Global scheme of sweeping cluster algorithm to sample among topological sectors

Local constraint is closely related to the gauge field, so constrained models are usually effective low energy descriptions and important in condensed matter physics. On the other hand, local restriction hinders the application of numerical algorithms. In addition to the computational difficulties of the constraints, the various topological sectors which cannot be connected through local operators are also one of the key computational difficulties. Taking a quantum dimer model as an example in this paper, we construct a global scheme based on a sweeping cluster Monte Carlo method, which can sample among different topological sectors. In principle, this method can be generalized to other models.

Figure 1

For the square lattice, the winding number of the $x$ axis is defined as $W_x=N_x\left(A\right)-N_x\left(B\right)$, where $N_x\left(A\right)$ and $N_x\left(B\right)$ are the number of dimers that the dashed line cuts on $A$ or $B$ links; the same as does the winding number of the $y$ axis. For the triangular lattice, the winding number is defined as 0 (even)/1 (odd), when the number of dimers that are cut by the dashed line is even/odd.

Figure 2

Left: A global loop update crosses through the boundary to change the winding number. Right: A local loop cannot change the winding number.

Figure 3

(I) Schematic diagram of an update for quantum dimer models. Each imaginary time surface is a classical dimer configuration. Red lines are update lines of world-line QMC. The blue loops are the intersection of all imaginary time update lines and each imaginary time surface which are the same as the loop in right part of Fig. 2. (II) Some examples of the vertices and their update prescriptions. The horizontal bar represents the full plaquette operator $H_p$ and the lines of the squares represent the dimer states (thick and thin lines for dimer 1 or 0) on either side of the operator. Update lines are shown as lines with an arrow. (c) and (d) are different updates of the same configuration. This figure is from Ref. [25].

Figure 4

(a) Schematic diagram of a global scheme for quantum dimer models. Each imaginary-time surface is a classical dimer configuration. Here we ignore drawing update- lines of world-line QMC for convenience. The blue loops are the intersection of all imaginary-time update lines and each imaginary-time surface, the same as the loop in the left part of Fig. 2. The evolution of the loop via operators can be seen clearly. (b) Configurations of QDM in imaginary-time space. Each picture is a dimer configuration at a certain imaginary time, and the arrows indicate the increasing imaginary time. The D and O means a diagonal and off-diagonal operator which acts on the configuration. First row and second row stand for dimer configuration snapshots before and after the global sweeping cluster algorithm update, respectively. The red D or O means the operators after update. The dashed line here means an update line exists on the link, i.e., the dimer has to be toggled on/off. The updated dimers are blue.

Figure 5

Compare energy per plaquette of QDM by three methods on a $4\times 4$ square lattice: exact diagonalization (ED) method which works in all topological sectors, original SCA QMC which only works in a certain sector [we choose the (0,0) sector here, i.e., columnar sector], and the GSCA which samples in all sectors. (a) At $T=0.01$, we see the ground state of ED and GSCA transfers from the columnar sector to the staggered sector in $V>1$ while the SCA always stays in the columnar sector. Both ED and GSCA work well near the topological first-order phase transition point ($V=1$). (b) At $V=0.5$, the energy of the GSCA which contains information of all sectors matches well with the correct result of ED, but the SCA is not right at finite temperature because the algorithm is confined in the columnar sector of the initial state. Note: The error bars are small and many are obscured by data points.

Figure 6

Left: the correlation function of QDM on a $4\times 4$ square lattice at $V=2,T=0.01$. The data are rounded to three decimal places, and the errors are in/after the fourth decimal place. It can be decomposed into those staggered configurations on the right. It is worth noting that there are two kinds of staggered configurations corresponding to different winding sectors $(0,2)$ and $(1,1)$. For convenience, here we do not distinguish between the positive and negative of the winding number, and do not distinguish between $(a,b)$ and $(b,a)$. In the $40000$ Monte Carlo steps we simulated, they accounted for 31.33% and 68.67%, respectively.

Figure 7

Because the size of a Hilbert space of different sectors is different, it is easy to randomly walk from the staggered sector to the columnar, but hard reversely, especially in large system sizes.

Figure 8

In different sizes, the relationship of energy/kinetic energy per plaquette and parameter $V$. Error bars are smaller than the data points. In order to avoid overlapping of data of different sizes, we have made corresponding shifts according to sizes. The shift value is $+0.01\times (28-L)/4$. (a) shows a clear turn at $V=1$, it means the ground state jumps between columnar and staggered sectors. Energy is equal to 0 in the error bar region when $V\ge 1$. (b) shows the kinetic energy is equal to 0 exactly while $V>1$, which is strong evidence for a staggered phase without any flippable plaquette. At $V=1$, the kinetic energy is not 0 obviously and the energy of (a) is 0, which matches with exact solution of ground state at the RK point.

Figure 9

The probability that all the update lines of the global loop close after one round along imaginary time. We simulate different sizes and temperature cases at $V=0.5$. The inset uses the same data but changes the $x$ axis as $T$ instead of $1/L$.

Figure 10

The accepted probability of a global cluster of different temperature and sizes at $V=0.5$.

Figure 11

The nearest way to connect two staggered configurations is to update the dimers along the clino-diagonal line.