Statistics of the two-point transmission at Anderson localization transitions

At Anderson critical points, the statistics of the two-point transmission $T_L$ for disordered samples of linear size $L$ is expected to be multifractal with the following properties [Janssen {\it et al} PRB 59, 15836 (1999)] : (i) the probability to have $T_L \sim 1/L^{\kappa}$ behaves as $L^{\Phi(\kappa)}$, where the multifractal spectrum $\Phi(\kappa)$ terminates at $\kappa=0$ as a consequence of the physical bound $T_L \leq 1$; (ii) the exponents $X(q)$ that govern the moments $\overline{T_L^q} \sim 1/L^{X(q)}$ become frozen above some threshold: $X(q \geq q_{sat}) = - \Phi(\kappa=0)$, i.e. all moments of order $q \geq q_{sat}$ are governed by the measure of the rare samples having a finite transmission ($\kappa=0$). In the present paper, we test numerically these predictions for the ensemble of $L \times L$ power-law random banded matrices, where the random hopping $H_{i,j}$ decays as a power-law $(b/| i-j |)^a$. This model is known to present an Anderson transition at $a=1$ between localized ($a>1$) and extended ($a<1$) states, with critical properties that depend continuously on the parameter $b$. Our numerical results for the multifractal spectra $\Phi_b(\kappa)$ for various $b$ are in agreement with the relation $\Phi(\kappa \geq 0) = 2 [ f(\alpha= d+ \frac{\kappa}{2}) -d ]$ in terms of the singularity spectrum $f(\alpha)$ of individual critical eigenfunctions, in particular the typical exponents are related via the relation $\kappa_{typ}(b)= 2 (\alpha_{typ}(b)-d)$. We also discuss the statistics of the two-point transmission in the delocalized phase and in the localized phase.Comment: v2=final version with two new appendices with respect to v1; 12 pages, 10 figure

A critical Dyson hierarchical model for the Anderson localization transition

A Dyson hierarchical model for Anderson localization, containing non-random hierarchical hoppings and random on-site energies, has been studied in the mathematical literature since its introduction by Bovier [J. Stat. Phys. 59, 745 (1990)], with the conclusion that this model is always in the localized phase. Here we show that if one introduces alternating signs in the hoppings along the hierarchy (instead of choosing all hoppings of the same sign), it is possible to reach an Anderson localization critical point presenting multifractal eigenfunctions and intermediate spectral statistics. The advantage of this model is that one can write exact renormalization equations for some observables. In particular, we obtain that the renormalized on-site energies have the Cauchy distributions for exact fixed points. Another output of this renormalization analysis is that the typical exponent of critical eigenfunctions is always $\alpha_{typ}=2$, independently of the disorder strength. We present numerical results concerning the whole multifractal spectrum $f(\alpha)$ and the compressibility $\chi$ of the level statistics, both for the box and the Cauchy distributions of the random on-site energies. We discuss the similarities and differences with the ensemble of ultrametric random matrices introduced recently by Fyodorov, Ossipov and Rodriguez [J. Stat. Mech. L12001 (2009)].Comment: 21 pages, 11 figures; v2=final versio

Zero-temperature spinglass-ferromagnetic transition : scaling analysis of the domain-wall energy

For the Ising model with Gaussian random coupling of average $J_0$ and unit variance, the zero-temperature spinglass-ferromagnetic transition as a function of the control parameter $J_0$ can be studied via the size-$L$ dependent renormalized coupling defined as the domain-wall energy $J^R(L) \equiv E_{GS}^{(AF)}(L)-E_{GS}^{(F)}(L)$ (i.e. the difference between the ground state energies corresponding to AntiFerromagnetic and and Ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension $d=2,3,4,5,6$. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent $\theta^c$ is clearly bigger than the spin-glass stiffness exponent $\theta^{SG}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.Comment: v2=final version, 19 pages, 8 figure

Chaos properties of the one-dimensional long-range Ising spin-glass

For the long-range one-dimensional Ising spin-glass with random couplings decaying as $J(r) \propto r^{-\sigma}$, the scaling of the effective coupling defined as the difference between the free-energies corresponding to Periodic and Antiperiodic boundary conditions $J^R(N) \equiv F^{(P)}(N)-F^{(AP)}(N) \sim N^{\theta(\sigma)}$ defines the droplet exponent $\theta(\sigma)$. Here we study numerically the instability of the renormalization flow of the effective coupling $J^R(N)$ with respect to magnetic, disorder and temperature perturbations respectively, in order to extract the corresponding chaos exponents $\zeta_H(\sigma)$, $\zeta_J(\sigma)$ and $\zeta_T(\sigma)$ as a function of $\sigma$. Our results for $\zeta_T(\sigma)$ are interpreted in terms of the entropy exponent $\theta_S(\sigma) \simeq 1/3$ which governs the scaling of the entropy difference $S^{(P)}(N)-S^{(AP)}(N) \sim N^{\theta_S(\sigma)}$. We also study the instability of the ground state configuration with respect to perturbations, as measured by the spin overlap between the unperturbed and the perturbed ground states, in order to extract the corresponding chaos exponents $\zeta^{overlap}_H(\sigma)$ and $\zeta^{overlap}_J(\sigma)$.Comment: 14 pages, 15 figure

Random Transverse Field Ising model in d=2d=2 : analysis via Boundary Strong Disorder Renormalization

To avoid the complicated topology of surviving clusters induced by standard Strong Disorder RG in dimension $d>1$, we introduce a modified procedure called 'Boundary Strong Disorder RG' where the order of decimations is chosen a priori. We apply numerically this modified procedure to the Random Transverse Field Ising model in dimension $d=2$. We find that the location of the critical point, the activated exponent $\psi \simeq 0.5$ of the Infinite Disorder scaling, and the finite-size correlation exponent $\nu_{FS} \simeq 1.3$ are compatible with the values obtained previously by standard Strong Disorder RG.Our conclusion is thus that Strong Disorder RG is very robust with respect to changes in the order of decimations. In addition, we analyze in more details the RG flows within the two phases to show explicitly the presence of various correlation length exponents : we measure the typical correlation exponent $\nu_{typ} \simeq 0.64$ in the disordered phase (this value is very close to the correlation exponent $\nu^Q_{pure}(d=2) \simeq 0.63$ of the {\it pure} two-dimensional quantum Ising Model), and the typical exponent $\nu_h \simeq 1$ within the ordered phase. These values satisfy the relations between critical exponents imposed by the expected finite-size scaling properties at Infinite Disorder critical points. Within the disordered phase, we also measure the fluctuation exponent $\omega \simeq 0.35$ which is compatible with the Directed Polymer exponent $\omega_{DP}(1+1)=1/3$ in $(1+1)$ dimensions.Comment: 10 pages, 10 figure

Random polymers and delocalization transitions

In these proceedings, we first summarize some general properties of phase transitions in the presence of quenched disorder, with emphasis on the following points: the need to distinguish typical and averaged correlations, the possible existence of two correlation length exponents $\nu$, the general bound $\nu_{FS} \geq 2/d$, the lack of self-averaging of thermodynamic observables at criticality, the scaling properties of the distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples of size $L$. We then review our recent works on the critical properties of various delocalization transitions involving random polymers, namely (i) the bidimensional wetting (ii) the Poland-Scheraga model of DNA denaturation (iii) the depinning transition of the selective interface model (iv) the freezing transition of the directed polymer in a random medium.Comment: 20 pages, Conference Proceedings "Inhomogeneous Random Systems", I.H.P., Paris, France, January 200

Typical versus averaged overlap distribution in Spin-Glasses : Evidence for the droplet scaling theory

We consider the statistical properties over disordered samples of the overlap distribution $P_{\cal J}(q)$ which plays the role of an order parameter in spin-glasses. We show that near zero temperature (i) the {\it typical} overlap distribution is exponentially small in the central region of $-1<q<1$: $P^{typ}(q) = e^{\bar{\ln P_{\cal J}(q)}} \sim e^{- \beta N^{\theta} \phi(q)}$, where $\theta$ is the droplet exponent defined here with respect to the total number $N$ of spins (in order to consider also fully connected models where the notion of length does not exist); (ii) the rescaled variable $v = - (\ln P_{\cal J}(q))/N^{\theta}$ remains an O(1) random positive variable describing sample-to sample fluctuations; (iii) the averaged distribution $\bar{P_{\cal J}(q)}$ is non-typical and dominated by rare anomalous samples. Similar statements hold for the cumulative overlap distribution $I_{\cal J}(q_0) \equiv \int_{0}^{q_0} dq P_{\cal J}(q)$. These results are derived explicitly for the spherical mean-field model with $\theta=1/3$, $\phi(q)=1-q^2$, and the random variable $v$ corresponds to the rescaled difference between the two largest eigenvalues of GOE random matrices. Then we compare numerically the typical and averaged overlap distributions for the long-ranged one-dimensional Ising spin-glass with random couplings decaying as $J(r) \propto r^{-\sigma}$ for various values of the exponent $\sigma$, corresponding to various droplet exponents $\theta(\sigma)$, and for the mean-field SK-model (corresponding formally to the $\sigma=0$ limit of the previous model). Our conclusion is that future studies on spin-glasses should measure the {\it typical} values of the overlap distribution or of the cumulative overlap distribution to obtain clearer conclusions on the nature of the spin-glass phase.Comment: v2=final revised version (in particular new sections IIE,IIIC and Appendix B w.r.t. v1

Matching between typical fluctuations and large deviations in disordered systems : application to the statistics of the ground state energy in the SK spin-glass model

For the statistics of global observables in disordered systems, we discuss the matching between typical fluctuations and large deviations. We focus on the statistics of the ground state energy $E_0$ in two types of disordered models : (i) for the directed polymer of length $N$ in a two-dimensional medium, where many exact results exist (ii) for the Sherrington-Kirkpatrick spin-glass model of $N$ spins, where various possibilities have been proposed. Here we stress that, besides the behavior of the disorder-average $E_0^{av}(N)$ and of the standard deviation $\Delta E_0(N) \sim N^{\omega_f}$ that defines the fluctuation exponent $\omega_f$, it is very instructive to study the full probability distribution $\Pi(u)$ of the rescaled variable $u= \frac{E_0(N)-E_0^{av}(N)}{\Delta E_0(N)}$ : (a) numerically, the convergence towards $\Pi(u)$ is usually very rapid, so that data on rather small sizes but with high statistics allow to measure the two tails exponents $\eta_{\pm}$ defined as $\ln \Pi(u \to \pm \infty) \sim - | u |^{\eta_{\pm}}$. In the generic case $1< \eta_{\pm} < +\infty$, this leads to explicit non-trivial terms in the asymptotic behaviors of the moments $\bar{Z_N^n}$ of the partition function when the combination $[| n | N^{\omega_f}]$ becomes large (b) simple rare events arguments can usually be found to obtain explicit relations between $\eta_{\pm}$ and $\omega_f$. These rare events usually correspond to 'anomalous' large deviation properties of the generalized form $R(w_{\pm} = \frac{E_0(N)-E_0^{av}(N)}{N^{\kappa_{\pm}}}) \sim e^{- N^{\rho_{\pm}} {\cal R}_{\pm}(w_{\pm})}$ (the 'usual' large deviations formalism corresponds to $\kappa_{\pm}=1=\rho_{\pm}$).Comment: 10 pages, 4 figure