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.
Highlights
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
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
8In the case of U(N ) gauge theory at V = ∞, we find that the free energy density is independent of N, which is due to the fact that the singular value for the trivial representation is given by σtrv = C(0)N 2 for arbitrary N in the strong coupling phase λ > 2 as one can see from figure 10 (Bottom)
Summary
We consider the 2D U(N ) and SU(N ) gauge theories and rewrite the partition function in the tensor network representation. The partition function of the lattice theory is given as. Where Dr(U ) is the representation matrix of U for the representation r, and the indices α, β, γ, δ are summed over implicitly It can be seen that any given link variable. The tensor (2.23) can be considered to be associated with each link on the lattice as depicted in figure 1 (Left). After reassigning the factor βr dr for each link to the two plaquettes sharing it and reassigning the factor dr for the site n to the plaquette Pn. The indices of the fundamental tensor (T )pqrs in (2.27) are assumed to be contracted appropriately as depicted, where we impose periodic boundary conditions. Which agrees with the known exact result in ref. [30]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
