Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1206.5539 · Journal of Statistical Mechanics Theory and Experiment

Universality inp-spin glasses with correlated disorder

Valentin Bonzom, Razvan Gurău, Matteo Smerlak

Published Feb 27, 2013• 35 citations
arXivHTMLPDF

Abstract

We introduce a new method, based on the recently developed random tensor theory, to study the p-spin glass model with non-Gaussian, correlated disorder. Using a suitable generalization of Gurau's theorem on the universality of the large N limit of the p-unitary ensemble of random tensors, we exhibit an infinite family of such non-Gaussian distributions which leads to same low temperature phase as the Gaussian distribution. While this result is easy to show (and well known) for uncorrelated disorder, its robustness with respect to strong quenched correlations is surprising. We show in detail how the critical temperature is renormalized by these correlations. We close with a speculation on possible applications of random tensor theory to finite-range spin glass models.

DOI
10.1088/1742-5468/2013/02/l02003

Related papers

Correlation decay in low-dimensional spin glasses

ARMEN N. KOCHARIAN, A. Sogomonian

A new method to investigate the two-point correlation function decay in a finite size random spin system and at finite temperatures is proposed. It is applicable to systems with cyclic boundary conditions and in any local or uniform magnetic field. The efficiency of this method is presented in the case of an isotropic XY spin glass model with Gaussian bond distribution in external field. Using unitary U(1) gauge transformation and operator inequalities we have obtained explicit upper bounds for generalized spin-glass susceptibility κ(T,R) that describes off-diagonal transverse magnetization in two dimensional random XY spin lattices at finite temperatures. It is found that the slowest possible decay for susceptibility in random XY spin glass is of the Kosterlitz–Thouless-type with power-law decay in 2D, and the exponential type in 1D, at low temperatures. These upper bounds rule out the possibility of the corresponding magnetic ordering in 1D and 2D isotropic XY spin glasses at finite temperatures.

Journal of Applied Physics•Nov 15, 1994•1 citation

Critical behavior and universality in Lévy spin glasses

Juan Carlos Andresen, Katharina Janzen, Helmut G. Katzgraber

Using large-scale Monte Carlo simulations that combine parallel tempering with specialized cluster updates, we show that Ising spin glasses with L\'evy-distributed interactions share the same universality class as Ising spin glasses with Gaussian or bimodal-distributed interactions. Corrections to scaling are large for L\'evy spin glasses. To overcome these and show that the critical exponents agree with the bimodal and Gaussian case, we perform an extended scaling of the two-point finite-size correlation length and the spin-glass susceptibility. Furthermore, we compute the critical temperature and compare its dependence on the disorder distribution width with recent analytical predictions [J. Stat. Mech. (2008) P04006].

arXiv:1102.1661•Physical Review B•May 13, 2011•10 citations

Low-temperature behavior of the infinite-range Ising spin-glass: Exact statistical mechanics for small samples

A. P. Young, Scott Kirkpatrick

The low-temperature behavior of the infinite-range Ising spin-glass model with a Gaussian distribution of interactions is studied by evaluating the statistical mechanics exactly for small systems (up to $N=20$ spins). Our results can be well understood by a simple picture of elementary cluster excitations from the ground state which overturn, on average, of order ${N}^{\frac{1}{2}}$ spins. The conventional definition of the spin-glass order parameter is shown to be unsatisfactory and a new definition is proposed. It is speculated that ensemble averages may give different results from time averages for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ even in the infinite time limit.

Physical review. B, Condensed matter•Jan 1, 1982•85 citations

The SYK model and random tensors: Gaussian universality

Thomas Krajewski, Matteo Laudonio, Romain Pascalie +1 more

The SYK model involves N Majorana fermions in 1+0 dimensions with quenched Gaussian dis- order and proves to be exactly solvable in the large N limit at strong coupling. It has been initially proposed by Sachdev and Ye as a model of condensed matter and later gained some interest as a toy model of AdS/CFT correspondence, thanks to the work of Kitaev. On the other side, random tensors are generalisations of random matrices to objets that carry more than two indices. It turns out that the SYK model and random tensors involve a special class of Feynman graphs known as "melons". We will briefly review both constructions. Then, we will show how non Gaussian disorder can be reduced to a Gaussian one, treating the coupling as a random tensor, thanks to Gurau’s Gaussian universality result.

Corfu Summer Institute School and Workshops on Elementary Particle Physics and Gravity (CORFU)•Aug 18, 2020•1 citation

Infinite-ranged models of spin-glasses

Scott Kirkpatrick, David Sherrington

A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and $\mathrm{XY}$ spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for $T$ greater than ${T}_{c}$, the spin-glass transition temperature. At and below ${T}_{c}$, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.

World Scientific lecture notes in physics•Nov 1, 1986•1 citation

Infinite-ranged models of spin-glasses

Scott Kirkpatrick, David Sherrington

A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and $\mathrm{XY}$ spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for $T$ greater than ${T}_{c}$, the spin-glass transition temperature. At and below ${T}_{c}$, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.

Physical review. B, Condensed matter•Jun 1, 1978•1,007 citations
We introduce a new method, based on the recently developed random tensor theory, to study the p-spin glass model with non-Gaussian, correlated disorder. Using a suitable generalization of Gurau's theorem on the universality of the large N limit of the p-unitary ensemble of random tensors, we exhibit an infinite family of such non-Gaussian distributions which leads to same low temperature phase as the Gaussian distribution. While this result is easy to show (and well known) for uncorrelated disorder, its robustness with respect to strong quenched correlations is surprising. We show in detail how the critical temperature is renormalized by these correlations. We close with a speculation on possible applications of random tensor theory to finite-range spin glass models.
A new method to investigate the two-point correlation function decay in a finite size random spin system and at finite temperatures is proposed. It is applicable to systems with cyclic boundary conditions and in any local or uniform magnetic field. The efficiency of this method is presented in the case of an isotropic XY spin glass model with Gaussian bond distribution in external field. Using unitary U(1) gauge transformation and operator inequalities we have obtained explicit upper bounds for generalized spin-glass susceptibility κ(T,R) that describes off-diagonal transverse magnetization in two dimensional random XY spin lattices at finite temperatures. It is found that the slowest possible decay for susceptibility in random XY spin glass is of the Kosterlitz–Thouless-type with power-law decay in 2D, and the exponential type in 1D, at low temperatures. These upper bounds rule out the possibility of the corresponding magnetic ordering in 1D and 2D isotropic XY spin glasses at finite temperatures.
Using large-scale Monte Carlo simulations that combine parallel tempering with specialized cluster updates, we show that Ising spin glasses with L\'evy-distributed interactions share the same universality class as Ising spin glasses with Gaussian or bimodal-distributed interactions. Corrections to scaling are large for L\'evy spin glasses. To overcome these and show that the critical exponents agree with the bimodal and Gaussian case, we perform an extended scaling of the two-point finite-size correlation length and the spin-glass susceptibility. Furthermore, we compute the critical temperature and compare its dependence on the disorder distribution width with recent analytical predictions [J. Stat. Mech. (2008) P04006].
The low-temperature behavior of the infinite-range Ising spin-glass model with a Gaussian distribution of interactions is studied by evaluating the statistical mechanics exactly for small systems (up to N=20N=20N=20 spins). Our results can be well understood by a simple picture of elementary cluster excitations from the ground state which overturn, on average, of order N12{N}^{\frac{1}{2}}N21​ spins. The conventional definition of the spin-glass order parameter is shown to be unsatisfactory and a new definition is proposed. It is speculated that ensemble averages may give different results from time averages for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ even in the infinite time limit.
The SYK model involves N Majorana fermions in 1+0 dimensions with quenched Gaussian dis- order and proves to be exactly solvable in the large N limit at strong coupling. It has been initially proposed by Sachdev and Ye as a model of condensed matter and later gained some interest as a toy model of AdS/CFT correspondence, thanks to the work of Kitaev. On the other side, random tensors are generalisations of random matrices to objets that carry more than two indices. It turns out that the SYK model and random tensors involve a special class of Feynman graphs known as "melons". We will briefly review both constructions. Then, we will show how non Gaussian disorder can be reduced to a Gaussian one, treating the coupling as a random tensor, thanks to Gurau’s Gaussian universality result.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.