Quantum Magnetism in Wannier-Obstructed Mott Insulators

TL;DR

This work introduces a nonorthogonal Wannier basis to address strong-coupling magnetism in Wannier obstructed bands, enabling a systematic diagrammatic projection to an effective spin model that incorporates novel exchange channels arising from orbital overlap. By constructing localized, nonorthogonal orbitals via Hartree-energy minimization and projecting the Hamiltonian onto a spin space, the authors derive a spin Hamiltonian with Heisenberg and chiral terms, plus higher-order interactions, in which nonorthogonality can dominate near the FM-AFM crossover. Applying the framework to a Kagome lattice with both Chern and fragile topological bands demonstrates stable ferromagnetism up to a finite bandwidth $W \sim U g$, and reveals a potential for frustrated or chiral spin phases near the crossover, offering a mechanism to explain robust ferromagnetism in moiré systems such as twisted bilayer graphene. The approach provides a concrete, calculable route to explore magnetism in Wannier obstructed bands and paves the way for studying intermediate quantum phases in Moiré magnets.

Abstract

We develop a strong coupling approach towards quantum magnetism in Mott insulators for Wannier obstructed bands. Despite the lack of Wannier orbitals, electrons can still singly occupy a set of exponentially-localized but nonorthogonal orbitals to minimize the repulsive interaction energy. We develop a systematic method to establish an effective spin model from the electron Hamiltonian using a diagrammatic approach. The nonorthogonality of the Mott basis gives rise to multiple new channels of spin-exchange (or permutation) interactions beyond Hartree-Fock and superexchange terms. We apply this approach to a Kagome lattice model of interacting electrons in Wannier obstructed bands (including both Chern bands and fragile topological bands). Due to the orbital nonorthogonality, as parameterized by the nearest neighbor orbital overlap $g$, this model exhibits stable ferromagnetism up to a finite bandwidth $W\sim U g$, where $U$ is the interaction strength. This provides an explanation for the experimentally observed robust ferromagnetism in Wannier obstructed bands. The effective spin model constructed through our approach also opens up the possibility for frustrated quantum magnetism around the ferromagnet-antiferromagnet crossover in Wannier obstructed bands.

Figures (11)

• Figure 1: The lattice model in Eq. \ref{['eq:H']} is defined in the full many-body Hilbert space spanned by the orthogonal Fock states of electrons, which includes a low-energy subspace $\mathcal{H}$ spanned by the nonorthogonal trial states $|\Psi_{\bm{\sigma}} \rangle$ in Eq. \ref{['eq:ketsigma']}. A spin Hilbert space $\tilde{\mathcal{H}}$ spanned by the orthogonal Ising basis $|{\bm{\sigma}} \rangle$ is introduced to represent the effective spin model. The linear map $A$ (and $A^\dagger$) connects $\mathcal{H}$ and $\tilde{\mathcal{H}}$.
• Figure 2: (a) Kagome lattice model with imaginary hopping. Bound directions are specified by arrows. The gray hexagon marks out the unit cell. The Wyckoff positions $a$ and $b$ are respectively the hexagon and triangle centers. (b) The band structure of the Kagome lattice model, with the band Chern number $C$ labeled. The inset shows the Brillouin zone and high-symmetry momentum points.
• Figure 3: Localized orbitals found by minimizing objective energy $H_{()}$ (the constant piece of the effective spin Hamiltonian $H_c$). The phase of the orbital wave function is specified by the colorbar.
• Figure 4: Logarithmic norm of orbital wave functions against distance in units of bond length for (a) $(b,p_-)$ orbital as in Fig.\ref{['fig:orbitals']}(c,d); (b) $(a,d_+)$ orbital as in Fig.\ref{['fig:orbitals']}(b).
• Figure 5: The triangular lattice formed by the $(b,p_-)$ orbitals (in the bottom Chern band) arranged on the Kagome lattice. The effective spin model contains the Heisenberg interaction $J_1$ across the bonds and the chiral spin interaction $K_{\vartriangle/\triangledown}$ around the up/down triangles. It respects the translation $T_{1,2}$, three-fold rotation $C_3$ and anti-unitary mirror $\mathcal{M}$ symmetries.

Theorems & Definitions (8)

• Lemma C.1
• Definition E.1
• Theorem E.1: Mielke
• Corollary E.1: Mielke
• Theorem E.2
• proof
• Theorem E.3
• proof