Large momentum part of a strongly correlated Fermi gas (Q2518408)

The paper investigates the momentum distribution of a two-component Fermi-gas with large scattering length demonstrating a tail proportional to 1/k in fourth power, where \(k\) is the wave vector. The paper states that two quantities, independently measured in experiments, the magnitude of the above-mentioned tail and the adiabatic derivative of energy are connected by a simple, exact and universal relationship named by the author the adiabatic sweep theorem. This theorem holds whenever the effective interaction radius is negligible compared to the other relevant length scales of the problem: the scattering length, the average interparticle spacing, the thermal de Broglie wave length, and the characteristic length scale over which the external potential is inhomogeneous. In this limit, the interaction between two fermions in opposite spin states is a \(s\)-wave contact interaction, characterized by the scattering length only, and there is no interaction between fermions in the same spin state because of Fermi statistics and a centrifugal barrier. First, based on the system's Hamiltonian this theorem is proved in the case of any pure energy eigenstate. The conventional proof of the theorem is demonstrated in the two-body case, and then a sketched the extension to the \(N\)-body case. Further, the implications of the adiabatic sweep theorem are presented in the cases of (i) Bose-Einstein condensate (BEC) Bardeen-Cooper-Schrieffer superfluid (BCS) crossover by considering the uniform Fermi gas with equal populations of the two spin states at zero temperature, (ii) large momentum part of the trapped gas with discussion how to directly test the adiabatic sweep theorem in a trap at any temperature, (iii) consideration of the thermodynamic properties at any temperature. In the last case, it is proved the pressure theorem which states a relationship between the pressure of the uniform Fermi gas with large scattering length and equal populations of the two spin states in thermal equilibrium. Finally, it is proved the dynamic sweep theorem showing that the energy expectation value of the system changes at a rate. This theorem holds for any populations of the two spin states, any nonzero scattering length, any quantum state, and any sweep rate, provided that the s-wave zero-range interaction model is still valid.