From generating functions to the geometric Binder cumulant.

Area of Science:

• Quantum Mechanics
• Condensed Matter Physics
• Statistical Mechanics

Background:

• Generating functions are crucial for deriving moments and cumulants, which characterize probability distribution fluctuations.
• Quantum systems exhibit fluctuations that can be analyzed using these statistical tools.
• The study focuses on the modern theory of polarization and quantum phase transitions.

Purpose of the Study:

• To provide an overview of generating functions in quantum mechanical contexts, specifically polarization and quantum phase transitions.
• To extend the formalism for geometric phases to include quasi-adiabatic cycles with degeneracy points.
• To demonstrate the utility of geometric Binder cumulants in identifying quantum phase transitions.

Main Methods:

• Utilizing generalized Bargmann invariants as generating functions for quasi-adiabatic cycles.
• Forming geometric Binder cumulants from generated cumulants, analogous to statistical mechanics schemes.
• Calculating fidelity susceptibility to complement geometric Binder cumulant results.

Main Results:

• Generating functions, particularly generalized Bargmann invariants, facilitate the extension of geometric phase formalism to quasi-adiabatic cycles.
• Geometric Binder cumulants are sensitive to gap closure, proving effective in identifying metal-insulator, localization, and quantum phase transitions.
• Example calculations on model systems validate the approach, showing agreement with known localization properties and fidelity susceptibility.

Conclusions:

• Generating functions offer a powerful framework for analyzing quantum system fluctuations and geometric properties.
• Geometric Binder cumulants provide a robust tool for detecting and characterizing various quantum phase transitions.
• The presented methods offer a unified approach to understanding quantum geometry and phase transitions in condensed matter systems.