The order parameter for spin glasses: a function on the interval 0-1

Many-body localization in the age of classical computing. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2025;88:026502. [PMID: 39591655 DOI: 10.1088/1361-6633/ad9756] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBLregimegive rise to an MBLphase, in which the thermalization does not occur even in theasymptoticlimit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.

Triadic interaction in the background of a pairwise spin-glass

Developing an equilibrium solution for a pairwise spin-glass with a quenched random infinite range shows a continuous phase transition. Models with p-spin interactions have been studied and the exact solution was provided that shows a continuous phase transition for p=2 and a first-order one for p>2. Although the p-spin interactions were studied individually without considering lower-order interactions, is it always feasible to ignore the lower ones? Here, we are interested in finding an analytical solution for considering a triadic interaction as a perturbation in the background of a pairwise interaction in the Sherrington-Kirkpatrick spin-glass model. Our results indicate a sudden phase transition as a consequence of considering triadic interactions that signal a switch from a continuous to an explosive phase transition.

Metastate analysis of the ground states of two-dimensional Ising spin glasses

Using an efficient polynomial-time ground-state algorithm we investigate the Ising spin glass state at zero temperature in two dimensions. For large sizes, we show that the spin state in a central region is independent of the interactions far away, indicating a "single-state" picture, presumably the droplet model. Surprisingly, a single power law describes corrections to this result down to the smallest sizes studied.

Quantum annealing: an overview

In this review, after providing the basic physical concept behind quantum annealing (or adiabatic quantum computation), we present an overview of some recent theoretical as well as experimental developments pointing to the issues which are still debated. With a brief discussion on the fundamental ideas of continuous and discontinuous quantum phase transitions, we discuss the Kibble-Zurek scaling of defect generation following a ramping of a quantum many body system across a quantum critical point. In the process, we discuss associated models, both pure and disordered, and shed light on implementations and some recent applications of the quantum annealing protocols. Furthermore, we discuss the effect of environmental coupling on quantum annealing. Some possible ways to speed up the annealing protocol in closed systems are elaborated upon: we especially focus on the recipes to avoid discontinuous quantum phase transitions occurring in some models where energy gaps vanish exponentially with the system size. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

EEG-based spatio-temporal relation signatures for the diagnosis of depression and schizophrenia

The diagnosis of psychiatric disorders is currently based on a clinical and psychiatric examination (intake). Ancillary tests are used minimally or only to exclude other disorders. Here, we demonstrate a novel mathematical approach based on the field of p-adic numbers and using electroencephalograms (EEGs) to identify and differentiate patients with schizophrenia and depression from healthy controls. This novel approach examines spatio-temporal relations of single EEG electrode signals and characterizes the topological structure of these relations in the individual patient. Our results indicate that the relational topological structures, characterized by either the personal universal dendrographic hologram (DH) signature (PUDHS) or personal block DH signature (PBDHS), form a unique range for each group of patients, with impressive correspondence to the clinical condition. This newly developed approach results in an individual patient signature calculated from the spatio-temporal relations of EEG electrodes signals and might help the clinician with a new objective tool for the diagnosis of a multitude of psychiatric disorders.

Nematic ordering in the Heisenberg spin-glass system in three dimensions

Nematic ordering, where the spins globally align along a spontaneously chosen axis irrespective of direction, occurs in spin-glass systems of classical Heisenberg spins in d=3. In this system where the nearest-neighbor interactions are quenched randomly ferromagnetic or antiferromagnetic, instead of the locally randomly ordered spin-glass phase, the system orders globally as a nematic phase. This nematic ordering of the Heisenberg spin-glass system is dramatically different from the spin-glass ordering of the Ising spin-glass system. The system is solved exactly on a hierarchical lattice and, equivalently, Migdal-Kadanoff approximately on a cubic lattice. The global phase diagram is calculated, exhibiting this nematic phase, and ferromagnetic, antiferromagnetic, disordered phases. The nematic phase of the classical Heisenberg spin-glass system is also found in other dimensions d>2: We calculate nematic transition temperatures in 24 different dimensions in 2<d≤4.

High Entropy van der Waals Materials

By breaking the restrictions on traditional alloying strategy, the high entropy concept has promoted the exploration of the central area of phase space, thus broadening the horizon of alloy exploitation. This review highlights the marriage of the high entropy concept and van der Waals systems to form a new family of materials category, namely the high entropy van der Waals materials (HEX, HE = high entropy, X = anion clusters) and describes the current issues and next challenges. The design strategy for HEX has integrated the local feature (e.g., composition, spin, and valence states) of structural units in high entropy materials and the holistic degrees of freedom (e.g., stacking, twisting, and intercalating species) in van der Waals materials, and is successfully used for the discovery of high entropy dichalcogenides, phosphorus tri-chalcogenides, halogens, and MXene. The rich combination and random distribution of the multiple metallic constituents on the nearly regular 2D lattice give rise to a flexible platform to study the correlation features behind a range of selected physical properties, e.g., superconductivity, magnetism, and metal-insulator transition. The deliberate design of structural units and their stacking configuration can also create novel catalysts to enhance their performance in a bunch of chemical reactions.

Force Correlations in Disordered Magnets

We present a proof of principle for the validity of the functional renormalization group, by measuring the force correlations in Barkhausen-noise experiments. Our samples are soft ferromagnets in two distinct universality classes, differing in the range of spin interactions, and the effects of eddy currents. We show that the force correlations have a universal form predicted by the functional renormalization group, distinct for short-range and long-range elasticity, and mostly independent of eddy currents. In all cases correlations grow linearly at small distances, as in mean-field models, but in contrast to the latter are bounded at large distances. As a consequence, avalanches are anti-correlated. We derive bounds for these anticorrelations, which are saturated in the experiments, showing that the multiple domain walls in our samples effectively behave as a single wall.

Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.

Treating random sequential addition via the replica method

While many physical processes are non-equilibrium in nature, the theory and modeling of such phenomena lag behind theoretical treatments of equilibrium systems. The diversity of powerful theoretical tools available to describe equilibrium systems has inspired strategies that map non-equilibrium systems onto equivalent equilibrium analogs so that interrogation with standard statistical mechanical approaches is possible. In this work, we revisit the mapping from the non-equilibrium random sequential addition process onto an equilibrium multi-component mixture via the replica method, allowing for theoretical predictions of non-equilibrium structural quantities. We validate the above approach by comparing the theoretical predictions to numerical simulations of random sequential addition.

Complexity as information in spin-glass Gibbs states and metastates: Upper bounds at nonzero temperature and long-range models

In classical finite-range spin systems, especially those with disorder such as spin glasses, a low-temperature Gibbs state may be a mixture of a number of pure or ordered states; the complexity of the Gibbs state has been defined in the past roughly as the logarithm of this number, assuming the question is meaningful in a finite system. As nontrivial pure-state structure is lost in finite size, in a recent paper [Phys. Rev. E 101, 042114 (2020)2470-004510.1103/PhysRevE.101.042114] Höller and the author introduced a definition of the complexity of an infinite-size Gibbs state as the mutual information between the pure state and the spin configuration in a finite region, and applied this also within a metastate construction. (A metastate is a probability distribution on Gibbs states.) They found an upper bound on the complexity for models of Ising spins in which each spin interacts with only a finite number of others, in terms of the surface area of the region, for all T≥0. In the present paper, the complexity of a metastate is defined likewise in terms of the mutual information between the Gibbs state and the spin configuration. Upper bounds are found for each of these complexities for general finite-range (i.e., short- or long-range, in a sense we define) mixed p-spin interactions of discrete or continuous spins (such as m-vector models), but only for T>0. For short-range models, the bound reduces to the surface area. For long-range interactions, the definition of a Gibbs state has to be modified, and for these models we also prove that the states obtained within the metastate constructions are Gibbs states under the modified definition. All results are valid for a large class of disorder distributions.

Mean-field model for the Curie-Weiss temperature dependence of coherence length in metallic liquids

The coherence length of the medium-range order (MRO) in metallic liquids is known to display a Curie-Weiss temperature dependence; its inverse is linearly related to temperature, and when extrapolated from temperatures above the glass transition, the coherence length diverges at a negative temperature with a critical exponent of unity. We propose a mean-field pseudospin model that explains this behavior. Specifically, we model the atoms and their local environment as Ising spins with antiferromagnetic exchange interactions. We further superimpose an exchange interaction between dynamical heterogeneities, or clusters of atoms undergoing cooperative motion. The coherence length in the metallic liquid is thus the correlation length between dynamical heterogeneities. Our results reaffirm the idea that the MRO coherence length is a measure of point-to-set correlations, and that local frustrations in the interatomic interactions are prominent in metallic liquids.

Unexpected Upper Critical Dimension for Spin Glass Models in a Field Predicted by the Loop Expansion around the Bethe Solution at Zero Temperature

The spin-glass transition in a field in finite dimension is analyzed directly at zero temperature using a perturbative loop expansion around the Bethe lattice solution. The loop expansion is generated by the M-layer construction whose first diagrams are evaluated numerically and analytically. The generalized Ginzburg criterion reveals that the upper critical dimension below which mean-field theory fails is D_{U}≥8, at variance with the classical result D_{U}=6 yielded by finite-temperature replica field theory. Our expansion around the Bethe lattice has two crucial differences with respect to the classical one. The finite connectivity z of the lattice is directly included from the beginning in the Bethe lattice, while in the classical computation the finite connectivity is obtained through an expansion in 1/z. Moreover, if one is interested in the zero temperature (T=0) transition, one can directly expand around the T=0 Bethe transition. The expansion directly at T=0 is not possible in the classical framework because the fully connected spin glass does not have a transition at T=0, being in the broken phase for any value of the external field.

Existence of Replica-Symmetry Breaking in Quantum Glasses

By controlling quantum fluctuations via the Falk-Bruch inequality we give the first rigorous argument for the existence of a spin-glass phase in the quantum Sherrington-Kirkpatrick model with a "transverse" magnetic field if the temperature and the field are sufficiently low. The argument also applies to the generalization of the model with multispin interactions, sometimes dubbed as the transverse p-spin model.

Nontrivial maturation metastate-average state in a one-dimensional long-range Ising spin glass: Above and below the upper critical range

Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations following a quench from infinite temperature to a temperature well below the spin-glass transition temperature T_{c} for a one-dimensional Ising spin-glass model with diluted long-range interactions. In this model, the probability P_{ij} that an edge {i,j} has nonvanishing interaction falls as a power law with chord distance, P_{ij}∝1/R_{ij}^{2σ}, and we study a range of values of σ with 1/2<σ<1. We consider a correlation function C_{4}(r,t). A dynamic correlation length that shows power-law growth with time ξ(t)∝t^{1/z} can be identified in the data and, for large time t, C_{4}(r,t) decays as a power law r^{-α_{d}} with distance r when r≪ξ(t). The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent α_{d} differentiates between a trivial MMAS (α_{d}=0), as expected in the droplet picture of spin glasses, and a nontrivial MMAS (α_{d}≠0), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero α_{d} even in the regime σ>2/3 which corresponds to short-range systems below six dimensions. For σ<2/3, the decay exponent α_{d} follows the RSB prediction for the decay exponent α_{s}=3-4σ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches α_{d}=1-σ in the regime 2/3<σ<1; however, it deviates from both lines in the vicinity of σ=2/3.

Equilibrium phase diagram and thermal responses of charged DNA-virus rod-suspensions at low ionic strengths

The collective behavior of DNA is important for exploring new types of bacteria in the means of detection, which is greatly interested in the understanding of interactions between DNAs in living systems. How they self-organize themselves is a physical common phenomenon for broad ranges of thermodynamic systems. In this work, the equilibrium phase diagrams of charged chiral rods (fd viruses) at low ionic strengths (below a few mM) are provided to demonstrate both replicas of (or self-organized) twist orders and replica symmetry breaking near high concentration glass-states. By varying the ionic strengths, it appears that a critical ionic strength is obtained below 1-2 mM salt, where the twist and freezing of nematic domains diverge. Also, the microscopic relaxation is revealed by the ionic strength-dependent effective Debye screening length. At a fixed low ionic strength, the local orientations of twist are shown by two different length scales of optical pitch, in the chiral-nematic N* phase and the helical domains [Formula: see text], for low and high concentration, respectively. RSB occurs in several cases of crossing phase boundary lines in the equilibrium phase diagram of DNA-rod concentration and ionic strength, including long-time kinetic arrests in the presence of twist orders. The different pathways of PATH I, II and III are due to many-body effects of randomized orientations for charged fd rods undergoing long-range electrostatic interactions in bulk elastic medium. In addition, the thermal stability are shown for chiral pitches of the N* phase and the abnormal cooling process of a specific heat in a structural glass. Here, the concentration-driven twist-effects of charged DNA rods are explored using various experimental methods involving image-time correlation, microscopic dynamics in small angle dynamic light scattering, optical activity in second harmonic generation, and differential scanning calorimetry for the glass state.

Machine learning in physics: the pitfalls of poisoned training sets

Known for their ability to identify hidden patterns in data, artificial neural networks are among the most powerful machine learning tools. Most notably, neural networks have played a central role in identifying states of matter and phase transitions across condensed matter physics. To date, most studies have focused on systems where different phases of matter and their phase transitions are known, and thus the performance of neural networks is well controlled. While neural networks present an exciting new tool to detect new phases of matter, here we demonstrate that when the training sets are poisoned (i.e. poor training data or mislabeled data) it is easy for neural networks to make misleading predictions.

Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

We present a numerical study of ground states of the dilute versions of the Sherrington-Kirkpatrick (SK) mean-field spin glass. In contrast to so-called "sparse" mean-field spin glasses that have been studied widely on random networks of finite (average or regular) degree, the networks studied here are randomly bond diluted to an overall density p, such that the average degree diverges as ∼pN with the system size N. Ground state energies are obtained with high accuracy for random instances over a wide range of fixed p. Since this is an NP-hard combinatorial problem, we employ the extremal optimization heuristic to that end. We find that the exponent describing the finite-size corrections ω varies continuously with p, a somewhat surprising result, as one would not expect that gradual bond dilution would change the T=0 universality class of a statistical model. For p→1, the familiar result of ω(p=1)≈2/3 for the SK model is obtained.

One-step replica-symmetry-breaking phase below the de Almeida-Thouless line in low-dimensional spin glasses

The de Almeida-Thouless (AT) line is the phase boundary in the temperature-magnetic field plane of an Ising spin glass at which a continuous (i.e., second-order) transition from a paramagnet to a replica-symmetry-breaking (RSB) phase occurs, according to mean-field theory. Here, using field-theoretic perturbative renormalization group methods on the Bray-Roberts reduced Landau-Ginzburg-type theory for a short-range Ising spin glass in space of dimension d, we show that at nonzero magnetic field the nature of the corresponding transition is modified as follows: (a) For d-6 small and positive, with increasing field on the AT line, first, the ordered phase just below the transition becomes the so-called one-step RSB, instead of the full RSB that occurs in mean-field theory; the transition on the AT line remains continuous with a diverging correlation length. Then at a higher field, a tricritical point separates the latter transition from a quasi-first-order one, that is one at which the correlation length does not diverge, and there is a jump in part of the order parameter, but no latent heat. The location of the tricritical point tends to zero as d→6^{+}. (b) For d≤6, we argue that the quasi-first-order transition could persist down to arbitrarily small nonzero fields, with a transition to full RSB still expected at lower temperature. Whenever the quasi-first-order transition occurs, it is at a higher temperature than the AT transition would be for the same field, preempting it as the temperature is lowered. These results may explain the reported absence of a diverging correlation length in the presence of a magnetic field in low-dimensional spin glasses in some simulations and in high-temperature series expansions. We also draw attention to the similarity of the "dynamically frozen" state, which occurs at temperatures just above the quasi-first-order transition, and the "metastate-average state" of the one-step RSB phase, and discuss the issue of the number of pure states in either.

Spin Glasses in a Field Show a Phase Transition Varying the Distance among Real Replicas (And How to Exploit It to Find the Critical Line in a Field)

We discuss a phase transition in spin glass models that have been rarely considered in the past, namely, the phase transition that may take place when two real replicas are forced to be at a larger distance (i.e., at a smaller overlap) than the typical one. In the first part of the work, by solving analytically the Sherrington-Kirkpatrick model in a field close to its critical point, we show that, even in a paramagnetic phase, the forcing of two real replicas to an overlap small enough leads the model to a phase transition where the symmetry between replicas is spontaneously broken. More importantly, this phase transition is related to the de Almeida-Thouless (dAT) critical line. In the second part of the work, we exploit the phase transition in the overlap between two real replicas to identify the critical line in a field in finite dimensional spin glasses. This is a notoriously difficult computational problem, because of considerable finite size corrections. We introduce a new method of analysis of Monte Carlo data for disordered systems, where the overlap between two real replicas is used as a conditioning variate. We apply this analysis to equilibrium measurements collected in the paramagnetic phase in a field, h > 0 and T c ( h ) < T < T c ( h = 0 ) , of the d = 1 spin glass model with long range interactions decaying fast enough to be outside the regime of validity of the mean field theory. We thus provide very reliable estimates for the thermodynamic critical temperature in a field.

Effective one-component model of binary mixture: molecular arrest induced by the spatially correlated stochastic dynamics

Spatially correlated noise (SCN), i.e. the thermal noise that affects neighbouring particles in a similar manner, is ubiquitous in soft matter systems. In this work, we apply the over-damped SCN-driven Langevin equations as an effective, one-component model of the dynamics in dense binary mixtures. We derive the thermodynamically consistent fluctuation-dissipation relation for SCN to show that it predicts the molecular arrest resembling the glass transition, i.e. the critical slow-down of dynamics in the disordered phases. We show that the mechanism of singular dissipation is embedded in the dissipation matrix, accompanying SCN. We are also able to identify the characteristic length of collective dissipation, which diverges at critical packing. This novel physical quantity conveniently describes the difference between the ergodic and non-ergodic dynamics. The model is fully analytically solvable, one-dimensional and admits arbitrary interactions between the particles. It qualitatively reproduces several different modes of arrested disorder encountered in binary mixtures, including e.g. the re-entrant arrest. The model can be effectively compared to the mode coupling theory.

Solving Statistical Mechanics Using Variational Autoregressive Networks

We propose a general framework for solving statistical mechanics of systems with finite size. The approach extends the celebrated variational mean-field approaches using autoregressive neural networks, which support direct sampling and exact calculation of normalized probability of configurations. It computes variational free energy, estimates physical quantities such as entropy, magnetizations and correlations, and generates uncorrelated samples all at once. Training of the network employs the policy gradient approach in reinforcement learning, which unbiasedly estimates the gradient of variational parameters. We apply our approach to several classic systems, including 2D Ising models, the Hopfield model, the Sherrington-Kirkpatrick model, and the inverse Ising model, for demonstrating its advantages over existing variational mean-field methods. Our approach sheds light on solving statistical physics problems using modern deep generative neural networks.

Physical foundations of biological complexity

Biological systems reach hierarchical complexity that has no counterpart outside the realm of biology. Undoubtedly, biological entities obey the fundamental physical laws. Can today's physics provide an explanatory framework for understanding the evolution of biological complexity? We argue that the physical foundation for understanding the origin and evolution of complexity can be gleaned at the interface between the theory of frustrated states resulting in pattern formation in glass-like media and the theory of self-organized criticality (SOC). On the one hand, SOC has been shown to emerge in spin-glass systems of high dimensionality. On the other hand, SOC is often viewed as the most appropriate physical description of evolutionary transitions in biology. We unify these two faces of SOC by showing that emergence of complex features in biological evolution typically, if not always, is triggered by frustration that is caused by competing interactions at different organizational levels. Such competing interactions lead to SOC, which represents the optimal conditions for the emergence of complexity. Competing interactions and frustrated states permeate biology at all organizational levels and are tightly linked to the ubiquitous competition for limiting resources. This perspective extends from the comparatively simple phenomena occurring in glasses to large-scale events of biological evolution, such as major evolutionary transitions. Frustration caused by competing interactions in multidimensional systems could be the general driving force behind the emergence of complexity, within and beyond the domain of biology.

Binomial Spin Glass

To establish a unified framework for studying both discrete and continuous coupling distributions, we introduce the binomial spin glass, a class of models where the couplings are sums of m identically distributed Bernoulli random variables. In the continuum limit m→∞, the class reduces to one with Gaussian couplings, while m=1 corresponds to the ±J spin glass. We demonstrate that for short-range Ising models on d-dimensional hypercubic lattices the ground-state entropy density for N spins is bounded from above by (sqrt[d/2m]+1/N)ln2, and further show that the actual entropies follow the scaling behavior implied by this bound. We thus uncover a fundamental noncommutativity of the thermodynamic and continuous coupling limits that leads to the presence or absence of degeneracies depending on the precise way the limits are taken. Exact calculations of defect energies reveal a crossover length scale L^{*}(m)∼L^{κ} below which the binomial spin glass is indistinguishable from the Gaussian system. Since κ=-1/(2θ), where θ is the spin-stiffness exponent, discrete couplings become irrelevant at large scales for systems with a finite-temperature spin-glass phase.

Analysis and optimization of population annealing

Population annealing is an easily parallelizable sequential Monte Carlo algorithm that is well suited for simulating the equilibrium properties of systems with rough free-energy landscapes. In this work we seek to understand and improve the performance of population annealing. We derive several useful relations between quantities that describe the performance of population annealing and use these relations to suggest methods to optimize the algorithm. These optimization methods were tested by performing large-scale simulations of the three-dimensional (3D) Edwards-Anderson (Ising) spin glass and measuring several observables. The optimization methods were found to substantially decrease the amount of computational work necessary as compared to previously used, unoptimized versions of population annealing. We also obtain more accurate values of several important observables for the 3D Edwards-Anderson model.

Triviality of the ground-state metastate in long-range Ising spin glasses in one dimension

We consider the one-dimensional model of a spin glass with independent Gaussian-distributed random interactions, which have mean zero and variance 1/|i-j|^{2σ}, between the spins at sites i and j for all i≠j. It is known that, for σ>1, there is no phase transition at any nonzero temperature in this model. We prove rigorously that, for σ>3/2, any translation-covariant Newman-Stein metastate for the ground states (i.e., the frequencies with which distinct ground states are observed in finite-size samples in the limit of infinite size, for given disorder) is trivial and unique. In other words, for given disorder and asymptotically at large sizes, the same ground state, or its global spin flip, is obtained (almost) always. The proof consists of two parts: One is a theorem (based on one by Newman and Stein for short-range two-dimensional models), valid for all σ>1, that establishes triviality under a convergence hypothesis on something similar to the energies of domain walls and the other (based on older results for the one-dimensional model) establishes that the hypothesis is true for σ>3/2. In addition, we derive heuristic scaling arguments and rigorous exponent inequalities which tend to support the validity of the hypothesis under broader conditions. The constructions of various metastates are extended to all values σ>1/2. Triviality of the metastate in bond-diluted power-law models for σ>1 is proved directly.

Stability of the quantum Sherrington-Kirkpatrick spin glass model

I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).

Numerical Construction of the Aizenman-Wehr Metastate

Chaotic size dependence makes it extremely difficult to take the thermodynamic limit in disordered systems. Instead, the metastate, which is a distribution over thermodynamic states, might have a smooth limit. So far, studies of the metastate have been mostly mathematical. We present a numerical construction of the metastate for the d=3 Ising spin glass. We work in equilibrium, below the critical temperature. Leveraging recent rigorous results, our numerical analysis gives evidence for a dispersed metastate, supported on many thermodynamic states.

de Almeida-Thouless instability in short-range Ising spin glasses

We use high-temperature series expansions to study the ±J Ising spin glass in a magnetic field in d-dimensional hypercubic lattices for d=5-8 and in the infinite-range Sherrington-Kirkpatrick (SK) model. The expansions are obtained in the variable w=tanh^{2}J/T for arbitrary values of u=tanh^{2}h/T complete to order w^{10}. We find that the scaling dimension Δ associated with the ordering-field h^{2} equals 2 in the SK model and for d≥6. However, in agreement with the work of Fisher and Sompolinsky [Phys. Rev. Lett. 54, 1063 (1985)PRLTAO0031-900710.1103/PhysRevLett.54.1063], there is a violation of scaling in a finite field, leading to an anomalous h-T dependence of the de Almeida-Thouless (AT) [J. Phys. A 11, 983 (1978)JPHAC50305-447010.1088/0305-4470/11/5/028] line in high dimensions, whereas scaling is restored as d→6. Within the convergence of our series analysis, we present evidence supporting an AT line in d≥6. In d=5, the exponents γ and Δ are substantially larger than mean-field values, but we do not see clear evidence for the AT line in d=5.

Numerical simulations of Ising spin glasses with free boundary conditions: The role of droplet excitations and domain walls

The relative importance of the contributions of droplet excitations and domain walls on the ordering of short-range Edwards-Anderson spin glasses in three and four dimensions is studied. We compare the spin overlap distribution functions of periodic and free boundary conditions using population annealing Monte Carlo. For system sizes up to about 1000 spins, spin glasses show nontrivial spin overlap distributions. Periodic boundary conditions may trap diffusive domain walls which can contribute to small spin overlaps, and the other contribution is the existence of low-energy droplet excitations within the system. We use free boundary conditions to minimize domain-wall effects, and show that low-energy droplet excitations are the major contribution to small overlaps in numerical simulations. Free boundary conditions has stronger finite-size effects, and is likely to have the same thermodynamic limit with periodic boundary conditions.

Instability of spin glass phase in divalent iron phosphate glass under a magnetic field

The spin glass behaviour of 50FeO · 50P₂O₅ (in mol%) glass has been examined under finite magnetic fields. The Sherrington-Kirkpatrick (SK) model, i.e. the mean field theory, is unsuitable for the interpretation of the frequency dependence of the ac magnetic susceptibility observed under an external field of 0.1 T; the critical exponent derived from the SK model is unphysically large. On the other hand, the droplet model explains well the frequency and field dependence of the spin-freezing temperature and the exponent of the thermally activated process is within the range defined by the droplet model. The results indicate that the spin glass phase of the 50FeO · 50P₂O₅ glass is unstable against magnetic fields.

Interface free-energy exponent in the one-dimensional Ising spin glass with long-range interactions in both the droplet and broken replica symmetry regions

The one-dimensional Ising spin-glass model with power-law long-range interactions is a useful proxy model for studying spin glasses in higher space dimensions and for finding the dimension at which the spin-glass state changes from having broken replica symmetry to that of droplet behavior. To this end we have calculated the exponent that describes the difference in free energy between periodic and antiperiodic boundary conditions. Numerical work is done to support some of the assumptions made in the calculations and to determine the behavior of the interface free-energy exponent of the power law of the interactions. Our numerical results for the interface free-energy exponent are badly affected by finite-size problems.

Quantum annealing for the number-partitioning problem using a tunable spin glass of ions

Exploiting quantum properties to outperform classical ways of information processing is an outstanding goal of modern physics. A promising route is quantum simulation, which aims at implementing relevant and computationally hard problems in controllable quantum systems. Here we demonstrate that in a trapped ion setup, with present day technology, it is possible to realize a spin model of the Mattis-type that exhibits spin glass phases. Our method produces the glassy behaviour without the need for any disorder potential, just by controlling the detuning of the spin-phonon coupling. Applying a transverse field, the system can be used to benchmark quantum annealing strategies which aim at reaching the ground state of the spin glass starting from the paramagnetic phase. In the vicinity of a phonon resonance, the problem maps onto number partitioning, and instances which are difficult to address classically can be implemented.

Spin models appear in several fields of physics and beyond, but solving many of them is a task for which no general efficient classical algorithm is known to exist. Here the authors demonstrate how a variety of spin glass models can be implemented and solved, via quantum simulation, in a system of trapped ions.

Finite-size critical scaling in Ising spin glasses in the mean-field regime

We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.

Maximum-Entropy Inference with a Programmable Annealer

Optimisation problems typically involve finding the ground state (i.e. the minimum energy configuration) of a cost function with respect to many variables. If the variables are corrupted by noise then this maximises the likelihood that the solution is correct. The maximum entropy solution on the other hand takes the form of a Boltzmann distribution over the ground and excited states of the cost function to correct for noise. Here we use a programmable annealer for the information decoding problem which we simulate as a random Ising model in a field. We show experimentally that finite temperature maximum entropy decoding can give slightly better bit-error-rates than the maximum likelihood approach, confirming that useful information can be extracted from the excited states of the annealer. Furthermore we introduce a bit-by-bit analytical method which is agnostic to the specific application and use it to show that the annealer samples from a highly Boltzmann-like distribution. Machines of this kind are therefore candidates for use in a variety of machine learning applications which exploit maximum entropy inference, including language processing and image recognition.

Universal dynamic scaling in three-dimensional Ising spin glasses

We use a nonequilibrium Monte Carlo simulation method and dynamical scaling to study the phase transition in three-dimensional Ising spin glasses. The transition point is repeatedly approached at finite velocity v (temperature change versus time) in Monte Carlo simulations starting at a high temperature. This approach has the advantage that the equilibrium limit does not have to be strictly reached for a scaling analysis to yield critical exponents. For the dynamic exponent we obtain z=5.85(9) for bimodal couplings distribution and z=6.00(10) for the Gaussian case. Assuming universal dynamic scaling, we combine the two results and obtain z=5.93±0.07 for generic 3D Ising spin glasses.

Spin glasses: redux: an updated experimental/materials survey

This article reviews the 40+ year old spin-glass field and one of its earliest model interpretations as a spin density wave. Our description is from an experimental phenomenological point of view with emphasis on new spin glass materials and their relation to topical problems and strongly correlated materials in condensed matter physics. We first simply define a spin glass (SG), give its basic ingredients and explain how the spin glasses enter into the statistical mechanics of classical phase transitions. We then consider the four basic experimental properties to solidly characterize canonical spin glass behavior and introduce the early theories and models. Here the spin density wave (SDW) concept is used to explain the difference between a short-range SDW, i.e. a SG and, in contrast, a long-range SDW, i.e. a conventional magnetic phase transition. We continue with the present state of SG, its massive computer simulations and recent proposals of chiral glasses and quantum SG. We then collect and mention the various SG 'spin-off's'. A major section uncovers the fashionable unconventional materials that display SG-like freezing and glassy ground states, such as (high temperature) superconductors, heavy fermions, intermetallics and Heuslers, pyrochlor and spinels, oxides and chalogenides and exotics, e.g. quasicrystals. Some conclusions and future directions complete the review.

Spin glass in a field: a new zero-temperature fixed point in finite dimensions

By using real-space renormalization group (RG) methods, we show that spin glasses in a field display a new kind of transition in high dimensions. The corresponding critical properties and the spin-glass phase are governed by two nonperturbative zero-temperature fixed points of the RG flow. We compute the critical exponents and discuss the RG flow and its relevance for three-dimensional systems. The new spin-glass phase we discovered has unusual properties, which are intermediate between the ones conjectured by droplet and full replica symmetry-breaking theories. These results provide a new perspective on the long-standing debate about the behavior of spin glasses in a field.

Glassy slowdown and replica-symmetry-breaking instantons

Glass-forming liquids exhibit a dramatic dynamical slowdown as the temperature is lowered. This can be attributed to relaxation proceeding via large structural rearrangements whose characteristic size increases as the system cools. These cooperative rearrangements are well modeled by instantons in a replica effective field theory, with the size of the dominant instanton encoding the liquid's cavity point-to-set correlation length. Varying the parameters of the effective theory corresponds to varying the statistics of the underlying free-energy landscape. We demonstrate that, for a wide range of parameters, replica-symmetry-breaking instantons dominate. The detailed structure of the dominant instanton provides a rich window into point-to-set correlations and glassy dynamics.

Short-range Ising spin glasses: the metastate interpretation of replica symmetry breaking

Parisi's formal replica-symmetry-breaking (RSB) scheme for mean-field spin glasses has long been interpreted in terms of many pure states organized ultrametrically. However, the early version of this interpretation, as applied to the short-range Edwards-Anderson model, runs into problems because as shown by Newman and Stein (NS) it does not allow for chaotic size dependence, and predicts non-self-averaging that cannot occur. NS proposed the concept of the metastate (a probability distribution over infinite-size Gibbs states in a given sample that captures the effects of chaotic size dependence) and a nonstandard interpretation of the RSB results in which the metastate is nontrivial and is responsible for what was called non-self-averaging. In this picture, each state drawn from the metastate has the ultrametric properties of the old theory, but when the state is averaged using the metastate, the resulting mixed state has little structure. This picture was constructed so as to agree both with the earlier RSB results and with rigorous results. Here we use the effective field theory of RSB, in conjunction with the rigorous definitions of pure states and the metastate in infinite-size systems, to show that the nonstandard picture follows directly from the RSB mean-field theory. In addition, the metastate-averaged state possesses power-law correlations throughout the low-temperature phase; the corresponding exponent ζ takes the value 4 according to the field theory in high dimensions d, and describes the effective fractal dimension of clusters of spins. Further, the logarithm of the number of pure states in the decomposition of the metastate-averaged state that can be distinguished if only correlations in a window of size W can be observed is of order W(d-ζ). These results extend the nonstandard picture quantitatively; we show that arguments against this scenario are inconclusive. More generally, in terms of Parisi's function q(x), if q(0)≠∫(0)(1)dxq(x), then the metastate is nontrivial. In an Appendix, we also prove rigorously that the metastate-averaged state of the Sherrington-Kirkpatrick model is a uniform distribution on all spin configurations at all temperatures.

Singular-potential random-matrix model arising in mean-field glassy systems

We consider an invariant random matrix ensemble where the standard Gaussian potential is distorted by an additional single pole of arbitrary fixed order. Potentials with first- and second-order poles have been considered previously and found applications in quantum chaos and number theory. Here we present an application to mean-field glassy systems. We derive and solve the loop equation in the planar limit for the corresponding class of potentials. We find that the resulting mean or macroscopic spectral density is generally supported on two disconnected intervals lying on the two sides of the repulsive pole, whose edge points can be completely determined imposing the additional constraint of traceless matrices on average. For an unbounded potential with an attractive pole, we also find a possible one-cut solution for certain values of the couplings, which is ruled out when the traceless condition is imposed. Motivated by the calculation of the distribution of the spin-glass susceptibility in the Sherrington-Kirkpatrick spin-glass model, we consider in detail a second-order pole for a zero-trace model and provide the most explicit solution in this case. In the limit of a vanishing pole, we recover the standard semicircle. Working in the planar limit, our results apply to matrices with orthogonal, unitary, and symplectic invariance. Numerical simulations and an independent analytical Coulomb fluid calculation for symmetric potentials provide an excellent confirmation of our results.

Correlated cluster mean-field theory for spin-glass systems

The competition between cluster spin glass (CSG) and ferromagnetism or antiferromagnetism is studied in this work. The model considers clusters of spins with short-range ferromagnetic or antiferromagnetic (FE-AF) interactions (J_{0}) and long-range disordered couplings (J) between clusters. The problem is treated by adapting the correlated cluster mean-field theory of D. Yamamoto [Phys. Rev. B 79, 144427 (2009)]. Phase diagrams T/J×J_{0}/J are obtained for different cluster sizes n_{s}. The results show that the CSG phase is found below the freezing temperature T_{f} for lower intensities of J_{0}/J. The increase of short-range FE interaction can favor the CSG phase, while the AF one reduces the CSG region by decreasing the T_{f}. However, there are always critical values of J_{0} where AF or FE orders become stable. The results also indicate a strong influence of the cluster size in the competition of magnetic phases. For AF cluster, the increase of n_{s} diminishes T_{f} reducing the CSG phase region, which indicates that the cluster surface spins can play an important role in the CSG arising.

Theories of glass formation and the glass transition

This key-issues review is a plea for a new focus on simpler and more realistic models of glass-forming fluids. It seems to me that we have too often been led astray by sophisticated mathematical models that beautifully capture some of the most intriguing features of glassy behavior, but are too unrealistic to provide bases for predictive theories. As illustrations of what I mean, the first part of this article is devoted to brief summaries of imaginative, sensible, but disparate and often contradictory ideas for solving glass problems. Almost all of these ideas remain alive today, with their own enthusiastic advocates. I then describe numerical simulations, mostly by H Tanaka and coworkers, in which it appears that very simple, polydisperse systems of hard disks and spheres develop long range, Ising-like, bond-orientational order as they approach glass transitions. Finally, I summarize my recent proposal that topologically ordered clusters of particles, in disordered environments, tend to become aligned with each other as if they were two-state systems, and thus produce the observed Ising-like behavior. Neither Tanaka's results nor my proposed interpretation of them fit comfortably within any of the currently popular glass theories.

Temperature chaos and quenched heterogeneities

We present a treatable generalization of the Sherrington-Kirkpatrick (SK) model which introduces correlations in the elements of the coupling matrix through multiplicative disorder on the single variables and investigate the consequences on the phase diagram. We define a generalized qEA parameter and test the structural stability of the SK results in this correlated case evaluating the de Almeida-Thouless line of the model. As a main result we demonstrate the increase of temperature chaos effects due to heterogeneities.

Interplay between spin-glass clusters and geometrical frustration

The presence of spin-glass (SG) order in highly geometrically frustrated systems is analyzed in a cluster SG model. The model considers infinite-range disordered interactions among cluster magnetic moments and the J(1)-J(2) model couplings between Ising spins of the same cluster. This model can introduce two sources of frustration: one coming from the disordered interactions and another coming from the J(1)-J(2) intracluster interactions (intrinsic frustration). The framework of one-step replica symmetry breaking is adopted to obtain a one-cluster problem that is exactly solved. As a main result we create phase diagrams of the temperature T versus intensity of the disorder J, where the paramagnetic-SG phase transition occurs at T(f) when T decreases for high-J values. For low-J values, the SG order is absent for antiferromagnetic clusters without intrinsic frustration. However, the SG order can be observed within the intracluster intrinsic frustration regime even for lower intensity of disorder. In particular, the results indicate that the presence of small clusters in geometrically frustrated antiferromagnetic systems can help stabilize the SG order within a weak disorder.

Inverse freezing in a cluster Ising spin-glass model with antiferromagnetic interactions

Inverse freezing is analyzed in a cluster spin-glass (SG) model that considers infinite-range disordered interactions between magnetic moments of different clusters (intercluster interaction) and short-range antiferromagnetic coupling J(1) between Ising spins of the same cluster (intracluster interaction). The intercluster disorder J is treated within a mean-field theory by using a framework of one-step replica symmetry breaking. The effective model obtained by this treatment is computed by means of an exact diagonalization method. With the results we build phase diagrams of temperature T/J versus J(1)/J for several sizes of clusters n(s) (number of spins in the cluster). The phase diagrams show a second-order transition from the paramagnetic phase to the SG order at the freezing temperature T(f) when J(1)/J is small. The increase in J(1)/J can then destroy the SG phase. It decreases T(f)/J and introduces a first-order transition. In addition, inverse freezing can arise at a certain range of J(1)/J and large enough n(s). Therefore, the nontrivial frustration generated by disorder and short-range antiferromagnetic coupling can introduce inverse freezing spontaneously.

Long-range quantum Ising spin glasses at t=0: gapless collective excitations and universality

We solve the Sherrington-Kirkpatrick model in a transverse field Γ deep in its quantum glass phase at zero temperature. We show that the glass phase is critical everywhere, exhibiting collective excitations with a gapless Ohmic spectral function. Using an effective potential approach, we interpret the latter as arising from disordered collective excitations behaving like weakly coupled, underdamped oscillators. For a small transverse field Γ, the low-frequency spectrum takes a form independent of the fluctuation strength Γ.

1/m expansion in spin glasses and the de Almeida-Thouless line

It is shown by means of a 1/m expansion about the large-m limit of the m-vector spin glass that the value of the intercept of the de Almeida-Thouless line on the zero temperature axis h(AT) is proportional to T(c)(d), the zero-field transition temperature of the large-m model in d dimensions. Since numerical studies indicate that T(c)(d)=0 for d≤6, it follows that there should be no Almeida-Thouless line for d≤6.

Inverse freezing in the Ghatak-Sherrington model with a random field

The present work studies the Ghatak-Sherrington (GS) model in the presence of a magnetic random field (RF). Previous results obtained from the GS model without a RF suggest that disorder and frustration are the key ingredients to produce spontaneous inverse freezing (IF). However, in this model, the effects of disorder and frustration always appear combined. In that sense, the introduction of RF allows us to study the IF under the effects of a disorder which is not a source of frustration. The problem is solved within the one step replica symmetry approximation. The results show that the first order transition between the spin glass and the paramagnetic phases, which is related to the IF for a certain range of crystal field D, is gradually suppressed when the RF is increased.