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The tensor renormalization group method is a promising approach to latticefield theories, which is free from the sign problem unlike standard Monte Carlomethods. 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 higherrank, it becomes highly nontrivial to restrict the number of representations inthe character expansion to be used in constructing the fundamental tensor. Wepropose 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, weobtain the singular-value spectrum of the fundamental tensor, which turns outto have a definite profile in the large-$N$ limit. For the U($N$) case, inparticular, we show that the large-$N$ behavior of the singular-value spectrumchanges qualitatively at the critical coupling of the Gross-Witten-Wadia phasetransition. As an interesting consequence, we find a new type of volumeindependence 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-Kawaireduction.

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