Dynamical Quantum Phase Transitions: A Geometric Picture

Abstract: The Loschmidt echo (LE) is a purely quantum-mechanical quantity whose determination for large quantum many-body systems requires an exceptionally precise knowledge of all eigenstates and eigenenergies. One might therefore be tempted to dismiss the applicability of any approximations to the underlying time evolution as hopeless. However, using the fully connected transverse-field Ising model (FC-TFIM) as an example, we show that this indeed is not the case, and that a simple semiclassical approximation to systems well described by mean-field theory (MFT) is in fact in good quantitative agreement with the exact quantum-mechanical calculation. Beyond the potential to capture the entire dynamical phase diagram of these models, the method presented here also allows for an intuitive geometric interpretation of the fidelity return rate at any temperature, thereby connecting the order parameter dynamics and the Loschmidt echo in a common framework. Videos of the post-quench dynamics provided in the supplemental material visualize this new point of view.

• The paper introduces a novel geometric understanding of dynamical quantum phase transitions using a semiclassical approximation validated on the fully connected transverse-field Ising model.
• The semiclassical method provides good quantitative agreement with exact calculations and identifies critical times via cusps in the fidelity return rate.
• This geometric approach offers valuable insights for guiding experiments and understanding nonequilibrium quantum dynamics in interacting systems.

Overview of "Dynamical Quantum Phase Transitions: A Geometric Picture"

The paper entitled "Dynamical Quantum Phase Transitions: A Geometric Picture" by Lang, Frank, and Halimeh presents a novel approach to understanding dynamical quantum phase transitions (DPTs) using a semiclassical approximation. The work primarily focuses on the fully connected transverse-field Ising model (FC-TFIM), a paradigmatic model of spin physics. By demonstrating the efficacy of a semiclassical approach, the authors aim to bridge the gap between exact quantum mechanical calculations and simpler models that are applicable even at finite temperatures.

Key Contributions

1. Semiclassical Approximation: By approximating the quantum many-body dynamics of the FC-TFIM using classical trajectories on the Bloch sphere, the authors provide an intuitive geometrical framework. They show that by extending mean-field theory (MFT) into a semiclassical regime, the essential features of dynamical phase diagrams can be captured effectively.
2. Loschmidt Echo and Fidelity Return Rate: The Loschmidt echo (LE) is studied as a measure of DPTs, specifically the second type of DPTs (DPT-II), characterized by non-analytic behavior in the LE. The authors introduce an equation for the fidelity return rate that is valid at any temperature and connect it with the order parameter dynamics of the system.
3. Geometric Interpretation: The paper provides a geometric interpretation of DPT-II transitions in terms of spin dynamics on the Bloch sphere. This allows for a visualization of the return function, whereby critical times correspond to specific geometric configurations.

Numerical Results and Implications

• The semiclassical method derived shows good quantitative agreement with exact quantum mechanical calculations. This is significant because it implies that even for large system sizes, where exact solutions become computationally prohibitive, reliable estimates of quantum dynamics are possible.
• Cusps in Fidelity Return Rate: One of the main numerical observations is the characterization of "cusps" in the fidelity return rate at critical times, which reflect transitions akin to classical phase transitions but occurring in real-time evolution rather than equilibrium scenarios.
• Finite Temperature Extensions: The paper extends the semiclassical approximation to finite temperatures, demonstrating that it can encompass thermal fluctuations, which significantly alter DPT dynamics. In particular, at high temperatures near the equilibrium critical point, the dynamics are described well by classical thermal fluctuations.

Practical and Theoretical Implications

• Theoretically, this work opens avenues for understanding nonequilibrium quantum dynamics beyond the limitations of exactly solvable models by using simpler semiclassical techniques. This is particularly useful in cases where quantum fluctuations can be dominating yet are challenging to resolve analytically.
• Practically, in experimental contexts such as ultracold atom and ion-trap systems where dynamical phase transitions can now be observed, the geometric and semiclassical insights provided here can guide the design and interpretation of experiments.

Future Directions

This study suggests several promising directions for future research. These include extending the semiclassical framework to more complex interacting systems with richer phase structures or exploring the implications for entanglement dynamics in nonequilibrium settings. Additionally, the authors suggest that this approach could be adapted to other quantifiers of quantum phases, such as entanglement measures or Fisher information, thereby broadening the scope of semiclassical methodologies in quantum many-body physics.

In conclusion, "Dynamical Quantum Phase Transitions: A Geometric Picture" provides a comprehensive and mathematically robust exploration into semiclassical dynamics in quantum phase transitions, potentially impacting the understanding of quantum dynamics in a variety of physical systems.