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arXiv:0712.1828 · Physical Review B

Superfluid-insulator transition in Fermi-Bose mixtures and the orthogonality catastrophe

Gil Refael, Eugene Demler

Published Apr 10, 2008• 27 citations
arXivHTMLPDF

Abstract

The superfluid-insulator transition of bosons is strongly modified by the presence of fermions. Through an imaginary-time path-integral approach, we derive the self-consistent mean-field transition line, and account for both the static and dynamic screening effects of the fermions. We find that an effect akin to the fermionic orthogonality catastrophe, arising from the fermionic screening fluctuations, suppresses superfluidity. We analyze this effect for various mixture parameters and temperatures, and consider possible signatures of the orthogonality-catastrophe effect in other measurables of the mixture.

DOI
10.1103/physrevb.77.144511

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The superfluid-insulator transition of bosons is strongly modified by the presence of fermions. Through an imaginary-time path-integral approach, we derive the self-consistent mean-field transition line, and account for both the static and dynamic screening effects of the fermions. We find that an effect akin to the fermionic orthogonality catastrophe, arising from the fermionic screening fluctuations, suppresses superfluidity. We analyze this effect for various mixture parameters and temperatures, and consider possible signatures of the orthogonality-catastrophe effect in other measurables of the mixture.
It is well known that bosons on an optical lattice undergo a second-order superfluid-insulator transition (SIT) when the lattice potential increases. In this paper, we study SIT when fermions coexist with the bosons. We find that the critical properties of particle-hole symmetric SIT with dynamical exponent z=1z=1z=1 is modified when fermions are present; it either becomes a fluctuation-driven first-order transition or a different second-order transition. On the other hand, the more generic particle-hole asymmetric (with z=2z=2z=2) SIT is stable against coupling with fermions. We also discuss pairing interaction between fermions mediated by quantum critical fluctuations near SIT and show that the fermion pairing and/or superfluid transition temperature peaks near SIT when a single species of fermion is present.
We develop a strong-coupling theory for the superfluidity of fermion pairing phase in a Bose-Fermi mixture. Dynamical screening, self-energy renormalization, and a pairing gap function are included self-consistently within the adiabatic limit (i.e., the phonon velocity is much smaller than the Fermi velocity). An analytical solution for the transition temperature (T(c)) is derived within reasonable approximations. Using typical parameters of a 40K-87Rb mixture, we find that the calculated T(c) is several times larger than that obtained in the weak coupling theory, and can be up to several percent of the Fermi temperature.
We examine the effect of boson-fermion interaction in a one-dimensional Bose-Fermi mixture by using the density matrix renormalization group method. We show that the boson superfluidity is enhanced by fermions for a weak boson-fermion coupling at an approximate integer boson filling factor (e.g., 0.935≤ρb≤1.00.935\le ρ_b \le 1.00.935≤ρb​≤1.0), and this enhancement is produced both in a fermion metallic state and in a fermion insulating state. A metal-insulator phase transition of fermions induced by boson-fermion interaction is observed even though there is no fermion-fermion interaction in the parent Hamiltonian. Furthermore, we find that the boson superfluid order and density wave order can coexist in a deep fermion Mott region. All these features could be measured in future experiments and open up the possibility of detecting the new physical effect in the Bose-Fermi mixture.
Building on the work of Fisher et al. [Phys. Rev. B 40, 546 (1989)], we develop the perturbation theory for the Bose-Hubbard model and apply it to calculate the effects of a degenerate gas of spin-polarized fermions interacting by contact interactions with the constituent bosons. For the single-band Bose-Hubbard model, we find that the net effect of the screening of the boson on-site interaction by the fermions is to suppress the Mott-insulating lobes in the Bose-Hubbard phase diagram. For the more general multiband model, we find that, in addition to the fermion screening effects, the virtual excitations of the bosons to the higher Bloch bands, coupled with the contact interactions with the fermions, result in an effective increase (decrease) of the boson on-site repulsion (hopping parameter). If the higher-band renormalization of the boson parameters is dominant over the fermion screening of the interaction, the Mott-insulating lobes in the Bose-Hubbard phase diagram are enhanced for either sign of the Bose-Fermi interactions, consistent with the recent experiments.
The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc{d}_{c}dc​ which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on 4He^{4}\mathrm{He}4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.
An effective interaction between fermions in a Bose-Fermi mixture is derived. It is induced by density fluctuations of the bosonic background. The contributions from states containing both one and two virtual phonons are taken into account self-consistently. The time dependence of the effective interaction has been removed by assuming that the velocity of the fermions at the Fermi surface is much larger than the sound velocity in the Bose gas. This assumption is more appropriate for the actual experimental situations than the usual approximation of neglecting retardation effects. The effective interaction turns out to be attractive and, as a consequence, can give rise to a superconducting phase in the Fermi component of the mixture. The fermions are considered in only one magnetic state, so that pairing can be effective only in odd-lll channels. It has been found that the onset of the superconducting phase can occur at temperatures (>100nK)(>100\mathrm{nK})(>100nK) of the same order of magnitude as the Fermi temperature $(\ensuremath{\sim}300\mathrm{nK}),$ and the energy gap in the excitation spectrum is a small fraction $(\ensuremath{\sim}1%)$ of the Fermi energy.