Disentangling dynamical phase transitions from equilibrium phase transitions

Dynamical phase transitions (DPTs) occur after quenching some global parameters in quantum systems, and are signalled by the nonanalytical time evolution of the dynamical free energy, which is calculated from the Loschmidt overlap between the initial and time evolved states. In a recent Letter [M. Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)], it was suggested that DPTs are closely related to equilibrium phase transitions (EPTs) for the transverse field Ising model. By studying a minimal model, the XY chain in a transverse magnetic field, we show analytically that this connection does not hold generally. We present examples where DPT occurs without crossing any equilibrium critical lines by the quench, and a nontrivial example with no DPT but crossing a critical line by the quench. Although the nonanalyticities of the dynamical free energy on the real time axis do not indicate the presence or absence of an EPT, the structure of Fisher lines for complex times reveals a qualitative difference.

Figure 1

The phase diagram of the XY model in a magnetic field. The three studied phases (I, II, III) are marked on the plot. These gapless phases are separated by critical lines that form an H letterlike shape. DPTs can occur in quenches within the same phase. The domains $\mathcal{D}(h_0,{\gamma }_0)$ of the final parameters where DPTs appear are shown for four given initial conditions $(h_0,{\gamma }_0)$. Except from the region $h_1<-1$, the domains are determined from Eq. (7). Note that when quenching from II to $h_1<-1$, nonanalyticities only show up in the top left corner of the phase diagram and remain absent otherwise, in spite of crossing several critical lines.

Figure 2

The flow of Fisher lines $z_n\left(k\right)$ [$n=(-3,\cdots ,2)$] for various types of quenches discussed in the main text.

Figure 3

The dynamical free energy is nonanalytical at Fisher times $t_{i,n}=t_i^{*}(n+1/2)$, $i=1,2$ (solid and dashed lines, respectively). The time unit was chosen to be $t_1^{*}$. The longitudinal magnetization also shows two time scales: In (b) the zeros of the magnetization approximately lie at the Fisher times, and in (d) the relation between them is more involved. Quench parameters for (a) and (b) are $(h_0=0,{\gamma }_0=1)$ $\to$ $(h_1=0.6,{\gamma }_0=-1)$ and for (c) and (d) they are $(h_0=0,{\gamma }_0=0.1)$ $\to$ $(h_1=0.6,{\gamma }_0=0.1)$.