Quantum geometry and $X$-wave magnets with $X=p,d,f,g,i$

TL;DR

This work develops and applies a comprehensive framework of quantum geometry, including Zeeman and non-Hermitian generalizations, to a family of $X$-wave magnets ($p,d,f,g,i$). It provides compact two-band expressions for the quantum metric, Berry curvature, and their Zeeman counterparts, extends the formalism to multi-band and non-Hermitian settings, and connects these geometric quantities to a wide range of observables such as TKNN Hall conductance, optical responses, bulk photovoltaic effects, and nonlinear conductivities. The paper then applies this machinery to $X$-wave magnets, detailing symmetry classifications, model Hamiltonians, and explicit geometric tensors, and analyzes transport phenomena with and without Rashba interactions, including spin currents, spin-Nernst effects, anomalous and planar Hall effects, and tunneling magnetoresistance in various material realizations. Overall, it reveals universal geometric structures underlying the spin-split band topology of altermagnets and related $X$-wave systems, and shows how Zeeman geometry and quantum-information geometry enrich the toolbox for predicting and interpreting electromagnetic and transport responses. The results offer analytic pathways to characterize and distinguish $X$-wave materials and guide experimental probes of their spintronic and optoelectronic functionalities.

Abstract

Quantum geometry is a differential geometry based on quantum mechanics. It is related to various transport and optical properties in condensed matter physics. The Zeeman quantum geometry is a generalization of quantum geometry including the spin degrees of freedom. It is related to electromagnetic cross responses. Quantum geometry is generalized to non-Hermitian systems and density matrices. Especially, the latter is quantum information geometry, where the quantum Fisher information naturally arises as quantum metric. We apply these results to the $X$-wave magnets, which include $d$-wave, $g$-wave and $i$-wave altermagnets as well as $p$-wave and $f$-wave magnets. They have universal physics for anomalous Hall conductivity, tunneling magneto-resistance and planar Hall effect. We obtain various analytic formulas based on the two-band Hamiltonian.

Figures (5)

• Figure 1: (a) The energy spectrum of the massive Dirac fermion with the mass $m$. (b) Hall conductance as a function of the chemical potential for the massive Dirac Hamiltonian. (c) The temperature dependence of the Hall conductance at the Fermi energy $\mu =0$.
• Figure 2: Fermi surfaces in two and three dimensions. (a1), (a3) $s$-wave magnet; (b1), (b3) $p$-wave magnet; (c1), (c3) $d$-wave altermagnet; (c2) $d^{\prime }$-wave altermagnet; (d1), (d3) $f$-wave magnet; (d2) $f^{\prime }$-wave magnet; (e1), (e3) $g$-wave altermagnet; (e2) $g^{\prime }$-wave$\ $altermagnet; ((f1) and (f3) $i$-wave altermagnet. Red (blue) curves indicate up (down)-spin Fermi surfaces.
• Figure 3: Illustration of a bilayer magnetic tunneling junction made of magnets. (a) Parallel configuration, where the spin directions are identical at each lattice site between the two layers. (b) Antiparallel configuration, where the spin directions are opposite at each lattice site between the two layers. The green arrow indicates the tunneling current, which is larger in the parallel configuration.
• Figure 4: (a) Spin density $\left\langle S_{z}\right\rangle$ when $J>0$, and (b) that when $J<0$ in one layer. Red (blue) color indicates up (down) spin. (c) Overlap $\mathcal{O}_{\text{P}}$ for the parallel configuration, and (d) overlap $\mathcal{O}_{\text{AP}}$ for the antiparallel configuration. Red color indicates a large overlap.
• Figure 5: Illustration for (a) Hall effect and (b) planar Hall effect. In the planar Hall effect, when the electric field is applied along the $x$ axis and the magnetic field $(B\cos \Phi ,B\sin \Phi ,0)$ is applied parallel to the system, the Hall current flows along the $y$ axis. The Hall conductivity is predicted to be given by the formula (\ref{['pHall']}).