1/4 is the new 1/2 when topology is intertwined with Mottness

In non-interacting systems, bands from non-trivial topology emerge strictly at half-filling and exhibit either the quantum anomalous Hall or spin Hall effects. Here we show using determinantal quantum Monte Carlo and an exactly solvable strongly interacting model that these topological states now shift to quarter filling. A topological Mott insulator is the underlying cause. The peak in the spin susceptibility is consistent with a possible ferromagnetic state at T = 0. The onset of such magnetism would convert the quantum spin Hall to a quantum anomalous Hall effect. While such a symmetry-broken phase typically is accompanied by a gap, we find that the interaction strength must exceed a critical value for this to occur. Hence, we predict that topology can obtain in a gapless phase but only in the presence of interactions in dispersive bands. These results explain the recent quarter-filled quantum anomalous Hall effects seen in moiré systems.


Non-interacting results for the KM-HH model at lower temperatures
Here we show the non-interacting compressibility and TRI compressibility for the KM-HH model at lower temperature β = 20 in Supplementary Fig. 2 for the flat-band case (a) and the original dispersive case (b).In either case, the integer quantum Hall effects become sharper at this lower temperature.For the TRI compressibility, the inverse slope of the leading middle line crossing ⟨n⟩ = 2 gives the spin Chern number C s = 2.

a b c d
If the temperature is too high, we would not observe the non-trivial topology emerging at quarter-filling, no matter how large U is.In the Fig. 1 of the main text, at U = 12, we need to reach β = 3 to discern the dip in the compressibility and hence the QSH feature.If for a smaller U , we can expect the onset temperature for such a crossover (from a featureless metal to a high-temperature QAH/QSH effect) to decrease.For example, Supplementary Fig. 3 shows that β = 4 is needed to observe the high-temperature QAH behavior at U = 3. DQMC results for the original (dispersive) KM-HH model Here we show the DQMC simulation results in Supplementary Fig. 6 for the original (dispersive) KM-HH model at t ′ = 0.3, ψ = 0.5.Comparing panels (a) and (e), we find that the strong correlation induces non-trivial topology at zero-field for 1/4− and 3/4−filling with a "TRI" Chern number C TRI = 1, which is akin to that obtained for the spinful Haldane-Hofstader-Hubbard model [? ].This means QSH effects emerge at the zero-field 1/4− and 3/4−filled KM-HH systems with a spin Chern number C s = 1.The compressibility and spin susceptibility for the KM-HH model in Supplementary Fig. 6(f,g) are essentially the same as those for the BHZ-HH model in Fig. 3(b,c) of the main text.In the presence of strong electron-electron interaction, the compressibility displays a pair of zero-mode LLs at quarter-filling where the spin susceptibility has ridges.The relatively noticeable differences between the KM-HH and BHZ-HH models lie in the magnetization at the band edge as is evident from panels Supplementary Fig. 6h relative to Fig. 3d in the main text.Note the sign of the magnetization is not particularly important as a discrepancy already arises with the non-interacting case and hence likely arises from the difference in the lattices.In short, we observe the same QSH effects driven by Mottness in the quarter-filled KM-HH model with large enough Hubbard interaction and adds to the ubiquity of this correlation-driven effect.Unlike the flat-band case, here W −0 = ∆ = 2.There is no such intermediate region of U where a QAH effect emerges at high temperatures as in Fig. 1(d-f).As U becomes sufficiently large, the QSH effect appears.Here we calculate the average sign from DQMC simulations for the BHZ-HH model at zero field in Supplementary Fig. 7(a,b).An interesting observation is that half-filling is not sign-problem-free, unlike the KM-Hubbard case in Supplementary Fig. 1b.Previous studies[???] on the BHZ-Hubbard model at half-filling used dynamical mean field theory (DMFT) and found no sign problem when using quantum Monte Carlo as the impurity solver.To make a connection with these results, we further check the average sign in DCA simulations at different cluster sizes shown in Supplementary Fig. 7c.DCA is a cluster version of DMFT and DCA reduces to DMFT when the cluster size N = 1.In Supplementary Fig. 7c, we find that in the DMFT limit (L = 1), half-filling is exactly sign-problem-free, agreeing with the previous study.However, its average sign decreases dramatically as L increases (even at L = 2 the average sign deviates slightly from 1).Another observation is that if we turn on a staggered sublattice potential C ν in the KM-Hubbard model, there would be a sign problem at half-filling despite time-reversal symmetry.Note that all previous quantum Monte-Carlo studies[??] focused on the half-filled KM-Hubbard model with C ν = 0 and found it to be sign-problem-free.
The explanation for this is as follows.Recall that a famous sign-problem-free example is the half-filled single-band Hubbard model on a square lattice with only nearest-neighbor hopping.This arises because the Hamiltonian is unchanged under a particle-hole transformation c jσ → d † jσ (−1) j with j = ±1 for sub-lattice A and B. This particle-hole symmetry is broken if we introduce the next-nearest-neighbor hopping t ′ which leads to a minus sign under this transformation.In the KM-Hubbard model with C v = 0, the Hamiltonian under this transformation is equivalent to flipping a spin (↑→↓, ↓→↑).The Hamiltonian is effectively unchanged regarding the sign problem.If C v is finite, then it acquires a minus sign under this transformation and thus the Hamiltonian becomes different.Therefore, a sign problem is present in Supplementary Fig. 8 for the half-filling KM-Hubbard model when C v is finite, despite the fact that time-reversal symmetry is still maintained.In the BHZ-Hubbard model, the transformation involving j = ±1 involves different orbitals rather than sub-lattices because each unit cell contains two orbitals.Then under this particle-hole transformation, the diagonal term M +t cos(k x )+t cos(k y ) changes a sign, leading to a change in the Hamiltonian.Due to the lack of this symmetry, there is a sign problem at half-filling as shown in Supplementary Fig. 7 for DQMC simulations.That being said, this problem may be studied using the sign-problem-free quantum Monte Carlo method in Majorana representation [? ].Real-space spin correlation at half-filling under strong correlation In the presence of strong correlation, at half-filling the non-interacting QSH order is destroyed and turns into a topologically trivial Mott insulator.We expect the Mott insulator to show anti-ferromagnetism (AF) robust to an external magnetic field in a bipartite lattice.However, Fig. 3(c) in the main text shows a peak in the spin correlation (Q = 0) at half-filling, which is inconsistent with this expectation.To further explore this issue, we look into its zero-frequency spin correlation in real and momentum space at two temperatures β = 3 and β = 5, shown in Supplementary Fig. 15.Even at the relatively higher temperature of β = 3, the system already shows an AF pattern in the central region.As the temperature decreases, the AF region enlarges and further dominates the cluster.Thus, eventually at low enough temperature, AF order prevails in the Mott insulator, as expected.Note that here the smallest magnetic flux ϕ/ϕ 0 = 1/36 is used to reduce the finite-size effect at lower temperatures [? ].This does not affect our conclusion because the influence from the magnetic field is negligible compared to the AF order.We conduct the DQMC simulation for the corresponding bilayer flat-band KM-HH model at an intermediate Hubbard interaction U = 1.5t, β = 12/t and present the compressibility, spin susceptibility and magnetization in Supplementary Fig. 18(d-f) respectively.The system exhibits a QAH effect with spin polarization at 1/8-filling (⟨n⟩ = 1) and a QSH effect at quarter-filling (⟨n⟩ = 1), as expected given the similarity between single-layer and bilayer KM-HH models.Since the Hubbard interaction mixes the non-interacting bands and the resulting QAH state is spin polarized, this necessarily yields valley-coherence given the valley assignment in Supplementary Fig. 17(a).
The presence of an emergent topologically non-trivial state at 3/8-filling (⟨n⟩ = 3) is special to the bilayer model.It has the feature of a QAH effect, namely the single Landau level in the compressibility (Supplementary Fig. 18(d)) accompanied by the peak in the spin susceptibility (Supplementary Fig. 18(e)).It is also likely to have the QSH feature.This obtains because the system can not be fully polarized at ⟨n⟩ = 3.Some band must be doubly occupied, thereby explaining the white region in the magnetization in (Supplementary Fig. 18(f)).Also, the helical currents from the ⟨n⟩ = 2 QSH state probably play a role in the ⟨n⟩ = 3 edge state because their edge dispersion extends to the upper bands, which would not be filled until ⟨n⟩ > 4. Therefore, for the state at ⟨n⟩ = 3, we have either C ↑ = 2, C ↓ = −1 or C ↑ = 1, C ↓ = −2 depending on the polarization while the ⟨n⟩ = 1 state has either C ↑ = 1, C ↓ = 0 or C ↑ = 0, C ↓ = −1.
Supplementary Figure 3: The compressibility for the generalized flat-band KM-HH model (t ′ = 0.3, ψ = 0.81) as a function of magnetic flux and density with fixed for different temperatures a β = 3, b β = 4, c β = 5.The crossover from QAH to QSH features at high temperatures In Fig. 1 of the main text, we jump directly from the QAH effect at U = 3t (second row) to the QSH effect at U = 12t (third row) and also change the temperature.Here we fill in the gap to see how this happens by gradually increasing U while keeping β = 3/t in Supplementary Fig. 4. As U increases, the right Landau levels start to appear while the left Landau levels become more prominent, suggesting a crossover instead of a transition.compressibility for the generalized flat-band KM-HH model (t ′ = 0.3, ψ = 0.81) as a function of magnetic flux and density at a U = 3, b U = 6, c U = 9 and d U = 12.The inverse temperature for all cases is β = 3.The comparison between DCA and DQMC simulations In Supplementary Fig. 5, we show a good benchmark between DCA and DQMC on the ⟨n⟩ versus µ relation for the KM-Hubbard model (t ′ = 0.3, ψ = 0.63) at U = 2, β = 8 and U = 5, β = 4, supporting Fig. 5 in the main text.a b Supplementary Figure 5: The comparison on ⟨n⟩ versus µ relation between DQMC on a 6 × 6 × 2 cluster and DCA on a 2 × 2 × 2 cluster at for the generalized KM-Hubbard model (t ′ = 0.3, ψ = 0.63) at a U = 2, β = 8 and b U = 5, β = 4.
Figure 6: DQMC results for the KM-HH-TRI and KM-HH models at U = 0 (first row) and U/t = 12 (second row).The first column shows the compressibility χ as a function of magnetic flux and electron density for the KM-HH-TRI model.The second to fourth columns show χ, spin susceptibility χs and magnetization ⟨mz⟩ respectively, for the KM-HH model.The temperature is β = 3/t.SUPPLEMENTAL DQMC AND DCA RESULTS FOR THE BHZ-HH MODEL Sign problem of DQMC and DCA simulations for the BHZ-HH model Supplementary Figure 7: Average sign of DQMC simulations as a function of density for the BHZ-HH model (M = 1) at zero field and a U = 8,β = 4; b U = 12,β = 3.The DQMC simulation is conducted on a N = 6 × 6 cluster.Panel c shows the average sign of DCA simulations as a function of cluster size at half-filling and U = 8, β = 5.
Supplementary Figure 15: The static spin correlation at half-filling (⟨n⟩ = 2) in real spaces respectively for BHZ-HH (a, c) at M = 1 and KM-HH (b, d) models at ψ = 0.5, t ′ = 0.3 both with U = 12 (in the model-specific energy scale) and zero magnetic field.The inverse temperatures are β = 3 for panels (a, b) and β = 5 for panels (c, d).