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.

Figures (5)

• Figure 1: (Color online) (a) The schematic diagram of SSH model. (b) The $\mathcal{PT}$-symmetry phase diagram in non-Hermitian SSH model. In the $\mathcal{PT}$-symmetric phase (yellow areas), the system energy is real. The $\mathcal{PT}$-symmetry-broken phase can be divided into two types: Phase I (green areas), characterized by a purely imaginary energy spectrum, and Phase II (blue areas), which exhibits a complex energy spectrum. In the phase diagram, $q=J_2/J_1$ and $\eta=\mu/J_1$.
• Figure 2: (Color online) Lines of Fisher zeros (a1-b1), the time evolution of the real parts of the rate function $\mathrm{Re}r(t)$ [red curves in (a2) and (b2)] and the winding number $\mathrm{Re}\nu(t)$ [blue curves in (b1) and (b2)] for quenches $q=0.5\rightarrow q'=2$ (a1-a2) and $q=1.5\rightarrow q'=2$ (b1-b2). The other parameters are $\eta=\eta'=0.4$, and $l=0$.
• Figure 3: (Color online) Lines of Fisher zeros (a), the time evolution of the real parts of rate function $\mathrm{Re}r(t)$ [red curves in (b)], the winding number $\mathrm{Re}\nu(t)$ [blue curves in (b)] for the quench of $q$ from $q=1$ to $q'=2$. The other parameters are $\eta=\eta'=0$ and $l=0$.
• Figure 4: (Color online) Lines of Fisher zeros (a), the time evolution of the real parts of rate function $\mathrm{Re}r(t)$ [red curves in (b)], the winding number $\mathrm{Re}\nu(t)$ [blue curves in (b)] for the quench of $\eta$ from $\eta=2$ to $\eta'=0.2$. The other parameters are $q=q'=0.5$ and $l=0$.
• Figure 5: (Color online) Lines of Fisher zeros (a1-b1), the time evolution of the real parts of the rate function $\mathrm{Re}r(t)$ [red curves in (a2) and (b2)] and the winding number $\mathrm{Re}\nu(t)$ [blue curves in (b1) and (b2)] for quenches $\eta=1\rightarrow \eta'=0$ with $q=q'=0.5$ (a1-a2) and $q=0.9\rightarrow q'=2$ with $\eta=\eta'=0.4$ (b1-b2). The Fisher zeros are calculated by considering $l=0$.