The adiabatic theorem is one of the most interesting and significant theorems in quantum mechanics. However, the adiabatic theorem can fail for general non-Hermitian quantum systems. In this paper, by utilizing the complex geometric phase, the functional calculus for biorthogonal systems and the Grönwall inequality, we prove rigorously that the adiabatic theorem is still valid for diagonalizable non-Hermitian systems with real eigenvalues. The proof also justifies the definition of a complex Berry phase for non-Hermitian systems, in both Abelian and non-Abelian cases.The adiabatic theorem is one of the most interesting and significant theorems in quantum mechanics. However, the adiabatic theorem can fail for general non-Hermitian quantum systems. In this paper, by utilizing the complex geometric phase, the functional calculus for biorthogonal systems and the Grönwall inequality, we prove rigorously that the adiabatic theorem is still valid for diagonalizable non-Hermitian systems with real eigenvalues. The proof also justifies the definition of a complex Berry phase for non-Hermitian systems, in both Abelian and non-Abelian cases.

quant-pharXiv:2511.00968Nov 4, 20250 citations

Nonadiabatic non-Abelian geometric phase in non-Hermitian quantum systems

Xu-Ying Lou, Yu-Shi Cheng, P. Z. Zhao +1 more

With the growing interest in non-Hermitian quantum systems, the concepts of geometric phase such as Berry phase, adiabatic non-Abelian geometric phase, and nonadiabatic Abelian geometric phase have been gradually extended from Hermitian to non-Hermitian quantum systems. However, extending the most general form, the nonadiabatic non-Abelian geometric phase, to non-Hermitian systems has remained an open problem. In this paper, we address this issue by proposing a formalism of the nonadiabatic non-Abelian geometric phase in non-Hermitian quantum systems. Our formalism is consistent with the previous extensions in non-Hermitian quantum systems, and can also naturally revert to the concepts of geometric phase in Hermitian quantum systems. It provides a general formulation of geometric phase for both Hermitian and non-Hermitian quantum systems.With the growing interest in non-Hermitian quantum systems, the concepts of geometric phase such as Berry phase, adiabatic non-Abelian geometric phase, and nonadiabatic Abelian geometric phase have been gradually extended from Hermitian to non-Hermitian quantum systems. However, extending the most general form, the nonadiabatic non-Abelian geometric phase, to non-Hermitian systems has remained an open problem. In this paper, we address this issue by proposing a formalism of the nonadiabatic non-Abelian geometric phase in non-Hermitian quantum systems. Our formalism is consistent with the previous extensions in non-Hermitian quantum systems, and can also naturally revert to the concepts of geometric phase in Hermitian quantum systems. It provides a general formulation of geometric phase for both Hermitian and non-Hermitian quantum systems.

Physical Review AAug 14, 20255 citations

Non-Hermitian quantum systems and their geometric phases

Qi Zhang, Biao Wu

We discuss the basic theoretical framework for non-Hermitian quantum systems with particular emphasis on the diagonalizability of non-Hermitian Hamiltonians and their general linear $\text{GL}(1,\mathbb{C})$ gauge freedom, which are relevant to the adiabatic evolution of non-Hermitian quantum systems. We find that the adiabatic evolution is possible only when the eigenenergies are real. The accompanying geometric phase is found to be generally complex and associated with not only the phase of a wave function but also its amplitude. The condition for the real geometric phase is laid out. Our results are illustrated with two examples of non-Hermitian $\mathcal{PT}$ -symmetric systems, the two-dimensional non-Hermitian Dirac fermion model and bosonic Bogoliubov quasiparticles.We discuss the basic theoretical framework for non-Hermitian quantum systems with particular emphasis on the diagonalizability of non-Hermitian Hamiltonians and their general linear $\text{GL}(1,\mathbb{C})$ gauge freedom, which are relevant to the adiabatic evolution of non-Hermitian quantum systems. We find that the adiabatic evolution is possible only when the eigenenergies are real. The accompanying geometric phase is found to be generally complex and associated with not only the phase of a wave function but also its amplitude. The condition for the real geometric phase is laid out. Our results are illustrated with two examples of non-Hermitian $\mathcal{PT}$-symmetric systems, the two-dimensional non-Hermitian Dirac fermion model and bosonic Bogoliubov quasiparticles.

arXiv:1810.10729Physical Review AMar 26, 201930 citations

Geometry of quantal adiabatic evolution driven by a non-Hermitian Hamiltonian

Zhaoyan Wu, Ting Yu, Hongwei Zhou

Physics Letters AMar 1, 19947 citations

We discuss the basic theoretical framework for non-Hermitian quantum systems with particular emphasis on the diagonalizability of non-Hermitian Hamiltonians and their general linear

\text{GL}(1,\mathbb{C})

gauge freedom, which are relevant to the adiabatic evolution of non-Hermitian quantum systems. We find that the adiabatic evolution is possible only when the eigenenergies are real. The accompanying geometric phase is found to be generally complex and associated with not only the phase of a wave function but also its amplitude. The condition for the real geometric phase is laid out. Our results are illustrated with two examples of non-Hermitian

\mathcal{PT}

-symmetric systems, the two-dimensional non-Hermitian Dirac fermion model and bosonic Bogoliubov quasiparticles.