A Refined Biorthogonal Framework for Non-Hermitian Quantum Theory and Its Application in Dynamical Phase Transition

Abstract

The description of states and dynamics in non-Hermitian systems is fundamentally linked to the choice of an appropriate theoretical framework--a point of ongoing debate in the field. This work addresses this issue by proposing a consistent formulation that reconciles existing controversies and establishes a unified theoretical understanding. Our approach rests on a foundational premise: The dynamics of both left- and right-vectors of a non-Hermitian system must satisfy the Schrödinger equation. Building on this physically motivated assumption, we refine the biorthogonal framework, leading to a consistent reformulation of non-Hermitian quantum theory. This refined framework can naturally reduce to standard quantum mechanics in the Hermitian limit. As a concrete application, we analyze the dynamical phase transition in a one-dimensional Su-Schrieffer-Heeger (SSH) model within this refined framework. Notably, our formulation naturally generalizes the known condition for such transitions in Hermitian two-band systems, namely, $\mathbf{d}_{k}^i\cdot\mathbf{d}_{k}^f=0$, to the non-Hermitian case, where it takes the form $\mathrm{Re}\Bigl[\frac{\mathbf{d}_{k}^i}{d_{k}^i}\cdot\frac{\mathbf{d}_{k}^f}{d_{k}^f}\Bigr]=0$. Furthermore, we identify entirely new dynamical phase transitions that cannot be characterized by the winding number. We hope that this refined framework will find broad applications in the study of non-Hermitian systems.