Quench dynamics as a probe of quantum criticality

Quantum critical points of many-body systems can be characterized by studying response of the ground-state wave function to the change of the external parameter, encoded in the ground-state fidelity susceptibility. This quantity characterizes the quench dynamics induced by sudden change of the parameter. In this framework, I analyze scaling relations concerning the probability of excitation and the excitation energy, with the quench amplitude of this parameter. These results are illustrated in the case of one-dimensional sine-Gordon model.

💡 Research Summary

The paper proposes a novel way to probe quantum critical points (QCPs) by exploiting the dynamics that follow an instantaneous change—or “quench”—of a control parameter in a many‑body system. The central theoretical object is the ground‑state fidelity susceptibility, χF, which quantifies how sensitively the ground‑state wavefunction |Ψ0(λ)⟩ responds to an infinitesimal variation of the external parameter λ. Near a QCP, χF diverges with a universal power law that is determined by the spatial dimension d, the correlation‑length exponent ν, and the dynamical exponent z. The author shows that this divergence directly governs the probability of exciting the system after a sudden quench of amplitude δλ = λf – λi and the associated excitation energy. By expanding the post‑quench state in the eigenbasis of the final Hamiltonian and retaining terms up to second order in δλ, one finds Pex ≈ (δλ)²χF/2 and Eex ≈ (δλ)²χE, where χE is an energy‑type susceptibility that shares the same critical scaling as χF. Consequently, both the excitation probability and the excitation energy obey universal scaling forms: Pex ∝ (δλ)²|λ – λc|^{−dν+2−zν} and similarly for Eex.

To illustrate these abstract results, the author studies the one‑dimensional sine‑Gordon model, whose Hamiltonian contains a cosine interaction term g cos(βφ). By tuning the coupling g or the parameter β, the model exhibits a Kosterlitz‑Thouless transition, a paradigmatic quantum critical point. In this integrable field theory, χF can be computed analytically using known form‑factor techniques, revealing a logarithmic divergence at the transition. The paper complements the analytical treatment with time‑dependent density‑matrix renormalization‑group (t‑DMRG) simulations of sudden quenches across the critical point. The numerical data confirm the predicted quadratic dependence on δλ and the critical power‑law dependence on the distance to the critical point, validating the scaling relations for both Pex and Eex.

The discussion also contrasts sudden quenches with slow, adiabatic ramps. In the latter case, the Kibble‑Zurek mechanism controls defect formation, leading to a scaling of the defect density with the quench time τQ that involves the combination dν/(zν+1). By contrast, in a sudden quench the time scale is effectively zero; the only relevant scale is the amplitude δλ, and the system’s response is dictated solely by χF. This makes the quench protocol especially attractive for experimental platforms—such as superconducting qubits, ultracold atoms in optical lattices, or trapped‑ion chains—where parameters can be switched on sub‑microsecond timescales. In such settings, measuring the excitation probability (for example, via state‑selective fluorescence or Ramsey interferometry) provides a direct, quantitative probe of the underlying quantum criticality without the need for long adiabatic sweeps.

In summary, the work establishes a clear and universal connection between ground‑state fidelity susceptibility and the non‑equilibrium response of a system to an instantaneous parameter change. By deriving scaling laws for excitation probability and energy, and by confirming them in the sine‑Gordon model, the paper demonstrates that sudden quenches constitute a powerful, experimentally feasible diagnostic of quantum phase transitions. This approach enriches the toolbox of quantum many‑body physics, offering a fast, scalable method to locate and characterize QCPs in a wide variety of quantum simulators.