Edwards-Anderson Order Parameter Research Articles

Griffiths-Type Theorems for Short-Range Spin Glass Models

We establish relations between different characterizations of order in spin glass models. We first prove that the broadening of the replica overlap distribution indicated by a nonzero standard deviation of the replica overlap R1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{1,2}$$\\end{document} implies the non-differentiability of the two-replica free energy with respect to the replica coupling parameter λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}. In Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_2$$\\end{document} invariant models such as the standard Edwards–Anderson model, the non-differentiability is equivalent to the spin glass order characterized by a nonzero Edwards–Anderson order parameter. This generalization of Griffiths’ theorem is proved for any short-range spin glass models with classical bounded spins. We also prove that the non-differentiability of the two-replica free energy mentioned above implies replica symmetry breaking in the literal sense, i.e., a spontaneous breakdown of the permutation symmetry in the model with three replicas. This is a general result that applies to a large class of random spin models, including long-range models such as the Sherrington-Kirkpatrick model and the random energy model. There is a 25-minute video that explains the main results of the present work:https://youtu.be/BF3hJiY1xvI

Nishimori's Cat: Stable Long-Range Entanglement from Finite-Depth Unitaries and Weak Measurements.

In the field of monitored quantum circuits, it has remained an open question whether finite-time protocols for preparing long-range entangled states lead to phases of matter that are stable to gate imperfections, that can convert projective into weak measurements. Here, we show that in certain cases, long-range entanglement persists in the presence of weak measurements, and gives rise to novel forms of quantum criticality. We demonstrate this explicitly for preparing the two-dimensional Greenberger-Horne-Zeilinger cat state and the three-dimensional toric code as minimal instances. In contrast to random monitored circuits, our circuit of gates and measurements is deterministic; the only randomness is in the measurement outcomes. We show how the randomness in these weak measurements allows us to track the solvable Nishimori line of the random-bond Ising model, rigorously establishing the stability of the glassy long-range entangled states in two and three spatial dimensions. Away from this exactly solvable construction, we use hybrid tensor network and MonteCarlo simulations to obtain a nonzero Edwards-Anderson order parameter as an indicator of long-range entanglement in the two-dimensional scenario. We argue that our protocol admits a natural implementation in existing quantum computing architectures, requiring only a depth-3 circuit on IBM's heavy-hexagon transmon chips.

Emergent Glassy Behavior in a Kagome Rydberg Atom Array.

We present large-scale quantum MonteCarlo simulation results on a realistic Hamiltonian of kagome-lattice Rydberg atom arrays. Although the system has no intrinsic disorder, intriguingly, our analyses of static and dynamic properties on large system sizes reveal emergent glassy behavior in a region of parameter space located between two valence bond solid phases. The extent of this glassy region is demarcated using the Edwards-Anderson order parameter, and its phase transitions to the two proximate valence bond solids-as well as the crossover towards a trivial paramagnetic phase-are identified. We demonstrate the intrinsically slow (imaginary) time dynamics deep inside the glassy phase and discuss experimental considerations for detecting such a quantum disordered phase with numerous nearly degenerate local minima. Our proposal paves a new route to the study of real-time glassy phenomena and highlights the potential for quantum simulation of a distinct phase of quantum matter beyond solids and liquids in current-generation Rydberg platforms.

Haake–Lewenstein–Wilkens approach to spin-glasses revisited

We revisit the Haake–Lewenstein–Wilkens approach to Edwards–Anderson (EA) model of Ising spin glass (SG) (Haake et al 1985 Phys. Rev. Lett. 55 2606). This approach consists in evaluation and analysis of the probability distribution of configurations of two replicas of the system, averaged over quenched disorder. This probability distribution generates squares of thermal copies of spin variables from the two copies of the systems, averaged over disorder, that is the terms that enter the standard definition of the original EA order parameter, . We use saddle point/steepest descent (SPSD) method to calculate the average of the Gaussian disorder in higher dimensions. This approximate result suggest that at in 3D and 4D. The case of 2D seems to be a little more subtle, since in the present approach energy increase for a domain wall competes with boundary/edge effects more strongly in 2D; still our approach predicts SG order at sufficiently low temperature. We speculate, how these predictions confirm/contradict widely spread opinions that: (i) There exist only one (up to the spin flip) ground state in EA model in 2D, 3D and 4D; (ii) there is (no) SG transition in 3D and 4D (2D). This paper is dedicated to the memories of Fritz Haake and Marek Cieplak.

Higgs phase as a spin glass and the transition between varieties of confinement

We propose that the Higgs phase of a gauge Higgs theory is the phase of spontaneously broken custodial symmetry, and present a new gauge invariant order parameter for custodial symmetry breaking which is very closely analogous to the Edwards-Anderson order parameter for spin glasses. Custodial symmetry is a global symmetry acting on the Higgs field alone, and we show here that the spin glass transition in gauge Higgs theories, from a QCD-like phase to a Higgs phase of broken custodial symmetry, coincides with the transition between two distinct types of confinement. These are color confinement in the Higgs phase, and a stronger version of confinement, which we have termed "separation-of-charge" confinement, in the QCD-like phase.

Monte Carlo Modeling of Magnetic Phase Transitions in Amorphous Alloys of the Re–Gd System

A Monte Carlo study of the magnetic properties of simulated amorphous alloys of the Re–Gd system is performed using the Heisenberg model. The temperature dependences of spontaneous magnetization, magnetic susceptibility, and the Edwards–Anderson order parameter are calculated. The dependence of the temperature of the transition to the spin-glass state on the concentration of gadolinium atoms is constructed. The transition to the spin-glass state in Re–Gd amorphous alloys occurs only above the percolation threshold in this system.

Accessing eigenstate spin-glass order from reduced density matrices

Many-body localized phases may not only be characterized by their ergodicity breaking, but can also host ordered phases such as the many-body localized spin-glass (MBL-SG). The MBL-SG is challenging to access in a dynamical measurement and therefore experimentally since the conventionally used Edwards-Anderson order parameter is a two-point correlation function in time. In this work, we show that many-body localized spin-glass order can also be detected from two-site reduced density matrices, which we use to construct an eigenstate spin-glass order parameter. We find that this eigenstate spin-glass order parameter captures spin-glass phases in random Ising chains both in many-body eigenstates as well as in the nonequilibrium dynamics from a local in time measurement. We discuss how our results can be used to observe MBL-SG order within current experiments in Rydberg atoms and trapped ion systems.

Spin glass transition in AuFe, CuMn, AuMn, AgMn and AuCr systems

Abstract The Monte Carlo method was applied to the study of some metallic magnetic alloys which display spin glass behavior. The energies of the alloys are modelled considering the RKKY interaction among the magnetic moments, and the range of the interaction is controlled by introducing a cutoff radius. In the case of very diluted spin glasses (SG), the spin glass transition is only clearly found when distant next neighbors are considered. In that case of very diluted alloys, the Edward-Anderson order parameter shows that first and second neighbor Heisenberg-like Hamiltonians would not present a SG phase transition. The occurrence of the spin-glass condition is due to the RKKY long-range interaction. The presented Monte Carlo results allows a semi-quantitative prediction of the spin glass freezing temperature.

Spin glass in the bond-diluted J1−J2 Ising model on the square lattice

We use Monte Carlo (MC) methods to simulate a two-dimensional (2D) bond-diluted Ising model on the square lattice which has frustration between the nearest-neighbor interaction J1 and the next-nearest-neighbor interaction J2. In this paper, we use the parallel tempering algorithm to study the thermodynamics for different diluted ratio x and give the phase diagram. The presence of both frustration and disorder results in a spin-glass phase, which exists between the stripe antiferromagnetic phase and the Neel phase. We present the ground-state energy of T -> 0 and the size-dependence of Edwards-Anderson (EA) order parameter for the spin glass phase. By scaling the mean energy and the EA order parameter from the simulated annealing with the Kibble-Zurek (KZ) mechanism, we obtain two different dynamic exponents z_E and z_q for the spin glass phase. Experimentally, this model has close implication with the FeAs plane of the iron-based superconductor BaFe2(As(1-x)P(x))2, where a spin-glass like phase was found.

On the TAP Free Energy in the Mixed p-Spin Models

In [Physical Magazine, 35(3):593-601, 1977], Thouless, Anderson, and Palmer derived a representation for the free energy of the Sherrington-Kirkpatrick model, called the TAP free energy, written as the difference of the energy and entropy on the extended configuration space of local magnetizations with an Onsager correction term. In the setting of mixed $p$-spin models with Ising spins, we prove that the free energy can indeed be written as the supremum of the TAP free energy over the space of local magnetizations whose Edwards-Anderson order parameter (self-overlap) is to the right of the support of the Parisi measure. Furthermore, for generic mixed $p$-spin models, we prove that the free energy is equal to the TAP free energy evaluated on the local magnetization of any pure state.

Dynamic scaling in the two-dimensional Ising spin glass with normal-distributed couplings

We carry out simulated annealing and employ a generalized Kibble-Zurek scaling hypothesis to study the two-dimensional Ising spin glass with normal-distributed couplings. The system has an equilibrium glass transition at temperature T=0. From a scaling analysis when T→0 at different annealing velocities v, we find power-law scaling in the system size for the velocity required in order to relax toward the ground state, v∼L^{-(z+1/ν)}, the Kibble-Zurek ansatz where z is the dynamic critical exponent and ν the previously known correlation-length exponent, ν≈3.6. We find z≈13.6 for both the Edwards-Anderson spin-glass order parameter and the excess energy. This is different from a previous study of the system with bimodal couplings [Rubin etal., Phys. Rev. E 95, 052133 (2017)2470-004510.1103/PhysRevE.95.052133] where the dynamics is faster (z is smaller) and the above two quantities relax with different dynamic exponents (with that of the energy being larger). We argue that the different behaviors arise as a consequence of the different low-energy landscapes: for normal-distributed couplings the ground state is unique (up to a spin reflection), while the system with bimodal couplings is massively degenerate. Our results reinforce the conclusion of anomalous entropy-driven relaxation behavior in the bimodal Ising glass. In the case of a continuous coupling distribution, our results presented here also indicate that, although Kibble-Zurek scaling holds, the perturbative behavior normally applying in the slow limit breaks down, likely due to quasidegenerate states, and the scaling function takes a different form.

Clustering of Nonergodic Eigenstates in Quantum Spin Glasses.

The two primary categories for eigenstate phases of matter at a finite temperature are many-body localization (MBL) and the eigenstate thermalization hypothesis (ETH). We show that, in the paradigmatic quantum p-spin models of the spin-glass theory, eigenstates violate the ETH yet are not MBL either. A mobility edge, which we locate using the forward-scattering approximation and replica techniques, separates the nonergodic phase at a small transverse field from an ergodic phase at a large transverse field. The nonergodic phase is also bounded from above in temperature, by a transition in configuration-space statistics reminiscent of the clustering transition in the spin-glass theory. We show that the nonergodic eigenstates are organized in clusters which exhibit distinct magnetization patterns, as characterized by an eigenstate variant of the Edwards-Anderson order parameter.

On the quantum spin glass transition on the Bethe lattice

We investigate the ground-state properties of a disorderd Ising model with uniform transverse field on the Bethe lattice, focusing on the quantum phase transition from a paramagnetic to a glassy phase that is induced by reducing the intensity of the transverse field. We use a combination of quantum Monte Carlo algorithms and exact diagonalization to compute Rényi entropies, quantum Fisher information, correlation functions and order parameter. We locate the transition by means of the peak of the Rényi entropy and we find agreement with the transition point estimated from the emergence of finite values of the Edwards–Anderson order parameter and from the peak of the correlation length. We interpret the results by means of a mean-field theory in which quantum fluctuations are treated as massive particles hopping on the interaction graph. We see that the particles are delocalized at the transition, a fact that points towards the existence of possibly another transition deep in the glassy phase where these particles localize, therefore leading to a many-body localized phase.

Measuring the Edwards-Anderson order parameter of the Bose glass: A quantum gas microscope approach

With the advent of spatially resolved fluorescence imaging in quantum gas microscopes, it is now possible to directly image glassy phases and probe the local effects of disorder in a highly controllable setup. Here we present numerical calculations using a spatially-resolved local mean-field theory, show that it captures the essential physics of the disordered system and use it to simulate the density distributions seen in single-shot fluorescence microscopy. From these simulated images we extract local properties of the phases which are measurable by a quantum gas microscope and show that unambiguous detection of the Bose glass is possible. In particular, we show that experimental determination of the Edwards-Anderson order parameter is possible in a strongly correlated quantum system using existing experiments. We also suggest modifications to the experiments which will allow further properties of the Bose glass to be measured.

Colloids exposed to random potential energy landscapes: From particle number density to particle-potential and particle-particle interactions.

Colloidal particles were exposed to a random potential energy landscape that has been created optically via a speckle pattern. The mean particle density as well as the potential roughness, i.e., the disorder strength, were varied. The local probability density of the particles as well as its main characteristics were determined. For the first time, the disorder-averaged pair density correlation function g((1))(r) and an analogue of the Edwards-Anderson order parameter g((2))(r), which quantifies the correlation of the mean local density among disorder realisations, were measured experimentally and shown to be consistent with replica liquid state theory results.

Magnetic properties of (La0.56Ce0.14)Sr0.30MnO3 perovskite

We investigate in this paper magnetic properties of the perovskite compound (La0.56Ce0.14)Sr0.30MnO3. The method we use here is Monte Carlo simulation, in which we take into account different kinds of interactions between nearest and between next-nearest magnetic ions Mn3+ (S=2), Mn4+ (S=3/2) and Ce3+ (S=1/2). Using a classical spin model, we have calculated the internal energy, the magnetization per ion type and their corresponding magnetic susceptibility, as well as the Edwards–Anderson order parameter for each ion kind. We also studied the applied-field effect on the system magnetization. Our results show a good agreement with experiments.

BaZr0.5Ti0.5O3: Lead-free relaxor ferroelectric or dipolar glass

Glassy freezing dynamics was investigated in ${\mathrm{BaZr}}_{0.5}{\mathrm{Ti}}_{0.5}{\mathrm{O}}_{3}$ (BZT50) ceramic samples by means of dielectric spectroscopy in the frequency range 0.001 Hz--1 MHz at temperatures $10lTl300$ K. From measurements of the quasistatic dielectric polarization in bias electric fields up to $\ensuremath{\sim}28$ kV/cm it has been found that a ferroelectric state cannot be induced, in contrast to the case of typical relaxors. This suggests that---at least for the above field amplitudes---BZT50 effectively behaves as a dipolar glass, which can be characterized by a negative value of the static third order nonlinear permittivity. The relaxation spectrum has been analyzed by means of the frequency-temperature plot, which shows that the longest relaxation time obeys the Vogel-Fulcher relation $\ensuremath{\tau}={\ensuremath{\tau}}_{0}\phantom{\rule{0.16em}{0ex}}\mathrm{exp}[{E}_{0}/(T\ensuremath{-}{T}_{0})]$ with the freezing temperature of 48.1 K, whereas the corresponding value for the shortest relaxation time is $\ensuremath{\sim}0$ K, implying an Arrhenius type behavior. By applying a standard expression for the static linear permittivity of dipolar glasses and/or relaxors the value of the Edwards-Anderson order parameter $q(T)$ has been evaluated. It is further shown that $q(T)$ can be described by the spherical random bond-random field model of relaxors.

Magnetic properties of perovskites La0.7Sr0.3[formula omitted][formula omitted]TixO3: Monte Carlo simulation versus experiments

This work presents a Monte Carlo study of the phase transition in the perovskites La0.7Sr0.3Mn0.73+Mn0.3-x4+ TixO3 (x=0.1, 0.2, and 0.25). We take into account nearest-neighbor (NN) interactions between magnetic ions Mn3+(S=2) and Mn4+(S=3/2) using a spin model describing a strong anisotropy on the z-axis. We have calculated the uniform and staggered magnetizations as well as the Edwards–Anderson order parameter as functions of temperature, with and without an applied magnetic field. Fitting the experimental Curie temperature at x=0, we estimated values of various exchange interactions in the system. The dominant one is that between Mn3+ and Mn4+ which is at the origin of the ferromagnetic ordering. Effects of the very small interaction J2 between NN Mn3+ is analyzed: we show that it can cause an antiferromagnetic phase above Tc which disappears at smaller J2 or at Mn3+ concentrations smaller than 0.55. Our results show a good agreement with experiments on the magnetizations for substitution concentration x=0.1, 0.2 and 0.3. We also studied the applied-field effect on the magnetization and our obtained results are compared with experiments performed at x=10%.

Joint min-max distribution and Edwards-Anderson's order parameter of the circular 1/f-noise model

We calculate the joint min-max distribution and the Edwards-Anderson's order parameter for the circular model of 1/f-noise. Both quantities, as well as generalisations, are obtained exactly by combining the freezing-duality conjecture and Jack-polynomial techniques. Numerical checks come with significantly improved control of finite-size effects in the glassy phase, and the results convincingly validate the freezing-duality conjecture. Application to diffusive dynamics is discussed. We also provide a formula for the pre-factor ratio of the joint/marginal Carpentier-Le Doussal tail for minimum/maximum which applies to any logarithmic random energy model.

Critical properties of a two-dimensional Ising magnet with quasiperiodic interactions.

We address the study of quasiperiodic interactions on a square lattice by using an Ising model with ferromagnetic and antiferromagnetic exchange interactions following a quasiperiodic Fibonacci sequence in both directions of a square lattice. We applied the Monte Carlo method, together with the Metropolis algorithm, to calculate the thermodynamic quantities of the system. We obtained the Edwards-Anderson order parameter q_{EA}, the magnetic susceptibility χ, and the specific heat c in order to characterize the universality class of the phase transition. We also use the finite size scaling method to obtain the critical temperature of the system and the critical exponents β,γ, and ν. In the low-temperature limit we obtained a spin-glass phase with critical temperature around T_{c}≈2.274, and the critical exponents β,γ, and ν, indicating that the quasiperiodic order induces a change in the universality class of the system. Also, we discovered a spin-glass ordering in a two-dimensional system which is rare and, as far as we know, the unique example is an under-frustrated Ising model.