Anatomy of Non-Hermitian Dynamical Quantum Phase Transitions

TL;DR

This work develops a unified framework for dynamical quantum phase transitions in non-Hermitian systems, encompassing biorthogonal and non-biorthogonal formulations and extending to mixed-state dynamics under quantum quenches. It provides explicit expressions for Loschmidt amplitudes and echoes, and identifies a universal geometric signature in a two-band model via orthogonality of two 2D vectors, with Fisher zeros signaling DQPTs. The analysis reveals a winding-number structure in non-Hermitian settings, including a half-integer non-Hermitian winding number and topological DQPTs under chiral symmetry, demonstrated concretely in non-Hermitian SSH quenches. The framework is further connected to experimental platforms through dissipation-controlled quenches and circuit realizations, offering routes to observe dynamical criticality in open quantum systems and to explore dissipative many-body physics.

Abstract

We establish a unified framework for dynamical quantum phase transitions (DQPTs) in non-Hermitian systems that encompasses both biorthogonal and self-norm non-biorthogonal 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 non-biorthogonal 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.

Figures (5)

• Figure 1: The topological characteristic of the non-Hermitian SSH model under a quantum quench from $\gamma=1.5$ to $0$, are analyzed with fixed parameters $t_{1}=0.6$ [panels (a), (c), and (e)] or $2.2$ [panels (b), (d), and (f)], and $t_{2}=1$. (a) and (b) The rate function $\lambda'(t)$ as a function of time $t$, with the DQPTs indicated by the non-analytical cusps marked by red points along the $t$ axis. These critical times correspond precisely to the intersections of Fisher zero flows with the imaginary axis in the complex plane, as illustrated in panels (c) and (d) by the Fisher zeros $z'_{n}$ ($n=0,1,2,3,4$) arranged from left to right. Panels (e) and (f) depict the momentum-resolved flows of the vectors $\mathcal{R} e^{i\phi^{0}}$ and $\hat{d}_{1}$ associated with the initial and final Hamiltonians. As $k$ varies in $[-\pi, \pi]$, the winding number differences $\Delta \nu=1/2$ and $-1/2$ emerge. Notably, the blue flow in (f) evolves from $-\pi$, descends to the lower endpoint, then ascends to the upper endpoint, and finally returns downward to $\pi$ (identical to $-\pi$), resulting in a vanishing winding number. Red pints in (e) and (f) mark the critical momenta $k_c$, where the two vectors are orthogonal (as indicated by the gray arrows), signaling the onset of DQPTs.
• Figure 2: The intuitive schematic of parallel transport on principal bundles. (a) This illustrates the total space $\mathbb{E}$, the base space $\mathbb{M}$, and the fibre $\mathbb{F}$, which is equivalent to the structure group $\mathbb{G}$. A point $u$ on a fibre is projected to the point $b$ on $\mathbb{M}$ via the surjection $p$. A curve $\gamma$ in $\mathbb{M}$ is horizontal lifted to $\tilde{\gamma}$ in $\mathbb{E}$. The tangent vector at $u$ along $\tilde{\gamma}$ belongs to the horizontal subspace $\mathbb{H}_{u}\mathbb{E}$, which projects to the tangent vector at $b$ along $\gamma$. $\mathbb{H}_{u}\mathbb{E}$ is visually orthogonal to the vertical subspace $\mathbb{V}_{u}\mathbb{E}$ along $\mathbb{F}$. The structure group $\mathbb{G}$ acts on $u$ from the right, mapping it to another point on the same fibre. (b) A physical realization of parallel transport in the Hilbert space of the density matrix $\rho$ and its amplitude $\omega$, with the structure group $\mathbb{U}(N)$.
• Figure 3: The visualization of gauge transformation for parallel transport on the principal bundle $\mathcal{H}_{\omega}\rightarrow \mathcal{H}_{\rho}$, where $\tilde{\gamma}'=\tilde{\gamma}U_{g}(0)$ and $\sqrt{\rho(t)}$ is a global section.
• Figure 4: Schematic diagram of dissipation-controlled quantum quench. (a) The effective short-time evolution governed by $H_{eff}$ before the quantum jump $2L_{\mu}\rho L_{\mu}^{\dagger}$, derived from the Lindblad master equation. (b) The flux-controlled Lindblad operators realizing the quantum quench in the non-Hermitian SSH model.
• Figure 5: Schematic of the simple circuit design for the non-Hermitian SSH model. Resistors $R_{1,2}$ and grounded inductors $L$ are connected such that the nodes correspond to the sublattices $A$ and $B$ at each site $n$. The non-reciprocal hoppings are realized via INICs.