Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics

Abstract

Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this formalism, Hamiltonians can be interpreted as a Christoffel symbol-like operators, and the Schroedinger equation as a parallel transport in this formalism. We then derive the evolution equations for the states and metrics along the emergent dimensions and find that the curvature of the Hilbert space bundle for any given closed system is locally flat. Finally, we show that the fidelity susceptibilities and the Berry curvatures of states are related to these emergent parallel transports.

arXiv comment20 pages, 1 figure

References in corpus (45)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Berry Phase Effects on Electronic Properties
• Making Sense of Non-Hermitian Hamiltonians
• Exceptional Topology of Non-Hermitian Systems
• Non-Hermitian Physics
• Loss-induced suppression and revival of lasing
• Pseudo-Hermitian Representation of Quantum Mechanics
• Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems
• PT-Symmetric Phonon Laser
• Fidelity approach to quantum phase transitions
• Biorthogonal Quantum Mechanics
• Exceptional Points of Non-hermitian Operators
• Second-Order Topological Phases in Non-Hermitian Systems
• Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries
• Resolution of Gauge Ambiguities in Ultrastrong-Coupling Cavity QED
• Topological non-Hermitian origin of surface Maxwell waves
• Non-Hermitian Hamiltonians and no-go theorems in quantum information
• Time-dependent version of cryptohermitian quantum theory
• Hamiltonian for the zeros of the Riemann zeta function
• $\mathcal{P}\mathcal{T}$-symmetric circuit-QED
• Three-Hilbert-Space Formulation of Quantum Mechanics
• The C Operator in PT-Symmetric Quantum Theories
• Geometric Phase for Non-Hermitian Hamiltonians and Its Holonomy Interpretation
• Fidelity susceptibility made simple: A unified quantum Monte Carlo approach
• Topological band theory for non-Hermitian systems from the Dirac equation
• Berry phase theory of planar Hall effect in Topological Insulators
• Hunting for the non-Hermitian exceptional points with fidelity susceptibility
• Gauge invariance of the Dicke and Hopfield models
• Flattening the Curve with Einstein's Quantum Elevator: Hermitization of Non-Hermitian Hamiltonians via a Generalized Vielbein Formalism
• General properties of fidelity in non-Hermitian quantum systems with PT symmetry
• Gauge freedom, quantum measurements, and time-dependent interactions in cavity and circuit QED
• Gauge Principle and Gauge Invariance in Two-Level Systems
• Gauge Principle and Gauge Invariance in Quantum Two-Level Systems
• Gauge-Independent Emission Spectra and Quantum Correlations in the Ultrastrong Coupling Regime of Open System Cavity-QED
• Majorana's approach to nonadiabatic transitions validates the adiabatic-impulse approximation
• Time-Dependent Hilbert Spaces, Geometric Phases, and General Covariance in Quantum Mechanics
• The Fermionic influence superoperator: a canonical derivation for the development of methods to simulate the influence of a Fermionic environment on a quantum system with arbitrary parity symmetry
• An introduction to PT-symmetric quantum mechanics -- time-dependent systems
• Entanglement Dynamics in Anti-$\mathcal{P}\mathcal{T}$-Symmetric Systems
• Passage through exceptional point: Case study
• Quantum state discrimination in a PT-symmetric system
• Quantum geometric tensor away from Equilibrium
• Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions
• Paths of unitary access to exceptional points
• Non-Hermitian Quantum Fermi Accelerator

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