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Physical Review

Phonons and the Properties of a Bose System

L. Reatto, Geoffrey V. Chester

Published Mar 5, 1967• 281 citations
Publisher

Abstract

In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where $\mathrm{ln}f(r)$ has an infinite range. The other factor is assumed to have this same form but with $\mathrm{ln}f(r)$ of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors $k$, the structure factor $S(k)$ is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature $T$, $S(k)$ tends to the constant value $\frac{{k}_{B}T}{m{c}^{2}}$, where $m$ is the mass of the particles and $c$ the velocity of propagation of the phonons. The momentum distribution ${n}_{k}$ has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-$k$ behavior of both $S(k)$ and ${n}_{k}$ is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.

DOI
10.1103/physrev.155.88

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In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form \ensuremathΠi<jf(rij){\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})\ensuremathΠi<j​f(rij​), where lnf(r)\mathrm{ln}f(r)lnf(r) has an infinite range. The other factor is assumed to have this same form but with lnf(r)\mathrm{ln}f(r)lnf(r) of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors kkk, the structure factor S(k)S(k)S(k) is equal to \ensuremathℏk2mc\frac{\ensuremath{\hbar}k}{2mc}2mc\ensuremathℏk​, a result that was first derived by Feynman. At a finite temperature TTT, S(k)S(k)S(k) tends to the constant value kBTmc2\frac{{k}_{B}T}{m{c}^{2}}mc2kB​T​, where mmm is the mass of the particles and ccc the velocity of propagation of the phonons. The momentum distribution nk{n}_{k}nk​ has a k\ensuremath−1{k}^{\ensuremath{-}1}k\ensuremath−1 singularity at absolute zero and a stronger, k\ensuremath−2{k}^{\ensuremath{-}2}k\ensuremath−2 singularity at finite temperatures. The small-kkk behavior of both S(k)S(k)S(k) and nk{n}_{k}nk​ is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.
At any finite temperature, the correlation function $〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(\mathcal{r})\ensuremath{\psi}(0)〉$ of a two-dimensional Bose system is known to approach zero as the separation r\mathcal{r}r approaches infinity. This paper is devoted to supporting the point of view that superfluid phenomena may still persist in such a system. To describe the low-temperature behavior of a condensed two-dimensional Bose system, we use a coherent-state representation of the low-lying phonon excitations. The expectation of the particle field operator $〈\ensuremath{\psi}(\mathcal{r})〉$ in a coherent phonon state of the system is shown to be a well-defined function. This function has only small fluctuations in its magnitude and, therefore, it is possible to make the usual arguments for superfluid behavior based on the existence of the condensate wave function. The large phase fluctuations of the function, when the coherentphonon states are collected into the thermal ensemble, give rise to the spatial decay of the correlation function or reduced density matrix $〈{\ensuremath{\psi}}^{+}(\mathcal{r})\ensuremath{\psi}(0)〉$. It is concluded that two-dimensional Bose systems are superfluid at sufficiently low temperatures, even in the inifinite-volume limit. An analogous argument would indicate the possibility of superconductivity in a two-dimensional Fermi system.
After a brief summary of some basic properties of ideal gases of bosons and of fermions, two many-body Hamiltonians are cited for which ground-state wave functions allow the generation of excited states. Because of the complexity of ground-state many-body wave functions, we then consider properties of reduced density matrices, and in particular, the diagonal element of the second-order density matrix. For both the homogeneous correlated electron liquid and for an assembly of charged bosons, the ground-state pair correlation function g(r) has fingerprints of the zero-point energy of the plasmon modes. These affect crucially the static structure factor, S(k), in the long wavelength limit. This is best understood by means of the Ornstein–Zernike direct correlation function c(r), which plays an important role throughout this article. Turning from such charged liquids, both boson and fermion, to superfluid 4He, the elevated temperature (T) structure factor S(k, T) is related, albeit approximately, to its zero-temperature counterpart, via the velocity of sound, reflecting the collective phonon excitations and the superfluid density. Finally, some future directions are pointed.
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After a brief summary of some basic properties of ideal gases of bosons and of fermions, two many-body Hamiltonians are cited for which ground-state wave functions allow the generation of excited states. But because of the complexity of ground-state many-body wave functions, we then consider properties of reduced density matrices, and in particular, the diagonal element of the second-order density matrix. For both the homogeneous correlated electron liquid and for an assembly of charged bosons, the ground-state pair correlation function g(r) has fingerprints of the zero-point energy of the plasmon modes. These affect crucially the static structure factor S(k), in the long wavelength limit. This is best understood by means of the Ornstein-Zernike direct correlation function c(r), which plays an important role throughout this article. Turning from such charged liquids, both boson and fermion, to superfluid 4 He, the elevated temperature (T) structure factor S(k, T) is related, albeit approximately, to its zero-temperature counterpart, via the velocity of sound, reflecting the collective phonon excitations, and the superfluid density. Finally some future directions are pointed.
Motivated by the fundamental question of the fate of interacting bosons in flat bands, we consider a two-dimensional Bose gas at zero temperature with an underlying quartic single-particle dispersion in one spatial direction. This type of band structure can be realized using the NIST scheme of spin-orbit coupling [Y.-J. Lin, K. Jim\'enez-Garcia, and I. B. Spielman, Nature (London) 471, 83 (2011)], in the regime where the lower-band dispersion has the form ${\ensuremath{\varepsilon}}_{\mathbit{k}}\ensuremath{\sim}{k}_{x}^{4}/4+{k}_{y}^{2}+...$, or using the shaken lattice scheme of Parker et al. [C. V. Parker, L.-C. Ha, and C. Chin, Nat. Phys. 9, 769 (2013)]. We numerically compare the ground-state energies of the mean-field Bose-Einstein condensate (BEC) and various trial wave functions, where bosons avoid each other at short distances. We discover that, at low densities, several types of strongly correlated states have an energy per particle ($\ensuremath{\epsilon}$), which scales with density (nnn) as $\ensuremath{\epsilon}\ensuremath{\sim}{n}^{4/3}$, in contrast to $\ensuremath{\epsilon}\ensuremath{\sim}n$ for the weakly interacting Bose gas. These competing states include a Wigner crystal, quasicondensates described in terms of properly symmetrized fermionic states, and variational wave functions of Jastrow type. We find that one of the latter has the lowest energy among the states we consider. This Jastrow-type state has a strongly reduced, but finite, condensate fraction, and true off-diagonal long-range order, which suggests that the ground state of interacting bosons with quartic dispersion is a strongly correlated condensate reminiscent of superfluid helium-4. Our results show that even for weakly interacting bosons in higher dimensions, one can explore the crossover from a weakly coupled BEC to a strongly correlated condensate by simply tuning the single-particle dispersion or density.