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

TL;DR

This work investigates the continuity between the weakly coupled Ünsal-Dunne regime at small $L_0$ and the strongly coupled large-$L_0$ regime in the two-dimensional $SU(N)\times SU(N)$ Principal Chiral Model using first-principles lattice simulations with both periodic and $ZN$-symmetric twisted boundary conditions. By measuring the static correlation length $\xi$, mean energy $E$, and specific heat $C$ across varying $L_0$ (and thus $\rho=NL_0$ for twisted BC) and applying Gradient Flow to expose non-perturbative saddles, the study reveals a weak crossover or transition in both setups, with a clear $N$-dependence for periodic BC and a universal $\rho$-scaling under twists. The findings identify uniton- and fracton-like saddles and demonstrate emergent topology under twisted boundary conditions, supporting, but not conclusively proving, adiabatic continuity between regimes and offering insights into volume independence and large-$N$ dynamics in the PCM. These results inform the applicability of resurgent trans-series and the continuity conjecture to PCM-like theories and illuminate the role of boundary conditions in non-perturbative dynamics.

Abstract

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.

Figures (8)

• Figure 1: Relative change of correlation length $\Delta \xi/ \xi_0$ as a function of compactification length $L_0$ for different boundary conditions and values of $N$. In the plots at the top, we illustrate the dependence of the correlation length on the natural compactification scales: $L_0$ for PBC and $N L_0$ for TBC. The plots at the bottom illustrate how the peak in the correlation length with TBC shifts to smaller $L_0$ as $N$ is increased. For periodic boundary conditions, extrapolations to infinite $N$ are obtained using the fits of the form (\ref{['eq:observable_N_fit']}).
• Figure 2: Volume dependence of the correlation length $\xi$ in the vicinity of the peak for periodic and twisted boundary conditions for $N = 18$.
• Figure 3: Relative changes of the mean energy $\Delta E / E_0$ (left plot) and the specific heat $\Delta C / C_0$ (right plot) as functions of compactification length $L_0$ for different values of $N$ and boundary conditions. The $N \rightarrow \infty$ data for the mean energy are obtained using extrapolations of the form $E(N) = \tilde{E} + c_1 / N^2 + c_2 / N^4$ at fixed $L_0$.
• Figure 4: Relative change of mean energy $\Delta E/ E_0$ for twisted boundary conditions and different values of $N$ represented as a function of $\rho \equiv N L_0$.
• Figure 5: Dependence of the total action $S$ of smoothed configurations $U \left( \mathbf{x}, \tau \right)$ (\ref{['eq:gradient_flow']}) on the Gradient Flow time $\tau$ with periodic boundary conditions for $N = 6, \, 9$. Multiple solid lines represent independent Gradient Flows with different initial conditions $U \left( \mathbf{x}, \tau = 0 \right) = U \left( \mathbf{x} \right)$ chosen randomly from field configurations generated by Monte-Carlo process. The black dashed line represents the continuum action $S_u = 8 \pi \beta N$ of the uniton.