Quench dynamics near a quantum critical point: Application to the sine-Gordon model

We discuss the quench dynamics near a quantum critical point focusing on the sine-Gordon model as a primary example. We suggest a unified approach to sudden and slow quenches, where the tuning parameter $\lambda \left(t\right)$ changes in time as $\lambda \left(t\right)\sim \upsilon t^r$, based on the adiabatic expansion of the excitation probability in powers of $\upsilon$. We show that the universal scaling of the excitation probability can be understood through the singularity of the generalized adiabatic susceptibility ${\chi }_{2r+2}\left(\lambda \right)$, which for sudden quenches $\left(r=0\right)$ reduces to the fidelity susceptibility. In turn this class of susceptibilities is expressed through the moments of the connected correlation function of the quench operator. We analyze the excitations created after a sudden quench of the cosine potential using a combined approach of form-factors expansion and conformal perturbation theory for the low-energy and high-energy sector, respectively. We find the general scaling laws for the probability of exciting the system, the density of excited quasiparticles, the entropy and the heat generated after the quench. In the two limits where the sine-Gordon model maps to hard-core bosons and free massive fermions we provide the exact solutions for the quench dynamics and discuss the finite temperature generalizations.

Figure 1

$B_1{B}_1$ (top) and $B_2{B}_2$ (bottom) breather form factor for $\xi =1/41$ as a function of the rapidity $\theta$. The saturation lines are given by ${\left(F_{B_1}\right)}^2$ and ${\left(F_{B_2}\right)}^2$ respectively.