Tensor renormalization group approach to classical dimer models

We analyze classical dimer models on a square and a triangular lattice using a tensor network representation of the dimers. The correlation functions are numerically calculated using the recently developed “tensor renormalization group” (TRG) technique. The partition function for the dimer problem can be calculated exactly by the Pfaffian method, which is used here as a platform for comparing the numerical results. The TRG approach turns out to be a powerful tool for describing gapped systems with exponentially decaying correlations very efficiently due to its fast convergence. This is the case for the dimer model on the triangular lattice. However, the convergence becomes very slow and unstable in the case of the square lattice where the model has algebraically decaying correlations. We highlight these aspects with numerical simulations and critically appraise the robustness of the TRG approach by contrasting the results for small and large system sizes against the exact calculations. Furthermore, we benchmark our TRG results with the classical Monte Carlo method.

Figure 1

The square lattice with diagonal bonds and a two-site (black and red circles) unit cell. Also shown are the directions of the Kasteleyn arrows which satisfy the clockwise-odd rule described in the text.

Figure 2

(a) Dimer covering on square lattice. The arrow denotes the first move taken in the Monte Carlo approach. (b) Two defects, joined by the string, are created at the sites having no dimer and two dimers. (c) The defects move farther apart. (d) The final configuration achieved as a consequence of the successive moves becomes a valid dimer configuration when the two defects merge. The arrows indicate successive dimer moves.

Figure 3

(a, c) Tensor network representation for two-dimensional lattices. Example of a dimer state (of a site) represented by tensor network on (b) a triangular lattice and (d) a square lattice.

Figure 4

(a) Decomposing the tensor $T$ into $S_a, S_b, S_c$, and $T_a$. (b) Combining the three tensors in a dashed triangle $S_a, S_b$, and $S_c$ to form a new tensor $T_b$. The triangular lattice thus deforms into a hexagonal lattice with $T_a$ (white) and $T_b$ (black) tensors on the respective sublattices. Further grouping of the two tensors in every basis of the hexagonal lattice generates a new square lattice network of rank-4 tensors.

Figure 5

Schematic of the tensor renormalization for calculating dimer-dimer correlation function at a distance $r$. (a, c–e) The black dots represent the impurity tensors and white dots represent the uniform tensors. (b) The red dots represent the impurity tensors while white (black) dots represent the uniform tensors on respective sublattices.

Figure 6

Top: The dimer-dimer correlation on the triangular lattice as a function of the distance $r=|R_i-R_j|$. The plot shows data for a $128\times 128$ cluster as extracted from the Green function, TRG method, and MC method. In the tensor network representation the bond dimension has been truncated at ${\chi }_c=20$. Bottom: The same function is plotted for a $128\times 128$ square lattice cluster using the Green function, MC method, and TRG method with ${\chi }_c=20,48,64$. That the cutoff is kept very high for required accuracy is attributed to the critical nature of the model on the square lattice.

Figure 7

Top: The dimer concentration as a function of the cutoff for different lattice sizes on the triangular lattice. The red dotted line indicates the average value of $1/6$. Bottom: The above quantity for the square lattice. The average value in this case is $1/4$. Note the deviations from the dashed line for larger systems. The convergence for them requires a larger cutoff which grows with the system size.