Topological Transitions of Gapless Paired States in Mixed-Geometry Lattices

We propose a mixed-geometry system of fermionic species selectively confined in lattices of different geometry. We investigate how such asymmetry can lead to exotic multiband fermion pairing in an example system of honeycomb and triangular lattices. A rich phase diagram of interband pairing with gapped and gapless excitations is found at zero temperature. We find that the two-band contribution of the honeycomb lattices to the paired state helps to stabilize the gapless phase with one or two Fermi surfaces. We also show that the Fermi surface topology further divides the gapless phase into subclasses between which the system undergoes density-driven Lifshitz transitions.

Figure 1

The mixed-geometry lattice proposed and the phase diagram of exotic paired states at zero temperature. (a) Up- and down-spin components are selectively loaded on honeycomb (up-spin, $A$ and $B$) and triangular (down-spin, $A$) lattices. (b) Noninteracting energy dispersions are sketched along the $K-\Gamma$ line in the first Brillouin zone shown in (c). The gap in the up-spin bands is tuned by the on-site energy modulation $\widetilde{ϵ}$. (d) The phase diagram is plotted as a function of the up-spin density $n_{\uparrow }$ and the polarization $P=\frac{n_{\uparrow }-n_{\downarrow }}{n_{\uparrow }+n_{\downarrow }}$ for the case of $\widetilde{ϵ}=0$. The gapless states are characterized by the Fermi surface topology denoted as $z\mathrm{\text{−}}\mathrm{FS}\mathrm{\text{−}}(X)$ indicating $z$ Fermi surfaces being centered at $X\equiv \Gamma$ and/or $K$, as is illustrated in (c). Continuous (dotted lines) and discontinuous (solid lines) Lifshitz transitions are found, and the empty circle indicates the multicritical point.

Figure 2

Characterization of paired states in the gapless phase. Topologically distinct Fermi surface configurations are shown with momentum-resolved order parameter ${\Delta }_k\equiv ⟨{\tilde{a}}_{-\overrightarrow{k}\downarrow }{\tilde{a}}_{\overrightarrow{k}\uparrow }⟩$ in the first Brillouin zone (dashed line) for the cases (b)–(e) sampled from the phase diagram (a) for $\widetilde{ϵ}=0$. Momentum distributions further characterize the gapless paired states with (f) 2-FS and (g) 1-FS. The phase diagram in (a) presents the continuous (dotted lines) and discontinuous (solid lines) Lifshitz transitions as a function of chemical potentials ${\mu }_{\uparrow }$ and ${\mu }_{\downarrow }$, where the multicritical point (empty circle) at ${\mu }_{\downarrow }=0$ corresponds to a step-function-like singularity in the density of states appearing when a noninteracting down-spin band is fully filled.

Figure 3

Effects of the on-site energy modulation $\widetilde{ϵ}$. The phase diagrams are examined for $\widetilde{ϵ}=(a)$ 1.0, (b) 1.4, and (c) 1.8. While the gapped phase is insensitive to $\widetilde{ϵ}$, the gapless phase (shaded) shows different Fermi surface topologies with finite $\widetilde{ϵ}$. In particular, a dominant phase at a large $\widetilde{ϵ}$ is found to be the $2\mathrm{\text{−}}\mathrm{FS}\mathrm{\text{−}}(\Gamma ,K)$ state, which is distinguished from the one at $\widetilde{ϵ}=0$ by its different momentum distribution with a fully polarized area, as is illustrated in the inset of (c).

Figure 4

Schematics of pairing gap opening and Fermi surface formation. The diagrams are plotted along the $K-\Gamma$ line [see Fig. 1c]. The noninteracting particlelike up-spin (${\xi }_{\uparrow }$) and holelike down-spin ($-{\xi }_{\downarrow }$) bands sketched in (a) evolve into (b) the quasiparticle bands with interaction turned on. The boxes indicate the areas of gap formation. The Fermi surface formation depends on the position of the zero-excitation level ($E=0$), as shown in the spectral functions of the down-spin component ${\mathcal{A}}_{\downarrow }(k,-\omega )$ for all typical cases with (c) zero, (e) one, and (d), (f) two Fermi surfaces.