Correlation functions in the prethermalized regime after a quantum quench of a spin chain

Results are presented for a two-point correlation function of a spin chain after a quantum quench for an intermediate time regime where inelastic effects are weak. A Callan-Symanzik-like equation for the correlation function is explicitly constructed which is used to show the appearance of three distinct scaling regimes. One is for spatial separations within a light cone, the second is for spatial separations on the light cone, and the third is for spatial separations outside the light cone. In these three regimes, the correlation function is found to decay with power laws with nonequilibrium exponents that differ from those in equilibrium, as well as from those obtained from quenches in a quadratic Luttinger liquid theory. A detailed discussion is presented on how the existence of scaling depends on the properties of the initial state before the quench.

Figure 1

Ground-state phase diagram of the quantum sine-Gordon model. A critical line separates a gapless phase where the cosine term is irrelevant from a gapped phase where the cosine term is relevant. The critical line is located at $\delta =2\pi g$, where $\delta =K_{\mathrm{eq}}-2$, with $K_{\mathrm{eq}}=\frac{{\gamma }^{2}K}{4}$.

Figure 2

The equal-time correlator shows distinctly different behavior for the following three cases: spatial separations outside the light cone ($r>2T_m$), spatial separations on the light cone ($r=2T_m$), and spatial separations within the light cone ($r<2T_m$). The present paper gives results for correlation functions in these three regimes under the additional constraint that $T_m<1/\eta$ where $\eta$ is an inelastic scattering rate.

Figure 3

The dynamics after a quench is characterized by three different regimes: a short-time regime that depends on microscopic parameters ($T_m\ll 1/\Lambda$), an intermediate-time regime ($1/\Lambda \ll T_m\ll 1/\eta$) where inelastic effects are weak and the correlator shows universal scaling behavior, and a long-time thermal regime ($T_m\gg 1/\eta$) where inelastic-scattering effects are strong and cause the system to thermalize.

Figure 4

Plot of $I_C$ for a simultaneous lattice and interaction quench for points within the light cone and near the critical point $K_{\mathrm{neq}}=2$ where $K_{\mathrm{eq}}=\frac{{\gamma }^2{K}_0}{4}\sqrt{\frac{16}{{\gamma }^2{K}_0}-1}$.

Figure 5

The first-order correction to the equal-time correlation function $R^{\left(1\right)}(r=100,T_m)$ as a function of the time $T_m$ after the quench. Length and time are measured in units of the cutoff $\Lambda$. Three different quenches are considered: a pure lattice quench where $K_0=K=2$ (solid line), lattice and interaction quench corresponding to $K_0=1,K=\sqrt{3}$ (dashed line), and $K_0=3,K=\sqrt{3}$ (dotted line). The light cone is at $T_m=r/2=50$. We choose $\gamma =2$ and all quenches are to the critical point $K_{\mathrm{neq}}=2$.