Isoperimetric Inequalities in Quantum Geometry

Abstract

We reveal strong and weak inequalities relating two fundamental macroscopic quantum geometric quantities, the quantum distance and Berry phase, for closed paths in the Hilbert space of wavefunctions. We recount the role of quantum geometry in various quantum problems and show that our findings place new bounds on important physical quantities.

6 pages, 2 figures

References in corpus (20)

• Topological Insulators
• Berry Phase Effects on Electronic Properties
• Maximally-localized generalized Wannier functions for composite energy bands
• Anomalous Hall effect
• Type-II Weyl Semimetals
• The maximum speed of dynamical evolution
• Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control
• Spontaneous Quantum Hall States in Chirally-stacked Few-Layer Graphene Systems
• Field Induced Positional Shift of Bloch Electrons and its Dynamical Implications
• Superfluidity and Quantum Geometry in Twisted Multilayer Systems
• Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure
• Structured Weyl Points in Spin-Orbit Coupled Fermionic Superfluids
• End states, ladder compounds, and domain wall fermions
• Effective theory and emergent $SU(2)$ symmetry in the flat bands of attractive Hubbard models
• Geometric stability of topological lattice phases
• Fundamental bound on topological gap
• Pairing and superconductivity in the flat band: Creutz lattice
• Contrasting lattice geometry dependent versus independent quantities: ramifications for Berry curvature, energy gaps, and dynamics
• Nontrivial Quantum Geometry and the Strength of Electron-Phonon Coupling
• Relativistic resonances: Their masses, widths, lifetimes, superposition, and causal evolution