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arXiv:1709.00767 · Physical Review A

Hofstadter-Hubbard model with opposite magnetic fields: Bardeen-Cooper-Schrieffer pairing and superfluidity in the nearly flat butterfly bands

M. Iskin

Published Oct 27, 2017• 10 citations
arXivHTMLPDF

Abstract

Despite the multiband spectrum of the widely known Hofstadter butterfly, it turns out that the pairing correlations of the time-reversal-symmetric Hofstadter-Hubbard model are well described by a single order parameter that is uniform in real space. By exploiting a BCS mean-field theory for the nearly flat butterfly-bands regime of low magnetic-flux limit, here we reveal a number of unusual superfluid properties both in the ground state and at finite temperatures. Our thorough analysis includes but is not limited to the order parameter, condensate, and superfluid fractions, and the critical BCS and Berezinskii-Kosterlitz-Thouless transition temperatures.

DOI
10.1103/physreva.96.043628

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Despite the multiband spectrum of the widely known Hofstadter butterfly, it turns out that the pairing correlations of the time-reversal-symmetric Hofstadter-Hubbard model are well described by a single order parameter that is uniform in real space. By exploiting a BCS mean-field theory for the nearly flat butterfly-bands regime of low magnetic-flux limit, here we reveal a number of unusual superfluid properties both in the ground state and at finite temperatures. Our thorough analysis includes but is not limited to the order parameter, condensate, and superfluid fractions, and the critical BCS and Berezinskii-Kosterlitz-Thouless transition temperatures.
Despite the multi-band spectrum of the widely-known Hofstadter butterfly, it turns out that the pairing correlations of the time-reversal-symmetric Hofstadter-Hubbard model are well-described by a single order parameter that is uniform in real space. By exploiting a BCS mean-field theory for the nearly-flat butterfly-bands regime of low magnetic-flux limit, here we reveal a number of unusual superfluid properties both in the ground state and at finite temperatures. Our thorough analysis includes but is not limited to the order parameter, condensate and superfluid fractions, and the critical BCS and BKT transition temperatures.
Assuming that two-component Fermi gases with opposite artificial magnetic fields on a square optical lattice are well described by the so-called time-reversal-symmetric Hofstadter-Hubbard model, we explore the thermal superfluid properties along with the critical Berezinskii-Kosterlitz-Thouless (BKT) transition temperature in this model over a wide range of its parameters. In particular, since our self-consistent BCS-BKT approach takes the multiband butterfly spectrum explicitly into account, it unveils how dramatically the interband contribution to the phase stiffness dominates the intraband one with an increasing interaction strength for any given magnetic flux.
The competition between the length scales associated with the periodicity of a lattice potential and the cyclotron radius of a uniform magnetic field is known to have dramatic effects on the single-particle properties of a quantum particle, e.g., the fractal spectrum is known as the Hofstadter butterfly. Having this intricate competition in mind, we consider a two-component Fermi gas on a square optical lattice with opposite synthetic magnetic fields for the components, and study its effects on the many-body BCS-pairing phenomenon. By a careful addressing of the distinct superfluid transitions from the semimetal, quantum spin-Hall insulator, or normal phases, we explore the low-temperature phase diagrams of the model, displaying lobe structures that are reminiscent of the well-known Mott-insulator transitions of the Bose-Hubbard model.
We consider a Fermi gas that is loaded onto a square optical lattice and subjected to a perpendicular artificial magnetic field, and determine its superfluid transition boundary by adopting a BCS-like mean-field approach in momentum space. The multiband structure of the single-particle Hofstadter spectrum is taken explicitly into account while deriving a generalized pairing equation. We present the numerical solutions as functions of the artificial magnetic flux, interaction strength, Zeeman field, chemical potential, and temperature, with a special emphasis on the roles played by the density of single-particle states and center-of-mass momentum of Cooper pairs.
We consider multiband BCS superconductors that exhibit time-reversal symmetry and uniform pairing, and analyze their dynamic density and spin structure factors using linear-response theory within the mean-field BCS-BEC crossover framework at zero temperature. Our results for the multi-orbital Hubbard model satisfy the associated f-sum rules in several limits. In particular, in the strong-coupling limit, they coincide with those of a weakly-interacting Bose gas of Cooper pairs, where the low-energy collective Goldstone modes serve as Bogoliubov phonons. We further reveal that the quantum-geometric origin of the low-energy structure factors, along with related observables such as the superfluid-weight tensor and the effective-mass tensor of Cooper pairs, can be traced all the way back to the effective-mass theorem for Bloch bands in this limit. As an illustration, we investigate the pyrochlore-Hubbard model numerically and demonstrate that the Goldstone modes are the only relevant collective degrees of freedom in the flat-band regime.
This paper addresses the transition from the normal to the superfluid state in strongly correlated two dimensional fermionic superconductors and Fermi gases. We arrive at the Berezinskii-Kosterlitz-Thouless (BKT) temperature TBKTT_{\text{BKT}}TBKT​ as a function of \emph{attractive} pairing strength by associating it with the onset of quasi-condensation in the normal phase. Our approach builds on a criterion for determining the BKT transition temperature for atomic gases which is based on a well established Quantum Monte Carlo analysis of the phase space density. This latter quantity, when derived from BCS-BEC crossover theory for fermions, leads to non-monotonic behavior for TBKTT_{\text{BKT}}TBKT​ as a function of the attractive interaction or inverse scattering length. In Fermi gases, this implies a robust superconducting dome followed by a long tail from the flat BEC asymptote, rather similar to what is observed experimentally. For lattice systems we find that TBKTT_{\text{BKT}}TBKT​ has an absolute maximum of the order of 0.1EF0.1 E_F0.1EF​. We discuss how our results compare with those derived from the Nelson Kosterlitz criterion based on the mean field superfluid density and the approach to the transition from below. While there is agreement in the strict mean-field BCS regime at weak coupling, we find that at moderate pairing strength bosonic excitations cause a substantial increase in TBKTT_\text{BKT}TBKT​ followed by an often dramatic decrease before the system enters the BEC regime.