Quantum Geometry and Nonlinear Responses in Magnetic and Topological Quantum Materials

Abstract

This dissertation explores various nonlinear responses that arise from the rich interplay between quantum geometry, disorder, magnetism and topology in quantum materials. In addition to presenting generalizations of quantum kinetic theory, Kubo formulas and semiclassical Boltzmann transport theory to the nonlinear response regime, we discuss several predictions of novel transport effects and physical insights that emerge from these developments.

Figures (21)

• Figure 1: Schematic map of carrier densities at first and second order in the applied electric field and their physical origins, starting from the equilibrium (Fermi-Dirac) distribution $f_{0\mathbf{k}}$. The three blue central arrows represent, ordinary, side-jump and skew scattering in linear response theory, while the surrounding red arrows arise in quadratic response theory. The dashed lines separate different orders in the impurity density.
• Figure 2: Scalings of leading-order contributions to the nonlinear Hall conductivity in the presence of time reversal symmetry as a function of the Fermi energy, starting from the bottom of the conduction band. Parameters used: $m=0.1$ eV, $t_x = 0.1$ eV Å, $t_y = 0$, $v=1$ eV Å, and $n_I U_0^2=10^2$ eV$^2$ Å$^2$.
• Figure 3: Schematic summary of this chapter. Correlation functions corresponding to conductivity Kubo formulas are applied to simultaneously dress with dissipation the quantum geometry, carrier dynamics and momentum-space field equations. The latter two yield dressed versions of the two central equations of general relativity, resulting in dressed gravity in momentum space.
• Figure 4: Linear [(a) and (b)] and quadratic [(c)-(f)] conductivity diagrams. Loops and legs are electron and photon propagators, respectively, and vertices imply velocity operator insertions, with the index $\mu$ reserved for the output. The Feynman rules resulting in Eqs. (\ref{['eq_sigma1']}) and (\ref{['eq_sigma2']}) consist of the following factors and procedures: 1) A factor of $1/k!$ for every set of $k$ connected photons. 2) Factors of $e/\hbar$ for the output photon and $ie/\chi$ for each input photon, with $\chi=\omega^{^{\prime}},\omega^{^{\prime\prime}}$. 3) Trace over momentum and band indices of the loop. 4) Matsubara frequency summation $1/\beta \sum_n$.
• Figure 5: Schematics of the spin AH-UMR effect. (a) In a single ferromagnetic-metal (FM) layer, the spin accumulation $\mu_{s,F}$ (indicated by the small yellow arrows) has an antisymmetric distribution about the center line of the layer, with no net nonequilibrium spin density (spatially-averaged $\mu_{s,F}$) induced. So the overall spin AH-UMR is also zero. (b) The presence of a neighboring nonmagnetic-metal (NM) layer induces a net nonequilibrium spin density $\delta \mu_{s,F}$ (indicated by the large yellow arrow) in the FM layer, giving rise to a finite spin AH-UMR. (c) Reversing the electric field direction flips the direction of the AH current $\mathbf{j}_{\text{AH}}$ and thus the sign of $\delta \mu_{s,F}$, thereby changing the sign of the spin AH-UMR.