Generalized directed loop method for quantum Monte Carlo simulations

Efficient quantum Monte Carlo update schemes called directed loops have recently been proposed, which improve the efficiency of simulations of quantum lattice models. We propose to generalize the detailed balance equations at the local level during the loop construction by accounting for the matrix elements of the operators associated with open world-line segments. Using linear programming techniques to solve the generalized equations, we look for optimal construction schemes for directed loops. This also allows for an extension of the directed loop scheme to general lattice models, such as high-spin or bosonic models. The resulting algorithms are bounce free in larger regions of parameter space than the original directed loop algorithm. The generalized directed loop method is applied to the magnetization process of spin chains in order to compare its efficiency to that of previous directed loop schemes. In contrast to general expectations, we find that minimizing bounces alone does not always lead to more efficient algorithms in terms of autocorrelations of physical observables, because of the nonuniqueness of the bounce-free solutions. We therefore propose different general strategies to further minimize autocorrelations, which can be used as supplementary requirements in any directed loop scheme. We show by calculating autocorrelation times for different observables that such strategies indeed lead to improved efficiency; however, we find that the optimal strategy depends not only on the model parameters but also on the observable of interest.

Figure 1

The vertex state ${\mathbf{\Sigma }}_p$ is equal to the direct product of the local states on its four legs: ${\mathbf{\Sigma }}_p=\mid \sigma \left(1\right)⟩\otimes \mid \sigma \left(2\right)⟩\otimes \mid \sigma \left(3\right)⟩\otimes \mid \sigma \left(4\right)⟩$.

Figure 2

The two possible ways of inserting a pair of operators on a state $s$: (a) insertion of a pair $A_0{A}_0^{†}$, (b) insertion of a pair $A_0^{†}{A}_0$.

Figure 3

The two possibilities of moving the worm after insertion of an initial pair $A_0^{†}{A}_0$. (a) The operator $A_0$ is moved upwards in propagation direction. As a result, the transformation induced on the state $s$ is $T_0=A_0^{†}$ [the new state is ${\tilde{T}}_0\left(s\right)={\tilde{A}}_0^{†}\left(s\right)$]. (b) The operator $A_0^{†}$ is moved downwards in negative propagation direction. The transformation induced on the state $s$ is again $T_0=A_0^{†}$ [the new state being ${\tilde{T}}_0\left(s\right)={\tilde{A}}_0^{†}\left(s\right)$]. Dashed horizontal lines indicate where the operators were before they were moved.

Figure 4

The two possible closings of a worm. The initial pair inserted was $A_0^{†}{A}_0$ [Fig. 2b] and the first move the propagation of $A_0$ upward in propagation direction [Fig. 3a], so that the initial transformation is $T_0=A_0^{†}$. (a) For a bounce loop, the final transformation $T_N=T_0^{†}$. (b) For a normal loop, the final transformation $T_N=T_0$. The dashed horizontal line indicates the initial position of the operator $A_0$.

Figure 5

(Color online) Worm entering the $i\text{th}$ vertex during its construction.

Figure 6

(Color online) Worm leaving the $i\text{th}$ vertex during its construction.

Figure 7

(Color online) Algorithmic phase diagram for an easy-axis anisotropic spin-$S$ Heisenberg model in a magnetic field $h$ field, on a $d$-dimensional cubic lattice with nearest-neighbor exchange $J$. The easy-axis anisotropy is denoted $\Delta$. The shaded region indicates those parameters for which a bounce-free solution of the generalized directed loop equations can be found.

Figure 8

(Color online) Algorithmic phase diagram for the bosonic Hubbard model on a $d$-dimensional cubic lattice with nearest-neighbor hopping $t$, on-site interaction strength $U$, and a chemical potential $\mu$. $N_{\max }$ denotes the cutoff in the local occupation number. The shaded region indicates the regime of bounce-free solutions of the generalized directed loop equations.

Figure 9

(Color online) Bounce probabilities using the heat bath algorithm, standard directed loops, and generalized directed loops for a $L=64$ sites spin-$3∕2$ $XY$ chain in a magnetic field $h$ at $\beta =64∕J$. A different scale is used for the heat bath algorithm.

Figure 10

(Color online) Autocorrelation times for the uniform magnetization, measured using heat bath, standard directed loops, and generalized directed loops for a $L=64$ sites spin-$3∕2$ $XY$ chain in a magnetic field $h$ at $\beta =64∕J$.

Figure 11

(Color online) Autocorrelation times for the energy, measured using heat bath, standard directed loops, and generalized directed loops for a $L=64$ sites spin-$3∕2$ $XY$ chain in a magnetic field $h$ at $\beta =64∕J$.

Figure 12

(Color online) Bounce probabilities using the heat bath algorithm, standard directed loops, and generalized directed loops for a $L=64$ sites spin-2 antiferromagnetic Heisenberg chain in a magnetic field $h$ at $\beta =64∕J$. A different scale is used for the heat bath algorithm.

Figure 13

(Color online) Autocorrelation times for the uniform magnetization, measured using the heat bath algorithm, standard directed loops, and generalized directed loops for a $L=64$ sites spin-2 antiferromagnetic Heisenberg chain in a magnetic field $h$ at $\beta =64∕J$. The inset shows on a larger scale the autocorrelation times using directed loops at small values of the field.

Figure 14

(Color online) Autocorrelation times for the energy, measured using heat bath, standard directed loops, and generalized directed loops for a $L=64$ sites spin-2 antiferromagnetic Heisenberg chain in a magnetic field $h$ at $\beta =64∕J$.

Figure 15

(Color online) The different paths a worm can take across a vertex: “bounce,” “jump,” “straight,” or “turn.”

Figure 16

(Color online) Autocorrelation times for the uniform magnetization, measured using heat bath, standard directed loops, generalized directed loops, and the optimal solution (see text) for a $L=64$ sites spin-$3∕2$ $XY$ chain in a magnetic field $h$ at $\beta =64∕J$, for $ϵ=3∕4J$.