Competing orders in the Hofstadter $t–J$ model

The Hofstadter model describes noninteracting fermions on a lattice in the presence of an external magnetic field. Motivated by the plethora of solid-state phases emerging from electron interactions, we consider an interacting version of the Hofstadter model, including a Hubbard repulsion $U$. We investigate this model in the large-$U$ limit corresponding to a $t–J$ Hamiltonian with an external (orbital) magnetic field. By using renormalized mean-field theory supplemented by exact diagonalization calculations of small clusters, we find evidence for competing symmetry-breaking phases, exhibiting (possibly coexisting) charge, bond, and superconducting orders. Topological properties of the states are also investigated, and some of our results are compared to related experiments involving ultracold atoms loaded on optical lattices in the presence of a synthetic gauge field.

Figure 1

“Phase diagram” vs. electron filling $\rho$ and magnetic flux $\mathrm{\Phi }$ showing the various phases presented in Table 1. Circles are nonpolarized (singlet) states while squares represent ferromagnets. Black symbols correspond to uniform solutions. Red, green, and blue symbols encode symmetry-breaking supercells of size $4\times 4, \sqrt{2}\times \sqrt{2}$, and $2\times 2$, respectively.

Figure 2

Comparison between RMFT and ED energies (per magnetic $4\times 4$ unit cell). (a) Kinetic energy ($E_{\mathrm{kin}}$) and (b) magnetic (potential) energy ($E_{\mathrm{pot}}$) vs. inserted flux $\mathrm{\Phi }$. The doping level is fixed to $\delta =1/8$ and $J=0.3t$. The numerical values are given in the Appendix.

Figure 3

Schematic patterns and results for the states in Sec. 3. (a)–(d) show the current and hopping patterns of each state within the $4\times 4$ sublattice. The widths of the underlying orange bars and black arrows represent the magnitudes of hopping and current on each bond separately. The flows of current are indicated by the arrow directions. The numerical values are shown in Fig. 5.

Figure 4

Band structure for the three lowest energy bands for (a) ${\nu }^{*}=2/7$ and (b) ${\nu }^{*}=2/5$. At this doping, the first two bands are filled. Note that in (a) the first two bands are almost degenerate.

Figure 5

Distribution of the phases $A_{ij}$ on the bonds of $4\times 4$ and $2\times 2$ unit cells (on the 2-torus) for the flux densities $\mathrm{\Phi }$ considered in this work. Arrows again indicate the directions of current and negative signs stand for opposite flows. The flux density $\mathrm{\Phi }=1/4$ has only two different bonds (bond 1 and 2). The lower panel shows detailed numbers of variables for the patterns in Fig. 3.

Figure 6

(a) Vector potential gauge choice for $\mathrm{\Phi }=q/16,q=0,\cdots ,15$. Periodic boundary conditions are assumed. $A_{ij}$ in units of $F=2\pi \mathrm{\Phi }$ is given by the integer number shown between site $i$ and $j$, with positive sign if the respective arrow points from site $i$ to site $j$, and negative sign otherwise. (b) Spectrum $E\left(\mathbf{\varphi }\right)$ as a function of inserted flux for $\nu =1/5$. The Chern number evaluates to 6, however, there is no indication for a topological GSD.

Figure 7

Lanczos ED spectrum of $H$ for various values of $\nu$, with $\mathrm{\Phi }$ and $\rho$ as given by Table 4. When there is no magnetization, only the $S_z=0,\pm 1$ sector is shown.

Figure 8

RMFT energy spectrum as a function of staggered potential $\delta$ with Chern numbers for each band shown beside the figure. For $\mathrm{\Gamma }>2t$ the system is topologically trivial with the Chern number $C$ of the bands zero. At the transition point, the band gap closes and it becomes topologically nontrivial with $C=1$ for the lowest band. After passing the transition point, the gap opens again and the lowest band now possesses a Chern number of $-1$. Notice that within this chosen reduced BZ, each of the four bands originating from the $2\times 2$ modulation is folded into four sub-bands, producing a total of 16 bands.