Two-dimensional Edwards-Anderson Research Articles

Entanglement Entropy on the Boundary of the Square-Lattice ±J Ising Model

A classical analogue of the entanglement entropy is calculated on the system boundary of the two-dimensional Edwards-Anderson model, where the nearest-neighbor interaction is stochastically chosen from +J and -J. The boundary spin distribution is obtained by means of the time-evolving block decimation (TEBD) method, where the random ensemble is created from the successive multiplications of position-dependent transfer matrices, whose width is up to N = 300. The random average of the entanglement entropy is calculated on the Nishimori line, and it is confirmed that the entanglement entropy shows critical singularity at the Nishimori point. The central charge of the boundary state is estimated.

Boosting Monte Carlo simulations of spin glasses using autoregressive neural networks.

The autoregressive neural networks are emerging as a powerful computational tool to solve relevant problems in classical and quantum mechanics. One of their appealing functionalities is that, after they have learned a probability distribution from a dataset, they allow exact and efficient sampling of typical system configurations. Here we employ a neural autoregressive distribution estimator (NADE) to boost Markov chain Monte Carlo (MCMC) simulations of a paradigmatic classical model of spin-glass theory, namely, the two-dimensional Edwards-Anderson Hamiltonian. We show that a NADE can be trained to accurately mimic the Boltzmann distribution using unsupervised learning from system configurations generated using standard MCMC algorithms. The trained NADE is then employed as smart proposal distribution for the Metropolis-Hastings algorithm. This allows us to perform efficient MCMC simulations, which provide unbiased results even if the expectation value corresponding to the probability distribution learned by the NADE is not exact. Notably, we implement a sequential tempering procedure, whereby a NADE trained at a higher temperature is iteratively employed as proposal distribution in a MCMC simulation run at a slightly lower temperature. This allows one to efficiently simulate the spin-glass model even in the low-temperature regime, avoiding the divergent correlation times that plague MCMC simulations driven by local-update algorithms. Furthermore, we show that the NADE-driven simulations quickly sample ground-state configurations, paving the way to their future utilization to tackle binary optimization problems.

Ground-state phase-space structures of two-dimensional ±J spin glasses: A network approach.

We illustrate a complex-network approach to study the phase spaces of spin glasses. By mapping the whole ground-state phase spaces of two-dimensional Edwards-Anderson bimodal (±J) spin glasses exactly into networks for analysis, we discovered various phase-space properties. The Gaussian connectivity distribution of the phase-space networks demonstrates that both the number of free spins and the visiting frequency of all microstates follow the Gaussian distribution. The spectra of phase-space networks are Gaussian, which is proven to be exact when the system is infinitely large. The phase-space networks exhibit community structures. By coarse graining to the community level, we constructed a network representing the entropy landscape of the ground state and discovered its scale-free property. The phase-space networks exhibit fractal structures, as a result of the rugged entropy landscape. Moreover, we show that the connectivity distribution, community structures, and fractal structures change drastically at the ferromagnetic-to-glass phase transition. These quantitative measurements of the ground states provide new insight into the study of spin glasses. The phase-space networks of spin glasses share a number of common features with those of lattice gases and geometrically frustrated spin systems and form a new class of complex networks with unique topology.

Topologically invariant tensor renormalization group method for the Edwards-Anderson spin glasses model

Tensor renormalization group (TRG) method is a real space renormalization group approach. It has been successfully applied to both classical and quantum systems. In this paper, we study a disordered and frustrated system, the two-dimensional Edwards-Anderson model, by a new topological invariant TRG scheme. We propose an approach to calculate the local magnetizations and nearest pair correlations simultaneously. The Nishimori multicritical point predicted by the topological invariant TRG agrees well with the recent Monte Carlo results. The TRG schemes outperform the mean-field methods on the calculation of the partition function. We notice that it might obtain a negative partition function at sufficiently low temperatures. However, the negative contribution can be neglected if the system is large enough. This topological invariant TRG can also be used to study three-dimensional spin glass systems.

Exact algorithm for sampling the two-dimensional Ising spin glass

A sampling algorithm is presented that generates spin-glass configurations of the two-dimensional Edwards-Anderson Ising spin glass at finite temperature with probabilities proportional to their Boltzmann weights. Such an algorithm overcomes the slow dynamics of direct simulation and can be used to study long-range correlation functions and coarse-grained dynamics. The algorithm uses a correspondence between spin configurations on a regular lattice and dimer (edge) coverings of a related graph: Wilson's algorithm [D. B. Wilson, Proceedings of the Eighth Symposium on Discrete Algorithms (SIAM, Philadelphia, 1997), p 258] for sampling dimer coverings on a planar lattice is adapted to generate samplings for the dimer problem corresponding to both planar and toroidal spin-glass samples. This algorithm is recursive: it computes probabilities for spins along a "separator" that divides the sample in half. Given the spins on the separator, sample configurations for the two separated halves are generated by further division and assignment. The algorithm is simplified by using Pfaffian elimination rather than Gaussian elimination for sampling dimer configurations. For n spins and given floating point precision, the algorithm has an asymptotic run-time of O(n(3/2)); it is found that the required precision scales as inverse temperature and grows only slowly with system size. Sample applications and benchmarking results are presented for samples of size up to n=128(2), with fixed and periodic boundary conditions.

Correlation length of the two-dimensional Ising spin glass with bimodal interactions

We study the correlation length of the two-dimensional Edwards-Anderson Ising spin glass with bimodal interactions using a combination of parallel tempering Monte Carlo and a rejection-free cluster algorithm in order to speed up equilibration. Our results show that the correlation length grows ~ exp(2J/T) suggesting through hyperscaling that the degenerate ground state is separated from the first excited state by an energy gap ~4J, as would naively be expected.

Cluster Monte Carlo Algorithms for Diluted Spin Glasses

Recently a cluster Monte Carlo algorithm has been used very successfully in the two-dimensional Edwards-Anderson (EA) model. We show that this algorithm and a variant thereof can also be used successfully in models with a non-zero spin glass transition temperature. The application of such algorithms to the site-diluted EA model in three dimensions is discussed and the efficiency of the two algorithms is compared among each other and to parallel tempering. Finally, we give evidence for a spin glass transition in the three-dimensional site-diluted EA model with Gaussian couplings at a site occupation of p = 62.5 %.

Quantum properties of a strongly interacting frustrated disordered magnet

We compute the low-energy quantum states and low-frequency dynamical susceptibility of a quantum generalization of the two-dimensional Edwards-Anderson spin glass, obtaining exact numerical results for system sizes much larger than previously accessible. The ground state is a complex superposition of a substantial fraction of all the classical ground states, often having large connected regions of spins all exhibiting strong quantum fluctuations, and yet the dynamical susceptibility exhibits sharp resonances reminiscent of the behavior of single spins. The dependence of the energy spectra on system size differs qualitatively from that of the energy spectra of random undirected bipartite graphs with similar statistics, implying that strong interactions are giving rise to these unusual spectral properties.

Quasiequilibrium during aging of the two-dimensional Edwards-Anderson model

We test the quasiequilibrium picture of the aging dynamics--strictly valid in the asymptotic dynamical regime of aging systems--in the preasymptotic aging regime of the two-dimensional Edwards-Anderson spin glass model. We compare the fluctuation-dissipation characteristic for spin autocorrelation function and response with a corresponding one obtained for a suitably defined correlation function and its conjugated response. In agreement with the quasiequilibrium picture we find that after a short transient the two corresponding fluctuation-dissipation ratios (FDR's) coincide at equal times. Moreover we show that, as it happens for the usual FDR, the dynamic FDR at finite time coincides with the static one at finite size.

Ground states of two-dimensional±JEdwards-Anderson spin glasses

We present an exact algorithm for finding all the ground states of the two-dimensional Edwards-Anderson $\pm J$ spin glass and characterize its performance. We investigate how the ground states change with increasing system size and and with increasing antiferromagnetic bond ratio $x$. We find that that some system properties have very large and strongly non-Gaussian variations between realizations.

Fractal Droplets in Two-Dimensional Spin Glass

The two-dimensional Edwards-Anderson model with Gaussian bond distribution is investigated at T=0 with a numerical method. Droplet excitations are directly observed. It turns out that the averaged volume of droplets is proportional to l^D with D = 1.80(2) where l is the spanning length of droplets, revealing their fractal nature. The exponent characterizing the l dependence of the droplet excitation energy is estimated to be -0.42(4), clearly different from the stiffness exponent for domain wall excitations.

Dynamic weighting in simulations of spin systems

We establish the invariance property of a dynamically weighted Monte Carlo process and apply the method to the simulation of spin glasses and Ising models. For the two-dimensional Edwards-Anderson model, we obtain an ergodicity time of O( L 2.44), where L is the linear dimension of the lattice. The results suggest that dynamic weighting is a highly promising new tool for Monte Carlo simulation.

Q2R simulations of two-dimensional spin glasses

Q2R cellular automata are shown to attain thermal equilibration in the two-dimensional Edwards-Anderson model (1975) of a spin glass. Some problems associated with equilibration at low temperatures are identified as finite-size effects. The authors use a thermalization test due to Bhatt and Young (1988). Their results indicate the possibility of calculating Gibbs averages in this model by means of the Q2R algorithm. A rough efficiency comparison with the conventional (Metropolis) Monte Carlo algorithm is also made.

Static critical behavior of the two-dimensional Ising spin glass Rb2Cu1-xCoxF4.

The temperature and field dependences of the nonlinear susceptibility are examined in ${\mathrm{Rb}}_{2}$${\mathrm{Cu}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Co}}_{\mathrm{x}}$${\mathrm{F}}_{4}$ with x=0.218, which is an experimental realization of the two-dimensional Edwards-Anderson Ising spin glass with random nearest-neighbor bonds. From the divergence of the nonlinear terms in the expansion of the magnetization in the field as well as from static scaling, the spin-glass transition is shown to occur at zero temperature, which implies the lower critical dimensionality for Ising spin glasses to exceed 2. The critical exponents \ensuremath{\gamma}=4.5\ifmmode\pm\else\textpm\fi{}0.2 and \ensuremath{\beta}=0.0\ifmmode\pm\else\textpm\fi{}0.1 agree with recent theoretical estimates for the \ifmmode\pm\else\textpm\fi{}J Ising spin glass in two dimensions.

Static and dynamic magnetic response of spin-glass models with short-range interactions

The static and dynamic magnetic response of the two-dimensional Edwards-Anderson model with a nearest-neighbor Gaussian exchange distribution is investigated by Monte Carlo simulation. A plateau in the equilibrium (slowly field cooled) susceptibility $\frac{{M}_{\mathrm{eq}}}{H}$ is found, which diverges for small fields with a power law $\frac{{M}_{\mathrm{eq}}}{H}\ensuremath{\propto}{H}^{\frac{1}{\ensuremath{\Delta}}}$, $\ensuremath{\Delta}=3.5\ifmmode\pm\else\textpm\fi{}0.5$. The boundary of the plateau ${H}_{c}^{\mathrm{eq}}(T)$ tends to zero temperature as ${H}_{c}^{\mathrm{eq}}(T)\ensuremath{\propto}{T}^{\ensuremath{\Delta}}$, consistent with a scaling description appropriate for a static phase transition at $T=0$, and the associate scaling function is estimated. Surprisingly the data also are consistent with a scaling representation with nonzero freezing temperatures ${T}_{f}$, in striking similarity to experimental data, but ${T}_{f}=0$ is shown to be the correct choice. The zero-field-cooled susceptibility starts to differ from $\frac{{M}_{\mathrm{eq}}}{H}$ below a certain critical field ${H}_{c}(t)$, $t$ being the time scale over which the field is applied. In the $H\ensuremath{-}T$ plane ${H}_{c}(t)$ extrapolates to a time-dependent freezing temperature ${T}_{c}(t)$, and closely resembles the de Almeida---Thouless line. A tentative interpretation of these findings is attempted by combining scaling considerations with the reorientation of correlated clusters which have the size of the correlation length describing Edwards-Anderson order. The reduction of the free-energy barriers due to the magnetic field is treated in analogy with the interface free-energy reduction of ferromagnets in random fields.