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Joan A Austrich-Olivares1, Jose David Vergara1

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We present a quantum geometric tensor for curved spaces, incorporating quantum metric tensor and Berry curvature. This framework modifies standard inner products, impacting quantum geometry in various physical systems.

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Area of Science:

  • Quantum mechanics
  • Geometric phase
  • Condensed matter physics

Background:

  • The quantum geometric tensor is crucial for understanding geometric phases in quantum systems.
  • Standard formulations often assume flat parameter spaces.

Purpose of the Study:

  • To introduce a generalized quantum geometric tensor for parameter-dependent curved spaces.
  • To investigate the impact of curved metrics on quantum geometry and Berry curvature.

Main Methods:

  • Definition of quantum geometric tensor in curved space with parameter-dependent metric.
  • Modification of inner product and its effect on quantum metric tensor and Berry curvature.
  • Utilizing infinitesimal distance and fidelity susceptibility approaches.

Main Results:

  • The quantum geometric tensor comprises a quantum metric tensor and Berry curvature.
  • Curved metrics introduce parameter-derivative terms into the quantum metric tensor and Berry curvature.
  • The Berry connection gains an additional term, transforming it into a density of weight one.

Conclusions:

  • The developed framework provides a robust method for analyzing quantum geometry in curved spaces.
  • The study offers new insights into the behavior of quantum systems with non-trivial metrics.
  • Demonstrated applicability through diverse one- and two-dimensional examples.