← Jonas Köppl

Simulations

Interacting particle systems are easiest to get a feeling for by watching them. This page runs the driven clock and driven XY models of C. Maes and S. Shlosman, Rotating states in driven clock- and XY-models, J. Stat. Phys. 144 (2011) live in your browser. Every spin is a clock hand that interacts with its neighbours like an XY spin and is pushed to rotate by a constant drift. In two dimensions the collective rotation is destroyed by spiral defects; in three dimensions the whole system rotates coherently. The pictures follow the conventions of Y. Avni, M. Fruchart, D. Martin, D. Seara and V. Vitelli, Dynamical phase transitions in the nonreciprocal Ising model, Phys. Rev. E 111 (2025), whose nonreciprocal Ising model is closely related to the four-state clock.

Pick a preset or change the parameters yourself; β and d take effect immediately. Below the lattice views the page measures truncated correlations in space and in time and compares them with the spin-wave predictions, and the section at the end sketches the phase diagram conjectured by Maes and Shlosman.

time t0
|M| = s1.000
ω = d arg M / dt–
vortices N₊ / N₋0 / 0
phasestarting
rate–
σA = sgn cos θ
σB = sgn sin θ
θ
MA MB s = ((MA²+MB²)/2)½ N₊+N₋ (scaled)

Truncated correlations in space, at a fixed time t

Ct(r) = ⟨σ0(t)·σre(t)⟩ − ⟨σ0(t)⟩·⟨σre(t)⟩ with σ = (cos θ, sin θ), estimated from the configuration at time t by averaging over all sites and the d lattice directions e.

current time t mean over the last 0 times mean, finite-size corrected spin waves on the L-torus, A·GL(r) A·G∞(r) ≈ A/(4πr), the decay |x|2−d

On the torus, subtracting the magnetization removes the k = 0 spin wave, which bends Ct(r) down long before r = L/2. The rings add back exactly this spin-wave finite-size effect, A·[G∞(r) − GL(r)], so in 3D they should lie on the 1/r line out to r ≈ L/2. The amplitude A is fitted for r ≤ L/4. Let the run relax for a few hundred time units: the longest spin waves equilibrate on a time scale of order L².

Truncated correlations in time

S(t0, t) = L−d Σx σx(t0)·σx(t) − M(t0)·M(t), as in Chaté, Grinstein and Tang. It oscillates with the collective rotation; its envelope is predicted to decay like |t − t0|−(d−2)/2, that is t−1/2 in d = 3.

|S(t0, t)|, one reference time t0 co-rotating Srot, mean over 0 reference times spin waves on the L-torus, B·HL(D τ) ∝ τ−(d−2)/2 = τ−1/2

Srot measures every spin relative to the current direction of M, which removes the oscillation and leaves the envelope: Srot(t0, t) = Re L−d Σx δw̄x(t0) δwx(t), δw = eiθ M̄/|M| − |M|. HL(u) = L−d Σk≠0 e−uλk/λk is the spin-wave autocorrelation on the torus, with a fitted phase diffusion constant D. Reference times are taken every – time units and the panel resets when β or d changes.

The model

Each site of the periodic lattice carries a clock spin k ∈ {0,…,q−1} with angle φ = 2πk/q. It advances or retreats by one step at the rates of Maes and Shlosman (Eq. 4), simulated exactly in continuous time:

c±(x) = p± · exp{ (β/2) Σ_y~x [ cos(φx − φy ± 2π/q) − cos(φx − φy) ] } d = log(p+/p−), p+ + p− = 1

For d = 0 the dynamics is reversible with respect to the Gibbs measures of the clock model H = −Σ cos(φx − φy). For d > 0 every free spin turns counterclockwise, and the question is what the interaction makes of this drive. The XY option integrates the limit q → ∞ (Eq. 7), dφ = d·dt − Σ sin(φx − φy) dt + (2/β)½ dW, whose generator commutes with that of the undriven XY model. Its stationary translation-invariant measure is unique for almost all β, yet at low temperature the rotated XY Gibbs states rotate forever and never relax to it (Theorem 7).

Reading the views

Pictures use θ = 2π(k + ½)/q, a global half-step rotation. For q = 4 the four states then sit on the diagonals, so σA = sgn cos θ and σB = sgn sin θ are two Ising spins. Their cycle (++) → (−+) → (−−) → (+−) is the swap order of the nonreciprocal Ising model of Avni et al.. The driven 4-clock is their fully antisymmetric model with exponential instead of Glauber rates, with J̃ = dim·β and K̃ = d/2.

A vortex is a plaquette whose four corners wind once around the clock. In 3D, voxels are drawn semi-transparent with grey encoding depth (dark = front).

The conjectured phase diagram in d ≥ 3

Conjectured phase diagram of the driven q-clock model in three dimensions Inverse temperature beta on the horizontal axis, drift d on the vertical axis. Disordered for beta below beta_cr. Between beta_cr and beta_G every positive drift gives rotation. Above beta_G a critical drift d_cr grows from zero; below it the magnetized phases are static, above it they rotate. β d βcr βG disordered M = 0, exponential decay rotating soft modes, dcr = 0 rotating static order d = dcr(β, q) d = 0: q phases, algebraic decay
Schematic, after Conjectures 2–6 of Maes and Shlosman, for q large. Positions and curve shapes are not to scale. As q → ∞, βG → ∞: for the XY model every d > 0 sets the ordered phase rotating.

Equilibrium (d = 0, Conjecture 2). For q large the 3D clock model has two transitions. Below βcr the Gibbs state is unique and correlations decay exponentially. Between βcr and βG there are q magnetized phases whose truncated correlations decay only algebraically: the discreteness of the clock is invisible at these scales and the model behaves like the XY model, with massless spin waves (Goldstone modes). Above βG the Pirogov–Sinai theory applies and correlations decay exponentially again. Both βG and the Pirogov–Sinai threshold tend to infinity as q → ∞.

With drift (Conjectures 3–6). In the Pirogov–Sinai regime the q phases survive a small drift d < dcr(β, q): they stay static and close to equilibrium. Above dcr the magnetization rotates, |⟨σx⟩T − m(T)ei(Φ(T)+φk)| → 0 with periodic m and Φ, and the state never converges. In the soft-mode regime βcr < β < βG the critical drift vanishes: any d > 0 sets the spins rotating synchronously. The stationary translation-invariant distribution is then unique with zero magnetization, while the rotating states are not attracted to it.

q = 4. The four-state clock is two Ising models and has no soft-mode phase in equilibrium. For the corresponding fully antisymmetric nonreciprocal Ising model, Avni et al. find numerically that static order is unstable to droplet growth in any finite dimension and that the transition from disorder to the rotating (swap) phase is in the 3D XY universality class. The 3D preset sits just above this transition.

Correlations in the rotating phase. The rotation breaks the continuous time-translation symmetry, and the local phase of the oscillation is a soft mode. Following Grinstein, Mukamel, Seidin and Bennett, its fluctuations obey a KPZ equation whose smooth phase exists for d > 2. This predicts equal-time truncated correlations decaying like |x|2−d and autocorrelations whose envelope decays like |t|−(d−2)/2 (Chaté, Grinstein and Tang). For the driven XY model the spatial law follows from Theorem 7, since the rotating states are rotated equilibrium XY states with spin-wave correlations. The two correlation panels above test both predictions, together with their spin-wave forms on the finite torus.

References

  1. C. Maes and S. Shlosman, Rotating states in driven clock- and XY-models, J. Stat. Phys. 144 (2011), 1238–1246.
  2. Y. Avni, M. Fruchart, D. Martin, D. Seara and V. Vitelli, Dynamical phase transitions in the nonreciprocal Ising model, Phys. Rev. E 111 (2025), 034124.
  3. Y. Avni, M. Fruchart, D. Martin, D. Seara and V. Vitelli, Nonreciprocal Ising model, Phys. Rev. Lett. 134 (2025), 117103.
  4. H. Chaté, G. Grinstein and L.-H. Tang, Long-range correlations in systems with coherent (quasi)periodic oscillations, Phys. Rev. Lett. 74 (1995), 912–915.
  5. G. Grinstein, D. Mukamel, R. Seidin and C. H. Bennett, Temporally periodic phases and kinetic roughening, Phys. Rev. Lett. 70 (1993), 3607–3610.