Two-dimensional Ising Spin Glass Research Articles

Exact ground states of large two-dimensional planar Ising spin glasses

Studying spin-glass physics through analyzing their ground-state properties has a long history. Although there exist polynomial-time algorithms for the two-dimensional planar case, where the problem of finding ground states is transformed to a minimum-weight perfect matching problem, the reachable system sizes have been limited both by the needed CPU time and by memory requirements. In this work, we present an algorithm for the calculation of exact ground states for two-dimensional Ising spin glasses with free boundary conditions in at least one direction. The algorithmic foundations of the method date back to the work of Kasteleyn from the 1960s for computing the complete partition function of the Ising model. Using Kasteleyn cities, we calculate exact ground states for huge two-dimensional planar Ising spin-glass lattices (up to 3000;{2} spins) within reasonable time. According to our knowledge, these are the largest sizes currently available. Kasteleyn cities were recently also used by Thomas and Middleton in the context of extended ground states on the torus. Moreover, they show that the method can also be used for computing ground states of planar graphs. Furthermore, we point out that the correctness of heuristically computed ground states can easily be verified. Finally, we evaluate the solution quality of heuristic variants of the L. Bieche approach.

Energy gap of the bimodal two-dimensional Ising spin glass

An exact algorithm is used to compute the degeneracies of the excited states of the bimodal Ising spin glass in two dimensions. It is found that the specific heat at arbitrary low temperature is not a self-averaging quantity and has a distribution that is neither normal nor lognormal. Nevertheless, it is possible to estimate the most likely value and this is found to scale as L3T-2 exp (−4J/kT), for a L×L lattice. Our analysis also explains, for the first time, why a correlation length ξ∼exp (2J/kT) is consistent with an energy gap of 2J. Our method allows us to obtain results for up to 105 disorder realizations with L⩽96. Distributions of second and third excitations are also shown.

Renormalization Group Approach to Exact Sampling

In this Letter, we use a general renormalization-group algorithm to implement Propp and Wilson's "coupling from the past" approach to complex physical systems. Our algorithm follows the evolution of the entire configuration space under the Markov chain Monte Carlo dynamics from parts of the configurations (patches) on increasing length scales, and it allows us to generate "exact samples" of the Boltzmann distribution, which are rigorously proven to be uncorrelated with the initial condition. We validate our approach in the two-dimensional Ising spin glass on lattices of size 64 x 64.

Matching Kasteleyn cities for spin glass ground states

As spin glass materials have extremely slow dynamics, devious numerical methods are needed to study low-temperature states. A simple and fast optimization version of the classical Kasteleyn treatment of the Ising model is described and applied to two-dimensional Ising spin glasses. The algorithm combines the Pfaffian and matching approaches to directly strip droplet excitations from an excited state. Extended ground states in Ising spin glasses on a torus, which are optimized over all boundary conditions, are used to compute precise values for ground state energy densities.

Fractal dimension of domain walls in two-dimensional Ising spin glasses

We study domain walls in two-dimensional Ising spin glasses in terms of a minimum-weight path problem. Using this approach, large systems can be treated exactly. Our focus is on the fractal dimension ${d}_{f}$ of domain walls, which describes via $⟨\ensuremath{\ell}⟩\ensuremath{\sim}{L}^{{d}_{f}}$ the growth of the average domain-wall length with system size $L\ifmmode\times\else\texttimes\fi{}L$. Exploring systems up to $L=320$ we yield ${d}_{f}=1.274(2)$ for the case of Gaussian disorder, i.e., a much higher accuracy compared to previous studies. For the case of bimodal disorder, where many equivalent domain walls exist due to the degeneracy of this model, we obtain a true lower bound ${d}_{f}=1.095(2)$ and a (lower) estimate ${d}_{f}=1.395(3)$ as upper bound. Furthermore, we study the distributions of the domain-wall lengths. Their scaling with system size can be described also only by the exponent ${d}_{f}$, i.e., the distributions are monofractal. Finally, we investigate the growth of the domain-wall width with system size (``roughness'') and find a linear behavior.

Domain wall entropy of the bimodal two-dimensional Ising spin glass

We report calculations of the domain wall entropy for the bimodal two-dimensional Ising spin glass in the critical ground state. The $L\ifmmode\times\else\texttimes\fi{}L$ system sizes are large with $L$ up to $256$. We find that it is possible to fit the variance of the domain wall entropy to a power function of $L$. The data were obtained using an algorithm that works exactly in the ground state. The integrity of the floating point arithmetic was rigorously checked against exact results for correlation functions and some disorder realizations were run in arbitrary precision as necessary. Nevertheless, in spite of the excellent integrity of the results, the quality of the data distributions are unsatisfactory with large $L>96$. Consequently, it is not possible to reliably determine the fractal dimension of the domain walls.

Magnetic exponents of two-dimensional Ising spin glasses

The magnetic critical properties of two-dimensional Ising spin glasses are controversial. Using exact ground state determination, we extract the properties of clusters flipped when increasing continuously a uniform field. We show that these clusters have many holes but otherwise have statistical properties similar to those of zero-field droplets. A detailed analysis gives for the magnetization exponent � � 1.30 ± 0.02 using lattice sizes up to 80 × 80; this is compatible with the droplet model prediction � = 1.282. The reason for previous disagreements stems from the need to analyze both singular and analytic contributions in the low-field regime.

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.

Effective critical behavior of the two-dimensional Ising spin glass with bimodal interactions

Monte Carlo data of the two-dimensional Ising spin glass with bimodal interactions are presented with the aim of understanding the low-temperature physics of the model. An analysis of the specific heat, spin-glass susceptibility, finite-size correlation length, and the Binder ratio is performed to try to verify a recent proposal in which for large system sizes and finite but low temperatures the effective critical exponents are identical to the critical exponents of the two-dimensional Ising spin glass with Gaussian interactions. Our results show that with present system sizes the recently proposed scenario in which the two-dimensional Ising spin glass with bimodally distributed interactions is in the same universality class as the model with Gaussian-distributed disorder at low but finite temperatures cannot be reliably proven.

Conformal Invariance and Stochastic Loewner Evolution Processes in Two-Dimensional Ising Spin Glasses

We present numerical evidence that the techniques of conformal field theory might be applicable to two-dimensional Ising spin glasses with Gaussian bond distributions. It is shown that certain domain wall distributions in one geometry can be related to that in a second geometry by a conformal transformation. We also present direct evidence that the domain walls are stochastic Loewner (SLE) processes with kappa approximately 2.1. An argument is given that their fractal dimension d(f) is related to their interface energy exponent theta by d(f) - 1 = 3/[4(3 + theta)], which is consistent with the commonly quoted values d(f) approximately 1.27 and theta approximately -0.28.

Strong Universality and Algebraic Scaling in Two-Dimensional Ising Spin Glasses

At zero temperature, two-dimensional Ising spin glasses are known to fall into several universality classes. Here we consider the scaling at low but nonzero temperatures and provide numerical evidence that eta approximately equal 0 and nu approximately equal 3.5 in all cases, suggesting a unique universality class. This algebraic (as opposed to exponential) scaling holds, in particular, for the +/- J model, with or without dilutions, and for the plaquette diluted model. Such a picture, associated with an exceptional behavior at T = 0, is consistent with a real space renormalization group approach. We also explain how the scaling of the specific heat is compatible with the hyperscaling prediction.

Determining energy barriers by iterated optimization: The two-dimensional Ising spin glass

Energy barriers determine the dynamics in many physical systems like structural glasses, disordered spin systems or proteins. Here we present an approach, which is based on subdividing the configuration space in a hierarchical manner, leading to upper and lower bounds for the energy barrier separating two configurations. The fundamental operation is to perform a constrained energy optimization, where the degree of constraintness increases with the level in the hierarchy. As application, we consider Ising spin glasses, where the energy barrier which needs to be surmounted in order to flip a compact region of spins of linear dimension $L$ are expected to scale as $L^{\psi}$. The exponent $\psi$ is very hard to estimate from experimental and simulation studies. By using the new approach, applying efficient combinatorial matching algorithms, we are able to give the first non-trivial numerical bounds $0.25 < \psi < 0.54$ for the two-dimensional Ising spin glass.

Multicritical point of Ising spin glasses on triangular and honeycomb lattices

The behavior of two-dimensional Ising spin glasses at the multicritical point on triangular and honeycomb lattices is investigated, with the help of finite-size scaling and conformal-invariance concepts. We use transfer-matrix methods on long strips to calculate domain-wall energies, uniform susceptibilities, and spin-spin correlation functions. Accurate estimates are provided for the location of the multicritical point on both lattices, which lend strong support to a conjecture recently advanced by Takeda, Sasamoto, and Nishimori. Correlation functions are shown to obey rather strict conformal-invariance requirements, once suitable adaptations are made to account for geometric aspects of the transfer-matrix description of triangular and honeycomb lattices. The universality class of critical behavior upon crossing the ferro-paramagnetic phase boundary is probed, with the following estimates for the associated critical indices: $\nu=1.49(2)$, $\gamma=2.71(4)$, $\eta_1= 0.183(3)$, distinctly different from the percolation values.

Reduction of two-dimensional dilute Ising spin glasses

The recently proposed reduction method is applied to the Edwards-Anderson model on bond-diluted square lattices. This allows, in combination with a graph-theoretical matching algorithm, to calculate numerically exact ground states of large systems. Low-temperature domain-wall excitations are studied to determine the stiffness exponent y_2. A value of y_2=-0.281(3) is found, consistent with previous results obtained on undiluted lattices. This comparison demonstrates the validity of the reduction method for bond-diluted spin systems and provides strong support for similar studies proclaiming accurate results for stiffness exponents in dimensions d=3,...,7.

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.

Energy size effects of two-dimensional Ising spin glasses

We analyze exact ground-state energies of two-dimensional Ising spin glasses with either Gaussian or bimodal nearest-neighbor interactions for large system sizes and for three types of boundary conditions: free on both axes, periodic on both axes, and free on one axis and periodic on the other. We find accurate values for bulk-, edge-, and corner-site energies. Fits for the system with Gaussian bonds are excellent for all types of boundary conditions over the whole range of system sizes. In particular, the leading behavior for non-free boundary conditions is governed by the stiffness exponent ~ -0.28 describing the scaling of domain-wall and droplet excitations. For the system with a bimodal distribution of bonds the fit is good for free boundary conditions but worse for other geometries, particularly for periodic-free boundary conditions where there appear to be unorthodox corrections to scaling up to large sizes. Finally, by introducing hard bonds we test explicitly for the Gaussian case the relationship between domain walls and the standard scaling behavior.

Low-temperature behavior of two-dimensional Gaussian Ising spin glasses

We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\beta=1/T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state and the specific heat. We confirm that $T_c=0$. All results are compatible with an algebraic divergence of the correlation length with an exponent $\nu$. We find $-1/\nu=-0.295(30)$, which is compatible with the value for the domain-wall and droplet exponent $\theta\approx-0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.

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

We study the correlation length of the two-dimensional Ising spin glass with a Gaussian distribution of interactions, using an efficient Monte Carlo algorithm, proposed by Houdayer, that allows larger sizes and lower temperatures to be studied than was possible before. We find that the ``effective'' value of the bulk correlation length exponent $\ensuremath{\nu}$ increases as the temperature is lowered, and, at low temperatures, apparently approaches $\ensuremath{-}1∕\ensuremath{\theta}$, where $\ensuremath{\theta}\ensuremath{\simeq}\ensuremath{-}0.29$ is the stiffness exponent obtained at zero temperature. This means scaling is satisfied and earlier results at higher temperatures that find a smaller value for $\ensuremath{\nu}$ are affected by corrections to scaling.

Generating droplets in two-dimensional Ising spin glasses using matching algorithms

We study the behavior of droplets for two dimensional Ising spin glasses with Gaussian interactions. We use an exact matching algorithm which enables study of systems with linear dimension L up to 240, which is larger than is possible with other approaches. But the method only allows certain classes of droplets to be generated. We study single-bond, cross and a category of fixed volume droplets as well as first excitations. By comparison with similar or equivalent droplets generated in previous works, the advantages but also the limitations of this approach are revealed. In particular we have studied the scaling behavior of the droplet energies and droplet sizes. In most cases, a crossover of the data can be observed such that for large sizes the behavior is compatible with the one-exponent scenario of the droplet theory. Only for the case of first excitations, no clear conclusion can be reached, probably because even with the matching approach the accessible system sizes are still too small.

Energetics of clusters in the two-dimensional Gaussian Ising spin glass

We study numerically the properties of local low-energy excitations in the two-dimensional Ising spin glass. Given the ground state, we determine the lowest-lying connected cluster of flipped spins containing one given spin, either with a fixed volume, or with a volume constrained to lie in a certain range. Our aim is to understand corrections to the scaling predicted by the droplet picture of spin glasses and to resolve contradictory results reported in the literature for the stiffness exponent. We find no clear trace of corrections to scaling, and the obtained stiffness exponent is in relatively good agreement with standard domain wall calculations.