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arXiv:0902.0172 · Physical Review B

Effects of fermions on the superfluid-insulator phase diagram of the Bose-Hubbard model

Sumanta Tewari, Roman M. Lutchyn, S. Das Sarma

Published Aug 14, 2009• 16 citations
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Abstract

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.

DOI
10.1103/physrevb.80.054511

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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.
We study the phase diagram of a degenerate gas of bosons in an optical lattice, described by the boson Hubbard Hamiltonian, interacting with spin-polarized fermions. Virtual transitions of the bosons to the higher Bloch bands, coupled with the contact interspecies interactions of either sign, result in an effective increase of the bosonic on-site repulsion. If this renormalization of the on-site potential is dominant over the fermionic screening of the boson interactions, superfluid coherence of the Bose-Hubbard system will be reduced by the fermions. We discuss experimental implications for cold atomic systems.
The experimentally observed loss of superfluidity by introducing fermions to the boson Hubbard system on an optical lattice is explained. We show that the virtual transitions of bosons to higher Bloch bands, coupled with contact boson-fermion interactions of either sign, result in an effective increase of the boson on-site repulsion. If this renormalization of the on-site potential is dominant over the fermion screening of the boson interactions, the Mott insulating lobes of the Bose-Hubbard phase diagram will be enhanced for either sign of the boson-fermion interactions. We discuss the implications for cold-atom experiments where expansion of the Mott lobes by fermions has been conclusively established.
Using an imaginary-time path integral approach, we develop the perturbation theory suited to the boson Hubbard model and apply it to calculate the effects of a dilute gas of spin-polarized fermions weakly interacting with the bosons. The full theory captures both the static and the dynamic effects of the fermions on the generic superfluid-insulator phase diagram. We find that, in a homogenous system described by a single-band boson Hubbard Hamiltonian, the intrinsic perturbative effect of the fermions is to generically suppress the insulating lobes and to enhance the superfluid phase.
The phase transitions in the Bose-Fermi-Hubbard model are investigated.The boson Green's function is calculated in the random phase approximation (RPA) and the formalism of the Hubbard operators is used.The regions of existence of the superfluid and Mott insulator phases are established and the (µ, t) (the boson chemical potential vs. hopping parameter) phase diagrams are built at different values of boson-fermion interaction constant (in the regimes of fixed chemical potential or fixed concentration of fermions).The effect of temperature change on this transition is analyzed and the phase diagrams in the (T, µ) plane are constructed.The role of thermal activation of the ion hopping is investigated by taking into account the temperature dependence of the transfer parameter.The reconstruction of the Mott-insulator lobes due to this effect is analyzed.
We study phase diagrams of one-dimensional bosons with contact interactions in the presence of a lattice. We use the worm algorithm in continuous space and focus on the incommensurate superfluid--Mott-insulator transition. Our results are compared to those from the one-band Bose-Hubbard model. When Wannier states are used to determine the Bose-Hubbard model parameters, the comparison unveils an apparent breakdown of the one-band description for strong interactions, even for the Mott-insulating state with an average of one particle per site (n=1n=1n=1) in deep lattices. We introduce an inverse confined scattering analysis to obtain the ratio U/JU/JU/J, with which the Bose-Hubbard model provides correct results for strong interactions, deep lattices, and n=1n=1n=1.
In this study, we perform a strong-coupling expansion up to third order of the hopping parameter ttt for the spin-1 Bose-Hubbard model with antiferromagnetic interaction. As expected from previous studies, the Mott-insulator phase is considerably more stable against the superfluid phase when filling with an even number of bosons than when filling with an odd number of bosons. The phase-boundary curves are consistent with the perturbative mean-field theory in the limit of infinite dimensions. The critical value of the hopping parameter tC{t}_{\mathrm{C}}tC​ at the peak of the Mott lobe depends on the antiferromagnetic interaction. This result indicates the reliability of the strong-coupling expansion when U2{U}_{2}U2​ possesses large (intermediate) values for Mott lobe with an even (odd) number of bosons. Moreover, in order to improve our results, we apply a few extrapolation methods up to infinite order in ttt. The fitting results of the phase-boundary curves agree better with those of the perturbative mean-field approximation. In addition, the linear fit error of tC{t}_{\mathrm{C}}tC​ is very small for the strong antiferromagnetic interaction.