Topological charge in 1+1 dimensional lattice $\phi^4$ theory

TL;DR

This paper explores the topological charge in 1+1 dimensional lattice $^4$ theory with Anti Periodic Boundary Conditions, proposing an order parameter and examining its role in phase transitions and topological charge invariance.

Contribution

It introduces a simple order parameter for lattice $^4$ theory with APBC and analyzes topological charge, linking kink condensation to phase transitions and demonstrating charge invariance after renormalization.

Findings

Proposed an effective order parameter for lattice $^4$ with APBC.

Showed topological charge invariance in the broken symmetry phase after renormalization.

Suggested kink condensation as a mechanism for phase transition.

Abstract

We investigate the topological charge in 1+1 dimensional $\phi^4$ theory on a lattice with Anti Periodic Boundary Condition (APBC) in the spatial direction. We propose a simple order parameter for the lattice theory with APBC and we demonstrate its effectiveness. Our study suggests that kink condensation is a possible mechanism for the order-disorder phase transition in the 1+1 dimensional $\phi^4$ theory. With renormalizations performed on the lattice with Periodic Boundary Condition (PBC), the topological charge in the renormalized theory is given as the ratio of the order parameters in the lattices with APBC and PBC. We present a comparison of topological charges in the bare and the renormalized theory and demonstrate invariance of the charge of the renormalized theory in the broken symmetry phase.