Biorthogonal-only Floquet Dynamical Quantum Phase Transitions

Technical Summary: Biorthogonal-only Floquet Dynamical Quantum Phase Transitions

Problem Statement
Non-Hermitian quantum systems exhibit unique phenomena such as exceptional points and the non-Hermitian skin effect, yet their far-from-equilibrium dynamics remain less explored than their spectral and topological properties. Dynamical Quantum Phase Transitions (DQPTs), characterized by nonanalyticities in the Loschmidt rate, provide a framework for studying nonequilibrium criticality. However, extending DQPTs to non-Hermitian systems is subtle due to nonunitary time evolution, which allows for inequivalent definitions of state overlap. Specifically, there is an ambiguity between the self-normal formulation (based solely on right eigenstates) and the biorthogonal formulation (based on associated left and right eigenstates). Previous studies suggested these two forms of dynamical criticality were concomitant, occurring in the same parameter regimes, differing only in the location of critical points. This raised a fundamental question: Can the choice of overlap structure determine not just where a dynamical singularity occurs, but whether it occurs at all?

Methodology
The authors investigate this question using an exactly solvable, periodically driven non-Hermitian Su-Schrieffer-Heeger (SSH) chain. The system is described by a time-dependent Hamiltonian $H(t)$ with dimerization parameter $\eta$, intracell nonreciprocity $\gamma$, and driving frequency $\omega$.

1. Effective Hamiltonian: By applying a rotating-frame transformation, the authors derive a time-independent effective Floquet Hamiltonian, $H_F(k) = \mathbf{d}_k \cdot \boldsymbol{\sigma}$.
2. Spectral Analysis: They analyze the eigenvalues $E_{k,\pm} = \pm E_k$ of $H_F(k)$ to identify real-part gap closings and exceptional lines (where eigenvalues and eigenvectors coalesce). The parameter space $(\eta, \gamma)$ is partitioned into four spectral regimes based on whether the real-part gap closes at momenta $k_c=0$ or $k_c=\pi$.
3. Loschmidt Echoes: The authors compute two distinct Loschmidt rates for the system initialized in the ground state of the effective Hamiltonian:
   • Biorthogonal Loschmidt Rate ($\lambda_B$): Derived from the normalized biorthogonal Loschmidt echo $L_B(t)$, which utilizes the associated state $\langle \tilde{\psi}(t) |$ and ensures a probabilistic interpretation ($0 \le L_B \le 1$).
   • Self-Normal Loschmidt Rate ($\lambda_S$): Derived from the conventional overlap $|\langle \psi(0) | \psi(t) \rangle|^2$ using only right eigenstates.
4. Critical Conditions: Analytical solutions are derived for the conditions under which the Loschmidt rates become nonanalytic (i.e., when the single-mode echo vanishes).

Key Contributions and Results
The study uncovers a finite biorthogonal-only (B-only) Floquet DQPT regime, demonstrating that biorthogonal and self-normal criticalities are not necessarily concomitant. The key findings are:

• Existence of B-only Regime: The authors identify a finite region in the $(\eta, \gamma)$ parameter space where the biorthogonal Loschmidt rate $\lambda_B(t)$ exhibits nonanalytic cusps (signaling a DQPT), while the self-normal rate $\lambda_S(t)$ remains smooth (analytic). This proves that the overlap structure can determine the very existence of a phase transition.
• Spectral Mechanisms:
  • Biorthogonal Criticality: The onset of biorthogonal DQPTs is strictly locked to the exceptional lines of the effective Floquet Hamiltonian. Specifically, critical momenta occur where the real-part spectral gap closes ($\Delta_{\text{Re}}(k_c) = 0$).
  • Self-Normal Criticality: This form of criticality is governed by a right-state overlap condition and has no corresponding spectral boundary (no direct link to exceptional lines or gap closings).
• Temporal Structure: The two formulations exhibit distinct intraperiod temporal structures. For each critical momentum, the biorthogonal DQPT generates a pair of critical times within every driving period ($t_{c,1}$ and $t_{c,2}$), symmetric around the period midpoint. In contrast, the self-normal DQPT yields only one critical time per period, located exactly at the midpoint.
• Global Phase Diagram: A complete parameter-space analysis reveals four distinct regimes:
  1. B-only: Biorthogonal DQPT exists; self-normal does not.
  2. Both: Both types of DQPTs exist.
  3. S-only: Self-normal DQPT exists; biorthogonal does not.
  4. None: Neither type exists.
     The B-only regime spans a finite area, indicating that this phenomenon is robust and not a result of fine-tuning.

Significance
The paper establishes a fundamental distinction between biorthogonal and self-normal DQPTs in non-Hermitian systems. It demonstrates that the choice of inner product (overlap structure) is a decisive factor in non-Hermitian quantum mechanics, capable of creating dynamical criticality that is entirely absent in descriptions relying exclusively on right eigenstates. This work highlights the biorthogonal framework as a source of novel nonequilibrium phenomena, suggesting that the left and right eigenstates carry independent information essential for characterizing the dynamics of non-Hermitian systems. The findings open a route toward identifying new nonequilibrium quantum phenomena that are specific to the biorthogonal formulation.