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arXiv:0902.0949 · Physical Review A

Highly polarized Fermi gases: One-dimensional case

S. Giraud, R. Combescot

Published Apr 15, 2009• 53 citations

Abstract

We consider the problem of a single particle interacting with $N$ identical fermions, at zero temperature and in one dimension. We calculate the binding energy as well as the effective mass of the single particle. We use an approximate method developed in the three-dimensional case, where the Hilbert space for the excited states of the $N$ fermions is restricted to have at most two particle-hole pairs. When the mass of the single particle is equal to the fermion mass, we find excellent agreement with the exact results of McGuire. When the mass of the single particle is infinite, we solve exactly the problem and find again excellent agreement between approximate results and exact ones. This overall agreement in one dimension gives a strong validation for the approximate method applied in three dimensions. Moreover it shows that our approximate treatment is excellent for the one-dimensional problem in the general case with respect to the mass of the single particle.

DOI: 10.1103/physreva.79.043615

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We consider the problem of a single particle interacting with

N

identical fermions, at zero temperature and in one dimension. We calculate the binding energy as well as the effective mass of the single particle. We use an approximate method developed in the three-dimensional case, where the Hilbert space for the excited states of the

N

fermions is restricted to have at most two particle-hole pairs. When the mass of the single particle is equal to the fermion mass, we find excellent agreement with the exact results of McGuire. When the mass of the single particle is infinite, we solve exactly the problem and find again excellent agreement between approximate results and exact ones. This overall agreement in one dimension gives a strong validation for the approximate method applied in three dimensions. Moreover it shows that our approximate treatment is excellent for the one-dimensional problem in the general case with respect to the mass of the single particle.

We present a full many-body analysis of the problem of a single ↓ atom resonantly interacting with a Fermi sea of ↑ atoms. A series of successive approximations permits us to clarify the quite mysterious agreement between Monte Carlo results and approximate calculations taking only into account single particle-hole excitations. We show that it results from a nearly perfect destructive interference of the contributions of states with more than one particle-hole pair. Our treatment provides, at the same time, an essentially exact solution to this problem.

We consider a single atom within a Fermi sea of atoms. We elucidate by a full many-body analysis the quite mysterious agreement between Monte Carlo results and approximate calculations taking only into account single particle-hole excitations. It results from a nearly perfect destructive interference of the contributions of states with more than one particle-hole pair. This is linked to the remarkable efficiency of the expansion in powers of hole wave vectors, the lowest order leading to perfect interference. Going up to two particle-hole pairs gives an essentially perfect agreement with known exact results. Hence our treatment amounts to an exact solution of this problem.

We consider the problem of a single downward arrow atom in the presence of a Fermi sea of upward arrow atoms, in the vicinity of a Feshbach resonance. We calculate the chemical potential and the effective mass of the downward arrow atom using two simple approaches: a many-body variational wave function and a T-matrix approximation. These two methods lead to the same results and are in good agreement with existing quantum Monte Carlo calculations performed at unitarity and, in one dimension, with the known exact solution. Surprisingly, our results suggest that, even at unitarity, the effect of interactions is fairly weak and can be accurately described using single particle-hole excitations. We also consider the case of unequal masses.

The theoretical treatment of Fermi systems consisting of particles with unequal masses is challenging. Even in one spatial dimension analytic solutions are limited to special configurations and numerical progress with Monte Carlo simulations is hindered by the sign-problem. To circumvent this issue, we exploit the Complex Langevin approach and study one-dimensional mass-imbalanced two-component Fermi gases with attractive and repulsive interactions. We find perfect agreement with results obtained by other methods in a range of parameter space. Promisingly, our approach is not limited to the specific model presented here and can easily be extended to finite spin polarization and, most notably, can also be applied in higher dimensions.

The theoretical treatment of Fermi systems consisting of particles with unequal masses is challenging. Even in one spatial dimension analytic solutions are limited to special configurations and numerical progress with Monte Carlo simulations is hindered by the sign-problem. To circumvent this issue, we exploit the Complex Langevin approach and study one-dimensional mass-imbalanced two-component Fermi gases with attractive and repulsive interactions. We find perfect agreement with results obtained by other methods in a range of parameter space. Promisingly, our approach is not limited to the specific model presented here and can easily be extended to finite spin polarization and, most notably, can also be applied in higher dimensions.

The theoretical treatment of Fermi systems consisting of particles with unequal masses is challenging. Even in one spatial dimension analytic solutions are limited to special configurations and numerical progress with Monte Carlo simulations is hindered by the sign-problem. To circumvent this issue, we exploit the Complex Langevin approach and study one-dimensional mass-imbalanced two-component Fermi gases with attractive and repulsive interactions. We find perfect agreement with results obtained by other methods in a range of parameter space. Promisingly, our approach is not limited to the specific model presented here and can easily be extended to finite spin polarization and, most notably, can also be applied in higher dimensions.