Spin-glass Order Parameter Research Articles

Equilibrium Dynamics of Infinite-Range Quantum Spin Glasses in a Field

We determine the low-energy spectrum and Parisi replica-symmetry-breaking function for the spin-glass phase of the quantum Ising model with infinite-range random exchange interactions and transverse and longitudinal (h) fields. We show that, for all h, the spin-glass state has full replica symmetry breaking and the local spin spectrum is gapless, with a spectral density that vanishes linearly with frequency. These results are obtained using an action functional—argued to yield exact results at low frequencies—that expands in powers of a spin-glass order parameter, which is bilocal in time, and a matrix in replica space. We also present the exact solution of the infinite-range spherical quantum p-rotor model at nonzero h: here, the spin-glass state has one-step replica symmetry breaking and gaplessness only appears after imposition of an additional marginal stability condition. Possible connections to experiments on random arrays of trapped Rydberg atoms are noted. Published by the American Physical Society 2024

Gauge theory for mixed p-spin glasses

Physical quantities in the mixed p-spin glasses are evaluated with Nishimori’s gauge theory and several variance inequalities. The -symmetry breaking and the replica-symmetry breaking (RSB) are studied in finite and infinite dimensions. Obtained bounds on the expectation of the square of the magnetization and spontaneous magnetization enable us to clarify properties of paramagnetic and spin glass phases. It is proven that variances of ferromagnetic and spin glass order parameters vanish on the Nishimori line in the infinite volume limit. These results imply the self-averaging of these order parameters on the Nishimori line. The self-averaging of the spin glass order parameter rigorously justifies already argued absence of RSB on the Nishimori line.

Antiferromagnetic Ising model with frustration on Graphenylene lattice

Frustration effects due to competition between nearest-neighbor and next-nearest-neighbor spins interaction on the graphenylene lattice, also known as biphenylene carbon lattice, were investigated by Monte Carlo simulations. We identified two possible phases from our results of the entropy at zero temperature, namely an ordered spin glass and other an antiferromagnetic phase. Despite the complexity of the lattice, a finite size scaling analysis for the antiferromagnetic regime shows that it can be classified into the same universality class of the usual Ising model. The canonical probability distribution as a function of the spin glass order parameter was computed and the continuous spin glass phase transition confirmed.

Role of quantum fluctuation ininducing ergodicity inthe spin glass phase and its effect inquantum annealing.

We review the numerical studies on the critical behaviour of the quantum Sherrington-Kirkpatrick (SK) spin glass model, which indicate that aquantum critical behaviour is observed up to a low but non-zero value of temperature. We revisit the numerical investigations on the spin glass order parameter distributions, which identify a low temperature along with high transverse field spin glass phase where the order parameter distribution becomes a delta function in the thermodynamic limit indicating the restoration of replica symmetry and ergodic nature of the system. In the remaining spin glass phase associated with high temperature and low transverse field, the observed distribution is broad akin to the Parisi order parameter distribution. This essentially indicates the non-ergodic behaviour of the system. We further discuss the annealing dynamics studies on the quantum SK model. Such investigations reveal the system size independence of annealing time when the annealing paths go through the ergodic spin glass region. Interestingly, when such dynamics are performed in the non-ergodic spin glass phase the annealing time becomes an increasing function of the system size. Spin autocorrelation shows faster relaxation in the ergodic spin glass region compared with that found in the non-ergodic spin glass region. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

Critical behavior of the three-dimensional random-anisotropy Heisenberg model.

We have studied the critical properties of the three-dimensional random anisotropy Heisenberg model by means of numerical simulations using the Parallel Tempering method. We have simulated the model with two different disorder distributions, cubic and isotropic ones, with two different anisotropy strengths for each disorder class. For the case of the anisotropic disorder, we have found evidence of universality by finding critical exponents and universal dimensionless ratios independent of the strength of the disorder. In the case of isotropic disorder distribution the situation is very involved: we have found two phase transitions in the magnetization channel which are merging for larger lattices remaining a zero magnetization low-temperature phase. Studying this region using a spin-glass order parameter we have found evidence for a spin-glass phase transition. We have estimated effective critical exponents for the spin-glass phase transition for the different values of the strength of the isotropic disorder, discussing the crossover regime.

Random anisotropy magnet at finite temperature

We present finite-temperature Monte Carlo studies of a 2D random-anisotropy (RA) magnet on lattices containing one million spins. The correlated spin-glass state predicted by analytical theories is reproduced in simulations, as are the field-cooled and zero-field-cooled magnetization curves observed in experiments. The orientations of lattice spins begin to freeze when the temperature is lowered. The freezing transition is due to the energy barriers generated by the RA rather than due to random interactions in conventional spin-glasses. We describe freezing by introducing the time-dependent spin-glass order parameter q and the spin-melting time τ M defined via q = τ M/t above freezing, where t is the time of the experiment represented by the number of Monte Carlo steps.

Nonlocal discrete time crystals in periodically driven surface codes

Discrete time crystals (DTCs) are nonequilibrium phases of matter characterized by robust subharmonic order parameter dynamics. We report a type of DTC in a periodically driven surface code, the subharmonic signature of which is only observable with respect to some nonlocal order parameter. By modifying the commonly used metrics for characterizing ordinary DTCs, i.e., the spectral functions, spin-glass order parameters, and two-point correlators, we further numerically demonstrate the rich phases of the proposed system. Specifically, depending on the system parameters and boundary conditions, we find that the system may fall into either a ``surface code'' phase, trivial paramagnet phase, nonlocal period-doubling DTC phase, or nonlocal period-quadrupling DTC phase. Our work thus demonstrates the prospect of exploring topological codes for discovering novel phases of matter.

Localization dynamics in a centrally coupled system

In systems where interactions couple a central degree of freedom and a bath, one would expect signatures of the bath's phase to be reflected in the dynamics of the central degree of freedom. This has been recently explored in connection with many-body localized baths coupled with a central qubit or a single cavity mode -- systems with growing experimental relevance in various platforms. Such models also have an interesting connection with Floquet many-body localization via quantizing the external drive, although this has been relatively unexplored. Here we adapt the multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) method, a well-known tree tensor network algorithm, to numerically simulate the dynamics of a central degree of freedom, represented by a $d$-level system (qudit), coupled to a disordered interacting 1D spin bath. ML-MCTDH allows us to reach $\approx 10^2$ lattice sites, a far larger system size than what is feasible with exact diagonalization or kernel polynomial methods. From the intermediate time dynamics, we find a well-defined thermodynamic limit for the qudit dynamics upon appropriate rescaling of the system-bath coupling. The spin system shows similar scaling collapse in the Edward-Anderson spin glass order parameter or entanglement entropy at relatively short times. At longer time scales, we see slow growth of the entanglement, which may arise from dephasing mechanisms in the localized system or long-range interactions mediated by the central degree of freedom. Similar signs of localization are shown to appear as well with unscaled system-bath coupling.

Inequality for Local Energy of Ising Model with Quenched Randomness and Its Application

In this study, we extend the lower bound on the average of the local energy of the Ising model with quenched randomness [J. Phys. Soc. Jpn. 76, 074711 (2007)] obtained for a symmetric distribution to an asymmetric one. Compared with the case of symmetric distribution, our bound has a non-trivial term. By applying the acquired bound to a Gaussian distribution, we obtain the lower bounds on the expectation of the square of the correlation function. Thus, we demonstrate that in the Ising model in a Gaussian random field, the spin-glass order parameter generally has a finite value at any temperature, regardless of the forms of the other interactions.

Correspondence between phase oscillator network and classical XY model with the same random and frustrated interactions.

We study correspondence between a phase oscillator network with distributed natural frequencies and a classical XY model at finite temperatures with the same random and frustrated interactions used in the Sherrington-Kirkpatrick model. We perform numerical calculations of the spin glass order parameter q and the distributions of the local fields P(R), where R is the amplitude of the local field. As a result, we find that the parameter dependences of P(R) in both models agree fairly well in all ranges of parameters in the spin glass phase and those of q agree at least for lower values of parameters in the spin glass phase, if parameters are normalized by using the previously obtained correspondence relation between two models with the same other types of interactions. Furthermore, we numerically calculate the time evolution of quantities such as the instantaneous local field in the phase oscillator network in order to study the roles of synchronous and asynchronous oscillators. We also study the self-consistent equation of the local fields in the oscillator network and XY model derived by the mean-field approximation.

Deriving the order parameters of a spin-glass model using principal component analysis.

We investigate the relationship between the order parameters of spin models and principal component analysis (PCA). PCA is applied to the spin configurations generated from the Boltzmann distribution for the cases of uniformly interacting and randomly interacting spin models, and the datasets obtained for various specific temperatures are analyzed. In the case of the uniformly interacting spin model, the first principal component is found to be equivalent to the magnetization in the ordered phase, which is the order parameter. For the randomly interacting spin model, we apply the analysis to datasets generated by the Sherrington-Kirkpatrick model. When PCA is performed under the assumption that the Hadamard product of the spin configuration is taken as a new dataset, it is found that the first principal component coincides with the spin-glass order parameter. By analytically treating the limit in which the number of datasets is infinite, it is shown that the first principal component agrees with the order parameter.

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.

Phase transitions in quantum annealing of an NP-hard problem detected by fidelity susceptibility

Quantum annealing (QA) for the NP-hard maximum independent set problem with a unique solution is studied using the quantum Monte Carlo method. A fraction of the samples exhibit first-order phase transitions in terms of the spin-glass order parameter q. Moreover, while no singularities could be observed for the sample-averaged , a divergence in the sample-averaged fidelity susceptibility is found. Since all of the first-order transitions of q occur with smaller transverse field strength compared to the divergence point of , it is likely that the phase detected by induces the first-order transitions. This suggests that failures of QA due to exponentially small energy gaps are phenomena within this underlying phase.

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.

Spin glass transition in a simple variant of the Ising model on multiplex networks

Multiplex networks consist of a fixed set of nodes connected by several sets of edges which are generated separately and correspond to different networks (“layers”). Here, the Ising model on multiplex networks with two layers is considered, with spins located in the nodes and edges corresponding to ferromagnetic or antiferromagnetic interactions between them. Critical temperatures for the spin glass and ferromagnetic transitions are evaluated for the layers in the form of random Erdös–Rényi graphs or heterogeneous scale-free networks using the replica method, from the replica symmetric solution. Stability of this solution is investigated and location of the de Almeida–Thouless line is determined. For the Ising model on multiplex networks with scale-free layers it is shown that the critical temperature is finite if the distributions of the degrees of nodes within both layers have a finite second moment, and that depending on the model parameters the transition can be to the ferromagnetic or spin glass phase. It is also shown that the correlation between the degrees of nodes within different layers significantly influences the critical temperatures and the phase diagram. The scaling behavior for the spin glass order parameter is determined and it is shown that for the Ising model on multiplex networks with scale-free layers the scaling exponents can depend on the distributions of the degrees of nodes within layers. The analytic results are partly confirmed by Monte Carlo simulations using the parallel tempering algorithm.

Critical properties of the antiferromagnetic Ising model on rewired square lattices

The effect of randomness on critical behavior is a crucial subject in condensed matter physics due to the the presence of impurity in any real material. We presently probe the critical behaviour of the antiferromagnetic (AF) Ising model on rewired square lattices with random connectivity. An extra link is randomly added to each site of the square lattice to connect the site to one of its next-nearest neighbours, thus having different number of connections (links). Average number of links (ANOL) κ is fractional, varied from 2 to 3, where κ = 2 associated with the native square lattice. The rewired lattices possess abundance of triangular units in which spins are frustrated due to AF interaction. The system is studied by using Monte Carlo method with Replica Exchange algorithm. Some physical quantities of interests were calculated, such as the specific heat, the staggered magnetization and the spin glass order parameter (Edward-Anderson parameter). We investigate the role played by the randomness in affecting the exisiting phase transition and its interplay with frustration to possibly bring any spin glass (SG) properties. We observed the low temperature magnetic ordered phase (Néel phase) preserved up to certain value of κ and no indication of SG phase for any value of κ.

Spin-glass-like transition in the majority-vote model with anticonformists

Majority-vote model on scale-free networks and random graphs is investigated in which a randomly chosen fraction p of agents (called anticonformists) follows an antiferromagnetic update rule, i.e., they assume, with probability governed by a parameter q (0 < q < 1∕2), the opinion opposite to that of the majority of their neighbors, while the remaining 1 − p fraction of agents (conformists) follows the usual ferromagnetic update rule assuming, with probability governed by the same parameter q, the opinion in accordance with that of the majority of their neighbors. For p = 1 it is shown by Monte Carlo simulations and using the Binder cumulants method that for decreasing q the model undergoes second-order phase transition from a disordered (paramagnetic) state to a spin-glass-like state, characterized by a non-zero value of the spin-glass order parameter measuring the overlap of agents’ opinions in two replicas of the system, and simultaneously by the magnetization close to zero. In the case of the model on scale-free networks the critical value of the parameter q weakly depends on the details of the degree distribution. As p is decreased, the critical value of q falls quickly to zero and only the disordered phase is observed. On the other hand, for p close to zero for decreasing q the usual ferromagnetic transition is observed.

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.

Magnetic phases of spin-1 lattice gases with random interactions

A spin-1 atomic gas in an optical lattice, in the unit-filling Mott Insulator (MI) phase and in the presence of disordered spin-dependent interaction, is considered. In this regime, at zero temperature, the system is well described by a disordered rotationally-invariant spin-1 bilinear-biquadratic model. We study, via the density matrix renormalization group algorithm, a bounded disorder model such that the spin interactions can be locally either ferromagnetic or antiferromagnetic. Random interactions induce the appearance of a disordered ferromagnetic phase characterized by a non-vanishing value of spin-glass order parameter across the boundary between a ferromagnetic phase and a dimer phase exhibiting random singlet order. The study of the distribution of the block entanglement entropy reveals that in this region there is no random singlet order.

Dual time scales in simulated annealing of a two-dimensional Ising spin glass.

We apply a generalized Kibble-Zurek out-of-equilibrium scaling ansatz to simulated annealing when approaching the spin-glass transition at temperature T=0 of the two-dimensional Ising model with random J=±1 couplings. Analyzing the spin-glass order parameter and the excess energy as functions of the system size and the annealing velocity in Monte Carlo simulations with Metropolis dynamics, we find scaling where the energy relaxes slower than the spin-glass order parameter, i.e., there are two different dynamic exponents. The values of the exponents relating the relaxation time scales to the system length, τ∼L^{z}, are z=8.28±0.03 for the relaxation of the order parameter and z=10.31±0.04 for the energy relaxation. We argue that the behavior with dual time scales arises as a consequence of the entropy-driven ordering mechanism within droplet theory. We point out that the dynamic exponents found here for T→0 simulated annealing are different from the temperature-dependent equilibrium dynamic exponent z_{eq}(T), for which previous studies have found a divergent behavior: z_{eq}(T→0)→∞. Thus, our study shows that, within Metropolis dynamics, it is easier to relax the system to one of its degenerate ground states than to migrate at low temperatures between regions of the configuration space surrounding different ground states. In a more general context of optimization, our study provides an example of robust dense-region solutions for which the excess energy (the conventional cost function) may not be the best measure of success.