3D Spin Glass Research Articles

Sampling diverse near-optimal solutions via algorithmic quantum annealing.

Sampling a diverse set of high-quality solutions for hard optimization problems is of great practical relevance in many scientific disciplines and applications, such as artificial intelligence and operations research. One of the main open problems is the lack of ergodicity, or mode collapse, for typical stochastic solvers based on Monte Carlo techniques leading to poor generalization or lack of robustness to uncertainties. Currently, there is no universal metric to quantify such performance deficiencies across various solvers. Here, we introduce a new diversity measure for quantifying the number of independent approximate solutions for NP-hard optimization problems. Among others, it allows benchmarking solver performance by a required time-to-diversity (TTD), a generalization of often used time-to-solution (TTS). We illustrate this metric by comparing the sampling power of various quantum annealing strategies. In particular, we show that the inhomogeneous quantum annealing schedules can redistribute and suppress the emergence of topological defects by controlling space-time separated critical fronts, leading to an advantage over standard quantum annealing schedules with respect to both TTS and TTD for finding rare solutions. Using path-integral Monte Carlo simulations for up to 1600 qubits, we demonstrate that nonequilibrium driving of quantum fluctuations, guided by efficient approximate tensor network contractions, can significantly reduce the fraction of hard instances for random frustrated 2D spin glasses with local fields. Specifically, we observe that by creating a class of algorithmic quantum phase transitions, the diversity of solutions can be enhanced by up to 40% with the fraction of hard-to-sample instances reducing by more than 25%.

Searching for spin glass ground states through deep reinforcement learning

Spin glasses are disordered magnets with random interactions that are, generally, in conflict with each other. Finding the ground states of spin glasses is not only essential for understanding the nature of disordered magnets and many other physical systems, but also useful to solve a broad array of hard combinatorial optimization problems across multiple disciplines. Despite decades-long efforts, an algorithm with both high accuracy and high efficiency is still lacking. Here we introduce DIRAC – a deep reinforcement learning framework, which can be trained purely on small-scale spin glass instances and then applied to arbitrarily large ones. DIRAC displays better scalability than other methods and can be leveraged to enhance any thermal annealing method. Extensive calculations on 2D, 3D and 4D Edwards-Anderson spin glass instances demonstrate the superior performance of DIRAC over existing methods. The presented framework will help us better understand the nature of the low-temperature spin-glass phase, which is a fundamental challenge in statistical physics. Moreover, the gauge transformation technique adopted in DIRAC builds a deep connection between physics and artificial intelligence. In particular, this opens up a promising avenue for reinforcement learning models to explore in the enormous configuration space, which would be extremely helpful to solve many other hard combinatorial optimization problems.

2D spin glass MnIn2Se4: application of liquid-phase exfoliation to a layered structure with seven-atom-thick layers

Liquid-phase exfoliation of a layered-structure semiconductor, MnIn2Se4, followed by sequential centrifugation, affords nanosheets of variable thicknesses. Both the bulk material and nanosheets exhibit spin-glass behavior with spin freezing at ∼3 K.

Non-equilibrium criticality and efficient exploration of glassy landscapes with memory dynamics

Spin glasses are notoriously difficult to study both analytically and numerically due to the presence of frustration and metastability. Their highly non-convex landscapes require collective updates to explore efficiently. Currently, most state-of-the-art algorithms rely on stochastic spin clusters to perform non-local updates, but such “cluster algorithms” lack general efficiency. Here, we introduce a non-equilibrium approach for simulating spin glasses based on classical dynamics with memory. By simulating various classes of 3d spin glasses (Edwards–Anderson, partially-frustrated, and fully-frustrated models), we find that memory dynamically promotes critical spin clusters during time evolution, in a self-organizing manner. This facilitates an efficient exploration of the low-temperature phases of spin glasses.

Determination of the Phase Diagram for the 3d Spin Glass ± J Model Using the Zeros of the Energy Probability Distribution

In this work we use the zeros of the energy probability distribution to obtain the critical diagram of the three dimensional (L 3) ±J Ising model. Starting with a complete ferromagnetic system we introduce a number of p × L 3 antiferromagnetic bonds in the system. The phase diagram, temperature against antiferromagnetic bond concentration (p), is estimated for several values of p. The goal of this preliminary study is to show how it is possible to determine the phase diagram of such a complex model without the need to define any order parameter.

An experiment-oriented analysis of 2D spin-glass dynamics: a twelve time-decades scaling study

Recent high precision experimental results on spin-glass films ask for a detailed understanding of the domain-growth dynamics of two-dimensional spin glasses. To achieve this goal, we numerically simulate the out-equilibrium dynamics of the Ising spin glass for a time that spans close to twelve orders of magnitude (from picoseconds to order of a second), in systems large enough to avoid finite-size effects. We find that the time-growth of the size of the glassy domains is excellently described by a single scaling function. A single time-scale controls the dynamics. diverges upon approaching the T = 0 critical point. The divergence of is Arrhenius-like, with a barrier height that depends very mildly on temperature. The growth of this barrier-height is best described by critical dynamics. As a side product we obtain an impressive confirmation of universality of the equilibrium behavior of two-dimensional spin-glasses.

Effective optimization using sample persistence: A case study on quantum annealers and various Monte Carlo optimization methods.

We present and apply a general-purpose, multistart algorithm for improving the performance of low-energy samplers used for solving optimization problems. The algorithm iteratively fixes the value of a large portion of the variables to values that have a high probability of being optimal. The resulting problems are smaller and less connected, and samplers tend to give better low-energy samples for these problems. The algorithm is trivially parallelizable since each start in the multistart algorithm is independent, and could be applied to any heuristic solver that can be run multiple times to give a sample. We present results for several classes of hard problems solved using simulated annealing, path-integral quantum Monte Carlo, parallel tempering with isoenergetic cluster moves, and a quantum annealer, and show that the success metrics and the scaling are improved substantially. When combined with this algorithm, the quantum annealer's scaling was substantially improved for native Chimera graph problems. In addition, with this algorithm the scaling of the time to solution of the quantum annealer is comparable to the Hamze-de Freitas-Selby algorithm on the weak-strong cluster problems introduced by Boixo etal. Parallel tempering with isoenergetic cluster moves was able to consistently solve three-dimensional spin glass problems with 8000 variables when combined with our method, whereas without our method it could not solve any.

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.

Parallel tempering and 3D spin glass models

We review parallel tempering schemes and examine their main ingredients for accuracy and efficiency. We discuss two selection methods of temperatures and some alternatives for the exchange of replicas, including all-pair exchange methods. We measure specific heat errors and round-trip efficiency using the two-dimensional (2D) Ising model, and also test the efficiency for the ground state production in 3D spin glass models. We find that the optimization of the GS problem is highly influenced by the choice of the temperature range of the PT process. Finally, we present numerical evidence concerning the universality aspects of an anisotropic case of the 3D spin-glass model.

Comparative study of selected parallel tempering methods

We review several parallel tempering schemes and examine their main ingredients for accuracy and efficiency. The present study covers two selection methods of temperatures and several choices for the exchange of replicas, including a recent novel all-pair exchange method. We compare the resulting schemes and measure specific heat errors and efficiency using the two-dimensional (2D) Ising model. Our tests suggest that an earlier proposal for using numbers of local moves related to the canonical correlation times is one of the key ingredients for increasing efficiency, and protocols using cluster algorithms are found to be very effective. Some of the protocols are also tested for efficiency and ground state production in 3D spin-glass models where we find that a simple nearest-neighbor approach using a local n-fold-way algorithm is the most effective. Finally, we present evidence that the asymptotic limits of the ground state energy for the isotropic case and for an anisotropic case of the 3D spin-glass model are very close and may even coincide.

Spin-glass in the spinel-type CuCrTiS4

It is difficult to find a typical compound to understand the spin-glass (SG). Only a few model compounds for the SG have been investigated; Fe0.50Mn0.50TiO3 as a short-range interacting three-dimensional (3D) Ising SG, Dy0.103Y0.897Ru2Si2 as a long-range 3D Ising SG, and AgMn as a long-range 3D Heisenberg SG. A spinel-type compound CuCrTiS4 will be provided in this study as a realistic candidate for a short-range interacting 3D Heisenberg SG. This CuCrTiS4 can cause a SG phase below T g = 7.91 K, which is obtained by a random non-magnetic 50% substitution of Ti for Cr on B-sites in the ferromagnetic mother spinel CuCr2S4. The localized magnetic moment of CuCrTiS4 comes from Cr3+ with S = 3/2, where a valence state of Cu+Cr3+Ti4+S4 2− is built up. Experimental data will be focused only on the static properties of dc magnetization M(T) under various magnetic fields, without any dynamical magnetic feature. The spin-glass phase transition T g is well defined at low field, nevertheless hardly in higher fields than approximately 40 Oe. An appearance of the field dependence of M(T) of CuCrTiS4 is similar to that of Fe0.50Mn0.50TiO3 with the short-range Ising SG, and also these of AgMn and CuMn with the long-range interacting Heisenberg SGs. This general nature in M(T) seems to originate from the common mechanism under magnetic fields for all the spin-glasses.

Magnetic Order and Frustrated Dynamics in Li (Ni0.8Co0.1Mn0.1)O2: A Study by μ+SR and SQUID Magnetometry

Recently, the mixed transition metal oxides of the form Li (Ni 1−y−z Co y Mn z )O 2 , havebecome the center of attention as promising candidates for novel battery material. These materials have also revealed very interesting magnetic properties due to the alternate stacking of planes of metal oxides on a 2D triangular lattice and the Li-layers. The title compound, Li (Ni 0.8 Co 0.1 Mn 0.1 ) O 2 , has been investigated by both magnetometry and measurements and μ + SR. We find the evolution of localized magnetic moments with decreasing temperature below 70 K. The magnetic ground state ( T = 2K) is, however, shown to be a frustrated system in 3D, followed by a transition into a possible 2D spin-glass above 22 K.With further increasing temperature the compound show the presence of remaining correlations with increasing effective dimensionality all the way up to the ferrimagnetic transition at T C = 70 K.

Entropic long range order in a 3D spin glass model

We uncover a new kind of entropic long range order in finite dimensional spin glasses. We studythe link-diluted version of the Edwards–Anderson spin glass model with bimodal couplings (J = ± 1) on a 3D lattice. By using exact reduction algorithms, we prove that there exists a regionof the phase diagram (at zero temperature and low enough link density), where spins arelong range correlated, even if the ground state energy stiffness is null. In other words, inthis region twisting the boundary conditions costs no energy, but spins are long rangecorrelated by means of pure entropic effects.

Magnetic order, aging, and spin frustration in a percolating spin system, LiNi0.8Co0.1Mn0.1O2

Structural and magnetic properties of the Li layered oxide, LiNi0.8Co0.1Mn0.1O2, have been studied by means of x-ray diffraction, dc magnetization, ac linear, and nonlinear susceptibility as well as by magnetic aging and temperature cycling experiments. A percolating spin system interacting via antiferromagnetic and ferromagnetic superexchange interactions of different strength induce a ferrimagnetic quasilong-range order with a transition temperature Tc≈70 K, a Weiss temperature of −25 K and an effective magnetic moment of 2.07 μB. On cooling below Tc the two-dimensional (2D) triangular lattice built up of edge-sharing metal oxide octahedras first exhibits a 2D spin glass like behavior followed by complete spin frustration in three dimensions below 30 K. The findings correlate well with expectations for a ferrimagnetic reentrant cluster glass system as well as with predictions of the percolating cluster model.

The overlap parameter across an inverse first-order phase transition in a 3D spin-glass

We investigate the thermodynamic phase transition taking place in the Blume–Capel model in the presence of quenched disorder in three dimensions. In particular, performing exchange Monte Carlo simulations, we study the behaviour of the order parameters across the first-order phase transition and its related coexistence region. This transition is inverse freezing.

Two models of spin glasses — Ising versus Heisenberg

Brief review is given on recent numerical research of the ordering of two typical models of spin glasses (SGs), the three-dimensional (3D) Ising SG and the 3D Heisenberg SG models. Particular attention is paid to the questions of whether there is a thermodynamic transition in zero field, what are the associated critical properties, what is the nature of the ordered state, particularly of a possible replica-symmetry breaking, and whether there is a thermodynamic transition in applied fields. The properties of the two models are contrasted, and possible relation to experiments is discussed.

Thermodynamic First Order Transition and Inverse Freezing in a 3D Spin Glass

We present a numerical study of the random Blume-Capel model in three dimensions. The phase diagram is characterized by spin-glass-paramagnet phase transitions of both first and second order in the thermodynamic sense. Numerical simulations are performed using the exchange Monte Carlo algorithm, providing clear evidence for inverse freezing. The main features at criticality and in the phase coexistence region are investigated. We are not privy to other 3D short-range systems with quenched disorder undergoing inverse freezing.

Logarithmic growth law in the two-dimensional Ising spin glass state resulting from the electron doping in single-layered manganites

The ac susceptibility of the electron-doped single-layered manganite ${\mathrm{La}}_{1.1}{\mathrm{Sr}}_{0.9}\mathrm{Mn}{\mathrm{O}}_{4}$ is analyzed in detail. A quasi-two-dimensional antiferromagnetic (AFM) order with Ising anisotropy is stabilized below ${T}_{N}\ensuremath{\sim}80\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. We show that below ${T}_{N}$, a rare two-dimensional (2D) spin-glass (SG) correlation develops with the same Ising anisotropy as the AFM state. Using simple scaling arguments of the droplet model, we derive a scaling form for the ac susceptibility data of a 2D SG, which our experimental data follow fairly well. Due to simplifications in this 2D case, the proposed scaling form only contains two unknown variables, $\ensuremath{\psi}\ensuremath{\nu}$ and ${\ensuremath{\tau}}_{0}$. Hence, the logarithmic growth law of the SG correlation predicted by the droplet model is convincingly evidenced by the scaling of our experimental data. The origin and nature of this 2D SG state are also discussed.

Experimental first order reversal curves (FORC) on Ising and Heisenberg spin glasses

The first order reversal curve (FORC) method is employed to analyse the low temperature hysteresis behaviour of two model 3d spin glasses: the Ising system Fe 0.5 Mn 0.5 TiO 3 and the near isotropic Heisenberg like system Ag(11 at% Mn). The hysteresis loops of the two samples are strikingly different (in accord with results from earlier investigations on spin glasses with different anisotropy). The derived FORC distribution of the Ising system is wide, centred around 0-bias field and in parts similar to what has been observed for interacting magnetic nanoparticle systems with wide particle size distributions. The FORC distribution for the Ag(Mn) spin glass on the other hand shows a sharp peak at a weak negative bias field and a coercive field of about - 80 and 200 Oe, respectively.

Ordering of a frustrated Josephson-coupled network system of sub-micron-scale grains

Ordering characteristics of a superconductive ceramic of YBa 2Cu 4O 8 are newly examined experimentally by analyzing the nonlinear magnetic responses, in order to discuss intergrain ordering of frustrated Josephson-coupled network system of sub-micron-scale grains below the superconductive critical temperature. The derived nonlinear magnetic susceptibility χ 2 shows a symmetric negative divergence at the temperature corresponding to the intergrain ordering point T c2. The result indicates that the ordering of a 3D XY-type spin glass is associated with the same singularity of nonlinear chiral susceptibility at the temperature.