Theory Of Spin Glasses Research Articles

Storage properties of a quantum perceptron.

Driven by growing computational power and algorithmic developments, machine learning methods have become valuable tools for analyzing vast amounts of data. Simultaneously, the fast technological progress of quantum information processing suggests employing quantum hardware for machine learning purposes. Recent works discuss different architectures of quantum perceptrons, but the abilities of such quantum devices remain debated. Here, we investigate the storage capacity of a particular quantum perceptron architecture by using statistical mechanics techniques and connect our analysis to the theory of classical spin glasses. Specifically, we focus on one concrete quantum perceptron model and explore its storage properties in the limit of a large number of inputs.

Replica symmetry breaking in 1D Rayleigh scattering system: theory and validations

Spin glass theory, as a paradigm for describing disordered magnetic systems, constitutes a prominent subject of study within statistical physics. Replica symmetry breaking (RSB), as one of the pivotal concepts for the understanding of spin glass theory, means that under identical conditions, disordered systems can yield distinct states with nontrivial correlations. Random fiber laser (RFL) based on Rayleigh scattering (RS) is a complex disordered system, owing to the disorder and stochasticity of RS. In this work, for the first time, a precise theoretical model is elaborated for studying the photonic phase transition via the platform of RS-based RFL, in which we clearly reveal that, apart from the pump power, the photon phase variation in RFL is also an analogy to the temperature term in spin-glass phase transition, leading to a novel insight into the intrinsic mechanisms of photonic phase transition. In addition, based on this model and real-time high-fidelity detection spectral evolution, we theoretically predict and experimentally observe the mode-asymmetric characteristics of photonic phase transition in RS-based RFL. This finding contributes to a deeper understanding of the photonic RSB regime and the dynamics of RS-based RFL.

Spin glass theory and its new challenge: structured disorder

This paper first describes, from a high-level viewpoint, the main challenges that had to be solved in order to develop a theory of spin glasses in the last fifty years. It then explains how important inference problems, notably those occurring in machine learning, can be formulated as problems in statistical physics of disordered systems. However, the main questions that we face in the analysis of deep networks require to develop a new chapter of spin glass theory, which will address the challenge of structured data.

Local algorithms for maximum cut and minimum bisection on locally treelike regular graphs of large degree

AbstractGiven a graph of degree over vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth , we develop a local message passing algorithm whose complexity is , and that achieves near optimal cut values among all ‐local algorithms. Focusing on max‐cut, the algorithm constructs a cut of value , where is the value of the Parisi formula from spin glass theory, and (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, that is, graphs whose girth becomes after removing a small fraction of vertices. Earlier work established that, for random ‐regular graphs, the typical max‐cut value is . Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max‐cut, and nearly maximum min‐bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near‐Ramanujan property of random regular graphs.

Double-replica theory for evolution of genotype-phenotype interrelationship

The relationship between genotype and phenotype plays a crucial role in determining the function and robustness of biological systems. Here the evolution progresses through the change in genotype, whereas the selection is based on the phenotype, and the genotype-phenotype relation also evolves. The theory for such phenotypic evolution remains poorly developed, in contrast to evolution under the fitness landscape determined by genotypes. Here we provide a statistical-physics formulation of this problem by introducing replicas for genotype and phenotype. We apply it to an evolution model in which phenotypes are given by spin configurations; genotypes are an interaction matrix for spins to give the Hamiltonian, and the fitness depends only on the configuration of a subset of spins called the target. We describe the interplay between the genetic variations and phenotypic variances by noise in this model by our approach that extends the replica theory for spin glasses to include a spin replica for phenotypes and a coupling replica for genotypes. Within this framework we obtain a phase diagram of the evolved phenotypes against the noise and selection pressure, where each phase is distinguished by the fitness and overlaps for genotypes and phenotypes. Among the phases, a robust fitted phase, relevant to biological evolution, is achieved under the intermediate level of noise (temperature), where robustnesses to noise and to genetic mutation are correlated, as a result of replica symmetry. We also find a tradeoff between maintaining a high fitness level of phenotype and acquiring a robust pattern of genes as well as the dependence of this tradeoff on the ratio between the size of the functional (target) part to that of the remaining nonfunctional (non-target) one. The selection pressure needed to achieve high fitness increases with the fraction of target spins.

Tunable replica symmetry breaking in random laser

Abstract Replica symmetry breaking (RSB) has been widely recognized as a statistical analysis approach to understand the disorder and nonlinear interactions in complex systems ranging from atoms to the cosmic scale. However, it is challenging to analyze the nonlinear optical characteristics of random laser (RL) in disordered gain medium via RSB due to the lack of a general RSB-based statistical analysis framework. In this work, we report the tunable RSB in polymer fiber RL, where the effects of temperature and different structures on RSB are investigated experimentally and theoretically. It experimentally proves that RSB in RL is not robust, and disorder and temperature are responsible for tunable RSB in RL, which contributes to the improvement of the statistical analysis framework for investigating the optical principles of RL using RSB. And the finding of the tunable RSB allows to investigate the dynamical differences for various RL systems, which broadens the directions for the use of spin-glass theory to explore the physical mechanism of RL.

Transmission matrix inference via pseudolikelihood decimation

Recently, significant efforts in medical imaging are towards the exploitation of disordered media as optics tools. Among several approaches, the transmission matrix description is promising for characterizing complex structures and, currently, has enabled imaging and focusing through disorder. In the present work, we report a statistical mechanics description of the transmission problem. We convert a linear input–output transmission recovery into the statistical inference of an effective interaction matrix. We do this by relying on a pseudolikelihood maximization process based on random intensity observations. Our aim is to bridge results from spin-glass theory to the field of disordered photonics, uncovering insights from the scattering problem and encouraging the development of novel imaging techniques for better medical investigations.

Sparse high-dimensional linear regression. Estimating squared error and a phase transition

We consider a sparse high-dimensional regression model where the goal is to recover a k-sparse unknown binary vector β∗ from n noisy linear observations of the form Y=Xβ∗+W∈Rn where X∈Rn×p has i.i.d. N(0,1) entries and W∈Rn has i.i.d. N(0,σ2) entries. In the high signal-to-noise ratio regime and sublinear sparsity regime, while the order of the sample size needed to recover the unknown vector information-theoretically is known to be n∗:=2klogp/log(k/σ2+1), no polynomial-time algorithm is known to succeed unless n>nalg:=(2k+σ2)logp. In this work, we offer a series of results investigating multiple computational and statistical aspects of the recovery task in the regime n∈[n∗,nalg]. First, we establish a novel information-theoretic property of the MLE of the problem happening around n=n∗ samples, which we coin as an “all-or-nothing behavior”: when n>n∗ it recovers almost perfectly the support of β∗, while if n<n∗ it fails to recover any fraction of it correctly. Second, at an attempt to understand the computational hardness in the regime n∈[n∗,nalg], we prove that at order nalg samples there is an Overlap Gap Property (OGP) phase transition occurring at the landscape of the MLE: for constants c,C>0 when n<cnalg OGP appears in the landscape of MLE while if n>Cnalg OGP disappears. OGP is a geometric “disconnectivity” property, which initially appeared in the theory of spin glasses and is known to suggest algorithmic hardness when it occurs. Finally, using certain technical results obtained to establish the OGP phase transition, we additionally establish various novel positive and negative algorithmic results for the recovery task of interest, including the failure of LASSO with access to n<cnalg samples and the success of a simple local search method with access to n>Cnalg samples.

High-order correlations in species interactions lead to complex diversity-stability relationships for ecosystems.

How ecosystems maintain stability is an active area of research. Inspired by applications of random matrix theory in nuclear physics, May showed decades ago that in an ecosystem model with many randomly interacting species, increasing species diversity decreases the stability of the ecosystem. There have since been many additions to May's efforts, one being an improved understanding the effect of mutualistic, competitive, or predator-prey-like correlations between pairs of species. Here we extend a random matrix technique developed in the context of spin-glass theory to study the effect of high-order correlations among species interactions. The resulting analytically solvable models include next-to-nearest-neighbor correlations in the species interaction network, such as the enemy of my enemy is my friend, as well as higher-order correlations. We find qualitative differences from May and others' models, including nonmonotonic diversity-stability relationships. Furthermore, inclusion of particular next-to-nearest-neighbor correlations in predator-prey as opposed to mutualist-competitive networks causes the former to transition to being more stable at higher species diversity. We discuss potential applicability of our results to microbiota engineering and to the ecology of interpredator interactions, such as cub predation between lions and hyenas as well as companionship between humans and dogs.

Parisi’s Theory of Spin Glass

The spin-glass phase is characterized by the existence of many pure states due to random exchange interactions between spins. Parisi established the novel concept of replica symmetry breaking (RSB) from Sherrington Kirkpatrick’s mean-field theory via an abstract replica trick. In this article, his RSB scheme is reviewed from the view point of infinitely many pure states.

Параллельные вычисления модели Эдвардса–Андерсона

An algorithm for parallel exact calculation of the ground state of a two-dimensional Edwards–Anderson model with free boundary conditions is given. The running time of the algorithm grows exponentially as the side of the lattice square increases. If one side of the lattice is fixed, the running time grows polynomially with increasing size of the other side. The method may find application in the theory of spin glasses, in the field of quantum computing. Performance data for the bimodal distribution is given. The distribution of spin bonds can be either bimodal or Gaussian. The method makes it possible to compute systems up to a size of 40x40.

Network dismantling on factor graphs: break long loops and spare local structures

A new solution framework for the task of network dismantling is recently developed, based on a two-scale bipartite factor-graph representation of the original graph where local structures are abstracted as factor nodes. This technique leads to advancement of extant dismantling algorithms, among which the belief-propagation decimation (BPD) algorithm has an efficient counterpart (factor BPD, i.e., FBPD) on the factor graph, building upon a mean-field spin-glass theory developed for the underlying long-loop feedback vertex set (FVS) problem. In this paper, I (1) demonstrate the advantage as well as disadvantage of the new factor-graph approach, and investigate the varying choice of factors, (2) show that the method can be supported by an alternative microscopic picture, and the two distinct spin-glass theories derive equivalent outcomes, whose analytical results serve as lower bounds for the FVS size on random regular factor graphs, besides (3) an extra mathematical lower bound from the result on random regular (original) graphs. Performances of graph/factor-graph algorithms are compared on various real networks. It shows empirically and analytically that the factor-graph approach does not interfere with what we could achieve without applying this technique; the new approach does a good job where traditional algorithms may perform poorly.

Symmetry enriched phases of quantum circuits

Quantum circuits consisting of random unitary gates and subject to local measurements have been shown to undergo a phase transition, tuned by the rate of measurement, from a state with volume-law entanglement to an area-law state. From a broader perspective, these circuits generate a novel ensemble of quantum many-body states at their output. In this paper, we characterize this ensemble and classify the phases that can be established as steady states. Symmetry plays a nonstandard role in that the physical symmetry imposed on the circuit elements does not on its own dictate the possible phases. Instead, it is extended by dynamical symmetries associated with this ensemble to form an enlarged symmetry. Thus, we predict phases that have no equilibrium counterpart and could not have been supported by the physical circuit symmetry alone. We give the following examples. First, we classify the phases of a circuit operating on qubit chains with Z2 symmetry. One striking prediction, corroborated with numerical simulation, is the existence of distinct volume-law phases in one dimension, which nonetheless support true long-range order. We furthermore argue that owing to the enlarged symmetry, this system can in principle support a topological area-law phase, protected by the combination of the circuit symmetry and a dynamical permutation symmetry. Second, we consider a Gaussian fermionic circuit that only conserves fermion parity. Here the enlarged symmetry gives rise to a U(1) critical phase at moderate measurement rates and a Kosterlitz–Thouless transition to area-law phases. We comment on the interpretation of the different phases in terms of the capacity to encode quantum information. We discuss close analogies to the theory of spin glasses pioneered by Edwards and Anderson as well as crucial differences that stem from the quantum nature of the circuit ensemble.

Replica Symmetry Breaking in Cholesteric Liquid Crystal Bandgap Lasing

AbstractSpin glass theory is an interdisciplinary statistical theory that has been widely used to describe the complex physical systems in diverse fields. In photonic platforms, the characteristic phenomenon of the phase transition between paramagnetic and spin glass state, i.e., replica symmetry breaking (RSB), has been recently observed in random laser (RL) systems via analogies between optical modes and magnetic spins. Here, the RSB phenomenon in cholesteric liquid crystal (CLC) bandgap lasing is reported. The CLC lasing arises from the edge of bandgap when the reflection spectrum edge of CLC spectrally overlaps with the fluorescence spectrum of dye. Different from the RL systems, where the RSB phenomenon is frequently observed, bandgap laser shows weak disorder due to its self‐assembled helical structure. Unexpectedly, based on the statistical study of the lasing spectra, the RSB can still be found when the bandgap laser operates at the threshold point. This RSB phenomenon is robust when the bandgap laser works at different temperatures and wavelengths. These results extend the scope of the feasibility of the spin glass theory in the photonic field, which boost the understanding of the interplay between system disorder and lasing statistical behavior.

Statistical mechanics of the directed 2-distance minimal dominating set problem

The directed L-distance minimal dominating set (MDS) problem has wide practical applications in the fields of computer science and communication networks. Here, we study this problem from the perspective of purely theoretical interest. We only give results for an Erdós Rényi (ER) random graph and regular random (RR) graph, but this work can be extended to any type of network. We develop spin glass theory to study the directed 2-distance MDS problem. First, we find that the belief propagation (BP) algorithm does not converge when the inverse temperature β exceeds a threshold on either an ER random network or RR network. Second, the entropy density of replica symmetric theory has a transition point at a finite β on a regular random graph when the arc density exceeds 2 and on an ER random graph when the arc density exceeds 3.3; there is no entropy transition point (or ) in other circumstances. Third, the results of the replica symmetry (RS) theory are in agreement with those of BP algorithm while the results of the BP decimation algorithm are better than those of the greedy heuristic algorithm.

Two-Dimensional Coulomb Glass as a Model for Vortex Pinning in Superconducting Films

A glass model of vortex pinning in highly disordered thin superconducting films in magnetic fields $B \ll H_{c2}$ at low temperatures is proposed. Strong collective pinning of a vortex system realized in disordered superconductors that are close to the quantum phase transition to the insulating phase -- such as $\mathrm{In O}_x$, $\mathrm{Nb N}$, $\mathrm{Ti N}$, $\mathrm{Mo Ge}$, nano-granular aluminium, and others -- is considered theoretically for the first time. Utilizing the replica trick developed for the spin glass theory, we demonstrate that such vortex system is in non-ergodic state of glass type with large kinetic inductance per square $L_K$. Distribution function of local pinning energies is calculated, and it is shown that it possesses a wide gap, i.e. the probability to find a weakly pinned vortex is extremely low.

Phases of a matrix model with non-pairwise index contractions

Abstract Recently a matrix model with non-pairwise index contractions has been studied in the context of the canonical tensor model, a tensor model for quantum gravity in the canonical formalism. This matrix model also appears in the same form with different ranges of parameters and variables, when the replica trick is applied to the spherical $p$-spin model ($p=3$) in spin glass theory. Previous studies of this matrix model suggested the presence of a continuous phase transition around $R\sim N^2/2$, where $N$ and $R$ designate its matrix size $N\times R$. This relation between $N$ and $R$ intriguingly agrees with a consistency condition of the tensor model in the leading order of $N$, suggesting that the tensor model is located near or on the continuous phase transition point and therefore its continuum limit is automatically taken in the $N\rightarrow \infty$ limit. In the previous work, however, the evidence for the phase transition was not satisfactory due to the slowdown of the Monte Carlo simulations. In this work, we provide a new setup for Monte Carlo simulations by integrating out the radial direction of the matrix. This new strategy considerably improves the efficiency, and allows us to clearly show the existence of the phase transition. We also present various characteristics of the phases, such as dynamically generated dimensions of configurations, cascade symmetry breaking and a parameter zero limit, and discuss their implications for the canonical tensor model.

Boosting Monte Carlo simulations of spin glasses using autoregressive neural networks.

The autoregressive neural networks are emerging as a powerful computational tool to solve relevant problems in classical and quantum mechanics. One of their appealing functionalities is that, after they have learned a probability distribution from a dataset, they allow exact and efficient sampling of typical system configurations. Here we employ a neural autoregressive distribution estimator (NADE) to boost Markov chain Monte Carlo (MCMC) simulations of a paradigmatic classical model of spin-glass theory, namely, the two-dimensional Edwards-Anderson Hamiltonian. We show that a NADE can be trained to accurately mimic the Boltzmann distribution using unsupervised learning from system configurations generated using standard MCMC algorithms. The trained NADE is then employed as smart proposal distribution for the Metropolis-Hastings algorithm. This allows us to perform efficient MCMC simulations, which provide unbiased results even if the expectation value corresponding to the probability distribution learned by the NADE is not exact. Notably, we implement a sequential tempering procedure, whereby a NADE trained at a higher temperature is iteratively employed as proposal distribution in a MCMC simulation run at a slightly lower temperature. This allows one to efficiently simulate the spin-glass model even in the low-temperature regime, avoiding the divergent correlation times that plague MCMC simulations driven by local-update algorithms. Furthermore, we show that the NADE-driven simulations quickly sample ground-state configurations, paving the way to their future utilization to tackle binary optimization problems.

Spin pumping into a spin glass material

Spin pumping is a recently established means for generating a pure spin current, whereby spins are pumped from a magnet into the adjacent target material under the ferromagnetic resonance condition. We theoretically investigate the spin pumping from an insulating ferromagnet into spin glass materials. Combining a dynamic theory of spin glasses with the linear-response formulation of the spin pumping, we calculate temperature dependence of the spin pumping near the spin glass transition. The analysis predicts that a characteristic peak appears in the spin pumping signal, reflecting that the spin fluctuations slow down upon the onset of spin freezing.

The endpoint distribution of directed polymers

Probabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this paper, we introduce some new computational tools that do not require integrability. We begin by defining a new kind of abstract limit object, called “partitioned subprobability measure,” to describe the limits of endpoint distributions of directed polymers. Inspired by a recent work of Mukherjee and Varadhan on large deviations of the occupation measure of Brownian motion, we define a suitable topology on the space of partitioned subprobability measures and prove that this topology is compact. Then using a variant of the cavity method from the theory of spin glasses, we show that any limit law of a sequence of endpoint distributions must satisfy a fixed point equation on this abstract space, and that the limiting free energy of the model can be expressed as the solution of a variational problem over the set of fixed points. As a first application of the theory, we prove that in an environment with finite exponential moment, the endpoint distribution is asymptotically purely atomic if and only if the system is in the low temperature phase. The analogous result for a heavy-tailed environment was proved by Vargas in 2007. As a second application, we prove a subsequential version of the longstanding conjecture that in the low temperature phase, the endpoint distribution is asymptotically localized in a region of stochastically bounded diameter. All our results hold in arbitrary dimensions, and make no use of integrability.