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arXiv:1307.0328 · Physical Review Letters

Parallelized Quantum Monte Carlo Algorithm with Nonlocal Worm Updates

Akiko Masaki-Kato, Takafumi Suzuki, Kenji Harada, Synge Todo, Naoki Kawashima

Published Apr 10, 2014• 3 citations
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Abstract

Based on the worm algorithm in the path-integral representation, we propose a general quantum Monte Carlo algorithm suitable for parallelizing on a distributed-memory computer by domain decomposition. Of particular importance is its application to large lattice systems of bosons and spins. A large number of worms are introduced and its population is controlled by a fictitious transverse field. For a benchmark, we study the size dependence of the Bose-condensation order parameter of the hard-core Bose-Hubbard model with $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}\ensuremath{\beta}t=10240\ifmmode\times\else\texttimes\fi{}10240\ifmmode\times\else\texttimes\fi{}16$, using 3200 computing cores, which shows good parallelization efficiency.

DOI
10.1103/physrevlett.112.140603

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Based on the worm algorithm in the path-integral representation, we propose a general quantum Monte Carlo algorithm suitable for parallelizing on a distributed-memory computer by domain decomposition. Of particular importance is its application to large lattice systems of bosons and spins. A large number of worms are introduced and its population is controlled by a fictitious transverse field. For a benchmark, we study the size dependence of the Bose-condensation order parameter of the hard-core Bose-Hubbard model with L\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fi\ensuremathβt=10240\ifmmode×\else\texttimes\fi10240\ifmmode×\else\texttimes\fi16L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}\ensuremath{\beta}t=10240\ifmmode\times\else\texttimes\fi{}10240\ifmmode\times\else\texttimes\fi{}16L\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fi\ensuremathβt=10240\ifmmode×\else\texttimes\fi10240\ifmmode×\else\texttimes\fi16, using 3200 computing cores, which shows good parallelization efficiency.
We present a novel and open-source implementation of the worm algorithm, which is an algorithm to simulate Bose-Hubbard and sign-positive spin models using a path-integral representation of the partition function. The code can deal with arbitrary lattice structures and assumes spin-exchange terms, or bosonic hopping amplitudes, between nearest-neighbor sites, and local or nearest-neighbor interactions of the density-density type. We explicitly demonstrate the near-linear scaling of the algorithm with respect to the system volume and the inverse temperature and analyze the autocorrelation times in the vicinity of a U(1) U ( 1 ) second order phase transition. The code is written in such a way that extensions to other lattice models as well as closely-related sign-positive models can be done straightforwardly on top of the provided framework.
A quantum Monte Carlo method with a nonlocal update scheme is presented. The method is based on a path-integral decomposition and a worm operator which is local in imaginary time. It generates states with a fixed number of particles and respects other exact symmetries. Observables like the equal-time Green's function can be evaluated in an efficient way. To demonstrate the versatility of the method, results for the one-dimensional Bose-Hubbard model and a nuclear pairing model are presented. Within the context of the Bose-Hubbard model the efficiency of the algorithm is discussed.
Simulating long-range interacting systems is a challenging task due to its computational complexity that the computational effort for each local update is of order \mathcal{O}(N) 𝒪 ( N ) , where N N is the size of the system. In this work, we introduce the clock factorized quantum Monte Carlo method, an efficient technique for simulating long-range interacting quantum systems. The method is developed by generalizing the clock Monte Carlo method for classical systems [Phys. Rev. E 99 010105 (2019)] to the path-integral representation of long-range interacting quantum systems, with some specific treatments for quantum cases and a few significant technical improvements in general. We first explain how the clock factorized quantum Monte Carlo method is implemented to reduce the computational overhead from \mathcal{O}(N) 𝒪 ( N ) to \mathcal{O}(1) 𝒪 ( 1 ) . In particular, the core ingredients, including the concepts of bound probabilities and bound rejection events, the recursive sampling procedure, and the fast algorithms for sampling an extensive set of discrete and small probabilities, are elaborated. Next, we show how the clock factorized quantum Monte Carlo method can be flexibly implemented in various update strategies, like the Metropolis and worm-type algorithms. Finally, we demonstrate the high efficiency of the clock factorized quantum Monte Carlo algorithms using examples of three typical long-range interacting quantum systems, including the transverse field Ising model with long-range z z - z z interaction, the extended Bose-Hubbard model with long-range density-density interactions, and the XXZ Heisenberg model with long-range spin interactions. We expect that the clock factorized quantum Monte Carlo method would find broad applications in statistical and condensed-matter physics.
We present a new approach to path integral Monte Carlo (PIMC) simulations based on the worm algorithm, originally developed for lattice models and extended here to continuous-space many-body systems. The scheme allows for efficient computation of thermodynamic properties, including winding numbers and off-diagonal correlations, for systems of much greater size than that accessible to conventional PIMC simulations. As an illustrative application of the method, we simulate the superfluid transition of 4He in two dimensions.
We provide a detailed description of the path-integral Monte Carlo worm algorithm used to exactly calculate the thermodynamics of Bose systems in the canonical ensemble. The algorithm is fully consistent with periodic boundary conditions, which are applied to simulate homogeneous phases of bulk systems, and it does not require any limitation in the length of the Monte Carlo moves realizing the sampling of the probability distribution function in the space of path configurations. The result is achieved by adopting a representation of the path coordinates where only the initial point of each path is inside the simulation box, the remaining ones being free to span the entire space. Detailed balance can thereby be ensured for any update of the path configurations without the ambiguity of the selection of the periodic image of the particles involved. We benchmark the algorithm using the non-interacting Bose gas model for which exact results for the partition function at finite number of particles can be derived. Convergence issues and the approach to the thermodynamic limit are also addressed for interacting systems of hard spheres in the regime of high density.
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.