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Bulletin of the American Physical Society

Virial expansion for the Tan contact and Beth-Uhlenbeck formula from 2D SO(2,1) anomalies

W. S. Daza, Chris L. Lin, CARLOS R. ORDÓÑEZ

Published Mar 9, 2018• 0 citations
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The relationship between two-dimensional SO(2,1)\text{SO}(2,1)SO(2,1) conformal anomalies in nonrelativistic systems and the virial expansion is explored using recently developed path-integral methods. In the process, the Beth-Uhlenbeck formula for the shift of the second virial coefficient $\ensuremath{\delta}{b}_{2}$ is obtained, as well as a virial expansion for the Tan contact. A possible extension of these techniques for higher orders in the virial expansion is discussed.
In a non-degenerated gas with low density and high temperature, the fugacity is a small parameter. The equation of state can be thus efficiently approached by the virial expansion. Since the n-order virial coefficient is known from the solution of the n-body problem, the virial expansion built a bridge between thermodynamics and few-body physics. In this talk we study the 2nd order virial coefficient for a gas trapped into a 1D or a 2D harmonic waveguide (shape of a cigar or a pancake). Interactions are modeled by a quantitative two-channel model that include the coupling between atoms and Feshbach molecules. We defined in this context the scattering phase-shift and we derive from a diagrammatic approach a Beth-Uhlenbeck formula that takes into account the presence of the molecules and the external waveguide.
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 ϵB→0 (where ϵB 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.