Dynamical phase transitions after quenches in nonintegrable models

We investigate the dynamics following sudden quenches across quantum critical points belonging to different universality classes. Specifically, we use matrix product state methods to study the quantum Ising chain in the presence of two additional terms which break integrability. We find that in all models the rate function for the return probability to the initial state becomes a nonanalytic function of time in the thermodynamic limit. This so-called “dynamical phase transition” was first observed in a recent work by Heyl, Polkovnikov, and Kehrein [Phys. Rev. Lett. 110, 135704 (2013)] for the exactly-solvable quantum Ising chain, which can be mapped to free fermions. Our results for “interacting theories” indicate that nonanalytic dynamics is a generic feature of sudden quenches across quantum critical points. We discuss potential connections to the dynamics of the order parameter.

Figure 1

Rate function $l\left(t\right)$ defined in Eq. (10) for a quench from the PM to the FM phase in the Ising model of Eq. (4). $l\left(t\right)$ characterizes the overlap with the initial state in the thermodynamic limit. The DMRG data agree well with the analytic result obtained by mapping the model to free fermions [Eq. (5); see also Ref. 6]. The rate function shows nonanalytic behavior at the times $t_n^{*}$ indicated by the vertical dashed lines [Eq. (11)].

Figure 2

The same as in Fig. 1 but for quenches from the FM to the PM phase in the Ising model. In the thermodynamic limit, the ground state $|\pm \rangle$ within the FM phase is two-fold degenerate. (a) Quench performed starting from the polarized state $\left|+\right.〉$. The rate function shows nonanalytic behavior which, however, does not occur at the times $t_n^{*}$ defined in Eq. (11). Inset: Time evolution of the order parameter $\langle {\sigma }_i^z\left(t\right)\rangle$, which at sufficiently late times oscillates with the frequency $2t^{*}$. (b) Quench starting from the mixed state $\left|\mathrm{NS}\right.〉=(\left|+\right.〉-\left|-\right.〉)/\sqrt{2}$. The DMRG data agree well with the analytic result obtained for the corresponding quench in the fermionic model[6] of Eq. (5). The rate function shows nonanalytic behavior at the times $t_n^{*}$. Inset: Comparison of the rate functions starting from the mixed and polarized states $\left|\mathrm{NS}\right.〉$ and $\left|+\right.〉$, respectively.

Figure 3

Sketch of the phase diagram[15, 16] of the ANNNI model defined in Eq. (16) as a function of $\Delta$ and $g$. There are four phases: a paramagnetic (PM) phase, a ferromagnetic (FM) phase, an anti phase (AP), and a floating phase (FP). The PM and FM phases are separated by an Ising transition located at $g_c\left(\Delta \right)$ defined in Eq. (17). We study quenches across the Ising transition as indicated by the solid arrow [see Fig. 4a], the dashed arrow [see Fig. 4b], and the dashed-dotted arrow (see Fig. 5).

Figure 4

Rate function for a quench from the PM to the FM phase of the quantum Ising model in the presence of integrability-breaking next-nearest-neighbor interactions [the so-called ANNNI model; see Eq. (16)]. (a) Quench in the transverse field $g$ (indicated by the solid arrow in the phase diagram shown in Fig. 3). (b) Quench in the next-nearest-neighbor interaction $\Delta$ (dashed arrow in Fig. 3). In complete analogy with the integrable “noninteracting” Ising chain ($\Delta =0$), the rate function exhibits nonanalytic behavior as a function of time in the thermodynamic limit if one quenches across a critical point [note that the curve in (b) for ${\Delta }_1=0.2$ corresponds to a quench within the PM phase].

Figure 5

The same as in Fig. 4 but for a quench from the polarized ground state of the FM phase to the PM phase (dashed-dotted arrow in Fig. 3). The rate function consistently features nonanalyticities. Inset: The order parameter oscillates periodically at sufficiently late times, but the precise connection of its dynamics to $l\left(t\right)$ remains elusive.

Figure 6

Rate function for quenches within the generalized Ising model defined in Eq. (20). The model is critical at $g=1$ for all $\varphi \ge 0$. For $\varphi >0$, the model has a unique ground state for any $g$, is not integrable, and the quantum critical point is no longer in the Ising universality class. One consistently observes nonanalyticities in the time evolution; they occur periodically only at $\varphi =0$. We note that for large angles $\varphi$ the nonanalyticities appear at rather late times (see the inset) while the behavior for short times is absolutely smooth.

Figure 7

DMRG calculation of the rate function per site $l\left(t\right)$ for a quench across the phase transition of the generalized Ising model of Eq. (20) with ${\varphi }_0={\varphi }_1=\pi /32$. As the system size $L$ is increased one successively approaches the result obtained directly in the thermodynamic limit $L=\infty$ via an infinite-system DMRG algorithm.[22]