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.
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.
We have computed the one-body density matrix ${\ensuremath{\rho}}_{1}$ in solid 4He at 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 ${\ensuremath{\rho}}_{1}$ for a perfect hcp and bcc solid 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. 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 consider a lattice of bosonic atoms, whose number N may be smaller than the number of lattice sites M. We study the Hartree-Fock wave function built up from localized wave functions w(r) of single atoms, with nearest-neighboring overlap. The zero-momentum particle number is expressed in terms of permanents of matrices. In one dimension, it is analytically calculated to be $\ensuremath{\alpha}N(M\ensuremath{-}N+1)/M$, with $\ensuremath{\alpha}={|\ensuremath{\int}w(\mathbf{r})d\ensuremath{\Omega}|}^{2}/[(1+2a)l]$, where a is the nearest-neighboring overlap and l is the lattice constant. $\ensuremath{\alpha}$ is on the order of 1. The result indicates that the condensate fraction is proportional to and of the same order of magnitude as that of the vacancy concentration, hence there is off-diagonal long-range order or Bose-Einstein condensation of atoms when the number of vacancies $M\ensuremath{-}N$ is a finite fraction of the number of the lattice sites M.
We propose a many-body wavefunction that exhibits both diagonal and off-diagonal long-range order. Incorporating short-range correlations due to interatomic repulsion, this wavefunction is shown to allow condensation of zero-point lattice vibrations and phase rigidity. In the presence of an external velocity field, such a perfect crystal will develop non-classical rotational inertia, exhibiting supersolid behaviour. In a sample calculation we show that the superfluid fraction in this state can be as large as of order 0.01 in a reasonable range of microscopic parameters. The relevance to the recent experimental evidence of a supersolid state given by Kim and Chan is discussed.
The supersolid phase of soft-core bosons in two dimensions is investigated using the self-consistent Hartree-Fock and quantum Monte Carlo methods. An approximate phase diagram at finite temperatures is initially constructed using the mean-field approach, which is subsequently validated through precise path-integral simulations, enabling a microscopic characterization of the various phases. Superfluid and melting/freezing transitions are analyzed through the superfluid density and the long-range behavior of correlation functions associated with positional and orientational order, in accordance with the general picture of Berezinskii-Kosterlitz-Thouless transitions. A broad region at low temperatures is identified where the supersolid phase exists, separating the uniform superfluid phase from the normal quasicrystal phase. Additionally, a potential intermediate hexatic phase with quasi-long-range orientational order is identified in a narrow region between the normal solid and fluid phases. These findings establish self-consistent Hartree-Fock theory beyond the local density approximation as an effective tool, complementary to computationally intensive quantum Monte Carlo simulations, for investigating the melting of the supersolid phase and the possible emergence of the hexatic superfluid phase in bosonic systems with various interaction potentials.
For a strongly interacting boson system, it has been proposed that the concept of off-diagonal long-range ordering (ODLRO) is equivalent to Bose-Einstein condensation for a free boson system. Therefore, the existence of ODLRO in the ground state implies superfluidity. The author checks the validity of this proposition from another point of view. He rigorously shows that, for a hard-core lattice-boson system with a short-ranged interaction ODLRO is suppressed when a charged excitation gap develops and, hence, the boson system becomes a Mott-insulator. Finally, by using his theorem, he shows some interesting properties of the antiferromagnetic Heisenberg model, which can be taken as a hard-core lattice-boson system.
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.