Discrete Scale-Invariant Boson-Fermion Duality in One Dimension

TL;DR

This paper introduces exactly solvable models of one-dimensional many-body systems that exhibit a phase transition from continuous to discrete scale invariance, providing explicit spectra and S-matrices for both bosons and fermions.

Contribution

It classifies scale-invariant two-body interactions, establishes a criterion for scale invariance breaking, and solves the many-body problem exactly in the broken phase.

Findings

Exact $n$-body spectra with discrete scale invariance

Explicit $n$-body S-matrix elements

Phase diagram for scale-invariance breaking

Abstract

We introduce models of one-dimensional $n(\geq3)$-body problems that undergo phase transition from a continuous scale-invariant phase to a discrete scale-invariant phase. In this paper, we focus on identical spinless particles that interact only through two-body contacts. Without assuming any particular cluster-decomposition property, we first classify all possible scale-invariant two-body contact interactions that respect unitarity, permutation invariance, and translation invariance in one dimension. We then present a criterion for the breakdown of continuous scale invariance to discrete scale invariance. Under the assumption that the criterion is met, we solve the many-body Schr\"{o}dinger equation exactly; we obtain the exact $n$-body bound-state spectrum as well as the exact $n$-body S-matrix elements for arbitrary $n\geq3$, all of which enjoy discrete scale invariance or log-periodicity. Thanks to the boson-fermion duality, these results can be applied equally well to both bosons and fermions. Finally, we demonstrate how the criterion is met in the case of $n=3$; we determine the exact phase diagram for the scale-invariance breaking in the three-body problem of identical bosons and fermions. The zero-temperature transition from the unbroken phase to the broken phase is the Berezinskii-Kosterlitz-Thouless-like transition discussed in the literature.