Jonathan Machta

Graphical representations and cluster algorithms II

We continue the study, initiated in Part I, of graphical representations and cluster algorithms for various models in (or related to) statistical mechanics. For certain models, e.g. the Blume–Emery–Griffths model and various generalizations, we develop Fortuin Kasteleyn-type representations which lead immediately to Swendsen Wang-type algorithms. For other models, e.g. the random cluster model, that are defined by a graphical representation, we develop cluster algorithms without reference to an underlying spin system. In all cases, phase transitions are related to percolation (or incipient percolation) in the graphical representation which, via the IC algorithm, allows for the rapid simulation of these systems at the transition point. Pertinent examples include the (continuum) Widom–Rowlinson model, the restricted 1-step solid-on-solid model and the XY model.

Erratum: Glassy Chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines

Recently, a programmable quantum annealing machine has been built that minimizes the cost function of hard optimization problems by adiabatically quenching quantum fluctuations. Tests performed by different research teams have shown that, indeed, the machine seems to exploit quantum effects. However experiments on a class of random-bond instances have not yet demonstrated an advantage over classical optimization algorithms on traditional computer hardware. Here we present evidence as to why this might be the case. These engineered quantum annealing machines effectively operate coupled to a decohering thermal bath. Therefore, we study the finite-temperature critical behavior of the standard benchmark problem used to assess the computational capabilities of these complex machines. We simulate both random-bond Ising models and spin glasses with bimodal and Gaussian disorder on the D-Wave Chimera topology. Our results show that while the worst-case complexity of finding a ground state of an Ising spin glass on the Chimera graph is not polynomial, the finite-temperature phase space is likely rather simple: Spin glasses on Chimera have only a zero-temperature transition. This means that benchmarking optimization methods using spin glasses on the Chimera graph might not be the best benchmark problems to test quantum speedup. We propose alternative benchmarks by embedding potentially harder problems on the Chimera topology. Finally, we also study the (reentrant) disorder-temperature phase diagram of the random-bond Ising model on the Chimera graph and show that a finite-temperature ferromagnetic phase is stable up to 19.85(15)% antiferromagnetic bonds. Beyond this threshold the system only displays a zero-temperature spin-glass phase. Our results therefore show that a careful design of the hardware architecture and benchmark problems is key when building quantum annealing machines.

Long time tails in stationary random media. I. Theory

Diffusion of moving particles in stationary disordered media is studied using a phenomenological mode-coupling theory. The presence of disorder leads to a generalized diffusion equation, with memory kernels having power law long time tails. The velocity autocorrelation function is found to decay like t−(d/2+1), while the time correlation function associated with the super-Burnett coefficient decays liket−d/2 for long times. The theory is applicable to a wide variety of dynamical and stochastic systems including the Lorentz gas and hopping models. We find new, general expressions for the coefficients of the long time tails which agree with previous results for exactly solvable hopping models and with the low-density results obtained for the Lorentz gas. Finally we mention that if the moving particles are charged, then the long time tails imply that there is an ωd/2 contribution to the low-frequency part of the frequency-dependent electrical conductivity.

Power law decay of correlations in a billiard problem

A billiard problem with boundary arcs that meet tangentially is studied both analytically and numerically. It is shown that the presence of tangential vertices leads to velocity correlations which decay like 1/n wheren is the number of collisions. This result contrasts with related billiard and Lorentz models where velocity correlations decay exponentially.

Graphical representations and cluster algorithms I. Discrete spin systems

Graphical representations similar to the FK representation are developed for a variety of spin-systems. In several cases, it is established that these representations have (FKG) monotonicity properties which enables characterization theorems for the uniqueness phase and the low-temperature phase of the spin system. Certain systems with intermediate phases and/or first-order transitions are also described in terms of the percolation properties of the representations. In all cases, these representations lead, in a natural fashion, to Swendsen-Wang-type algorithms. Hence, at least in the above-mentioned instances, these algorithms realize the program described by Kandel and Domany, Phys. Rev. B 43 (1991) 8539–8548. All of the algorithms are shown to satisfy a Li-Sokal bound which (at least for systems with a divergent specific heat) implies critical slowing down. However, the representations also give rise to invaded cluster algorithms which may allow for the rapid simulation of some of these systems at their transition points.

Invaded cluster algorithm for equilibrium critical points.

Cluster algorithms are reviewed and a new approach, the invaded cluster algorithm, is described. Invaded cluster algorithms sample critical points without input of the critical temperature. Instead, the critical temperature is an output of the method. Invaded cluster algorithms have less critical slowing than other cluster methods for the Ising model.

Long time tails in stationary random media II: Applications

In a previous paper we developed a mode-coupling theory to describe the long time properties of diffusion in stationary, statistically homogeneous, random media. Here the general theory is applied to deterministic and stochastic Lorentz models and several hopping models. The mode-coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse-grained diffusion coefficient and a coarse-grained free volume. For one-dimensional models these amplitudes can be evaluated, and the mode-coupling theory is shown to agree with exact solutions obtained for these models. For higher-dimensional Lorentz models the formal theory yields expressions which are difficult to evaluate. For these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers.

INVADED CLUSTER ALGORITHM FOR POTTS MODELS

The invaded cluster algorithm, a method for simulating phase transitions, is described in detail. Theoretical, albeit nonrigorous, justification of the method is presented and the algorithm is applied to Potts models in two and three dimensions. The algorithm is shown to be useful for both first-order and continuous transitions and evidently provides an efficient way to distinguish between these possibilities. The dynamic properties of the invaded cluster algorithm are studied. Numerical evidence suggests that the algorithm has no critical slowing for Ising models. @S1063-651X~96!09208-2#

Replica-exchange algorithm and results for the three-dimensional random field Ising model

The random field Ising model with Gaussian disorder is studied using a different Monte Carlo algorithm. The algorithm combines the advantages of the replica-exchange method and the two-replica cluster method and is much more efficient than the Metropolis algorithm for some disorder realizations. Three-dimensional systems of size 24(3) are studied. Each realization of disorder is simulated at a value of temperature and uniform field that is adjusted to the phase-transition region for that disorder realization. Energy and magnetization distributions show large variations from one realization of disorder to another. For some realizations of disorder there are three well separated peaks in the magnetization distribution and two well separated peaks in the energy distribution suggesting a first-order transition.

Monte Carlo Study of the Widom-Rowlinson Fluid Using Cluster Methods

The Widom-Rowlinson model of a fluid mixture is studied using a new cluster algorithm that is an adaptation of the invaded cluster method previously applied to Potts models. The algorithm overcomes the difficulties of treating continuum hard-core systems and has almost no critical slowing down. Our estimates of byn and gyn for the two-component fluid are consistent with the Ising universality class in two and three dimensions. We also present preliminary results for the three-component fluid. [S0031-9007(97)04175-6] PACS numbers: 05.50. + q, 64.60.Fr, 75.10.Hk Some years ago Widom and Rowlinson [1] introduced a simple continuum model that exhibits a phase transition [2]. The two-component version of this model consists of “black” and “white” particles; particles of the same type do not interact, but particles of differing type experience a hard-core repulsion at separations less than or equal to s [3]. We present a new Monte Carlo method for simulating the Widom-Rowlinson (WR) model and apply the method to study the demixing transition in two and three dimensions. There have been few Monte Carlo studies of the WR critical point because of the difficulties of treating hardcore systems and critical slowing down using standard Monte Carlo techniques. We discuss a new algorithm that overcomes these difficulties using cluster methods of the type introduced by Swendsen and Wang [4]. The algorithm employs the invaded cluster (IC) approach [5,6] to locate the critical point. We find that the algorithm has almost no critical slowing down and that we can obtain accurate values of the critical density and the exponent ratios byn and gyn with modest computational effort. The two-component WR model is expected to be in the Ising universality class. Our results for byn and gyn are consistent with this assumption, and our value for the critical density of the three-dimensional ( d › 3 )W R model agrees with recent results obtained in Ref. [7]. We also consider a WR model in which there are q components, any two of which interact via a hard-core repulsion [8,9]. Our algorithm easily extends to these q-component WR models, and we present results for the three-component model in d › 2, 3. Graphical representations of the WR model. — A configuration of the WR fluid consists of two sets of points, S and T , corresponding to the positions of the black and white particles. In the grand canonical ensemble, the probability density for finding the configuration sS, Td is