# Quantum anomaly and thermodynamics of one-dimensional fermions with antisymmetric two-body interactions

## TL;DR

This paper investigates a one-dimensional fermionic system with antisymmetric two-body interactions, revealing a scale anomaly with power-law behavior, and explores its effects on bound states, scattering, and thermodynamics using both quantum mechanics and field theory.

## Contribution

It introduces a novel one-dimensional fermion model with derivative-delta interactions exhibiting a scale anomaly and analyzes its bound states, scattering properties, and thermodynamics.

## Key findings

- The reflection and transmission coefficients match those of the regular delta potential.
- The second-order virial coefficient is derived analytically.
- Quantum anomaly influences the equation of state and universal relations.

## Abstract

A system of two-species, one-dimensional fermions, with an attractive two-body interaction of the derivative-delta type, features a scale anomaly. In contrast to the well-known two-dimensional case with contact interactions, and its one-dimensional cousin with three-body interactions (studied recently by some of us and others), the present case displays dimensional transmutation featuring a power-law rather than a logarithmic behavior. We use both the Schr\"{o}dinger equation and quantum field theory to study bound and scattering states, showing consistency between both approaches. We show that the expressions for the reflection $(R)$ and the transmission $(T)$ coefficients of the renormalized, anomalous derivative-delta potential are identical to those of the regular delta potential. The second-order virial coefficient is calculated analytically using the Beth-Uhlenbeck formula, and we make comments about the proper $\epsilon_B\rightarrow 0$ (where $\epsilon_B$ is the bound-state energy) limit. We show the impact of the quantum anomaly (which appears as the binding energy of the two-body problem, or equivalently as Tan's contact) on the equation of state and on other universal relations. Our emphasis throughout is on the conceptual and structural aspects of this problem.