Higher-Order Exchange Interactions

• Higher-order exchange interactions are multi-spin couplings beyond pairwise Heisenberg terms that stabilize complex noncollinear and topological spin textures in various systems.
• They arise from higher-order perturbation processes in models like the Hubbard model and are crucial for selecting unique ground states such as triple-Q and skyrmion phases.
• Computational methods such as DFT, Monte Carlo, and spectral analysis enable precise modeling of HOI, aiding the design of advanced magnetic and quantum devices.

Higher-order exchange interactions (HOI) are multi-spin couplings in quantum or classical systems that go beyond the simple pairwise terms of the Heisenberg model. These interactions—including terms such as biquadratic, three-site, and four-site four-spin couplings—emerge naturally in strongly correlated materials, frustrated magnets, low-dimensional magnets, ultracold gases, and engineered quantum systems. They play a critical role in stabilizing complex, often noncollinear, spin textures and can lead to novel phase transitions, emergent dynamics, and enhanced functionality in both natural and synthetic systems.

1. Mathematical Formulation and Physical Origin

HOI arise as higher-order terms in the expansion of the effective spin Hamiltonian, particularly when underlying degrees of freedom (e.g., induced moments, electronic correlations) lead to non-linear dependencies. The generic extended spin Hamiltonian incorporates:

$$\mathcal{H} = -\sum_{\langle ij \rangle} J_{ij}\; \mathbf{S}_i \cdot \mathbf{S}_j - \sum_{\langle ij \rangle} B_{ij} (\mathbf{S}_i \cdot \mathbf{S}_j)^2 - \sum_{\langle ijkl \rangle} K_{ijkl} \sum_{\text{sym}} (\mathbf{S}_i \cdot \mathbf{S}_j)(\mathbf{S}_k \cdot \mathbf{S}_l) + \ldots$$

• $J_{ij}$: Bilinear (Heisenberg) exchange
• $B_{ij}$: Biquadratic (two-site four-spin) exchange
• $K_{ijkl}$: Four-site four-spin exchange (often written as a symmetrized sum)
• Additional terms: Three-site four-spin, Dzyaloshinskii–Moriya, anisotropy, Zeeman, etc.

The physical mechanisms for HOI include:

• Nonlinear induction of magnetic moments (as in FeRh, where the Rh moment responds quadratically to the Fe Weiss field, giving rise to four-spin terms after integrating out the Rh degrees of freedom) (Barker et al., 2014).
• Fourth-order perturbation processes in the Hubbard model, leading to biquadratic and four-spin couplings, especially for $S \geq 1$ systems (Beyer et al., 5 Jun 2025, Gutzeit et al., 2021).
• Electron hopping paths involving more than two lattice sites (multi-site superexchange).

The explicit inclusion of HOI is necessary to capture states where bilinear exchange is insufficient or degenerate, such as noncollinear multi-$\mathbf{Q}$ states, triple-$Q$ states, and noncoplanar spin arrangements (Beyer et al., 5 Jun 2025, Kollwitz et al., 18 Aug 2025).

2. Role in Stabilizing Complex Magnetic States

HOI often select and stabilize exotic noncollinear or topological spin states that are degenerate or nearly degenerate in bilinear models:

• Triple-Q and Multi-Q States: In hexagonal or triangular lattices, the superposition of three symmetry-related spin spirals (triple-Q) leads to a noncoplanar, tetrahedral configuration. Pairwise models typically cannot distinguish between single-Q (spiral) and triple-Q states—HOI terms are required to select the ground state. For example, in a Mn monolayer or bilayer on noble metal substrates, a triple-Q state with tetrahedral nearest-neighbor angles is stabilized only with significant biquadratic and multi-site HOI, as shown by first-principles and atomistic modeling (Beyer et al., 5 Jun 2025).
• Row-Wise AFM and 2Q/uudd States: In systems like Rh/Fe/Ir(111), the stacking sequence dramatically changes the ground state: frustrated exchange in fcc-stacking yields spin spirals, while strong HOI in hcp-stacking stabilize collinear up-up-down-down ($\uparrow\uparrow\downarrow\downarrow$) (2Q) or canted states (Romming et al., 2016, Gutzeit et al., 2021).
• Skyrmion Stability and Topological Transitions: In 2D and ultrathin film magnets, the four-site four-spin term can substantially increase skyrmion collapse energy barriers (enhancing lifetimes), or, when negative, reduce stability and enable transitions to other topological or ferrimagnetic states. Notably, while DMI dominates the energy barrier at the saddle point for skyrmion collapse, HOI can induce new collapse pathways and topological transitions, such as the ferric transition observed in MnSeTe (Paul et al., 2019, Arya et al., 12 Sep 2025).

3. Thermodynamics, Phase Transitions, and Critical Behavior

The thermodynamics and phase behavior of systems with HOI often depart drastically from those described by pairwise-only interactions:

• First-Order Phase Transitions: The competition between bilinear and four-spin exchange terms can induce first-order metamagnetic transitions, as in FeRh, featuring thermal hysteresis and mixed-phase coexistence near the transition temperature. The different temperature scaling exponents of bilinear and higher-order terms (e.g., $M(T)^{3.48}$ for HOI vs. $M(T)^{1.5-2}$ for bilinear) govern the energetics and the thermal crossover (Barker et al., 2014).
• Entropy-Driven Transitions: In frustrated noncollinear antiferromagnets supporting triple-Q states, the free energy difference between triple-Q and row-wise AFM states is subtle, and a low-temperature phase transition can be driven by entropy, not just energy. Monte Carlo simulations and partition function analysis show that even energetically suboptimal states may become stable due to larger entropy contributions (Kollwitz et al., 18 Aug 2025).
• Discontinuous Magnetization and Even-Odd Effects: The inclusion of rotationally invariant HOI in Heisenberg models can cause discontinuities and even–odd effects in the ground state magnetization, even in isotropic or unfrustrated lattices. The parity of the HOI order $q$ controls the number and character of discontinuities, leading to profoundly different responses from those in pairwise-only systems (Konstantinidis, 2017).
• XY Models and Criticality: Inclusion of infinite or a finite number of HOI in generalized XY models alters the stiffness, critical exponents, and phase transition order (continuous vs. first order), depending on the sign, decay, and parity of the higher-order terms. Competing HOI can drive multiple transitions (e.g., into canted ferromagnetic phases) or suppress transition temperatures close to zero (Žukovič et al., 2017, Žukovič et al., 2021).

4. Computational and Analytical Methodologies

A variety of computational and analytical techniques are used to paper HOI:

• Density Functional Theory (DFT): First-principles DFT provides direct access to the total energy change for various collinear and noncollinear spin arrangements, allowing for extraction of both bilinear and HOI parameters. Methods include spin-spiral calculations (generalized Bloch theorem), mapping to effective spin Hamiltonians, and parametric studies as a function of stacking, band filling, or layer architecture (Beyer et al., 5 Jun 2025, Kartsev et al., 2020, Gutzeit et al., 2021).
• Atomistic Spin Dynamics and Monte Carlo: Time-dependent ASD simulations integrate the stochastic Landau–Lifshitz–Gilbert equation for Hamiltonians including HOI, enabling the paper of ultrafast magnetization dynamics (e.g., following laser excitation) and finite-temperature phase transitions (Barker et al., 2014, Kollwitz et al., 18 Aug 2025). Classical Monte Carlo/Langevin simulations are fundamental to probing equilibrium and nonequilibrium properties.
• Analytical Expansions: Perturbation theory in multi-orbital Hubbard models produces explicit expressions for multi-spin exchange terms (biquadratic, three-site, four-site), elucidating their scaling with superexchange (e.g., $t^4/U^3$) (Kartsev et al., 2020, Gutzeit et al., 2021). Harmonic and extended harmonic approximations can analytically model composition of free energies and entropy contributions near mean-field solutions (Kollwitz et al., 18 Aug 2025).
• Spectral and Information-Theoretic Approaches: For complex networks, advanced information-theoretic frameworks quantify non-pairwise synergy, redundancy, and collective constraints, providing computational tools for experimental or simulation data (Belloli et al., 6 Jan 2025, Faes et al., 2022).

5. Impact in Quantum Gases and Condensed Matter

HOI are not limited to magnetic crystals but play vital roles in cold atom and condensed matter systems:

• Bose–Einstein Condensates with HOI: In atomic BECs, modified Gross–Pitaevskii equations with higher-order interaction terms ($-\delta\nabla^2(|\psi|^2)\psi$) have been rigorously derived for low-dimensional and strongly confined geometries (Ruan et al., 2015, Bao et al., 2017). HOI alter mean-field coefficients, the shape of the condensate density (e.g., enforcing Thomas–Fermi “flat-top” profiles instead of Gaussians), and can regularize collapse instabilities, yielding novel scaling regimes and phase diagrams.
• Ecological and Biological Systems: Dynamical models generalized beyond pairwise Lotka–Volterra equations exhibit that HOI can stabilize diversity, prevent collapse, and produce realistic species abundance distributions not captured by classical models. The speed of onset for these interactions may control system stability and extinction dynamics, revealing new control variables in complex networks (Kang et al., 29 Jul 2025, Giel et al., 26 Aug 2024).
• Information Flow and Brain Networks: In neuroscience, covariational measures rooted in information theory generalize to HOI, enabling assessment of synergistic and redundant multi-way couplings. Such analyses, both static and spectral, are central to understanding integrated cognitive function and functional MRI dynamics (Belloli et al., 6 Jan 2025, Faes et al., 2022, Zhao et al., 27 Jul 2025).

6. Control, Tunability, and Materials Engineering

The realizability and tunability of HOI in real materials and devices is of increasing experimental and technological importance:

• Material Engineering: The magnitude and sign of HOI (biquadratic, three-site, four-site) in transition-metal multilayers, van der Waals magnets, and 2D materials can be tuned by band filling, stacking sequence, or introduction of asymmetry (as in Janus structures). This ability directly impacts the stability and switching of skyrmions, spin spirals, or more exotic spin textures (Paul et al., 2019, Gutzeit et al., 2021, Arya et al., 12 Sep 2025).
• Technological Applications: Systems with engineered HOI offer high thermal stability for skyrmion-based memory elements, robust multi-state logic architectures, and potentially new logic devices leveraging nontrivial topological transitions such as the HOI-induced ferric transitions in MnSeTe (Arya et al., 12 Sep 2025).
• Design in 2D Magnetism: HOI renormalize magnetic anisotropy, spin wave gaps, and magnon topology, thus providing handles for magnonic transport and topological protection in atomically thin magnets (Kartsev et al., 2020).

7. Outlook and Future Directions

The explicit inclusion, measurement, and control of HOI represent an expanding frontier in condensed matter, cold atom physics, and complex networks:

• Expanded Theoretical Frameworks: General frameworks combining DFT, atomistic modeling, information theory, and spectral analysis are enabling the systematic exploration of HOI consequences, including unified descriptions across quantum and classical platforms (Beyer et al., 5 Jun 2025, Belloli et al., 6 Jan 2025, Faes et al., 2022).
• Novel Noncollinear and Topological Phases: The discovery of entropy-driven and topologically unconventional transitions (e.g., ferric transitions, canted phases, multi-Q Lifshitz points) supported or selected by HOI underlines the need for joint theoretical-experimental studies.
• Application Beyond Magnetism: The principles identified in HOI studies—such as non-additivity, redundancy/synergy, nested interactions (as in pangraph and hypergraph formalisms)—are applicable to ecological, brain, epidemiological, and even engineered network systems (Iskrzyński et al., 14 Feb 2025, Belloli et al., 6 Jan 2025).
• Systematic Materials Search: High-throughput and machine-learning-guided materials searches targeting strong HOI regimes are promising for the rational discovery of functional quantum materials with complex, adjustable magnetic or collective phenomena.

In summary, higher-order exchange interactions are essential for the realistic modeling and control of magnetic and multi-component many-body systems, yielding phenomena not anticipated from pairwise models alone. Their inclusion is critical for predictive theoretical models, experimental interpretation, and material/device engineering across a range of classical and quantum systems.