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arXiv:2603.21281

Gauge-Invariant Non-Hermitian Quantum Theory: Foundation and Applications to Dynamical Phase Transitions

Fei Wang, Guoying Liang, Zecheng Zhao, Bao-Ming Xu

quant-phcond-mat.otherSubmitted Mar 22, 2026• Announced Mar 24, 2026• 0 citations• Revised May 6, 2026 (v2)
v1 · Mar 22, 2026v2 · May 6, 2026
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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 two foundational premises: (i) the dynamics of both left and right-vectors of a non-Hermitian system must satisfy the Schrödinger equation; (ii) the theoretical framework must preserve gauge invariance, ensuring that physical quantities are independent of unobservable phase choices. Building on these physically motivated assumptions, we refine the biorthogonal framework, leading to a gauge-invariant non-Hermitian quantum theory. Our framework naturally encompasses the open-system effective non-Hermitian evolution as a special case, and 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 gauge-invariant non-Hermitian quantum theory. 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 gauge-invariant non-Hermitian quantum theory will find broad applications in the study of non-Hermitian systems.

Comments
11 pages, 5 figures; Comments are welcome! Revised version

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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 two foundational premises: (i) the dynamics of both left and right-vectors of a non-Hermitian system must satisfy the Schrödinger equation; (ii) the theoretical framework must preserve gauge invariance, ensuring that physical quantities are independent of unobservable phase choices. Building on these physically motivated assumptions, we refine the biorthogonal framework, leading to a gauge-invariant non-Hermitian quantum theory. Our framework naturally encompasses the open-system effective non-Hermitian evolution as a special case, and 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 gauge-invariant non-Hermitian quantum theory. Notably, our formulation naturally generalizes the known condition for such transitions in Hermitian two-band systems, namely, dki⋅dkf=0\mathbf{d}_{k}^i\cdot\mathbf{d}_{k}^f=0dki​⋅dkf​=0, to the non-Hermitian case, where it takes the form Re[dkidki⋅dkfdkf]=0\mathrm{Re}\Bigl[\frac{\mathbf{d}_{k}^i}{d_{k}^i}\cdot\frac{\mathbf{d}_{k}^f}{d_{k}^f}\Bigr]=0Re[dki​dki​​⋅dkf​dkf​​]=0. Furthermore, we identify entirely new dynamical phase transitions that cannot be characterized by the winding number. We hope that this gauge-invariant non-Hermitian quantum theory will find broad applications in the study of non-Hermitian systems.
We establish a unified framework for dynamical quantum phase transitions (DQPTs) in non-Hermitian systems that encompasses both biorthogonal and self-norm nonbiorthogonal formulations for pure and mixed states under quantum quench protocols. Our framework provides explicit expressions for the Loschmidt amplitude, Loschmidt echo, and rate function, revealing a universal geometric signature of DQPTs in the two-band model: orthogonality of two related vectors in two-dimensional real space. Strikingly, we demonstrate that nonbiorthogonal quenches from non-Hermitian to Hermitian Hamiltonians under chiral symmetry exhibit emergent topological characteristics of DQPTs, unveiling their fundamental features beyond conventional Hermitian regimes. This work establishes fundamental geometric and topological principles governing quantum criticality in open systems, with implications for quantum sensing and many-body physics in dissipative environments.
By utilizing biorthogonal bases, we develop a comprehensive framework for studying biorthogonal dynamical quantum phase transitions in non-Hermitian systems. With the help of the previously overlooked associated state, we define the automatically normalized biorthogonal Loschmidt echo. This approach is capable of handling arbitrary non-Hermitian systems with complex eigenvalues and naturally eliminates the negative value of Loschmidt rate obtained without the biorthogonal bases. Taking the non-Hermitian Su-Schrieffer-Heeger model as a concrete example, a 1/2 change of dynamical topological order parameter in biorthogonal bases is observed which is not shown in self-normal bases. Furthermore, we discover that the periodicity of biorthogonal dynamical quantum phase transitions depends on whether the two-level subsystem at the critical momentum oscillates or reaches a steady state.
We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence ∼∣εμ∣−2\sim |ε_μ|^{-2}∼∣εμ​∣−2, while the Berry curvature shows a weaker, subleading divergence ∼∣εμ∣−3/2\sim |ε_μ|^{-3/2}∼∣εμ​∣−3/2, with εμε_μεμ​ denoting the parameter displacement from the EP along an individual parameter axis μμμ. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.
Abstract The dynamical and topological properties of non-Hermitian systems have attracted great attention in recent years. In this work, we establish an intrinsic connection between two classes of intriguing phenomena—topological phases and dynamical quantum phase transitions (DQPTs)—in non-Hermitian systems. Focusing on one-dimensional models with chiral symmetry, we find DQPTs following the quench from a trivial to a non-Hermitian topological phase. Moreover, the critical momenta and critical time of the DQPTs are found to be directly related to the topological invariants of the non-Hermitian system. We further demonstrate our theory in three prototypical non-Hermitian lattice models, the lossy Kitaev chain (LKC), the LKC with next-nearest-neighbor hoppings, and the nonreciprocal Su–Schrieffer–Heeger model. Finally, we suggest a proposal to experimentally verify the found connection by a nitrogen-vacancy center in diamond.
Dynamical quantum phase transitions in non-Hermitian systems pose fundamental challenges due to the intrinsic biorthogonality of their eigenstates. In this work, we extend a biorthogonal framework to investigate dynamical quantum phase transitions in non-Hermitian topological superconductors. Taking the non-Hermitian Kitaev chain as a prototypical model, we construct an associated-state formalism and reformulate the Loschmidt rate function, dynamical topological order parameter, and dynamical Fisher zeros. Within this framework, we find that the critical times at which dynamical quantum phase transitions occur differ from those based on the conventional self-normal approaches. We further analyze momentum-resolved subsystems at critical momenta and demonstrate the robustness of the biorthogonal framework. Our work highlights the essential role of biorthogonality in nonequilibrium dynamics and establishes a consistent theoretical framework for dynamical quantum phase transitions in non-Hermitian topological superconductors.
We demonstrate that genuinely non-Hermitian topological phases and corresponding topological phase transitions can be naturally realized in monitored quantum circuits, exemplified by the paradigmatic non-Hermitian Su-Schrieffer-Heeger model.We emulate this model by a 1D chain of spinless electrons evolving under unitary dynamics and subject to periodic measurements that are stochastically invoked.The non-Hermitian topology is visible in topological invariants adapted to the context of monitored circuits.For instance, the topological phase diagram of the monitored realization of the non-Hermitian Su-Schrieffer-Heeger model is obtained from the biorthogonal polarization computed from an effective Hamiltonian of the monitored system.Importantly, our monitored circuit realization allows direct access to steady-state biorthogonal expectation values of generic observables, and hence, to measure physical properties of a genuine non-Hermitian model.We expect our results to be applicable more generally to a wide range of models that host non-Hermitian topological phases.