Exactly Solvable Models Hosting Altermagnetic Quantum Spin Liquids

TL;DR

This work constructs two exactly solvable spin models that host orbital altermagnetic quantum spin liquids (OAMSLs) by mapping spins to Majorana fermions on square-octagon and checkerboard lattices. The spin-$3/2$ model yields a unique $g$-wave OAMSL, while the spin-$7/2$ model exhibits a rich phase diagram including a $d$-wave OAMSL and chiral spin liquids, controlled by competing interactions that set complex flux textures. Flux excitations include non-topological local flips and topological visons, whose properties are shaped by lattice symmetry and operator algebra, revealing a nuanced topological-symmetry landscape in altermagnetic QSLs. These results broaden the catalog of exactly solvable quantum spin liquids and point toward experimental platforms for simulating orbital altermagnetism and related phases.

Abstract

We construct spin-$3/2$ and spin-$7/2$ models on the square-octagon and checkerboard lattices that are exactly solvable with Majorana representations. They give rise to spin-liquid phases with full spin-rotation and lattice-translational symmetries but broken time-reversal symmetry. Although non-zero on elementary plaquettes, the net orbital magnetic moment is guaranteed to vanish as a result of point symmetries; due to the analogy to long-range ordered altermagnets, these types of phases were dubbed altermagnetic spin liquids in [Phys. Rev. Research 7, 023152 (2025)]. For the spin-$3/2$ model, we find that a $g$-wave altermagnetic spin liquid emerges as the unique ground state. In contrast, the spin-7/2 model exhibits a significantly richer phase diagram, involving different types of chiral spin liquids competing with a $d$-wave altermagnetic spin liquid. Finally, we identify and characterize the topological and non-topological excitations, illustrating the rich physics of altermagnetic spin liquids resulting from the interplay of non-trivial topological and symmetry aspects of this novel phase of matter.

Figures (8)

• Figure 1: Schematic representation of the spin-3/2 model defined in Eqs. (\ref{['eq:spin3/2model']}) and (\ref{['eq:HF3/2']}). (a) The chosen unit cell is illustrated with a gray square. The horizontal gray line indicates the $\sigma_v$ reflection plane. The blue, orange, red, and green circles denote lattice sites belonging to the $A$, $B$, $C$ and $D$ sublattices, respectively. The red arrows denote a particular instance of the interactions with coefficients $J_1$, $J_2$ and $J_5$. The numbers around each lattice site indicate which $\Gamma$-matrix appears for the given sublattice site in the interaction term along the bond on which the number is located. (b) Gauge flux structure of the $g$-wave OAMSL. The elementary plaquettes include one square, one octagon, and pentagons inside the latter, with fluxes $0$, $\pi$ and $\pi$ respectively. The arrows indicate the direction in which the respective $\hat{u}_{i_1,i_2}^{\mu_1,\mu_2}$ is positive.
• Figure 2: Schematic representation of the spin-7/2 model in Eqs. (\ref{['eq:spin7/2model']}) and (\ref{['eq:spin7/2modelmajor']}). (a) Lattice structure with the $\sigma_{d}$ reflection symmetry indicated by the diagonal dashed gray line. As in Fig. \ref{['fig:firstmodel']}a, the numbers around each lattice site indicate which $\gamma^{a}$-matrix appears in the interaction term for the given sublattice site along the corresponding bond. Each colored dot represents one Majorana fermion, as defined in Eq. (\ref{['eq:majorana7/2']}) and also shown in the zoomed-in unit cell. The red lines and symbols illustrate representative interactions with coefficients $J_{1}$, $J_{2}$, and $J_{7}$. (b) Phase diagram as a function of $J_{2}$ and $J_{7}$ for $J_{2}^{\prime}=J_{2}$ and $J_{1}^{\prime}=J_{1}=1$. In this regime, there are two chiral spin liquids (CSL), an orbital altermagnetic spin liquid (OAMSL) and an orbital stripy spin liquid (see main text for further details about each phase). (c) Flux configurations for each phase shown in (b). Elementary plaquettes (empty and crossed squares, and triangles inside the latter) are colored according to their gauge flux (oriented clockwise), with values $0$ (green) or $\pi$ (red). The white square in the CSL-I phase indicates the unit cell size considered in our analysis.
• Figure 3: (a) Schematic representation of flux excitations in the OAMSL phase of the spin-7/2 model. The non-topological excitation (A) consists of localized flux flips within a single plaquette, requiring no gauge string for creation or annihilation. Two topological excitations (visons B and C) are shown with their characteristic gauge strings (white); the gauge strings carry zero energy cost and enable free vison propagation. We indicate the modified flux patterns on triangular plaquettes relative to the staggered ground state configuration (not shown for clarity). (b) Type of the lowest flux excitation within the OAMSL domain as a function of $J_{2}$ and $J_{7}$ for $J_{2}^{\prime}=J_{2}$ and $J_{1}^{\prime}=J_{1}=1$. See also Fig. \ref{['fig:supvison']} for more details.
• Figure 4: Boundary conditions for the spin-3/2 (a) and spin-7/2 models (b). Each site is indicated through blue boxed labels. The representative bonds $\hat{u}_{ij}^{\alpha,\tau}$ for each model are represented and colored in accordance with Fig. \ref{['fig:firstmodel']}a and Fig. \ref{['fig:secondmodel']}a.
• Figure 5: (a) Extended phase diagram for the spin-$7/2$ model shown in Fig. \ref{['fig:secondmodel']}b. Schematic representations of the flux configurations for each phase $P_l$ ($l = 1, \dots, 15$) are shown in Fig. \ref{['fig:phasesext']}. (b) Gap $\Delta$ (log scale) as a function of the couplings for the phase diagram in panel (a). As indicated by the color bar, apart from the CSL-I and CSL-II phases, all remaining phases remain gapless over this parameter range.