Lattice study of continuity and finite-temperature transition in $2d$ $SU(N) \times SU(N)$ Principal Chiral Model
We present first-principle lattice study of continuity conjecture in $2d$ $SU(N) \times SU(N)$ Principal Chiral Model (PCM) on $R \times S^1$ with respect to circumference $L$ of $S^1$ in the presence of $Z(N)$-preserving twist. The twist can be considered as analogous to Twisted Eguchi-Kawai reduction in lattice gauge theory. We study static correlation length and find that it exhibits a peak at finite value of $\rho \equiv N L$, the shape of which shows no dependence on $N$ if considered as a function of $\rho$. The peak separates two regions: $\rho \rightarrow \infty$ where static correlation length matches zero temperature value with periodic boundary conditions and $\rho \rightarrow 0$ where it significantly decreases. Without twist we find a signature for large $N$ finite-temperature transition where correlation length demonstrates a peak enhancing with $N$. Using Gradient flow we study non-perturbative content of the theory and find that this transition sets up at the point where typical size of uniton, unstable saddle point of PCM, becomes comparable to $L$. After imposing the twist saddle points become stable and effectively $1d$ in the region $\rho \rightarrow 0$, whereas in the opposite limit they resemble to $2d$ profile of unitons with periodic boundary conditions. The position of the peak in correlation length with twisted boundary conditions seems to coincide with the moment when $2d$ saddle points transform into effectively $1d$. Our findings suggest possible crossover at finite value of $\rho$ which might have impact on continuity conjecture in twisted PCM.