Scale Invariance Breaking and Discrete Phase Invariance in Few-Body Problems

TL;DR

This paper explores how continuous scale invariance in quantum mechanics can be broken to discrete phase invariance, revealing new symmetry-breaking phenomena in few-body quantum systems with inverse-square potentials and magnetic fluxes.

Contribution

It introduces the concept of discrete phase invariance as a new form of symmetry breaking in quantum mechanics and provides multiple examples involving few-body problems with magnetic flux.

Findings

Continuous scale invariance can be broken to discrete phase invariance.

Discrete phase invariance appears as poles on Riemann sheets of the S-matrix.

Examples include Aharonov-Bohm problem and few-body systems with magnetic flux.

Abstract

Scale invariance in quantum mechanics can be broken in several ways. A well-known example is the breakdown of continuous scale invariance to discrete scale invariance, whose typical realization is the Efimov effect of three-body problems. Here we discuss yet another discrete symmetry to which continuous scale invariance can be broken: discrete phase invariance. We first revisit the one-body problem on the half line in the presence of an inverse-square potential -- the simplest example of nontrivial scale-invariant quantum mechanics -- and show that continuous scale invariance can be broken to discrete phase invariance in a small window of coupling constant. We also show that discrete phase invariance manifests itself as circularly distributed simple poles on Riemann sheets of the S-matrix. We then present three examples of few-body problems that exhibit discrete phase invariance. These examples are the one-body Aharonov-Bohm problem, a two-body problem of nonidentical particles in two dimensions, and a three-body problem of nonidentical particles in one dimension, all of which contain a codimension-two ``magnetic'' flux in configuration spaces.