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arXiv:2603.29984

Pattern Expansion of Spin Glasses

Mutian Shen, Zohar Nussinov, Yang‐Yu Liu

Published Mar 31, 2026• 0 citations
arXivHTMLPDF

Abstract

We introduce a systematic method for expanding general spin-glass Hamiltonians in terms of Mattis interactions, providing a novel perspective for understanding the fundamental differences between short-range Edwards-Anderson (EA) and mean-field Sherrington-Kirkpatrick (SK) spin glasses. By iteratively extracting patterns from the coupling matrix, we expand the original spin-glass system into a Hopfield-like model (a series of Mattis interactions) plus a residual system. Our analysis reveals profound distinctions between EA and SK models: while EA models in two and three dimensions break into isolated subconnected sections after expansion, the SK model exhibits remarkable self-similar behavior, with the residual system preserving the mean-field structure and Gaussian statistics throughout the expansion process. This self-similarity manifests in exponential decay of residual matrix norms and expansion coefficients, reflecting the inherent mean-field nature of the SK model. Furthermore, we demonstrate that pattern expansion can identify ultra-low energy excitations in EA models, revealing excitations with energies that decrease rapidly with expansion step. Through connected component analysis, we quantify the size-energy relationship of these independent excitation clusters, opening new avenues for understanding the low-energy landscape of spin glasses and providing insights into the nature of metastable states.

DOI
10.48550/arxiv.2603.29984

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arXiv:cond-mat/0002134•Physical Review Letters•Oct 2, 2000•173 citations
We introduce a systematic method for expanding general spin-glass Hamiltonians in terms of Mattis interactions, providing a novel perspective for understanding the fundamental differences between short-range Edwards-Anderson (EA) and mean-field Sherrington-Kirkpatrick (SK) spin glasses. By iteratively extracting patterns from the coupling matrix, we expand the original spin-glass system into a Hopfield-like model (a series of Mattis interactions) plus a residual system. Our analysis reveals profound distinctions between EA and SK models: while EA models in two and three dimensions break into isolated subconnected sections after expansion, the SK model exhibits remarkable self-similar behavior, with the residual system preserving the mean-field structure and Gaussian statistics throughout the expansion process. This self-similarity manifests in exponential decay of residual matrix norms and expansion coefficients, reflecting the inherent mean-field nature of the SK model. Furthermore, we demonstrate that pattern expansion can identify ultra-low energy excitations in EA models, revealing excitations with energies that decrease rapidly with expansion step. Through connected component analysis, we quantify the size-energy relationship of these independent excitation clusters, opening new avenues for understanding the low-energy landscape of spin glasses and providing insights into the nature of metastable states.
Spin glasses have been the subject of intense study and considerable controversy for decades, and the low-temperature phase of short-range spin glasses is still poorly understood. Our main goal is to improve our understanding in this area and find an answer to the following question: Are there only a single pair or a countable infinity of pure states in the low temperature phase of the EA spin glass? To that aim we first start by introducing spin glasses and provide a brief history of their research, then proceed to describe our method of simulation, the parallel tempering Monte Carlo algorithm. Next, we present the results of a large-scale numerical study of the equilibrium three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder. In order to understand how the parallel tempering algorithm works, we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the parallel tempering Monte Carlo method. We examine the correlation between static and dynamic observables for _ 5000 disorder realizations and up to 1000 spins down to temperatures at 20% of the critical temperature, and our results show that autocorrelation times are directly correlated with the roughness of the free-energy landscape. In the following chapters, the three- and four-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied again via large-scale Monte Carlo simulations at low temperatures, deep within the spinglass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. We arrive to the following conclusion: The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states. Finally we present results for several new observables, along with a few preliminary studies and suggestions for future research.
The three-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied via large-scale Monte Carlo simulations at low temperatures, deep within the spin-glass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states.
We have simulated Edwards-Anderson (EA) as well as Sherrington-Kirkpatrick systems of L3{L}^{3}L3 spins. After averaging over large sets of EA system samples of $3\ensuremath{\le}L\ensuremath{\le}10$, we obtain accurate numbers for distributions p(q)p(q)p(q) of the overlap parameter qqq at very low-temperature TTT. We find $p(0)/T\ensuremath{\rightarrow}0.233(4)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with the droplet scenario of spin glasses. We also study the number of mismatched links---between replica pairs---that come with large scale excitations. Contributions from small scale excitations are discarded. We thus obtain for the fractal dimension of outer surfaces of $q\ensuremath{\sim}0$ excitations in the EA model ${d}_{s}\ensuremath{\rightarrow}2.59(3)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with ${d}_{s}\ensuremath{\rightarrow}3$ as $T\ensuremath{\rightarrow}0$ that is predicted by mean-field theory for the macroscopic limit.
A recent interesting paper [Yucesoy et al. Phys. Rev. Lett. 109, 177204 (2012), arXiv:1206:0783] compares the low-temperature phase of the 3D Edwards-Anderson (EA) model to its mean-field counterpart, the Sherrington-Kirkpatrick (SK) model. The authors study the overlap distributions P_J(q) and conclude that the two models behave differently. Here we notice that a similar analysis using state-of-the-art, larger data sets for the EA model (generated with the Janus computer) leads to a very clear interpretation of the results of Yucesoy et al., showing that the EA model behaves as predicted by the replica symmetry breaking (RSB) theory.
An expansion for the free energy functional of quantum spin models is build around diagonalisable single-ion term of the system’s Hamiltonian. This expansion is obtained within the Matsubara time formalism by introducing recursive relations involving the spin correlation functions. Introduction Spin glass systems (SG) are collections of atoms or molecules with randomly distributed magnetic moments and frustrated interactions. They have gained much attentions of the scientific community since the seminal work by Edwards and Anderson [1]. Physical concepts and mathematical tools developed in the contest of SG are applicable not only in physics but also in biology, economy, informatics, e.g. in the study of optimization theory and neural networks. In quantum spin glass models with infinite-range interactions attention has been paid to description of the transition between disordered (paramagnetic) and ordered spin glass phases. However, even an approximate description of the phase change involves an expansion of the free energy functional in terms of the powers of the order parameters. In this work we present a method to built such an expansion in a systematic way. The general formulation requires that the Hamiltonian H of the system of interacting spins is composed of two parts: a single-site part H0 and an exchange, two-site part H1. The single-site term may represent an external magnetic field or single-ion anisotropies. The exchange two-site term we use is the quantum Sherrington-Kirkpatrick (SK) model [2]: ∑∑ ≠ − = α α α N j i j i ij S S J H 2 1 1 (1) where α i S are the spin operators Pauli matrices referred to the spin i of an ensemble of N spins z y x , , = α and . , , 1 N i = The random exchange interactions Please cite this article as: Zbigniew Domanski, An effective expansion for the free energy of quantum spin glass models with infinite-range interactions, Scientific Research of the Institute of Mathematics and Computer Science, 2009, Volume 8, Issue 2, pages 37-42. The website: http://www.amcm.pcz.pl/ Z. Domanski 38 { } ij J are governed by independent probability distributions taken to be Gaussian with zero mean and variance . / 1 N 1. Free energy In order to obtain the free energy averaged over the distribution of { } ij J we apply the replica trick and Matsubara ”time” representation [3]. The replica trick is used to overcome the difficulties of averaging a logarithm and it employs the identity [ ] [ ] ( ) n Z Z J n n J 1 lim ln 0 − = → (2) where [ ]J means an averaging over the random exchange interactions ij J . The Matsubara ”time” representation is based on the operator identity ( ) ( ) ( )( ) [ ] ( ) A t B A t t tA B d T t t 0 0 exp exp exp exp 0 ⋅ + − ⋅ − =     ∫ τ τ τ (3) ( ) ( ) ( ) tA B tA t B exp exp ⋅ ⋅ − = (4) Here the time ordering operator τ T rearranges the operator in the expansion of the exponential, in order of decreasing time arguments. This representation enables us to avoid the problem of the non-commutativity of the spin operators α S and to treat them as numbers. Applying (2)-(4) to the Hamiltonian (1) the replica free energy has the following form [4, 5] ( ) [ ] ( ) ∫ ∫ = Φ ⋅ = eff 0 Tr ln exp Tr ln H q H q e Dz T e Dz F τ (5) 3 2 1 Φ + Φ + Φ = Φ q z (6) ( ) ∫ ∑ = Φ 1 0 1 2 n n S d τ τ β (7) ( ) ( ) ∫ ∫ ∑ ′ ⋅ ′ − = Φ 1 0 1 0 , 2 m n m n S S d d τ τ τ τ β (8) ( ) ( ) ( ) ∫ ∫ ∑ ′ ′ ′ = Φ 1 0 1 0 3 , n n n S R S d d τ τ τ τ τ τ β (9) An effective expansion for the free energy of quantum spin glass models with infinite-range interactions 39 where T / 1 = β is the inverse temperature and the integration in Eq. (5) is over the field: ( ) ( )    − = 2 / / exp / 2 1 2 q z q z d Dzq π (10) In Equations (5)-(10), the quantity q is the Edwards-Anderson spin glass order parameter and ( ) τ τ ′ , R is the dynamic spin self-interaction. ( ) τ τ ′ , R describes the quantum fluctuations. The spin glass order parameter vanishes in the paramagnetic (disordered) phase whereas the dynamic spin self-interaction persists in both phases. In order to locate the phase change one needs to expand the free energy functional as a series in power of q. The phase transition takes place when the coefficient of 2 q becomes zero as a function of the temperature T and thus, one of the most useful information can be retrieve from an expansion of the functional F, Eqs. (5-9). 2. An expansion Consider the free energy F given by Equations (5)-(10). Using these equations we can compute the thermal average with respect to the effective Hamiltonian eff H defined in the exponent of Eq. (5). We denote this thermal average by the angular bracket, namely: ( ) ( ) ( ) ( ) ( ) ( ) Φ Φ τ τ τ τ τ τ τ exp Tr exp Tr 0 1 0 1 1 eff 1 T e S S T e S S T H p n n H p n n p p ⋅ ⋅ ⋅ = (11) From Eq. (10) we see that ( ) ( ) q q Dz z Dz q ν ν ν ν ν 2 2 2 ∂ ∂ = ∂ ∂ − (12) Using Eqs. (11) and (12) we can compute the first derivative of F with respect to the order parameter q. First, we do this by direct computation. Direct computation involves the straightforward evaluation of multiple thermal averages of Eq. (11). ( ) ( ) [ ] ( ) [ ] ∫ ∫ + + ⋅ ∂ + ⋅ ∂ = ∂ 3 2 1 exp Tr ln exp Tr ln
The nature of the spin glass state is investigated by studying changes to the ground state when a weak perturbation is applied to the bulk of the system. We consider short range models in three and four dimensions and the infinite range Sherrington-Kirkpatrick (SK) and Viana-Bray models. Our results for the SK and Viana-Bray models agree with the replica symmetry breaking picture. The data for the short range models fit naturally a picture in which there are large scale excitations which cost a finite energy but whose surface has a fractal dimension, d(s), less than the space dimension d. We also discuss the possible crossover to other behavior at larger length scales than the sizes studied.