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arXiv:2110.05800 · Journal of High Energy Physics

Tensor renormalization group and the volume independence in 2D U(N) and SU(N) gauge theories

Mitsuaki Hirasawa, Akira Matsumoto, Jun Nishimura, Atis Yosprakob

Published Dec 1, 2021• 27 citations

Abstract

Abstract The tensor renormalization group method is a promising approach to lattice field theories, which is free from the sign problem unlike standard Monte Carlo methods. One of the remaining issues is the application to gauge theories, which is so far limited to U(1) and SU(2) gauge groups. In the case of higher rank, it becomes highly nontrivial to restrict the number of representations in the character expansion to be used in constructing the fundamental tensor. We propose a practical strategy to accomplish this and demonstrate it in 2D U(N) and SU(N) gauge theories, which are exactly solvable. Using this strategy, we obtain the singular-value spectrum of the fundamental tensor, which turns out to have a definite profile in the large-N limit. For the U(N) case, in particular, we show that the large-N behavior of the singular-value spectrum changes qualitatively at the critical coupling of the Gross-Witten-Wadia phase transition. As an interesting consequence, we find a new type of volume independence in the large-N limit of the 2D U(N) gauge theory with the θ term in the strong coupling phase, which goes beyond the Eguchi-Kawai reduction.

DOI: 10.1007/jhep12(2021)011

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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.

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 propose a novel tensor network representation for two-dimensional Yang-Mills theories with arbitrary compact gauge groups. In this method, tensor indices are directly given by group elements with no direct use of the character expansion. We apply the tensor renormalization group method to this tensor network for

SU(2)

and

SU(3)

, and find that the free energy density and the energy density are accurately evaluated. We also show that the singular value decomposition of a tensor has a group theoretic structure and can be associated with the character expansion.

Abstract We make an analysis of the two-dimensional U(1) lattice gauge theory with a θ term by using the tensor renormalization group. Our numerical result for the free energy shows good consistency with the exact one at finite coupling constant. The topological charge density generates a finite gap at θ = π toward the thermodynamic limit. In addition finite size scaling analysis of the topological susceptibility up to V = L × L = 1024 × 1024 allows us to determine the phase transition at θ = π is the first order.

Abstract We propose a novel tensor network representation for two-dimensional Yang–Mills theories with arbitrary compact gauge groups. In this method, tensor indices are given directly by group elements with no direct use of the character expansion. We apply the tensor renormalization group method to this tensor network for SU(2) and SU(3), and find that the free energy density and the energy density are accurately evaluated. We also show that the singular value decomposition of a tensor has a group-theoretic structure and can be associated with the character expansion.

Abstract We perform a tensor renormalization group simulation of non-Abelian gauge theory in three dimensions using a formulation based on the “armillary sphere”. In this formulation, matrix indices are completely traced out, eliminating the degeneracy in the singular-value spectrum of the initial tensor. We demonstrate the usefulness of this technique by computing the average plaquette at zero temperature and the Polyakov-loop susceptibility at finite temperatures for the 2+1D SU(2) and SU(3) gauge theories. The deconfinement transition is identified for both gauge groups, with the SU(2) case being consistent with previous Monte Carlo results.