“Light-Cone” Dynamics After Quantum Quenches in Spin Chains

Signal propagation in the nonequilibrium evolution after quantum quenches has recently attracted much experimental and theoretical interest. A key question arising in this context is what principles, and which of the properties of the quench, determine the characteristic propagation velocity. Here we investigate such issues for a class of quench protocols in one of the central paradigms of interacting many-particle quantum systems, the spin-$1/2$ Heisenberg $XXZ$ chain. We consider quenches from a variety of initial thermal density matrices to the same final Hamiltonian using matrix product state methods. The spreading velocities are observed to vary substantially with the initial density matrix. However, we achieve a striking data collapse when the spreading velocity is considered to be a function of the excess energy. Using the fact that the $XXZ$ chain is integrable, we present an explanation of the observed velocities in terms of “excitations” in an appropriately defined generalized Gibbs ensemble.

Figure 1

(a) Quench protocol: The system is initially prepared in either the ground state of some Hamiltonian $H({\mathrm{\Delta }}_i)$ or in a thermal state $\rho =Z_{\beta }^{-1}\exp [-\beta H({\mathrm{\Delta }}_i)]$ with temperature $T$. At time 0, the anisotropy is quenched to ${\mathrm{\Delta }}_f$ and we let the system evolve in time for various initial values of ${\mathrm{\Delta }}_i$ and $T$. (b) Outline of the numerical procedure: The METTS projection loop generates an ensemble of wave function for some initial ${\mathrm{\Delta }}_i$ and temperature $T$. Each realization is evolved in time and expectation values are obtained by averaging over the ensemble.

Figure 2

Space-time plot of the ${\mathcal{S}}^z$ correlation functions for the quench from ${\mathrm{\Delta }}_i=4$ to ${\mathrm{\Delta }}_f=\cos (\pi /4)$. The upper panel shows ground state data whereas the lower panel shows data from a thermal density matrix at $T/J=1$. This illustrates that the light-cone effect in this observable persists also at finite temperatures.

Figure 3

(a) Extracted inflection points versus distance for different initial temperatures for the quench from $\mathrm{\Delta }=4$ to $\cos (\pi /4)$. The straight lines correspond to the velocities extracted from the GGE, where only the offset of the time axis has been fitted. The orange dashed line denotes the ground state Bethe ansatz velocity at ${\mathrm{\Delta }}_f$. (b) Rescaled averaged spin correlation functions for the quench from $\mathrm{\Delta }=4$ to $\cos (\pi /4)$ for $T/J=1$ and the ground state (dashed line) and different distances $j=3$, 5, 7, and 9. We omit the error bars for clarity of the figure. The time axis is relative to the first inflection point of the correlation functions for $j=3$. One can see that the signal is delayed as the initial temperature is increased.

Figure 4

Spreading velocity $v_s$ extracted from the spin correlation function ${\mathcal{S}}^z$ as a function of the final energy $e_f$ density for ${\mathrm{\Delta }}_f=1/2$ and $\cos \pi /4$. The symbols denote numerical results obtained from either thermal or pure initial states with different ${\mathrm{\Delta }}_i$. The blue and black solid lines denote the spreading velocities from TBA using only the energy density whereas the red line shows the results for the quench from ${\mathrm{\Delta }}_i=10$ to 0.5 using also the first conserved quantity. The corresponding velocities for the quench from ${\mathrm{\Delta }}_i=1.5$ lie on top of the black line; i.e., the GGE effects are smaller than the linewidth. The rightmost symbols denote $v_{{\mathrm{\Delta }}_f}$ at the energy density of the ground state whereas the right most ones denote $v_{1,\max }$. The inset shows the velocity at $\beta =0$, $v_{1,\max }$, extracted from the thermodynamic Bethe ansatz for $\mathrm{\Delta }=\cos (\pi /n)$.

Figure 5

Scheme for the extraction of the velocities from the GGE. See text for details.