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

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 $\theta$ term in the strong coupling phase, which goes beyond the Eguchi-Kawai reduction.

40 pages, 19 figures

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• GrassmannTN: a Python package for Grassmann tensor network computations
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• Improving efficiency of the path optimization method for a gauge theory
• Tensor renormalization group study of (1+1)-dimensional U(1) gauge-Higgs model at $\theta =\pi$ with Lüscher's admissibility condition
• Matrix product decomposition for two- and three-flavor Wilson fermions: Benchmark results in the lattice Gross-Neveu model at finite density
• Evidence of a CP broken deconfined phase in 4D SU(2) Yang-Mills theory at $\theta =\pi$ from imaginary $\theta$ simulations
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• Grassmann tensor renormalization group approach to $(1+1)$-dimensional two-color lattice QCD at finite density
• Spectroscopy with the tensor renormalization group method
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• Triad representation for the anisotropic tensor renormalization group in four dimensions
• Forward-mode automatic differentiation for the tensor renormalization group and its relation to the impurity method
• Grassmann tensor renormalization group for the massive Schwinger model with a $\theta$ term using staggered fermions