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Physical review. E

Tensor renormalization group study of hard-disk models on a triangular lattice

S. S. Akimenko, Vitaly A. Gorbunov, A. V. Myshlyavtsev, P. V. Stishenko

Published Aug 7, 2019• 20 citations
Publisher

Abstract

High accuracy and performance of the tensor renormalization group (TRG) method have been demonstrated for the model of hard disks on a triangular lattice. We considered a sequence of models with disk diameter ranging from a to 2sqrt[3]a, where a is the lattice constant. Practically, these models are good for approximate description of thermodynamics properties of molecular layers on crystal surfaces. Theoretically, it is interesting to analyze if and how this sequence converges to the continuous model of hard disks. The dependencies of the density and heat capacity on the chemical potential were calculated with TRG and transfer-matrix (TM) methods. We benchmarked accuracy and performance of the TRG method comparing it with TM method and with exact result for the model with nearest-neighbor exclusions (1NN). The TRG method demonstrates good convergence and turns out to be superior over TM with regard to considered models. Critical values of chemical potential (μ_{c}) have been computed for all models. For the model with next-nearest-neighbor exclusions (2NN) the TRG and TM produce consistent results (μ_{c}=1.75587 and μ_{c}=1.75398 correspondingly) that are also close to earlier Monte Carlo estimation by Zhang and Deng. We found that 3NN and 5NN models shows the first-order phase transition, with close values of μ_{c} (μ_{c}=4.4488 for 3NN and 4.4<μ_{c}<4.5 for 5NN). The 4NN model demonstrates continuous yet rapid phase transition with 2.65<μ_{c}<2.7.

DOI
10.1103/physreve.100.022108

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High accuracy and performance of the tensor renormalization group (TRG) method have been demonstrated for the model of hard disks on a triangular lattice. We considered a sequence of models with disk diameter ranging from a to 2sqrt[3]a, where a is the lattice constant. Practically, these models are good for approximate description of thermodynamics properties of molecular layers on crystal surfaces. Theoretically, it is interesting to analyze if and how this sequence converges to the continuous model of hard disks. The dependencies of the density and heat capacity on the chemical potential were calculated with TRG and transfer-matrix (TM) methods. We benchmarked accuracy and performance of the TRG method comparing it with TM method and with exact result for the model with nearest-neighbor exclusions (1NN). The TRG method demonstrates good convergence and turns out to be superior over TM with regard to considered models. Critical values of chemical potential (μ_{c}) have been computed for all models. For the model with next-nearest-neighbor exclusions (2NN) the TRG and TM produce consistent results (μ_{c}=1.75587 and μ_{c}=1.75398 correspondingly) that are also close to earlier Monte Carlo estimation by Zhang and Deng. We found that 3NN and 5NN models shows the first-order phase transition, with close values of μ_{c} (μ_{c}=4.4488 for 3NN and 4.4<μ_{c}<4.5 for 5NN). The 4NN model demonstrates continuous yet rapid phase transition with 2.65<μ_{c}<2.7.
The representation of complex lattice models in the form of a tensor network is a promising approach to the analysis of the thermodynamics of such systems. Once the tensor network is built, various methods can be used to calculate the partition function of the corresponding model. However, it is possible to build the initial tensor network in different ways for the same model. In this work, we have proposed two ways of constructing tensor networks and demonstrated that the construction process affects the accuracy of calculations. For demonstration purposes, we have done a brief study of the 4 nearest-neighbor (NN) and 5NN models, where adsorbed particles exclude all sites up to the fourth and fifth nearest neighbors from being occupied by another particle. In addition, we have studied a 4NN model with finite repulsions with a fifth neighbor. In a sense, this model is intermediate between 4NN and 5NN models, so algorithms designed for systems with hard-core interactions may experience difficulties. We have obtained adsorption isotherms, as well as graphs of entropy and heat capacity for all models. The critical values of the chemical potential were determined from the position of the heat capacity peaks. As a result, we were able to improve our previous estimate of the position of the phase transition points for the 4NN and 5NN models. And in the model with finite interactions, we found the presence of two first-order phase transitions and made an estimate of the critical values of the chemical potential for them.
The recently developed tensor renormalization-group (TRG) method provides a highly precise technique for deriving thermodynamic and critical properties of lattice Hamiltonians. The TRG is a local coarse-graining transformation, with the elements of the tensor at each lattice site playing the part of the interactions that undergo the renormalization-group flows. These tensor flows are directly related to the phase diagram structure of the infinite system, with each phase flowing to a distinct surface of fixed points. Fixed-point analysis and summation along the flows give the critical exponents, as well as thermodynamic functions along the entire temperature range. Thus, for the ferromagnetic triangular lattice Ising model, the free energy is calculated to better than 10(-5) along the entire temperature range. Unlike previous position-space renormalization-group methods, the truncation (of the tensor index range D) in this general method converges under straightforward and systematic improvements. Our best results are easily obtained with D=24, corresponding to 4624-dimensional renormalization-group flows.
We show that the Tensor Renormalization Group (TRG) method can be applied to O(N) spin models, principal chiral models and pure gauge theories (Z2, U(1) and SU(2)) on (hyper) cubic lattices.We explain that contrarily to some common belief, it is very difficult to write compact formulas expressing the blockspinning of lattice models.We show that in contrast to other approaches, the TRG formulation allows us to write exact blocking formulas with numerically controllable truncations.The basic reason is that the TRG blocking separates neatly the degrees of freedom inside the block and which are integrated over, from those kept to communicate with the neighboring blocks.We argue that the TRG is a method that can handle large volumes, which is crucial to approach quasi-conformal systems.The method can also get rid of some sign problems.We discuss recent results regarding the critical properties of the 2D O(2) nonlinear sigma model with complex β and chemical potential.As some of these results appeared in a recently published paper (PRD 88, 056005) and two recent preprints (
The tensor-network renormalization group (TNRG) is an accurate numerical real-space renormalization group method for studying phase transitions in both quantum and classical systems. Continuous phase transitions, as an important class of phase transitions, are usually accompanied by spontaneous breaking of various symmetries. However, the understanding of symmetries in the TNRG is well-established mainly for global on-site symmetries like U(1) and SU(2). In this paper, we demonstrate how to incorporate lattice symmetries (including reflection and rotation) and the PT symmetry in the TNRG in two dimensions (2D) through a case study of the hard-square lattice gas with nearest-neighbor exclusion. This model is chosen because it is well-understood and has two continuous phase transitions whose spontaneously-broken symmetries are lattice and PT symmetries. Specifically, we write down proper definitions of these symmetries in a coarse-grained tensor network and propose a TNRG scheme that incorporates these symmetries. We demonstrate the validity of the proposed method by estimating the critical parameters and the scaling dimensions of the two phase transitions of the model. The technical development in this paper has made the 2D TNRG a more well-rounded numerical method.
This thesis describes the Tensor Renormalization Group (TRG) method used to solve the Z5 model in lattice gauge theory. The TRG method provides exact blocking for the models and is much less expansive in terms of computational costs. Here we consider the Z5 model in two dimensions, we discuss the formulation of the tensor and the process we used to do the TRG blocking, in order to find the phase transition and critical behavior of the model. We give some simple results and plots, and some expectation for further research in the model.
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.