Spin-imbalanced Fermi superfluidity in a Hubbard model on a Lieb lattice

We obtain a phase diagram of the spin-imbalanced Hubbard model on the Lieb lattice, which is known to feature a flat band in its single-particle spectrum. Using the BCS mean-field theory for multiband systems, we find a variety of superfluid phases with imbalance. In particular, we find four different types of FFLO phases, i.e., superfluid phases with periodic spatial modulation. They differ by the magnitude and direction of the center-of-mass momentum of Cooper pairs. We also see a large region of stable Sarma phase, where the density imbalance is associated with zero Cooper pair momentum. In the mechanism responsible for the formation of those phases, the crucial role is played by the flat band, wherein particles can readjust their density at zero energy cost. The multiorbital structure of the unit cell is found to stabilize the Sarma phase by allowing for a modulation of the order parameter within a unit cell. We also study the effect of finite temperature and a lattice with staggered hopping parameters on the behavior of these phases.

Figure 1

(a) The Lieb lattice has three sites in the unit cell, which may assume different values of density or order parameter for spin-imbalanced superfluid phases. In Sec. 7, we consider hoppings $J_{\pm }^{x,y}$ that may have different values in different directions. (b) The band structure of the lattice: There are three bands, two dispersive bands, and a FB in the center. This band structure is symmetric with respect to the zero energy plane, where the FB resides. The FB and the Van Hove singularities give rise to a diverging density of states. If the hoppings $J_{\pm }^{x,y}$ are not all equal, a gap ${\delta }_{\mathrm{gap}}$ opens, thus isolating the FB.

Figure 2

Phase diagram at $U=-4J$ and $\beta J=10$ around the intersection of the Van Hove and the FB singularity lines. The area around this intersection features enhanced superfluidity and a region with nontrivial superfluid phases, wherein we can identify the following phases: FF-II with the FF momentum $\mathbf{q}$ directed along the diagonal of the unit cell, $\eta$-II where the momentum $\mathbf{q}$ saturates at the $M=(\pi ,\pi )$ point of the Brillouin zone, $\eta$-I where $\mathbf{q}$ saturates at the $X=(\pi ,0)$ point, and a Sarma phase (S), where the spin imbalance is present but no relative shift of the two FSs. Solid and dashed lines mark first order and continuous phase transitions, respectively. In the panels on the right-hand side, the value of the three components of the order parameter, $\mathbf{\Delta }$, are plotted. In the $\eta$-II and Sarma phases ${\mathbf{\Delta }}_A=0$, while in the $\eta$-I phase ${\mathbf{\Delta }}_B=0$. The order parameters close to the steep right-hand side boundary of the imbalanced superfluidity lobe are small and go to zero as we approach the continuous phase transition to the normal phase; therefore, at this boundary, there is visible numerical noise.

Figure 3

Thermodynamic potential, $\mathrm{\Omega }\left(\mathbf{q}\right)$, as a function of the FF momentum $\mathbf{q}$ at (a) $\mu =1.4J$ and $h=0.9J$, which is minimized by the FF-II phase and at (b) $\mu =0.8J$ and $h=1.05J$, which is minimized by the $\eta$-I phase.

Figure 4

Phase diagrams for interactions: (a) $U=-3J$ and (b) $U=-3.5J$. Unlike in Fig. 2, there is visible FF-I phase. Also, with decreasing interaction strength, the region with imbalanced superfluidity shrinks until it disappears completely. In panels (c) and (d), we show the configurations the order parameter assumes in the $\eta$-I and $\eta$-II phases, respectively.

Figure 5

(a) The thermodynamic potential as it changes across the $\eta$-I and $\eta$-II phases. At the transition point, the minimum for the parallel q vector goes below that for the diagonal $\mathbf{q}$ vector. The $\eta$-I phase becomes stable and the transition is clearly discontinuous. (b) The thermodynamic potential across the FF and $\eta$-II phases as a function of $q=\left|\mathbf{q}\right|$. The FF wave vector for which the minimum is attained converges continuously to $\pi$, and the double-well shape of the thermodynamic potential suggests a Landau mechanism of a continuous phase transition.

Figure 6

Pairing between various bands for the $\eta$-II phase in panel (a) and the small $\eta$-II region for $\mu =0$ and $h=2J$ in panel (b), see text. In (a) we can separate two contributions to the pairing: an interband pairing between the FB and II-DB and an intraband pairing, mostly within the II-DB. There is also a visible trace of deformation of the majority FS. In (b), the pairing is between particles of opposite spins residing on two different dispersive bands: I-DB and II-DB. The two FSs are identical, as they are formed at ${\mu }_{\uparrow \downarrow }=\pm 2J$. They are far away from the FB, which does not contribute to the pairing in this case.

Figure 7

(a) In the Sarma phase, the order parameter at the central site is zero, ${\mathbf{\Delta }}_A=0$, while it takes opposite but equal magnitude values at sites B and C, ${\mathbf{\Delta }}_B=-{\mathbf{\Delta }}_C$. In panel (b), we show the dependence of $\mathrm{\Omega }$ on the order parameters at fixed $\mu$ as the imbalance $h$ is being increased across the transition between the $\eta$-II phase and the Sarma phase. While the thermodynamic potential is always minimized by ${\mathbf{\Delta }}_A=0$, its symmetry with respect to ${\mathbf{\Delta }}_B$ changes from a symmetric function to a single minimum function. In this way, the sign of ${\mathbf{\Delta }}_B$ is determined as opposite to ${\mathbf{\Delta }}_C$. This intracell modulation of the order parameter allows for the existence of the stable Sarma phase. In (c), we plot the band resolved correlations, $\langle d_{\alpha \mathbf{k}\uparrow }{d}_{\beta \mathbf{q}-\mathbf{k}\downarrow }\rangle$, which show that the main contribution comes from the pairing between majority atoms in the II-DB and minority atoms in the FB (color scale as in Fig. 6). (d) Cumulative densities for majority and minority components along the high symmetry lines: $\mathrm{\Gamma }$–$M$–$X$–$\mathrm{\Gamma }$. There is a clear matching of the density profiles where the interband pairing takes place, accompanied by a jump in densities $n_{\uparrow }-n_{\downarrow }=1$, in accordance with the Luttinger theorem. Around the $M$ point (corners of the BZ) the densities are equal.

Figure 8

Real space configuration of the order parameter ${\mathbf{\Delta }}_{\mathbf{i}\alpha }$ obtained as a solution to Eq. (14). In (a), an eta-II phase is obtained for $\mu =1.4J$ and $h=1.05J$, and in (b) the solution converges to Sarma phase at $\mu =1.4J$ and $h=1.3J$. The shown central fragment comes from the simulation on a $32\times 32$ lattice.

Figure 9

Temperature dependence of the phase diagram at interaction $U=-4J$. The phase diagram is plotted against the temperature $kT/J$ and chemical potential $\mu /J$ along the FB singularity line $\mu =h$. Color intensity denotes the polarization (apart from the normal phase), and lines show the detected phase boundaries.

Figure 10

Order parameters (solid lines with markers), densities (dashed lines) and FF momentum components (dashed lines with markers) of (a) the BCS phase ($\mu =0.8J$ and $h=0.8J$) and (b) the Sarma phase ($\mu =1.4J$ and $h=1.3J$), as the staggering parameter ${\delta }_x$ is increased, while ${\delta }_y=0$. The nature of the phase can change with growing staggering of the hoppings, as detailed in the text. (c), (d) Thermodynamic potential, $\mathrm{\Omega }\left(\mathbf{q}\right)$, as a function of $\mathbf{q}$ for a dimerized lattice at $\mu =1.4J$ and $h=1.3J$, corresponding to the Sarma phase. With growing dimerization in the $x$ direction (${\delta }_y=0$), the initial configuration favoring the Sarma phase at ${\delta }_x=0.3$ (c) turns into one whose minimum does not depend on $q_x$, here ${\delta }_x=0.7$ (d).