Metric structures characterized by norm properties in non-Hermitian systems
Phys. Rev. A 114, 052201 – Published 2 November, 2026
DOI: https://doi.org/10.1103/6rg6-2n55
Abstract
For non-Hermitian quantum systems, metric operators provide a way to define the corresponding inner-product structures. In the entirely real-spectrum regime, a positive-definite metric can be introduced such that the Hamiltonian is -self-adjoint with respect to the corresponding inner product. In the complex-spectrum regime, different metric constructions can be characterized by their associated norm properties. For finite-dimensional non-Hermitian systems with a complete energy eigenbasis, we compare three representative metrics, namely, the cross-inner-product metric , the dynamical metric , and the pseudo-Hermitian metric , and analyze their relationships and differences. For time-independent Hamiltonians, the norm obeys a balance equation governed by the imaginary parts of the eigenvalues, with its long-time behavior dominated by the largest imaginary part among contributions with nonzero initial weights. In contrast, preserves the norm through its time dependence, while remains norm-preserving but becomes indefinite in the complex-spectrum regime. For an entirely real spectrum, all three metrics coincide under the same normalization convention when . At exceptional points, the energy eigenbasis is incomplete and is singular and positive semidefinite, whereas a positive-definite can be obtained from another complete initial basis, and a Hermitian invertible can still be obtained from its defining equation. For time-dependent Hamiltonians, the norm equation acquires a geometric contribution from the evolving biorthogonal basis, while imposing dynamical compatibility on makes it coincide with for the same initial metric. The corresponding exceptional-point properties are also discussed. A concrete model illustrates these distinctions.
