576 research outputs found

### 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

### Block Renormalization for quantum Ising models in dimension $d=2$ : 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

### 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

### 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

### 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

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