Published: October 25, 2019

Introduction {#sec1}
============

The exploration of topological matter constitutes a major theme in physics for the past few decades ([@bib22], [@bib48]) With rapid progress in the discovery and understanding of topological phases in solid-state materials, a challenging quest lies in extending the study of conventional topological matter to unconventional regimes. Prominent examples include the investigation of dynamic topological properties in out-of-equilibrium dynamics ([@bib5], [@bib6], [@bib7], [@bib8], [@bib9], [@bib11], [@bib12], [@bib21], [@bib24], [@bib25], [@bib26], [@bib28], [@bib29], [@bib30], [@bib34], [@bib36], [@bib44], [@bib47], [@bib51], [@bib57], [@bib58], [@bib59], [@bib63], [@bib66], [@bib72], [@bib73]) and the characterization of topological phases in non-Hermitian systems ([@bib10], [@bib14], [@bib18], [@bib19], [@bib20], [@bib23], [@bib27], [@bib33], [@bib35], [@bib37], [@bib38], [@bib39], [@bib41], [@bib42], [@bib43], [@bib49], [@bib50], [@bib52], [@bib64], [@bib67], [@bib68], [@bib69], [@bib74], [@bib75]). With the flexible controls afforded by synthetic systems such as ultracold atoms and engineered photonic configurations, the experimental implementation of these interesting scenarios is already within ([@bib15], [@bib16], [@bib31], [@bib46], [@bib54], [@bib56], [@bib62], [@bib65], [@bib70], [@bib71]).

An exemplary situation for the study of topological properties in out-of-equilibrium dynamics is the quantum quench of a topological system, where the ground state of the initial Hamiltonian *H*^i^ is subject to a unitary time evolution governed by the final Hamiltonian *H*^f^. The topological invariant characterizing the instantaneous state is unchanged during the unitary dynamics ([@bib6], [@bib9]), whereas previous studies have revealed the emergence of intriguing phenomena such as dynamic quantum phase transitions (DQPTs) ([@bib5], [@bib16], [@bib24], [@bib25], [@bib26], [@bib29], [@bib32], [@bib58]) and quantized non-equilibrium Hall responses in quench processes ([@bib7], [@bib28], [@bib63]). Furthermore, in a series of recent theoretical and experimental studies, it has been established that dynamic topological invariants can be defined in unitary quantum quenches, which are related to the topology of initial and final Hamiltonians in equilibrium ([@bib21], [@bib56], [@bib59], [@bib66]).

Here arises a series of interesting questions: Are dynamic topological phenomena robust in open systems where dynamics is effectively driven by non-Hermitian Hamiltonians? How does one characterize dynamic topological phenomena in non-Hermitian/non-unitary settings? What new results would non-unitarity bring? These questions are particularly relevant in light of recent studies on topological phenomena in parity-time $\left( {\mathcal{P}\mathcal{T}} \right)$-symmetric non-Hermitian systems ([@bib13], [@bib33], [@bib35], [@bib42], [@bib45], [@bib46], [@bib52], [@bib62], [@bib65], [@bib75]). Under $\mathcal{P}\mathcal{T}$ symmetry, eigenenergies of a non-Hermitan Hamiltonian are entirely real in the $\mathcal{P}\mathcal{T}$-symmetry-preserving regime, in contrast to regimes with spontaneously broken $\mathcal{P}\mathcal{T}$ symmetry ([@bib1], [@bib2], [@bib3]). It has been shown that $\mathcal{P}\mathcal{T}$ symmetry has profound impact on topological properties of static topological phases ([@bib33], [@bib35], [@bib42], [@bib46], [@bib52], [@bib62], [@bib65], [@bib75]) and that the interplay of non-Hermiticity and dynamics in topological systems leads to non-trivial consequences ([@bib50], [@bib70], [@bib71]). However, the role of $\mathcal{P}\mathcal{T}$ symmetry in dynamic topological phenomena associated with non-unitary dynamics has never been explored. A particular difficulty lies in the theoretical description of dynamics generated by non-Hermitian Hamiltonians, which is non-unitary even when the system is in the $\mathcal{P}\mathcal{T}$-symmetry-preserving regime with real eigen spectra. Searching for topological phenomena in dynamics of non-Hermitian systems thus requires theories beyond the conventional paradigm in Hermitian systems under unitary time evolution ([@bib9], [@bib21], [@bib56], [@bib59], [@bib63], [@bib66]).

In this work, we study the quench dynamics of a non-Hermitian Su-Schrieffer-Heeger (SSH) Hamiltonian with $\mathcal{P}\mathcal{T}$ symmetry. We construct a general theoretical framework based on biorthogonal quantum mechanics, which is particularly suitable for characterizing dynamic topological properties in non-Hermitian systems. We then demonstrate that $\mathcal{P}\mathcal{T}$ symmetry plays a key role in the robustness of dynamic topological phenomena, such as dynamic Chern numbers and DQPTs, under non-unitary dynamics. Importantly, these dynamic topological constructions exhibit both qualitative and quantitative differences from their counterparts in unitary dynamics: although the momentum-time submanifolds on which dynamic Chern numbers are defined undergo deformations, an additional timescale exists in the periodic occurrence of DQPTs, which necessitates the definition of two different dynamic topological order parameters. In both these cases, the underlying cause is traced back to fixed points in the non-unitary dynamics, which serve as building elements for the dynamic topological phenomena.

Results {#sec2}
=======

$\mathcal{P}\mathcal{T}$-Symmetric SSH Model {#sec2.1}
--------------------------------------------

As illustrated in [Figure 1](#fig1){ref-type="fig"}A, we consider the non-Hermitian SSH model with alternating gain and loss on adjacent sites under the periodic boundary condition ([@bib42], [@bib55]):$$H = \sum\limits_{j = 1}^{L - 1}\left( {va_{j}^{\dagger}b_{j} + wa_{j + 1}^{\dagger}b_{j} + \text{H}.\text{c}.} \right) + iu\sum\limits_{j = 1}^{L}\left( {a_{j}^{\dagger}a_{j} - b_{j}^{\dagger}b_{j}} \right)\text{,}$$where $a_{j}^{\dagger}$ $\left( b_{j}^{\dagger} \right)$ is the creation operator on *A* (*B*) sub-lattice at site *j*, *L* is the total number of unit cells, the tunneling amplitudes *v*,*w* and the gain-loss rate *u* satisfy *u*,*v*,*w* ≥ 0, and H.c. stands for Hermitian conjugation. Note that without loss of generality, we assume *u*,*v*,*w* to be non-negative real numbers.Figure 1Schematic illustration of the non-Hermitian SSH Hamiltonian and its phase diagram(A) Schematic illustration.(B) Topological phase diagram. The solid line is the topological phase boundary with *w* = *v*. Dashed lines are the boundaries between $\mathit{PT}$-symmetry-preserving and broken regimes with \|*w*−*v*\| = *u*.See also [Figure S1](#mmc1){ref-type="supplementary-material"}.

Hamiltonian (1) possess $\mathcal{P}\mathcal{T}$ symmetry, as $\mathcal{P}\mathcal{T}H\left( {\mathcal{P}\mathcal{T}} \right)^{- 1} = H$, with the parity operator $\mathcal{P}a\left( b \right)_{j}\mathcal{P} = b\left( a \right)_{L + 1 - j}$, and the time reversal operator $\mathcal{T}i\mathcal{T}^{- 1} = - i$, respectively. Note that our definition of the parity operator relies on a cyclic indexing of unit cells such that an inversion center can be defined on the lattice despite the periodic boundary condition. Under $\mathcal{PT}$ symmetry, the eigenspectrum of (1) is entirely real if all eigenstates are simultaneous eigenstates of the $\mathcal{PT}$-symmetry operator. In this case, the Hamiltonian is in the $\mathcal{PT}$-symmetry-preserving regime. Otherwise, the Hamiltonian is in the $\mathcal{PT}$-symmetry-broken regime, where some eigenstates spontaneously break $\mathcal{PT}$ symmetry and acquire imaginary eigenergies. The transition between the $\mathcal{PT}$-symmetry-preserving and broken regimes can be derived by examining the Bloch Hamiltonian *H*~*k*~ = ***h***(*k*)⋅***σ*** at momentum *k*, where ***σ***=(*σ*~1~,*σ*~2~,*σ*~3~) and *σ*~*α*~ (*α* = 1,2,3) are the Pauli matrices. The complex vector ***h***(*k*)=(*h*~1~,*h*~2~,*h*~3~), with *h*~1~ = *w* cos*k* + *v*, *h*~2~ = *w* sin*k*, and *h*~3~ = *iu*. As the eigenenergy is given by $\mathit{\epsilon}_{\mu} = \mu E_{k}$ (*μ* = ±) with $E_{k} = \sqrt{w^{2} + v^{2} + 2wv\text{cos}k - u^{2}}$, the Hamiltonian is in the $\mathcal{PT}$-symmetry-preserving regime when *u*\<\|*v*−*w*\|, where $\mathit{ɛ}_{\pm}$ is real for all *k*.

The $\mathcal{PT}$-symmetric SSH Hamiltonian possesses topological properties, guaranteed by the so-called pseudo-anti-Hermiticity with $\eta H^{\dagger}\eta^{- 1} = - H$ ([@bib14]), where *η* = *σ*~*z*~. As a result, $\mathcal{PT}$-symmetry-broken topological edge states with purely imaginary eigenenergies emerge at the boundary between bulks of different topological phases. Topologically inequivalent phases can be distinguished by the generalized winding number $\nu = \varphi_{\text{B}}/2\pi$. Here the global Berry phase $\varphi_{\text{B}}$ is ([@bib17], [@bib41], [@bib42]) (see [Methods](#sec4){ref-type="sec"}):$$\varphi_{B} = - i\sum\limits_{\mu = \pm}{\oint{dk}}\frac{\left\langle {\chi_{\mu}\left| \frac{\partial}{\partial k} \right|\psi_{\mu}} \right\rangle}{\left\langle \chi_{\mu} \middle| \psi_{\mu} \right\rangle},$$where the integral runs over the first Brillioun zone (1BZ) and the right (left) eigenvector is defined as $H_{k}\left. \left| \psi_{\mu} \right\rangle = ɛ_{\mu}|\psi_{\mu} \right\rangle$ $\left( {H_{k}^{\dagger}\left. \left| \chi_{\mu} \right\rangle = ɛ_{\mu}^{\text{∗}}|\chi_{\mu} \right\rangle} \right)$. When *v* \< *w*, the system is topologically non-trivial, as *ν* = 1. In contrast, when *v* \> *w*, we have *ν* = 0 and the system is topologically trivial. In [Figure 1](#fig1){ref-type="fig"}B, we show the topological phase diagram as well as the boundary between $\mathcal{PT}$-symmetry-preserving and broken regimes. Notably, the $\mathcal{PT}$-symmetry-broken regime lies in the vicinity of the topological phase boundary.

Here an important remark is in order. In recent years, a general class of non-Hermitian topological models have been identified where the conventional bulk-boundary correspondence breaks down ([@bib10], [@bib37], [@bib38], [@bib39], [@bib43], [@bib67], [@bib68], [@bib69]). Therein, topological invariants calculated from the homogeneous bulks fail to correctly predict topological edge states in a system with boundaries. Instead, one must invoke the definition of non-Bloch topological invariants, which take into account the localization (near system boundaries) of the bulk eigenstates under the non-Hermitian skin effect ([@bib10], [@bib37], [@bib38], [@bib43], [@bib67], [@bib68], [@bib69]). However, since we focus on the quench dynamics of a homogeneous system without boundaries, it is sufficient that we only consider quantum quenches of the bulk Hamiltonian (1), where the parameters (*u*,*v*,*w*) undergo abrupt changes at *t* = 0. Furthermore, we note that the non-Hermitian SSH model considered here does not suffer from non-Hermitian skin effects even in the presence of boundaries. For example, under the open-boundary condition, the bulk eigenstates are all extended and the Bloch topological invariant defined in [Equation 2](#fd2){ref-type="disp-formula"} is capable of correctly predicting topological edge states near the boundaries (see [Figure S1](#mmc1){ref-type="supplementary-material"} and [Methods](#sec4){ref-type="sec"}).

Visualizing Non-unitary Dynamics on the Bloch Sphere {#sec2.2}
----------------------------------------------------

To account for the non-unitary dynamics of quenched non-Hermitian systems, we invoke the notion of biorthogonal quantum mechanics ([@bib4]). Owing to the lattice translational symmetry of the Hamiltonian (1), dynamics in different *k*-sectors are decoupled and can be analyzed independently. Denoting the initial Hamiltonian by *H*^i^, the initial state in each *k*-sector $\left| \psi_{-}^{i} \right\rangle$, with $H_{k}^{\text{i}}\left. \left| \psi_{-}^{i} \right\rangle = ɛ_{-}^{\text{i}}|\psi_{-}^{i} \right\rangle$, evolves under the final Hamiltonian $H_{k}^{\text{f}}$ according to $\left| {\psi_{k}\left( t \right)} \right\rangle = \sum\limits_{\mu = \pm}{c_{\mu}e^{- i\mathit{ɛ}_{\mu}^{\text{f}}t}\left| \psi_{\mu}^{\text{f}} \right\rangle}$, where $ɛ_{\mu}^{\text{f}} = \mu E_{k}^{\text{f}}$ is the eigenenergy of $H_{k}^{\text{f}}$. We have defined $c_{\mu} = \left\langle \chi_{\mu}^{\text{f}} \middle| \psi_{-}^{\text{i}} \right\rangle$, where $\left\langle \chi_{\mu}^{\text{f}} \right|$ $\left( \left| \psi_{\mu}^{\text{f}} \right\rangle \right)$ is the left (right) eigenvector of $H_{k}^{\text{f}}$, with the biorthonormal conditions $\left\langle \chi_{\mu}^{\text{f}} \middle| \psi_{\nu}^{\text{f}} \right\rangle = \delta_{\mu\nu}$ and $\sum\limits_{\mu}{\left| \psi_{\mu}^{\text{f}} \right\rangle\left\langle \chi_{\mu}^{\text{f}} \right|} = 1$.

To characterize the non-unitary time evolution in the biorthogonal basis $\left\{ \left. \left| \psi_{\mu}^{\text{f}} \right\rangle,|\chi_{\mu}^{\text{f}} \right\rangle \right\}$, we define an associated state of $\left| {\psi_{k}\left( t \right)} \right\rangle$ as $\left\langle {\chi_{k}\left( t \right)}| = \sum\limits_{\mu}{c_{\mu}^{\text{∗}}e^{iɛ_{\mu}^{\text{f∗}}t}}\left\langle \chi_{\mu}^{\text{f}} \right| \right.$ ([@bib4]), with the normalization condition $\left\langle \chi_{k}\left( 0 \right) \middle| \psi_{k}\left( 0 \right) \right\rangle = 1$. The non-unitary time evolution of the system is then captured by the non-Hermitian density matrix $\rho\left( {k,t} \right) = \frac{\left| {\psi_{k}\left( t \right)} \right\rangle\left\langle {\chi_{k}\left( t \right)} \right|}{\left\langle \chi_{k}\left( t \right) \middle| \psi_{k}\left( t \right) \right\rangle}$, such that the expectation value of any operator *A*~*k*~ is expressed as Tr(*ρA*~*k*~) ([@bib4]). We further write$$\rho\left( {k,t} \right) = \frac{1}{2}\left\lbrack {\tau_{0} + \mathbf{d}\left( {k,t} \right) \cdot \mathbf{\tau}} \right\rbrack\text{,}$$where ***d***(*k*,*t*)=(*d*~1~,*d*~2~,*d*~3~), ***τ***=(*τ*~1~,*τ*~2~,*τ*~3~), and we have defined the matrices $\tau_{\gamma} = \sum\limits_{\mu,\nu = \pm}{\left| \psi_{\mu}^{\text{f}} \right\rangle\sigma_{\gamma}^{\mu\nu}\left\langle \chi_{\nu}^{\text{f}} \right|}$ $\left( {\gamma = 0,1,2,3} \right)$. Here $\sigma_{\gamma}^{\mu\nu}$ is the matrix element of $\sigma_{\gamma}$, and *σ*~0~ is the 2×2 identity matrix. Note that $\left\{ \tau_{\gamma} \right\}$ fulfill the standard ![](fx2.gif)$\left( 2 \right)$ algebra.

As a key ingredient of our theoretical construction, the choice of *ρ*(*k*,*t*) and the introduction of $\left\{ \tau_{\gamma} \right\}$ matrices make the corresponding ***d***(*k*,*t*) a real unit vector on the Bloch sphere *S*^2^, even as the dynamics is non-unitary. [Equation 3](#fd3){ref-type="disp-formula"} thus allows a geometrical picture for understanding non-unitary dynamics and forms the basis for subsequent characterization of dynamic topological properties. Note that the non-Hermitian density matrix *ρ*(*k*,*t*) is connected to the conventional Hermitian one $\rho^{\prime}\left( {k,t} \right) = \left| {\psi_{k}\left( t \right)} \right\rangle\left\langle {\psi_{k}\left( t \right)} \right|$ through the metric operator $g = \sum\limits_{\mu}\left| \chi_{\mu}^{\text{f}} \right\rangle\left\langle \chi_{\mu}^{\text{f}} \right|$ ([@bib4]), with $\rho\left( {k,t} \right) = \frac{\rho^{\prime}\left( {k,t} \right)g}{\text{Tr}\left\lbrack {\rho^{\prime}\left( {k,t} \right)g} \right\rbrack}$ ([@bib60]). Although either density matrix can be used for the characterization of dynamic topological phenomena ([@bib60]), it is formally more elegant and convenient to use *ρ*(*k*,*t*) (see [Methods](#sec4){ref-type="sec"}), not the least because it is then straightforward to visualize non-unitary dynamics on the Bloch sphere.

Fixed Points in Non-unitary Dynamics {#sec2.3}
------------------------------------

When $E_{k}^{\text{f}}$ is real, ***d***(*k*,*t*) and hence the density matrix *ρ*(*k*,*t*) are oscillatory in time, with a momentum-dependent period $t_{0} = \pi/E_{k}^{\text{f}}$ (see [Methods](#sec4){ref-type="sec"}). This corresponds to a periodic rotation of ***d***(*k*,*t*) around the poles of the Bloch sphere, as illustrated in [Figure 2](#fig2){ref-type="fig"}A. Importantly, when ***d***(*k*,0) is on the poles of the Bloch sphere, the density matrix becomes time independent. This occurs at momenta *k*~*m*~ with either *c*~−~(*k*~*m*~) = 0 (north pole) or *c*~+~(*k*~*m*~) = 0 (south pole), which are identified as two different kinds of fixed points. In contrast, when $E_{k}^{\text{f}}$ is imaginary, ***d***(*k*,*t*) always starts from the equator at *t* = 0 and approaches the north pole in the long-time limit ([Figure 2](#fig2){ref-type="fig"}B), i.e., *ρ*(*k*,*t*) exponentially approaches a steady-state value (see [Methods](#sec4){ref-type="sec"}). In this case, there are no fixed points in the dynamics. Detailed time evolutions of ***d***(*k*,*t*) under different parameters are shown in [Figure S2](#mmc1){ref-type="supplementary-material"}.Figure 2Schematic Illustrations of the Time Evolution of *d*(*k*,*t*) on the Bloch SphereTime evolution of ***d***(*k*,*t*) (black) in *k*-sectors with: (A) real $E_{k}^{\text{f}}$ and (B) imaginary $E_{k}^{\text{f}}$ (assuming $\text{Im}\left( E_{k}^{\text{f}} \right) > 0$). The orange and purple vectors correspond, respectively, to fixed points with *c*~−~ = 0 and *c*~+~ = 0. The green vector in (B) indicates the long-time steady state. See also [Figure S2](#mmc1){ref-type="supplementary-material"}.

Based on the aforementioned understanding, it is straightforward to show that the number of fixed points with *c*~+~ = 0 or *c*~−~ = 0 should be at least \|*ν*^i^−*ν*^f^\| each, provided both *H*^i^ and *H*^f^ belong to the $\mathcal{PT}$-symmetry-preserving regime with completely real eigenspectra (see [Methods](#sec4){ref-type="sec"}). Here *ν*^*β*^ (*β* = i,f) are the generalized winding numbers of *H*^*β*^. On the other hand, when *H*^f^ is in the $\mathcal{PT}$-symmetry-broken regime, the corresponding $E_{k}^{\text{f}}$ becomes imaginary for a certain range of *k*, and the existence of fixed points are no longer guaranteed. We note that, although in the unitary limit (*u* = 0), our conclusions agree with previous studies ([@bib21], [@bib66]), our theoretical formalism is quite different from previous cases even for unitary dynamics. This is because matrices $\tau_{\gamma}$ are not reduced to Pauli matrices for *u* = 0, such that our mapping from quantum-state dynamics to the Bloch sphere is different from previous studies. It is also worth mentioning that, when starting from the Hermitian density matrix *ρ*′(*k*,*t*), it becomes quite difficult to theoretically relate the number and type of fixed points to the static topological invariants of the non-Hermitian Hamiltonians.

In the following, we mainly focus on the case where both *H*^i^ and *H*^f^ are in the $\mathcal{PT}$-symmetry-preserving regime. When the system is quenched across the topological phase boundary, fixed points divide the BZ into a series of submanifolds, where the density matrices at two ends of each given submanifold do not evolve in time. Fixed points in non-unitary dynamics are protected by both $\mathcal{PT}$ symmetry and band topology of the pre- and post-quench Hamiltonians, as we detail in the remainder of the work, whereas their physical consequences are manifested as observable dynamic topological phenomena such as dynamic skyrmions and DQPTs.

Dynamic Chern Number {#sec2.4}
--------------------

When *H*^f^ is in the $\mathcal{PT}$-symmetry-preserving regime, the periodic oscillation of the density-matrix evolution gives rise to an *S*^1^ topology in the time evolution. In the presence of fixed points, each submanifold between two adjacent fixed points can be combined with the *S*^1^ topology in time to form a momentum-time manifold *S*^2^, which can be mapped to the Bloch sphere associated with the vector ***d***(*k*,*t*) ([@bib21], [@bib66]). These *S*^2^→*S*^2^ mappings define a series of dynamic Chern numbers$$C_{mn} = \frac{1}{4\pi}{\int_{k_{m}}^{k_{n}}{dk}}{\int_{0}^{t_{0}}{dt\left\lbrack {\mathbf{d}\left( {k,t} \right) \times \partial_{t}\mathbf{d}\left( {k,t} \right)} \right\rbrack \cdot \partial_{k}\mathbf{d}\left( {k,t} \right)}}\text{,}$$where *k*~*m*~ and *k*~*n*~ denote two neighboring fixed points. For quenches between Hamiltonians with different winding numbers, the dynamic Chern numbers are quantized, with values dependent on the nature of fixed points at *k*~*m*~ and *k*~*n*~ (see [Methods](#sec4){ref-type="sec"}): *C*~*mn*~ = 1 when *c*~+~(*k*~*m*~) = 0 and *c*~−~(*k*~*n*~) = 0; *C*~*mn*~ = −1 when *c*~−~(*k*~*m*~) = 0 and *c*~+~(*k*~*n*~) = 0. When the two fixed points are of the same kind, *C*~*mn*~ = 0.

The emergence of finite dynamic Chern numbers can be visualized in the spin textures of ***d***(*k*,*t*), which possess skyrmion-lattice structures in the momentum-time space ([@bib21], [@bib53]), with the skyrmion number equivalent to the dynamic Chern number in each *S*^2^ momentum-time submanifold. In [Figure 3](#fig3){ref-type="fig"}, we show these skyrmion-lattice structures for different quench parameters. When the system is quenched across the topological phase boundary with \|*ν*^i^−*ν*^f^\| = 1, as we illustrate in [Figures 3](#fig3){ref-type="fig"}A and 3B, two fixed points of different kinds exist in the BZ. In the unitary limit ([Figure 3](#fig3){ref-type="fig"}A), the fixed points are pinned at *k* = 0 and *π* ([@bib21], [@bib66]). In the more general non-unitary case ([Figure 3](#fig3){ref-type="fig"}B), fixed points deviate from 0 and *π* and need to be solved from *c*~±~(*k*~*m*~) = 0. Pairs of vortices with positive (yellow) or negative (blue) vorticity emerge in the spin texture between adjacent fixed points on the plane of *d*~1~-*d*~3~, with vortex cores given by *d*~2~(*k*,*t*) = ±1. These vortices can be mapped to a lattice of skyrmions, whose topological charges are essentially the dynamic Chern numbers. In contrast, skyrmions are absent when *H*^i^ and *H*^f^ belong to the same topological phase ([Figure 3](#fig3){ref-type="fig"}C).Figure 3Dynamics of Spin Textures **d**(*k*,*t*) in the *k*-*t* SpaceQuench processes are between *H*^i^ and *H*^f^ characterized by: (A) (*u*^i^ = 0,*w*^i^ = *v*/3) and (*u*^f^ = 0,*w*^f^ = 2*v*); (B) (*u*^i^ = *v*/2,*w*^i^ = *v*/3) and (*u*^f^ = *v*/2,*w*^f^ = 2*v*); (C) (*u*^i^ = *v*/2,*w*^i^ = 3*v*) and (*u*^f^ = *v*/2,*w*^f^ = 2*v*); (D) (*u*^i^ = *v*/2,*w*^i^ = *v*/3) and (*u*^f^ = 2*v*,*w*^f^ = 2*v*). We fix *v*^i^ = *v*^f^ = *v* in the quenches, whereas the dynamic is unitary in (A) and non-unitary in (B--D). The vertical dash-dotted lines in (A)--(C) indicate fixed points with *c*~+~ = 0 (purple) and *c*~−~ = 0 (orange), respectively. Skyrmion lattices associated with finite dynamic Chern numbers only emerge in (A) and (B). The solid red lines in the horizontal direction mark each period ${n\pi}/E_{k}^{\text{f}}$ (*n* = 1,2, ...) of spin oscillation in regions with real $E_{k}^{\text{f}}$. The spin dynamics is non-oscillatory in regions with imaginary $E_{k}^{\text{f}}$, as shown in (D). The color bar indicates the value of *d~2~*, while the arrows indicate the spin texture in the *d*~1~-*d~2~* plane. See also [Figure S3](#mmc1){ref-type="supplementary-material"}.

For comparison, in [Figure 3](#fig3){ref-type="fig"}D, we plot the spin texture when *H*^f^ is in the $\mathcal{PT}$-symmetry-broken regime. As expected, in the momentum range where $E_{k}^{\text{f}}$ is imaginary, the spin texture approaches a steady state in the long-time limit, in sharp contrast to the periodic spin dynamics in the momentum range with real $E_{k}^{\text{f}}$. We note that coincidental fixed points may still exist in the momentum range where $E_{k}^{\text{f}}$ is real (see [Figure S3](#mmc1){ref-type="supplementary-material"} and [Methods](#sec4){ref-type="sec"}), but their number is no longer directly related to the topology of *H*^i^ and *H*^f^.

Dynamic Quantum Phase Transition {#sec2.5}
--------------------------------

Fixed points in the non-unitary quench dynamics further give rise to DQPTs, where physical quantities become nonanalytic at critical times. Interestingly, we find that, although DQPTs occur biperiodically in time for non-unitary quench dynamics, critical points of DQPTs generically emerge as vortex cores in the momentum-time-space spin texture, which provides a crucial link between different dynamic topological phenomena.

The central object in the theory of DQPT is the Loschmidt amplitude defined as the inner product of a time-evolved state with the initial state ([@bib25]). In non-unitary time evolutions, we generalize the Loschmidt amplitude as:$$\mathcal{G}\left( t \right) = \prod\limits_{k \in 1\text{BZ}}{\mathcal{G}_{k}\left( t \right)} = \prod\limits_{k \in 1\text{BZ}}\left\langle \chi_{k}\left( 0 \right) \middle| \psi_{k}\left( t \right) \right\rangle\text{,}$$where $\left\langle {\chi_{k}\left( t \right)} \right|$ is the associated state of $\left| {\psi_{k}\left( t \right)} \right\rangle$ defined previously. It is straightforward to derive $\mathcal{G}_{k}\left( t \right) = \left| c_{-} \right|^{2}e^{iE_{k}^{\text{f}}t} + \left| c_{+} \right|^{2}e^{- iE_{k}^{\text{f}}t}$.

DQPTs occur when the rate function $g\left( t \right) = - \frac{1}{L}\text{ln}\left( \left| {\mathcal{G}\left( t \right)} \right|^{2} \right)$ exhibits nonanalyticities during the time evolution, which are caused by Fisher zeros ([@bib25]), where $\mathcal{G}_{k_{c}}\left( t_{c} \right) = 0$ at critical points *k*~*c*~ and *t*~*c*~ in the dynamics. According to the expression of $\mathcal{G}_{k}\left( t \right)$, Fisher zeros, and hence DQPTs, occur periodically at $t_{c} = {{\left( {n + \frac{1}{2}} \right)\pi}/E_{k_{c}}^{\text{f}}}$ $\left( {n \in N} \right)$ when $E_{k_{c}}^{\text{f}}$ is real. Here $k_{c}$ satisfies \|*c*~−~(*k*~*c*~)\| = \|*c*~+~(*k*~*c*~)\|. When the system is quenched between different topological phases in the $\mathcal{PT}$-symmetry-preserving regime, fixed points with *c*~+~ = 0 and those with *c*~−~ = 0 always emerge in pairs. As \|*c*~+~\|−\|*c*~−~\| are continuous functions of *k*, there must be at least one critical momentum satisfying \|*c*~+~(*k*~*c*~)\| = \|*c*~−~(*k*~*c*~)\| between two fixed points of different kinds. We thus conclude that DQPTs necessarily occur in this case.

Interestingly, at critical points, *d*~2~(*k*~*c*~,*t*~*c*~) = 1, which lie on the equator of the Bloch sphere (see [Methods](#sec4){ref-type="sec"}). The critical points (*k*~*c*~,*t*~*c*~) thus correspond to vortex cores with positive vorticity in the momentum-time spin texture ([Figures 3](#fig3){ref-type="fig"}A and 3B). As vortices are manifestations of skyrmions, whose topological charges are the dynamic Chern numbers, the dynamic topological phenomena are related through the spin texture.

DQPTs are characterized by the dynamic topological order parameter *ν*~*D*~, which is defined through the Pancharatnam geometric phase (PGP) $\phi_{k}^{G}\left( t \right)$ as:$$\nu_{D}\left( t \right) = \frac{1}{2\pi}{\int_{k_{m}}^{k_{n}}{\frac{\partial\phi_{k}^{G}\left( t \right)}{\partial k}dk}}\text{.}$$

Here, *k*~*m*~ and *k*~*n*~ are fixed points of different kinds, and $\phi_{k}^{G}\left( t \right) = \phi_{k}\left( t \right) - \phi_{k}^{\text{dyn}}\left( t \right)$, where *ϕ*~*k*~(*t*) is defined through $\mathcal{G}_{k}\left( t \right) = \left| {\mathcal{G}_{k}\left( t \right)} \right|e^{i\phi_{k}{(t)}}$ and the dynamic phase $\phi_{k}^{\text{dyn}}\left( t \right) = - {\int_{0}^{t}{\left\langle {\chi_{k}\left( {t'} \right)}|H_{k}^{\text{f}}|{\psi_{k}\left( {t'} \right)} \right\rangle dt'}}$. At critical points, $\mathcal{G}_{k}\left( t \right)$ vanishes, which leads to abrupt jumps in $\phi_{k}^{G}\left( t \right)$ and *ν*~*D*~(*t*). Furthermore, as $\phi_{k}^{G}\left( t \right)$ also vanishes at fixed points, *ν*~*D*~ characterizes the *S*^1^→*S*^1^ mapping from the momentum submanifold between *k*~*m*~ and *k*~*n*~ to $e^{i\phi_{k}^{G}{(t)}}$ on the same momentum submanifold. *ν*~*D*~(*t*) is therefore quantized despite the non-unitary time evolution.

In [Figures 4](#fig4){ref-type="fig"}A--4C, we show typical *g*(*t*), $\phi_{k}^{G}\left( t \right)$, and *ν*~*D*~(*t*) for a quench between different topological phases in the $\mathcal{PT}$-symmetry-preserving regime. The BZ is divided into two submanifolds by the fixed points *k*~1~ and *k*~2~, where each submanifold contains a critical momentum, labeled as $k_{c}^{(1)}$ and $k_{c}^{(2)}$, respectively. These critical momenta give rise to two distinct critical time scales $t_{c}^{(1)}$ and $t_{c}^{(2)}$, and thus an apparent biperiodicity in the occurrence of DQPTs ([Figure 4](#fig4){ref-type="fig"}A), in contrast to its single-period counterpart in the unitary limit (*u* = 0) ([@bib24], [@bib25]). This is due to the breaking of time reversal symmetry under the non-Hermitian SSH Hamiltonian (1), such that the double degeneracy of Fisher zeros in the Hermitian case is lifted ([@bib58]). Correspondingly, two distinct types of *ν*~*D*~(*t*) exist, accounting for DQPTs occurring with the period $t_{c}^{(1)}$ and $t_{c}^{(2)}$, respectively ([Figure 4](#fig4){ref-type="fig"}C). In contrast, when *H*^i^ and *H*^f^ have the same winding numbers, *g*(*t*) is a smooth function in time and *ν*~*D*~(*t*) = 0 (dashed lines in [Figures 4](#fig4){ref-type="fig"}A and 4C).Figure 4DQPTs and Dynamic Topological Order Parameters in Non-Unitary Quench Dynamics(A) The rate function *g*(*t*), (B) the PGP $\phi_{k}^{G}\left( t \right)$, and (C) the dynamic topological order parameter *ν*~*D*~ for the quench in [Figure 3](#fig3){ref-type="fig"}B. At fixed points *k*~*m*~ (*m* = 1,2), *c*~−~(*k*~1~) = 0, and *c*~+~(*k*~2~) = 0. $k_{c}^{(m)}$ and $t_{c}^{(m)} = {\pi/\left( {2E_{k_{c}^{(m)}}^{\text{f}}} \right)}$ are the corresponding critical points. Two distinct types of *ν*~*D*~(*t*) exist in (C), obtained by setting the range of integral in [Equation 6](#fd6){ref-type="disp-formula"} as (*k*~1~,*k*~2~) (red) and (*k*~2~,*k*~1~+2*π*) (blue), respectively. The black dashed lines in (A) and (C) correspond to the quench in [Figure 3](#fig3){ref-type="fig"}C, whereas the quench in (D) is the same as that in [Figure 3](#fig3){ref-type="fig"}D. The color bar in (B) indicates the value of PGP. See also [Figure S3](#mmc1){ref-type="supplementary-material"}.

Finally, when *H*^f^ is in the $\mathcal{PT}$-symmetry-broken regime, DQPT is not guaranteed (see [Figure S3](#mmc1){ref-type="supplementary-material"} and [Methods](#sec4){ref-type="sec"}), even if *H*^i^ and *H*^f^ possess different winding numbers. In [Figure 4](#fig4){ref-type="fig"}D, we demonstrate the typical rate function *g*(*t*) for the quench process in [Figure 3](#fig3){ref-type="fig"}D, where neither fixed points nor DQPTs are present.

Discussion {#sec3}
==========

We have shown that dynamic topological phenomena emerge in the quench dynamics of a $\mathcal{PT}$-symmetric non-Hermitian SSH model. These dynamic topological phenomena are connected with discrete fixed points in the post-quench dynamics and emerge in the non-unitary time evolution when the system is quenched between different topological phases in $\mathcal{PT}$-symmetry-preserving regime. Given the recent experimental observation of spontaneous $\mathcal{PT}$-symmetry breaking in cold atoms ([@bib40]) and topological edge states in $\mathcal{PT}$-symmetric quantum-walk dynamics ([@bib65]), we expect that the dynamic topological phenomena discussed here can be probed using cold atoms loaded into a superlattice with engineered on-site loss, or through discrete-time quantum-walk dynamics in photonic configurations ([@bib60], [@bib61]). For either the cold atom or the photonic setup, tomography of instantaneous time-evolved states is needed to probe the dynamic topological phenomena.

Limitations of the Study {#sec3.1}
------------------------

In this study, we only consider quench dynamics of a homogeneous, non-interacting SSH model with on-site gain and loss. The results obtained, namely, the presence of fixed points and the appearance of dynamic topological structures, may be different, either quantitatively or qualitatively, in a different setting. For example, the fate of the dynamic topological phenomena is unknown for systems with boundaries, or with interactions. Neither is the presence of fixed points and dynamic topological structures entirely clear for systems with different topological classifications, or with alternative forms of non-Hermiticity. We leave these unaddressed questions to future studies.

Methods {#sec4}
=======

All methods can be found in the accompanying [Transparent Methods supplemental file](#mmc1){ref-type="supplementary-material"}.

Supplemental Information {#appsec2}
========================

Document S1. Transparent Methods and Figures S1--S3

This work has been supported by the National Natural Science Foundation of China (Grant Nos. 11434007, 11474049, 11674056, 11522545, 11874038 and U1930402), the National Key Research and Development Program of China (Grant Nos. 2016YFA0301700, 2017YFA0304203), Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (Grant No. IRT13076), and the Natural Science Foundation of Jiangsu Province (Grant No. BK20160024). Y. H. also acknowledges support from the National Thousand-Young-Talents Program, and Shanxi 100 Talents Program.

Author Contributions {#sec5}
====================

W.Y., Y.H., and P.X. conceived the project; X.Q. and T.-S.D. performed the theoretical analysis; W.Y., Y.H., and P.X. wrote the paper with input from other authors. All authors contributed to revising the paper.

Declaration of Interests {#sec6}
========================

The authors declare no competing interests.

Supplemental Information can be found online at <https://doi.org/10.1016/j.isci.2019.09.037>.

[^1]: These authors contributed equally

[^2]: Lead Contact
