Real-space renormalization for the finite temperature statics and dynamics of the Dyson Long-Ranged Ferromagnetic and Spin-Glass models

14 pages, 2 figuresInternational audienceThe finite temperature dynamics of the Dyson hierarchical classical spins models is studied via real-space renormalization rules concerning the couplings and the relaxation times. For the ferromagnetic model involving Long-Ranged coupling $J(r) \propto r^{-1-\sigma}$ in the region $1/2<\sigma<1$ where there exists a non-mean-field-like thermal Ferromagnetic-Paramagnetic transition, the RG flows are explicitly solved: the characteristic relaxation time $\tau(L)$ follows the critical power-law $\tau(L)\propto L^{z_c(\sigma)}$ at the phase transition and the activated law $\ln \tau(L)\propto L^{\psi}$ with $\psi=1-\sigma$ in the ferromagnetic phase. For the Spin-Glass model involving random Long-Ranged couplings of variance $\overline{J^2(r)} \propto r^{-2\sigma}$ in the region $2/3<\sigma<1$ where there exists a non-mean-field-like thermal SpinGlass-Paramagnetic transition, the coupled RG flows of the couplings and of the relaxation times are studied numerically : the relaxation time $\tau(L)$ follows some power-law $\tau(L)\propto L^{z_c(\sigma)}$ at criticality and the activated law $\ln \tau(L)\propto L^{\psi}$ in the Spin-Glass phase with the dynamical exponent $\psi=1-\sigma=\theta$ coinciding with the droplet exponent governing the flow of the couplings $J(L) \propto L^{\theta}$

Many-Body Localization : construction of the emergent local conserved operators via block real-space renormalization

11 pagesInternational audienceA Fully Many-Body Localized (FMBL) quantum disordered system is characterized by the emergence of an extensive number of local conserved operators that prevents the relaxation towards thermal equilibrium. These local conserved operators can be seen as the building blocks of the whole set of eigenstates. In this paper, we propose to construct them explicitly via some block real-space renormalization. The principle is that each RG step diagonalizes the smallest remaining blocks and produces a conserved operator for each block. The final output for a chain of $N$ spins is a hierarchical organization of the $N$ conserved operators with $\left(\frac{\ln N}{\ln 2}\right)$ layers. The system-size nature of the conserved operators of the top layers is necessary to describe the possible long-ranged order of the excited eigenstates and the possible critical points between different FMBL phases. We discuss the similarities and the differences with the Strong Disorder RSRG-X method that generates the whole set of the $2^N$ eigenstates via a binary tree of $N$ layers. The approach is applied to the Long-Ranged Quantum Spin-Glass Ising model, where the constructed excited eigenstates are found to be exactly like ground states in another disorder realization, so that they can be either in the paramagnetic phase, in the spin-glass phase or critical

On the flux distribution in a one dimensional disordered system

We study some transport properties of a one dimensional disordered system of finite length N. In this system particles are subject to random forces resulting both from a thermal noise and from a quenched random force F(x) which models the inhomogeneous medium. The latter is distributed as a white noise with a non zero average bias. Imposing some fixed concentration of particles at the end points of the chain yields a steady current J(N) which depends on the environnent {F(x)}. The problem of computing the probabilility distribution P(J) over the environments is addressed. Our approchh is based on a path integral method and on a moment calculation. In the case of a non zero bias our results generalize those obtained recently by Oshanin et al

Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d=1,2,3d=1,2,3 via a Monte-Carlo procedure in the disorder

In order to probe with high precision the tails of the ground-state energy distribution of disordered spin systems, K\"orner, Katzgraber and Hartmann \cite{Ko_Ka_Ha} have recently proposed an importance-sampling Monte-Carlo Markov chain in the disorder. In this paper, we combine their Monte-Carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution $P_d(E_0)$ for the directed polymer in a random medium of dimension $d=1,2,3$. In $d=1$, we check the validity of the algorithm by a direct comparison with the exact result, namely the Tracy-Widom distribution. In dimensions $d=2$ and $d=3$, we measure the negative tail up to ten standard deviations, which correspond to probabilities of order $P_d(E_0) \sim 10^{-22}$. Our results are in agreement with Zhang's argument, stating that the negative tail exponent $\eta(d)$ of the asymptotic behavior $\ln P_d (E_0) \sim - | E_0 |^{\eta(d)}$ as $E_0 \to -\infty$ is directly related to the fluctuation exponent $\theta(d)$ (which governs the fluctuations $\Delta E_0(L) \sim L^{\theta(d)}$ of the ground state energy $E_0$ for polymers of length $L$) via the simple formula $\eta(d)=1/(1-\theta(d))$. Along the paper, we comment on the similarities and differences with spin-glasses.Comment: 13 pages, 16 figure

ON A DYNAMICAL MODEL OF GLASSES

We analyze a simple dynamical model of glasses, based on the idea that each particle is trapped in a local potential well, which itself evolves due to hopping of neighbouring particles. The glass transition is signalled by the fact that the equilibrium distribution ceases to be normalisable, and dynamics becomes non-stationary. We generically find stretching of the correlation function at low temperatures and a Vogel-Fulcher like behaviour of the terminal time.Comment: 11 pages, 2 figures (available upon request

Finite-size scaling of pseudo-critical point distributions in the random transverse-field Ising chain

We study the distribution of finite size pseudo-critical points in a one-dimensional random quantum magnet with a quantum phase transition described by an infinite randomness fixed point. Pseudo-critical points are defined in three different ways: the position of the maximum of the average entanglement entropy, the scaling behavior of the surface magnetization, and the energy of a soft mode. All three lead to a log-normal distribution of the pseudo-critical transverse fields, where the width scales as $L^{-1/\nu}$ with $\nu=2$ and the shift of the average value scales as $L^{-1/\nu_{typ}}$ with $\nu_{typ}=1$, which we related to the scaling of average and typical quantities in the critical region.Comment: 4 pages, 2 figure

Statistics of first-passage times in disordered systems using backward master equations and their exact renormalization rules

We consider the non-equilibrium dynamics of disordered systems as defined by a master equation involving transition rates between configurations (detailed balance is not assumed). To compute the important dynamical time scales in finite-size systems without simulating the actual time evolution which can be extremely slow, we propose to focus on first-passage times that satisfy 'backward master equations'. Upon the iterative elimination of configurations, we obtain the exact renormalization rules that can be followed numerically. To test this approach, we study the statistics of some first-passage times for two disordered models : (i) for the random walk in a two-dimensional self-affine random potential of Hurst exponent $H$, we focus on the first exit time from a square of size $L \times L$ if one starts at the square center. (ii) for the dynamics of the ferromagnetic Sherrington-Kirkpatrick model of $N$ spins, we consider the first passage time $t_f$ to zero-magnetization when starting from a fully magnetized configuration. Besides the expected linear growth of the averaged barrier $\bar{\ln t_{f}} \sim N$, we find that the rescaled distribution of the barrier $(\ln t_{f})$ decays as $e^{- u^{\eta}}$ for large $u$ with a tail exponent of order $\eta \simeq 1.72$. This value can be simply interpreted in terms of rare events if the sample-to-sample fluctuation exponent for the barrier is $\psi_{width}=1/3$.Comment: 8 pages, 4 figure

Numerical study of the disordered Poland-Scheraga model of DNA denaturation

We numerically study the binary disordered Poland-Scheraga model of DNA denaturation, in the regime where the pure model displays a first order transition (loop exponent $c=2.15>2$). We use a Fixman-Freire scheme for the entropy of loops and consider chain length up to $N=4 \cdot 10^5$, with averages over $10^4$ samples. We present in parallel the results of various observables for two boundary conditions, namely bound-bound (bb) and bound-unbound (bu), because they present very different finite-size behaviors, both in the pure case and in the disordered case. Our main conclusion is that the transition remains first order in the disordered case: in the (bu) case, the disorder averaged energy and contact densities present crossings for different values of $N$ without rescaling. In addition, we obtain that these disorder averaged observables do not satisfy finite size scaling, as a consequence of strong sample to sample fluctuations of the pseudo-critical temperature. For a given sample, we propose a procedure to identify its pseudo-critical temperature, and show that this sample then obeys first order transition finite size scaling behavior. Finally, we obtain that the disorder averaged critical loop distribution is still governed by $P(l) \sim 1/l^c$ in the regime $l \ll N$, as in the pure case.Comment: 12 pages, 13 figures. Revised versio

Energy dynamics in the Sinai model

We study the time dependent potential energy $W(t)=U(x(0)) - U(x(t))$ of a particle diffusing in a one dimensional random force field (the Sinai model). Using the real space renormalization group method (RSRG), we obtain the exact large time limit of the probability distribution of the scaling variable $w=W(t)/(T \ln t)$. This distribution exhibits a {\it nonanalytic} behaviour at $w=1$. These results are extended to a small non-zero applied field. Using the constrained path integral method, we moreover compute the joint distribution of energy $W(t)$ and position $x(t)$ at time $t$. In presence of a reflecting boundary at the starting point, with possibly some drift in the + direction, the RSRG very simply yields the one time and aging two-time behavior of this joint probability. It exhibits differences in behaviour compared to the unbounded motion, such as analyticity. Relations with some magnetization distributions in the 1D spin glass are discussed.Comment: 21 pages, 4 eps figure