Some approximate analytical methods in the study of the self-avoiding loop model with variable bending rigidity and the critical behaviour of the strong coupling lattice Schwinger model with Wilson fermions

TL;DR

This paper explores approximate analytical methods to study the critical behavior of a self-avoiding loop model with variable bending rigidity, relating it to the strong coupling lattice Schwinger model and confirming its second-order phase transition in the Ising universality class.

Contribution

It applies two approximate analytical methods to analyze the critical properties of the self-avoiding loop model with variable bending rigidity, validating results against Monte Carlo data.

Findings

The Schwinger model becomes critical at r pprox 0.38-0.39.

The phase transition is second order and in the Ising universality class.

The central charge at criticality is predicted to be 1/2.

Abstract

Some time ago Salmhofer demonstrated the equivalence of the strong coupling lattice Schwinger model with Wilson fermions to a certain 8-vertex model which can be understood as a self-avoiding loop model on the square lattice with bending rigidity $\eta = 1/2$ and monomer weight $z = (2\kappa)^{-2}$. The present paper applies two approximate analytical methods to the investigation of critical properties of the self-avoiding loop model with variable bending rigidity, discusses their validity and makes comparison with known MC results. One method is based on the independent loop approximation used in the literature for studying phase transitions in polymers, liquid helium and cosmic strings. The second method relies on the known exact solution of the self-avoiding loop model with bending rigidity $\eta = 1/\sqrt{2}$. The present investigation confirms recent findings that the strong coupling lattice Schwinger model becomes critical for $\kappa_{cr} \simeq 0.38-0.39$. The phase transition is of second order and lies in the Ising model universality class. Finally, the central charge of the strong coupling Schwinger model at criticality is discussed and predicted to be $c = 1/2$.