Physical Review A

Zero-Momentum Condensate Fraction in Ground-State Helium

J.F. Fernández, H. A. Gersch

Published Jan 1, 1973• 4 citations

Abstract

Estimates are given for the zero-momentum condensate fraction of ground-state $^{4}\mathrm{He}$ described by a Jastrow wave function at the solidification density of the fluid (2%) and at the melting density of the solid (\ensuremath{\sim}1%). Indications are given that these estimates may be sensitively dependent on the form of the pair function utilized.

DOI: 10.1103/physreva.7.239

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Ground State of Liquid He4

William Porter Francis, Geoffrey V. Chester, L. Reatto

Physical Review AJan 1, 197070 citations

Bose-Einstein condensation in solid He4

Davide Emilio Galli, M. Rossi, L. Reatto

arXiv:cond-mat/0503443Physical Review BApr 20, 200543 citations

Bose-Einstein Condensation in Solid He4

Souleymane Diallo, Jonathan V Pearce, R. T. Azuah +3 more

arXiv:cond-mat/0702347Physical Review LettersMay 17, 200745 citations

Bose-Einstein Condensation and Liquid Helium

O. Penrose, Lars Onsager

Physical ReviewNov 1, 19561,570 citations

Fluid-Solid Phase Transition of a Hard-Sphere Bose System

Jean-Pierre Hansen, D. Levesque, Daniel Schiff

Physical Review AFeb 1, 197159 citations

Quantum phase transition with a simple variational ansatz

Y. Lutsyshyn, G. E. Astrakharchik, Claudio Cazorla +1 more

arXiv:1410.0392Physical Review BDec 11, 20142 citations

Physical Review A

Zero-Momentum Condensate Fraction in Ground-State Helium

J.F. Fernández, H. A. Gersch

Published Jan 1, 1973• 4 citations

Publisher

Abstract

Estimates are given for the zero-momentum condensate fraction of ground-state

^{4}\mathrm{He}

described by a Jastrow wave function at the solidification density of the fluid (2%) and at the melting density of the solid (\ensuremath{\sim}1%). Indications are given that these estimates may be sensitively dependent on the form of the pair function utilized.

DOI: 10.1103/physreva.7.239

Related papers

Ground State of Liquid He4

William Porter Francis, Geoffrey V. Chester, L. Reatto

A trial wave function describing the ground state of a quantum system of $N$ interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid ${\mathrm{He}}^{4}$ , and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.A trial wave function describing the ground state of a quantum system of $N$ interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid ${\mathrm{He}}^{4}$, and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.

Physical Review AJan 1, 197070 citations

Bose-Einstein condensation in solid He4

Davide Emilio Galli, M. Rossi, L. Reatto

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.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.

arXiv:cond-mat/0503443Physical Review BApr 20, 200543 citations

Bose-Einstein Condensation in Solid He4

Souleymane Diallo, Jonathan V Pearce, R. T. Azuah +3 more

We present neutron scattering measurements of the atomic momentum distribution $n(\mathbf{k})$ in solid helium under a pressure $p=41\text{ }\mathrm{bar}$ (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, ${T}_{c}=200\text{ }\text{ }\mathrm{mK}$ , where a superfluid density has been observed. Assuming BEC appears as a macroscopic occupation of the $k=0$ state below ${T}_{c}$ , we find a condensate fraction of ${n}_{0}=(\ensuremath{-}0.10\ifmmode\pm\else\textpm\fi{}1.20)%$ at $T=80\text{ }\text{ }\mathrm{mK}$ and ${n}_{0}=(0.08\ifmmode\pm\else\textpm\fi{}0.78)%$ at $T=120\text{ }\text{ }\mathrm{mK}$ , consistent with zero. The shape of $n(\mathbf{k})$ also does not change on crossing ${T}_{c}$ within measurement precision.We present neutron scattering measurements of the atomic momentum distribution $n(\mathbf{k})$ in solid helium under a pressure $p=41\text{ }\mathrm{bar}$ (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, ${T}_{c}=200\text{ }\text{ }\mathrm{mK}$, where a superfluid density has been observed. Assuming BEC appears as a macroscopic occupation of the $k=0$ state below ${T}_{c}$, we find a condensate fraction of ${n}_{0}=(\ensuremath{-}0.10\ifmmode\pm\else\textpm\fi{}1.20)%$ at $T=80\text{ }\text{ }\mathrm{mK}$ and ${n}_{0}=(0.08\ifmmode\pm\else\textpm\fi{}0.78)%$ at $T=120\text{ }\text{ }\mathrm{mK}$, consistent with zero. The shape of $n(\mathbf{k})$ also does not change on crossing ${T}_{c}$ within measurement precision.

arXiv:cond-mat/0702347Physical Review LettersMay 17, 200745 citations

Bose-Einstein Condensation and Liquid Helium

O. Penrose, Lars Onsager

The mathematical description of B.E. (Bose-Einstein) condensation is generalized so as to be applicable to a system of interacting particles. B.E. condensation is said to be present whenever the largest eigenvalue of the one-particle reduced density matrix is an extensive rather than an intensive quantity. Some transformations facilitating the practical use of this definition are given.An argument based on first principles is given, indicating that liquid belium II in equilibrium shows B.E. condensation. For absolute zero, the argument is based on properties of the ground-state wave function derived from the assumption that there is no "long-range configurational order." A crude estimate indicates that roughly 8% of the atoms are "condensed" (note that the fraction of condensed particles need not be identified with $\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}}$). Conversely, it is shown why one would not expect B.E. condensation in a solid. For finite temperatures Feynman's theory of the lambda-transition is applied: Feynman's approximations are shown to imply that our criterion of B.E. condensation is satisfied below the lambda-transition but not above it.The mathematical description of B.E. (Bose-Einstein) condensation is generalized so as to be applicable to a system of interacting particles. B.E. condensation is said to be present whenever the largest eigenvalue of the one-particle reduced density matrix is an extensive rather than an intensive quantity. Some transformations facilitating the practical use of this definition are given.An argument based on first principles is given, indicating that liquid belium II in equilibrium shows B.E. condensation. For absolute zero, the argument is based on properties of the ground-state wave function derived from the assumption that there is no "long-range configurational order." A crude estimate indicates that roughly 8% of the atoms are "condensed" (note that the fraction of condensed particles need not be identified with $\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}}$). Conversely, it is shown why one would not expect B.E. condensation in a solid. For finite temperatures Feynman's theory of the lambda-transition is applied: Feynman's approximations are shown to imply that our criterion of B.E. condensation is satisfied below the lambda-transition but not above it.

Physical ReviewNov 1, 19561,570 citations

Fluid-Solid Phase Transition of a Hard-Sphere Bose System

Jean-Pierre Hansen, D. Levesque, Daniel Schiff

The ground-state energy of a system of hard-sphere bosons of diameter $\ensuremath{\sigma}$ is calculated by a variational method using a Jastrow wave-function for the fluid phase and a "localized" Jastrow times a Gaussian wave function for the solid phase. The transition densities are ${\ensuremath{\rho}}_{\mathrm{fluid}}=0.23\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$ and ${\ensuremath{\rho}}_{\mathrm{solid}}=0.25\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$.The ground-state energy of a system of hard-sphere bosons of diameter $\ensuremath{\sigma}$ is calculated by a variational method using a Jastrow wave-function for the fluid phase and a "localized" Jastrow times a Gaussian wave function for the solid phase. The transition densities are ${\ensuremath{\rho}}_{\mathrm{fluid}}=0.23\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$ and ${\ensuremath{\rho}}_{\mathrm{solid}}=0.25\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$.

Physical Review AFeb 1, 197159 citations

Quantum phase transition with a simple variational ansatz

Y. Lutsyshyn, G. E. Astrakharchik, Claudio Cazorla +1 more

We study the zero-temperature quantum phase transition between liquid and hcp solid $^{4}\mathrm{He}$ . We use the variational method with a simple yet exchange-symmetric and fully explicit wave function. It is found that the optimized wave function undergoes spontaneous symmetry breaking and describes the quantum solidification of helium at 22 atm. The explicit form of the wave function allows us to consider various contributions to the phase transition. We find that the employed wave function is an excellent candidate for describing both a first-order quantum phase transition and the ground state of a Bose solid.We study the zero-temperature quantum phase transition between liquid and hcp solid $^{4}\mathrm{He}$. We use the variational method with a simple yet exchange-symmetric and fully explicit wave function. It is found that the optimized wave function undergoes spontaneous symmetry breaking and describes the quantum solidification of helium at 22 atm. The explicit form of the wave function allows us to consider various contributions to the phase transition. We find that the employed wave function is an excellent candidate for describing both a first-order quantum phase transition and the ground state of a Bose solid.

arXiv:1410.0392Physical Review BDec 11, 20142 citations

Estimates are given for the zero-momentum condensate fraction of ground-state

^{4}\mathrm{He}

A trial wave function describing the ground state of a quantum system of

N

interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid

{\mathrm{He}}^{4}

, and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.

We have computed the one-body density matrix ${\ensuremath{\rho}}_{1}$ in solid

^{4}\mathrm{He}

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.

We present neutron scattering measurements of the atomic momentum distribution

n(\mathbf{k})

in solid helium under a pressure

p=41\text{ }\mathrm{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,

{T}_{c}=200\text{ }\text{ }\mathrm{mK}

, where a superfluid density has been observed. Assuming BEC appears as a macroscopic occupation of the

k=0

state below

{T}_{c}

, we find a condensate fraction of ${n}_{0}=(\ensuremath{-}0.10\ifmmode\pm\else\textpm\fi{}1.20)%$ at

T=80\text{ }\text{ }\mathrm{mK}

and ${n}_{0}=(0.08\ifmmode\pm\else\textpm\fi{}0.78)%$ at

T=120\text{ }\text{ }\mathrm{mK}

, consistent with zero. The shape of

n(\mathbf{k})

also does not change on crossing

{T}_{c}

within measurement precision.

We study the zero-temperature quantum phase transition between liquid and hcp solid

^{4}\mathrm{He}

. We use the variational method with a simple yet exchange-symmetric and fully explicit wave function. It is found that the optimized wave function undergoes spontaneous symmetry breaking and describes the quantum solidification of helium at 22 atm. The explicit form of the wave function allows us to consider various contributions to the phase transition. We find that the employed wave function is an excellent candidate for describing both a first-order quantum phase transition and the ground state of a Bose solid.