Ubiquity of zeros of Loschmidt amplitude for mixed states in different physical processes and their implications

TL;DR

This paper investigates the zeros of the Loschmidt amplitude in mixed quantum states across various processes, revealing their connection to dynamical and topological phase transitions and their impact on phases and state purifications.

Contribution

It demonstrates the ubiquity of Loschmidt amplitude zeros in mixed states during different processes and links these zeros to phase transitions and phase discontinuities.

Findings

Zeros of Loschmidt amplitude indicate phase transitions in mixed states.

Loschmidt amplitude zeros can occur without phase transitions in quasistatic processes.

Spinor representation provides insights into state purification changes.

Abstract

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.