Zeeman Quantum Geometry as a Probe of Unconventional Magnetism

TL;DR

This work addresses the challenge of probing unconventional magnets, which exhibit momentum-dependent spin splitting with zero net magnetization. By formulating the Zeeman quantum geometric tensor (ZQGT), encompassing both momentum translations and spin rotations, the authors uncover intrinsic gyrotropic magnetic currents that reveal hidden spin textures. Through analyses of a $d$-wave altermagnet and an unconventional $p$-wave magnet (and a mixed $d$-wave case) with Rashba SOC, they show that ZQGT components yield distinct conduction and displacement currents governed by symmetry, enabling direct experimental signatures via Hall transport and THz probes. The results establish ZQGT as both a diagnostic tool and a design principle for engineering novel unconventional magnetic materials with robust, symmetry-protected transport responses.

Abstract

Unconventional magnets with momentum-dependent spin-splitting but zero net magnetization form a recently identified class of collinear magnets that are challenging to probe via conventional means. We show that these systems can be distinguished through their intrinsic gyrotropic magnetic (IGM) currents, enabled by the Zeeman quantum geometry, which captures the coupled response of electronic states to momentum translation and spin rotation. Examining two prototypical two-dimensional unconventional magnets with Rashba spin-orbit coupling, a time-reversal-broken $d$-wave altermagnet and a time-reversal-symmetric $p$-wave magnet, we uncover a direct link between crystalline symmetry, spin-split band structures, and transport signatures. The $d_{x^2-y^2}$-wave altermagnet exhibits both transverse conduction and longitudinal displacement IGM currents, whereas the $p$-wave magnet supports only a transverse conduction IGM current. Remarkably, the mixed $d$-wave altermagnet supports all four types of IGM currents, including a longitudinal conduction current enabled by symmetric (Zeeman) Berry curvature that is forbidden in conventional quantum geometry. These responses, measurable via Hall transport and optical probes, persist even when conventional quantum geometry-driven linear responses vanish, offering unique access to hidden spin-split band structures. Our results establish Zeeman quantum geometry as both a diagnostic tool and a design principle for novel magnetic materials.

Figures (3)

• Figure 1: Schematic of Zeeman quantum geometry-driven linear response: The interplay between ZBC ($Z$) and ZQM ($Q$) drives both longitudinal and transverse (Hall-like) IGM currents in the linear response of unconventional magnets. Unlike conventional quantum geometry, which forbids such responses, the symmetric part of $Z$ and the antisymmetric part of $Q$ enable both conduction (C) and displacement (D) IGM currents, appearing parallel as well as perpendicular to the applied oscillating magnetic field. Here, $a$ and $b$ are spatial indices taking the values $x$ and $y$.
• Figure 2: Zeeman quantum geometry in unconventional magnets: Distribution of different components of the ZBC ($Z$) and ZQM ($Q$) obtained from the lattice model of the three different types of unconventional magnets. (a) and (b): $Z_{- +}^{yx}$ and $Q_{- +}^{xx}$ are shown for the $d_{x^2 - y^2}$-wave altermagnet. (c)--(f): $Z_{- +}^{xx}$, $Q_{- +}^{xx}$, $Z_{- +}^{yx}$, and $Q_{- +}^{yx}$ are shown for the mixed $d$-wave altermagnet. (g) and (h): $Z_{- +}^{xy}$ and $Q_{- +}^{xy}$ are shown for the $p_x$-wave unconventional magnet. Parameters used: $t = 0.35t'$, $\lambda = 0.2t'$, $t_{am} = 0.5t'$, $t_{am}^\prime = 0.2t'$, $T = 1 \, \text{K}$ and $t' = 1 \, \text{eV}$.
• Figure 3: Zeeman quantum geometry-induced intrinsic gyrotropic magnetic (IGM) conductivities: (a) In the $d_{x^2-y^2}$-wave altermagnet, a transverse (Hall-like) conduction IGM conductivity ($\sigma_{yx}^C = - \sigma_{xy}^C$) and a longitudinal displacement IGM conductivity ($\sigma_{xx}^D = -\sigma_{yy}^D$) appear. (b) In the $p$-wave magnet, only transverse conduction IGM conductivities ($\sigma_{xy}^C \neq \sigma_{yx}^C$) are present, with no displacement response. (c,d) In the mixed $d$-wave altermagnet, all four conductivity components emerge, reflecting the symmetry-mixed structure. The responses are computed using $\omega = 10^{12}\,\text{Hz}$ and other parameters are the same as in Fig. \ref{['Geometric_Quantities']}.