The $\theta$-vacuum from functional renormalisation

TL;DR

This paper applies the functional renormalisation group to analyze the topological properties of a quantum system with $U(1)$ symmetry, including vacuum energy and susceptibility, by complexifying the flow equations.

Contribution

It introduces a novel complexified fRG approach embedding $U(1)$ symmetry into the complex plane to compute topological sector potentials.

Findings

Results agree with Schrödinger equation benchmarks.

Effective potential constructed from sector potentials.

Method captures topological features accurately.

Abstract

We study topological properties of a quantum mechanical system with $U(1)$-symmetry within the functional renormalisation group (fRG) approach. These properties include the vacuum energy structure and the topological susceptibility. Our approach works with a complexification of the flow equation, and specifically we embed the original symmetry into the complex plane, $U(1)\rightarrow \mathbb{C}$. We compute the effective potential of a given topological sector by restricting ourselves to field configurations with a given generalised non-trivial Chern-Simons numbers. The full potential is directly constructed from these sector potentials. Our results compare well with the benchmark results obtained from solving the corresponding Schr\"odinger equation.