Quantum Geometry Defines, And, Wave Magnet Properties, Enabling Analysis of Anomalous Hall Conductivity

The interplay between geometry and magnetism forms the basis of many important phenomena in materials science, and recent work by Motohiko Ezawa from The University of Tokyo, and colleagues, significantly advances our understanding of this connection. Researchers now explore a broadened concept of geometry, extending it to encompass spin and quantum information, revealing links to electromagnetic responses within materials. This new framework allows scientists to analyse a wide range of magnetic materials, including those exhibiting unusual Hall effects and magnetoresistance, offering a unified explanation for their behaviour.

The team’s analytical formulas, derived from a two-band model, provide a powerful tool for predicting and controlling the properties of these complex magnetic systems, potentially paving the way for novel electronic devices.

Quantum Geometry Links Transport and Optical Properties Quantum geometry, a differential geometry rooted in the wave functions of quantum mechanics, is rapidly advancing our understanding of condensed matter physics and its relationship to fundamental material properties. This work establishes a framework where the quantum geometric tensor, derived from the fidelity of wave functions, provides insights into both transport and optical phenomena. The real component of this tensor defines the quantum metric, while its imaginary part corresponds to the Berry curvature, a measure of the geometric phase acquired by quantum particles. Integrating the Berry curvature yields the Chern number, a topological invariant crucial for characterizing materials like topological insulators. Researchers have extended this concept by generalizing quantum geometry to incorporate spin degrees of freedom, creating Zeeman quantum geometry, which directly links to electromagnetic responses within materials. This theoretical development is particularly relevant to understanding unconventional magnetism, specifically in a class of materials termed “X-wave magnets,” encompassing p-wave, d-wave, f-wave, g-wave, and i-wave systems. These magnets, regardless of whether they preserve or break time-reversal symmetry, exhibit universal behaviors in phenomena like anomalous Hall conductivity, tunneling magnetoresistance, and the planar Hall effect. This study delivers compact analytical formulas for the Zeeman quantum geometry, derived from a simplified two-band Hamiltonian, bypassing the need for complex eigenfunction calculations. These formulas allow scientists to predict and understand the electromagnetic responses of X-wave magnets with unprecedented efficiency. Furthermore, the research extends the framework to encompass multiband systems and investigates the impact of Rashba interactions on transport properties, revealing how these interactions influence anomalous and planar Hall effects. By connecting quantum geometry to material properties, this work provides a powerful new lens for designing and discovering advanced materials with tailored functionalities.,. Quantum Geometry from Wave Function Fidelity and Wannier Scientists developed a comprehensive framework for quantum geometry beginning with the fidelity of wave functions, a measure of similarity between two quantum states. The study defines quantum distance based on this fidelity and then expands this distance with infinitesimal momentum translation to obtain the quantum geometric tensor. This tensor decomposes into the quantum metric, representing geometric properties, and the Berry curvature, which describes the phase acquired by a wave function due to external forces. Integration of the Berry curvature yields the Chern number, an integer characterizing topological insulators. Researchers then transformed wave functions from momentum space to real space, obtaining the Wannier function, and demonstrated a relationship between the quantum metric and fluctuations in position. They derived simple formulas to directly calculate both the quantum metric and Berry curvature from the two-band Hamiltonian, circumventing the need for detailed knowledge of the system’s eigenfunctions. This approach was then generalized to encompass N-fold degenerate multiband systems, expanding the applicability of the quantum geometric framework.

The team further connected quantum geometry to observable phenomena, deriving the Thouless-Kohmoto-Nightingale-Nijs formula, which links Hall conductivity to the Chern number and explains the quantization observed in topological insulators. Applying these results to a two-dimensional Dirac system, scientists demonstrated optical dichroism, where optical absorption selectively occurs based on the chirality of the system. They also established a sum rule connecting optical conductivity to the quantum metric and related bulk photovoltaic effects, including injection and shift currents, to the quantum metric. Furthermore, the second-order nonlinear conductivity was shown to be directly linked to the quantum metric. Scientists extended the framework to include spin degrees of freedom, defining Zeeman quantum geometry and clarifying its relationship to electromagnetic cross responses. They derived simple formulas for the Zeeman quantum geometric tensor for two-band systems, relying solely on the Hamiltonian without requiring knowledge of the eigenfunctions, and generalized this approach to N-fold degenerate systems. The study also generalized quantum geometry to non-Hermitian systems and density matrices, relating quantum distance to quantum Fisher information and demonstrating the Cramer-Rao inequality, which establishes a lower bound on the covariance of physical observables. They showed that the quantum Fisher information reduces to the classical Fisher information for pure states and derived quantum geometry at thermal equilibrium.,.

Quantum Geometry Defines X-wave Magnetism This work presents a comprehensive study of quantum geometry and its application to a newly defined class of magnetic materials termed X-wave magnets, encompassing p-wave, d-wave, f-wave, g-wave, and i-wave systems. Researchers established a framework for quantum geometry rooted in the fidelity of wave functions, defining quantum distance as a measure of similarity between quantum states. This foundational work allows for the derivation of analytical formulas describing the Zeeman quantum geometry, a generalization incorporating spin degrees of freedom, based solely on a two-band Hamiltonian. The study demonstrates that the real component of the quantum geometric tensor corresponds to the quantum metric, while the imaginary component defines the Berry curvature, directly linking to the Thouless-Kohmoto-Nightingale-Nijs formula which describes Hall conductivity. Furthermore, the research connects quantum geometry to observable properties such as optical absorption and nonlinear conductivity, with experimental observations confirming these theoretical relationships. A key achievement is the development of compact analytical formulas for the Zeeman quantum geometry, applicable to multiband systems, and the investigation of electromagnetic cross responses within X-wave magnets, particularly when coupled with Rashba interaction.

The team also extended the quantum geometric framework to encompass non-Hermitian systems and density matrices, broadening its applicability.

This research establishes a powerful theoretical foundation for understanding and potentially manipulating the properties of these novel magnetic materials, offering new avenues for spintronics and magnetic memory technologies.,.

Zeeman Geometry Unifies Magnet Transport Properties This work establishes a geometric framework for understanding transport and optical properties in complex materials, extending conventional differential geometry to incorporate spin degrees of freedom and non-Hermitian systems. Researchers successfully applied this “Zeeman geometry” to investigate a range of magnets, including altermagnets and conventional magnetic materials, revealing universal behaviours governing anomalous Hall conductivity, tunneling magnetoresistance and the planar Hall effect. Through detailed analysis using a two-band Hamiltonian, the team derived analytic formulas describing nonlinear conductivity, demonstrating that higher-order nonlinear spin-Drude conductivities emerge depending on the material’s wave function symmetry. Specifically, they found linear spin current generation in d-wave altermagnets, second-order nonlinear spin current in f-wave magnets, and higher-order effects in other materials. Furthermore, the study extends to the spin Nernst effect, where spin current is generated perpendicular to a thermal gradient, providing an analytic result for this phenomenon in d-wave altermagnets. By solving the Boltzmann equation under specific conditions, the researchers derived expressions for the spin current driven by a temperature gradient, linking it to the material’s band structure and Fermi distribution. 👉 More information 🗞 Quantum geometry and -wave magnets with 🧠 ArXiv: https://arxiv.org/abs/2512.05477 Tags: