Topological Mott Insulator at Quarter Filling in the Interacting Haldane Model

While the recent advances in topology have led to a classification scheme for electronic bands described by the standard theory of metals, a similar scheme has not emerged for strongly correlated systems such as Mott insulators in which a partially filled band carries no current. By including interactions in the topologically non-trivial Haldane model, we show that a quarter-filled state emerges with a non-zero Chern number provided the interactions are sufficiently large. We establish this result first analytically by solving exactly a model in which interactions are local in momentum space. The exact same results obtain also for the Hubbard interaction, lending credence to the claim that both interactions lie in the same universality class. From the simulations with determinantal quantum Monte Carlo, we find that the spin structure at quarter filling is ferromagnetic for the topologically non-trivial case. Possible experimental realizations in cold-atom and solid state systems are discussed

1/4 is the new 1/2: Interaction-induced Unification of Quantum Anomalous and Spin Hall Effects

We introduce interactions into two general models for quantum spin Hall physics. Although the traditional picture is that such physics appears when the two lower spinful bands are occupied, that is, half-filling, we show using determinantal quantum Monte Carlo as well as from an exactly solvable model that in the presence of strong interactions, the quarter-filled state instead exhibits the quantum spin Hall effect at high temperature. 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. We argue that it is the consistency with the Lieb-Schultz-Mattis theorem\cite{lsm1,lsm2} for interacting systems with an odd number of charges per unit cell that underlies the emergence of the quantum anomalous Hall effect as a low-temperature symmetry-broken phase of the quantum spin 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 the gap to form using quantum Monte Carlo dynamical cluster approximation simulations. Hence, we predict that topology can obtain in a gapless phase but only in the presence of interactions in dispersive bands. These results are applied to recent experiments on moir\'e systems and shown to be consistent with valley-coherent quantum anomalous Hall physics.Comment: Figure 4e,f added as well as a referenc

Topological Phase Transition without Single-Particle-Gap Closing in Strongly Correlated Systems

We show here that numerous examples abound where changing topology does not necessarily close the bulk insulating charge gap as demanded in the standard non-interacting picture. From extensive determinantal and dynamical cluster quantum Monte Carlo simulations of the half-filled and quarter-filled Kane-Mele-Hubbard model, we show that for sufficiently strong interactions at either half- or quarter-filling, a transition between topological and trivial insulators occurs without the closing of a charge gap. To shed light on this behavior, we illustrate that an exactly solvable model reveals that while the single-particle gap remains, the many-body gap does in fact close. These two gaps are the same in the non-interacting system but depart from each other as the interaction turns on. We purport that for interacting systems, the proper probe of topological phase transitions is the closing of the many-body rather than the single-particle gap

Particle-hole asymmetric ferromagnetism and spin textures in the triangular Hubbard-Hofstadter model

In a lattice model subject to a perpendicular magnetic field, when the lattice constant is comparable to the magnetic length, one enters the "Hofstadter regime," where continuum Landau levels become fractal magnetic Bloch bands. Strong mixing between bands alters the nature of the resulting quantum phases compared to the continuum limit; lattice potential, magnetic field, and Coulomb interaction must be treated on equal footing. Using determinant quantum Monte Carlo (DQMC) and density matrix renormalization group (DMRG) techniques, we study this regime numerically in the context of the Hubbard-Hofstadter model on a triangular lattice. In the field-filling phase diagram, we find a broad wedge-shaped region of ferromagnetic ground states for filling factor $\nu \lesssim 1$, bounded by incompressible states at filling factor $\nu = 1$. For magnetic field strengths $\Phi/\Phi_0 \lesssim 0.4$, we observe signatures of SU(2) quantum Hall ferromagnetism in the lowest magnetic Bloch band; however, we find no numerical evidence for conventional quantum Hall skyrmions. At large fields $\Phi/\Phi_0 \gtrsim 0.4$, above the ferromagnetic wedge, we observe a low-spin metallic region with spin correlations peaked at small momenta. We argue that the phenomenology of this region likely results from exchange interaction mixing fractal Hofstadter subbands. The phase diagram derived beyond the continuum limit points to a rich landscape to explore interaction effects in magnetic Bloch bands.Comment: 15 pages, 15 figure

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

Abstract 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