On the possibility of complete revivals after quantum quenches to a critical point

In a recent letter [J. Cardy, Phys. Rev. Lett. 112, 220401 (2014)], the author made a very interesting observation that complete revivals of quantum states after quantum quench can happen in a period that is a fraction of the system size. This is possible for critical systems that can be described by minimal conformal field theories with central charge $c<1$. In this paper, we show that these complete revivals are impossible in microscopic realizations of those minimal models. We will prove the absence of the mentioned complete revivals for the critical transverse field Ising chain analytically, and present numerical results for the critical line of the XY chain. In particular, for the considered initial states, we will show that criticality has no significant effect in partial revivals. We also comment on the applicability of quasiparticle picture to determine the period of the partial revivals qualitatively. In particular, we detect a regime in the phase diagram of the XY chain in which one can not determine the period of the partial revivals using the quasiparticle picture.

Figure 1

Different regions in the phase diagram of the quantum XY chain. The critical XX chain has central charge $c=1$ and the critical XY line has $c=\frac{1}{2}$. The region $a^2+h^2<1$ is depicted with the yellow color.

Figure 2

Logarithmic fidelity for the periodic critical Ising chain starting from the all aligned spins ${\sigma }^z$ initial state.

Figure 3

Logarithmic fidelity for the open critical Ising chain starting from the all aligned spins ${\sigma }^z$ initial state.

Figure 4

Group velocity $v_{\varphi }$ with respect to $\varphi$ for different values of $a$ and $h$.

Figure 5

Maximum group velocity for different values of $a$ and $h$.

Figure 6

Loschmidt echo on the Ising line for periodic and open chains in the top and bottom panels, respectively ($L=80$). The revivals are compatible with the quasiparticle picture with maximum group velocity $v_g^{\max }=2h$.

Figure 7

Loschmidt echo on the XY critical line for periodic and open chains in the top and bottom panels, respectively, ($L=80$). The revivals are compatible with a quasiparticle picture with a Fermi group velocity $v_g^f=2a$.

Figure 8

Loschmidt echo on the line $h=0.8$ for periodic and open chains in the left and right panels, respectively ($L=80$). The revivals are compatible with a quasiparticle picture with maximum group velocity (see Fig. 5) as far as $a^2+h^2>1$. For the region $a^2+h^2\le 1$, there are no clean periodic revivals.