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

P. V. Buividovich, S. N. Valgushev|arXiv (Cornell University)|Jun 27, 2017

Theoretical and Computational Physics4 citations

TL;DR

This lattice study investigates the two-dimensional SU(N)×SU(N) Principal Chiral Model with periodic and Z_N-twisted boundary conditions to test the adiabatic continuity conjecture across compactification lengths L₀. Using thermodynamic observables and Gradient Flow, it finds evidence for a weak crossover—marked by a peak in static correlation length—at intermediate NL₀, with stronger N-dependence in periodic cases, suggesting a possible finite-temperature transition in the large-N limit, while twisted boundary conditions show stable, topology-emergent unitons and a non-trivial crossover at the Dunne-Ünsal regime transition.

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.

Motivation & Objective

• To test the adiabatic continuity conjecture in the 2D SU(N)×SU(N) Principal Chiral Model by studying the behavior across compactification lengths L₀.
• To investigate whether a phase transition or crossover separates the strong-coupling regime (large L₀) from the weak-coupling Dunne-Ünsal regime (small L₀).
• To compare the effects of periodic versus Z_N-twisted boundary conditions on thermodynamic and non-perturbative structure.
• To probe the nature of non-perturbative saddle points (unitons) using the Gradient Flow method and assess their role in the crossover.
• To determine whether the observed correlation length enhancement signals a true phase transition or a weak crossover, especially in the large-N limit.

Proposed method

• Perform first-principle lattice simulations of the 2D SU(N)×SU(N) PCM on a cylinder R×S¹ with variable compactification length L₀.
• Implement both periodic and Z_N-symmetric twisted boundary conditions to compare their effects on phase structure and thermodynamics.
• Compute universal observables: mean energy, specific heat, and static correlation length to detect signatures of crossover or phase transition.
• Apply the Gradient Flow to extract non-perturbative structures, identifying localized saddle points with quantized action scaling linearly with N.
• Analyze the spatial and N-dependence of correlation length peaks and IPR (inverse participation ratio) to detect structural changes in non-perturbative configurations.
• Map the transition region to the Dunne-Ünsal regime by comparing the behavior of unitons under twisted vs. periodic boundary conditions.

Experimental results

Research questions

• RQ1 Does a finite-temperature phase transition or weak crossover occur in the 2D SU(N)×SU(N) PCM as compactification length L₀ varies?
• RQ2 How do periodic and Z_N-twisted boundary conditions affect the thermodynamic and correlation properties of the model?
• RQ3 Is the observed enhancement in static correlation length at intermediate NL₀ indicative of a true phase transition or a weak crossover?
• RQ4 Do non-perturbative saddle points (unitons) undergo a structural transition at the same L₀ where correlation length peaks?
• RQ5 Can the adiabatic continuity conjecture between the strong-coupling and Dunne-Ünsal regimes be supported given the observed crossover behavior?

Key findings

• For periodic boundary conditions, the static correlation length exhibits a peak at intermediate NL₀ that grows in height and narrows with increasing N, suggesting a possible finite-temperature phase transition in the large-N limit.
• For twisted boundary conditions, the correlation length peak is independent of N and lattice volume, indicating a weak crossover rather than a true phase transition.
• The peak in the static correlation length for periodic boundary conditions becomes more pronounced with larger spatial lattice volumes, supporting the existence of a genuine transition in the thermodynamic limit.
• Gradient Flow reveals localized non-perturbative objects with quantized action scaling linearly with N, consistent with unitons, which become effectively stable under twisted boundary conditions.
• The geometric structure of non-perturbative saddles in the twisted case changes precisely at the compactification length where the correlation length peak occurs, indicating a non-trivial transition in the saddle-point structure.
• The coincidence of the IPR peak with the correlation length peak in the twisted case suggests a non-trivial rearrangement of non-perturbative configurations, though not conclusive evidence for a phase transition.