Classical Spin Glass 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.

Exact Numerical Solution of the Fully Connected Classical and Quantum Heisenberg Spin Glass.

We present the mean field solution of the quantum and classical Heisenberg spin glasses, using the combination of a high precision numerical solution of the Parisi full replica symmetry breaking equations and a continuous time quantum MonteCarlo algorithm. We characterize the spin glass order and its low-energy excitations down to zero temperature. The Heisenberg spin glass has a rougher energy landscape than its Ising analog, and exhibits a very slow temperature evolution of its dynamical properties. We extend our analysis to the doped, metallic Heisenberg spin glass, which displays unexpectedly slow spin dynamics, reflecting the proximity to the melting quantum critical point and its associated Sachdev-Ye-Kitaev Planckian dynamics.

Magnetic boson peak in classical spin glasses

We revisited spin dynamics in archetypical classical spin-glass (SG) systems, such as Cu1−xMnx (x=0.017, 0.034, and 0.067) dilute alloys and iron aluminosilicate glass, using a modern neutron scattering spectrometer with high neutron flux. The former is crystalline, and the latter is amorphous, where their SG state is well separated from magnetically ordered phases. Bose-scaled localized magnetic excitations were observed in both compounds below the spin-freezing temperature (Tf). The spectrum exhibits a maximum at low energy and a broad tail on the high-energy side. The excitation energy tends to be higher for the material with higher Tf. Above Tf, the spectrum considerably changes with temperature, thereby indicating the emergence of the magnetic relaxation process. The magnetic excitation in the SG state has much in common with the boson peak in structural glasses. We consider that the Bose-scaled broad excitation peak is an elementary excitation inherent in disordered systems. Published by the American Physical Society 2024

Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

Ising spin Hamiltonians are often used to encode a computational problem in their ground states. Quantum Annealing (QA) computing searches for such a state by implementing a slow time-dependent evolution from an easy-to-prepare initial state to a low energy state of a target Ising Hamiltonian of quantum spins, HI. Here, we point to the existence of an analytical solution for such a problem for an arbitrary HI beyond the adiabatic limit for QA. This solution provides insights into the accuracy of nonadiabatic computations. Our QA protocol in the pseudo-adiabatic regime leads to a monotonic power-law suppression of nonadiabatic excitations with time T of QA, without any signature of a transition to a glass phase, which is usually characterized by a logarithmic energy relaxation. This behavior suggests that the energy relaxation can differ in classical and quantum spin glasses strongly, when it is assisted by external time-dependent fields. In specific cases of HI, the solution also shows a considerable quantum speedup in computations.

Dynamical Approach to the TAP Equations for the Sherrington–Kirkpatrick Model

We present a new dynamical proof of the Thouless-Anderson-Palmer (TAP) equations for the classical Sherrington-Kirkpatrick spin glass at sufficiently high temperature. In our derivation, the TAP equations are a simple consequence of the decay of the two point correlation functions. The methods can also be used to establish the decay of higher order correlation functions. We illustrate this by proving a suitable decay bound on the three point functions from which we derive an analogue of the TAP equations for the two point functions.

Machine learning outperforms thermodynamics in measuring how well a many-body system learns a drive

Diverse many-body systems, from soap bubbles to suspensions to polymers, learn and remember patterns in the drives that push them far from equilibrium. This learning may be leveraged for computation, memory, and engineering. Until now, many-body learning has been detected with thermodynamic properties, such as work absorption and strain. We progress beyond these macroscopic properties first defined for equilibrium contexts: We quantify statistical mechanical learning using representation learning, a machine-learning model in which information squeezes through a bottleneck. By calculating properties of the bottleneck, we measure four facets of many-body systems’ learning: classification ability, memory capacity, discrimination ability, and novelty detection. Numerical simulations of a classical spin glass illustrate our technique. This toolkit exposes self-organization that eludes detection by thermodynamic measures: Our toolkit more reliably and more precisely detects and quantifies learning by matter while providing a unifying framework for many-body learning.

Glassy Dynamics in a Disordered Heisenberg Quantum Spin System

Understanding the dynamics of strongly interacting disordered quantum systems is one of the most challenging problems in modern science, due to features such as the breakdown of thermalization and the emergence of glassy phases of matter. We report on the observation of anomalous relaxation dynamics in an isolated XXZ quantum spin system realized by an ultracold gas of atoms initially prepared in a superposition of two-different Rydberg states. The total magnetization is found to exhibit sub-exponential relaxation analogous to classical glassy dynamics, but in the quantum case this relaxation originates from the build-up of non-classical correlations. In both experiment and semi-classical simulations, we find the evolution towards a randomized state is independent of the strength of disorder up to a critical value. This hints towards a unifying description of relaxation dynamics in disordered isolated quantum systems, analogous to the generalization of statistical mechanics to out-of-equilibrium scenarios in classical spin glasses.

Unfolding of phases and multicritical points in the classical anisotropic van Hemmen spin glass model with random field

We study magnetic properties of the 3-state spin ($S_{i}=0$ and $\pm 1$) spin glass (SG) van Hemmen model with ferromagnetic interaction $J_0$ under a random field (RF). The RF follows a bimodal distribution The combined effect of the crystal field $D$ and the special type of on-site random interaction of the van Hemmen model engenders the unfolding of the SG phases for strong enough RF, i. e., instead of one SG phase, we found two SG phases. Moreover, as $J_0$ is finite, there is also the unfolding of the mixed phase (with the SG order parameter and the spontaneous magnetization simultaneously finite) in four distinct phases. The emergence of these new phases separated by first and second order line transitions produces a multiplication of triple and multicritical points.

Magnetoelastic coupling at spin-glass-like transition in Sr3NiSb2O9

We report a characteristic spin-glass-like behavior in Sr3NiSb2O9, as confirmed from the low-field dc magnetometry and ac susceptibility measurements. The memory effects in the dc magnetometry exhibit similar characteristic features of the spin-glass systems. The ac susceptibility measurements provide that the frequency dependent peak-shift follows the conventional Vogel-Fulcher and dynamical scaling laws with the characteristic relaxation time to be ∼ 10−14 s. This relaxation time fits in the recommended range of ∼10−12−10−14 s for the classical spin-glasses. Low temperature synchrotron diffraction studies point to the strong magnetoelastic coupling close to the spin-glass-like transition. The transition is found to be associated with the step-like anomalous lattice contraction. Possible origin of magnetic frustration leading to the spin-glass ground state is discussed by correlating the microstructural results, as obtained from the analysis of synchrotron diffraction studies.

Quantum inverse freezing and mirror-glass order

It is well-known that spontaneous symmetry breaking in one spatial dimension is thermodynamically forbidden at finite energy density. Here we show that mirror-symmetric disorder in an interacting quantum system can invert this paradigm, yielding spontaneous breaking of mirror symmetry only at finite energy density and giving rise to "mirror-glass" order. The mirror-glass transition, which is driven by a finite density of interacting excitations, is enabled by many-body localization, and appears to occur simultaneously with the localization transition. This counterintuitive manifestation of localization-protected order can be viewed as a quantum analog of inverse freezing, a phenomenon that occurs, e.g., in certain models of classical spin glasses.

Computation of disordered system from the first principles of classical mechanics and ℕℙ hard problem

We study the classical 1D Heisenberg spin glasses in the framework of nearest-neighboring model. Based on the Hamilton equations we obtained the system of recurrence equations which allows to perform node-by-node calculations of a spin-chain. It is shown that calculations from the first principles of classical mechanics lead to ℕℙ hard problem, that however in the limit of the statistical equilibrium can be calculated by ℙ algorithm. For the partition function of the ensemble a new representation is offered in the form of one-dimensional integral of spin-chains’ energy distribution.

Error correction for encoded quantum annealing

Recently, Lechner, Hauke and Zoller [Science Advances, 1(9)e1500838, (2015)] have proposed a quantum annealing architecture, in which a classical spin glass with all-to-all connectivity is simulated by a spin glass with geometrically local interactions. We interpret this architecture as a classical error-correcting code, which is highly robust against weakly correlated bit-flip noise, and we analyze the code's performance using a belief-propagation decoding algorithm. Our observations may also apply to more general encoding schemes and noise models.

Many-body localization transition in Rokhsar-Kivelson-type wave functions

We construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the Random Energy Model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with subsystem size: scaling with entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a sub-extensive scaling between these limits. Near the phase transition point, the fluctuations of the R\'enyi entropies are non-Gaussian. We find that R\'enyi entropies with different R\'enyi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior.

New algorithm for simulation of 3D classical spin glasses under the influence of external electromagnetic fields

We study statistical properties of 3D classical spin-glass under the influence of external fields. It is proved that in the framework of the nearest-neighboring model 3D spin-glass problem at performing of Birkhoff’s ergodic hypothesis regarding to orientations of spins in the 3D space can be reduced to the problem of disordered 1D spatial spin-chains (SSC) ensemble where each spin-chain interacts with a random environment. The 1D SSC is defined as a periodic 1D lattice, where spins in nodes are randomly oriented in 3D space, in addition they all interact with each other randomly. For minimization of the Hamiltonian in an arbitrary node of the 1D lattice lattice, we obtained recurrent equations and corresponding Sylvester criterion, which allow to find energy local minimum. On the bases of these equations the high-performance parallel algorithm is developed which allows to calculate all statistical parameters of 3D spin glass, including distribution of a constant of spin-spin interaction, from the first principles of the classical mechanics.

Quantum annealing search of Ising spin glass ground state(s) with tunable transverse and longitudinal fields

Here we first discuss briefly the quantum annealing technique. We then study the quantum annealing of Sherrington-Kirkpatrick spin glass model with the tuning of both transverse and longitudinal fields. Both the fields are time-dependent and vanish adiabatically at the same time, starting from high values. We solve, for rather small systems, the time-dependent Schrodinger equation of the total Hamiltonian by employing a numerical technique. At the end of annealing we obtain the final state having high overlap with the exact ground state(s) of classical spin glass system (obtained independently).

Structural Disorder and Classical Spin-Glass Behaviour in Ba3Fe2SbO9

A new 6H-type perovskite Ba3Fe2SbO9 has been synthesised for the first time. Synchrotron and neutron powder diffraction data reveal complete structural disorder between Sb and Fe in the octahedral perovskite B sites. This results in classical spin-glass behaviour, which we characterise using magnetic susceptibility, magnetisation, and heat capacity measurements, although some evidence is seen for a transition to a partially ordered spin-glass like state below 24 K. The behaviour of Ba3Fe2SbO9 is compared with that of the 6H-type perovskite Ba3Fe2WO9, which displays antiferromagnetic character below TN = 290 K before entering a glassy state below Tf = 60 K. Differences between the magnetism in these two phases are discussed in terms of the complete structural disorder between the Fe and Sb ions in the former case, versus partial disorder (limited to the distribution and local orientation of Fe–W and Fe–Fe dimer units) in the latter.

Classical spin glass system in external field with taking into account relaxation effects

We study statistical properties of disordered spin systems under the influence of an external field with taking into account relaxation effects. For description of system the spatial 1D Heisenberg spin-glass Hamiltonian is used. In addition, we suppose that interactions occur between nearest-neighboring spins and they are random. Exact solutions which define angular configuration of the spin in nodes were obtained from the equations of stationary points of Hamiltonian and the corresponding conditions for the energy local minimum. On the basis of these recurrent solutions an effective parallel algorithm is developed for simulation of stabile spin-chains of an arbitrary length. It is shown that by way of an independent order of N2 numerical simulations (where N is number of spin in each chain) it is possible to generate ensemble of spin-chains, which is completely ergodic which is equivalent to full self-averaging of spin-chains’ vector polarization. Distributions of different parameters (energy, average polarization by coordinates, and spin-spin interaction constant) of unperturbed system are calculated. In particular, analytically is proved and numerically is shown, that for the Heisenberg nearest-neighboring Hamiltonian model, the distribution of spin-spin interaction constants as opposed to widely used Gauss-Edwards-Anderson distribution satisfies Levy alpha-stable distribution law. This distribution is nonanalytic function and does not have variance. In the work we have in detail studied critical properties of an ensemble depending on value of external field parameters (from amplitude and frequency) and have shown that even at weak external fields the spin-glass systemis strongly frustrated. It is shown that frustrations have fractal behavior, they are selfsimilar and do not disappear at scale decreasing of area. By the numerical computation is shown that the average polarization of spin-glass on a different coordinates can have values which can lead to catastrophes in the equation ofClausius-Mossotti for dielectric constant. In other words, for some values of external field parameter, a critical phenomenon occurs in the system which is impossible to describe by the real-valued Heisenberg spin-glass Hamiltonian. For the solution of this problem at first the complex-valued disordered Hamiltonian is used. Physically this type of extension of Hamiltonian allows to consider relaxation effects which occur in the system under the influence of an external field. On the basis of developed approach an effective parallel algorithm is developed for simulation of statistic parameters of spin-glass system under the influence of an external field.

Classical Spin Glass System in External Field with Taking into Account Relaxation Effects

Classical Spin Glass System in External Field with Taking into Account Relaxation Effects

Spin glass-like phase below ∼210 K in magnetoelectric gallium ferrite

In this letter we show the presence of a spin-glass like phase in single crystals of magnetoelectric gallium ferrite (GaFeO3) below ~210 K via temperature dependent ac and dc magnetization studies. Analysis of frequency dispersion of the susceptibility peak at ~210 K using the critical slowing down model and Vogel-Fulcher law strongly suggests the existence of a classical spin-glass like phase. This classical spin glass behavior of GaFeO3 is understood in terms of an outcome of geometrical frustration arising from the inherent site disorder among the antiferromagnetically coupled Fe ions located at octahedral Ga and Fe sites.

The Ground State Energy per Site of the Quantum and Classical Edwards-Anderson Spin Glass in the Thermodynamic Limit

We derive rigorous lower bounds for the average ground-state energy per site $e^{(d)}$ of the quantum and classical Edwards-Anderson spin-glass model in dimensions $d = 2$ and $d = 3$ in the thermodynamic limit. For the classical model they imply that $e^{(2)} \ge -3/2$ and $e^{(3)} \ge -2.204...$.