Quantum cluster approach to the spinful Haldane-Hubbard model

We study the spinful fermionic Haldane-Hubbard model at half-filling using a combination of quantum cluster methods: cluster perturbation theory, the variational cluster approximation, and cluster dynamical mean-field theory. We explore possible zero-temperature phases of the model as a function of onsite repulsive interaction strength and next-nearest-neighbor hopping amplitude and phase. Our approach allows us to access the regime of intermediate interaction strength, where charge fluctuations are significant and effective spin model descriptions may not be justified. Our approach also improves upon mean-field solutions of the Haldane-Hubbard model by retaining local quantum fluctuations and treating them nonperturbatively. We find a correlated topological Chern insulator for weak interactions and a topologically trivial Néel antiferromagnetic insulator for strong interactions. For intermediate interactions, we find that topologically nontrivial Néel antiferromagnetic insulating phases and/or a topologically nontrivial nonmagnetic insulating phase may be stabilized.

Figure 1

Schematic depiction of the spinful Haldane-Hubbard model on the honeycomb lattice: the real nearest-neighbor hopping $t$ and complex next-nearest-neighbor hopping $t^{\prime }{e}^{\pm i\varphi }$ give rise to the band structure of a Chern insulator, while the onsite repulsion $U>0$ introduces electronic correlations.

Figure 2

Six-site cluster used in the VCA calculations (shaded area), with Bravais lattice vectors ${\mathit{e}}_1,{\mathit{e}}_2$ (green arrows) and superlattice vectors ${\mathit{E}}_1,{\mathit{E}}_2$ (red arrows).

Figure 3

Ground-state phase diagram of the half-filled spinful Haldane-Hubbard model in the $U\text{−}{t}^{\prime }$ plane for $\varphi =\pi /2$, obtained in VCA. CI: Chern insulator, NMI: nonmagnetic insulator, AF: Néel antiferromagnetic insulator. The CI, NMI, and AF phases all have a nonzero one-particle (charge) gap. The CI-NMI phase boundary (blue circles) corresponds to a closing of the one-particle gap as determined from the one-particle density of states or the dependence of the electron density $n$ on the chemical potential $\mu$ (both methods closely agree). Shown for comparison, the solid gray line is the direct CI-AF transition found in the Hartree-Fock (HF) study of Ref. [16]. The thin vertical green lines at $t^{\prime }=\frac{1}{6},\frac{1}{5}$ are cuts through the phase diagram across which various quantities are plotted in following figures.

Figure 4

Ground-state phase diagram of the half-filled spinful Haldane-Hubbard model in the $U\text{−}\varphi$ plane for $t^{\prime }=\frac{1}{6}$, obtained in VCA. Phase boundaries are determined in the same way as in Fig. 3.

Figure 5

Quantum phase transitions in the half-filled spinful Haldane-Hubbard model, obtained in VCA. (a) One-particle gap $\mathrm{\Delta }$ and AF order parameter as a function of $U$ for $t^{\prime }=\frac{1}{5},\varphi =\pi /2$ and $t^{\prime }=\frac{1}{6},\varphi =0.8$. (b) Potthoff functional $\mathrm{\Omega }$ as a function of the Weiss field $M_z^{\prime }$ for $t^{\prime }=\frac{1}{6}$ and $\varphi =0.8$.

Figure 6

One-particle density of states $\rho \left(\omega \right)$ obtained in VCA for $t^{\prime }=\frac{1}{6},\varphi =0.8$, as a function of frequency $\omega$ in (a) the correlated CI $\left(U=1\right)$, NMI $\left(U=4.3\right)$, and AF $\left(U=5\right)$ phases; (b) at the CI-NMI transition $\left(U=3.81\right)$; and (c) at the NMI-AF transition $\left(U=4.72\right)$.

Figure 7

Generalized Chern number $N_2$ in the $U\text{−}{t}^{\prime }$ plane for $\varphi =\pi /2$, computed from the one-particle CPT Green's function obtained in VCA. The AF phase of Fig. 3 contains a narrow topologically nontrivial region (red region) near the AF-NMI phase boundary. We show for comparison the topologically nontrivial AF region (gray region) found in the Hartree-Fock study of Ref. [16].

Figure 8

Berry curvature as a function of wave vector for $t^{\prime }=0.2,\varphi =\pi /2$ and four values of $U$ associated with the following phases obtained in VCA: (a) CI, (b) NMI, (c) topological AF, and (d) nontopological AF. Blue means positive, red means negative. Only the spin-up contribution is shown. The spin-down contribution is obtained by inverting with respect to the origin, but the two spin contributions to $N_2$ are equal.

Figure 9

Left: cluster-bath system used in CDMFT. The cluster contains two sites (blue symbols) forming the unit cell of the model. The bath sites are indicated by gray squares. They have energies ${\varepsilon }_i \left(i=1,...,4\right)$ and are hybridized with the cluster sites as indicated (dashed lines) with hopping amplitudes ${\theta }_i$. Right: arrangement of this cluster to form a repeated pattern.

Figure 10

One-particle density of states $\rho \left(\omega \right)$ computed from CDMFT for a range of values of $U$, shifted for clarity. On the top panel $\left(t^{\prime }=0\right)$, we observe a transition at $U=5.6$ from the semimetal to the Mott insulator. On the middle panel $(t^{\prime }=0.2,\varphi =\pi /2)$, antiferromagnetism is suppressed by hand and a transition occurs from the CI to the Mott insulator at $U=6.1$. On the bottom panel, antiferromagnetism is allowed to develop and two transitions occur: a CI to a topological AF at $U=3.9$, followed by a topological transition to an ordinary AF at $U=5.8$.

Figure 11

Ground-state phase diagram of the half-filled spinful Haldane-Hubbard model in the $U\text{−}{t}^{\prime }$ plane for $\varphi =\pi /2$, according to CDMFT computations based on the cluster-bath system of Fig. 9. CI: Chern insulator, TAF: topological antiferromagnetic insulator, AF: ordinary antiferromagnetic insulator. The CI and TAF phases have topological charge $N_2=1$. The CI-TAF and TAF-AF phase boundaries are shown in red and blue, respectively. Also shown is the boundary between the CI and the NMI Mott phase when suppressing antiferromagnetism. In that case, the boundary also marks a topological change: the Mott phase has $N_2=-1$. Again, the topologically nontrivial AF region found in the Hartree-Fock study of Ref. [16] is shown (gray area).

Figure 12

Berry curvature as a function of wave vector for $t^{\prime }=0.2,\varphi =\pi /2$ and four values of $U$ associated with the following phases obtained in CDMFT: (a) CI, (b) TAF, (c) AF, and (d) NMI (normal phase). Blue means positive, red means negative. Only the spin-up contribution is shown. The spin-down contribution is obtained by inverting with respect to the origin, but the two spin contributions to $N_2$ are equal.