generated by the vector-fields $$\p {y_1} + \frac{b s (s-1)(s-2)y_2^{s-t} }{ a t (t-1)(t-2)y_1^{s-t}}
\p {y_2}.$$ If in addition $s= t$, then using Theorem \[theo:constant\] we obtain that there is no regular Lagrangian associated to the system. Let us suppose now that $s \neq t$. The image of the curvature $R$ is generated by the vector field $$\begin{aligned}
{1}
\p {y_1} + \frac{b}{a} \frac{
y_2^{s} y_{1}^{2 t-s+2}\, b s(2-s)
+ y_2^{t+1} y_{1}^{t+1} \, a(2s^2-4s+2t-st)
} {
y_2^{t+1} y_{1}^{t+1} \, a t(t-2)
+ y_2^s y_{1}^{2t-s+2} \, b (st-2t^2 + 2t) }
\left( \frac{y_2}{y_1} \right)^{s-t} \p {y_2}.\end{aligned}$$ If $s \neq t+1$, or $s \neq t-1$, then $\mathrm{Im} \, R \neq
\mathrm{Im} \, \P$. Since $ T^h \oplus \mathrm{Im} \, R \oplus
\mathrm{Im} \, \P \subset \L$ we obtain that $\L=T$. Using Theorem \[theo:L=T\] we find that $S$ is not Landsberg-metrizable.
If $s = t+1$ or $s = t-1$, then $\mathrm{Im} \, R = \mathrm{Im} \,
\P$. Computing the Lie-brackets of the horizontal vector-fields with the image of the Berwald curvature we find that $$\bigl[ T^h, \, \mathrm{Im} \, \P \bigl] \, \nsubseteq \, T^h \oplus
\mathrm{Im} \, \P.$$ We arrive at $\L=T$, and using Theorem \[theo:L=T\] we conclude again that $S$ is not Landsberg-metrizable.
[99]{} *Finsler metrics of constant positive curvature on the Lie group $S^3$*, Journal of the London Math Soc. **66** (2002), 453-467.
*Projectively Flat Finsler 2-spheres of Constant Curvature*. math.dg-ga/9611010
*Exterior Differential Systems*, Springer, Berlin (1991), 475.
*Theory of vector-valued differential forms*, Proc. Kon. Ned. Akad. A, **59**, (1956), 338-359.
*Structure presque-tangente et connexions I, II*, Ann. Inst. Fourier XXII **1** (1972), 287-334; XXII **3** (1972), 291-338.
*On the inverse problem of the variational calculus: existence of Lagrangians associated with a spray in the isotropic case*, Ann. Inst. Fourier **49** (4) (1999), 1387-1421.
*Variational Principles For Second-Order Differential Equations*, World Scientific, Singapore, 2000.
*On variational second order differential equations: polynomial case*, Differential Geometry and Its Applications, Proc. Conf. Aug. 24-28, Silesian Univ. Opava, 1993, 449-459.
*On Landsberg spaces and holonomy of Finsler manifolds*, Contemporary Mathematics, **196**, 1996, 177-185.
*On Holonomy groups of Landsberg manifolds*, Tensor, N.S. **62**, (2000), 87-90.
*Variational metric structures*, Publ. Math. Debrecen **62** (2003), no. 3-4, 461–495.
: *The inverse problem of the calculus of variations: the use of geometrical calculus in Douglas’s analysis*, Trans. Am. Math. Soc. 354, 2897-2919 (2002).
*Funk Metrics and R-Flat Sprays*, math.DG/0109037.
: *On the Finsler-metrizabilities of spray manifolds*, Period. Math. Hungar. 44 (2002), no. 1, 81-100.
[^1]: The equation (\[eq:E\_h\_metric\_loc\]) is a 3rd order PDE, taking into account of (\[eq:g\]).
[^2]: There is a slight problem of language here. In the works of Cartan, and more generally in the theory of exterior differential systems, “involutivity” means more than the existence of a quasi-regular basis and it refers to “integrability” (cf. [@BCGGG], p.107, 140). Here we are following the terminology of Goldschmidt (cf. [@BCGGG], p.409).
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---
abstract: 'The abstract should appear at the top of the left-hand column of text, about 0.5 inch (12 mm) below the title area and no more than 3.125 inches (80 mm) in length. Leave a 0.5 inch (12 mm) space between the end of the abstract and the beginning of the main text. The abstract should contain about 100 to 150 words, and should be identical to the abstract text submitted electronically along with the paper cover sheet. All manuscripts must be in English, printed in black ink.'
address: 'Author Affiliation(s)'
bibliography:
- 'strings.bib'
- 'refs.bib'
title: AUTHOR GUIDELINES FOR ICASSP 2016 PROCEEDINGS MANUSCRIPTS
---
One, two, three, four, five
Introduction {#sec:intro}
============
These guidelines include complete descriptions of the fonts, spacing, and related information for producing your proceedings manuscripts. Please follow them and if you have any questions, direct them to Conference Management Services, Inc.: Phone +1-979-846-6800 or email to\
`papers@icassp2016.org`.
Formatting your paper {#sec:format}
=====================
All printed material, including text, illustrations, and charts, must be kept within a print area of 7 inches (178 mm) wide by 9 inches (229 mm) high. Do not write or print anything outside the print area. The top margin must be 1 inch (25 mm), except for the title page, and the left margin must be 0.75 inch (19 mm). All [*text*]{} must be in a two-column format. Columns are to be 3.39 inches (86 mm) wide, with a 0.24 inch (6 mm) space between them. Text must be fully justified.
PAGE TITLE SECTION {#sec:pagestyle}
==================
The paper title (on the first page) should begin 1.38 inches (35 mm) from the top edge of the page, centered, completely capitalized, and in Times 14-point, boldface type. The authors’ name(s) and affiliation(s) appear below the title in capital and lower case letters. Papers with multiple authors and affiliations may require two or more lines for this information. Please note that papers should not be submitted blind; include the authors’ names on the PDF.
TYPE-STYLE AND FONTS {#sec:typestyle}
====================
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|
---
abstract: 'Stripping cross sections in nitrogen have been calculated using the classical trajectory approximation and the Born approximation of quantum mechanics for the outer shell electrons of 3.2GeV I$^{-}$ and Cs$^{+}$ ions. A large difference in cross section, up to a factor of six, calculated in quantum mechanics and classical mechanics, has been obtained. Because at such high velocities the Born approximation is well validated, the classical trajectory approach fails to correctly predict the stripping cross sections at high energies for electron orbitals with low ionization potential.'
author:
- 'Igor D. Kaganovich, Edward A. Startsev and Ronald C. Davidson'
title: 'Comparison of quantum mechanical and classical trajectory calculations of cross sections for ion-atom impact ionization of negative - and positive -ions for heavy ion fusion applications '
---
Introduction
============
Ion-atom ionizing collisions play an important role in many applications, such as heavy ion inertial fusion [@HIF; @reference], collisional and radiative processes in the Earth’s upper atmosphere [@atmosphere], ion-beam lifetimes in accelerators [@accelerators; @life; @time], atomic spectroscopy [@spectroscopy] and ion stopping in matter [@beam; @stopping], and are also of considerable academic interest in atomic physics [@Review; @atomic; @physics].
To estimate the ionization and stripping rates of fast ions propagating through gas or plasma, the values of ion-atom ionization cross sections are necessary. In contrast to the electron [@Voronov] and proton [@Rudd; @Rudd; @2] ionization cross sections, where experimental data or theoretical calculations exist for practically any ion and atom, the knowledge of ionization cross sections by fast complex ions and atoms is far from complete [@Shvelko; @book]. While specific values of the cross sections for various pairs of projectile ions and target atoms have been measured at several energies [@our; @PoP; @hif; @Olson; @exp; @Watson; @exp], the scaling of cross sections with energy and target or projectile nucleus charge has not been experimentally mapped.
There are several theoretical approaches to cross section calculations. These include: classical calculations that make use of a classical trajectory and the atomic electron velocity distribution functions given by quantum mechanics \[this approach is frequently referred to as the classical trajectory Monte Carlo (CTMC) approach\]; quantum mechanical calculations based on the Born, eikonal or quasiclassical approximations, and so forth [@Shvelko; @book]. All approaches are computationally intensive, and the error and range of validity have to be assessed carefully before making any approximations or applying the results.
Classical trajectory calculations are simpler to perform in comparison with quantum mechanical calculations. Moreover, in some cases the CTMC calculations yield results close to the quantum mechanical calculations [our PoP hif, Mueller new, Our new]{}. The reason for similar results lies in the fact that the Rutherford scattering cross section is identical in both classical and quantum mechanical derivations [@Landau; @book]. Therefore, when an ionizing collision is predominantly a consequence of electron scattering at small impact parameters close to the nucleus, the quantum mechanical uncertainty in the scattering angle is small compared with the angle itself, and the classical calculation can yield an accurate description [@Bohr; @my; @PAC; @Xsection]. But this is not always a case, as we demonstrate below. For fast projectile velocities and low ionization potentials, the difference between the classical and quantum mechanical calculations of ionization cross section can be as large as a factor of six for parameters to relevant to heavy ion fusion cross sections.
In the present analysis, we consider at first only the stripping cross section of loosely bound electron orbitals of $I^{-}$ and $Cs^{+}$ ions colliding with a neutral atom of nitrogen, or with a fully stripped nitrogen ion with $Z_{T}=7$ (for comparison). Atomic units are used throughout this paper with $e=\hbar
=m_{e}=1$, which corresponds to length normalized to $a_{0}=\hbar
^{2}/(m_{e}e^{2})=0.529\cdot 10^{-8}cm,$ velocity normalized to $%
v_{0}=e^{2}/\hbar =2.19\cdot 10^{8}cm/s$, and energy normalized to $%
E_{0}=m_{e}v_{0}^{2}=2Ry=27.2eV$, where $Ry$ is the Rydberg energy. The normalizing coefficients are retained in all equations for robust application of the formulas. For efficient manipulation of the formulas, it is worth noting that the normalized velocity is $v/v_{0}=0.2\sqrt{E[keV/amu]}
$, where $E$ is energy per nucleon in $keV/amu$. Therefore, $25keV/amu$ corresponds to the atomic velocity scale.
The typical scale for the electron orbital velocity with ionization potential $I_{nl}$ is $v_{nl}=v_{0}\sqrt{2I_{nl}/E_{0}}$. Here, $n,l$ is the standard notation for the main quantum number and the orbital angular momentum quantum number [@Landau; @book]. The collision dynamics is very different depending on whether $v$ is smaller or larger than $v_{nl}$.
Behavior of cross sections at large values of projectile velocity $%
v>v_{nl}$
====================================================================
When $v>>v_{nl}$, the projectile interaction with the target atom occurs for a very short time, and the interaction time decreases as the velocity increases. For $3.2GeV$ $I^{-}$ ions, envisioned for heavy ion fusion applications, the projectile velocity in atomic units is $32v_{0}$, while the electron orbital velocity is $v_{nl}=0.5v_{0}$ for the first ($3.06eV$) ionization potential of $I^{-}$, and $v_{nl}=1.3v_{0}$ for the first ($%
22.4eV $) ionization potential of $Cs^{+}$. Therefore, we shall use the limit $v>>v_{nl}$.
In the limit, where $v>v_{0}Z_{T}$ and $v>>v_{nl}$, the Born approximation of quantum mechanics can be used [@Landau; @book; @Mueller; @new]. The first inequality assures that the nitrogen atomic potential can be taken into account as a small perturbation (the Born approximation); the second inequality allows us to use the unperturbed atomic wave function.
![Shown in the figure is a comparison of the ionization probabilities \[$P_{quP}(q)$ in Eq.(\[P quantum\]), and $P_{clP}(q)$ in Eq.(\[P classical mechanics\])\] and the effective charges \[$Z_{quT}(q)$ $N_{eT}(q)$ in Eq.(\[Z quantum\]), and $Z_{clT}(q)$ in Eq.(\[Z cl effective\])\] in quantum and classical mechanics for $3.2GeV$ $I^{-}
$ ions colliding with a nitrogen atom. Ionization of only the outer electron shell is considered (here, $I_{nlP}=3eV$). []{data-label="Fig1"}](ImNFig1.eps)
In both classical mechanics and in the Born approximation, the ionization cross section can be recast in the form [@Shvelko; @book; @Bethe; @Bethe; @book; @Our; @new],$$\sigma =\int_{0}^{\infty }P_{P}(q)\frac{d\sigma }{dq}dq,
\label{sigma ionization as integral}$$where $P_{P}(q)$ is the probability of electron stripping from the projectile when the electron acquires the momentum $q$, and $%
d\sigma /dq$ is the differential cross section for scattering with momentum $%
q$.
![Plots of differential cross sections for stripping of $I^{-}$ ions by nitrogen atoms and fully stripped ions.[]{data-label="Fig.2"}](ImNFig2.eps)
In quantum mechanics, $P_{quP}(q)$ can be expressed by the square of the corresponding matrix element of transition from the initial state $|nl>$ to the state of the ejected electron $|\mathbf{k}>$ with momentum $\mathbf{k}$, integrated over all $\mathbf{k}$. This gives $$P_{quP}(q)=\int \left\vert <nl|e^{i\mathbf{q\cdot r}}|\mathbf{k}>\right\vert
^{2}d^{3}\mathbf{k.} \label{P quantum}$$The analytical form of $P_{quP}(q)$ for hydrogen-like electron functions is given in Ref. [@Bethe]. In classical mechanics, $P_{clP}(q)$ is given by the integral over the electron velocity distribution function $f(\mathbf{v}%
_{e}\mathbf{)}$ defined by$$P_{clP}(q)=\int \Theta \left( \mathbf{q\cdot v}_{e}+\frac{q^{2}}{2m_{e}}%
-I_{nl}\right) f(\mathbf{v}_{e}\mathbf{)dv}_{e}\mathbf{.}
\label{P classical mechanics}$$Classical mechanics prescribes the electron velocity distribution function (EVDF) for hydrogen-like orbitals as a microcanonical ensemble, where $$f\left( \mathbf{v}_{e}\right) =Cv_{e}^{2}\int \delta \left( \frac{%
m_{e}v_{e}^{2}}{2}-\frac{e^{2}Z_{T}}{r}+I_{nl}\right) r^{2}dr.$$Here, $C$ is a normalization constant defined so that $\int \,f\left(
v_{e}\right) dv_{e}=1$, and $\delta (...)$ denotes the Dirac delta-function. Interestingly, the EVDF for a hydrogen-like electron orbitals is identical in both the quantum mechanical and classical calculations [@Landau; @book], with $$\,f\left( v_{e}\right) \,=\frac{32v_{nl}^{7}}{\pi }\frac{v_{e}^{2}}{\left[
v_{e}^{2}+v_{nl}^{2}\right] ^{4}},$$where $v_{nl}$ is the scale of the electron orbital velocity defined by $$v_{nl}=v_{0}\sqrt{2I_{nl}/E_{0}}.$$In the Born approximation of quantum mechanics, $d\sigma /dq$ is given by [@Landau; @book; @Shevelko; @paper] $$\frac{d\sigma }{dq}=8\pi a_{0}^{2}\frac{v_{0}^{2}(m_{e}v_{0})^{2}}{v^{2}}%
\frac{Z_{quT}^{2}(q)+N_{eT}(q)}{q^{3}}, \label{d sigma dq qm}$$where $$Z_{quT}(q)=\left\vert Z_{T}-\sum_{nl}F_{nlT}(q)\right\vert
,\;N_{eT}(q)=[N_{eT}^{total}-\sum_{nlT}\left\vert F_{nlT}(q)\right\vert
^{2}]. \label{Z quantum}$$Here, $Z_{quT}(q)$ is the effective charge, subscript $qu$ stands for quantum mechanics, $F_{nlT}(q)=\int e^{i\mathbf{q\cdot r}}\rho _{nlT}(r)d^{3}%
\mathbf{r}$ is the form factor of the target atom’s orbital $nl$ with the electron density $\rho _{nlT}(r)$, and $N_{eT}^{total}$ is the total number of electrons in the target atom \[$N_{eT}(q\rightarrow \infty )=$ $%
N_{eT}^{total}$\].
In classical mechanics, $d\sigma /dq$ is given by $$\frac{d\sigma }{dq}=2\pi \rho \frac{d\rho }{dq}. \label{d sigma d ro}$$Here, $\rho (q)$ is the impact parameter for a collision resulting in the momentum transfer $q.$ For fast collisions, $q$ is mainly perpendicular to the projectile velocity, and $q$ is determined by integration of the electric field of the target atom on the electron, which gives$$q(\rho )=-\frac{2\rho }{v}\int_{\rho }^{\infty }\frac{dU_{T}}{dr}\frac{1}{%
\sqrt{r^{2}-\rho ^{2}}}dr, \label{q(r)}$$where $U_{T}(r)$ is the atomic potential of the target atom. To compare the classical calculation with the quantum mechanical calculation, we recast Eqs.(\[d sigma d ro\]) and (\[q(r)\]) into a form similar to Eq.(\[d sigma dq qm\]), introducing the effective charge $Z_{clT}(q)$ defined by$$Z_{clT}(q)=\frac{qv}{2m_{e}a_{0}v_{0}^{2}}\sqrt{-q\rho (q)\frac{d\rho }{dq}},
\label{Z cl effective}$$where subscript $cl$ stands for classical mechanics. Note that for the bare target ion, $U_{T}=-e^{2}Z_{T}/r$ and $Z_{clT}(q)=Z_{T}$. Finally, making use of the effective charge in Eq.(\[Z cl effective\]), the differential cross section in classical mechanics takes on a form similar to Eq.(\[d sigma dq qm\]) in quantum mechanics, i.e., $$\frac{d\sigma }{dq}=8\pi a_{0}^{2}\frac{v_{0}^{2}(m_{e}v_{0})^{2}}{v^{2}}%
\frac{Z_{clT}(q)^{2}+N_{eT}^{total}}{q^{3}}.$$Here, the final term accounts for ionization by the $N_{eT}^{total}$ target electrons.
Figure 1 shows a comparison of the ionization probabilities \[$P_{quP}(q)$ in Eq.(\[P quantum\]), and $P_{clP}(q)$ in Eq.(\[P classical mechanics\])\] and the effective charges \[$Z_{quT}(q)$ in Eq.(\[Z quantum\]), and $%
Z_{clT}(q)$ in Eq.(\[Z cl effective\])\] in quantum mechanics and classical mechanics for $3.2GeV$ $I^{-}$ ions colliding with a nitrogen atom. Ionization of only the outer electron shell is considered (here, $%
I_{nlP}=3.06eV$, approximating as a hydrogen-like orbital).
Figure 2 shows that for stripping by neutral atoms, the main contributions arise from intermediate momenta in the range $q=0.5-1$, while for stripping by the bare target nucleus, small values of $q$ make the largest contribution to the cross section, which corresponds to large impact parameters (due to the Coulomb long-range interaction). Because $%
P_{quP}>P_{clP}$ for $q<<1$, but $Z_{quT}<Z_{clT}(q)$, the quantum mechanical cross sections are larger than the classical stripping cross sections for stripping by the bare nucleus, but smaller than the classical stripping cross sections for the atoms. Carrying out the integration in Eq. (\[sigma ionization as integral\]) gives the stripping cross sections for only one electron from the outer electron shell for different ions with the same velocity $v=32v_{0}$ colliding with a nitrogen atom. The results are shown in Table 1 for $3.2GeV$ $I^{-}$ ions; in Table 2 for $3.35GeV$ $Cs^{+}$ ions; and in Table 3 for $25MeV$ $H^{-}$.
-------------------------- --------- -----------
$\sigma ,10^{-16}cm^{2}$ quantum classical
N 0.08 0.47
N$^{+7}$ 2.5 1.29
-------------------------- --------- -----------
**Table 1.** Cross section for stripping of $3.2GeV$ $I^{-}$ ions colliding with a nitrogen atom and a fully stripped nitrogen ion ( stripping of only one electron from the outer electron shell is considered here with $%
I_{nlP}=3.06eV$ ).
-------------------------- --------- -----------
$\sigma ,10^{-16}cm^{2}$ quantum classical
N 0.045 0.10
N$^{+7}$ 0.32 0.17
-------------------------- --------- -----------
**Table 2.** Cross section for stripping of $3.35GeV$ $Cs^{+}$ ions (the same velocity as $3.2GeV$ $I^{-}$) colliding with a nitrogen atom or a fully stripped nitrogen ion ( stripping of only one electron from the outer electron shell is considered here with $I_{nlP}=22.4eV$ ).
-------------------------- --------- -----------
$\sigma ,10^{-16}cm^{2}$ quantum classical
N 0.10 1.34
N$^{+7}$ 12.5 5.05
-------------------------- --------- -----------
**Table 3.** Cross section for stripping of $25MeV$ $H^{-}$ ions (the same velocity as $3.2GeV$ $I^{-}$) colliding with a nitrogen atom or a fully stripped nitrogen ion ( stripping of only one electron from the outer electron shell is considered here with $I_{nlP}=0.75eV$ ). Figure 3 shows the same results as in Fig.2, but the results are obtained for $3.35GeV$ $Cs^{+}$ ions (ionization of only one outer electron shell is considered here with $I_{nlP}=22.4eV$ ). Note that $3.35GeV$ $Cs^{+}$ is chosen to have the same velocity as a $3.2GeV$ $I^{-}$ ion.
In the limit $v>>v_{nl},$ the stripping cross section by a fully stripped ion can be analytically evaluated. The Bohr formula, derived by means of classical mechanics, neglects the electron atomic velocity, and gives for the cross section [@Bohr] $$\sigma ^{Bohr}(v,I_{nl},Z_{p})=2\pi Z_{p}^{2}a_{0}^{2}\,\,\frac{%
v_{0}^{2}E_{0}}{v^{2}I_{nl}}. \label{Bohr}$$Accounting for the electron atomic velocity gives an additional factor of $%
5/3$ [@Our; @new]. The Bethe formula [@Bethe] derived by means of the Born approximation of quantum mechanics gives $$\sigma ^{Bethe}=\sigma ^{Bohr}(v,I_{nl},Z_{p})\left[ 0.566\ln \left( \frac{v%
}{v_{nl}}\right) +1.261\right] . \label{Bethe}$$The results of cross sections calculations using Eq.(\[Bohr\]) with a factor $5/3$ and the result in Eq.(\[Bethe\]) coincide with the results in Tables 1, 2 and 3 of stripping cross sections by a fully stripped nitrogen ions calculated in classical trajectory approximation and the Born approximation of quantum mechanics, respectively.
The stripping cross sections calculated in classical trajectory approximation for $Cs^{+}$ and $I^{-}$ ions by fully stripped nitrogen ions is only factor 2-3 larger than the stripping cross sections by neutral nitrogen atoms, which is in qualitative agreement with the observations in Ref.[@Olson; @exp]. However, there is a large difference, up to a factor 30, in the stripping cross sections calculated in the Born approximation of quantum mechanics.
It is evident that the stripping of $Cs^{+}$ ions by fully stripped nitrogen ions decreases by a factor of $22.4eV/3eV=7.5$ compared with $I^{-}$ ions, which is in agreement with the Bohr \[Eq.(\[Bohr\])\] and Bethe \[Eq.([Bethe]{})\] formulas. The stripping cross sections for $Cs^{+}$ and $I^{-}$ions by neutral nitrogen atoms differ by only a factor of 2. In classical mechanics, because the interaction potential is a strong function of the separation, to transfer a considerably larger momentum requires a rather small decrease in impact parameter. This is why, notwithstanding the large difference in ionization potential by a factor of $7$, the difference between the two cross sections is only a factor of 2. Table 3 shows that the difference between the quantum and classical treatments increases for smaller ionization potentials (compare Table 3 with Table 1).
![Plots of the differential cross sections of ionization for $Cs^{+}$ and $I^{-}$ ions by nitrogen atoms and fully stripped ions.[]{data-label="Fig3"}](ImNFig3.eps)
The reason for such a large difference between the quantum mechanical and classical mechanical stripping cross sections for $I^{-}$ can be easily understood from the example of elastic electron scattering from the shielded Coulomb potential $U(r)=\exp (-r/a_{0})/r$. The differential cross section for elastic scattering is shown in Fig.4 .
![Plots of the differential cross sections for the shielded Coulomb potential for $v=32v_{0}$.[]{data-label="fig4"}](ImNFig4.eps)
For the shielded Coulomb potential, direct application of the Born approximation gives [@Landau; @book]$$\frac{d\sigma }{qdq}=8\pi a_{0}^{2}\frac{v_{0}^{2}(m_{e}v_{0})^{2}}{v^{2}}%
\frac{1}{(q^{2}+m_{e}^{2}\hbar ^{2}/a_{0}^{2})^{2}},$$and the total cross section is $\sigma =4\pi a_{0}^{2}v_{0}^{2}/v^{2}.$ The total classical cross section, obtained from integrating $\int \rho d\rho $, diverges because of the contributions from large $\rho $ (small $q)$. Evidently, the quantum mechanical cross section departs from the Rutherford scattering formula for $q/(m_{e}v_{0})<1$, whereas the classical mechanical cross section departs from the Rutherford scattering formula only for $%
q/(m_{e}v_{0})<2v_{0}/v$ \[see Eq.(\[q(r)\]) and Fig.4\]. Therefore, the classical differential cross section differs from the quantum mechanical result by a factor of $[v/(2v_{0})]^{4}$, which for $v=32v_{0}$ gives a difference in small-angle differential cross section of up to a factor of $%
10^{4}$ (see Fig.4).
Tables 4 and 5 are similar to Tables 1 and 2, but the calculations are carried out for ion energies 30 times smaller, in the range of $100MeV.$ Table 5 shows that the predictions of the classical and quantum mechanical theories are similar for 100MeV ions. However, they are a factor two different for $I^{-}$ ions, and the cross sections are the same within 10% accuracy for $Cs^{+}$ ions. The contribution from small $q$ to the stripping cross section by a neutral nitrogen atom is smaller for $Cs^{+}$ ions than for $I^{-}$ ions, thereby significantly reducing the stripping cross section of $Cs^{+}$ ions compared with $I^{-}$ ions, especially for the calculation in the classical trajectory approximation (see Tables 4 and 5, and Fig.5).
-------------------------- --------- -----------
$\sigma ,10^{-16}cm^{2}$ quantum classical
N 2.47 6.8
N$^{+7}$ 61 37
-------------------------- --------- -----------
**Table 4.** Cross section for the stripping of $105MeV$ $I^{-}$ ions ($%
v=5.75v_{0}$) colliding with a nitrogen atom and a fully stripped nitrogen ion (stripping of only one electron from the outer electron shell is considered here with $I_{nlP}=3eV$ ).
-------------------------- --------- -----------
$\sigma ,10^{-16}cm^{2}$ quantum classical
N 1.36 1.4
N$^{+7}$ 6.6 5.2
-------------------------- --------- -----------
**Table 5.** Cross section for the stripping of $110MeV$ $Cs^{+}$ ions ($v=5.75v_{0}$) colliding with a nitrogen atom and a fully stripped nitrogen ion (stripping of only one electron from the outer electron shell is considered here with $I_{nlP}=22.4eV$).
![Plots of the differential cross sections for stripping of 100MeV$%
Cs^{+}$ and 105MeV $I^{-}$ ions ($v=7.5v_{0}$) by nitrogen atoms.[]{data-label="Fig.5"}](ImNFig5.eps)
Calculation of total cross sections
===================================
The total stripping cross section is defined as $$\sigma ^{total}=\sum_{m}m\sigma _{m},$$where $\sigma _{m}$ is the cross section for stripping $m$ electrons in each collision. This cross section is convenient to use for electron production calculations. The stripping cross section for any degree of ionization is defined as $$\sigma =\sum_{m}\sigma _{m},$$which is a convenient expression to use to determine the ion confinement time in an accelerator. In the limit $v>>v_{nl}$, the calculation of the total stripping cross section can be performed assuming that the stripping from different electron orbitals occurs independently [@Shvelko; @book], i.e.,$$\sigma ^{total}=\sum_{nl}N_{nl}\sigma _{nl,} \label{summ of cross sections}$$where $\sigma _{nl}$ is the stripping cross section of only one electron from the electron orbital $nl$, and $N_{nl}$ is the number of electrons in the orbital. The structure of the electron orbitals for $I^{-}$ ions is shown in Table 6.
------------------------------- ------- ------- ------- ------- -------- -------- -------- -------- -------- --------
$nl$ 5p 5s 4d 4p 4s 3d 3p 3s 2p 2s
N$_{nl}$ 6 2 10 6 2 10 6 2 6 2
I$_{nl}$ 3.08 13.2 50.1 125.0 185.83 623.26 892.5 1.07e3 4.65e3 5.2e3
$\sigma _{nl}$($v=32v_{0})$ 0.080 0.054 0.030 0.018 0.013 5.5e-3 4.2e-3 3.6e-3 8.3e-4 7.3e-4
$\sigma _{nl}$($v=5.75v_{0})$ 2.45 1.65 0.92 0.52 0.39 0.12 0.078 0.062 5.8e-3 4.6e-3
------------------------------- ------- ------- ------- ------- -------- -------- -------- -------- -------- --------
**Table 6.** The structure of electron orbitals for $I^{-}$ ions and the individual cross sections avaluated for an orbital electron in units of $%
10^{-16}cm^{2}$.
Here, $nl$ denotes the atomic orbital quantum numbers, I$_{nl}$ is the ionization potential in eV, and $\sigma _{nl}$ denotes the individual cross section for an orbital electron in units of $10^{-16}cm^{2}$. The sum over all orbitals gives $\sigma ^{total}=1.1\cdot 10^{-16}cm^{2}$ for 3.2GeV $%
I^{-}$ ions. To correctly account for multiple ionization, the inclusion of multi-electron effects is necessary. This will be addressed in a future publication. However, it is clear that the stripping cross section for any degree of ionization by neutral atoms is limited by the geometrical cross section of the atom (the geometrical cross section of a nitrogen atom is much smaller than the geometrical cross section of a $Cs^{+}$ ion or a $I^{-}
$ion [@Periodic; @table]). The nitrogen atom geometric cross section is $%
\sigma _{N}=1.5\cdot 10^{-16}cm^{2}$[@Periodic; @table], and therefore $%
\sigma <\sigma _{N}$ is expected. Preliminary estimates suggest that single electron stripping is expected under these conditions.
For 105MeV $I^{-}$ ions, however, the sum over all orbitals gives $\sigma ^{total}=33\cdot 10^{-16}cm^{2}$, whereas $\sigma
_{N}=1.5\cdot 10^{-16}cm^{2}.$ This indicates that multi-electron ionization is expected. However, it is clear that the stripping cross section for any degree of ionization is limited from above by $\sigma _{N}=1.5\cdot 10^{-16}cm^{2}$.
The structure of the electron orbitals for $Cs^{+}$ ions, and the individual cross sections for an orbital electron in units of $10^{-16}cm^{2}$ are illustrated in Table 7. Note that a $Cs^{+}$ ion has the same number of electrons on each orbital as a $I^{-}$ ion.
------------------------------- ------- ------- ------- ------- ------- -------- -------- -------- -------- --------
$nl$ 5p 5s 4d 4p 4s 3d 3p 3s 2p 2s
N$_{nl}$ 6 2 10 6 2 10 6 2 6 2
I$_{nl}$ 22.4 34.0 88.3 176 242 742 1.03e3 1.2e3 5.1e3 5.7e3
$\sigma _{nl}$($v=32v_{0})$ 0.044 0.037 0.022 0.014 0.011 4.8e-3 3.7e-3 3.2e-3 7.4e-4 6.5e-4
$\sigma _{nl}$($v=5.75v_{0})$ 1.35 1.12 0.66 0.41 0.32 0.098 0.065 0.052 4.7e-3 3.8e-3
------------------------------- ------- ------- ------- ------- ------- -------- -------- -------- -------- --------
**Table 7**. The structure of electron orbitals for $Cs^{+}$ ions and the individual cross sections for an orbital electron in units of $%
10^{-16}cm^{2}$.
For 3.35GeV $Cs^{+}$ ions colliding with a nitrogen atom with velocity $%
v=32v_{0}$ ($25MeV/amu$), the summation in Eq.(\[summ of cross sections\]) over all orbitals gives $\sigma ^{total}=0.72\cdot 10^{-16}cm^{2}.$ This estimate of the cross section is consistent with Olson’s result in Ref.[Olson exp]{}, $\sigma =2\cdot 10^{-16}cm^{2}$ for $25MeV/amu$ $Xe^{+}.$ Note that the factor of three difference between the results presented in Table 7 and the results in Ref.[@Olson; @exp] is due to the fact that the cross sections in Table 7 are predicted by making use of quantum mechanics, whereas results in Ref.[@Olson; @exp] are classical trajectory calculations, not applicable at such high projectile velocities.
For $110MeV$ $Cs^{+}$ ions colliding with a nitrogen atom, $v=5.75v_{0}$ ($%
0.8Mev/amu$) and the summation over all orbitals in Eq.(\[summ of cross sections\]) gives $\sigma ^{total}=21\cdot 10^{-16}cm^{2},$ whereas the geometrical cross section of a nitrogen atom is only $\sigma _{N}=1.5\cdot
10^{-16}cm^{2}<<\sigma ^{total}.$ This indicates that multi-electron ionization is expected, similar to $I^{-}$ ions at the same velocity. As noted earlier, to correctly account for multiple ionization, multi-electron calculations are necessary. However, it is clear that the stripping cross section $\sigma $ for any degree ionization is limited by $\sigma
_{N}=1.5\cdot 10^{-16}cm^{2}$. This estimate of the cross section is consistent with Olson’s result [@Olson; @exp], $\sigma ^{total}=4\cdot
10^{-16}cm^{2}$ for $2MeV/amu$ $Xe^{+}$. The inequality $\sigma
^{total}>\sigma _{N}$ indicates the important effect of multi-electron events.
**Conclusions**
===============
For low ionization potential, where a small momentum transfer $q$ contributes to stripping, the classical approach is not valid. For $3.2GeV$ $%
I^{-}$ ions, the classical trajectory approach overestimates by a factor of six the stripping cross section by atomic nitrogen, and by a factor of two the stripping cross section of $3.35GeV$ $Cs^{+}$ ions. For $110MeV$ $Cs^{+}$ ions and $105MeV$ $I^{-}$ ions colliding with a nitrogen atom at velocity $%
v=5.75v_{0}$ ($0.8Mev/amu$), multi-electron ionization is expected. For a correct description of multiple ionization, multi-electron calculations are necessary. However, it is clear that the stripping cross section for any degree of ionization is limited from above by the geometrical cross section of nitrogen, with $\sigma _{N}=1.5\cdot 10^{-16}cm^{2}$, and should be be similar in magnitude for $I^{-}$ ions and $Cs^{+}$ ions at energies in the 100MeV range. (The geometrical cross section of a nitrogen atom is much smaller than the geometrical cross section of a $Cs^{+}$ ion or a $I^{-}$ion [@Periodic; @table]. This effect is similar to the hole produced by a bullet piercing a paper target, where the hole size is determined by the bullet cross section, *not* by the paper target.)
**Acknowledgments**
This research was supported by the U.S. Department of Energy. It is a pleasure to acknowledge the benefits of useful discussion with Christine Celata, Larry Grisham, Grant Logan and Art Molvik.
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