Parallelized Quantum Monte Carlo Algorithm with Nonlocal Worm Updates

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 $L\times L\times \beta t=10240\times 10240\times 16$, using 3200 computing cores, which shows good parallelization efficiency.

Figure 1

A vertex, its four legs, and worms (a), and creation or annihilation of worms on the temporal (b),(c),(d),(e) and on the spatial domain boundary (f). Circles and the triangle are worms whereas horizontal lines are vertices. (b) The configuration before the boundary update. (The red horizontal line is the temporal domain boundary.) (c) The initial intermediate state is $|s⟩$. (d) The intermediate state is updated. (e) The final configuration compatible to the new intermediate state is generated. (f) The vertex $w$ on the spatial domain boundary, the red vertical line.

Figure 2

Energy $E$ and order-parameter $Q$ as a function of the source field $\eta$. (a) and (b): Energy at fixed system size, temperature, and repulsive interaction ($L=128$, $\beta t=16$, $V/t=3.0$). The chemical potential is $\mu /t=4.2$ (CS phase) for (a), and $\mu /t=1.2$ (SF phase) for (b). The dashed curves represent a quadratic fitting for (a) and a linear fitting for (b). (c): Double logarithmic plot of $⟨Q⟩$ at $\beta t=16$, $\mu /t=1.2$ and $V/t=3.0$. The dashed line is the power-law fitting with $L=\infty$ data.

Figure 3

The estimated standard deviation (error) as functions of the number of domains $N$ in a SF state ($\mu /t=1.2$, $V/t=3.0$). (a) With fixed number of MC sweeps. ($L=\beta t=128$, $\eta =0.002$), (b) With fixed wall-clock time of ${10}^4$ seconds (not including thermalization). ($L=256$ and $\beta t=16$, $\eta =0.004$)