Altermagnetism without a long-range order

Abstract: The Kugel-Khomskii spin-pseudospin model, originally developed for transition-metal compounds with orbital degrees of freedom, has recently been reinterpreted in the context of altermagnetism. In this work, we theoretically investigate the emergence of altermagnetic behavior in the absence of long-range magnetic or orbital order. Using the rotation-invariant Green's function method for the SU(2) x SU(2) symmetric model on a square lattice and on a linear chain, we analyze spin-spin and spin-pseudospin correlation functions, excitation spectra, heat capacity, and susceptibilities. We show that beyond a critical intersubsystem exchange Kc(T), a composite state arises with nonzero spin-pseudospin correlations, even though the average spin and pseudospin at each site are zero. The excitation spectrum splits into acoustic and optical branches, with nodal lines along qx = qy - a direct signature of altermagnetic symmetry. A peak in heat capacity and a jump in susceptibility are observed at the phase boundary. In 1D, the phase boundary is nonmonotonic and demonstrates reentrant transition. These results establish the concept of an "altermagnetic paramagnet" or "altermagnetic liquid" without long-range order, relevant for low-dimensional and strongly fluctuating systems.

• The paper demonstrates that spin–pseudospin correlations can form an altermagnetic phase without nonzero site-averaged spin or orbital order, using the rotation-invariant Green’s function method in two- and one-dimensional Kugel–Khomskii models.
• The calculated excitation spectrum separates into acoustic and optical branches beyond a critical intersubsystem exchange, with splitting that vanishes along square-lattice zone diagonals and provides a dynamical altermagnetism signature.
• The results predict a two-dimensional boundary scaling approximately as T_c ≈ 0.55|K|^0.55 and a one-dimensional reentrant entangled regime beginning near K ≈ −2.44 at zero temperature, offering tests for neutron scattering, cold atoms, and spin–orbital chains.

Overview

This paper examines whether the defining signatures of altermagnetism can survive in the complete absence of long-range magnetic or orbital order (2607.15954). The authors, Valiulin, Mikheyenkov, and Kugel, revisit the $SU(2)\times SU(2)$-symmetric Kugel–Khomskii spin–pseudospin model on a square lattice and a linear chain, using the rotation-invariant Green's function method (RGM), an approach in which the Mermin–Wagner theorem holds explicitly: all single-site averages $\langle \mathbf{S}_i\rangle$ and $\langle \mathbf{T}_i\rangle$ vanish at any nonzero temperature. Their central finding is that beyond a temperature-dependent critical intersubsystem exchange $K_c(T)$, a composite "entangled" phase emerges, characterized by nonzero on-site and inter-site spin–pseudospin correlations, without breaking translational or spin-rotation symmetry. The paper interprets this phase as an "altermagnetic paramagnet" or "altermagnetic liquid," and identifies its excitation spectrum, with symmetry-protected nodal lines along the zone diagonal, as the direct dynamical analog of altermagnetic band splitting.

Motivation and physical picture

Altermagnetism is conventionally tied to ordered antiferromagnets such as RuO$_2$ and MnTe, where zero net magnetization coexists with momentum-dependent spin splitting of electronic bands, satisfying $$\varepsilon_\uparrow(\mathbf{k}) = \varepsilon_\downarrow(\hat{R}\mathbf{k})$$ for a non-primitive point-group rotation $\hat{R}$. The paper argues that this symmetry is fundamentally a property of the spatial arrangement of bonds and orbitals—orbitally selective hopping and sublattice-dependent orbital polarization—rather than of the ordered spin texture itself. In the Kugel–Khomskii framework, the pseudospin encodes orbital or bond-alternation degrees of freedom, and antiferromagnetic exchange locks spin to these orbitals, producing momentum-dependent spin–orbital locking (2607.15954). The authors' key conceptual move is to note that if the exchange symmetry is preserved while the order is destroyed—by low dimensionality or strong fluctuations—the altermagnetic imprint should persist in correlation functions and excitation spectra even when $$\langle \mathbf{S}\rangle = \langle \mathbf{T}\rangle = 0$$.

The RGM is well suited to this program because it enforces the disordered regime by construction. The composite bilinear $m_0 = \langle S_i^z T_i^z\rangle$ (together with the inter-site $m_g$) acts as an effective order parameter that locks spin and orbital sectors, breaking the full $\langle \mathbf{S}_i\rangle$0 internal symmetry down to a residual diagonal subgroup. The paper is careful to distinguish this from a genuine spin–orbital liquid, in which the full internal symmetry remains unbroken and only short-range entanglement exists.

Model and method

The Hamiltonian is the nearest-neighbor spin–pseudospin model with equal antiferromagnetic intrasubsystem couplings $\langle \mathbf{S}_i\rangle$1 and antiferromagnetic intersubsystem exchange $\langle \mathbf{S}_i\rangle$2:

$\langle \mathbf{S}_i\rangle$3

The ratio $\langle \mathbf{S}_i\rangle$4, $\langle \mathbf{S}_i\rangle$5 is chosen because it is known to maximize spin–orbital entanglement. The calculation follows the standard RGM algorithm with the approximation of Kagan et al. for the biquadratic $\langle \mathbf{S}_i\rangle$6 term, yielding coupled self-consistent equations for spin–spin correlators $\langle \mathbf{S}_i\rangle$7 (first, second, and third neighbors) and spin–pseudospin correlators $\langle \mathbf{S}_i\rangle$8, $\langle \mathbf{S}_i\rangle$9. Vertex corrections for the intersubsystem channel are set to unity; the authors note that experience with the RGM indicates this choice shifts results only quantitatively, not qualitatively.

Correlations and thermodynamics in 2D

On the square lattice, for $\langle \mathbf{T}_i\rangle$0 the spin–pseudospin correlators vanish identically while the spin–spin correlation $\langle \mathbf{T}_i\rangle$1 evolves smoothly. At $\langle \mathbf{T}_i\rangle$2, both $\langle \mathbf{T}_i\rangle$3 (negative, indicating on-site anticorrelation) and $\langle \mathbf{T}_i\rangle$4 (positive, indicating intersite correlation) switch on steeply, while $\langle \mathbf{T}_i\rangle$5 remains featureless. Below the transition, the correlators follow a power law $\langle \mathbf{T}_i\rangle$6 with $\langle \mathbf{T}_i\rangle$7–$\langle \mathbf{T}_i\rangle$8, weakly dependent on $\langle \mathbf{T}_i\rangle$9—behavior the authors characterize as resembling a second-order phase transition with a composite order parameter, though they are explicit that this is not a rigorous proof of an entangled state and that a proper entanglement measure would be needed for confirmation.

The thermodynamic consequences are consistent with this picture. The heat capacity exhibits a step and a peak at $K_c(T)$0, with the peak height growing with $K_c(T)$1; curves converge to a common high-temperature asymptote and satisfy the Nernst theorem as $K_c(T)$2. Both the spin–spin susceptibility $K_c(T)$3 and the spin–pseudospin susceptibility $K_c(T)$4 show jumps at the transition, and outside the transition region both are only weakly dependent on $K_c(T)$5 and $K_c(T)$6. The small-$K_c(T)$7 behavior of $K_c(T)$8 reproduces the known Heisenberg result.

Excitation spectrum as an altermagnetic diagnostic

The most consequential result concerns the collective modes. For $K_c(T)$9 the two subsystems are spectrally degenerate (given $_2$0). Beyond the transition, the spectrum splits into an acoustic branch $_2$1—always gapless at $_2$2, i.e., a Goldstone mode—and an optical branch $_2$3, with the ordering $_2$4 at $_2$5. The splitting is controlled by $_2$6 and the lattice form factor $_2$7, which plays the role of the $_2$8-wave or $_2$9-wave structure factor familiar from electronic altermagnets. Critically, the splitting vanishes along the nodal lines $$\varepsilon_\uparrow(\mathbf{k}) = \varepsilon_\downarrow(\hat{R}\mathbf{k})$$0—the zone diagonal—mirroring the symmetry-protected nodes of the altermagnetic band structure. The authors argue that this combination of features (nodal lines, sharp onset driven by $$\varepsilon_\uparrow(\mathbf{k}) = \varepsilon_\downarrow(\hat{R}\mathbf{k})$$1, and branch interconversion under $$\varepsilon_\uparrow(\mathbf{k}) = \varepsilon_\downarrow(\hat{R}\mathbf{k})$$2) distinguishes genuine altermagnetic symmetry from generic hybridization. The optical branch is identified as an "altermagnon," a propagating composite spin–orbital excitation that exists above any ordering temperature precisely because short-range entanglement lifts the degeneracy without condensation; it should be observable in neutron scattering as a dissipative mode.

At large $$\varepsilon_\uparrow(\mathbf{k}) = \varepsilon_\downarrow(\hat{R}\mathbf{k})$$3 the upper portions of both branches become nearly dispersionless. The 2D phase boundary is well fitted by $$\varepsilon_\uparrow(\mathbf{k}) = \varepsilon_\downarrow(\hat{R}\mathbf{k})$$4.

The 1D anomaly and reentrant behavior

The one-dimensional chain yields a qualitatively different and counterintuitive phase diagram. The boundary between zero and nonzero spin–pseudospin correlations does not emanate from $$\varepsilon_\uparrow(\mathbf{k}) = \varepsilon_\downarrow(\hat{R}\mathbf{k})$$5 at $$\varepsilon_\uparrow(\mathbf{k}) = \varepsilon_\downarrow(\hat{R}\mathbf{k})$$6; it begins at $$\varepsilon_\uparrow(\mathbf{k}) = \varepsilon_\downarrow(\hat{R}\mathbf{k})$$7 at $$\varepsilon_\uparrow(\mathbf{k}) = \varepsilon_\downarrow(\hat{R}\mathbf{k})$$8 and is nonmonotonic, implying a reentrant transition in which increasing temperature can drive the system into the entangled state. The authors connect this to previously reported reentrant entanglement in low-dimensional spin models, offering the qualitative explanation that at low temperatures lower-energy states need not be more correlated than slightly higher-energy ones—an entropy-stabilization mechanism. They suggest this prediction is testable in cold-atom simulators or organic spin–orbital chains. The RGM in 1D is calibrated against Bethe ansatz and finite-chain data for the pure Heisenberg limit, which lends credibility to the thermodynamics, although the reentrant boundary itself has not been verified by an exact method in this work.

Limitations and open questions

Several caveats are stated or implicit in the paper. The claim of an "entangled" phase rests on the behavior of correlation functions rather than on a computed entanglement measure; the authors themselves flag this as requiring further work, as does a detailed analysis of entanglement in the altermagnetic context. The RGM is a self-consistent approximate scheme: the treatment of the $$\varepsilon_\uparrow(\mathbf{k}) = \varepsilon_\downarrow(\hat{R}\mathbf{k})$$9 term follows Kagan et al.'s approximation, and intersubsystem vertex corrections are fixed at unity, so the precise location of $\hat{R}$0—particularly the 1D boundary and its reentrant segment—may shift under a more refined treatment. The identification of the composite correlator as an order parameter and the analogy to a second-order transition is argued by analogy, not proven. Finally, the paper does not address whether the short-range-order altermagnetic state produces the electronic consequences (spin-split bands, anomalous transport) associated with crystalline altermagnets; the mapping is made at the level of excitation spectra and symmetry, and the extension to itinerant or doped systems remains open.

Conclusion

This work demonstrates within the $\hat{R}$1 Kugel–Khomskii model that the essential signatures of altermagnetism—composite spin–pseudospin correlations, an acoustic/optical excitation splitting with symmetry-protected nodal lines, and thermodynamic anomalies at a sharp boundary—can exist entirely without long-range order, in both 2D and 1D at finite temperature (2607.15954). The resulting "altermagnetic liquid" concept extends the search for altermagnetic physics to low-dimensional and strongly fluctuating systems where conventional order is prohibited, and its reentrant 1D phase boundary constitutes a specific, falsifiable prediction for cold-atom and molecular-chain realizations.