Quantum-geometry-driven Mott transitions and magnetism

Abstract

Quantum geometry quantifies how the single-particle Bloch wavefunction changes in phase and amplitude across the Brillouin Zone. In multi-orbital systems where bands have strongly mixed orbital composition, quantum geometry plays a vital role in determining the ground state and low-energy properties of interacting electronic systems. In this work, we show that Mott metal-insulator transitions, as well as transitions between different magnetic orders within the Mott insulating phase, can be driven by the quantum geometry of the underlying Bloch band, thereby providing a mechanism complementary to conventional bandwidth-tuned Mott transitions. By studying the Kane-Mele-Hubbard model using exact diagonalization, we demonstrate that in in half-filled and topologically-trivial bands, quantum geometric properties of the Bloch states alone can act as a tuning knob for Mott metal-to-insulator and affect the competition between ferromagnetism and antiferromagnetism. We show that both transitions may be heuristically understood via non-local Coulomb scattering in a basis of exponentially localized Wannier functions. These results highlight the role of quantum geometry beyond topological settings as a governing principle for conventional Mott and magnetic physics in multi-orbital and moiré materials.

Figures (16)

• Figure 1: Wannier function evolution of the Kane-Mele model at fixed $t'=0.05$. a Representative real space spread of the Wannier function of the lower band of the Kane-Mele model at $M=1.2$. The size of the markers is proportional to the amplitude $\lvert W_{\mathbf{R}=\mathbf{0}}(\mathbf{r})\rvert$ on the original orbitals of the hexagonal lattice, with purple discs for orbital A and orange discs for orbital B. b$M$ dependence of Wannier function amplitude $\lvert W_{\mathbf{R}=\mathbf{0}}(\mathbf{r})\rvert$ at four $\mathbf{r}$ points annotated in a in order of increasing distance from the origin. The black dotted line marks $M = 1.2$.
• Figure 2: Quarter-filled Kane-Mele model results. a Schematic phase diagram of charge and spin order, as a function of bare dispersion $t_W$ and orbital imbalance $M$. Magenta line dashed denotes the FM-AFM magnetic transition. Green dashed line denotes the possible location of an AFM - remnant region (described in the main text) transition. Hatched region denotes approximate location of Mott transition. The hatched region is identical to the area between red dashed and yellow dotted line in b. The black and grey arrows indicate the position of the metal-to-insulator and magnetic ordering transitions, respectively, of the triangular lattice Hubbard model Szasz2020. b Ground state charge gap $\Delta(N_e)$. Blue triangles, bright green triangles, yellow stars, and red dashed line denote position of the MIT estimated by four different methods. Blue triangles denote when the ground state charge gap falls below a threshold of 0.1. Bright green triangles denote critical $t_W^*$ inferred by performing a linear fit to $\Delta(N_e, U) = a U - b$ and using the $x$-intercept of the fit as $U_c/t_W$Morales-Duran2021Botzung2024a (for example fits, see \ref{['fig:Uc-fit']}). Yellow stars denote when the charge gap averaged over twisted boundary conditions reaches 0 Koretsune2007 (see \ref{['fig:gap-vs-tW-KM']}). The red dashed line is an analytical estimate of the MIT based on \ref{['eq:Weff']}. c Ground state spin configuration. d Magnitude of band-projected spin correlation function $C^x(\mathbf{K})$ in the $S_z=0$ sector. Green dashed line in d (identical to that in a) is a guide to the eye. Identical magenta dashed lines in a, c and d denotes where the nearest neighbor exchange $J^{zz}_1(M,t_W)$ changes sign as per \ref{['eq:J-total']}. ED results are obtained on a hexagonal 12d4 Betts cluster Betts1996Betts1999Gustiani2015 with $t'=0.05$, $M > 3\sqrt{3}t' = 0.26$.
• Figure 3: Nearest-neighbor interaction matrix elements for Wannier orbitals displaced by $\mathbf{a}_1$ for the Kane-Mele model, expressed in terms of bare interaction strength $U$. Except $U_0$, all other interaction terms have been re-scaled to improve visibility. Effective nearest-neighbor repulsion and pair hopping are related to spin exchange matrix elements via $U_1 = - J^{zz}_{\mathrm{direct}}/4$, and $P_1 = -J^{zz}_{\mathrm{direct}}/2$.
• Figure S1: Band-projected spin correlation functions of Kane-Mele model in Mott AFM phase on 12d4 cluster, demonstrating $120^\circ$ in-plane Neél order. Parameters: $t'=0.05$, $M=1.4$, $t_W = 0.04$, $U=1$.
• Figure S2: Band-projected spin correlation functions of Kane-Mele model in metallic phase on 12d4 cluster, demonstrating paramagnetic spin correlations. Parameters: $t'=0.05$, $M=1$, $t_W = 0.1$, $U=1$.