Intertwined order in fractional Chern insulators from finite-momentum pairing of composite fermions

We investigate the problem of intertwined orders in fractional Chern insulators by considering lattice fractional quantum Hall (FQH) states arising from pairing of composite fermions in the square-lattice Hofstadter model. At certain filling fractions, magnetic translation symmetry ensures the composite fermions form Fermi surfaces with multiple pockets, leading to the formation of finite-momentum Cooper pairs in the presence of attractive interactions. We obtain mean-field phase diagrams exhibiting a rich array of striped and topological phases, establishing paired lattice FQH states as an ideal platform to investigate the intertwining of topological and conventional broken symmetry order.

Figure 1

Flux attachment on the square lattice. The Chern-Simons flux, $\mathrm{\Phi }\left(\mathit{x}\right)$, through the plaquette northeast of $\mathit{x}$ is attached to the fermion density $\rho \left(\mathit{x}\right)$ via Gauss' law, $\rho \left(\mathit{x}\right)=\theta \mathrm{\Phi }\left(\mathit{x}\right)$.

Figure 2

Composite Fermi surfaces for the period-two, -three, and -four configurations given in Table 1.

Figure 3

Unit cell used in the mean-field analysis. The net flux per unit cell is ${\varphi }_{*}$ out of the page. Here we take ${\varphi }_{*}=2\pi \left(\frac{5}{8}-\frac{3}{8}\right)=2\pi \frac{1}{4}$ so that the unit cell contains $q_{*}\times q_{*}=4\times 4$ lattice sites. The arrows represent our choice of the Landau gauge, with the net mean-field gauge field taking the form ${\mathit{a}}_{*}=(0,{\varphi }_{*}\alpha )$. Lastly, $(\alpha ,\beta )$ represent the horizontal and vertical coordinates of the lattice sites within a unit cell.

Figure 4

Schematic mean-field phase diagrams as functions of the NN attraction, $\left|V\right|=-V$, and NNN repulsion, $g$, for the fermionic configurations of Table 1. Solid (dashed) black lines correspond to first-order (continuous) transitions. The dotted line separating the stripe-I and -II regions in (b) indicates a crossover. Gapped phases are labeled by the Chern number $C$ of the BdG bands. The gray regions indicate where the energies of the saddle-point equation solutions are too close to numerically deduce which is the ground state. Details of the phases are presented in the main text and illustrated in Figs. 5, 7, and 8.

Figure 5

(a, b) Period-two mean-field configurations. In this and the following figures, the color of the sites indicates the charge density, with darker (lighter) sites corresponding to higher (lower) density. Likewise, the width of the links represents the magnitude of the bond density, $B_{\mathit{x},j}$. The blue arrows represent the pair fields ${\mathrm{\Delta }}_{\mathit{x},j}=\left|{\mathrm{\Delta }}_{\mathit{x},j}\right|e^{i{\theta }_{\mathit{x},j}}$, with length proportional to $|{\mathrm{\Delta }}_{\mathit{x},j}|$ and angle relative to the horizontal given by ${\theta }_{\mathit{x},j}$. The link currents all vanish. (c) Spectrum of the BdG Hamiltonian for mean-field configuration (b). The left panel depicts the two bands closest to $E=0$. The black circle highlights the presence of two Majorana cones along the line $k_y=0$, which are depicted in more detail in the right panel.

Figure 6

Dispersions of the (top row) nodal and (bottom row) gapped stripe phases on finite-size systems with different boundary conditions. In (c), we also plot a horizontal line at $E=0$, representing the topological invariant $\mathcal{M}\left(k_x\right)$ defined in Appendix pp2-s1. Purple (yellow) indicates $\mathcal{M}\left(k_x\right)=-1(+1)$.

Figure 7

Period-three mean-field configurations. For each configuration, the left figure depicts the link currents, $j_k$, as red arrows, while the right figure depicts the pair fields in the same manner as Fig. 5.

Figure 8

Period-four mean-field configurations. The link currents in the stripe configurations (a,b) all vanish.

Figure 9

Composite Fermi surfaces for the period-two, -three, and -four configurations given in Table 2.

Figure 10

Schematic mean-field phase diagrams as functions of the NN attraction, $\left|V\right|=-V$, and NNN repulsion, $g$, for the bosonic configurations listed in Table 2. The bidirectional stripe-I phase in (a) is the same as the bidirectional stripe phase present in Fig. 4.

Figure 11

Real-space configuration of the bidirectional stripe (a) IIA and (b) IIB phases in the bosonic FCI period-two phase diagram [Fig. 10]. The link currents vanish in both configurations. (c) The two BdG bands closest to $E=0$ for IIB (the spectrum for IIA is similar). Despite appearances, there is a very small gap, as the pair fields are nonzero but small.