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arXiv:cond-mat/0306145 · Physical Review B

Monte Carlo simulations of two-dimensional charged bosons

S. De Palo, Sergio Conti, Saverio Moroni

Published Jan 22, 2004• 32 citations
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Abstract

Quantum Monte Carlo methods are used to calculate various ground-state properties of charged bosons in two dimensions, throughout the whole density range where the fluid phase is stable. Wigner crystallization is predicted at ${r}_{s}\ensuremath{\simeq}60.$ Results for the ground-state energy and the momentum distribution are summarized in analytic interpolation formulas embodying known asymptotic behaviors. Near freezing, the condensate fraction is less than 1%. The static structure factor $S(k)$ and susceptibility $\ensuremath{\chi}(k)$ are obtained from the density-density correlation function in imaginary time, $F(\mathbf{k},\ensuremath{\tau}).$ An estimate of the energy of elementary excitations, given in terms of an upper bound involving $S(k)$ and $\ensuremath{\chi}(k),$ is compared with the result obtained via analytic continuation from $F(\mathbf{k},\ensuremath{\tau}).$

DOI
10.1103/physrevb.69.035109

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Quantum Monte Carlo methods are used to calculate various ground-state properties of charged bosons in two dimensions, throughout the whole density range where the fluid phase is stable. Wigner crystallization is predicted at rs\ensuremath≃60.{r}_{s}\ensuremath{\simeq}60.rs​\ensuremath≃60. Results for the ground-state energy and the momentum distribution are summarized in analytic interpolation formulas embodying known asymptotic behaviors. Near freezing, the condensate fraction is less than 1%. The static structure factor S(k)S(k)S(k) and susceptibility \ensuremathχ(k)\ensuremath{\chi}(k)\ensuremathχ(k) are obtained from the density-density correlation function in imaginary time, F(k,\ensuremathτ).F(\mathbf{k},\ensuremath{\tau}).F(k,\ensuremathτ). An estimate of the energy of elementary excitations, given in terms of an upper bound involving S(k)S(k)S(k) and \ensuremathχ(k),\ensuremath{\chi}(k),\ensuremathχ(k), is compared with the result obtained via analytic continuation from F(k,\ensuremathτ).F(\mathbf{k},\ensuremath{\tau}).F(k,\ensuremathτ).
Variational and fixed-node Green's-function Monte Carlo calculations have been performed to find the ground-state properties of the two-dimensional electron gas in the density range 1\ensuremath{\le}rs{r}_{s}rs​\ensuremath{\le}100. Our calculations predict a Wigner crystallization at the density rs{r}_{s}rs​\ensuremath{\simeq}37\ifmmode\pm\else\textpm\fi{}5. The electron system is found to be in the normal- (paramagnetic) fluid state below the transition density, but the fully polarized state is very close in energy. We have tabulated the values of pair distribution function g(r), the static structure factor S(k), and the momentum distribution n(k) at several densities of interest both in the normal and the polarized phases. An estimate of the spin susceptibility \ensuremath{\chi} is also given.
The diffusion Monte Carlo (DMC) method is used to analyze various properties of the three-dimensional plasma of charged bosons in the homogeneous fluid regime, over the density range 1\ensuremath{\le}rs{\mathit{r}}_{\mathit{s}}rs​\ensuremath{\le}160 extending from the so-called uniform limit (${\mathit{r}}_{\mathit{s}}1)tothecrystallinephase(1) to the crystalline phase (1)tothecrystallinephase({\mathit{r}}_{\mathit{s}}$\ensuremath{\gtrsim}160). The data on the static density response function \ensuremath{\chi}(k) extend to intermediate and high wave number k the work of Sugiyama, Bowen, and Alder [Phys. Rev. B 46, 13 042 (1992)] and allow us to extract the static local-field factor G(k) for exchange and correlation. The DMC results for the momentum distribution n(k) show that the condensate fraction decreases with increasing coupling strength from 83% at rs{\mathit{r}}_{\mathit{s}}rs​=1 to less than 1% at rs{\mathit{r}}_{\mathit{s}}rs​=160. We also present results for the ground-state energy Eg{\mathit{E}}_{\mathit{g}}Eg​(rs{\mathit{r}}_{\mathit{s}}rs​) and for the structure factor S(k). The results for Eg{\mathit{E}}_{\mathit{g}}Eg​(rs{\mathit{r}}_{\mathit{s}}rs​), n(k), and \ensuremath{\chi}(k) are summarized in analytic interpolation formulas embodying the known asymptotic behaviors. A rigorous upper bound on the plasmon dispersion curve is obtained from the DMC data through a sum-rule argument. \textcopyright{} 1996 The American Physical Society.
We study the two-dimensional gas of charged bosons interacting via a logarithmic potential within the quantal hypernetted-chain approximation. The pair correlation function, the static structure factor, the ground-state energy, and the spectrum of collective excitations are calculated over a range of the density parameter rs.{r}_{s}.rs​. We find that the static and dynamical properties are described rather well in this approach in comparison to previous theories. Our results are in good agreement with the available quantum Monte Carlo simulations.
The ground state of a two-dimensional charged-boson system is investigated over the range of densities 1\ensuremath{\lesssim}Rs{R}_{s}Rs​\ensuremath{\lesssim}10 in the self-consistent-field approximation; Rs{R}_{s}Rs​=(a02{a}_{0}^{2}a02​\ensuremath{\pi}\ensuremath{\rho}${)}^{\mathrm{\ensuremath{-}}1/2}$, where a0{a}_{0}a0​ is the Bohr radius and \ensuremath{\rho} is the number density. Starting with numerical self-consistent calculations of the static structure factor, the elementary excitation, pair-correlation functions, pressure, and ground-state energy are evaluated. These results are compared with those of the two-dimensional and three-dimensional systems obtained from other methods. The ground-state energy is given as ${\mathit{E}}_{0=\mathrm{\ensuremath{-}}1.2918{\mathit{R}}_{\mathit{s}}}^{\mathrm{\ensuremath{-}}2/3}$+0.03, which improves that result from the ring-diagram approximation.
We present results of a ground-state study, via the diffusion Monte Carlo method, of a two-dimensional charged boson liquid. We observe three distinct behaviors in this system. At the lowest densities a close-packed Wigner crystal forms. At ${r}_{s}\ensuremath{\approx}12$ the crystal melts into a noncondensed liquid exhibiting algebraic off-diagonal long-range order; the single-particle density matrix decays as ${r}^{\ensuremath{-}\frac{{r}_{s}}{4}}$. Above a threshold density (rs<8{r}_{s}<8rs​<8), the long-wavelength momentum distribution diverges as ${k}^{\frac{({r}_{s}\ensuremath{-}8)}{4}}$, although no condensate forms.
Using a Jastrow trial wave function and the hypernetted-chain integral equation, as well as Monte Carlo computations, to calculate energy expectation values, we have obtained a variational upper bound to the ground-state energy of the charged Bose gas over an extensive range of densities. In the high-density region our data are in good agreement with previous weak-coupling results. In the low-density (strong-coupling) limit, comparison of our fluid energies with the self-consistent phonon calculations of Glyde and Keech in the solid allow us to locate the Wigner transition for charged bosons at rs=139{r}_{s}=139rs​=139, where rs{r}_{s}rs​ is the usual density parameter. We propose a simple and accurate equation of state based on our numerical data. Furthermore, we find that the Bose condensate fraction drops from a value of about 80% at rs=2{r}_{s}=2rs​=2, to a small but finite value (\ensuremath{\sim}1%) at crystallization. Finally we use our results for the charged Bose gas to derive a heuristic equation of state of the electron gas in the strong-coupling limit (${r}_{s}\ensuremath{\gg}1$).