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

Journal of Physics A Mathematical and General

Fluctuations of correlation functions in disordered spin systems

A. Crisanti, Stam Nicolis, G. Paladin, Angelo Vulpiani

Published Jul 7, 1990• 15 citations
Publisher

Abstract

The authors study the fluctuations of the two-point correlation function in one-dimensional disordered spin models. These survive even in the thermodynamic limit and, in order to reconstruct their probability distribution from the moments, they study a set of generalised correlation lengths zeta q . These moments may also be calculated within the transfer matrix formalism and provide insight on disorder-induced fluctuations. They show that the zeta q can be computed in Monte Carlo simulations. They discuss the crossover of the correlation decay rate at large distances to dominance by the most probable value given by zeta 0 , and the relation with the finite-volume fluctuations of the free energy. Finally they sketch how to extend their arguments to dimensions two and three.

DOI
10.1088/0305-4470/23/13/042

Related papers

2D Potts-Model Correlation Lengths: Numerical Evidence for ξ o = ξ d at β t

W Janke, S Kappler

We have studied spin-spin correlation functions in the ordered phase of the two-dimensional q -state Potts model with q = 10, 15, and 20 at the first-order transition point β t . Through extensive Monte Carlo simulations we obtain strong numerical evidence that the correlation length in the ordered phase agrees with the exactly known and recently numerically confirmed correlation length in the disordered phase: ξ o (β t ) = ξ d (β t ). As a by-product we find the energy moments in the ordered phase at β t in very good agreement with a recent large q -expansion.

arXiv:hep-lat/9509056•Europhysics Letters (EPL)•Sep 1, 1995•15 citations

Correlation functions, free energies, and magnetizations in the two-dimensional random-field Ising model

S. L. A. de Queiroz, Robin B. Stinchcombe

Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).

arXiv:cond-mat/0105326•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 28, 2001•5 citations

Quantum Correlation in One Dimensional Spin Model

Sarnava Datta

We investigate the behaviour of quantum correlations (entanglement and quantum discord) for homogeneous XY model at absolute zero and at finite temperature. Study of this correlation measures of quenched disordered quantum spin models like the XY spin glass and random-field XY models is also reviewed. We show that quenched averaged quantum correlations can showcase the order-from-disorder phenomenon for finite-size systems and also when the system is in the thermodynamic limit. Though we find some regions with positive enhancement score. This order from disorder phenomena is more prominent when the system is in the finite temparature with some definite values of external parameter. We also look into correlation length. Correlation functions of ground states typically decay exponentially with the distance for quantum spin systems. In such systems, entanglement does not survive when |i-j| ≳ 3. However, quantum discord survives long lattice distances as it is not monogamous for this system. We investigate the effects of quenched disorder on quantum correlation lengths of quenched averaged entanglement and quantum discord, in the anisotropic XY spin glass chains. We realize that there is practically neither diminution nor growth in entanglement length while quantum discord length increases significantly with the introduction of quenched disorder except near factorization point. Apart from that, we also investigate the scaling behaviour of quantum entanglement and quantum discord. We find that though entanglement follows finite size scaling close to quantum phase transition, scaling can't be done for quantum discord.

•Apr 1, 2017•0 citations

Classical statistical physics: One dimension

Jean Zinn‐Justin

This chapter, within the framework of classical statistical mechanics, discusses a family of models defined on one-dimensional lattices. It studies the simplest local examples: models that involve only interactions between nearest neighbours on the lattice. For such models, correlation functions can be calculated by a transfer matrix formalism. The chapter first describes some general properties of transfer matrices in one-dimensional models. This formalism is used to establish various properties of correlation functions, like the thermodynamic or infinite volume limit, the large-distance behaviour of the two-point correlation function, and introduces the very important concept of correlation length. Connected correlation functions, cumulants of the distribution, play a particularly important role. Indeed, these functions satisfy the cluster property, which characterizes their decay at large distance. The transfer matrix formalism is applied to the example of a Gaussian Boltzmann weight, which is studied in detail. The chapter calculates the partition function and correlation functions explicitly, and observes that , the correlation length diverges, making it possible to define a continuum limit. It shows that results of the continuum limit can be reproduced directly by solving a partial differential equation in which all traces of the initial lattice structure have disappeared. Finally, it exhibits a slightly more general class of models which share the same properties: divergent correlation length and continuum limit. Exercises are provided at the end of the chapter.

Oxford University Press eBooks•Jul 5, 2007•0 citations

Phase transitions in systems with correlated disorder

J.C. Lee, Richard L. Gibbs

For random systems with quenched disorders distributed randomly in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions and perfectly correlated in the remaining ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ dimensions, the renormalization-group theory has been performed in the past in two different ways. In the single-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as finite and expands the perturbation terms only for \ensuremath{\epsilon}=${\mathit{d}}_{\mathit{c}}$-d, where ${\mathit{d}}_{\mathit{c}}$ is the upper critical dimension. In the double-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as infinitesimal and performs an additional expansion for ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$. The former predicts smeared phase transitions, while the latter predicts sharp second-order phase transitions. We wish to find out which of the two predictions holds for a three-dimensional Ising model for which ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$=2 and ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$=1. We argue, using the replica method, that the spin correlation is isotropic for large distances as long as the fluctuation among the random bonds (in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions) is sufficiently smaller than the thermal energy ${\mathit{k}}_{\mathit{B}}$T. With parameters chosen to satisfy this condition in the model, a Monte Carlo simulation has been performed. The results favor a sharp second-order phase transition.

Physical review. B, Condensed matter•Feb 1, 1992•17 citations

Zeros in Partition Function and Critical Behavior of Disordered Three Dimensional Ising Model

Andrey N. Vakilov

We used a Monte Carlo simulation of the structurally disordered three dimensional Ising model.For the systems with spin concentrations p = 0.95, 0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling.Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5), ν(0.8) = 0.711(6), ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6).The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.

Journal of Siberian Federal University Mathematics & Physics•Mar 1, 2017•0 citations
The authors study the fluctuations of the two-point correlation function in one-dimensional disordered spin models. These survive even in the thermodynamic limit and, in order to reconstruct their probability distribution from the moments, they study a set of generalised correlation lengths zeta q . These moments may also be calculated within the transfer matrix formalism and provide insight on disorder-induced fluctuations. They show that the zeta q can be computed in Monte Carlo simulations. They discuss the crossover of the correlation decay rate at large distances to dominance by the most probable value given by zeta 0 , and the relation with the finite-volume fluctuations of the free energy. Finally they sketch how to extend their arguments to dimensions two and three.
We have studied spin-spin correlation functions in the ordered phase of the two-dimensional q -state Potts model with q = 10, 15, and 20 at the first-order transition point β t . Through extensive Monte Carlo simulations we obtain strong numerical evidence that the correlation length in the ordered phase agrees with the exactly known and recently numerically confirmed correlation length in the disordered phase: ξ o (β t ) = ξ d (β t ). As a by-product we find the energy moments in the ordered phase at β t in very good agreement with a recent large q -expansion.
Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).
We investigate the behaviour of quantum correlations (entanglement and quantum discord) for homogeneous XY model at absolute zero and at finite temperature. Study of this correlation measures of quenched disordered quantum spin models like the XY spin glass and random-field XY models is also reviewed. We show that quenched averaged quantum correlations can showcase the order-from-disorder phenomenon for finite-size systems and also when the system is in the thermodynamic limit. Though we find some regions with positive enhancement score. This order from disorder phenomena is more prominent when the system is in the finite temparature with some definite values of external parameter. We also look into correlation length. Correlation functions of ground states typically decay exponentially with the distance for quantum spin systems. In such systems, entanglement does not survive when |i-j| ≳ 3. However, quantum discord survives long lattice distances as it is not monogamous for this system. We investigate the effects of quenched disorder on quantum correlation lengths of quenched averaged entanglement and quantum discord, in the anisotropic XY spin glass chains. We realize that there is practically neither diminution nor growth in entanglement length while quantum discord length increases significantly with the introduction of quenched disorder except near factorization point. Apart from that, we also investigate the scaling behaviour of quantum entanglement and quantum discord. We find that though entanglement follows finite size scaling close to quantum phase transition, scaling can't be done for quantum discord.
This chapter, within the framework of classical statistical mechanics, discusses a family of models defined on one-dimensional lattices. It studies the simplest local examples: models that involve only interactions between nearest neighbours on the lattice. For such models, correlation functions can be calculated by a transfer matrix formalism. The chapter first describes some general properties of transfer matrices in one-dimensional models. This formalism is used to establish various properties of correlation functions, like the thermodynamic or infinite volume limit, the large-distance behaviour of the two-point correlation function, and introduces the very important concept of correlation length. Connected correlation functions, cumulants of the distribution, play a particularly important role. Indeed, these functions satisfy the cluster property, which characterizes their decay at large distance. The transfer matrix formalism is applied to the example of a Gaussian Boltzmann weight, which is studied in detail. The chapter calculates the partition function and correlation functions explicitly, and observes that , the correlation length diverges, making it possible to define a continuum limit. It shows that results of the continuum limit can be reproduced directly by solving a partial differential equation in which all traces of the initial lattice structure have disappeared. Finally, it exhibits a slightly more general class of models which share the same properties: divergent correlation length and continuum limit. Exercises are provided at the end of the chapter.
For random systems with quenched disorders distributed randomly in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions and perfectly correlated in the remaining ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ dimensions, the renormalization-group theory has been performed in the past in two different ways. In the single-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as finite and expands the perturbation terms only for \ensuremath{\epsilon}=dc{\mathit{d}}_{\mathit{c}}dc​-d, where dc{\mathit{d}}_{\mathit{c}}dc​ is the upper critical dimension. In the double-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as infinitesimal and performs an additional expansion for ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$. The former predicts smeared phase transitions, while the latter predicts sharp second-order phase transitions. We wish to find out which of the two predictions holds for a three-dimensional Ising model for which ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$=2 and ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$=1. We argue, using the replica method, that the spin correlation is isotropic for large distances as long as the fluctuation among the random bonds (in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions) is sufficiently smaller than the thermal energy kB{\mathit{k}}_{\mathit{B}}kB​T. With parameters chosen to satisfy this condition in the model, a Monte Carlo simulation has been performed. The results favor a sharp second-order phase transition.