Altermagnetism Without Crystal Symmetry

Abstract

Altermagnetism is a collinear magnetic order in which opposite spin species are exchanged under a real-space rotation. Hence, the search for physical realizations has focussed on crystalline solids with specific rotational symmetry. Here, we show that altermagnetism can also emerge in non-crystalline systems, such as amorphous solids, despite the lack of global rotational symmetries. We construct a Hamiltonian with two directional orbitals per site on an amorphous lattice with interactions that are invariant under spin rotation. Altermagnetism then arises due to spontaneous symmetry breaking in the spin and orbital degrees of freedom around each atom, displaying a common point group symmetry. This form of altermagnetism exhibits anisotropic spin transport and spin spectral functions, both experimentally measurable. Our mechanism generalizes to any lattice and any altermagnetic order, opening the search for altermagnetic phenomena to non-crystalline systems.

Figures (4)

• Figure 1: (a) Adjacent orbitals are coupled with an orbital-dependent hopping parametrised by two terms. A strong $t_1$ acts along the orbital's axis ($\sigma$-bonding) and a weak $t_2$ acts perpendicular ($\pi$-bonding). At intermediate angles the hopping interpolates between the types of bonding via a matrix $T(\theta)$. (b) A section of an amorphous lattice with an altermagnetic state at half filling. Each site hosts two orbitals, occupied by spin-up and spin-down electrons, respectively. The state is invariant under a combined $\mathcal{T}$ spin flip and local$C_4$ rotation of the orbitals around each site. (c) A phase diagram for a three-coordinated amorphous lattice in filling $n$ and the interaction coupling strength $J$. The filling is parametrised such that $n=1$ is a fully filled system, with four fermions per lattice site. Three phases are shown: a trivial metal, an amorphous altermagnet, and a charge-density wave. The colouring represents the mean alter-magnetization $m$, normalized to its maximal value $m_{\textit{max}}=2-|4n-2|$. Labelled points are discussed in the text.
• Figure 2: Spectral functions for three example points in the phase diagram, labelled by three polygons that correspond to the points in \ref{['fig:1']}c. The first row ($\triangle$) shows the trivial metal phase, with weak $J/t$ at half filling. The second row ($\Square$) shows an altermagnetic phase with strong $J/t$ at low filling, where the states at the Fermi level are close to $\Gamma$. The third row ($\pentagon$) shows an altermagnetic phase with strong $J/t$ at high filling. Here the states close to the Fermi level are far from $\Gamma$ so are not well-approximated by plane waves. (a) Spectral density as a function of $\textbf{p}$ at the Fermi level $\epsilon_F$ (shown as dashed line in panel b). Four cases are shown, the total density, the densities of spin-up and spin-down respectively, and the difference between spin-up and spin-down. The difference vanishes in the non-altermagnetic case. (b) Spectral density as a function of energy and momentum on a path around the quasi-Brillouin zone (shown in the top right panel of subfig. a. Two cases are shown, the overall spectral density and the spin-difference density. The quasi-Brillouin zone $[-\pi/\bar{a},\pi/\bar{a}]^2$ and its high symmetry points are defined via the average lattice spacing $\bar{a}$, which is the relevant length scale in an amorphous solid Ciocys_2024.
• Figure 3: Spin conductance of an amorphous altermagnet. (a) The set-up to evaluate the spin conductance consists of the mean-field-converged amorphous Hamiltonian, onto which we attach metallic leads. The spin-transmission, or equivalently the spin-conductance $\sigma^s_{\phi \phi}$, is determined for a given direction $\phi$ at filling $\langle n \rangle$. (b) The spin-resolved conductance as a function of angle $\phi$. In a trivial amorphous metal ($\triangle$) the conductance is isotropic and equally shared between the 2 spin species, whereas in the amorphous, metallic altermagnet ($\Square$ and $\pentagon$) the spin-resolved conductance is anisotropic. Total conductance is shown in grey. (c) The spin splitter angle $\alpha$ quantifies the relative strength of spin conductance $G_\uparrow-G_\downarrow$ versus the total conductance $G_\uparrow+G_\downarrow$. At half-filling the system becomes insulating. Above half filling the sign of $\alpha$ is reversed because the upper two bands consist of $\ket{\uparrow y}$ and $\ket{\downarrow x}$ states, which have negative alter-magnetisation $m$.
• Figure 4: (a) The procedure for determining the hopping matrix $T(\theta)$. We start with a pair of orbitals rotated to align with the bond direction, assigning the hopping parameters $t_1$ and $t_2$. Rotating back to a canonical basis ($x$ and $y$ orbitals) we obtain the matrix elements of $T(\theta)$. (b) Spectral function for a series of higher-order altermagnetic phases, parametrised by $n$, where the altermagnetic symmetry of each phase is $C_{4n}\mathcal{T}$. Each phase was constructed and solved using the mean field prescription detailed in the main text.