Discrete Scale-Invariant Boson-Fermion Duality in One Dimension

Abstract

We introduce models of one-dimensional $n(\ge 3)$-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 Schrödinger equation exactly; we obtain the exact $n$-body bound-state spectrum as well as the exact $n$-body S-matrix elements for arbitrary $n\ge 3$, 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.

15 pages, 6 eepic figures; typos corrected, references added, discussions improved

References in corpus (1)

• A Generalization of the One-Dimensional Boson-Fermion Duality Through the Path-Integral Formalism

Cited by in corpus (4)

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• Efimov effect for two particles on a semi-infinite line
• Duality as a Method to Derive a Gauge Invariant Massive Electrodynamics and New Interactions