General relations for quantum gases in two and three dimensions. Two-component fermions
Résumé
We derive exact relations for $N$ spin-1/2 fermions with zero-range or short-range interactions, in continuous space or on a lattice, in $2D$ or in $3D$, in any external potential. Some of them generalize known relations between energy, momentum distribution $n(k)$, pair distribution function $g^{(2)}(r)$, derivative of the energy with respect to the scattering length $a$. Expressions are found for the second order derivative of the energy with respect to $1/a$ (or to $\ln a$ in $2D$). Also, it is found that the leading energy corrections due to a finite interaction range, are proportional to the effective range $r_e$ in $3D$ (and to $r_e^2$ in $2D$) with exprimable model-independent coefficients, that give access to the subleading short distance behavior of $g^{(2)}(r)$ and to the subleading $1/k^6$ tail of $n(k)$. This applies to lattice models for some magic dispersion relations, an example of which is given. Corrections to exactly solvable two-body and three-body problems are obtained. For the trapped unitary gas, the variation of the finite-$1/a$ and finite $r_e$ energy corrections within each $SO(2,1)$ energy ladder is obtained; it gives 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 experimental uncertainty on the Bertsch parameter due to a finite $r_e$.
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