Multipole order in two-dimensional altermagnets

TL;DR

The paper addresses how hidden magnetic multipole orders manifest in two-dimensional altermagnets. It develops low-energy envelope-function Hamiltonians for a minimal 2D altermagnet and a monolayer FeSe model, and defines band-structure indicators for magnetic octupole and hexadecapole orders. It finds that the minimal model exhibits a nonzero magnetic-octupole density with $\mathcal{I}^{(\rm{m},3)}(\boldsymbol{k}, \sigma) = \sigma (k_x^2 - k_y^2)$, while the FeSe model has vanishing octupole but finite hexadecapole density with $\mathcal{I}^{(\rm{m},4)}(\boldsymbol{k}, \tau, \sigma) = \tau \sigma (k_x^2 - k_y^2)$. The altermagnetic spin splitting in FeSe arises from the interplay with a sublattice pseudospin, highlighting a new mechanism for altermagnetism in 2D. This work broadens the multipole-based classification of magnetic order in low dimensions and points to ways to detect these hidden orders experimentally.

Abstract

We theoretically investigate the magnetic-multipole orders in two-dimensional (2D) altermagnets, focusing on two representative models: a generic minimal three-site model, and a four-site model representative of monolayer FeSe. We construct low-energy effective Hamiltonians for both systems and calculate their respective multipole indicators to characterize the underlying magnetic order. Our analysis reveals an intriguing contrast between the two systems. We find that the generic minimal model exhibits the expected non-zero magnetic-octupole order. In the monolayer-FeSe model, however, the magnetic-octupole order vanishes globally, and a magnetic-hexadecapole order is present instead. The emergence of altermagnetic splitting in the band structure then arises via the interplay with a sublattice-isospin degree of freedom. Our work demonstrates how the classification and comprehensive understanding of 2D altermagnetic materials transcends bulk descriptions.

Figures (7)

• Figure 1: Minimal model for a 2D altermagnet Brekke2023. (a) Square lattice with sites having a localized spin-up (red), spin-down (blue), or no magnetization (white). Primitive lattice vectors $\vb*{a}_1, \vb*{a}_2$ are shown as black arrows, and a (next-)nearest-neighbor hopping process with parameter $t$ ($t_2$) is depicted by gray arrows. (b) First Brillouin zone of the 2D square lattice.
• Figure 2: Structure representing monolayer FeSe. The unit cell (indicated by the cyan square) contains two Fe atoms (red and blue) and two Se atoms (white). In the tight-binding model, nearest-neighbor ($t$) and second-nearest-neighbor ($t_2$) hopping parameters are included. The Fe sites are assumed to have localized spins $\pm S$. By applying a perpendicular electric field $\vb*{E}$, the two Se sites acquire different potentials $\pm\Delta$.
• Figure 3: Band structure for the minimal 2D-altermagnet model. The six bands plotted as the solid lines are calculated from the original Hamiltonian \ref{['eq:original-hamil']}. The dispersions shown as dashed lines are obtained from the effective Hamiltonian \ref{['eq:eff-hamil']}. The red (blue) color indicates spin-up (spin-down) bands. The hopping parameters are set to be $J/t=0.2,t_2/t=0.01$.
• Figure 4: (a) Fermi surfaces for spin-up (red) and spin-down (blue) states for the effective low-energy Hamiltonian \ref{['eq:eff-hamil']} derived for the generic 2D-altermagnet model Brekke2023 with parameters $J / 4t_2 = 0.1$. Purple shading indicates the four regions with spin-polarized occupation. (b) Expectation value of the magnetic-octupole indicator $\langle \mathcal{I}^{(\rm{m},3)}\rangle$. The black (red) curve plots the numerical (analytical) result.
• Figure 5: Band structure of the four-site model for monolayer FeSe. The eight bands depicted by the solid lines are calculated from the original Hamiltonian \ref{['eq:4x4-hamil']}, while the four bands shown by the dashed lines are obtained from the effective Hamiltonian \ref{['eq:eff-hamil-4x4']}. Red (blue) color represents the spin-up (spin-down) dispersion. The right panel is a magnification around the $M$ point. Parameters used in the calculation are $J/t=1$ ,$t_2/t=0$, and $\Delta/t=0.1$.