Ubiquity of zeros of the Loschmidt amplitude for mixed states in different physical processes and its implication

The Loschmidt amplitude of the purified states of mixed-state density matrices is shown to have zeros when the system undergoes a quasistatic, quench, or Uhlmann process. While the Loschmidt-amplitude zero of a quench process corresponds to a dynamical quantum phase transition (DQPT) accompanied by the diverging dynamical free energy, the Loschmidt-amplitude zero of the Uhlmann process corresponds to a topological phase transition (TQPT) accompanied by a jump of the Uhlmann phase. Although the density matrix remains intact in a quasistatic process, the Loschmidt amplitude can have zeros not associated with a phase transition. We present examples of two-level and three-level systems exhibiting finite- or infinite-temperature DQPTs and finite-temperature TQPTs associated with the Loschmidt-amplitude zeros. Moreover, the dynamical phase or geometrical phase of mixed states can be extracted from the Loschmidt amplitude. Those phases may become quantized or exhibit discontinuity at the Loschmidt-amplitude zeros. A spinor representation of the purified states of a general two-level system is presented to offer more insights into the change of purification in different processes. The quasistatic process, for example, is shown to cause a rotation of the spinor.

Figure 1

(Top panel) The geometrical generating function (49) versus $T$ and $m$ for the Creutz ladder going through an Uhlmann process, showing diverging peaks at the finite-temperature TQPTs. The surface with peaks on the right (left) corresponds to $\mathrm{\Theta }=\frac{\pi }{3}$ and $\frac{\pi }{8}$, respectively. (Bottom panel) The Uhlmann phase as a function of $T$ and $m$ for $\mathrm{\Theta }=\frac{\pi }{3}$. Here ${\theta }_U=\pi$ in the red shaded (lower) region and ${\theta }_U=0$ in the blue (upper) region. The dashed line at $m=1.0$ indicates where the winding number ${\omega }_1$ jumps from 1 to 0.

Figure 2

Dynamical free energy density $f$, defined in Eq. (37), as a function of $t$ and $T$ for the three-level system in a quench process. The diverging peaks show where the finite-temperature DQPTs occur. The brown and blue surfaces correspond to $\theta =\frac{\pi }{5}$ and $\frac{2\pi }{5}$, respectively.

Figure 3

The geometrical generating function $g$ (49) of the three-level system as a function of $T$. The red dotted and blue solid lines correspond to $R=0.5$ and 1.0, respectively. The diverging peaks indicate where the finite-temperature TQPTs occur. The inset shows the jump of the Uhlmann phase at the critical temperature.