High density limit of the Fermi polaron with infinite mass
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We analyze the ground state energy for N identical fermions in a two-dimensional box of volume \(L^2\) interacting with an external point scatterer. Since the point scatterer can be considered as an impurity particle of infinite mass, this system is a limit case of the Fermi polaron. We prove that its ground state energy in the limit of high density \(N/L^2\gg 1\) is given by the polaron energy. The polaron energy is an energy estimate based on trial states up to first order in particle-hole expansion, which was proposed by Chevy (Phys Rev A 74:063628, 2006) in the physics literature. The relative error in our result is shown to be small uniformly in L. Hence, we do not require a gap of fixed size in the spectrum of the Laplacian on the box. The strategy of our proof relies on a twofold Birman–Schwinger type argument applied to the many-particle Hamiltonian of the system.
KeywordsPoint interaction Fermi polaron Energy asymptotics Thermodynamic limit
Mathematics Subject ClassificationPrimary 81V70 Secondary 35P15
We thank Marcel Griesemer for extended discussions and helpful remarks. Our work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group 1838: Spectral Theory and Dynamics of Quantum Systems.
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