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In systems where values diminish by a constant proportion at each stage, the resulting sequence follows a geometric structure. Each new value in the sequence is obtained by applying a fixed multiplier to the preceding term. This regular, proportional decline type is often used to represent processes involving gradual loss, such as energy dissipation or reduction in amplitude over time.When analyzing the total effect of such a process across unlimited iterations, the series of values is referred...
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From generating functions to the geometric Binder cumulant.

Balazs Hetenyi1,2

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Generating functions are key tools in quantum mechanics for understanding polarization and quantum phase transitions. They help derive moments and cumulants, revealing quantum system fluctuations and enabling the identification of various phase transitions.

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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.