General relations for quantum gases in two and three dimensions: Two-component fermions

Abstract

We derive exact general relations between various observables for N spin-1/2 fermions with zerorange or short-range interactions, in continuous space or on a lattice, in two or three dimensions, in an arbitrary external potential. Some of our results generalize known relations between the large-momentum behavior of the momentum distribution, the short-distance behaviors of the pair distribution function and of the one-body density matrix, the norm of the regular part of the wavefunction, the derivative of the energy with respect to the scattering length or to time, and the interaction energy (in the case of finite-range interactions). The expression relating the energy to a functional of the momentum distribution is also generalized. Moreover, expressions are found (in terms of the regular part of the wavefunction) for the derivative of the energy with respect to the effective range re in 3D, and to the effective range squared in 2D. They express the fact that the leading corrections to the eigenenergies due to a finite interaction-range are linear in the effective range in 3D (and in its square in 2D) with model-independent coefficients. There are subtleties in the validity condition of this conclusion, for the 2D continuous space (where it is saved by factors that are only logarithmically large in the zero-range limit) and for the 3D lattice models (where it applies only for some magic dispersion relations on the lattice, that sufficiently weakly break Galilean invariance and that do not have cusps at the border of the first Brillouin zone; an example of such relations is constructed). Furthermore, the subleading short distance behavior of the pair distribution function and the subleading 1/k 6 tail of the momentum distribution are related to ∂E/∂re (or to ∂E/∂(r 2 e ) in 2D). The second order derivative of the energy with respect to the inverse (or the logarithm in the two-dimensional case) of the scattering length is found to be expressible, for any eigenstate, in terms of the eigenwavefunctions' regular parts; this implies that, at thermal equilibrium, this second order derivative, taken at fixed entropy, is negative. Applications of the general relations are presented: We compute corrections to exactly solvable two-body and three-body problems and find agreement with available numerics; for the unitary gas in an isotropic harmonic trap, we determine how the finite-1/a and finite range energy corrections vary within each energy ladder (associated to the SO(2, 1) dynamical symmetry) and we deduce the frequency shift and the collapse time of the breathing mode; for the bulk unitary gas, we compare to fixed-node Monte Carlo data, and we estimate the deviation from the Bertsch parameter due to the finite interaction range in typical experiments.PACS numbers: 67.85. Lm,31.15.ac I. GENERAL INTRODUCTIONThe experimental breakthroughs of 1995 having led to the first realization of a Bose-Einstein condensate in an atomic vapor [1][2][3] have opened the era of experimental studies of ultracold gases with non-negli...

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