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Physical review. B, Condensed matter

Theory of boson mixtures

Michael David Miller

Published Nov 1, 1978• 34 citations
Publisher

Abstract

We investigate the thermodynamic ground state of binary boson mixtures using a variational ansatz. With a pair-independent trial function we map the isotopic boson mixture onto a quantum theorem of corresponding-states picture which enables us to conclude that all such isotopic boson mixtures completely phase-separate at zero temperature. We also investigate mixtures of spin-aligned hydrogen (H\ensuremath{\uparrow}) and spin-aligned tritium (T\ensuremath{\uparrow}) with $^{4}\mathrm{He}$. We find that these systems also prefer to completely phase-separate. However, due to important differences in the sizes of "cores" of the interaction, allowance must be made for different correlations between different types of pairs. We then find that at zero pressure the zero-concentration chemical potential for H\ensuremath{\uparrow} in $^{4}\mathrm{He}$ is 40 K and for T\ensuremath{\uparrow} in $^{4}\mathrm{He}$ it is 0.5 K.

DOI
10.1103/physrevb.18.4730

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We investigate the thermodynamic ground state of binary boson mixtures using a variational ansatz. With a pair-independent trial function we map the isotopic boson mixture onto a quantum theorem of corresponding-states picture which enables us to conclude that all such isotopic boson mixtures completely phase-separate at zero temperature. We also investigate mixtures of spin-aligned hydrogen (H\ensuremath{\uparrow}) and spin-aligned tritium (T\ensuremath{\uparrow}) with 4He^{4}\mathrm{He}4He. We find that these systems also prefer to completely phase-separate. However, due to important differences in the sizes of "cores" of the interaction, allowance must be made for different correlations between different types of pairs. We then find that at zero pressure the zero-concentration chemical potential for H\ensuremath{\uparrow} in 4He^{4}\mathrm{He}4He is 40 K and for T\ensuremath{\uparrow} in 4He^{4}\mathrm{He}4He it is 0.5 K.
Experimental realizations of a variety of atomic binary Bose-Fermi mixtures have brought opportunities for studying composite quantum systems with different spin statistics. The binary atomic mixtures can exhibit a structural transition from a mixture into phase separation as the boson-fermion interaction increases. By using a path-integral formalism to evaluate the grand partition function and the thermodynamic grand potential, we obtain the effective potential of binary Bose-Fermi mixtures. Thermodynamic quantities in a broad range of temperatures and interactions are also derived. The structural transition can be identified as a loop of the effective potential curve, and the volume fraction of phase separation can be determined by the lever rule. For $^{6}\mathrm{Li}\ensuremath{-}^{7}\mathrm{Li}$ and $^{6}\mathrm{Li}\ensuremath{-}^{41}\mathrm{K}$ mixtures, we present the phase diagrams of the mixtures in a box potential at zero and finite temperatures. Due to the flexible densities of atomic gases, the construction of phase separation is more complicated when compared to conventional liquid or solid mixtures where the individual densities are fixed. For harmonically trapped mixtures, we use the local density approximation to map out the finite-temperature density profiles and present typical trap structures, including the mixture, partially separated phases, and fully separated phases.
We review the recent activity in the theoretical description of spin-polarized atomic hydrogen and its isotopes at very low temperatures. Spin-polarized hydrogen is the only system in nature that remains stable in the gas phase even in the zero temperature limit due to its small mass and weak interatomic interaction. Hydrogen and its heavier isotope tritium are bosons, the heavier mass of tritium producing a self-bound (liquid) system at zero temperature. The other isotope, deuterium, is a fermion with nuclear spin one making possible the study of three different quantum systems depending on the population of the three degenerate spin states. From the theoretical point of view, spin-polarized hydrogen is specially appealing because its interatomic potential is very accurately known making possible its precise quantum many-body study. The experimental study of atomic hydrogen has been very difficult due to its high recombination rate, but it finally led to its Bose-Einstein condensate state in 1998. Degeneracy has also been observed in thin films of hydrogen adsorbed on the 4He surface allowing for the possibility of observing the Berezinskii–Kosterlitz–Thouless superfluid transition.
We have investigated self-bound ultradilute bosonic binary mixtures at zero temperature within density functional theory using a local density approximation. We provide the explicit expression of the Lee-Huang-Yang correction in the general case of heteronuclear mixtures, and investigate the general thermodynamic conditions which lead to the formation of self-bound systems. We have determined the conditions for stability against the evaporation of one component, as well as the mechanical and diffusive spinodal lines. We have also calculated the surface tension of the self-bound state as a function of the interspecies interaction strength. We find that relatively modest variations of the latter result in order-of-magnitude changes in the calculated surface tension. We suggest experimental realizations which might display the metastability and phase separation of the mixture when entering regions of the phase diagram characterized by negative pressures. Finally, we show that these droplets may sustain stable vortex and vortex pairs.
We consider a dilute homogeneous mixture of bosons and spin-polarized fermions at zero temperature. We first construct the formal scheme for carrying out systematic perturbation theory in terms of single particle Green's functions. We especially focus on the description of the boson-fermion interaction. To do so we need to introduce the renormalized boson-fermion TTT matrix, which we determine to second order in the boson-fermion sss-wave scattering length. We also discuss how to incorporate the usual boson-boson TTT matrix in mean field approximation to obtain the total ground-state properties of the system. The next-order term beyond mean field stems from the boson-fermion interaction and is proportional to aBFkF.{a}_{\mathrm{BF}}{k}_{\mathrm{F}}.aBF​kF​. The total ground-state energy density to this order is the sum of the kinetic energy of the free fermions, the boson-boson mean-field interaction, the usual mean-field contribution to the boson-fermion interaction energy, and the first boson-fermion correction beyond mean field. We also compute the bosonic and the fermionic chemical potentials, the compressibilities, and the modification to the induced fermion-fermion interaction. We discuss the behavior of the total ground-state energy and the importance of the correction beyond mean field for various parameter regimes, in particular considering mixtures of 6Li{}^{6}\mathrm{Li}6Li and 7Li{}^{7}\mathrm{Li}7Li and of 3He{}^{3}\mathrm{He}3He and 4He.{}^{4}\mathrm{He}.4He. Moreover, we determine the modification of the induced fermion-fermion interaction due to the effects beyond mean field. We show that there is no effect on the depletion of the Bose condensate to first order in the boson-fermion scattering length aBF.{a}_{\mathrm{BF}}.aBF​.
We study the thermodynamic properties of binary Bose mixtures by developing a beyond mean-field Popov theory which properly includes the effects of quantum and thermal fluctuations in both the density and spin channels. Results for key thermodynamic quantities, such as the isothermal compressibility and the magnetic susceptibility, are derived from a perturbative calculation of the grand-canonical potential. We find that thermal fluctuations can play a crucial role on the miscibility condition of a binary mixture, favoring phase separation at finite temperature even if the mixture is soluble at zero temperature, as already anticipated in a previous work [M. Ota et al., Phys. Rev. Lett. 123, 075301 (2019)]. We further investigate the miscibility condition for binary mixtures in the presence of asymmetry in the intraspecies interactions, as well as in the masses of the two components. Furthermore, we discuss the superfluid behavior of the mixture and the temperature dependence of the Andreev-Bashkin effect.