Topological charge in1+1dimensional latticeϕ4theory

Abstract

We investigate the topological charge in 1 1 dimensional 4 theory on a lattice with antiperiodic 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 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.

Effects of boundary conditions and gradient flow in ( 1+1 )-dimensional lattice ϕ4 theory

“…For the scalar field, for example, one can have anti-periodic boundary condition in the spatial direction (APBC). In the latter case one can study quantum kinks [2]. Lattice Quantum ChromoDynamics (LQCD) conventionally uses PBC in both the temporal and spatial directions for the gauge field.…”

Layered Phase Investigations

“…It would be interesting to study the fluctuations of this charge, as expressed by its susceptibility, for example, by taking into account the coupling of the scalar field with the fermions, along the lines, for instance, of ref. [9]. This brings us naturally to consider scalar fields interacting with anisotropic gauge fields and this has, indeed, been done in the context of the Abelian Higgs model [10], where we mapped the phase diagram and used the susceptibility to find a continuous phase transition between the bulk and layered Higgs phases.…”