Fate of dynamical phase transitions at finite temperatures and in open systems

When a quantum system is quenched from its ground state, the time evolution can lead to nonanalytic behavior in the return rate at critical times $t_c$. Such dynamical phase transitions (DPTs) can occur, in particular, for quenches between phases with different topological properties in Gaussian models. In this paper we discuss Loschmidt echos generalized to density matrices and obtain results for quenches in closed Gaussian models at finite temperatures as well as for open-system dynamics described by a Lindblad master equation. While cusps in the return rate are always smoothed out by finite temperatures we show that dissipative dynamics can be fine-tuned such that DPTs persist.

Figure 1

The return rate $l\left(t\right)$ for the Ising chain in the thermodynamic limit for a quench from $g=0.5$ to $g=1.5$ at different temperatures $T$. (a) Projection onto the ground state, Eq. (3.11) (note that the curves for $T=0$ and $T=0.05$ are almost on top of each other). (b) Generalized Loschmidt echo, Eqs. (2.1) and (3.15).

Figure 2

The return rate $l\left(t\right)$ for the SSH chain in the thermodynamic limit for a quench from $\delta =-0.5$ to $\delta =0.5$ at different temperatures $T$. (a) Projection onto the ground state, Eq. (3.11). (b) Generalized Loschmidt echo, Eq. (3.15). Note that the curves for $T=0$ and $T=0.2$ are almost on top of each other.

Figure 3

The return rate $l\left(t\right)$ for the Ising chain in the thermodynamic limit for a quench from $g=0.5$ to $g=1.5$ at $T=0$ for different particle loss rates $\gamma ={\gamma }_k={\gamma }_{-k}$. Inset: Broadening of the first cusp at $t=t_c$.

Figure 4

The return rate $l\left(t\right)$ for the Ising chain in the thermodynamic limit for a quench from $g=0.5$ to $g=1.5$ at $T=0$ for various equal particle loss and creation rates $\gamma ={\gamma }_k={\overline{\gamma }}_{-k}$.