Valence-bond quantum Monte Carlo algorithms defined on trees

We present a class of algorithms for performing valence-bond quantum Monte Carlo of quantum spin models. Valence-bond quantum Monte Carlo is a projective $T=0$ Monte Carlo method based on sampling of a set of operator strings that can be viewed as forming a treelike structure. The algorithms presented here utilize the notion of a worm that moves up and down this tree and changes the associated operator string. In quite general terms, we derive a set of equations whose solutions correspond to a whole class of algorithms. As specific examples of this class of algorithms, we focus on two cases. The bouncing worm algorithm, for which updates are always accepted by allowing the worm to bounce up and down the tree, and the driven worm algorithm, where a single parameter controls how far up the tree the worm reaches before turning around. The latter algorithm involves only a single bounce where the worm turns from going up the tree to going down. The presence of the control parameter necessitates the introduction of an acceptance probability for the update.

Figure 1

The branching tree of length 5 for the selection of an operator string in a system with a Hamiltonian of $N_B$ (here 6) terms. The operator that acts on the state first is chosen at the first node on the left. This node is called the root and the direction towards the root we define to be up. The operator that acts on the state last is at the end of the string. The two colored paths differ in the choice of the last three operators. The last three branches, thus, contribute different operators and weights $(s_i,t_i)$. The resulting strings $\mathsf{S}$ and $\mathsf{T}$ are different.

Figure 2

A possible path that contains one bounce and connects the string $\mathsf{S}$ and the string $\mathsf{T}$. The worm first goes up to the node 2 where it turns to go down to node 3. The worm bounces back and goes all the way to node 1. Then the worm turns around and does not bounce again.

Figure 3

The autocorrelation time of the energy as a function of system size $N$. The lines indicate tentative power-law fits to the data at large $N$ with a power of $2.21$ for $b_u=0.26$ and $2.16$ for $b_u=0.275$. Operator strings of $20000$ operators were used.

Figure 4

The scaled autocorrelation time of the energy as a function of system size $N$. The data from Fig. 3 were multiplied with $\langle l/2\rangle$ which was estimated in independent calculations. Operator strings of $20000$ operators were used.

Figure 5

With a bigger probability to bounce $b_u$, the bouncing worm algorithm yields a better approximation of the ground-state energy. The ground-state energy was calculated using the Bethe ansatz and is indicated by a dotted line. Since some operators in the string are never updated, calculations with different $b_u$ were effectively done with different trial states. The calculation was done for a chain with 50 sites. Operator strings of $20000$ operators were used and for each point ${10}^7$ measurements taken.

Figure 6

The mean and the maximal penetration depths grow as $\alpha$ approaches one. The mean penetration depth is always smaller than $\langle r\rangle =1/(1-\alpha )$. This behavior is independent of the system size. For the chain with 50 sites $3\times {10}^8$ and for the chain with 1000 sites ${10}^7$ updates were performed. Operator strings of $20000$ operators were used.

Figure 7

The bigger the penetration probability $\alpha$, the better the approximation of the ground-state energy given by the driven worm algorithm. The ground-state energy was calculated using the Bethe ansatz. It is indicated by a dotted line. Since some operators in the string are never updated, calculations with different $\alpha$ were effectively done with different trial states. The calculation was done for a chain with 50 sites. Operator strings of $20000$ operators were used and for each point ${10}^7$ measurements taken.

Figure 8

The scaled autocorrelation time decreases with $\langle r\rangle$, if typical updates are smaller than the system size. Since bigger $\langle r\rangle$ means better projection, this implies that the autocorrelation time decreases as the quality of the projection is improved. Operator strings of $20000$ operators were used.

Figure 9

Upon increasing the quality of the projection by using longer operator strings, the estimate of the ground-state energy converges to the correct value in the same way for worm and conventional VBQMC algorithms as long as the string is penetrated sufficiently deeply. The driven worm algorithm was run with the probability to go up the tree $\alpha =0.995$, which corresponds to a mean penetration depth of about 90 and full penetration of the string. The bouncing worm algorithm was run with a bounce probability $b_u=0.2675$, which corresponds to a mean penetration depth of roughly 8 and full penetration of the string. The calculation was done for a chain with 50 sites. For the worm algorithms ${10}^7$ and for the conventional VBQMC ${10}^6$ measurements were taken at each point.

Figure 10

The autocorrelation function versus scaled number of updates for the two worm algorithms and conventional VBQMC. The number of updates was scaled by the number of operators attempted to be changed in an update $\langle l/2\rangle$. Hence, data for conventional VBQMC updates, the driven worm algorithm, and the bouncing worm algorithm were multiplied by 4, 200, and $63.758$, respectively. For the worm algorithms, the same parameters as in Fig. 9 were used. This means that $\alpha =0.995$ and $b_u=0.2675$. An operator string of length 1000 was used for all three algorithms. The horizontal line at $0.1$ was added to allow for easy visual estimation of the scaled autocorrelation time.

Figure 11

The deviation $\left|\Delta E\right|$ from the exact Bethe-ansatz results for a chain with 100 sites. The results are shown for the driven and bouncing worm algorithms versus the mean penetration depth and for conventional VBQMC updates versus the projective power (length of operator string). The worm algorithms reach the same small value of $\left|\Delta E\right|$ with a mean penetration depth an order of magnitude smaller than the projective power used for the calculation with conventional VBQMC updates. The bars on the markers indicated the statistical uncertainty. The colored (dark) surfaces are due to overlapping error bars. Operator strings of $20000$ operators were used for the calculations with the worm algorithms. For each point, ${10}^5$ measurements were taken.

Figure 12

The scaled autocorrelation time for the driven and bouncing worm algorithms as a function of system size. A fixed operator string of length $20000$ was used in the calculations. In the calculations shown, the bouncing worm algorithm was run with $b_u=0.275$ and the driven worm algorithm was run with $\alpha =0.995$. For the scaling we use $\langle l/2\rangle =200$ for the driven worm algorithm and an $\langle l/2\rangle$ between 60 for $N=10$ and 221 for $N=1000$ for the bouncing worm algorithm.