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arXiv:cond-mat/0206071 · Physical Review Letters

Zero-Temperature Equation of State of Two-Dimensional H3e

V. Grau, Jordi Boronat, J. Casulleras

Published Jul 3, 2002• 29 citations
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Abstract

The equation of state of two-dimensional 3He at zero temperature has been calculated using the diffusion Monte Carlo method. By means of a combination of the fixed-node and released-node techniques, it is shown that backflow correlations provide a very accurate equation of state. The results prove unambiguously the non-self-bound character of two-dimensional 3He due to its Fermi statistics. We present solid evidence that the gas phase, predicted for the two-dimensional system, can be extrapolated to the case of 3He adsorbed on a strong substrate such as graphite.

DOI
10.1103/physrevlett.89.045301

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The equation of state of two-dimensional 3He at zero temperature has been calculated using the diffusion Monte Carlo method. By means of a combination of the fixed-node and released-node techniques, it is shown that backflow correlations provide a very accurate equation of state. The results prove unambiguously the non-self-bound character of two-dimensional 3He due to its Fermi statistics. We present solid evidence that the gas phase, predicted for the two-dimensional system, can be extrapolated to the case of 3He adsorbed on a strong substrate such as graphite.
The correlation energy of the homogeneous three-dimensional interacting electron gas is calculated using the variational and fixed-node diffusion Monte Carlo methods, with trial functions that include backflow and three-body correlations. In the high-density regime (rs< 5){(r}_{s}<~5)(rs​< 5) the effects of backflow dominate over those due to three-body correlations, but the relative importance of the latter increases as the density decreases. Since the backflow correlations vary the nodes of the trial function, this leads to improved energies in the fixed-node diffusion Monte Carlo calculations. The effects are comparable to those found for the two-dimensional electron gas, leading to much improved variational energies and fixed-node diffusion energies similar to the released-node energies of Ceperley and Alder.
We present recent results from a diffusion Monte Carlo study of small 3 He drops in the limit of zero temperature. Using the same methodology than in previous calculations carried out for bulk 3 He , we have obtained accurate results for their energy and structure properties. A relevant concern of this analysis has been the determination of the stability threshold for self binding. Our results show that the smallest self-bound drop is formed by N = 30 atoms, with preferred orbitals for open shells corresponding to maximum value of the spin. The quality of the upper bound energies provided by the fixed node approximation is analyzed by performing released-node estimations for short released times. Possible improvements of the nodal surface by including BCS-like correlation functions are also discussed.
Helium systems at zero temperature are strongly correlated neutral systems which can only be described by means of quantum mechanics. The existence of the two isotopes 4He and 3He allows the comparison between the quantum effects due to different statistics, bosonic and fermionic. In this Thesis, an analysis of Helium droplets has been done. Throughout this work, both energetic and structural properties of systems with different number of particles N are obtained by means of the Quantum Monte Carlo techniques (Variational and Diffusion Monte Carlo - VMC and DMC-). Several studies on Helium droplets have been published in the last years, most of them using the mean field theory (as the Density Functional) and variational approaches (as VMC). In the case of 4He, also exact DMC results have already been reported, actually, a great number of works have described with high accuracy the ground state of this bosonic system. The appearance of magic numbers from experimental point of view has motivated the study of 4He droplets, and one chapter of this Thesis is focused on this bosonic system. We have carried out a one-by-one energetic study in the range from 20 to 30 particles using the DMC technique to discuss the existence of magic numbers. On the other hand, a brief study about vortices is also done; using out code, one can calculate the energy of a droplet with one vortex inside (EV) and, therefore, the energy of a vortex (EV -E0). About 3He droplets, the sign problem makes their study quite more difficult, for this reason, the number of works is considerably smaller than in the bosonic case. One aim of this thesis was to write a DMC code using both Fixed and Released Node approximations (FN and RN) able to perform high accurate calculations of energetic and structural properties of these finite fermionic systems.
Accurate diffusion Monte Carlo technique within the fixed-node approximation has been used to determine the equation of state of the two-dimensional electron gas in three phases: the normal paramagnetic liquid, the ferromagnetic (spin-polarized) liquid, and the Wigner crystal. Our backflow fixed-node calculation, which is a significant improvement over two previous conflicting Slater-Jastrow calculations, shows that there is only one clear phase transition from the paramagnetic liquid to the Wigner crystal as the density decreases. Unlike the three-dimensional case, we do not see the stability of the polarized liquid phase at intermediate densities before the crystallization.
The equation of state of a weakly interacting two-dimensional Bose gas is studied at zero temperature by means of quantum Monte Carlo methods. Going down to as low densities as $n{a}^{2}\ensuremath{\propto}{10}^{\ensuremath{-}100}$ permits us to obtain agreement on beyond mean-field level between predictions of perturbative methods and direct many-body numerical simulation, thus providing an answer to the fundamental question of the equation of state of a two-dimensional dilute Bose gas in the universal regime (i.e., entirely described by the gas parameter na2n{a}^{2}na2). We also show that the measure of the frequency of a breathing collective oscillation in a trap at very low densities can be used to test the universal equation of state of a two-dimensional Bose gas.