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arXiv:cond-mat/0211262 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Cluster Monte Carlo algorithm for the quantum rotor model

Fabien Alet, Erik Schwartz Sørensen

Published Jan 31, 2003• 83 citations
arXivHTMLPDF

Abstract

We propose a highly efficient ``worm''-like cluster Monte Carlo algorithm for the quantum rotor model in the link-current representation. We explicitly prove detailed balance for the algorithm even in the presence of disorder. For the pure quantum rotor model with $\ensuremath{\mu}=0,$ the algorithm yields high- precision estimates for the critical point ${K}_{c}=0.33305(5)$ and the correlation length exponent $\ensuremath{\nu}=0.670(3).$ For the disordered case, $\ensuremath{\mu}=\frac{1}{2}\ifmmode\pm\else\textpm\fi{}\frac{1}{2},$ we find $\ensuremath{\nu}=1.15(10).$

DOI
10.1103/physreve.67.015701

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arXiv:2005.10066•Physical review. B./Physical review. B•Sep 2, 2020•20 citations
We propose a highly efficient ``worm''-like cluster Monte Carlo algorithm for the quantum rotor model in the link-current representation. We explicitly prove detailed balance for the algorithm even in the presence of disorder. For the pure quantum rotor model with \ensuremathμ=0,\ensuremath{\mu}=0,\ensuremathμ=0, the algorithm yields high- precision estimates for the critical point Kc=0.33305(5){K}_{c}=0.33305(5)Kc​=0.33305(5) and the correlation length exponent \ensuremathν=0.670(3).\ensuremath{\nu}=0.670(3).\ensuremathν=0.670(3). For the disordered case, \ensuremathμ=12\ifmmode±\else\textpm\fi12,\ensuremath{\mu}=\frac{1}{2}\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ensuremathμ=21​\ifmmode±\else\textpm\fi21​, we find \ensuremathν=1.15(10).\ensuremath{\nu}=1.15(10).\ensuremathν=1.15(10).
We discuss the implementation of a directed geometrical worm algorithm for the study of quantum link-current models. In this algorithm the Monte Carlo updates are made through the biased reptation of a worm through the lattice. A directed algorithm is an algorithm where, during the construction of the worm, the probability for erasing the immediately preceding part of the worm, when adding a new part, is minimal. We introduce a simple numerical procedure for minimizing this probability. The procedure only depends on appropriately defined local probabilities and should be generally applicable. Furthermore, we show how correlation functions C(r,tau) can be straightforwardly obtained from the probability of a worm to reach a site (r,tau) away from its starting point independent of whether or not a directed version of the algorithm is used. Detailed analytical proofs of the validity of the Monte Carlo algorithms are presented for both the directed and undirected geometrical worm algorithms. Results for autocorrelation times and Green's functions are presented for the quantum rotor model.
Efficient schemes called directed loops for cluster updates in Quantum Monte Carlo simulations have recently been proposed, which improve both the efficiency and precision for simulations of quantum models. In this work we show that local detailed balance is not necessary in order to fulfill global detailed balance during the construction of these loops. We therefore propose to insert additional degrees of freedom into the directed loop equations, resulting in even more efficient algorithms. Our approach works directly in the natural representation of an extended space where the matrix elements of the operators defining the worm (broken worldline segments) are taken into account.
We systematically test the performance of several Monte Carlo update schemes for the (2 + 1)d XY phase transition of quantum rotor model. By comparing the local Metropolis (LM), LM plus over-relaxation (OR), Wolff-cluster (WC), hybrid Monte Carlo (HM), hybrid Monte Carlo with Fourier acceleration (FA) schemes, it is clear that among the five different update schemes, at the quantum critical point, the WC and FA schemes acquire the smallest autocorrelation time and cost the least amount of CPU hours in achieving the same level of relative error, and FA enjoys a further advantage of easily implementable for more complicated interactions such as the long-range ones. These results bestow one with the necessary knowledge of extending the quantum rotor model, which plays the role of ferromagnetic/antiferromagnetic critical bosons or Z 2 topological order, to more realistic and yet challenging models such as Fermi surface Yukawa-coupled to quantum rotor models.
We apply a meron cluster algorithm to the XY spin chain, which describes a quantum rotor. This is a multi-cluster simulation supplemented by an improved estimator, which deals with objects of half-integer topological charge. This method is powerful enough to provide precise results for the model with a θ-term — it is therefore one of the rare examples, where a system with a complex action can be solved numerically. In particular we measure the correlation length, as well as the topological and magnetic susceptibility. We discuss the algorithmic efficiency in view of the critical slowing down. Due to the excellent performance that we observe, it is strongly motivated to work on new applications of meron cluster algorithms in higher dimensions.
We apply a meron cluster algorithm to the XY spin chain, which describes a quantum rotor. This is a multi-cluster simulation supplemented by an improved estimator, which deals with objects of half-integer topological charge. This method is powerful enough to provide precise results for the model with a θ-term — it is therefore one of the rare examples, where a system with a complex action can be solved numerically. In particular we measure the correlation length, as well as the topological and magnetic susceptibility. We discuss the algorithmic efficiency in view of the critical slowing down. Due to the excellent performance that we observe, it is strongly motivated to work on new applications of meron cluster algorithms in higher dimensions.
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.