Non-Hermitian topological phases and dynamical quantum phase transitions: a generic connection

We start with a non-Hermitian Hamiltonian H ≠ H^†, which describes particles in a 1D lattice subjecting to gains, losses and/or nonreciprocal effects. Under the periodic boundary condition, we can express the Hamiltonian of the system as $H={\sum }_{k}{{\Psi}}_{k}^{{\dagger}}H\left(k\right){{\Psi}}_{k}$, where k ∈ [−π, π) is the quasimomentum, ${{\Psi}}_{k}^{{\dagger}}$ (Ψ_k ) is the two-component creation (annihilation) operator in momentum representation, and the Bloch Hamiltonian H(k) takes the general form

$$\begin{equation}H\left(k\right)=\left[{h}_{a}\left(k\right)-\mathrm{i}{g}_{a}\left(k\right)\right]{\sigma }_{a}+\left[{h}_{b}\left(k\right)-\mathrm{i}{g}_{b}\left(k\right)\right]{\sigma }_{b}.\end{equation} \tag{ 1 }$$

Here h_a,b (k) and g_a,b (k) are real-valued functions of the quasimomentum k, i denotes the imaginary unit, and σ_a,b are any two of the three Pauli matrices σ_x , σ_y , and σ_z , with {σ_a , σ_b } = 0 for a ≠ b. We will also denote the 2 × 2 identity matrix as σ₀.

The non-Hermiticity of H implies that H(k) ≠ H^†(k) at a generic quasimomentum k. The dispersion relation of non-Hermitian Bloch Hamiltonian H(k) is given by

$$\begin{align}{E}_{{\pm}}\left(k\right)=& {\pm}\sqrt{{\left[{h}_{a}\left(k\right)-\mathrm{i}{g}_{a}\left(k\right)\right]}^{2}+{\left[{h}_{b}\left(k\right)-\mathrm{i}{g}_{b}\left(k\right)\right]}^{2}}\\ =& {\pm}E\left(k\right).\end{align} \tag{ 2 }$$

It is clear that ±E(k) are in general complex numbers. The spectrum of H(k) becomes gapless at zero energy if E(k) = 0. According to equation (2), this is achieved when both the following conditions are satisfied

$$\begin{equation}{h}_{a}^{2}\left(k\right)+{h}_{b}^{2}\left(k\right)-{g}_{a}^{2}\left(k\right)-{g}_{b}^{2}\left(k\right)=0,\end{equation} \tag{ 3 }$$

$$\begin{equation}{h}_{a}\left(k\right){g}_{a}\left(k\right)+{h}_{b}\left(k\right){g}_{b}\left(k\right)=0.\end{equation} \tag{ 4 }$$

By solving these equations, we could obtain the quasimomentum k₀ at which the spectrum gap closes, and find the boundaries separating different gapped phases in the parameter space, which could also be the boundaries among different bulk topological phases of the system.

To characterize the topological properties of the gapped phases of H(k) (i.e., E(k) ≠ 0 for all k), the usual recipe is to identify the symmetries of the system and construct the relevant topological invariants. From the commutation relation of Pauli matrices [σ_a , σ_b ] = 2i_abc σ_c , it is clear that the H(k) in equation (1) possesses the chiral (sublattice) symmetry $\mathcal{S}={\sigma }_{c}$ (c ≠ a, b), in the sense that ${\mathcal{S}}^{2}={\sigma }_{0}$ and $\mathcal{S}H\left(k\right)\mathcal{S}=-H\left(k\right)$. The bulk topological phases of a non-Hermitian Bloch Hamiltonian with sublattice symmetry $\mathcal{S}$ can usually be characterized by a winding number w, defined as

$$\begin{equation}w=\underset{-\pi }{\overset{\pi }{\int }}\frac{\mathrm{d}k}{2\pi }{\partial }_{k}\phi \left(k\right),\qquad \phi \left(k\right)\equiv \mathrm{arctan}\left[\frac{{h}_{b}\left(k\right)-\mathrm{i}{g}_{b}\left(k\right)}{{h}_{a}\left(k\right)-\mathrm{i}{g}_{a}\left(k\right)}\right],\end{equation} \tag{ 5 }$$

which describes the accumulated change of winding angle ϕ(k) throughout the first Brillouin zone (BZ). Note that the value of w is always real even though ϕ(k) is in general complex, as the imaginary part of ϕ(k) has no winding in the first BZ (see reference [43] for a proof). Furthermore, the w as defined in equation (5) can take either integer or half-integer values, depending on the relative locations between the EPs of H(k) on the h_a –h_b plane and the trajectory of vector h(k) = [h_a (k), h_b (k)] versus k. That is, if h(k) encircles an even (odd) number of EPs, we would have $w\in \mathbb{Z}$ ($w\in \left(2\mathbb{Z}+1\right)/2$) [43]. Within a gapped topological phase of H(k), the value of w is a constant, whereas it takes a quantized (or half-quantized jump) when a phase boundary determined by equations (3) and (4) is crossed. Therefore, the invariant w yields a characterization for all the bulk non-Hermitian topological phases of H(k). Experimentally, the winding number w can be obtained by measuring the mean chiral displacements of wave packets [40] or the dynamical winding numbers of time-averaged spin textures [43].

DQPTs are characterized by nonanalytic behaviors of system observables as functions of time. They are usually found in the dynamics following a quench across the equilibrium phase transition point of a quantum many-body system (see references [48–51] for reviews). The central object in the description of DQPTs is the return amplitude G(t) = ⟨Ψ|U(t)|Ψ⟩, where |Ψ⟩ is the initial many-particle state (usually taken as the equilibrium ground state of the system before the quench) and U(t) is the evolution operator of the system following a quantum quench (or some other nonequilibrium protocols). Formally, G(t) mimics the dynamical partition function of the post-quench evolution. When G(t) = 0 at a critical time t_c, the initial state evolves into its orthogonal state. The rate function of return probability g(t) = −lim_N→∞ N⁻¹ ln|G(t)|² (N is the number of degrees of freedom of the system) or its time derivatives would then become nonanalytic at t = t_c, signifying a DQPT. Accompanying theoretical discoveries [53–55], DQPTs have been observed in cold atoms [56–59], trapped ions [60, 61], superconducting qubits [62], nanomechanical and photonic systems [63–65]. Recent studies further extend DQPTs to periodically driven (Floquet) systems [66–71], accompanied by an experimental realization in the NV center setup [66].

To relate DQPTs with topological phases in non-Hermitian systems, we focus on a unique class of quench protocol, in which the system is initialized with equal populations but no coherence on the two bands of the non-Hermitian Bloch Hamiltonian H(k) in equation (1), i.e., an infinite-temperature initial state ∏_k∈BZ ρ₀, with ρ₀ = σ₀/2 being the single-particle density matrix. The evolution of ρ₀ at time t > 0 is governed by H(k), and the return amplitude G(k, t), defined as the expectation value of evolution operator U(k, t) = e^−iH(k)t over the initial state ρ₀ reads

$$\begin{equation}G\left(k,t\right)=\mathrm{Tr}\left[{\rho }_{0}U\left(k,t\right)\right]=\mathrm{cos}\left[E\left(k\right)t\right],\end{equation} \tag{ 6 }$$

where equations (1) and (2) have been used to reach the second equality. When possible DQPTs happen, we would have cos[E(k)t] = 0, leading to the critical times

$$\begin{equation}{t}_{n}\left(k\right)=\left(n-\frac{1}{2}\right)\frac{\pi }{E\left(k\right)},\quad n\in \mathbb{Z}.\end{equation} \tag{ 7 }$$

This seemingly innocent expression yields rather different predictions for Hermitian and non-Hermitian systems. In a Hermitian system, where the dispersion relation E(k) is always real and positive, we would have a set of critical times t_n (k) for each quasimomentum k. However, the resulting non-analyticity in the rate function g(t) is simply originated from the oscillatory dynamics of a single Bloch state rather then an actual phase transition, which only happens in thermodynamic limit (N → ∞). On the other hand, when H(k) is non-Hermitian, we have $E\left(k\right)\in \mathbb{C}$ in general, and real critical times t_n (k) emerge only at the critical momenta k_c where $E\left({k}_{\mathrm{c}}\right)\in \mathbb{R}$. According to equation (2), this is equivalent to the fulfillment of the following two conditions:

$$\begin{equation}{h}_{a}^{2}\left(k\right)+{h}_{b}^{2}\left(k\right)-{g}_{a}^{2}\left(k\right)-{g}_{b}^{2}\left(k\right){ >}0,\end{equation} \tag{ 8 }$$

$$\begin{equation}{h}_{a}\left(k\right){g}_{a}\left(k\right)+{h}_{b}\left(k\right){g}_{b}\left(k\right)=0,\end{equation} \tag{ 9 }$$

which only yield solutions at isolated values of k. For a critical momentum k_c satisfying both the equations (8) and (9), we would have G(k_c, t_n ) = 0 for all $n\in \mathbb{Z}$. In the thermodynamic limit, the rate function of return probability for the many-particle initial state ∏_k∈BZ ρ₀ is given by

$$\begin{equation}g\left(t\right)=-\underset{N\to \infty }{\mathrm{lim}}\;\frac{1}{N}\enspace \mathrm{ln}\vert G\left(k,t\right){\vert }^{2}=-{\int }_{\mathrm{B}\mathrm{Z}}\frac{\mathrm{d}k}{2\pi }\enspace \mathrm{ln}\vert G\left(k,t\right){\vert }^{2},\end{equation} \tag{ 10 }$$

which will have discontinuous first-order time derivatives at all t_n (k_c). Note that by taking the limit N → ∞, the distribution of t_n (k) on the complex time plane changes from isolated points to a continuous line, whose crossings along the real-time axis correspond to the critical time of genuine DQPTs in the sense of Fisher zeros [48].

The connection between DQPTs and NHTPs in chiral-symmetric 1D systems becomes transparent at this stage. First, we note that the equation (4), which determines the gapless quasimomenta {k₀} is identical to equation (9). This implies that the critical momenta {k_c} of DQPTs can only be a subset of {k₀}. Second, plugging {k₀} into equation (3), we obtain an expression for the boundaries separating different NHTPs in the parameter space. Combining this with equation (8) further suggests that DQPTs can only be observed in certain regimes that are distinguished from the others by the topological phase boundaries of H(k). Third, since the number of gapless quasimomenta k₀ is closely related to the change of topological invariant w in equation (5) across the corresponding topological phase transition point, DQPTs with different numbers of k_c are expected to happen following quenches to different NHTPs. As each critical momentum k_c determines a unique period T(k_c) = π/E(k_c) for the DQPTs, the number of critical period T(k_c) is determined by the number of distinct critical momenta. The third point then suggests a way to distinguish different NHTPs through the quantitative difference of the critical time-periods of DQPTs therein.

Putting together, we have uncovered an intrinsic relation between the topological phases and DQPTs in non-Hermitian systems, which not only bridges the gap between these two diverse fields, but also provides a way to probe the NHTPs through nonequilibrium dynamics. To make the connection more explicit, we will study the DQPTs in a couple of prototypical 1D non-Hermitian lattice models in the following section. Besides the rate function g(t), we will also investigate the real-valued, noncyclic geometric phase of the return amplitude G(k, t) [52], which is defined as

$$\begin{equation}{{\Phi}}_{\mathrm{G}}\left(k,t\right)={\Phi}\left(k,t\right)-{{\Phi}}_{\mathrm{D}}\left(k,t\right),\end{equation} \tag{ 11 }$$

where the total phase

$$\begin{equation}{\Phi}\left(k,t\right)\equiv -\mathrm{i}\enspace \mathrm{ln}\frac{G\left(k,t\right)}{\vert G\left(k,t\right)\vert },\end{equation} \tag{ 12 }$$

and the dynamical phase

$$\begin{equation}{{\Phi}}_{\mathrm{D}}\left(k,t\right)\equiv -{\int }_{0}^{t}\mathrm{d}{t}^{\prime }\mathrm{R}\mathrm{e}\left\{\frac{\mathrm{Tr}\left[{\tilde {U}}^{{\dagger}}\left(k,{t}^{\prime }\right){\rho }_{0}U\left(k,{t}^{\prime }\right)H\left(k\right)\right]}{\mathrm{Tr}\left[{\tilde {U}}^{{\dagger}}\left(k,{t}^{\prime }\right){\rho }_{0}U\left(k,{t}^{\prime }\right)\right]}\right\}\end{equation} \tag{ 13 }$$

(see appendix A for more details about these phase factors). The noncyclic geometric phase has been shown to contain important information about DQPTs in both Hermitian [72–76] and non-Hermitian [52] systems. At a given time, the winding number of the geometric phase in the first BZ can be further employed to construct a dynamical topological order parameter (DTOP), which is defined as

$$\begin{equation}\nu \left(t\right)=\int \frac{\mathrm{d}k}{2\pi }{\partial }_{k}{{\Phi}}_{\mathrm{G}}\left(k,t\right).\end{equation} \tag{ 14 }$$

It takes a quantized jump whenever the evolution of the system passes through a critical time of the DQPT. Note that the range of integration over k depends on the symmetry of Φ_G(k, t) in k-space. For example, if Φ_G(k, t) has the inversion symmetry with respect to k = 0, i.e., Φ_G(k, t) = Φ_G(−k, t), we can perform the integral over a reduced BZ with k ∈ [0, π] in the evaluation of ν in equation (14).

We first consider a non-Hermitian variant of the Kitaev chain, which describes a 1D topological superconductor with onsite particle loss. In momentum representation, the Hamiltonian of the model takes the form $H=\frac{1}{2}{\sum }_{k\in \mathrm{B}\mathrm{Z}}{{\Psi}}_{k}^{{\dagger}}H\left(k\right){{\Psi}}_{k}$, where ${{\Psi}}_{k}^{{\dagger}}=\left({c}_{k}^{{\dagger}},{c}_{-k}\right)$ is the Nambu spinor operator and ${c}_{k}^{{\dagger}}$ is the creation operator of an electron with quasimomentum k. The Bloch Hamiltonian H(k) in Nambu basis is given by

$$\begin{equation}H\left(k\right)={h}_{y}\left(k\right){\sigma }_{y}+\left[{h}_{z}\left(k\right)-\mathrm{i}v\right]{\sigma }_{z},\end{equation} \tag{ 15 }$$

where

$$\begin{equation}{h}_{y}\left(k\right)={\Delta}\enspace \mathrm{sin}\enspace k,\quad {h}_{z}\left(k\right)=u+J\enspace \mathrm{cos}\enspace k.\end{equation} \tag{ 16 }$$

Here the real parameters J, Δ and u denote the nearest-neighbor hopping amplitude, superconducting pairing amplitude and chemical potential. $v\in \mathbb{R}$ characterizes the strength of onsite particle loss. Following the discussions of subsection 2.1, we see that H(k) possesses the sublattice symmetry $\mathcal{S}={\sigma }_{x}$, i.e., $\mathcal{S}H\left(k\right)\mathcal{S}=-H\left(k\right)$. Furthermore, it also has the generalized particle–hole symmetry $\mathcal{C}={\sigma }_{x}$ and time-reversal symmetry $\mathcal{T}={\sigma }_{0}$, in the sense that $\mathcal{C}{H}^{\top }\left(k\right){\mathcal{C}}^{-1}=-H\left(-k\right)$ and $\mathcal{T}{H}^{\top }\left(k\right){\mathcal{T}}^{-1}=H\left(-k\right)$, where ⊤ performs matrix transposition. H(k) thus belongs to an extension of the symmetry class BDI in the periodic table of non-Hermitian topological phases [14]. In the meantime, H(k) possesses the inversion symmetry $\mathcal{P}={\sigma }_{z}$ as $\mathcal{P}H\left(k\right){\mathcal{P}}^{-1}=H\left(-k\right)$, which guarantees the correspondence between its bulk topological invariant w (as defined in equation (5)) and the number of Majorana edge modes under the open boundary condition [14].

According to subsection 2.1, the complex energy spectrum of LKC takes the form

$$\begin{equation}{E}_{{\pm}}\left(k\right)={\pm}\sqrt{{h}_{y}^{2}\left(k\right)+{\left[{h}_{z}\left(k\right)-\mathrm{i}v\right]}^{2}}={\pm}E\left(k\right),\end{equation} \tag{ 17 }$$

which will become gapless when

$$\begin{equation}{\Delta}\enspace \mathrm{sin}\enspace k={\pm}v,\end{equation} \tag{ 18 }$$

$$\begin{equation}u+J\enspace \mathrm{cos}\enspace k=0.\end{equation} \tag{ 19 }$$

Combining these equations, we find the gapless quasimomenta

$$\begin{equation}{\pm}{k}_{0}={\pm}\mathrm{arccos}\left(-u/J\right)\end{equation} \tag{ 20 }$$

for |u| < |J|, and the boundary between different NHTPs as

$$\begin{equation}\frac{{u}^{2}}{{J}^{2}}+\frac{{v}^{2}}{{{\Delta}}^{2}}=1.\end{equation} \tag{ 21 }$$

Geometrically, the trajectory of vector h(k) ≡ [h_y (k), h_z (k)] forms an ellipse on the h_y –h_z plane, which is centered at (0, u). When the gapless condition equation (21) is satisfied, the spectrum E_±(k) hold a pair of EPs at (±v, 0) on the h_y –h_z plane, which are passed through by the vector h(k). Whether the two EPs are encircled or not by the trajectory of h(k) when k scans through the first BZ then distinguishes two possible NHTPs. With equations (17) and (21), it is not hard to show that when u²/J² + v²/Δ² < 1 (>1), the two EPs are encircled (not encircled) by the trajectory of h(k). The topological invariant that distinguish these two phases has the form of equation (5), where the winding angle ϕ(k) for the LKC is explicitly given by

$$\begin{equation}\phi \left(k\right)=\mathrm{arctan}\left[\frac{{h}_{z}\left(k\right)-\mathrm{i}v}{{h}_{y}\left(k\right)}\right].\end{equation} \tag{ 22 }$$

In figure 1, we show the topological phase diagram of the LKC versus the real and imaginary parts of chemical potential u and v, with J = Δ = 1. The winding numbers equation (5) of the non-Hermitian topological and trivial phases are found to be w = 1 and w = 0 for u²/J² + v²/Δ² < 1 and >1, respectively. A loss-induced topological phase transition, which is unique to non-Hermitian systems, can be observed with the increase of v.

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To link the NHTPs of LKC with the DQPTs, we employ the protocol introduced in subsection 2.2, with the initial state ρ₀ = σ₀/2 and the dynamics being governed by the Hamiltonian H(k) in equation (15). According to equation (6), the return amplitude at a later time t > 0 is given by G(k, t) = cos[E(k)t], where E(k) is the dispersion relation of LKC in equation (17). From equations (7) and (9), we find the critical momenta and times to be

$$\begin{equation}{\pm}{k}_{\mathrm{c}}={\pm}\mathrm{arccos}\left(-u/J\right)={\pm}{k}_{0},\end{equation} \tag{ 23 }$$

$$\begin{equation}{t}_{n}\left({\pm}{k}_{\mathrm{c}}\right)=\left(n-\frac{1}{2}\right)\frac{\pi }{\vert {\Delta}\vert \sqrt{1-\left({u}^{2}/{J}^{2}+{v}^{2}/{{\Delta}}^{2}\right)}}.\end{equation} \tag{ 24 }$$

It is clear that when u²/J² + v²/Δ² < 1, there are real solutions of t_n (k_c) for all $n\in \mathbb{Z}$, and the two critical momenta ±k_c are coincide with the gapless quasimomenta ±k₀, yielding the same critical period $T\left({k}_{\mathrm{c}}\right)=\pi /\sqrt{{{\Delta}}^{2}\left(1-{u}^{2}/{J}^{2}\right)-{v}^{2}}$. On the other hand, there is no critical momenta and t_n is always imaginary when u²/J² + v²/Δ² > 1, yielding no DQPTs at any real time t. When u²/J² + v²/Δ² = 1, which corresponds to a gapless post-quench phase, we will have t_n (k_c) → ∞ for any solutions of critical momenta ±k_c, and the resulting DQPTs are not observable. For completeness, we numerically compute the return rates and geometric phases with the help of equations (10) and (11) for the cases with and without DQPTs in figures 2(a)–(d), respectively. As expected, DQPTs are only observed when the system is quenched to a nontrivial NHTP with the winding number w = 1.

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Combining the analysis in this subsection, we obtain a one-to-one correspondence between the NHTPs and DQPTs in the LKC, which is summarized in table 1. This connection not only unifies the NHTPs and DQPTs in the system, but also provides a way to dynamically distinguishing the different NHTPs of LKC and detecting the gapless quasimomenta, as exemplified by figures 2(a) and (c).

Table 1. The connection between NHTPs and DQPTs of the LKC model. The notation '/' in the table means 'ill-defined'. k₀ refers to the gapless quasimomenta of the model.

Condition	Geometric picture	Winding number	Critical time and momenta
$\frac{{u}^{2}}{{J}^{2}}+\frac{{v}^{2}}{{{\Delta}}^{2}}{< }1$	Two EPs are encircled by h(k)	w = 1	DQPTs at tn (kc)$\forall n\in \mathbb{Z}$, kc = k0
$\frac{{u}^{2}}{{J}^{2}}+\frac{{v}^{2}}{{{\Delta}}^{2}}=1$	Two EPs are crossed by h(k)	/	/
$\frac{{u}^{2}}{{J}^{2}}+\frac{{v}^{2}}{{{\Delta}}^{2}}{ >}1$	No EPs are encircled by h(k)	w = 0	No kc and tn , no DQPTs

We next consider the LKC with NNN hoppings and pairings, which could possesses NHTPs with larger topological invariants. In the momentum space and Nambu spinor basis, the NNN LKC is described by the Hamiltonian $H=\frac{1}{2}{\sum }_{k\in \mathrm{B}\mathrm{Z}}{{\Psi}}_{k}^{{\dagger}}H\left(k\right){{\Psi}}_{k}$, where H(k) takes the same form as equation (15), with

$$\begin{align}{h}_{y}\left(k\right)& ={{\Delta}}_{1}\enspace \mathrm{sin}\enspace k+{{\Delta}}_{2}\enspace \mathrm{sin}\enspace 2k,\\ {h}_{z}\left(k\right)& =u+{J}_{1}\enspace \mathrm{cos}\enspace k+{J}_{2}\enspace \mathrm{cos}\enspace 2k.\end{align} \tag{ 25 }$$

Here u is the real part of chemical potential, (J₁, Δ₁) and (J₂, Δ₂) are the nearest-neighbor and next-nearest-neighbor hopping and pairing amplitudes, respectively. It is not hard to verify that the H(k) here belongs to the same symmetry class as the LKC, with the same set of time-reversal, particle–hole, sublattice and inversion symmetries. The dispersion relations E_±(k) of H(k) share the same form with equation (17), yielding the gapless conditions

$$\begin{equation}{{\Delta}}_{1}\enspace \mathrm{sin}\enspace k+{{\Delta}}_{2}\enspace \mathrm{sin}\enspace 2k={\pm}v,\end{equation} \tag{ 26 }$$

$$\begin{equation}u+{J}_{1}\enspace \mathrm{cos}\enspace k+{J}_{2}\enspace \mathrm{cos}\enspace 2k=0.\end{equation} \tag{ 27 }$$

By solving equation (27), we could obtain at most four possible gapless quasimomenta ${\pm}{k}_{0}^{{\pm}}$ as

$$\begin{equation}{\pm}{k}_{0}^{{\pm}}={\pm}\mathrm{arccos}\left[\frac{-{J}_{1}{\pm}\sqrt{{J}_{1}^{2}+8{J}_{2}\left({J}_{2}-u\right)}}{4{J}_{2}}\right].\end{equation} \tag{ 28 }$$

According to equation (26), the boundary between different NHTPs is then determined by

$$\begin{equation}\enspace \mathrm{sin}\enspace {k}_{0}^{{\pm}}\left({{\Delta}}_{1}+2{{\Delta}}_{2}\enspace \mathrm{cos}\enspace {k}_{0}^{{\pm}}\right)={\pm}v.\end{equation} \tag{ 29 }$$

The explicit expression of the phase boundary in terms of system parameters is tedious, and will be left for numerical calculations. Geometrically, the trajectory of real vector h(k) = [h_y (k), h_z (k)] has the shape of a centered trochoid, which could encircle the two EPs of E(k) at (±v, 0) twice, once or zero times when k is scanned over the first BZ. These three possibilities then distinguish three different NHTPs, which are characterized by the topological winding number w in equation (5). In figure 3, we present the topological phase diagram of the NNN LKC model versus the real and imaginary parts of chemical potential u and v, with other system parameters set as J₁ = Δ₁ = 1 and J₂ = Δ₂ = 1.5. The three topological phases are discriminated by different colored regions in the figure, with the phase boundary curve (black solid line) determined by equation (29), and the value of w denoted explicitly within each phase. The quantized changes of w with the increase of the lossy strength v again signify non-Hermiticity induced topological phase transitions in the system.

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To build the connection between the NHTPs and the DQPTs of NNN LKC, we again employ the protocol introduced in subsection 2.2, with the initial state ρ₀ = σ₀/2 and the dynamics being governed by the Hamiltonian H(k) of the NNN LKC. The return amplitude at a later time t > 0 is then given by G(k, t) = cos[E(k)t], with E(k) being the dispersion of NNN LKC. The critical momenta and time are further obtained from equations (7) and (9) as

$$\begin{equation}{\pm}{k}_{\mathrm{c}}^{{\pm}}={\pm}\mathrm{arccos}\left[\frac{-{J}_{1}{\pm}\sqrt{{J}_{1}^{2}+8{J}_{2}\left({J}_{2}-u\right)}}{4{J}_{2}}\right]={\pm}{k}_{0}^{{\pm}},\end{equation} \tag{ 30 }$$

$$\begin{equation}{t}_{n}\left({k}_{\mathrm{c}}^{{\pm}}\right)=\left(n-\frac{1}{2}\right)\frac{\pi }{\sqrt{{\left({{\Delta}}_{1}\enspace \mathrm{sin}\enspace {k}_{\mathrm{c}}^{{\pm}}+{{\Delta}}_{2}\enspace \mathrm{sin}\enspace 2{k}_{\mathrm{c}}^{{\pm}}\right)}^{2}-{v}^{2}}},\end{equation} \tag{ 31 }$$

where $n\in \mathbb{Z}$. In parallel with the discussions of subsection 3.1, we could summarize the relationship between NHTPs and DQPTs in the NNN LKC model by table 2. Again, we obtain a one-to-one correspondence between these two nonequilibrium phenomena, which also provides us with a way to detect NHTPs with large winding numbers and to locate the phase boundaries between them.

Table 2. The connection between NHTPs and DQPTs of the NNN LKC model.

Condition	Geometric picture	Winding number	Critical times and momenta
${h}_{y}^{2}\left({k}_{\mathrm{c}}^{{\pm}}\right){ >}{v}^{2}$	Two EPs are encircled twice by h(k)	w = 2	DQPTs at ${t}_{n}\left({k}_{\mathrm{c}}^{{\pm}}\right)$ $\forall \enspace n\in \mathbb{Z}$, ${k}_{\mathrm{c}}^{{\pm}}={k}_{0}^{{\pm}}$
${h}_{y}^{2}\left({k}_{\mathrm{c}}^{+/-}\right){ >}{v}^{2}$ & ${h}_{y}^{2}\left({k}_{\mathrm{c}}^{-/+}\right){< }{v}^{2}$	Two EPs are encircled once by h(k)	w = 1	DQPTs at ${t}_{n}\left({k}_{\mathrm{c}}^{+/-}\right)$ $\forall \enspace n\in \mathbb{Z}$, ${k}_{\mathrm{c}}^{{\pm}}={k}_{0}^{{\pm}}$
${h}_{y}^{2}\left({k}_{\mathrm{c}}^{{\pm}}\right){< }{v}^{2}$	No EPs are encircled by h(k)	w = 0	No kc and tn , no DQPTs

For completeness, we present three numerical examples for the DQPTs in the NNN LKC in figure 4. The system parameters are chosen as J₁ = Δ₁ = 1, J₂ = Δ₂ = 1.5, u = 0.5 and v = 0.4, 1.4, 2.4 for figure panels 4(a), (d), 4(b), (e) and 4(c), (f). In figures 4(a) and (d) the post-quench system is in an NHTP with w = 2, and DQPTs are observed at two different sets of critical periods $T\left({k}_{\mathrm{c}}^{{\pm}}\right)=\pi /\sqrt{{h}_{y}^{2}\left({k}_{\mathrm{c}}^{{\pm}}\right)-{v}^{2}}$ of g(t) in figure 4(a), with the two pairs of critical momenta ${\pm}{k}_{\mathrm{c}}^{{\pm}}$ given by equation (30) and imaged by the 2π-jumps of geometric phase Φ_G(k, t) in figure 4(d). In figures 4(b) and (e), the post-quench system is in an NHTP with w = 1, and DQPTs are repeated at only one critical period $T\left({k}_{\mathrm{c}}^{+}\right)=\pi /\sqrt{{h}_{y}^{2}\left({k}_{\mathrm{c}}^{+}\right)-{v}^{2}}$ of g(t) in figure 4(b), with 2π-jumps of geometric phase Φ_G(k, t) observed at the critical momenta ${\pm}{k}_{\mathrm{c}}^{+}$ in figure 4(e). In figures 4(c) and (f), the post-quench system is in a trivial phase with w = 0, and no signatures of DQPTs are observed in the rate function g(t) and geometric phase Φ_G(k, t). Putting together, our numerical results confirm the connection between the NHTPs and DQPTs of the NNN LKC model, as summarized in table 2. Furthermore, the results presented here should be directly extendable to non-Hermitian models in the same symmetry class as the LKC, but with even longer-range hopping and pairing amplitudes.

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In the last part of this section, we consider a nonreciprocal variant of the SSH model, which possesses a different set of symmetries compared with the LKC. In momentum representation, the Hamiltonian of NRSSH model takes the form $H={\sum }_{k\in \mathrm{B}\mathrm{Z}}{{\Psi}}_{k}^{{\dagger}}H\left(k\right){{\Psi}}_{k}$, where ${{\Psi}}_{k}^{{\dagger}}=\left({a}_{k}^{{\dagger}},{b}_{k}^{{\dagger}}\right)$ is the creation operator on the two sublattices a and b of the SSH model, and k ∈ [−π, π) is the quasimomentum. The Bloch Hamiltonian H(k) is explicitly given by

$$\begin{equation}H\left(k\right)={h}_{x}\left(k\right){\sigma }_{x}+\left[{h}_{y}\left(k\right)-\mathrm{i}\gamma \right]{\sigma }_{y},\end{equation} \tag{ 32 }$$

with

$$\begin{equation}{h}_{x}\left(k\right)={J}_{1}+{J}_{2}\enspace \mathrm{cos}\enspace k,\quad {h}_{y}\left(k\right)={J}_{2}\enspace \mathrm{sin}\enspace k.\end{equation} \tag{ 33 }$$

Here J₁ ± γ and J₂ are the intracell and intercell hopping amplitudes. A finite γ makes the intracell hopping asymmetric, leading to a non-Hermitian H(k). From now on, we assume J₂, γ > 0 without loss of generality. It is clear that the system possesses the sublattice symmetry $\mathcal{S}={\sigma }_{z}$, in the sense that $\mathcal{S}H\left(k\right)\mathcal{S}=-H\left(k\right)$. This allows us to characterize the bulk topological phases of H(k) by the winding number w. Moreover, H(k) has the time reversal symmetry $\mathcal{T}={\sigma }_{0}$ and particle-hole symmetry $\mathcal{C}={\sigma }_{z}$, i.e., $\mathcal{T}{H}^{{\ast}}\left(k\right){\mathcal{T}}^{-1}=H\left(-k\right)$ and $\mathcal{C}{H}^{{\ast}}\left(k\right){\mathcal{C}}^{-1}=-H\left(-k\right)$. Therefore, the NRSSH model belongs to the same BDI symmetry class as the Hermitian SSH model. Nevertheless, H(k) in equation (32) does not have the inversion symmetry of the Hermitian SSH model, but instead possesses the PT-symmetry, i.e., $\mathcal{P}\mathcal{T}{H}^{{\ast}}\left(k\right){\left(\mathcal{P}\mathcal{T}\right)}^{-1}=H\left(k\right)$. This allows the bulk spectrum of H(k) to be very different under periodic and open boundary conditions, leading to the breakdown of conventional bulk-boundary correspondence [79].

The bulk spectrum of H(k) takes the form

$$\begin{equation}{E}_{{\pm}}\left(k\right)={\pm}\sqrt{{h}_{x}^{2}\left(k\right)+{\left[{h}_{y}\left(k\right)-\mathrm{i}\gamma \right]}^{2}}={\pm}E\left(k\right).\end{equation} \tag{ 34 }$$

With equation (33), we see that the dispersion is gapless at zero energy when the following two conditions are met

$$\begin{equation}{J}_{1}{\pm}{J}_{2}={\pm}\gamma ,\end{equation} \tag{ 35 }$$

$$\begin{equation}\mathrm{sin}\enspace k=0,\end{equation} \tag{ 36 }$$

which directly yield the phase boundary curves and the gapless quasimomenta k₀ = 0, π. Geometrically, the trajectory of vector h(k) = [h_x (k), h_y (k)] forms a circle with radius J₂ and centered at (J₁, 0) on the h_x –h_y plane, while the EPs of the spectrum are located at (±γ, 0). When J₁ − J₂ < −γ and J₁ + J₂ > γ, both two EPs are encircled by h(k) when k scans over the first BZ. When J₁ − J₂ < −γ (J₁ + J₂ > γ) and |J₁ + J₂| < γ (|J₁ − J₂| < γ), the EP at (−γ, 0) ((γ, 0)) is encircled by h(k). Otherwise no EPs are encircled by h(k). These three different situations distinguish three different types of bulk non-Hermitian topological phases, with each of them being characterized by the topological invariant w in equation (5), where the winding angle

$$\begin{equation}\phi \left(k\right)=\mathrm{arctan}\left[\frac{{h}_{y}\left(k\right)-\mathrm{i}\gamma }{{h}_{x}\left(k\right)}\right].\end{equation} \tag{ 37 }$$

In figure 5, we present the topological phase diagram of the NRSSH model versus the intercell hopping amplitude J₂ and asymmetric hopping parameter γ, with the intracell hopping amplitude J₁ = 0.5. Each colored region in the phase diagram corresponds to an NHTP, with the value of topological winding number w denoted therein. The phase boundaries separating different regions are determined by equation (35). Despite non-Hermiticity-induced topological phase transitions, the NRSSH model also features a unique NHTP with winding number w = 1/2, which corresponds to the case in which only a single EP is encircled by h(k).

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The connection between DQPTs and NHTPs in the NRSSH model can be built as follows. Choosing the initial state to be ρ₀ = σ₀/2 as in subsection 2.2, the dynamics of the system at t > 0 is governed by the Hamiltonian H(k) of the NRSSH model. The return amplitude at time t is given by G(k, t) = cos[E(k)t], with E(k) being the dispersion of H(k). The critical momenta and time are then obtained from equations (7) and (9), i.e.,

$$\begin{equation}{k}_{\mathrm{c}}=\left(0,\pi \right)={k}_{0},\end{equation} \tag{ 38 }$$

$$\begin{equation}{t}_{n}\left(0\right)=\left(n-\frac{1}{2}\right)\frac{\pi }{\sqrt{\left({J}_{1}+{J}_{2}-\gamma \right)\left({J}_{1}+{J}_{2}+\gamma \right)}}\equiv {t}_{n}^{0},\end{equation} \tag{ 39 }$$

$$\begin{equation}{t}_{n}\left(\pi \right)=\left(n-\frac{1}{2}\right)\frac{\pi }{\sqrt{\left({J}_{1}-{J}_{2}-\gamma \right)\left({J}_{1}-{J}_{2}+\gamma \right)}}\equiv {t}_{n}^{\pi },\end{equation} \tag{ 40 }$$

where $n\in \mathbb{Z}$. Combining these equations with the gapless conditions in equations (35) and (36), we could immediately identify the relationship between NHTPs and DQPTs in the NRSSH model, as listed in table 3.

Table 3. The connection between NHTPs and DQPTs of the NRSSH model.

Condition	Geometric picture	Winding number	Critical times and momenta
J1 − J2 < −γ & J1 + J2 > γ	Two EPs are encircled by h(k)	w = 1	DQPTs at ${t}_{n}^{0,\pi }$ $\forall n\in \mathbb{Z}$, kc = 0, π
J1 − J2 < −γ & |J1 + J2| < γ	One EP is encircled by h(k)	w = 1/2	DQPTs at ${t}_{n}^{\pi }$ $\forall \enspace n\in \mathbb{Z}$, kc = π
J1 + J2 > γ & |J1 − J2| < γ			DQPTs at ${t}_{n}^{0}$ $\forall \enspace n\in \mathbb{Z}$, kc = 0
|J1 ± J2| < γ	No EPs are encircled by h(k)	w = 0	No kc and tn , no DQPTs
|J1 ± J2| > γ			DQPTs at ${t}_{n}^{0,\pi }$ $\forall \enspace n\in \mathbb{Z}$, kc = 0, π

From the table, we observe that since there is only a single critical momentum for the NHTPs with w = 1/2, there is also a unique set of critical times (${t}_{n}^{0}$ or ${t}_{n}^{\pi }$ for $n\in \mathbb{Z}$) for the DQPTs in this case. This is in contrast with the NHTPs having w = 1, for which both k_c = k₀ = 0 and π are the critical momenta, and DQPTs at two different critical time periods T(k_c = 0, π) are expected in the post-quench dynamics. In the meantime, we also observe an anomalous case as shown in the last row of table 3. In this case, DQPTs are found when the post-quench system is in a trivial phase with w = 0. Therefore, even though a nontrivial topological phase of the NRSSH model always lead to a unique set of DQPTs following the quench to that phase, the reverse is not true in general. Such a breakdown of the one-to-one correspondence between the DQPTs and NHTPs in the NRSSH model might be due to the absence of inversion symmetry, as compared with the situations in the LKC and its NNN extension. Nevertheless, the most intriguing phase of the NRSSH model, i.e., the one with w = 1/2 can still be distinguished from the other phases through the DQPTs. Therefore, the connection between NHTPs and DQPTs we discovered can still be used as a powerful tool to probe the details of the NHTPs in the NRSSH model.

For completeness, we present the DQPTs in three typical post-quench phases of the NRSSH model in figure 6. In figures 6(a) and (d), the post-quench phase has winding number w = 1, and DQPTs are observed as cusps in the rate function g(t) at two sets of critical times ${t}_{n}^{0,\pi }$. At each critical time, a 2π-jump in the geometric phase Φ_G(k, t) is observed at both the critical momenta k_c = 0, π. In figures 6(b) and (e), the post-quench phase has winding number w = 1/2, and DQPTs are found at a unique set of critical time ${t}_{n}^{0}$, where a 2π-jump in the geometric phase Φ_G(k, t) is observed around k_c = 0. In figures 6(c) and (f), the post-quench phase is trivial and no DQPTs are found in the post-quench dynamics. Putting together, we found that the NHTPs and DQPTs in the NRSSH model are also two closely related phenomena, and the later can be employed to dynamically probe the properties of the former.

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