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Physics Procedia

Worms exploring geometrical features of phase transitions

Wolfhard Janke E, Thomas Neuhaus, Adriaan M. J. Schakel

Published Jan 1, 2011• 0 citations
Publisher

Abstract

The loop-gas approach to statistical physics provides an alternative, geometrical description of phase transitions in terms of line-like objects. The resulting statistical random-graph ensemble composed of loops and (open) chains can be e_ciently generated by Monte Carlo simulations using the so-called “worm” update algorithm. Concepts from percolation theory and the theory of self-avoiding random walks are used to derive estimators of physical observables that utilize the nature of the worm algorithm. The fractal structure of random loops and chains as well as their scaling properties encode the critical behavior of the statistical system. The general approach is illustrated for the hightemperature series expansion of the Ising model, or O(1) loop model, on a honeycomb lattice, with its known exact results as valuable benchmarks.

DOI
10.1016/j.phpro.2011.05.059

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The loop-gas approach to statistical physics provides an alternative, geometrical description of phase transitions in terms of line-like objects. The resulting statistical random-graph ensemble composed of loops and (open) chains can be e_ciently generated by Monte Carlo simulations using the so-called “worm” update algorithm. Concepts from percolation theory and the theory of self-avoiding random walks are used to derive estimators of physical observables that utilize the nature of the worm algorithm. The fractal structure of random loops and chains as well as their scaling properties encode the critical behavior of the statistical system. The general approach is illustrated for the hightemperature series expansion of the Ising model, or O(1) loop model, on a honeycomb lattice, with its known exact results as valuable benchmarks.
The exact solution of the Ising model on the complete graph (CG) provides an important, though mean-field, insight for the theory of continuous phase transitions. Besides the original spin, the Ising model can be formulated in the Fortuin-Kasteleyn random cluster and the loop representation, in which many geometric quantities have no correspondence in the spin representations. Using a lifted-worm irreversible algorithm, we study the CG-Ising model in the loop representation and, based on theoretical and numerical analyses, obtain a number of exact results including volume fractal dimensions and scaling forms. Moreover, by combining with the loop-cluster algorithm, we demonstrate how the loop representation can provide an intuitive understanding to the recently observed rich geometric phenomena in the random-cluster representation, including the emergence of two configuration sectors, two length scales, and two scaling windows.
We apply a worm algorithm to simulate the quantum transverse-field Ising model in a path-integral representation of which the expansion basis is taken as the spin component along the external-field direction. In such a representation, a configuration can be regarded as a set of nonintersecting loops constructed by ``kinks'' for pairwise interactions and spin-down (or -up) imaginary-time segments. The wrapping probability for spin-down loops, a dimensionless quantity characterizing the loop topology on a torus, is observed to exhibit small finite-size corrections and yields a high-precision critical point in two dimensions (2D) as hc=3.044330(6){h}_{c}=3.044\phantom{\rule{0.16em}{0ex}}330(6)hc​=3.044330(6), significantly improving over the existing results and nearly excluding the central value of the previous result hc=3.04438(2){h}_{c}=3.044\phantom{\rule{0.16em}{0ex}}38(2)hc​=3.04438(2). At criticality, the fractal dimensions of the loops are estimated as ${d}_{\ensuremath{\ell}\ensuremath{\downarrow}}(1\mathrm{D})=1.37(1)\ensuremath{\approx}\frac{11}{8}$ and ${d}_{\ensuremath{\ell}\ensuremath{\downarrow}}(2\mathrm{D})=1.75(3)$, consistent with those for the classical 2D and 3D O(1) loop model, respectively. An interesting feature is that in one dimension (1D), both the spin-down and -up loops display the critical behavior in the whole disordered phase ($0\ensuremath{\le}h<{h}_{c}$), having a fractal dimension ${d}_{\ensuremath{\ell}}=1.750(7)$ that is consistent with the hull dimension dH=74{d}_{\mathrm{H}}=\frac{7}{4}dH​=47​ for critical 2D percolation clusters. The current worm algorithm can be applied to simulate other quantum systems like hard-core boson models with pairing interactions.
We study the fractal geometry of O( n ) loop configurations in two dimensions by means of scaling and a Monte Carlo method, and compare the results with predictions based on the Coulomb gas technique. The Monte Carlo algorithm is applicable to models with noninteger n and uses local updates. Although these updates typically lead to nonlocal modifications of loop connectivities, the number of operations required per update is only of order 1. The Monte Carlo algorithm is applied to the honeycomb O( n ) model for several values of n , including noninteger ones. We thus determine scaling exponents that describe the fractal nature of O( n ) loops at criticality. The results of the numerical analysis agree with the theoretical predictions.
Theoretical studies often involve mappings of the original system onto an equivalent description in terms of abstract mathematical/graphical objects, where ‘equivalent’ means one obtains the same final answer for some particular property. Path integrals, high-temperature expansions, and Feynman diagrams are the well-known examples considered in this chapter. Under such mappings, one has to deal with the infinite-dimensional configuration space having complex topology and non-local constraints which severely reduce efficiency of Monte Carlo simulations based on standard local updates. This sometimes leads to ergodicity problems in large systems when the entire configuration space can not be sampled in a reasonable computation time. A somewhat related difficulty facing conventional Monte Carlo schemes is the computation of off-diagonal correlation functions since they have no direct relation to the configuration space of the partition function. In what follows we consider path integrals for lattice and continuous systems, high-temperature expansions, and Feynman diagrams, explain the general idea of how Worm Algorithms (WAs) deal with the topological constraints by going to the enlarged configuration space, and present illustrative results for several physics problems.
We show that high-temperature expansions provide a basis for the novel approach to efficient Monte Carlo simulations. "Worm" algorithms utilize the idea of updating closed-path configurations (produced by high-temperature expansions) through the motion of end points of a disconnected path. An amazing result is that local, Metropolis-type schemes using this approach appear to have dynamical critical exponents close to zero (i.e., their efficiency is comparable to the best cluster methods) as proved by finite-size scaling of the autocorrelation time for various universality classes.