Quantum anomaly and thermodynamics of one-dimensional fermions with antisymmetric two-body interactions
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 Schrödinger 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 $ε_B\rightarrow 0$ (where $ε_B$ 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.