BCS Theory of Time-Reversal-Symmetric Hofstadter-Hubbard Model

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.

Figure 1

Phase diagrams for $\alpha =p/q$ with $p=1$ and even-$q$ denominators (a) $q=2$, (b) $q=4$, and (c) $q=6$. Critical interaction strength $|U_c|/t$ is shown as a function of $\mu /t$ by a thick black curve separating $V$ (particle vacuum), $N$ (normal), SM (semimetal), quantum SHI (spin-Hall insulator), and SF (superfluid) phases at $k_{B}T={10}^{-4}t$. Total filling $F$ of an SHI phase is shown in parentheses. Red dot at $\mu =0$ shows the triple point at which the $N$, SM, and SF phases meet. Density of states $D(ϵ)$ is displayed in arbitrary units by a thin red curve ($ϵ$ and $\mu$ axes coincide). Horizontal dashed lines mark the band edges including $ϵ=0$.

Figure 2

Similar to Fig. 1 but for odd-$q$ denominators (a) $q=3$, (b) $q=5$, and (c) $q=7$.

Figure 3

SM-SF transition thresholds at $k_{B}T={10}^{-4}t$ for all unique $\alpha =p/q\le 1/2$ ratios with even-$q$ denominators up to 100. (a) While the red points are our self-consistent solutions, the vertical blue lines correspond to $(2W/t{)}^{0.44}$, where $W$ is the total bandwidth of the 2 central bands for a given $\alpha$. In (b) the same data are grouped according to the numerator $p$, where the connecting lines are drawn as a guide, showing the oscillatory dependence.

Figure 4

(Left) Ratios of the critical temperature $k_B{T}_c$ and $\mathrm{\Delta }$ at $k_{B}T={10}^{-4}t$ for $\alpha =1/4$. (Right) $\mathrm{\Delta }/t$ as dash-dotted red lines. Vertical dashed lines correspond to the $|U_c|/t$ thresholds for the (a) SM-SF, (b) $N$-SF, and (c) SHI-SF transitions discussed in the text.