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TENSOR RENORMALIZATION GROUP METHOD STUDY IN Z5 MODEL

Zhuohao Liu

Published Dec 15, 2017• 0 citations
OpenAlex

Abstract

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.

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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 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 (
We report recent progress on the application of the tensor renormalization group (TRG) to quantum field theories pursued by the Tsukuba group. We explain how to treat the scalar, fermion, and gauge theories with the TRG method presenting the results for the phase transitions in the (3+1)-dimensional ((3+1)ddd) complex ϕ4\phi^4ϕ4 theory at finite density, (1+1)ddd pure U(1) lattice gauge theory with a θ\thetaθ term, (3+1)ddd Nambu--Jona-Lasinio model at finite density and (1+1)ddd and (2+1)ddd Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.
The phase structure of the lattice CP(N−1N-1N−1) model in two dimensions is analyzed by the tensor renormalization group (TRG) method. We focus on the case N=2N=2N=2 and compare the numerical result of the TRG method with that of the strong-coupling analysis in the presence of the θ\thetaθ term and investigate the nature of the phase transition at θ=π\theta=\piθ=π.
Abstract The tensor renormalization group (TRG) is a highly effective method for calculating the free energy of lattice models and has demonstrated exceptional accuracy for several low-dimensional systems. However, its application to higher-dimensional models is often hampered by a large memory footprint and high computational cost. In this paper, we propose a new algorithm, “tt-lattice”, which approximates the tensor at each lattice vertex using a tensor train (tt) decomposition. The algorithm is designed to maintain the tt-format throughout the TRG computation, which significantly reduces both memory usage and computational cost. For a uniform lattice model, the computational cost of tt-lattice scales as O(d2log⁡(L)χ6)\mathcal {O}(d^2\log (L)\chi ^6)O(d2log(L)χ6), with the largest intermediate tensor size of O(χ4)\mathcal {O}(\chi ^4)O(χ4), where d is the lattice dimension, L is the linear size of each lattice axis, and χ\chiχ is the cutoff of the number of lattice edge states. To validate our algorithm, we conducted numerical experiments on a Gaussian field model, for which the exact free energy is computable in any lattice dimensions. The results demonstrate that the tt-lattice method can calculate the free energy with reasonable accuracy even for a 10D lattice model with modern computational architectures.
The analysis of the error of perturbative series by comparing it to the exact solution is an important tool to understand the non-perturbative physics of statistical models. For some toy models, a new method can be used to calculate higher order weak coupling expansion and modified perturbation theory can be constructed. However, it is nontrivial to generalize the new method to understand the critical behavior of high dimensional spin and gauge models. Actually, it is a big challenge in both high energy physics and condensed matter physics to develop accurate and efficient numerical algorithms to solve these problems. In this thesis, one systematic way named tensor renormalization group method is discussed. The applications of the method to several spin and gauge models on a lattice are investigated. theoretically, the new method allows one to write an exact representation of the partition function of models with local interactions. E.g. O(N) models, Z2 gauge models and U(1) gauge models. Practically, by using controllable approximations, results in both finite volume and the thermodynamic limit can be obtained. Another advantage of the new method is that it is insensitive to sign problems for models with complex coupling and chemical potential. Through the new approach, the Fisher's zeros of the 2D O(2) model in the complex coupling plane can be calculated and the finite size scaling of the results agrees well with the Kosterlitz-Thouless assumption. Applying the method to the O(2) model with a chemical potential, new phase diagram of the models can be obtained. The structure of the tensor language may provide a new tool to understand phase transition properties in general.
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.