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Isoperimetric Inequalities in Quantum Geometry.

Praveen Pai1, Fan Zhang1

  • 1University of Texas at Dallas, Department of Physics, Richardson, Texas 75080, USA.

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|April 3, 2026
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Summary
This summary is machine-generated.

We discovered a quantum isoperimetric inequality linking Berry phase and quantum distance in wave functions. This fundamental geometric principle sets bounds on key physical quantities in quantum systems.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Information Science
  • Quantum Geometry

Background:

  • The Berry phase and quantum distance are key geometric measures of wave functions in Hilbert space.
  • Understanding the relationship between these geometric quantities is crucial for characterizing quantum states.

Purpose of the Study:

  • To uncover a quantum analog of the classical isoperimetric inequality.
  • To reveal a fundamental link between Berry phase and quantum distance for closed paths in Hilbert space.
  • To establish bounds on physical quantities derived from quantum geometry.

Main Methods:

  • Derivation of a strong isoperimetric inequality for two-band systems, analogous to the spherical isoperimetric problem.
  • Proof of a weak isoperimetric inequality for multiband systems.
  • Analysis of the relationship between Berry phase and quantum distance.

Main Results:

  • A quantum isoperimetric inequality is established, linking Berry phase and quantum distance.
  • For two-band systems, a strong inequality is proven.
  • For multiband systems, it is shown that the Berry phase never exceeds the quantum distance.

Conclusions:

  • The study introduces new quantum geometric principles with fundamental bounds.
  • These principles have broad implications for understanding Wannier function spread, quantum speed, electron-phonon coupling, and superfluid weight.
  • The findings are relevant across condensed matter physics and quantum information science.