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.
AI Summary
This dissertation examines how quantum geometry, disorder and magnetic order jointly generate nonlinear charge transport in topological materials. It develops nonlinear extensions of quantum-kinetic, Kubo and semiclassical approaches, with applications to Dirac fermions and magnetized topological-insulator interfaces.
The quantum-kinetic treatment solves the disorder-averaged single-particle Liouville equation through second order in electric field, retaining density-matrix contributions scaling as $n_I^{-2}$, $n_I^{-1}$ and $n_I^0$ with impurity density. It classifies mixed and higher-order side-jump and skew-scattering channels. Reexpressing interband-coherence contributions in terms of band populations yields modified semiclassical velocities: alongside Berry-curvature and quantum-metric terms, four additional extrinsic velocities enter quadratic transport. For tilted massive Dirac fermions, analytical conductivities distinguish ordinary-scattering, Berry-curvature-dipole and disorder contributions. Restoring time-reversal symmetry cancels the leading ordinary-scattering response; competition among the remaining terms produces Fermi-energy-dependent sign changes.
A separate Kubo analysis predicts unidirectional magnetoresistance and a nonlinear planar Hall effect from interfacial spin–orbit disorder in topological-insulator/ferromagnetic-insulator bilayers, without modifying the minimal Dirac Hamiltonian. Predicted magnetoresistance coefficients are approximately $10$–$100\,\mathrm{nm/V}$. Angular quasiplateaus and additional sign reversals occur when the Fermi energy approaches the exchange scale under specified disorder conditions.
The kinetic derivation assumes weak, Gaussian disorder, the Born and Markov approximations, and a steady state. The tilted-Dirac example assumes weak tilt, a conduction-band Fermi level and low temperature; it does not evaluate the formally derived $n_I^0$ transport channels, and uses an isotropic transport time for subleading analytical results.
No sign of AI-generated textStrong assumptions
Central claim may not hold: The quadratic-density solution uses the linear-response transport time for angular harmonics with different relaxation rates, so it does not generally solve the stated collision equation, even for the isotropic Dirac model.
The disorder strength specified for Figure fig_qkt_fig2 gives ℏ/(τ_0 h_k) between 25 and 100, placing the plotted example outside the weak-disorder regime assumed in its derivation.
The two asymptotic conductivity ratios have the tilt-angle dependence inverted: the preceding conductivities and stated angle definition imply tanφ_t/3 and tanφ_t/11, not cotφ_t/3 and cotφ_t/11.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
Comments
184 pages. PhD thesis (also available at http://rave.ohiolink.edu/etdc/view?acc_num=case1758043426955896)
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.
This dissertation examines how quantum geometry, disorder and magnetic order jointly generate nonlinear charge transport in topological materials. It develops nonlinear extensions of quantum-kinetic, Kubo and semiclassical approaches, with applications to Dirac fermions and magnetized topological-insulator interfaces.
The quantum-kinetic treatment solves the disorder-averaged single-particle Liouville equation through second order in electric field, retaining density-matrix contributions scaling as nI−2, nI−1 and nI0 with impurity density. It classifies mixed and higher-order side-jump and skew-scattering channels. Reexpressing interband-coherence contributions in terms of band populations yields modified semiclassical velocities: alongside Berry-curvature and quantum-metric terms, four additional extrinsic velocities enter quadratic transport. For tilted massive Dirac fermions, analytical conductivities distinguish ordinary-scattering, Berry-curvature-dipole and disorder contributions. Restoring time-reversal symmetry cancels the leading ordinary-scattering response; competition among the remaining terms produces Fermi-energy-dependent sign changes.
A separate Kubo analysis predicts unidirectional magnetoresistance and a nonlinear planar Hall effect from interfacial spin–orbit disorder in topological-insulator/ferromagnetic-insulator bilayers, without modifying the minimal Dirac Hamiltonian. Predicted magnetoresistance coefficients are approximately 10–100nm/V. Angular quasiplateaus and additional sign reversals occur when the Fermi energy approaches the exchange scale under specified disorder conditions.
The kinetic derivation assumes weak, Gaussian disorder, the Born and Markov approximations, and a steady state. The tilted-Dirac example assumes weak tilt, a conduction-band Fermi level and low temperature; it does not evaluate the formally derived nI0 transport channels, and uses an isotropic transport time for subleading analytical results.
Quantum geometry - including both the quantum metric and Berry curvature - arises from the non-zero overlap of well-defined eigenstates and their adiabatic evolution across the Brillouin zone. It has revolutionized condensed matter physics and material science by explaining quantum Hall effects, establishing the modern theory of polarization, and enabling a systematic search for topological materials on a large scale. Yet, despite these advances, we lack a framework for leveraging quantum geometry in the strongly interacting regime. Bridging this gap is critical: if we can harness geometric responses in correlated metals, we stand to engineer desirable transport and optical properties using existing material platforms, thus bypassing the complex task of designing materials with tailored quantum geometry from scratch. In this thesis, we take steps toward such a framework. First, we analyze the resilience of topological boundary modes in the presence of electronic correlations, identifying when interactions preserve, diminish, or destroy boundary modes. Second, we reveal geometric fingerprints of fluctuations in magnetically ordered systems, tying the electric quantum metric to the formation of instabilities and chiral quasi-particle excitations. Third, we generalize quantum geometry to describe families of many-body wave functions, providing a new algorithm to compute state-manifold curvatures suited to interacting phases. Our approach combines analytical theory with numerical methods, including density functional theory and tensor-network simulations, and is supported by open-source software developed during the PhD. Together, these results advance the topological classification of interacting phases and extend quantum geometry from single-particle bands to correlation functions, providing tools to design materials and devices with targeted geometric responses.
Quantum geometry - including both the quantum metric and Berry curvature - arises from the non-zero overlap of well-defined eigenstates and their adiabatic evolution across the Brillouin zone. It has revolutionized condensed matter physics and material science by explaining quantum Hall effects, establishing the modern theory of polarization, and enabling a systematic search for topological materials on a large scale. Yet, despite these advances, we lack a framework for leveraging quantum geometry in the strongly interacting regime. Bridging this gap is critical: if we can harness geometric responses in correlated metals, we stand to engineer desirable transport and optical properties using existing material platforms, thus bypassing the complex task of designing materials with tailored quantum geometry from scratch. In this thesis, we take steps toward such a framework. First, we analyze the resilience of topological boundary modes in the presence of electronic correlations, identifying when interactions preserve, diminish, or destroy boundary modes. Second, we reveal geometric fingerprints of fluctuations in magnetically ordered systems, tying the electric quantum metric to the formation of instabilities and chiral quasi-particle excitations. Third, we generalize quantum geometry to describe families of many-body wave functions, providing a new algorithm to compute state-manifold curvatures suited to interacting phases. Our approach combines analytical theory with numerical methods, including density functional theory and tensor-network simulations, and is supported by open-source software developed during the PhD. Together, these results advance the topological classification of interacting phases and extend quantum geometry from single-particle bands to correlation functions, providing tools to design materials and devices with targeted geometric responses.
Quantum geometry – including both the quantum metric and Berry curvature – arises from the non-zero overlap of well-defined eigenstates and their adiabatic evolution across the Brillouin zone. It has revolutionized condensed matter physics and material science by explaining quantum Hall effects, establishing the modern theory of polarization, and enabling a systematic search for topological materials on a large scale. Yet, despite these advances, we lack a framework for leveraging quantum geometry in the strongly interacting regime. Bridging this gap is critical: if we can harness geometric responses in correlated metals, we stand to engineer desirable transport and optical properties using existing material platforms, thus bypassing the complex task of designing materials with tailored quantum geometry from scratch. In this thesis, we take steps toward such a framework. First, we analyze the resilience of topological boundary modes in the presence of electronic correlations, identifying when interactions preserve, diminish, or destroy boundary modes. Second, we reveal geometric fingerprints of fluctuations in magnetically ordered systems, tying the electric quantum metric to the formation of instabilities and chiral quasi-particle excitations. Third, we generalize quantum geometry to describe families of many-body wave functions, providing a new algorithm to compute state-manifold curvatures suited to interacting phases. Our approach combines analytical theory with numerical methods, including density functional theory and tensor-network simulations, and is supported by open-source software developed during the PhD. Together, these results advance the topological classification of interacting phases and extend quantum geometry from single-particle bands to correlation functions, providing tools to design materials and devices with targeted geometric responses.
This dissertation studies quantum entanglement in relation to the geometric response known as Hall viscosity. We begin by reviewing geometric response in the quantum Hall effect in comparison to its well-known electromagnetic response, Hall conductivity. We develop an understanding of momentum transport due to Hall viscosity in analogy to charge transport under Hall conductivity. We apply our momentum transport argument to continuum and lattice models of the quantum Hall effect. We also leverage this insight to reveal a previously-unrecognized manifestation of Hall viscosity: the acoustic Faraday effect in superfluid Helium-3 B. We suggest that the acoustic Faraday effect is a new platform for the direct observation of Hall viscosity in Helium-3 B and other systems. We then turn our focus to the entanglement spectrum. We calculate the momentum polarization of lattice models of the quantum Hall effect to determine the Hall viscosity based on the entanglement spectrum, revealing the close connection between geometric response and entanglement. Finally, we turn away from the quantum Hall effect to consider other topological insulators; we use the entanglement spectrum to develop a topological classification of composite systems comprising components of topological phases in different dimensions. Our composite topological index generalizes classification of weak and antiferromagnetic topological insulators. We predict the presence of topological bound states localized to defects in systems that are trivial under all other topological classifications.
This paper investigates the interplay between nonlinear geometric topology and quantum correlation, proposing a novel analysis framework to understand the behavior of quantum systems. We explore how quantum entanglement, driven by geometric topology, can illuminate fundamental aspects of quantum phenomena. The core mechanism centers around establishing a method to quantitatively assess and predict correlations arising from this topological structure. The investigation aims to provide a robust and insightful approach to analyzing complex quantum systems, potentially offering a pathway to novel quantum phenomena. The research incorporates a deep dive into the interplay between geometric topology, quantum entanglement, and the potential for emergent correlations. The paper presents a theoretical framework, outlines experimental considerations, and discusses the implications for future research in quantum geometry and quantum information theory.
Classical nonlinear theories are highly successful in describing far-from-equilibrium dynamics of magnets, encompassing phenomena such as parametric resonance, ultrafast switching, and even chaos. However, at ultrashort length and time scales, where quantum correlations become significant, these models inevitably break down. While numerous methods exist to simulate quantum many-body spin systems, they are often limited to near-equilibrium conditions, capture only short-time dynamics, or obscure the intuitive connection between nonlinear behavior and its geometric origin in the su(2) spin algebra. To advance nonlinear magnetism into the quantum regime, we develop a theory in which semiclassical spin correlations, rather than individual spins, serve as the fundamental dynamical variables. Defined on the bonds of a bipartite lattice, these correlations are inherently nonlocal, with dynamics following through a semiclassical mapping that preserves the original spin algebra. The resulting semiclassical theory captures nonlinear dynamics that are entirely nonclassical and naturally accommodates phenomenological damping at the level of correlations, which is typically challenging to include in quantum methods. As an application, we focus on Heisenberg antiferromagnets, which feature significant quantum effects. We predict nonlinear scaling of the mean frequency of quantum oscillations in the Néel state with the spin quantum number S. These have no classical analog and exhibit features reminiscent of nonlinear parametric resonance, fully confirmed by exact diagonalization. The predicted dynamical features are embedded in the geometric structure of the semiclassical phase space of spin correlations, making their physical origin much more transparent than in full quantum methods. With this, semiclassical spin-correlation dynamics provide a foundation for exploring nonlinear quantum magnetism.