Tensor renormalization group approach to classical dimer models

TL;DR

This paper applies tensor renormalization group techniques to classical dimer models on square and triangular lattices, comparing numerical results with exact and Monte Carlo methods to evaluate TRG's effectiveness in different correlation regimes.

Contribution

It demonstrates the effectiveness and limitations of TRG in analyzing classical dimer models, especially highlighting its efficiency for gapped systems and challenges with algebraic correlations.

Findings

TRG efficiently describes gapped systems with exponential decay.

TRG convergence is slow and unstable for algebraic correlations.

Results are benchmarked against exact Pfaffian and Monte Carlo methods.

Abstract

We analyze classical dimer models on the square and 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. TRG 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 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 TRG approach by contrasting the results for small and large system sizes against the exact calculations. Furthermore, we benchmark our TRG results with classical Monte Carlo (MC) method.