Sudden interaction quench in the quantum sine-Gordon model

We study a sudden interaction quench in the weak-coupling regime of the quantum sine-Gordon model. The real time dynamics of the bosonic mode occupation numbers is calculated using the flow equation method. While we cannot prove results for the asymptotic long time limit, we can establish the existence of an extended regime in time where the mode occupation numbers relax to twice their equilibrium values. This factor two indicates a non-equilibrium distribution and is a universal feature of weak interaction quenches. The weak-coupling quantum sine-Gordon model therefore turns out to be on the borderline between thermalization and non-thermalization.

💡 Research Summary

The paper investigates the non‑equilibrium dynamics that follow a sudden switch‑on of the interaction term in the quantum sine‑Gordon model, focusing on the weak‑coupling regime where the cosine potential is a small perturbation to a free bosonic field. The authors consider an initial state that is the ground state of the free theory (β = 0, V = 0) and at time t = 0 abruptly turn on the cosine interaction H_int = V∫dx cos(β φ). The central observable is the occupation number of each bosonic mode, n_k(t)=⟨a_k† a_k⟩_t, which directly reflects how energy is redistributed among the modes after the quench.

To treat the time evolution analytically, the authors employ the flow‑equation (continuous unitary transformation) method. In this approach the Hamiltonian is gradually diagonalized by a sequence of infinitesimal unitary transformations generated by η(l), where l is a flow parameter. As l→∞ the Hamiltonian becomes effectively quadratic, containing a renormalized mass term Δ and a residual interaction that is perturbatively small. Simultaneously the bosonic operators a_k are transformed into l‑dependent operators a_k(l), whose time dependence can be extracted once the flow is completed. By expanding the flow equations to second order in the small coupling g ∝ β²V, the authors obtain an explicit expression for n_k(t) that is valid for times longer than the microscopic scale set by the inverse mass Δ⁻¹ but still before any possible eventual thermalization.

The analytical solution exhibits two distinct regimes. Immediately after the quench (t ≲ Δ⁻¹) the occupations display rapid oscillations with frequency set by the gap Δ, reflecting coherent excitation of the newly opened cosine potential. As time progresses, these oscillations are damped by the weak non‑linearities generated in the flow, and the occupations settle into a plateau. Remarkably, in this plateau region the occupations are twice the values they would have in a thermal equilibrium state with the same final Hamiltonian, i.e. n_k(t) ≈ 2 n_k^eq. This “factor‑two” result is independent of the specific mode k and of the precise values of β and V, provided the coupling remains weak.

The factor‑two phenomenon has been reported previously in other weak‑interaction quenches, such as the Luttinger‑Sutherland model and certain fermionic systems, and is interpreted as a universal signature of a pre‑thermalized state: the system has redistributed energy among its quasiparticles but has not yet explored the full phase space required for true thermalization. The authors emphasize that their flow‑equation treatment does not guarantee convergence for arbitrarily long times; residual interactions that are not fully eliminated may eventually drive the system toward genuine thermal equilibrium, but this long‑time behavior lies beyond the scope of the present analysis. Consequently, the quantum sine‑Gordon model in the weak‑coupling limit sits at the borderline between thermalizing and non‑thermalizing dynamics.

Beyond the theoretical insight, the work suggests concrete experimental routes. In ultracold atomic gases, a sine‑Gordon potential can be engineered by coupling two one‑dimensional condensates with a tunable tunneling amplitude; a rapid change of this tunneling amplitude implements the quench. Time‑resolved measurements of mode occupations—via Bragg spectroscopy or matter‑wave interferometry—could directly test the predicted factor‑two plateau. Similar protocols could be envisaged in superconducting circuits or trapped‑ion chains where effective cosine potentials arise.

In summary, the paper provides a clear analytical demonstration that a sudden interaction quench in the weakly coupled quantum sine‑Gordon model leads to a robust pre‑thermal regime where bosonic mode occupations relax to twice their equilibrium values. This result enriches the broader understanding of non‑equilibrium quantum field dynamics, highlighting how integrability‑breaking perturbations can generate universal non‑thermal signatures while still leaving the ultimate fate of the system—thermalization versus persistent memory of the initial state—an open question.