Universal Postquench Prethermalization at a Quantum Critical Point

We consider an open system near a quantum critical point that is suddenly moved towards the critical point. The bath-dominated diffusive nonequilibrium dynamics after the quench is shown to follow scaling behavior, governed by a critical exponent that emerges in addition to the known equilibrium critical exponents. We determine this exponent and show that it describes universal prethermalized coarsening dynamics of the order parameter in an intermediate time regime. Implications of this quantum critical prethermalization are: (i) a power law rise of order and correlations after an initial collapse of the equilibrium state and (ii) a crossover to thermalization that occurs arbitrarily late for sufficiently shallow quenches.

Figure 1

(a) Schematic description of the setup and quench protocol. (b) Schematic phase diagram as a function of temperature $T$ and mass $\delta r_i=r_{0,i}-r_{0,c}$. Red arrows describe the quench protocol. Dynamics exhibits three time regimes: $t<t_{\gamma }={\gamma }^{-z/(2(z-1))}$ with nonuniversal dynamics, the universal prethermalized regime $t_{\gamma }<t<t^{*}\propto \delta r_i^{-\nu z/\kappa }$ which we study, and a quasiadiabatic regime $t>t^{*}$ described by equilibrium critical exponents. Here, $\gamma$ is the system-bath coupling and $\kappa /\nu$ the scaling dimension of $\delta r_i$. (c) Correlation length collapse and light-cone-like revival following a quench with initial length ${\xi }_i$. Inset: sketches of order parameter configurations with domains of typical size $\xi (t)$.

Figure 2

Schematic order parameter dynamics $⟨\phi (t)⟩$. In the prethermalized regime $t_{\gamma }<t<t^{*}$ (blue) it is governed by a new universal critical exponent $\theta$. At longer times, $⟨\phi (t)⟩$ decays to zero quasiadiabatically as described by equilibrium exponents.

Figure 3

(a) Prethermalization exponent $\theta$ as a function of dynamic critical exponent $z$. Plot shows ${\mathcal{C}}_z=\theta (N+8)/(N+2)(1/\epsilon )$, where $N$ is the number of components of $\mathit{\phi }$ and $\epsilon =4-d-z$. Blue dot indicates the analytical result of Eq. (14). (b) Free Keldysh scaling function $f_0^K(q^{z}t)-f_{\mathrm{eq},0}$ after the quantum quench for different dynamic critical exponents $z=1.2$ (red dotted), 1.4 (yellow dashed), 2 (blue dot-dashed), and 2.5 (green). Inset shows the exponential decay of the envelope towards the equilibrium distribution, which becomes algebraic in the presence of interactions.