دانلود رایگان مقاله رویکرد مونت کارلو شبکه غیر یکنواخت برای گاز فرمی در تله هارمونیک

Abstract

We present in detail two variants of the lattice Monte Carlo method aimed at tackling systems in external trapping potentials: a uniform-lattice approach with hard-wall boundary conditions, and a non-uniform Gauss–Hermite lattice approach. Using those two methods, we compute the ground-state energy and spatial density profile for systems of N=4–8 harmonically trapped fermions in one dimension. From the favorable comparison of both energies and density profiles (particularly in regions of low density), we conclude that the trapping potential is properly resolved by the hard-wall basis. Our work paves the way to higher dimensions and finite temperature analyses, as calculations with the hard-wall basis can be accelerated via fast Fourier transforms; the cost of unaccelerated methods is otherwise prohibitive due to the unfavorable scaling with system size. To illustrate this point, we show a brief performance comparison of accelerated versus unaccelerated methods across spatial dimensions.

6. Summary and conclusions

In this paper, we have presented two methods to address the problem of interacting fermions in harmonic traps: a uniformlattice method with hard-wall boundary conditions, and a nonuniform Gauss–Hermite lattice method (which we had used in previous work). While the latter has many attractive features (it diagonalizes the noninteracting Hamiltonian exactly), it is not amenable to Fourier acceleration (or at least not easily), which makes it practically unfeasible for higher dimensions (especially away from zero temperature). The hard-wall method, on the other hand, shares some of the positive features and can be Fourier accelerated. We showed the benefits of acceleration by comparing the performance across dimensions, and conclude that it is essential for d ≥ 2. To test the methods against each other, we compared here calculations for 1D attractively interacting fermions in a harmonic trap. Specifically, we computed the ground-state energy and density profiles of unpolarized systems of N = 4 and 8 particles. Our results show that for both the ground-state energy and the density profiles, the methods agree satisfactorily. For the density profiles, in particular, we note that the expected Gaussian decay is reproduced with the hard-wall basis over multiple orders of magnitude before breaking down at large distances due to the presence of the wall. From our calculations we conclude that it is possible to obtain high-quality results using uniform bases with hard-wall boundaries.