Lattice study of continuity and finite-temperature transition in two-dimensional SU(N) x SU(N) Principal Chiral Model

We present first-principle lattice study of the two-dimensional SU(N) x SU(N) Principal Chiral Model (PCM) on the cylinder R x S1 with variable compactification length L0 of S1 and with both periodic and ZN-symmetric twisted boundary conditions. For both boundary conditions our numerical results can be interpreted as signatures of a weak crossover or phase transition between the regimes of small and large L0. In particular, at small L0 thermodynamic quantities exhibit nontrivial dependence on L0, and the static correlation length exhibits a weak enhancement at some "critical" value of L0. We also observe important differences between the two boundary conditions, which indicate that the transition scenario is more likely in the periodic case than in the twisted one. In particular, the enhancement of correlation length for periodic boundary conditions becomes more pronounced at large N, and practically does not depend on N for twisted boundary conditions. Using Gradient Flow we study non-perturbative content of the theory and find that the peaks in the correlation length appear when the length L0 becomes comparable with the typical size of unitons, unstable saddle points of PCM. With twisted boundary conditions these saddle points become effectively stable and one-dimensional in the regime of small N L0, whereas at large N L0 they are very similar to the two-dimensional unitons with periodic boundary conditions. In the context of adiabatic continuity conjecture for PCM with twisted boundary conditions, our results suggest that while the effect of the compactification is clearly different for different boundary conditions, one still cannot exclude the possibility of a weak crossover separating the strong-coupling regime at large N L0 and the Dunne-Unsal regime at small N L0 with twisted boundary conditions.