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arXiv:1609.08342 · Communications in Mathematical Physics

Stability of a Fermionic N + 1 Particle System with Point Interactions

Thomas Moser, Robert Seiringer

Published Aug 14, 2017• 22 citations
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Abstract

We prove that a system of N fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical m *. The value of m * is independent of N and turns out to be less than 1. This fact has important implications for the stability of the unitary Fermi gas. We also characterize the domain of the Hamiltonian of this model, and establish the validity of the Tan relations for all wave functions in the domain.

DOI
10.1007/s00220-017-2980-0

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We prove that a system of N fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical m *. The value of m * is independent of N and turns out to be less than 1. This fact has important implications for the stability of the unitary Fermi gas. We also characterize the domain of the Hamiltonian of this model, and establish the validity of the Tan relations for all wave functions in the domain.
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We give a bound on the ground-state energy of a system of N non-interacting fermions in a three-dimensional cubic box interacting with an impurity particle via point interactions. We show that the change in energy compared to the system in the absence of the impurity is bounded in terms of the gas density and the scattering length of the interaction, independently of N. Our bound holds as long as the ratio of the mass of the impurity to the one of the gas particles is larger than a critical value m∗∗≈0.36{m^{**}}\approx 0.36m∗∗≈0.36 , which is the same regime for which we recently showed stability of the system.
We give a lower bound on the ground state energy of a system of two fermions of one species interacting with two fermions of another species via point interactions. We show that there is a critical mass ratio m2 ≈ 0.58 such that the system is stable, i.e., the energy is bounded from below, for m∈[m2,m2−1]m \in [m_{2}, m_{2}^{-1}]m∈[m2​,m2−1​] . So far it was not known whether this 2 + 2 system exhibits a stable region at all or whether the formation of four-body bound states causes an unbounded spectrum for all mass ratios, similar to the Thomas effect. Our result gives further evidence for the stability of the more general N + M system.
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The energy density of an interacting Fermi gas in a harmonic potential is derived within localdensity approximation framework,in which the gas is assumed at each point to be locally homogenous. Based on the derived expressions,the thermodynamic stability conditions of trapped gas at zero temperature are studied,and the analytical expressions of stability conditions for d=l,2,3 dimensions are given. Furthermore,the stability phase diagrams are plotted for different dimension.It is shown that it is always stable when the system has repulsive interactions or has no interactions in condition of certain density of particle number,no mater what the dimension is.However,the system with attractive interactions is stable only under the condition of weak interaction strength.For d=1 and d = 3 cases,the critical values of interaction strength have close relationship with particle number density,but as for d= 2 case,the corresponding critical value does not depend on the particle number density explicitly.