Active XY Model: Nonequilibrium U(1) Theories

• Active XY model is a family of nonequilibrium U(1) theories where classical XY spins are driven by persistent torques, intrinsic frequencies, and self-propulsion.
• These models modify equilibrium Berezinskii–Kosterlitz–Thouless physics by altering quasi-long-range order and defect dynamics through various active driving mechanisms.
• Research employs diverse methods—including numerical simulations, renormalization group analysis, and tensor-network techniques—to benchmark and explore the novel ordering phenomena in active XY systems.

In the literature, the term active XY model refers not to a single canonical Hamiltonian but to a family of nonequilibrium extensions of the classical XY model in which the local $U(1)$ angle field is driven by persistent torques, intrinsic frequencies, self-propelled particle motion, coupling to conserved diffusive species, or coupling to flow fields. Across these variants, the common structural element is an XY phase or spin variable $\theta$, while the central physical question is how activity modifies Berezinskii–Kosterlitz–Thouless (BKT) physics, quasi-long-range order (QLRO), and the dynamics of topological defects (Shi et al., 2024, Rouzaire et al., 2021, Inoue et al., 23 Jul 2025, Haldar et al., 2021, Haldar et al., 2019, Maes et al., 2011).

1. Equilibrium reference model and defect taxonomy

The equilibrium reference point is the two-dimensional XY model with planar spins $\mathbf{S}_i=(\cos\theta_i,\sin\theta_i)$ and nearest-neighbor Hamiltonian

$$H_{\text{XY}} = -J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).$$

In two dimensions, the low-temperature phase has QLRO rather than true long-range order: the spin correlation decays algebraically, $C(r)\sim r^{-\eta(T)}$, and the BKT transition occurs at $\eta(T_{\mathrm{KT}})=1/4$. The topological defects are integer vortices, defined by a winding number $m\in\mathbb{Z}$ around a loop (Longia et al., 2023).

A useful extension is the generalized XY model with a nematic interaction of period $q=2$,

$$H = - \left( \sum_{\langle i j \rangle} \Delta \cos(\theta_i - \theta_j) + \sum_{\langle i j \rangle} (1 - \Delta) \cos\big(2(\theta_i - \theta_j)\big) \right),$$

which interpolates between the usual vector XY limit $\Delta=1$ and a purely nematic model $\theta$0. This model supports both integer and half-integer vortices and exhibits ferromagnetic, nematic, and disordered phases separated by BKT-like and Ising-like transition lines (Samlodia et al., 2024). That defect taxonomy is directly relevant to active polar and active nematic generalizations, because activity changes defect energetics and kinetics before it changes the underlying topological classification.

2. Principal nonequilibrium formulations

One common construction is the persistent-noise XY model, in which passive XY spins are driven by an Ornstein–Uhlenbeck angular velocity. On a triangular lattice, the microscopic dynamics can be written as

$\theta$1

with exponentially correlated torque noise

$\theta$2

For $\theta$3, the model reduces to the equilibrium XY Langevin dynamics, whereas any finite $\theta$4 breaks detailed balance (Shi et al., 25 Feb 2026).

A second formulation is the self-driven rotor or short-range noisy Kuramoto version,

$\theta$5

with intrinsic frequencies $\theta$6 drawn from a Gaussian of variance $\theta$7. After rescaling, the relevant activity parameter is $\theta$8. Here the nonequilibrium drive is the sitewise energy injection by the intrinsic frequencies, which persists even at $\theta$9 (Rouzaire et al., 2021).

A third class uses mobile, self-propelled XY particles on a lattice with exclusion and vacancies. The spin sector is governed by

$\mathbf{S}_i=(\cos\theta_i,\sin\theta_i)$0

while particle motion is biased along the local spin direction by a self-propulsion parameter $\mathbf{S}_i=(\cos\theta_i,\sin\theta_i)$1. At $\mathbf{S}_i=(\cos\theta_i,\sin\theta_i)$2 the model reduces to the standard XY model; for $\mathbf{S}_i=(\cos\theta_i,\sin\theta_i)$3, it becomes a diluted XY system with spin-dependent motility (Inoue et al., 23 Jul 2025).

A fourth formulation couples the XY phase to additional hydrodynamic fields. For active spinners on a substrate with a conserved diffusing species $\mathbf{S}_i=(\cos\theta_i,\sin\theta_i)$4, the hydrodynamic equations are

$\mathbf{S}_i=(\cos\theta_i,\sin\theta_i)$5

$\mathbf{S}_i=(\cos\theta_i,\sin\theta_i)$6

where the active terms violate detailed balance (Haldar et al., 2021). Closely related hydrodynamic theories couple $\mathbf{S}_i=(\cos\theta_i,\sin\theta_i)$7 directly to a two-dimensional flow field $\mathbf{S}_i=(\cos\theta_i,\sin\theta_i)$8, with advection, vorticity, and compressibility couplings in the phase equation and an active Stokes equation for the flow (Haldar et al., 2019).

A fifth formulation is the uniformly driven three-dimensional rotor model,

$\mathbf{S}_i=(\cos\theta_i,\sin\theta_i)$9

which represents a spatially extended XY medium under a uniform non-conservative torque (Maes et al., 2011).

3. Ordering phenomena under activity

Activity does not modify XY ordering in a universal way. In the persistent-noise model, the long-wavelength spin-wave sector remains Gaussian but acquires an effective large-scale temperature $$H_{\text{XY}} = -J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).$$0. The equal-time spectrum becomes

$$H_{\text{XY}} = -J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).$$1

so that

$$H_{\text{XY}} = -J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).$$2

This produces QLRO with exponents that grow linearly with persistence and can exceed the equilibrium BKT value $$H_{\text{XY}} = -J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).$$3 without immediate defect proliferation (Shi et al., 2024). A later analysis concluded that the order–disorder transition remains of BKT type, with $$H_{\text{XY}} = -J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).$$4 and $$H_{\text{XY}} = -J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).$$5, but with a persistence-dependent critical exponent $$H_{\text{XY}} = -J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).$$6 that increases with $$H_{\text{XY}} = -J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).$$7; for example, the reported values are $$H_{\text{XY}} = -J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).$$8 at $$H_{\text{XY}} = -J\sum_{\langle ij\rangle}\cos(\theta_i-\theta_j).$$9, $C(r)\sim r^{-\eta(T)}$0 at $C(r)\sim r^{-\eta(T)}$1, $C(r)\sim r^{-\eta(T)}$2 at $C(r)\sim r^{-\eta(T)}$3, and $C(r)\sim r^{-\eta(T)}$4 at $C(r)\sim r^{-\eta(T)}$5 (Shi et al., 25 Feb 2026). This resolves an earlier open question only partially: the essential singularities remain BKT-like, but the equilibrium relation $C(r)\sim r^{-\eta(T)}$6 is lost.

The self-driven rotor model gives the opposite outcome. For any $C(r)\sim r^{-\eta(T)}$7, equilibrium QLRO is destroyed: at low temperature the correlation function becomes exponential,

$C(r)\sim r^{-\eta(T)}$8

and the system breaks into a mosaic of ordered domains with $C(r)\sim r^{-\eta(T)}$9. A finite-size crossover line survives in the density of free vortices,

$\eta(T_{\mathrm{KT}})=1/4$0

with $\eta(T_{\mathrm{KT}})=1/4$1, but in the thermodynamic limit the true BKT transition survives only at $\eta(T_{\mathrm{KT}})=1/4$2 (Rouzaire et al., 2021). The contrast with persistent-noise driving is structural: intrinsic-frequency disorder directly destroys the global phase field, whereas persistent Ornstein–Uhlenbeck forcing separates large-scale spin-wave fluctuations from local defect nucleation.

Hydrodynamic active XY theories on substrates and in flows produce additional ordered regimes. For active XY spins coupled to a conserved diffusing species, the stable phase-ordered states exhibit generalized QLRO in which

$\eta(T_{\mathrm{KT}})=1/4$3

with nonuniversal $\eta(T_{\mathrm{KT}})=1/4$4. Depending on the nonlinear couplings, the ordered state can be stronger than equilibrium QLRO or weaker than equilibrium QLRO, with a corresponding suppression or enhancement of density fluctuations (Haldar et al., 2021). Flow-coupled live XY spins show an analogous classification: in the Malthusian case, sufficiently strong active flow–phase couplings can yield QLRO, stronger-than-QLRO, or weaker-than-QLRO, whereas other parameter choices lead to disorder of the same qualitative type as immobile active XY spins or the two-dimensional Kardar–Parisi–Zhang equation (Haldar et al., 2019).

In three dimensions, a uniform torque produces yet another pattern. The driven XY model has essentially a unique translation-invariant stationary distribution for almost all temperatures, but at low temperature it also supports rotating states in which the ordered phase does not relax pointwise in time: the macroscopic magnetization simply rotates at the imposed drift rate (Maes et al., 2011).

4. Topological defects and defect-mediated dynamics

Topological defects remain central in all active XY variants, but their role becomes strongly model dependent. In equilibrium and generalized equilibrium XY models, vortices are classified by the topology of the angle field: integer vortices in the standard model, and both integer and half-integer vortices in the $\eta(T_{\mathrm{KT}})=1/4$5 generalized model (Longia et al., 2023).

In the self-driven rotor model, activity qualitatively changes defect motion. At equilibrium, an isolated vortex has a Gaussian displacement distribution and a mean-squared displacement

$\eta(T_{\mathrm{KT}})=1/4$6

For $\eta(T_{\mathrm{KT}})=1/4$7, the displacement distribution remains Gaussian but the defects become superdiffusive,

$\eta(T_{\mathrm{KT}})=1/4$8

and vortex–antivortex pairs generically unbind at any $\eta(T_{\mathrm{KT}})=1/4$9 once $m\in\mathbb{Z}$0. Their motion is guided by domain boundaries created by the active mosaic (Rouzaire et al., 2021).

In the lattice gas of self-propelled XY particles, defects become direct organizers of phase separation. Vortex charge is measured by the plaquette winding

$m\in\mathbb{Z}$1

and the total vorticity is $m\in\mathbb{Z}$2. Positive-charge vortices can act as aggregation centers: particles accumulate into dense clusters around specific $m\in\mathbb{Z}$3 defects, whereas $m\in\mathbb{Z}$4 defects dissipate. Cluster growth proceeds by defect-mediated merging, and the late-stage relaxation follows a two-stage exponential form,

$m\in\mathbb{Z}$5

with a characteristic timescale

$m\in\mathbb{Z}$6

The resulting motility-induced phase separation is therefore explicitly defect mediated (Inoue et al., 23 Jul 2025).

Persistent-noise XY models modify defect physics less dramatically at low temperature. Their key mechanism is that large-scale spin waves are controlled by $m\in\mathbb{Z}$7 while local defect nucleation remains controlled by $m\in\mathbb{Z}$8. That separation permits very large algebraic exponents without immediate vortex proliferation (Shi et al., 2024). In the later finite-$m\in\mathbb{Z}$9 analysis, vortices still drive the transition, but the critical fugacity–stiffness balance is shifted by persistence, producing a BKT-like transition with non-universal $q=2$0 (Shi et al., 25 Feb 2026).

5. Nonequilibrium field theory and effective temperatures

A recurring theme in active XY models is the breakdown of any single-temperature equilibrium description. In the persistent-noise formulation, the exact spin-wave stationary functional is

$q=2$1

The first term is equilibrium-like and identifies the large-scale effective temperature $q=2$2; the second is a nonequilibrium ultraviolet regularization that generates the $q=2$3 tail at high wave number and has no equilibrium counterpart (Shi et al., 25 Feb 2026). Earlier coarse graining reached the same qualitative conclusion in a slightly different notation: ultraviolet fluctuations remain controlled by $q=2$4, while long-wavelength spin waves are controlled by $q=2$5, so local defect physics and large-scale orientational fluctuations decouple (Shi et al., 2024).

The substrate and flow-coupled theories make the nonequilibrium structure explicit at the hydrodynamic level. The phase equations contain KPZ-like nonlinearities, density- or flow-dependent frequencies, and active currents, none of which can be derived from a free-energy functional satisfying fluctuation–dissipation. In the substrate model, the sign of the renormalized logarithmic correction controls both orientational order and density statistics: stronger-than-equilibrium QLRO is paired with miniscule or hyperuniform number fluctuations, while weaker-than-equilibrium QLRO is paired with giant number fluctuations (Haldar et al., 2021). In the flow-coupled model, after eliminating the Stokes field, the phase sector reduces to a KPZ-like theory with additional marginal nonlinearities. Depending on the dimensionless active couplings $q=2$6 or $q=2$7, the effective nonlinearity can flow to zero, yielding QLRO, stronger-than-QLRO, or weaker-than-QLRO, or to strong coupling, yielding disorder (Haldar et al., 2019).

Driven-dissipative XY-type systems can also be organized around fluxes rather than order parameters. A modified XY model with amplitude fluctuations and projected nearest-neighbor coupling supports a nonequilibrium steady state with inverse energy cascade, constant negative flux $q=2$8, and energy spectrum

$q=2$9

showing that “active XY” behavior can include cascade transfer and non-Kolmogorov universality rather than only defect-mediated ordering transitions (Tanogami et al., 2021).

6. Methods, benchmarks, and scope of the term

The numerical and analytical toolkit is correspondingly diverse. Persistent-noise XY models have been simulated on triangular lattices of about $$H = - \left( \sum_{\langle i j \rangle} \Delta \cos(\theta_i - \theta_j) + \sum_{\langle i j \rangle} (1 - \Delta) \cos\big(2(\theta_i - \theta_j)\big) \right),$$0 sites using explicit Euler integration with $$H = - \left( \sum_{\langle i j \rangle} \Delta \cos(\theta_i - \theta_j) + \sum_{\langle i j \rangle} (1 - \Delta) \cos\big(2(\theta_i - \theta_j)\big) \right),$$1 for $$H = - \left( \sum_{\langle i j \rangle} \Delta \cos(\theta_i - \theta_j) + \sum_{\langle i j \rangle} (1 - \Delta) \cos\big(2(\theta_i - \theta_j)\big) \right),$$2 steps, measuring the polar order parameter, $$H = - \left( \sum_{\langle i j \rangle} \Delta \cos(\theta_i - \theta_j) + \sum_{\langle i j \rangle} (1 - \Delta) \cos\big(2(\theta_i - \theta_j)\big) \right),$$3, and $$H = - \left( \sum_{\langle i j \rangle} \Delta \cos(\theta_i - \theta_j) + \sum_{\langle i j \rangle} (1 - \Delta) \cos\big(2(\theta_i - \theta_j)\big) \right),$$4 (Shi et al., 2024). The later BKT analysis on the same class of models used system sizes up to $$H = - \left( \sum_{\langle i j \rangle} \Delta \cos(\theta_i - \theta_j) + \sum_{\langle i j \rangle} (1 - \Delta) \cos\big(2(\theta_i - \theta_j)\big) \right),$$5, very long runs up to $$H = - \left( \sum_{\langle i j \rangle} \Delta \cos(\theta_i - \theta_j) + \sum_{\langle i j \rangle} (1 - \Delta) \cos\big(2(\theta_i - \theta_j)\big) \right),$$6–$$H = - \left( \sum_{\langle i j \rangle} \Delta \cos(\theta_i - \theta_j) + \sum_{\langle i j \rangle} (1 - \Delta) \cos\big(2(\theta_i - \theta_j)\big) \right),$$7 time steps, susceptibility finite-size scaling, and quenches from perfectly ordered initial conditions to extract $$H = - \left( \sum_{\langle i j \rangle} \Delta \cos(\theta_i - \theta_j) + \sum_{\langle i j \rangle} (1 - \Delta) \cos\big(2(\theta_i - \theta_j)\big) \right),$$8, $$H = - \left( \sum_{\langle i j \rangle} \Delta \cos(\theta_i - \theta_j) + \sum_{\langle i j \rangle} (1 - \Delta) \cos\big(2(\theta_i - \theta_j)\big) \right),$$9, $\Delta=1$0, and $\Delta=1$1 (Shi et al., 25 Feb 2026).

Self-driven rotor studies combine steady-state simulations with explicit vortex tracking on square lattices up to $\Delta=1$2; free vortices are identified by winding number and paired using the Hungarian algorithm (Rouzaire et al., 2021). The active lattice-gas XY model uses asynchronous Monte Carlo updates, Hoshen–Kopelman cluster analysis, and time-averaged vorticity heatmaps over the $\Delta=1$3 plane (Inoue et al., 23 Jul 2025). Hydrodynamic active-spin theories rely on one-loop dynamic renormalization group, in which the marginal couplings remain visible at two dimensions and generate logarithmic rather than simple power-law corrections (Haldar et al., 2021, Haldar et al., 2019).

Equilibrium tensor-network calculations remain important because they define the passive baseline for defect content and transition structure. Higher-order tensor renormalization group and HOTRG calculations of generalized XY models resolve the nematic, ferromagnetic, and disordered phases, track integer and half-integer susceptibilities, and locate the BKT and Ising lines in the $\Delta=1$4 model (Longia et al., 2023, Samlodia et al., 2024). Programmable photonic simulators implement classical XY Hamiltonians directly through optical vector–matrix multiplication and provide a hardware benchmark for BKT physics and ground-state search in finite XY systems (Ouyang et al., 2024).

Taken together, these studies establish that the active XY model is best understood as a class of nonequilibrium $\Delta=1$5 theories rather than a single model. The decisive distinction from the equilibrium XY model is not merely the presence of noise, but the way nonequilibrium drive reorganizes the relation among spin waves, defects, transport, and density. Depending on the mechanism—persistent torque, intrinsic-frequency disorder, self-propulsion, coupling to conserved species, or coupling to flow—activity can extend QLRO beyond equilibrium bounds, destroy it immediately, generate rotating non-ergodic states, or reorganize defects into phase-separating structures (Shi et al., 2024, Rouzaire et al., 2021, Inoue et al., 23 Jul 2025, Haldar et al., 2021, Haldar et al., 2019, Maes et al., 2011).