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Acta Physica Sinica

Asymptotic expressions of path curve for a class of Fermi gases in nonlinear disturbed mechanism

Shi Lan-Fang, Xianfeng Chen, Han Xiang-Lin, Xu Yong-Hong, Mo Jia-Qi

Published Jan 1, 2014• 0 citations
Publisher

Abstract

The model of nonlinear disturbed mechanism for one-dimensional Fermi gas is investigated. Firstly, the corresponding functional is constructed; secondly, its Lagrange operator is selected; using the modified generalized variational iteration method, the approximate analytic solutions of corresponding path curves are obtained. A simple example is given, and the approximation accuracy obtained by using the modified generalized variational iteration method is shown to be better. The aim of this article is to provide a valid method of solving the nonlinear physical problems.

DOI
10.7498/aps.63.060204

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The model of nonlinear disturbed mechanism for one-dimensional Fermi gas is investigated. Firstly, the corresponding functional is constructed; secondly, its Lagrange operator is selected; using the modified generalized variational iteration method, the approximate analytic solutions of corresponding path curves are obtained. A simple example is given, and the approximation accuracy obtained by using the modified generalized variational iteration method is shown to be better. The aim of this article is to provide a valid method of solving the nonlinear physical problems.
This thesis is concerned with the quantum mechanical system of a single particle interacting with an ideal gas of identical fermions by point interaction. In the physics literature this system is often referred to as Fermi polaron. We investigate the two-dimensional Fermi polaron. Unlike the one-dimensional case, point interactions in two or three dimensions cannot be implemented as perturbation of the quadratic form of the Laplacian. Either they are obtained as self-adjoint extensions of the Laplacian restricted to functions that vanish when the coordinates of two particles coincide, or they are constructed by a suitable limiting process. Choosing the second approach, a many-body operator with two-particle point interaction has firstly been rigorously defined by Dell'Antonio, Figari and Teta. We consider the Fermi polaron confined to a box with periodic boundary conditions and we identify a broad class of regularization schemes that approximate the Hamiltonian of the Fermi polaron as limit operator in the strong resolvent sense. The Hamiltonian is not given by a closed form, which could be conveniently used in standard variational principles. We establish a novel variational principle that characterizes all bound states, i.e. all energy eigenstates below the bottom of the spectrum of the kinetic energy. This variational principle turns out to be very useful for the following purposes. The ground state of the Fermi polaron is expected to be well approximated by the polaron and the molecule ansatz in the regime of weak and strong coupling between the impurity and the Fermi gas, respectively. In the physics literature, these two classes of trial states are used for variational computations with the (ultraviolet) regularized Hamiltonian. Although the implicit expressions for the minimal energy of both classes allow for the removal of the ultraviolet cutoff, it remains unclear whether the results are upper bounds to the ground state energy of the Fermi polaron. We show that the minimization of energy over polaron and molecule trial states can be reformulated in a natural way with the help of our variational principle. By doing so, the classes of trial states simplify considerably, and since there is no reference to regularized quantities, we can prove that the expressions for the polaron and the molecule energy in the physics literature are indeed upper bounds to the ground state energy of the Fermi polaron. As a further application of the variational principle, we prove analytically that to first order in a particle-hole expansion the molecule ansatz yields a better approximation to the ground state energy than the polaron ansatz if the coupling between the impurity and the Fermi gas is strong enough. So far, this had only been done numerically. The concluding chapter is devoted to the derivation of a lower bound to the ground state energy of the Fermi polaron in two-dimensional space. We show that the ground state energy can be bounded from below by a quantity that does not depend on the number of fermions in the Fermi gas. This result is correct under the assumption that the ratio of the mass of the impurity and the mass of a fermion exceeds 1.225. We also present a method which might yield a similar result for lower mass ratios. This method gives an estimate for the quadratic form of the regularized Hamiltonian in position space representation. In this connection, we present an inequality that bounds a singular potential of a two-dimensional Fermi gas depending only on the minimal distance between two fermions by the kinetic energy of the Fermi gas uniformly in the number of fermions. This inequality also applies to a potential with singularity 1/r^2, for which the Hardy inequality does not hold in two dimensions. Therefore, the full antisymmetry of the wave function has to be taken into account.
Based on the Gross-Pitaevskii(G-P) energy functional and variational method, we present a new method to find the ground state wave function for Bose-condensed gas in a harmonic trap at zero temperature. With this method we are able to find the analytic expression of ground state wave function and explore the relevant quantities, such as energy, chemical potential and so on. These results agree well with previous ground state numerical solutions of the G-P equation given by Dalfovo et al. Under the large number of particles N limit for repulsive interaction, our results and Thomas-Fermi approximation model approach to reaching unanimity. This new method is simple compared to other methods used to solve numerically the G-P equation, and one can obtained the analytic and reliable results.
A nonlinear disturbed model for a class of Fermi gases in optical lattices was investigated. Firstly, in the nondisturbed case, the exact solution of the model path curves of Fermi gases in optical lattices was given. Secondly, the generalized functional analysis homotopic mapping was introduced and an iterative system was constructed, the arbitrary order asymptotic solution to the nonlinear disturbed model for the path curves of Fermi gases in optical lattices was obtained. Finally, a nonlinear small disturbance system was studied. With the proposed method, the asymptotic expressions of the path curves can be conveniently formulated and further extended.