Extended s-wave altermagnets

TL;DR

The paper introduces extended $s$-wave altermagnets ($s$AMs), a class of fully gapped, spin-compensated magnets with spin-polarized bands stabilized by valley-exchange symmetries. Through a two-valley continuum model and a minimal lattice realization, it demonstrates isotropic spin splitting and spin-selective transport, while predicting a descendant pair-density wave in superconducting channels. Mean-field and one-loop RG analyses identify staggered spin polarization (SSP, Δ^SSP ∝ τ^z σ^z) as a leading instability under typical interactions, with FRG supporting SSP and suggesting an effective negative Hund’s-like coupling. The work further develops design principles for minimal $s$AM models, analyzes node structures and symmetry protections, and extends the framework to 1D and hexagonal lattices, paving the way for spintronic applications and new correlated phases.

Abstract

We propose extended s-wave altermagnets (sAMs) as a class of magnetic states which are fully gapped, spin-compensated, and feature spin-polarized bands. sAMs are formed through valley-exchange symmetries, which act as momentum-space translations beyond standard crystallographic spin-group classifications. Using an effective two-valley model, we demonstrate that sAMs exhibit isotropic spin splitting, enable spin-selective transport in tailored heterostructures, and give rise to descendant pair density wave order. From a microscopic sAM minimal model, we develop the guiding principles to identify sAMs in quantum magnets.

Figures (9)

• Figure 1: Sketch of an $s$-wave altermagnetic two-valley system. Left: schematic single-particle spectrum of the two valleys ($\nu=\Gamma,M$) described by \ref{['eqn:H0_valley']}. Right: spin-split Fermi surfaces for the ferromagnetic (top) and (bottom) order parameters, $\Delta^\text{FM}\tau^0\sigma^z$ and $\Delta^\text{\sAM}\tau^z\sigma^z$, respectively. In the case, valley symmetry enforces magnetic compensation through $\tau^z$, so the spin-up (red, $\color{bdiv2-7}\uparrow$) and spin-down (blue, $\color{bdiv2-2}\downarrow$) Fermi surfaces remain equal in size.
• Figure 2: Spin splitter effect in an -N junction. (a) Setup of the -N heterostructure with the Fermi surfaces and their respective radii $\kappa_{(\uparrow/\downarrow)}$ indicated. Due to the reduced modes available on the N side of the junction, only electronic eigenstates in the shaded areas can propagate through the interface and contribute to the current. Panels (b,c) display the spin polarized currents $j_\sigma$ (red, blue) and the spin conversion factor $R=|(j_\uparrow-j_\downarrow)/(j_\uparrow+j_\downarrow)|$ (gray) as a function of (b) Fermi surface mismatch $r = 2\kappa / (\kappa_\uparrow+\kappa_\downarrow)$ and (c) order parameter $\Delta^\text{\sAM}$ in units of the Fermi energy $E_\mathrm{F}$. See Supplemental Material SM for simulation parameters.
• Figure 3: $s$-wave altermagnetic lattice model. Panel (a) displays the altermagnetic band structure along the irreducible path, with spin-up/down indicated by red/blue, respectively. The dotted lines demonstrate how the lattice model \ref{['latticemodel']} is approximated by the two-valley model \ref{['eqn:H0_valley']}. The top-right inset shows the Fermi surface in the Brillouin zone. Panel (b) illustrates the real-space hopping processes that lead to the band structure in panel (a). We set $t_\parallel=1$, $t_\perp=0.5$, $t_\perp^\prime=0.1$, and $\Delta^\text{\sAM} = 0.3$. The chemical potential is close to the band bottom of the nonmagnetic system, i.e., $E_\mathrm{F}=-3$.
• Figure S1: Renormalization of the interaction parameter to logarithmic accuracy. (a) one-loop particle particle renormalization of the initial couplings given in \ref{['fig:bare_interaction']}. The light gray lines represent derivatives of the pp bubble with respect to the cutoff $\Lambda$. (b) Exemplary flow of the MF couplings as a function of the low energy cutoff $\Lambda$. We have chosen $\rho U = \rho V = 1$, $J^\prime = 0$ and $J/U = -0.5$. (c) Flow of the MF couplings for the lattice model defined in \ref{['latticemodel']} using unbiased FRG calculations at $\rho U = 1.2$ and $V = J = J^\prime = 0$.
• Figure S1: Fermi surface of the for the orbital model defined in \ref{['eqn:hamiltonian_orbital']} for different model parameter. The FS of the normal state is indicated by gray dotted lines. (a-d) Spin split FS for $t_z = 1$, $t_x = t_x^\prime = 0.5$, $t_0 = 0$ and a magnetic order $\Delta = 0.3$ for $E_\text{F} = [-2, -1.5, -1, -0.5]$. (f) Same as (a) but with $t_0 = 0.5$ breaking $C_4$ symmetry. In all cases, the momentum dependent spin splitting exhibits a symmetry protected node at $\cos(k_x) + \cos(k_y) = 0$ indicated by solid gray lines.