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arXiv:cond-mat/0503443 · Physical Review B

Bose-Einstein condensation in solid He4

Davide Emilio Galli, M. Rossi, L. Reatto

Published Apr 20, 2005• 43 citations
arXivHTMLPDF

Abstract

We have computed the one-body density matrix ${\ensuremath{\rho}}_{1}$ in solid $^{4}\mathrm{He}$ at $T=0\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ using the shadow wave-function (SWF) variational technique. The accuracy of the SWF has been tested with an exact projector method. We find that off-diagonal long-range order is present in ${\ensuremath{\rho}}_{1}$ for a perfect hcp and bcc solid $^{4}\mathrm{He}$ for a range of densities above the melting one, at least up to 54 bars. This is first microscopic indication that Bose Einstein condensation (BEC) is present in perfect solid $^{4}\mathrm{He}$. At melting, the condensate fraction in the hcp solid is $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and it decreases by increasing the density. The key process giving rise to BEC is the formation of vacancy interstitial pairs. We also present values for Leggett's upper bound on the superfluid fraction deduced from the exact local density.

DOI
10.1103/physrevb.71.140506

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We have computed the one-body density matrix \ensuremathρ1{\ensuremath{\rho}}_{1}\ensuremathρ1​ in solid 4He^{4}\mathrm{He}4He at T=0KT=0\phantom{\rule{0.3em}{0ex}}\mathrm{K}T=0K using the shadow wave-function (SWF) variational technique. The accuracy of the SWF has been tested with an exact projector method. We find that off-diagonal long-range order is present in \ensuremathρ1{\ensuremath{\rho}}_{1}\ensuremathρ1​ for a perfect hcp and bcc solid 4He^{4}\mathrm{He}4He for a range of densities above the melting one, at least up to 54 bars. This is first microscopic indication that Bose Einstein condensation (BEC) is present in perfect solid 4He^{4}\mathrm{He}4He. At melting, the condensate fraction in the hcp solid is 5\ifmmode×\else\texttimes\fi10\ensuremath−65\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}5\ifmmode×\else\texttimes\fi10\ensuremath−6 and it decreases by increasing the density. The key process giving rise to BEC is the formation of vacancy interstitial pairs. We also present values for Leggett's upper bound on the superfluid fraction deduced from the exact local density.
Measurements of the moment of inertia by Kim and Chan have found that solid (4)He acts like a supersolid at low temperatures. To understand the order in solid 4(He), we have used path integral Monte Carlo simulations to calculate the off-diagonal long-range order (ODLRO) [equivalent to Bose-Einstein condensation (BEC)]. We do not find ODLRO in a defect-free hcp crystal of (4)He at the melting density. We discuss these results in relation to proposed quantum solid trial functions. We conclude that the solid (4)He wave function has correlations which suppress both vacancies and BEC.
Several microscopic theories point out that Bose-Einstein condensation (BEC), i.e., a macroscopic occupation of the lowest energy single particle state in many-boson systems, may appear also in quantum fluids and solids and that it is at the origin of the phenomenon of superfluidity. Nevertheless, the connection between BEC and superfluidity is still matter of debate, since the experimental evidences indicating a non zero condensate fraction in superfluid helium are only indirect. In the theoretical study of BEC in quantum fluids and solids, perturbative approaches are useless because of the strong correlations between the atoms, arising both from the interatomic potential and from the quantum nature of the system. Microscopic Quantum Monte Carlo simulations provide a reliable description of these systems. In particular, the Path Integral Monte Carlo (PIMC) method is very suitable for this purpose. This method is able to provide exact results for the properties of the quantum system, both at zero and finite temperature, only with the definition of the Hamiltonian and of the symmetry properties of the system, giving an easy picture for superfluidity and BEC in many-boson systems. In this thesis, we apply PIMC methods to the study of several quantum fluids and solids. We describe in detail all the features of PIMC, from the sampling methods to the estimators of the physical properties. We present also the most recent techniques, such as the high-order approximations for the thermal density matrix and the worm algorithm, used in PIMC to provide reliable simulations. We study the liquid phase of condensed 4He, providing unbiased estimations of the one-body density matrix g1(r). We analyze the model for g1(r) used to fit the experimental data, highlighting its merits and its faults. In particular we see that, even if it presents some difficulties in the description of the overall behavior of g1(r), it can provide an accurate estimation of the kinetic energy K and of the condensate fraction n0 of the system. Furthermore, we show that our results for n0 as a function of the pressure are in a good agreement with the most recent experimental results. The study of the solid phase of 4He is the most significant part of this thesis. The recent observation of non classical rotational inertia (NCRI) effects in solid helium has generated big interest in the study of an eventual supersolid phase, characterized at the same time by crystalline order and superfluidity. Nevertheless, until now it has been impossible to give a theoretical model able to describe all the experimental evidences. In this work, we perform PIMC simulations of 4He at high densities, according to different microscopic configurations of the atoms. In commensurate crystals we see that BEC does not appear, our model being able to reproduce the momentum distribution obtained form neutron scattering experiments. In a crystal with vacancies, we have been able to see a transition to a superfluid phase at temperatures in agreement with experimental results if the vacancy concentration is low enough. In amorphous solids, superfluid effects are enhanced but appear at temperatures higher than the experimental estimation for the transition temperature. Finally, we study also metastable disordered configurations in molecular para-hydrogen at low temperature. The aim of this study is to investigate if a Bose liquid other than helium can display superfluidity. Choosing accurately a ¿quantum liquid¿ initial configuration and the dimensions of the simulation box, we have been able to frustrate the formation of the crystal and to calculate the temperature dependence of the superfluid density, showing a transition to a superfluid phase at temperatures close to 1 K.
We give a variational Monte Carlo description at T = 0 K of 4He filling a porous glass within the shadow wave function technique. The confining media is modeled by a smooth cylindrical pore, and we have considered two different pore radii: R = 13 A and R = 22.5 A. The radial density profiles show a strong layering of the 4He atoms. In the R = 13 A pore, as the density is increased, solidification takes place layer by layer, starting from the pore wall, and the radius is too small to allow a bulk‐like solid to nucleate in the small liquid region at the center of the pore. For the R = 22.5 A pore we find an hcp lattice accommodated in the central region surrounded by four layers which turn out to be solid at the density of the computation. Computing the one‐body density matrix (in the R = 13 A case) we are able to estimate the Bose‐Einstein condensate fraction, which is still non‐zero even when the whole system is in the solid phase.
We give a variational Monte Carlo description at T = 0 K of 4He filling a porous glass within the shadow wave function technique. The confining media is modeled by a smooth cylindrical pore, and we have considered two different pore radii: R = 13 Å and R = 22.5 Å. The radial density profiles show a strong layering of the 4He atoms. In the R = 13 Å pore, as the density is increased, solidification takes place layer by layer, starting from the pore wall, and the radius is too small to allow a bulk‐like solid to nucleate in the small liquid region at the center of the pore. For the R = 22.5 Å pore we find an hcp lattice accommodated in the central region surrounded by four layers which turn out to be solid at the density of the computation. Computing the one‐body density matrix (in the R = 13 Å case) we are able to estimate the Bose‐Einstein condensate fraction, which is still non‐zero even when the whole system is in the solid phase.
We present neutron scattering measurements of the atomic momentum distribution n(k)n(\mathbf{k})n(k) in solid helium under a pressure p=41 barp=41\text{ }\mathrm{bar}p=41 bar (molar volume ${V}_{m}=20.01\ifmmode\pm\else\textpm\fi{}0.02\text{ }\text{ }{\mathrm{cm}}^{3}/\mathrm{mol}$) and at temperatures between 80 and 500 mK. The aim is to determine whether there is Bose-Einstein condensation (BEC) below the critical temperature, Tc=200  mK{T}_{c}=200\text{ }\text{ }\mathrm{mK}Tc​=200  mK, where a superfluid density has been observed. Assuming BEC appears as a macroscopic occupation of the k=0k=0k=0 state below Tc{T}_{c}Tc​, we find a condensate fraction of ${n}_{0}=(\ensuremath{-}0.10\ifmmode\pm\else\textpm\fi{}1.20)%$ at T=80  mKT=80\text{ }\text{ }\mathrm{mK}T=80  mK and ${n}_{0}=(0.08\ifmmode\pm\else\textpm\fi{}0.78)%$ at T=120  mKT=120\text{ }\text{ }\mathrm{mK}T=120  mK, consistent with zero. The shape of n(k)n(\mathbf{k})n(k) also does not change on crossing Tc{T}_{c}Tc​ within measurement precision.
The properties of the many-particle Schr\"odinger wave function \ensuremath{\Psi} are examined in the presence of Bose-Einstein condensation (BEC). It is shown that it is possible to define, in terms of \ensuremath{\Psi}, a function $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s}),$ which can be regarded as the single-particle wave function of an arbitrary particle for a fixed configuration $\stackrel{\ensuremath{\rightarrow}}{s}$ of all other particles. It is shown that $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ plays an analogous role to the field operator of standard field-theoretical treatments of superfluidity. It is shown that in the presence of a Bose-Einstein condensate fraction f, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ must be nonzero and phase coherent within at least a fraction f of the total volume of the N-particle system for essentially all $\stackrel{\ensuremath{\rightarrow}}{s}.$ Examination of the form of variational many-particle wave functions shows that in liquid 4He,{}^{4}\mathrm{He},4He, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ extends throughout the spaces left between the hard cores of the other atoms at $\stackrel{\ensuremath{\rightarrow}}{s}.$ By contrast, in the absence of BEC, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ in the ground state must be nonzero only over a localized region of space. It is shown that in order for long-range phase coherence in $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ to be maintained in the presence of velocity fields, any circulation must be quantized over macroscopic length scales. Some numerical calculations of the properties and fluctuations of liquid helium are presented. These suggest that the approach outlined in this paper may have significant advantages for the numerical calculations of the properties of Bose-Einstein condensed systems. The properties of $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ are used to show that there is no general connection between the static structure factor and the size of the Bose-Einstein condensate fraction in a Bose fluid. It is suggested that the observed connection in liquid 4He{}^{4}\mathrm{He}4He is due to the creation of vacancies in the liquid structure, which are required so that $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ can delocalize, in the presence of hard-core interactions. It is shown that this suggestion is quantitatively consistent with observations on liquid 4He.{}^{4}\mathrm{He}.4He. It is suggested that the superfluid transition in 4He{}^{4}\mathrm{He}4He occurs when the vacancies in the liquid structure form a connected network.