Non-equilibrium steady states : maximization of the Shannon entropy associated to the distribution of dynamical trajectories in the presence of constraints

Filyokov and Karpov [Inzhenerno-Fizicheskii Zhurnal 13, 624 (1967)] have proposed a theory of non-equilibrium steady states in direct analogy with the theory of equilibrium states : the principle is to maximize the Shannon entropy associated to the probability distribution of dynamical trajectories in the presence of constraints, including the macroscopic current of interest, via the method of Lagrange multipliers. This maximization leads directly to generalized Gibbs distribution for the probability distribution of dynamical trajectories, and to some fluctuation relation of the integrated current. The simplest stochastic dynamics where these ideas can be applied are discrete-time Markov chains, defined by transition probabilities $W_{i \to j}$ between configurations $i$ and $j$ : instead of choosing the dynamical rules $W_{i \to j}$ a priori, one determines the transition probabilities and the associate stationary state that maximize the entropy of dynamical trajectories with the other physical constraints that one wishes to impose. We give a self-contained and unified presentation of this type of approach, both for discrete-time Markov Chains and for continuous-time Master Equations. The obtained results are in full agreement with the Bayesian approach introduced by Evans [Phys. Rev. Lett. 92, 150601 (2004)] under the name 'Non-equilibrium Counterpart to detailed balance', and with the 'invariant quantities' derived by Baule and Evans [Phys. Rev. Lett. 101, 240601 (2008)], but provide a slightly different perspective via the formulation in terms of an eigenvalue problem.Comment: v4=final versio

Dyson Hierarchical Long-Ranged Quantum Spin-Glass via real-space renormalization

We consider the Dyson hierarchical version of the quantum Spin-Glass with random Gaussian couplings characterized by the power-law decaying variance $\overline{J^2(r)} \propto r^{-2\sigma}$ and a uniform transverse field $h$. The ground state is studied via real-space renormalization to characterize the spinglass-paramagnetic zero temperature quantum phase transition as a function of the control parameter $h$. In the spinglass phase $h<h_c$, the typical renormalized coupling grows with the length scale $L$ as the power-law $J_L^{typ}(h) \propto \Upsilon(h) L^{\theta}$ with the classical droplet exponent $\theta=1-\sigma$, where the stiffness modulus vanishes at criticality $\Upsilon(h) \propto (h_c-h)^{\mu}$, whereas the typical renormalized transverse field decays exponentially $h^{typ}_L(h) \propto e^{- \frac{L}{\xi}}$ where the correlation length diverges at the transition $\xi \propto (h_c-h)^{-\nu}$. At the critical point $h=h_c$, the typical renormalized coupling $J_L^{typ}(h_c)$ and the typical renormalized transverse field $h^{typ}_L(h_c)$ display the same power-law behavior $L^{-z}$ with a finite dynamical exponent $z$. The RG rules are applied numerically to chains containing $L=2^{12}=4096$ spins in order to measure these critical exponents for various values of $\sigma$ in the region $1/2<\sigma<1$.Comment: 9 pages, 7 figure

Many-Body-Localization Transition : sensitivity to twisted boundary conditions

For disordered interacting quantum systems, the sensitivity of the spectrum to twisted boundary conditions depending on an infinitesimal angle $\phi$ can be used to analyze the Many-Body-Localization Transition. The sensitivity of the energy levels $E_n(\phi)$ is measured by the level curvature $K_n=E_n"(0)$, or more precisely by the Thouless dimensionless curvature $k_n=K_n/\Delta_n$, where $\Delta_n$ is the level spacing that decays exponentially with the size $L$ of the system. For instance $\Delta_n \propto 2^{-L}$ in the middle of the spectrum of quantum spin chains of $L$ spins, while the Drude weight $D_n=L K_n$ studied recently by M. Filippone, P.W. Brouwer, J. Eisert and F. von Oppen [arxiv:1606.07291v1] involves a different rescaling. The sensitivity of the eigenstates $\vert \psi_n(\phi) >$ is characterized by the susceptibility $\chi_n=-F_n"(0)$ of the fidelity $F_n =\vert < \psi_n(0) \vert \psi_n(\phi) >\vert$. Both observables are distributed with probability distributions displaying power-law tails $P_{\beta}(k) \simeq A_{\beta} \vert k \vert^{-(2+\beta)}$ and $Q(\chi) \simeq B_{\beta} \chi^{-\frac{3+\beta}{2}}$, where $\beta$ is the level repulsion index taking the values $\beta^{GOE}=1$ in the ergodic phase and $\beta^{loc}=0$ in the localized phase. The amplitudes $A_{\beta}$ and $B_{\beta}$ of these two heavy tails are given by some moments of the off-diagonal matrix element of the local current operator between two nearby energy levels, whose probability distribution has been proposed as a criterion for the Many-Body-Localization transition by M. Serbyn, Z. Papic and D.A. Abanin [Phys. Rev. X 5, 041047 (2015)].Comment: v2= revised version with many improvements , 11 page

Low-temperature dynamics of Long-Ranged Spin-Glasses : full hierarchy of relaxation times via real-space renormalization

We consider the long-ranged Ising spin-glass with random couplings decaying as a power-law of the distance, in the region of parameters where the spin-glass phase exists with a positive droplet exponent. For the Metropolis single-spin-flip dynamics near zero temperature, we construct via real-space renormalization the full hierarchy of relaxation times of the master equation for any given realization of the random couplings. We then analyze the probability distribution of dynamical barriers as a function of the spatial scale. This real-space renormalization procedure represents a simple explicit example of the droplet scaling theory, where the convergence towards local equilibrium on larger and larger scales is governed by a strong hierarchy of activated dynamical processes, with valleys within valleys.Comment: v2=final versio

Block Renormalization for quantum Ising models in dimension d=2d=2 : applications to the pure and random ferromagnet, and to the spin-glass

For the quantum Ising chain, the self-dual block renormalization procedure of Fernandez-Pacheco [Phys. Rev. D 19, 3173 (1979)] is known to reproduce exactly the location of the zero-temperature critical point and the correlation length exponent $\nu=1$. Recently, Miyazaki and Nishimori [Phys. Rev. E 87, 032154 (2013)] have proposed to study the disordered quantum Ising model in dimensions $d>1$ by applying the Fernandez-Pacheco procedure successively in each direction. To avoid the inequivalence of directions of their approach, we propose here an alternative procedure where the $d$ directions are treated on the same footing. For the pure model, this leads to the correlation length exponents $\nu \simeq 0.625$ in $d=2$ (to be compared with the 3D classical Ising model exponent $\nu \simeq 0.63$) and $\nu \simeq 0.5018$ (to be compared with the 4D classical Ising model mean-field exponent $\nu =1/2$). For the disordered model in dimension $d=2$, either ferromagnetic or spin-glass, the numerical application of the renormalization rules to samples of linear size $L=4096$ yields that the transition is governed by an Infinite Disorder Fixed Point, with the activated exponent $\psi \simeq 0.65$, the typical correlation exponent $\nu_{typ} \simeq 0.44$ and the finite-size correlation exponent $\nu_{FS} \simeq 1.25$. We discuss the similarities and differences with the Strong Disorder Renormalization results.Comment: v2=final version (21 pages, 6 figures

One-dimensional Ising spin-glass with power-law interaction : real-space renormalization at zero temperature

For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance $r$ as $J(r) \sim r^{-\sigma}$ and distributed with the L\'evy symmetric stable distribution of index $1 <\mu \leq 2$ (including the usual Gaussian case $\mu=2$), we consider the region $\sigma>1/\mu$ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings $J_L \propto L^{\theta_{\mu}(\sigma)}$ is found to be $\theta_{\mu}(\sigma)=\frac{2}{\mu}-\sigma$ whenever the long-ranged couplings are relevant $\theta_{\mu}(\sigma)=\frac{2}{\mu}-\sigma \geq -1$. For the statistics of the ground state energy $E_L^{GS}$ over disordered samples, we obtain that the droplet exponent $\theta_{\mu}(\sigma)$ governs the leading correction to extensivity of the averaged value $\overline{E_L^{GS}} \simeq L e_0 +L^{\theta_{\mu}(\sigma)} e_1$. The characteristic scale of the fluctuations around this average is of order $L^{\frac{1}{\mu}}$, and the rescaled variable $u=(E_L^{GS}-\overline{E_L^{GS}})/L^{\frac{1}{\mu}}$ is Gaussian distributed for $\mu=2$, or displays the negative power-law tail in $1/(-u)^{1+\mu}$ for $u \to -\infty$ in the L\'evy case $1<\mu<2$.Comment: v2=revised version (17 pages) with new section VII concerning the Dyson hierarchical Spin-Glass mode

On the localization of random heteropolymers at the interface between two selective solvents

To study the localization of random heteropolymers at an interface separating two selective solvents within the model of Garel, Huse, Leibler and Orland, Europhys. Lett. {\bf 8} 9 (1989), we propose an approach based on a disorder-dependent real space renormalization procedure. This approach allows to recover that a chain with a symmetric distribution in hydrophobic/hydrophilic components is localized at any temperature in the thermodynamic limit, whereas a dissymmetric distribution in hydrophobic/hydrophilic components leads to a delocalization phase transition. It yields in addition explicit expressions for the thermodynamic quantities as well as a very detailed description of the statistical properties of the behaviors of the heteropolymers in the high temperature limit. For the case of a small dissymmetry in hydrophobic/hydrophilic components, the renormalization approach yields explicit predictions for the delocalization transition temperature and for the critical behaviors of various quantities : in particular, the free energy presents an essential singularity at the transition, the typical length of blobs in the preferred solvent diverges with an essential singularity, whereas the typical length of blobs in the other solvent diverges algebraically. Finite-size properties are also characterized in details for both cases. In particular, we give the probability distribution of the delocalization temperature for the ensemble of chains of finite (large) length $L$. Finally, we discuss the non-equilibrium dynamics at temperature $T$ starting from a zero-temperature initial condition.Comment: 29 pages, Latex, 1 eps figure. Final revised version, to appear in EPJ

Many Body Localization Transition in the strong disorder limit : entanglement entropy from the statistics of rare extensive resonances

The space of one-dimensional disordered interacting quantum models displaying a Many-Body-Localization Transition seems sufficiently rich to produce critical points with level statistics interpolating continuously between the Poisson statistics of the Localized phase and the Wigner-Dyson statistics of the Delocalized Phase. In this paper, we consider the strong disorder limit of the MBL transition, where the critical level statistics is close to the Poisson statistics. We analyse a one-dimensional quantum spin model, in order to determine the statistical properties of the rare extensive resonances that are needed to destabilize the MBL phase. At criticality, we find that the entanglement entropy can grow with an exponent $0<\alpha < 1$ anywhere between the area law $\alpha=0$ and the volume law $\alpha=1$, as a function of the resonances properties, while the entanglement spectrum follows the strong multifractality statistics. In the MBL phase near criticality, we obtain the simple value $\nu=1$ for the correlation length exponent. Independently of the strong disorder limit, we explain why for the Many-Body-Localization transition concerning individual eigenstates, the correlation length exponent $\nu$ is not constrained by the usual Harris inequality $\nu \geq 2/d$, so that there is no theoretical inconsistency with the best numerical measure $\nu = 0.8 (3)$ obtained by D. J. Luitz, N. Laflorencie and F. Alet, Phys. Rev. B 91, 081103 (2015).Comment: v3= 22 pages with NEW SECTION V on the multifractality of the entanglement spectru