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arXiv:2610.36521

Anomalous Structures of Fisher Zero Lines in Biorthogonal Dynamical Quantum Phase Transitions

Wen-Hui Nie, Chong Li, Lin-Cheng Wang

quant-phSubmitted Oct 27, 2026• Announced Oct 28, 2026• 0 citations
arXivHTMLPDF

Abstract

Fisher zero lines of the Loschmidt amplitude in non-Hermitian quenches can exhibit anomalous structures in the complex time plane, such as divergence, separation, and the restriction of all branches or a specific branch to the real or imaginary axis. Here, we use the biorthogonal framework to reveal the origin of these structures and determine the corresponding critical momentum conditions by analyzing the values of $m_k=\hat{\mathbf{h}}_0\cdot\hat{\mathbf{h}}_1$, the dot product between the normalized prequench and postquench complex Bloch vectors. When Fisher zeros continuously cover the real-time axis, the density of Fisher zeros is further introduced to identify the critical times within these critical regions, at which the rate function becomes nonanalytic. We verify our results in a generalized non-Hermitian SSH model by quenching the Peierls phase and the non-Hermitian gain-loss strength, and also use the Jordan form to analyze the dynamics at postquench exceptional points, where the conventional Fisher zero formula is not applicable.

AI Summary

The paper studies when anomalous Fisher-zero trajectories in non-Hermitian quenches produce dynamical quantum phase transitions (DQPTs), particularly when zeros occupy continuous real-time intervals rather than intersecting the time axis at isolated points. It uses the biorthogonal Loschmidt echo for translationally invariant two-band systems, with an initial state occupying the prequench lower band. Away from exceptional points (EPs), the analysis starts from $z_n(k)=[i(n+1/2)\pi-\operatorname{atanh}m_k]/h_k^{(1)}$, where $m_k=\hat{\mathbf h}_k^{(0)}\cdot\hat{\mathbf h}_k^{(1)}$. Classifying real, imaginary, and generic complex $m_k$ explains divergent, separated, and axis-confined zero trajectories. In particular, $m_k\to\pm1$ drives zeros to infinity; real $|m_k|>1$ can confine one branch to an axis, whereas purely imaginary $m_k$ confines every branch when the postquench spectrum is real or purely imaginary. Thus the Hermitian condition $m_{k_c}=0$ is not generally sufficient to characterize non-Hermitian critical momenta. For zeros continuously covering real-time intervals, the paper introduces their line density and identifies rate-function nonanalyticities with density singularities, including singularities inside those intervals. Analytical conditions and numerical examples are presented for Peierls-phase and gain–loss quenches in a generalized non-Hermitian SSH model. Examples include DQPTs entirely within the fully PT-broken phase, without crossing a PT boundary. The treatment requires the initial Hamiltonian to remain away from EPs. The conventional zero formula excludes postquench EPs; isolated exceptional momenta are handled separately using the Jordan-form evolution $e^{-iH_{\mathrm{EP}}t}=I-iH_{\mathrm{EP}}t$. The model demonstrations concern one-dimensional two-band systems without a non-Hermitian skin effect.

Little sign of AI-generated textIncrementalUnclear novelty

Error found: The displayed Fisher-zero density formula omits the restriction Re z_n(k)=0; interpreted as an unrestricted branch sum, it diverges in the paper's BB example by counting infinitely many off-axis zeros with identical imaginary coordinates.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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Fisher zero lines of the Loschmidt amplitude in non-Hermitian quenches can exhibit anomalous structures in the complex time plane, such as divergence, separation, and the restriction of all branches or a specific branch to the real or imaginary axis. Here, we use the biorthogonal framework to reveal the origin of these structures and determine the corresponding critical momentum conditions by analyzing the values of mk=h^0⋅h^1m_k=\hat{\mathbf{h}}_0\cdot\hat{\mathbf{h}}_1mk​=h^0​⋅h^1​, the dot product between the normalized prequench and postquench complex Bloch vectors. When Fisher zeros continuously cover the real-time axis, the density of Fisher zeros is further introduced to identify the critical times within these critical regions, at which the rate function becomes nonanalytic. We verify our results in a generalized non-Hermitian SSH model by quenching the Peierls phase and the non-Hermitian gain-loss strength, and also use the Jordan form to analyze the dynamics at postquench exceptional points, where the conventional Fisher zero formula is not applicable.
The paper studies when anomalous Fisher-zero trajectories in non-Hermitian quenches produce dynamical quantum phase transitions (DQPTs), particularly when zeros occupy continuous real-time intervals rather than intersecting the time axis at isolated points. It uses the biorthogonal Loschmidt echo for translationally invariant two-band systems, with an initial state occupying the prequench lower band.
Away from exceptional points (EPs), the analysis starts from zn(k)=[i(n+1/2)π−atanh⁡mk]/hk(1)z_n(k)=[i(n+1/2)\pi-\operatorname{atanh}m_k]/h_k^{(1)}zn​(k)=[i(n+1/2)π−atanhmk​]/hk(1)​, where mk=h^k(0)⋅h^k(1)m_k=\hat{\mathbf h}_k^{(0)}\cdot\hat{\mathbf h}_k^{(1)}mk​=h^k(0)​⋅h^k(1)​. Classifying real, imaginary, and generic complex mkm_kmk​ explains divergent, separated, and axis-confined zero trajectories. In particular, mk→±1m_k\to\pm1mk​→±1 drives zeros to infinity; real ∣mk∣>1|m_k|>1∣mk​∣>1 can confine one branch to an axis, whereas purely imaginary mkm_kmk​ confines every branch when the postquench spectrum is real or purely imaginary. Thus the Hermitian condition mkc=0m_{k_c}=0mkc​​=0 is not generally sufficient to characterize non-Hermitian critical momenta.
For zeros continuously covering real-time intervals, the paper introduces their line density and identifies rate-function nonanalyticities with density singularities, including singularities inside those intervals. Analytical conditions and numerical examples are presented for Peierls-phase and gain–loss quenches in a generalized non-Hermitian SSH model. Examples include DQPTs entirely within the fully PT-broken phase, without crossing a PT boundary.
The treatment requires the initial Hamiltonian to remain away from EPs. The conventional zero formula excludes postquench EPs; isolated exceptional momenta are handled separately using the Jordan-form evolution e−iHEPt=I−iHEPte^{-iH_{\mathrm{EP}}t}=I-iH_{\mathrm{EP}}te−iHEP​t=I−iHEP​t. The model demonstrations concern one-dimensional two-band systems without a non-Hermitian skin effect.
Dynamical quantum phase transitions (DQPTs), characterized by non-analytic behavior in rate function and abrupt changes in dynamic topological order parameters (DTOPs) over time, have garnered enormous attention in recent decades. However, in non-Hermitian systems, the special biorthogonality of the bases makes the definition of DQPTs complex. In this work, we delve into the comprehensive investigation of self-normal DQPTs (originally used in Hermitian systems) to compare them with their biorthogonal counterpart, within the context of non-Hermitian quantum walks (QWs). We present a detailed analysis of the behaviors of Loschmidt rate functions and DTOPs under these two distinct theoretical approaches. While both self-normal and biorthogonal methods can be used to detect DQPTs in quench dynamics between different topological phases, we theoretically present their differences in the definition of critical momenta and critical times by analyzing the Fisher zeros and fixed points. Finally, we present an experiment that observes both types of DQPTs using one-dimensional discrete-time QWs with single photons.
Non-Hermitian dynamical quantum phase transitions (DQPTs) are intrinsically sensitive to the choice of inner product under nonunitary time evolution. Although the biorthogonal formulation based on associated states provides a normalized Loschmidt echo with a probabilistic interpretation, previous studies have found biorthogonal and self-normal DQPTs to occur in the same parameter regimes, suggesting that the two forms of dynamical criticality are concomitant. Here we demonstrate that this is not the case. In an exactly solvable periodically driven non-Hermitian Su-Schrieffer-Heeger chain, we uncover a finite biorthogonal-only Floquet DQPT regime, where the biorthogonal Loschmidt rate becomes nonanalytic while the self-normal Loschmidt rate remains smooth. The critical conditions are obtained analytically, showing that the onset of biorthogonal Floquet DQPTs is locked to the exceptional lines of the effective Floquet Hamiltonian, whereas self-normal criticality has no corresponding spectral boundary. Moreover, for each critical momentum, the biorthogonal DQPT exhibits a pair of critical times within every driving period, whereas the self-normal DQPT exhibits only one. Our results establish a fundamental distinction between biorthogonal and self-normal DQPTs, thereby opening a route toward new nonequilibrium quantum phenomena in non-Hermitian systems.
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.
In quasi-Hermitian mixed-state quenches, the initial and final Hamiltonians generally possess different positive-definite metrics, so post-quench evolution is not quasi-unitary in the initial physical space. Choosing the preparation/measurement space as reference, we derive from first principles a normalized mixed-state Loschmidt amplitude using metric-compatible purifications. The resulting biorthogonal trace representation contains a normalization factor given by the norm of the evolved purification in the initial metric; this factor is essential for comparing states from different physical Hilbert spaces and directly encodes the metric mismatch. We prove that this normalization function is strictly positive on the real-time axis, so real-time criticality arises solely from destructive interference in the dynamical overlap. In complex time, zeros of the normalization factor generate branch singularities distinct from Fisher zeros. For a general two-level system, we derive analytic expressions for the Fisher and normalization zeros and analyze exceptional-point scaling together with the temperature- and quasi-Hermiticity-dependent DQPT critical times in two solvable quench protocols. In contrast to metric-stationary quenches, where non-Hermitian effects may be absorbed into an equivalent Hermitian description, the metric mismatch here produces genuine non-Hermitian dynamics: a time-dependent normalization factor and complex-time branch structure that cannot be removed by any global similarity transformation.
The short time evolution of a topological system after a sudden quench is considered, calculating the Loschmidt rate and studying in detail the distribution of Fisher zeros. Singular behavior in the Loschmidt rate has been predicted and found in several systems and, in particular, in one dimension (1D) one finds sharp features in the time evolution of the Loschmidt rate. These are associated with critical times that may be seen as the imaginary part of the Fisher zeros, under the condition that their real part vanishes. In the complex plane one finds lines of zeros that cross the imaginary axis at certain points. In a 2D system, in general, the Fisher zeros occupy areas instead of lines, as shown before, with the consequences that a continuous set of critical times results along the imaginary axis and that one does not find a well defined singular behavior in the Loschmidt rate. Here we show that several 2D topological systems also display very narrow regions (effectively lines) of zeros, in addition to finite extent areas of zeros. As a consequence, well defined sharp critical times are found, under appropriate conditions. An example is a 2Dp2\text{D}\phantom{\rule{4pt}{0ex}}p2Dp-wave superconductor in the presence of a magnetic field, but other systems share the same property, such as a Haldane insulator, a system with an anomalous Hall effect, and topological insulators. Details on the momentum distribution of the real and imaginary parts of the Fisher zeros are discussed, clarifying the existence of the lines of zeros and taking as an example the ppp-wave superconductor in magnetic field. The time dependence of the Loschmidt rate phase and of the Pancharatnam geometrical phase is considered, for different times after the quench and as a function of momentum. Pairs of vortices and antivortices in the phase associated with the geometric phase and the Loschmidt rate phase, that have been proposed for some 2D systems, are also shown to appear in the regime where we find lines of zeros.
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.