Three-Spin Interactions in Quantum Systems

Updated 22 September 2025

Three-spin interactions are higher-order spin terms in lattice models that depend on the collective state of three spins, enabling exotic quantum phases and frustration mechanisms.
They are encoded through various Hamiltonian terms—such as scalar chiral, cluster, and triple Ising interactions—with applications in quantum simulation, cold atomic gases, and engineered spin systems.
Their inclusion drives novel phase transitions, modifies entanglement structures, and stabilizes unique topological orders and reentrant thermal behaviors in quantum magnets.

Three-spin interactions are higher-order interaction terms in lattice spin models whereby the energy depends on the collective state of three spins, often leading to complex quantum and classical phenomena that are distinct from those found in standard two-spin (bilinear) models. They play a central role in the stabilization of exotic quantum phases, encode new mechanisms of frustration, and yield unique universal behaviors across quantum phase transitions. Three-spin couplings appear naturally in effective Hamiltonians derived from higher-order perturbative processes or can be present as primary interactions in engineered spin systems, cold atomic gases, or quantum simulation platforms.

1. Mathematical Structures and Physical Realizations

Three-spin interactions are encoded in Hamiltonians through terms involving products of spin operators from three distinct sites. Canonical examples include:

Scalar chiral terms: $S_i \cdot (S_j \times S_k)$ , characteristic of chiral three-spin physics, as on the Kagome lattice (Bauer et al., 2013).
Cluster (e.g., XZX or YZY) terms: $\sigma^x_{i-1}\sigma^z_i\sigma^x_{i+1}$ (F et al., 18 Mar 2025), relevant for cluster-state quantum information.
Triple Ising terms: $\sigma^z_i \sigma^z_{i+1} \sigma^z_{i+2}$ , as in generalizations of the quantum Ising chain (Wang et al., 2011, Iglói et al., 24 Mar 2025).
Effective interactions among atoms in optical lattices and spinor Bose-Einstein condensates, often modeled via effective three-body operators derived from second-order perturbation theory (Mahmud et al., 2013, Mestrom et al., 2021, Nabi et al., 2017).

The physical realization of three-spin interactions spans quantum magnets, cluster-state computation, ultracold atom systems (where higher-order collisions or band effects are relevant), and electronic systems where they arise as corrections in strong-coupling expansions of multiorbital Hubbard-type models (Michaud et al., 2013).

2. Quantum Phase Transitions and Symmetry Breaking

Three-spin interactions drive rich phase diagrams, often introducing new symmetry-breaking patterns and critical points:

Bifurcation of Ground-State Fidelity: In one-dimensional models with both two-spin and three-spin interactions, ground-state fidelity per site exhibits bifurcation as a function of interaction strength, sharply signaling quantum phase transitions (QPTs). The critical point, determined by the bifurcation, coincides with spontaneous breaking of translation or more complex discrete symmetries such as $Z_3$ or $G=Z_2 \times Z_2 \times Z_2/Z_2$ (Wang et al., 2011).
Order Parameters and Central Charge: Local order parameters can be constructed from reduced density matrices, distinguishing phases with degenerate ground states. For instance, in phases with $Z_3$ symmetry breaking:

$O_{M_1} = \frac{\langle\sigma_z^A\rangle - 2\langle\sigma_z^B\rangle + \langle\sigma_z^C\rangle}{4}$

Finite-entanglement scaling of the entanglement entropy yields central charges characteristic of different universality classes, e.g., three-state (Potts, $c\approx0.8$ ) vs. four-state Potts ( $c\approx1$ ).

Tricriticality and Reentrant Transitions: In two-dimensional spin-1 quantum Ising models, three-spin couplings induce rich phase diagrams with lines of continuous and discontinuous transitions, tricritical points, and reentrant (double reentrant) thermal behaviors, controlled by the interplay of three-body interactions, crystal field, and biquadratic couplings (Jiang et al., 2011).
New Universality Classes in Disordered Systems: The random quantum Ising chain with three-spin couplings exhibits an infinite disorder fixed point distinct from the standard random Ising chain, with typical correlation-length exponent $\nu_{typ} = (\ln2)/(\ln3) \approx 0.631$ , and activated dynamical scaling $\ln\epsilon \sim L^\psi$ with $\psi = (3\ln3)/(2\ln2) - 1$ (Iglói et al., 24 Mar 2025).

3. Entanglement Structures and Indicators

Three-spin interactions engender nontrivial patterns of quantum correlations:

Multipartite vs. Bipartite Entanglement: Genuine multipartite entanglement measures such as the generalized geometric measure (GGM) serve as robust indicators of QPTs, particularly in regimes governed by three-spin terms. For the model

$H = -\frac{J}{4}\sum_n \left[\sigma^x_n \sigma^x_{n+1} + \sigma^y_n \sigma^y_{n+1} + \frac{\alpha}{2}(\sigma^x_{n-1}\sigma^z_n\sigma^y_{n+1} - \sigma^y_{n-1}\sigma^z_n\sigma^x_{n+1}) \right]$

the GGM signals a transition at $\alpha_c=1$ , unlike the two-site concurrence which shows extra (spurious) discontinuities (Bera et al., 2012).

Cluster State Correlations: Certain three-spin ("cluster") models yield ground states with vanishing nearest-neighbor quantum correlations but nontrivial next-nearest neighbor measures such as concurrence, mutual information, and discord, especially in regions proximate to degeneracy critical points (F et al., 18 Mar 2025).
Entanglement Entropy and Topological Order: In models with multi-spin chirality (e.g., on the Kagome lattice), chiral three-spin terms stabilize either topological spin liquids (gapped, with entanglement entropy showing area law with topological correction) or gapless non-Fermi liquid phases, depending on the patterning of the three-spin couplings (Bauer et al., 2013).

4. Frustration, Chiral Order, and Exotic Phases

Three-spin interactions naturally promote frustration and chiral order:

Chiral Spin Liquids: SU(2)-invariant chiral three-spin interactions, such as $S_i \cdot (S_j \times S_k)$ on each triangle, can stabilize gapped (Kalmeyer-Laughlin type) and gapless quantum spin liquid phases on the Kagome lattice, with distinctive ground-state degeneracy and edge excitations (Bauer et al., 2013).
Scalar Chirality and Orbital Antiferroelectricity: In exactly solvable XX chains with both XZX+YZY and XZY–YZX three-spin couplings, the ground state may exhibit spontaneous magnetoelectric order (coexisting nonzero magnetization and chirality) and orbital antiferroelectricity—characterized by alternating plaquette current loops inducing orbital electric dipole moments (Thakur et al., 2023).

Interaction Form	Symmetry Broken	Exemplary Ground State
$S_i \cdot (S_j \times S_k)$	Parity, Time-reversal	Kalmeyer-Laughlin spin liquid (Bauer et al., 2013)
$\sigma^x_{i-1}\sigma^z_i\sigma^x_{i+1}$	Dualities/Cluster symmetries	1D cluster state, measurement-based quantum info (F et al., 18 Mar 2025)
$\sigma^z_i\sigma^z_{i+1}\sigma^z_{i+2}$	$Z_3$ or $G$ discrete symmetry	Symmetry breaking with multiple degenerate ground states (Wang et al., 2011)

Nematic and Quadrupolar Phases: In alternating spin chains with three-body exchange, critical non-magnetic phases with dominant quadrupolar (nematic) correlations emerge, evidenced by the structure of the low-lying energy spectrum and entanglement entropy scaling (Ivanov et al., 2014).
Frustrated Magnetic Orders: Competition between two- and three-spin terms can generate large ground-state degeneracy, intricate phase diagrams featuring helical, up–up–down–down, and multipolar orders (Michaud et al., 2013, Ivanov et al., 2016).

5. Thermodynamic and Dynamical Properties

Three-spin interactions deeply impact thermodynamic and response properties:

Magnetocaloric Effect Enhancement: In exactly solvable spin-1/2 XX chains with next-nearest neighbor three-spin interactions, the enhancement of the magnetocaloric effect (MCE) is particularly marked near quantum critical points, and is even stronger (by ~30%) near quantum triple points where the excitation spectrum is unusually soft (quartic in momentum) (Topilko et al., 2012).
Thermal Reentrance and Tricriticality: In two-dimensional spin-1 Ising models with crystal field and biquadratic interaction, three-spin terms induce reentrant phase transitions and shift the location of tricritical points (Jiang et al., 2011).
Classical Integrability: The classical Heisenberg spin triangle serves as a rare exactly integrable many-body system with three-spin dynamics, allowing explicit calculation of the density of states, specific heat, and long-time correlation functions, demonstrating the geometric and topological origin of singularities in thermodynamic and dynamical observables (Schmidt et al., 2022).

6. Three-Spin Interactions in Quantum Simulators and Cold Atoms

Effective and native three-spin interactions are directly realizable and observable in state-of-the-art quantum simulation experiments:

Optical Lattices and Virtual Excitations: In spin-1 optical lattices, virtual excitation processes yield effective three-body (spin-dependent and spin-independent) on-site interactions, substantially modifying spin-mixing dynamics and phase coherence collapses and revivals. The effective Hamiltonian is

$H_{3B,eff} = \frac{V_0}{6}\,\hat{n}(\hat{n}-1)(\hat{n}-2) + \frac{V_2}{6}\,[\mathcal{F}^2 - 2\hat{n}](\hat{n}-2)$

where $V_2 = 2(U_2/U_0)V_0$ , and $U_2$ is the spin-dependent two-body interaction (Mahmud et al., 2013).

Spinor Bose Gases and Scattering Hypervolumes: Three-body scattering hypervolumes $D_{F_{3b}}$ parameterize effective three-body interaction strengths in spin-1 BECs; at high densities or near two-body Feschbach resonances, three-body spin mixing governs the spin dynamics and can dominate over two-body processes (Mestrom et al., 2021).
Mott Insulator Phases with Three-Body Repulsion/Attraction: In spin-1 ultracold Bose gases, repulsive three-body interactions stabilize higher occupation Mott lobes and preserve odd-even asymmetries (nematic vs. singlet phases), while attractive three-body terms can merge multiple Mott lobes, reshaping the phase diagram (Nabi et al., 2017).
Entanglement as a Diagnostic: In cluster models with three-spin interactions, the spatial structure of correlations and entanglement is highly sensitive to the presence and type of three-spin term, yielding vanishing nearest-neighbour but nonzero next-nearest neighbour measures—directly accessible in cold atom experiments (F et al., 18 Mar 2025).

7. Multispin Interactions and Beyond: Chiral Magnetism and Field Theory Constraints

The extension of spin Hamiltonians to include three-spin and higher multispin interactions is motivated both by first-principles calculations and fundamental field-theoretic consistency requirements:

Ab Initio Calculation and SOC/Topology Effects: First-principles calculations show that three-spin Dzyaloshinskii–Moriya-like (TDMI) and chiral (TCI) interactions emerge due to relativistic spin-orbit coupling (SOC) and topological orbital susceptibility (TOS)—essential for capturing the correct magnetic ground state in reduced-symmetry systems (Mankovsky et al., 2019, Mankovsky et al., 2021).
Antisymmetric Tensor Structure and Discrete Symmetries: The three-spin chiral interaction appears as an antisymmetric rank-3 tensor, with invariance properties fixed by fundamental discrete ( $C$ , $R$ , $T$ ) symmetries and the dimension of spacetime, governing the allowed interactions in Dirac–higher-spin field theory. Explicit mod-2 ( $D$ mod 4) classification constrains coupling structures and intrinsic parity (Chakraborty et al., 19 Feb 2024).
Impact on Magnetic Texture Stabilization: Three-spin terms can stabilize noncoplanar textures (such as skyrmions, spin liquids, or topological chiral states) by providing a spin-chirality energy cost that cannot be mimicked by pairwise or symmetric multispin terms alone (Mankovsky et al., 2019, Thakur et al., 2023).

The inclusion of three-spin interactions is thus essential not only for reproducing complex phenomena in quantum magnetism, topological matter, and quantum simulations, but also for constructing theoretically consistent and experimentally relevant models of strongly correlated quantum systems.