arXiv subjects

Thomas Garel

Publications and source records attributed to Thomas Garel.

At least 19 recordsLinked to original sources

arXiv2014

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 $θ^c$ is clearly bigger than the spin-glass stiffness exponent $θ^{SG}$, 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.

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arXiv2014

Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin-glass

The long-range one-dimensional Ising spin-glass with random couplings decaying as $J(r) \propto r^{-σ}$ presents a spin-glass phase $T_c(σ)>0$ for $0 \leq σ<1$ (the limit $σ=0$ corresponds to the mean-field SK-model). We use the eigenvalue method introduced in our previous work [C. Monthus and T. Garel, J. Stat. Mech. P12017 (2009)] to measure the equilibrium time $t_{eq}(N)$ at temperature $T=T_c(σ)/2$ as a function of the number $N$ of spins. We find the activated scaling $\ln t_{eq}(N) \sim N^ψ$ with the same barrier exponent $ψ\simeq 0.33$ in the whole region $0\leqσ<1$.

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arXiv2013

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 $J(r) \propto r^{-σ}$, the scaling of the effective coupling defined as the difference between the free-energies corresponding to Periodic and Antiperiodic boundary conditions $J^R(N) \equiv F^{(P)}(N)-F^{(AP)}(N) \sim N^{θ(σ)}$ defines the droplet exponent $θ(σ)$. 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 $ζ_H(σ)$, $ζ_J(σ)$ and $ζ_T(σ)$ as a function of $σ$. Our results for $ζ_T(σ) $ are interpreted in terms of the entropy exponent $θ_S(σ) \simeq 1/3$ which governs the scaling of the entropy difference $ S^{(P)}(N)-S^{(AP)}(N) \sim N^{θ_S(σ)}$. 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 $ζ^{overlap}_H(σ)$ and $ζ^{overlap}_J(σ)$.

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arXiv2013

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^{- βN^θ ϕ(q)} $, where $θ$ 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 $v = - (\ln P_{\cal J}(q))/N^θ$ 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 $θ=1/3$, $ϕ(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 $J(r) \propto r^{-σ}$ for various values of the exponent $σ$, corresponding to various droplet exponents $θ(σ)$, and for the mean-field SK-model (corresponding formally to the $σ=0$ 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.

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arXiv2013

Dynamical barriers for the random ferromagnetic Ising model on the Cayley tree : traveling-wave solution of the real space renormalization flow

We consider the stochastic dynamics near zero-temperature of the random ferromagnetic Ising model on a Cayley tree of branching ratio $K$. We apply the Boundary Real Space Renormalization procedure introduced in our previous work (C. Monthus and T. Garel, J. Stat. Mech. P02037 (2013)) in order to derive the renormalization rule for dynamical barriers. We obtain that the probability distribution $P_n(B)$ of dynamical barrier for a subtree of $n$ generations converges for large $n$ towards some traveling-wave $P_n(B) \simeq P^*(B-nv) $, i.e. the width of the probability distribution remains finite around an average-value that grows linearly with the number $n$ of generations. We present numerical results for the branching ratios K=2 and K=3. We also compute the weak-disorder expansion of the velocity $v$ for K=2.

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arXiv2013

Dynamical barriers of pure and random ferromagnetic Ising models on fractal lattices

We consider the stochastic dynamics of the pure and random ferromagnetic Ising model on the hierarchical diamond lattice of branching ratio $K$ with fractal dimension $d_f=(\ln (2K))/\ln 2$. We adapt the Real Space Renormalization procedure introduced in our previous work [C. Monthus and T. Garel, J. Stat. Mech. P02037 (2013)] to study the equilibrium time $t_{eq}(L)$ as a function of the system size $L$ near zero-temperature. For the pure Ising model, we obtain the behavior $t_{eq}(L) \sim L^α e^{β2J L^{d_s}} $ where $d_s=d_f-1$ is the interface dimension, and we compute the prefactor exponent $α$. For the random ferromagnetic Ising model, we derive the renormalization rules for dynamical barriers $B_{eq}(L) \equiv (\ln t_{eq}/β)$ near zero temperature. For the fractal dimension $d_f=2$, we obtain that the dynamical barrier scales as $ B_{eq}(L)= c L+L^{1/2} u$ where $u$ is a Gaussian random variable of non-zero-mean. While the non-random term scaling as $L$ corresponds to the energy-cost of the creation of a system-size domain-wall, the fluctuation part scaling as $L^{1/2}$ characterizes the barriers for the motion of the system-size domain-wall after its creation. This scaling corresponds to the dynamical exponent $ψ=1/2$, in agreement with the conjecture $ψ=d_s/2$ proposed in [C. Monthus and T. Garel, J. Phys. A 41, 115002 (2008)]. In particular, it is clearly different from the droplet exponent $θ\simeq 0.299$ involved in the statics of the random ferromagnet on the same lattice.

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arXiv2013

Dynamics of Ising models near zero temperature : Real Space Renormalization Approach

We consider the stochastic dynamics of Ising ferromagnets (either pure or random) near zero temperature. The master equation satisfying detailed balance can be mapped onto a quantum Hamiltonian which has an exact zero-energy ground state representing the thermal equilibrium. The largest relaxation time $t_{eq}$ governing the convergence towards this Boltzmann equilibrium in finite-size systems is determined by the lowest non-vanishing eigenvalue $E_1=1/t_{eq}$ of the quantum Hamiltonian $H$. We introduce and study a real-space renormalization procedure for the quantum Hamiltonian associated to the single-spin-flip dynamics of Ising ferromagnets near zero temperature. We solve explicitly the renormalization flow for two cases. (i) For the one-dimensional random ferromagnetic chain with free boundary conditions, the largest relaxation time $t_{eq}$ can be expressed in terms of the set of random couplings for various choices of the dynamical transition rates. The validity of these RG results in $d=1$ is checked by comparison with another approach. (ii) For the pure Ising model on a Cayley tree of branching ratio $K$, we compute the exponential growth of $t_{eq}(N)$ with the number $N$ of generations.

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arXiv2012

Dynamical Barriers in the Dyson Hierarchical model via Real Space Renormalization

The Dyson hierarchical one-dimensional Ising model of parameter $σ>0$ contains long-ranged ferromagnetic couplings decaying as $1/r^{1+σ}$ in terms of the distance $r$. We study the stochastic dynamics near zero-temperature via the Real Space Renormalization introduced in our previous work (C. Monthus and T. Garel, arxiv:1212.0643) in order to compute explicitly the equilibrium time $t_{eq}(L)$ as a function of the system size $L$. For $σ<1$ where the static critical temperature for the ferromagnetic transition is finite $T_c>0$, we obtain that dynamical barriers grow as the power-law: $\ln t_{eq}(L) \simeq β(\frac{4 J_0}{3(2^{1-σ}-1)}) L^{1-σ}$. For $σ=1$ where the static critical temperature vanishes $T_c=0$, we obtain that dynamical barriers grow logarithmically as : $\ln t_{eq}(L) \simeq [β(\frac{4 J_0}{3 \ln 2}) -1] \ln L $. We also compute finite contributions to the dynamical barriers that can depend on the choice of transition rates satisfying detailed balance.

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arXiv2012

Random Transverse Field Ising model on the Cayley Tree : analysis via Boundary Strong Disorder Renormalization

Strong Disorder Renormalization for the Random Transverse Field Ising model leads to a complicated topology of surviving clusters as soon as $d>1$. Even if one starts from a Cayley tree, the network of surviving renormalized clusters will contain loops, so that no analytical solution can been obtained. Here we introduce a modified procedure called 'Boundary Strong Disorder Renormalization' that preserves the tree structure, so that one can write simple recursions with respect to the number of generations. We first show that this modified procedure allows to recover exactly most of the critical exponents for the one-dimensional chain. After this important check, we study the RG equations for the quantum Ising model on a Cayley tree with a uniform ferromagnetic coupling $J$ and random transverse fields with support $[h_{min},h_{max}]$. We find the following picture (i) for $J>h_{max}$, only bonds are decimated, so that the whole tree is a quantum ferromagnetic cluster (ii) for $J<h_{min}$, only sites are decimated, so that no quantum ferromagnetic cluster is formed, and the ferromagnetic coupling to the boundary coincides with the partition function of a Directed Polymer model in a random medium (iii) for $h_{min}<J<h_{max}$, both sites and bonds can be decimated : the quantum ferromagnetic clusters can either remain finite (the physics is then similar to (ii), with a quantitative mapping to a modified Directed Polymer model) or an infinite quantum ferromagnetic cluster appears. We find that the quantum transition can be of two types : (a) either the quantum transition takes place in the region where quantum ferromagnetic clusters remain finite, and the singularity of the ferromagnetic coupling to the boundary involves the typical correlation length exponent $ν_{typ}=1$ (b) or the quantum transition takes place at the point where an extensive quantum ferromagnetic cluster appears.

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arXiv2012

Random Transverse Field Ising model in $d=2$ : 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 $ψ\simeq 0.5$ of the Infinite Disorder scaling, and the finite-size correlation exponent $ν_{FS} \simeq 1.3$ 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 $ν_{typ} \simeq 0.64$ in the disordered phase (this value is very close to the correlation exponent $ν^Q_{pure}(d=2) \simeq 0.63$ of the {\it pure} two-dimensional quantum Ising Model), and the typical exponent $ν_h \simeq 1$ 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 $ω\simeq 0.35$ which is compatible with the Directed Polymer exponent $ω_{DP}(1+1)=1/3$ in $(1+1)$ dimensions.

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arXiv2012

Strong Disorder RG principles within a fixed cell-size real space renormalization : application to the Random Transverse Field Ising model on various fractal lattices

Strong Disorder Renormalization is an energy-based renormalization that leads to a complicated renormalized topology for the surviving clusters as soon as $d>1$. In this paper, we propose to include Strong Disorder Renormalization ideas within the more traditional fixed cell-size real space RG framework. We first consider the one-dimensional chain as a test for this fixed cell-size procedure: we find that all exactly known critical exponents are reproduced correctly, except for the magnetic exponent $β$ (because it is related to more subtle persistence properties of the full RG flow). We then apply numerically this fixed cell-size procedure to two types of renormalizable fractal lattices (i) the Sierpinski gasket of fractal dimension $D=\ln 3/\ln 2$, where there is no underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens in $d=1$ (ii) a hierarchical diamond lattice of fractal dimension $D=4/3$, where there is an underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens on hypercubic lattices of dimension $d>1$. In both cases, we find that the transition is governed by an Infinite Disorder Fixed Point : besides the measure of the activated exponent $ψ$, we analyze the RG flow of various observables in the disordered and ordered phases, in order to extract the 'typical' correlation length exponents of these two phases which are different from the finite-size correlation length exponent.

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arXiv2012

Random Transverse Field Ising Model in dimension $d>1$ : scaling analysis in the disordered phase from the Directed Polymer model

For the quantum Ising model with ferromagnetic random couplings $J_{i,j}>0$ and random transverse fields $h_i>0$ at zero temperature in finite dimensions $d>1$, we consider the lowest-order contributions in perturbation theory in $(J_{i,j}/h_i)$ to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as : $\ln C(r) \sim - \frac{r}{ξ_{typ}} +r^ω u$, where $ξ_{typ} $ is the typical correlation length, $u$ is a random variable, and $ω$ coincides with the droplet exponent $ω_{DP}(D=d-1)$ of the Directed Polymer with $D=(d-1)$ transverse directions. Our main conclusions are (i) whenever $ω>0$, the quantum model is governed by an Infinite-Disorder fixed point : there are two distinct correlation length exponents related by $ν_{typ}=(1-ω)ν_{av}$ ; the distribution of the local susceptibility $χ_{loc}$ presents the power-law tail $P(χ_{loc}) \sim 1/χ_{loc}^{1+μ}$ where $μ$ vanishes as $ξ_{av}^{-ω} $, so that the averaged local susceptibility diverges in a finite neighborhood $0<μ<1$ before criticality (Griffiths phase) ; the dynamical exponent $z$ diverges near criticality as $z=d/μ\sim ξ_{av}^ω$ (ii) in dimensions $d \leq 3$, any infinitesimal disorder flows towards this Infinite-Disorder fixed point with $ω(d)>0$ (for instance $ω(d=2)=1/3$ and $ω(d=3) \sim 0.24$) (iii) in finite dimensions $d > 3$, a finite disorder strength is necessary to flow towards the Infinite-Disorder fixed point with $ω(d)>0$ (for instance $ω(d=4) \simeq 0.19$), whereas a Finite-Disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension $d=\infty$ where $ω=0$, we discuss the similarities and differences with the case of finite dimensions.

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arXiv2011

Random Transverse Field Ising Model in dimension $d=2,3$ : Infinite Disorder scaling via a non-linear transfer approach

The 'Cavity-Mean-Field' approximation developed for the Random Transverse Field Ising Model on the Cayley tree [L. Ioffe and M. Mézard, PRL 105, 037001 (2010)] has been found to reproduce the known exact result for the surface magnetization in $d=1$ [O. Dimitrova and M. Mézard, J. Stat. Mech. (2011) P01020]. In the present paper, we propose to extend these ideas in finite dimensions $d>1$ via a non-linear transfer approach for the surface magnetization. In the disordered phase, the linearization of the transfer equations correspond to the transfer matrix for a Directed Polymer in a random medium of transverse dimension $D=d-1$, in agreement with the leading order perturbative scaling analysis [C. Monthus and T. Garel, arxiv:1110.3145]. We present numerical results of the non-linear transfer approach in dimensions $d=2$ and $d=3$. In both cases, we find that the critical point is governed by Infinite Disorder scaling. In particular exactly at criticality, the one-point surface magnetization scales as $\ln m_L^{surf} \simeq - L^{ω_c} v$, where $ω_c(d)$ coincides with the droplet exponent $ω_{DP}(D=d-1)$ of the corresponding Directed Polymer model, with $ω_c(d=2)=1/3$ and $ω_c(d=3) \simeq 0.24$. The distribution $P(v)$ of the positive random variable $v$ of order O(1) presents a power-law singularity near the origin $P(v) \propto v^a$ with $a(d=2,3)>0$ so that all moments of the surface magnetization are governed by the same power-law decay $\bar{(m_L^{surf})^k} \propto L^{- x_S}$ with $x_S=ω_c (1+a)$ independently of the order $k$.

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arXiv2011

Random field Ising model : statistical properties of low-energy excitations and of equilibrium avalanches

With respect to usual thermal ferromagnetic transitions, the zero-temperature finite-disorder critical point of the Random-field Ising model (RFIM) has the peculiarity to involve some 'droplet' exponent $θ$ that enters the generalized hyperscaling relation $2-α= ν(d-θ)$. In the present paper, to better understand the meaning of this droplet exponent $θ$ beyond its role in the thermodynamics, we discuss the statistics of low-energy excitations generated by an imposed single spin-flip with respect to the ground state, as well as the statistics of equilibrium avalanches i.e. the magnetization jumps that occur in the sequence of ground-states as a function of the external magnetic field. The droplet scaling theory predicts that the distribution $dl/l^{1+θ}$ of the linear-size $l$ of low-energy excitations transforms into the distribution $ds/s^{1+{θ/d_f}}$ for the size $s$ (number of spins) of excitations of fractal dimension $d_f$ ($s \sim l^{d_f}$). In the non-mean-field region $d d_c$, droplets have a fractal dimension $d_f=2 θ$ leading to the well-known mean-field result $ds/s^{3/2}$. Zero-field equilibrium avalanches are expected to display the same distribution $ds/s^{1+{θ/d_f}}$. We also discuss the statistics of equilibrium avalanches integrated over the external field and finite-size behaviors. These expectations are checked numerically for the Dyson hierarchical version of the RFIM, where the droplet exponent $θ(σ)$ can be varied as a function of the effective long-range interaction $J(r) \sim 1/r^{d+σ}$ in $d=1$.

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arXiv2011

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 $α_{typ}=2$, independently of the disorder strength. We present numerical results concerning the whole multifractal spectrum $f(α)$ and the compressibility $χ$ 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)].

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arXiv2011

Anderson localization on the Cayley tree : multifractal statistics of the transmission at criticality and off criticality

In contrast to finite dimensions where disordered systems display multifractal statistics only at criticality, the tree geometry induces multifractal statistics for disordered systems also off criticality. For the Anderson tight-binding localization model defined on a tree of branching ratio K=2 with $N$ generations, we consider the Miller-Derrida scattering geometry [J. Stat. Phys. 75, 357 (1994)], where an incoming wire is attached to the root of the tree, and where $K^{N}$ outcoming wires are attached to the leaves of the tree. In terms of the $K^{N}$ transmission amplitudes $t_j$, the total Landauer transmission is $T \equiv \sum_j | t_j |^2$, so that each channel $j$ is characterized by the weight $w_j=| t_j |^2/T$. We numerically measure the typical multifractal singularity spectrum $f(α)$ of these weights as a function of the disorder strength $W$ and we obtain the following conclusions for its left-termination point $α_+(W)$. In the delocalized phase $W 0$ and is associated with a moment index $q_+(W)>1$. At criticality, it vanishes $α_+(W_c)=0$ and is associated with the moment index $q_+(W_c)=1$. In the localized phase $W>W_c$, $α_+(W)=0$ is associated with some moment index $q_+(W)<1$. We discuss the similarities with the exact results concerning the multifractal properties of the Directed Polymer on the Cayley tree.

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arXiv2011

Random elastic networks : strong disorder renormalization approach

For arbitrary networks of random masses connected by random springs, we define a general strong disorder real-space renormalization (RG) approach that generalizes the procedures introduced previously by Hastings [Phys. Rev. Lett. 90, 148702 (2003)] and by Amir, Oreg and Imry [Phys. Rev. Lett. 105, 070601 (2010)] respectively. The principle is to eliminate iteratively the elementary oscillating mode of highest frequency associated with either a mass or a spring constant. To explain the accuracy of the strong disorder RG rules, we compare with the Aoki RG rules that are exact at fixed frequency.

cond-mat.dis-nn↗

arXiv2010

Anderson localization transition with long-ranged hoppings : analysis of the strong multifractality regime in terms of weighted Levy sums

For Anderson tight-binding models in dimension $d$ with random on-site energies $ε_{\vec r}$ and critical long-ranged hoppings decaying typically as $V^{typ}(r) \sim V/r^d$, we show that the strong multifractality regime corresponding to small $V$ can be studied via the standard perturbation theory for eigenvectors in quantum mechanics. The Inverse Participation Ratios $Y_q(L)$, which are the order parameters of Anderson transitions, can be written in terms of weighted Lévy sums of broadly distributed variables (as a consequence of the presence of on-site random energies in the denominators of the perturbation theory). We compute at leading order the typical and disorder-averaged multifractal spectra $τ_{typ}(q)$ and $τ_{av}(q)$ as a function of $q$. For $q<1/2$, we obtain the non-vanishing limiting spectrum $τ_{typ}(q)=τ_{av}(q)=d(2q-1)$ as $V \to 0^+$. For $q>1/2$, this method yields the same disorder-averaged spectrum $τ_{av}(q)$ of order $O(V)$ as obtained previously via the Levitov renormalization method by Mirlin and Evers [Phys. Rev. B 62, 7920 (2000)]. In addition, it allows to compute explicitly the typical spectrum, also of order $O(V)$, but with a different $q$-dependence $τ_{typ}(q) \ne τ_{av}(q)$ for all $q>q_c=1/2$. As a consequence, we find that the corresponding singularity spectra $f_{typ}(α)$ and $f_{av}(α)$ differ even in the positive region $f>0$, and vanish at different values $α_+^{typ} > α_+^{av}$, in contrast to the standard picture. We also obtain that the saddle value $α_{typ}(q)$ of the Legendre transform reaches the termination point $α_+^{typ}$ where $f_{typ}(α_+^{typ})=0 $ only in the limit $q \to +\infty$.

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