PHYSICAL REVIEW RESEARCH 2, 023043 (2020)

Dynamic winding number for exploring band

Bo Zhu,1,2 Yongguan Ke,1,3 Honghua Zhong,4 and Chaohong Lee 1,2,* 1Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing & School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China 2State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University (Guangzhou Campus), Guangzhou 510275, China 3Nonlinear Physics Centre, Research School of Physics, Australian National University, Canberra, Australian Capital Territory 2601, Australia 4Institute of and Physics, Central South University of Forestry and Technology, Changsha 410004, China

(Received 29 August 2019; revised manuscript received 16 March 2020; accepted 18 March 2020; published 15 April 2020)

Topological invariants play a key role in the characterization of topological states. Because of the existence of exceptional points, it is a great challenge to detect topological invariants in non-Hermitian systems. We put forward a dynamic winding number, the winding of realistic observables in long-time average, for exploring band topology in both Hermitian and non-Hermitian two-band models via a unified approach. We build a concrete relation between dynamic winding numbers and conventional topological invariants. In one dimension, the dynamic winding number directly gives the conventional winding number. In two dimensions, the Chern number is related to the weighted sum of all the dynamic winding numbers of phase singularity points. This work opens a new avenue to measure topological invariants via time-averaged spin textures without requesting any prior knowledge of the system topology.

DOI: 10.1103/PhysRevResearch.2.023043

I. INTRODUCTION However, these quench schemes request prior knowledge of topology before and after quench. Topological invariant, a global quantity defined with static In non-Hermitian systems, due to the emergence of com- Bloch functions, has been widely used for classifying and plex spectrum and exceptional points (EPs) [31–33], novel characterizing topological states in various systems, includ- topological states have been theoretically predicted [34–50] ing insulators, superconductors, semimetals, and waveguides and experimentally observed [51–54]. Conventional topolog- [1–6]. Nontrivial topological invariants lead to novel topolog- ical invariants such as winding number and Chern number ical effects, such as winding number for quantized geometric have been generalized to those in non-Hermitian systems phase [7,8], and Chern number for both quantum [8,37,55–57], and new topological invariants such as vorticity Hall effect [9,10] and the Thouless pumping [11–13]. Mea- have also been introduced [58]. Because of the EPs, the wind- suring topological invariants provides unambiguous evidence ing number may be half of integer in non-Hermitian systems for topological states, which apply to precision measurement [8,37,59,60]. Besides, non-Bloch definition of Chern number [14,15], error-resistant spintronics [16,17], and quantum com- strictly gives the numbers of chiral edge modes [61–64]. The puting [18,19]. way to measure these topological invariants in non-Hermitian Most of the existing methods for measuring topologi- systems is more challenging than that in Hermitian systems. cal invariants are mainly based on adiabatic band sweeping For example, the Hall conductivity is no longer quantized [20–24]. However, these methods may fail for imperfect initial despite being classified as a Chern insulator based on non- states and small energy gaps and become invalid for non- Hermitian topological band theory [44,65]. In one dimension, Hermitian systems. Recently, it has been demonstrated that the winding number in a non-Hermitian system has been topological invariants can be measured via linking numbers determined via the mean displacement in long-time quantum and band-inversion surfaces in quench dynamics [25–30]. walk [66,67], which does not work for measuring Chern numbers and half-integer winding numbers. Is there a unified dynamic approach for measuring topological invariants in both Hermitian and non-Hermitian systems? *Corresponding author: [email protected]; In this work, we study a generic two-band model which [email protected] supports nontrivial topological invariants in both Hermitian and non-Hermitian regions. We define a dynamic winding Published by the American Physical Society under the terms of the number (DWN) based on the time-averaged spin textures, Creative Commons Attribution 4.0 International license. Further which is robust against various initial states. In one di- distribution of this work must maintain attribution to the author(s) mension, we prove that the DWNs directly give the con- and the published article’s title, journal citation, and DOI. ventional winding numbers in both chiral-symmetric and

2643-1564/2020/2(2)/023043(12) 023043-1 Published by the American Physical Society ZHU, KE, ZHONG, AND LEE PHYSICAL REVIEW RESEARCH 2, 023043 (2020) non-chiral-symmetric systems. In two dimensions, the Chern σ j will form a trajectory in the polarization . The DWN number is related to the weighted sum of the DWNs around of the spin vectors (ψ˜ (k, t )|σ |ψ(k, t ), ψ˜ (k, t )|σ |ψ(k, t )) + i j all singularity points (SPs), where the weight is 1forthe is defined as north pole and −1 for the south pole. When the system is  changed from Hermiticity to non-Hermiticity, each singularity 1 wd = ∂kη ji(k) · dk, (3) will be split into two EPs (which are also SPs); yet, the 2π S Chern number can still be extracted via the DWNs of all EPs. where S is a closed loop in the parameter space k, and η ji(k) Without requesting any prior knowledge of their topology, our is the dynamical azimuthal angle, approach provides general guidance for measuring topologi-   cal invariants in both Hermitian and non-Hermitian systems. ψ˜ (k, t )|σ j|ψ(k, t ) η ji(k) = arctan . (4) The paper is organized as follows. In Sec. II,weintro- ψ˜ , |σ |ψ , duce our physical model and give the definition of DWN. (k t ) i (k t ) In Sec. III, we consider one-dimensional systems and give On the other hand, we also define the equilibrium azimuthal the relationship between conventional winding number and angle φ ji(k) in the parameter plane (hi(k), h j (k)), DWN. In Sec. IV, we consider two-dimensional systems and φ = / . give the relationship between Chern number and DWN. In ji(k) arctan[h j (k) hi(k)] (5) 2 2 Sec. V, we give a brief conclusion and discussion. In the conditions |c+(k)| =|c−(k)| for Hermitian systems 2 2 and |c+(k)| = 0 ∧|c−(k)| = 0 for non-Hermitian systems, II. DYNAMIC WINDING NUMBER one can easily prove that η ji(k) converges to φ ji(k); see Appendix A. Therefore, in the conditions mentioned above, We consider a general two-band model in d dimension. The the DWN can be rewritten as Hamiltonian in the momentum space is composed of the three  Pauli matrices, 1 wd = ∂kφ ji(k) · dk. (6) 2π S H(k) = hx (k)σx + hy(k)σy + hz(k)σz. (1) For Hermitian systems, |ψ˜ (k, t )=|ψ(k, t ), and the dy- Here, k is the quasimomentum, and hx(y,z)(k) are peri- η odic functions of k. The Hamiltonian could be Hermitian namical azimuthal angle ji(k) is a real angle, so that the H †(k) = H(k) or non-Hermitian H †(k) = H(k). Then, the DWN can be directly probed via the long-time average of spin textures ψ(k, t )|σ j|ψ(k, t ) in experiment. For non- right and left eigenvectors are given by H(k)|ϕμ(k)= † ∗ Hermitian systems, |ψ˜ (k, t ) =|ψ(k, t ) and η ji(k) is a com- εμ(k)|ϕ (k) and H (k)|χ (k)=ε (k)|χ (k), respec- μ  μ μ μ plex angle which cannot be directly observed in experiment. ε = μ 2 + 2 + 2 tively, where μ(k) [hx (k)] [hy(k)] [hz(k)] with Interestingly, this problem can be fixed by decomposing the μ =±are the eigenvalues. For Hermitian systems, the right azimuthal angle into real and imaginary parts. We find that and left eigenvectors become the same and hence their cor- only the real part of η ji(k) contributes to the DWN and it responding energies, that is, |ϕμ(k)=|χ μ(k) and εμ(k) = satisfies ε∗ μ(k). For non-Hermitian systems, neither the eigenvec- π {|ϕ } {χ |} η = 1 φRR + φLL + , tors μ(k) nor μ(k) are orthogonal. To account Re[ ji(k)] ji (k) ji (k) n for the nonunitary dynamics of the non-Hermitian systems, 2  2  we invoke the notion of biorthogonal quantum mechanics ψ(k, t )|σ |ψ(k, t ) φRR(k) = arctan j , [68], and adopt the biorthogonal eigenvectors which fulfill ji ψ , |σ |ψ , χ |ϕ =δ |ϕ χ |= (k t ) i (k t ) ν (k) μ(k) ν,μ and μ μ(k) μ(k) 1bynormal-   |ϕ =|ϕ / χ |=χ |/ izing μ(k) μ(k) Nμ(k) and μ(k) μ(k) Nμ(k) ψ˜ (k, t )|σ |ψ˜ (k, t )  φLL = j . with Nμ(k) = χ μ(k)|ϕμ(k). ji (k) arctan (7) ψ˜ (k, t )|σ |ψ˜ (k, t )  Considering an arbitrary initial state |ψ(k, 0)= i μ |ϕμ ψ˜ , |= RR μ c (k) (k) and its associated state (k 0) Here, Re[η ji(k)] represents the real part of η ji(k), φ (k) and ∗ ji μ cμ(k)χμ(k)|, the time evolution of |ψ(k, t ) and φLL ji (k) are both real, and n is an integer; see Appendix B. ψ˜ (k, t )| respectively satisfy The superscript RR (LL) indicates that the azimuthal angle  is calculated in the basis of the right (left) biorthogonal |ψ , = −iεμ (k)t |ϕ , (k t ) cμ(k)e μ(k) eigenvectors. Thus, we can decompose the DWN as μ  ∗ ε∗ 1 RR LL ψ˜ , |= i μ (k)t χ |, w = w + w , (k t ) cμ(k)e μ(k) (2) d 2 d d μ  τ 1 τ =χ |ψ , w = ∂kφ (k) · dk,τ ∈ RR, LL. (8) where cμ(k) μ(k) (k 0) . According to the d 2π jl biorthogonal quantum mechanics, the spin textures are S given by the expectation values of the Pauli matrices, The above results mean that the DWN can also be observed in ψ˜ (k, t )|σ j|ψ(k, t ), where j ∈ x, y, z. Here, we are non-Hermitian systems via the time evolution of the left-left and right-right spin textures whose dynamics are respectively interested in their long-time averages, ψ˜ (k, t )|σ |ψ(k, t )= †  j governed by H(k) and H (k). In the following sections, we 1 T ψ˜ (k,t )|σ j |ψ(k,t ) lim →∞ dt. As the quasimomentum T T 0 ψ˜ (k,t )|ψ(k,t ) show how to utilize the DWN to uncover the topology in both continuously varies, the spin textures of Pauli matrices σi and Hermitian and non-Hermitian systems. For simplicity, we will

023043-2 DYNAMIC WINDING NUMBER FOR EXPLORING BAND … PHYSICAL REVIEW RESEARCH 2, 023043 (2020)

use σ j and σ j to denote the long-time average of spin textures

ψ(k, t )|σ j|ψ(k, t ) and ψ˜ (k, t )|σ j|ψ˜ (k, t ), respectively.

III. CONNECTION BETWEEN CONVENTIONAL WINDING NUMBER AND DYNAMIC WINDING NUMBER

In one dimension, if hz = 0, the Hamiltonian (1) has chiral symmetry H =−H with  = iσxσy and k → k. The conventional winding number w± for the Hamiltonian (1) reads  1 hx∂khy − hy∂khx w± = dk 2π (ε±)2 S 1 = ∂kφyxdk, (9) 2π S which is known as the Zak phase. According to Eqs. (6) 2 2 and (9), if the initial states satisfy |c+(k)| =|c−(k)| for 2 2 Hermitian systems and |c+(k)| = 0 ∧|c−(k)| = 0 for non- = FIG. 1. Extracting conventional winding number via dy- Hermitian systems, one can find that w± wd . δ = = namic winding number. Hermitian case ( 0): Panels (a) and If hz 0, the Hamiltonian (1) breaks the chiral symmetry. (b) respectively show the time evolution of the spin textures The winding numbers w± are given by ψ , |σ |ψ , ψ , |σ |ψ ,  (k t ) x (k t ) and (k t ) y (k t ) . (c) Long-time average ∂ − ∂ of spin textures σ (black line) and σ (red line) as a of k,and = 1 hx khy hy khx . x y w± dk (10) (d) the dynamical azimuthal angle ηyx (k) as a function of k,where 2π S ε±(ε± − hz ) k1 and k2 are discontinuity points. Non-Hermitian case (δ = 0.3): φRR φLL Unlike the systems with chiral symmetry, the conventional yx (k)and yx (k) as a function of k for (e) the chiral-symmetric winding number for each band is not a quantized number, system with hz = 0 and (f) the non-chiral-symmetric system with which indicates that w± are no longer topological invariants. hz = 0.5. The other parameters are chosen as J1 = 1andT = 80. However, the sum of two conventional winding numbers,  1 hx∂khy − hy∂khx When δ = 0, the system becomes non-Hermitian and one w = w+ + w− = dk t π 2 + 2 φRR φLL S hx hy always needs to measure both ji (k) and ji (k) to extract  the DWN. For a chiral-symmetric system, the two dynamical 1 = ∂ φ dk, azimuthal angles satisfy a universal relation φRR(k) = φLL(k), π k yx (11) ji ji S which can be easily proven. For the chosen parameters (J0 = = δ = . = = / RR = has been demonstrated to be a topological invariant, which J1 1, 0 3, hz 0, and w± 1 2), we have wd LL = 1 takes an integer [35,59]. Similarly, if the initial states satisfy wd 2 ; see Fig. 1(e). It means that we only need to measure | |2 =| |2 | |2 = φRR φLL c+(k) c−(k) for Hermitian systems and c+(k) ji (k)or ji (k) in experiments. For a non-chiral-symmetric 2 0 ∧|c−(k)| = 0 for non-Hermitian systems, The topological system (whose parameters are chosen as J0 = J1 = 1, δ = . = . = φRR = φLL RR = invariant wt is related to the dynamic winding number via 0 3, hz 0 5, and wt 1), we find ji (k) ji (k), wd = wt 2wd . 1, and wLL = 0; see Fig. 1(f). Nevertheless, the conventional = + d As an example, we consider a system with hx J0 winding number can be obtained by measuring w = (wRR + = − δ = d d J1 cos(k), hy J1 sin(k) i , and hz 0. In the Hermitian wLL)/2 in both chiral and nonchiral symmetric systems. δ = = d case, the parameters are chosen as 0 and J1 1. The In Appendix C, we provide more examples about the = | | < conventional winding number w± 1for J0 J1, and measurement of conventional winding number in the presence = | | > w± 0for J0 J1. We first calculate the time evolution of or absence of chiral symmetry. For non-Hermitian systems ψ , |σ |ψ , spin textures (k t ) x(y) (k t ) and their long-time aver- with chiral symmetry, the winding number for non-Hermitian = . ages with J0 0 5J1; see Figs. 1(a)–1(c). The spin textures chiral systems may take value of a half-integer. This is because ψ , |σ |ψ , (k t ) x(y) (k t ) oscillate with a momentum-dependent the winding number is equal to half of the summation of two = π/|εμ | σ period tk (k) , and their long-time averages x(y) de- winding numbers (v1 and v2) associated with two EPs, where pend on quasimomentum k; see the black and red lines in v1 and v2 can only take [8]. Fig. 1(c). With σx and σy, we calculate ηyx(k) as a function of k in Fig. 1(d), where two discontinuous points k1 and k2 appear. The DWN can be obtained via the of a piecewise IV. CONNECTION BETWEEN CHERN NUMBER AND function, DYNAMIC WINDING NUMBER    π  1 k1 k2 By generalizing the concept of gapped band structures wd = ∂kηyxdk + ∂kηyxdk + ∂kηyxdk . from Hermitian to non-Hermitian systems, the Chern num- 2π −π k1 k2 ber for an energy-separable band can be constructed in We find that the DWN is equal to 1, the same as the conven- a similar way [57]. In contrast to Hermitian systems, tional winding number w±. there are left-right, right-right, left-left, and right-left Chern

023043-3 ZHU, KE, ZHONG, AND LEE PHYSICAL REVIEW RESEARCH 2, 023043 (2020) numbers in non-Hermitian systems, depending on the def- LR = χ |∂ |ϕ RR = initions of Berry connection, Ak i μ k μ , Ak ϕ |∂ |ϕ LL = χ |∂ |χ RL = ϕ |∂ |χ i μ k μ , Ak i μ k μ , and Ak i μ k μ .Al- though the corresponding Berry curvatures are locally differ- ent quantities, the four kinds of Chern numbers are the same [57]. Here, we only focus on analyzing the Chern number LR = χ |∂ |ϕ defined with left-right Berry connection Ak i μ k μ , which naturally reduces to iχμ|∂k|χμ in Hermitian systems. We map the Hamiltonian to a normalized vector, n (k) = [sin(θi ) cos(φ jl ), sin(θi )sin(φ jl ), cos(θi )], which reduces to a Bloch vector in Hermitian systems. Here, θi(k) denotes the angle between the vector and the axis i, and φ jl (k) denotes the equilibrium azimuthal angle in the j − l plane. We can freely choose the reference axis without affecting the validity of the dynamic approach. Then, the left and right eigenvectors for the lower energy band are given by FIG. 2. Topologically nontrivial phase with Chern number C = iφ / θ −iφ / θ χ−(k)|= −e jl 2 cos i , e jl 2 sin i , 1. Panels (a) and (b) respectively show dynamical azimuthal angle  2  2 − φ / θ ηxz(k) obtained in the evolved time T = 10 and T = 80, and panel − i jl 2 i e cos( 2 ) φ |ϕ−(k)= φ / θ . (12) (c) displays equilibrium azimuthal angle xz(k). Panels (d) and (e) ei jl 2 sin( i ) 2 show how ηxz(k) (black dots) and φxz(k) (red line) change along the trajectory around the north and south poles in panels (b) and (c), The right and left eigenstates have phase singularities at = , = n (k ) = (0, 0, ±1), in which + and − respectively correspond respectively. The other parameters are chosen as Jx(y,z) 1 mz 1, 0 and δ = 0. to north and south poles. In the parameter space k → (kx, ky ), 2 2 the locations k0 of the poles satisfy h j (k0 ) + hl (k0 ) = 0. The left-right Berry connections ALR of the lower energy kx(y) φLL band are given by and jl (k), which can be respectively extracted via right- ψ , |σ |ψ , θ ∂φ right spin textures (k t ) j(l) (k t ) and the left-left spin LR cos( i ) jl ψ˜ , |σ |ψ˜ , A = iχ−|∂k |ϕ−= . textures (k t ) j(l) (k t ) . kx(y) x(y) 2 ∂ kx(y) As an example, we consider hx = Jx sin(kx ), hy = Jy sin(ky ), and hz = mz − Jz cos(kx ) − Jz cos(ky ) − iδ. Here, We discretize the parameter space (kx, ky )asN × M mesh grids in the first Briliouin zone [69,70]. For each grid, a direct Jx(y,z) denote spin-orbit coupling parameters, mz is the effec- δ application of the two-dimensional Stokes theorem implies tive magnetization, and is the gain or loss strength. When δ = that the Chern number reads 0, the system supports quantum anomalous Hall effect [71], which has been realized in recent experiments [27,72]. In N M  J , = , m = ,δ= = 1 LR + LR , the Hermitian case ( x(y z) 1 z 1 0), the north and C Ak dkx Ak dky (13) , 2π x y south poles in the parameter space (kx ky ) can be determined = = Slx ,ly lx 1 ly 1 as follows. Since the poles are related to the chosen axis, θ k = h /|h | φ k = h /h where Slx ,ly represents the clockwise path integration for we select y( ) arccos( y ) and xz( ) arctan( x z ). the (lx, ly ) grid. Then, we find that the Chern number is However, the validity of our dynamic approach is independent determined by the winding numbers for all SPs, where of the choice of reference axis; see Appendix D.Inthe , 2 + 2 = cos[θ (k )] = h (k )/|h(k )|=sgn(Re[h (k )]) = 1forthe parameter space (kx ky ), by solving hx hz 0, we find i 0 i 0 0 i 0 = , = north SPs and cos[θi(k0 )] =−1 for the south SPs, that the north and south poles are located at k0 (kx ky ) , ±π/ Re[hi(k0 )] represents the real part of hi(k0 ), and h(k0 ) = (0 2). Then, we need to extract the DWNs around the [h (k ), h (k ), h (k )]. At last, we can deduce the Chern two poles. We calculate the time evolution of spin textures x 0 y 0 z 0 ψ , |σ |ψ , ψ , |σ |ψ , number as (k t ) x (k t ) and (k t ) z (k t ) , and the dynam-  ical azimuthal angle ηxz(k) can be extracted via long-time 1 σ C = sgn(Re[hi(k0 )])w(k0 ), (14) average values x(z); see Figs. 2(a) and 2(b). We also calculate 2 φ k0∈SPs the equilibrium azimuthal angle xz(k) via the eigenstates;  see Fig. 2(c). Even in the relatively short time [Fig. 2(a)], the = 1 ∂ φ where w(k0 ) π k jldk is the winding number for 2 Slx ,ly profiles of dynamical azimuthal angle are close to those of the the SP at k0. equilibrium azimuthal angle. The difference between ηxz(k) 2 2 Accordingly, if the initial states satisfy |c+(k)| =|c−(k)| and φxz(k) gradually disappears with the increase of total time 2 2 for Hermitian systems and |c+(k)| = 0 ∧|c−(k)| = 0for T . Thus, we can obtain the DWNs for the north and south non-Hermitian systems, one can also find that the DWN poles via integrating the gradient of ηxz(k)inFigs.2(d) and wd (k0 ) = w(k0 ). In the Hermitian case, wd is related to the 2(e), respectively. The DWNs for the north and south poles dynamical azimuthal angle η jl (k), which can be extracted are respectively given by wd =±1. Applying Eq. (14), one via the right-right spin textures ψ(k, t )|σ j(l)|ψ(k, t ) due can obtain the Chern number as 1, which is consistent with to |ψ(k, t )=|ψ˜ (k, t ). In the non-Hermitian case, wd is the one calculated via integrating the static Berry curvature φRR related to the two real dynamical azimuthal angles jl (k) over the whole Brillouin zone.

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In addition, our approach can also apply to extract larger Chern number without extra efforts, but it is very hard to access by adiabatic band sweeping; see Appendix D.

V. CONCLUSION AND DISCUSSION We put forward a new concept of dynamic winding number (DWN) and uncover its connections to conventional topolog- ical invariants in both Hermitian and non-Hermitian models. Given a time-averaged spin texture in the parameter space, a DWN is given by a loop integral of the dynamical azimuthal angle gradient enclosing a single singularity point. We find FIG. 3. (a) Topological phase diagram. [(b)–(d)] The bulk bands that (i) the conventional winding numbers in one-dimensional (red and blue regions) and edge modes (black line) in the com- systems can be directly given by the corresponding DWNs and plex energy plane, corresponding to the parameter points b–d in (ii) the Chern numbers in two-dimensional systems is related panel (a). (e) North EPs (blue dots) and south EPs (red dots) to the weighted sum of all corresponding DWNs. Our scheme = , = δ = . with Jx(y,z) 1 mz 1, and 0 5. [(f)–(h)] Dynamical azimuthal has two main advantages. First, in contrast to the quench η φRR φLL angles Re[ xz(k)], xz (k), and xz (k), which are defined with left- schemes via measuring linking numbers [25,29] and band- right, right-right, and left-left spin textures at T = 80. inversion surfaces [27,28], which request prior knowledge of topology before and after a quench, our scheme does not request any prior knowledge. Second, our scheme can be used For more general cases, we first show the topological phase to measure half-integer winding numbers in one-dimensional ,δ = diagram in the parameter plane (mz ) by setting Jx(y,z) 1; non-Hermitian systems, which cannot be measured via previ- see Fig. 3(a). When the energy bands are separable (i.e., the ous methods. real parts of energy bands are gapped), the Chern number Our scheme is readily realized in various systems, rang- can be constructed via a method similar to that used in ing from cold atoms in optical lattices, optical waveguide Hermitian systems [57]. In this case, the Chern numbers arrays, to optomechanical devices. In Appendix E, we provide of the first band are C = 0 and 1 in the green and gray more details about the experimental realization of our scheme regions, respectively. However, the Chern number is not well via cold atoms and optical waveguide arrays. In the future, defined in the white region, because the energy bands are it would be interesting to extend our scheme to measure − 2 + δ2 < not separable. The boundaries satisfy (mz 1) 1for topological invariants in high-dimensional systems, multiband > δ the gray region and mz 2 for the green region. Varying systems, periodically driven systems, and disordered systems. along the dashed red arrow, we explore the correspondence between the Chern number and the edge modes under open boundary condition in Figs. 3(b)–3(d), corresponding to the ACKNOWLEDGMENTS parameter points b–d. For the topologically nontrivial phase, This work is supported by the Key-Area Research and one can see that the complex bulk bands are still gapped and Development Program of GuangDong Province under Grant the edge-state modes are still preserved in the real energy axis; No. 2019B030330001; the National Natural Science Foun- see Fig. 3(b). Now we consider δ = 0.5 and keep the other dation of China (NNSFC) under Grants No. 11874434, No. parameters the same as those in Fig. 2(b). We find that the 11574405, No. 11805283, and No. 11904419; and the Sci- two poles in Hermitian case are split into four EPs if the non- ence and Technology Program of Guangzhou (China) un- Hermiticity is present; see Fig. 3(e). The blue and red points der Grant No. 201904020024. Y.K. is partially supported represent north and south EPs, respectively. In Fig. 3(f),we by the Office of China Postdoctoral Council (Grant No. η , give Re[ xz(k)] in the parameter space (kx ky ), and the DWNs 20180052), the National Natural Science Foundation of China can be obtained by integrating the gradient of the dynamical (Grant No. 11904419), and the Australian Research Coun- azimuthal angle along the trajectory enclosing the north and cil (DP200101168). H.Z. is partially supported by the Hu- = south EPs. Here, DWNs for the north and south EPs are wd nan Provincial Natural Science Foundation under Grant No. 1/2 and −1/2, respectively. Although the SPs are doubled, 2019JJ30044. We are deeply grateful to our anonymous re- as each DWN is reduced by half, the Chern number remains viewers for their thoughtful suggestions that have helped to unchanged. For completeness, we also show the dynamical improve this paper substantially. azimuthal angles defined with the right-right and left-left spin B.Z. and Y.K. contributed equally to the work. textures; see Figs. 3(g) and 3(h). The dynamical azimuthal angles are quite different from each other, corresponding to φRR φLL APPENDIX A: CONVERGENCE OF DYNAMIC WINDING xz (k) and xz (k), respectively. Nevertheless, the north and south EPs are the same as those in Fig. 3(e). Around an NUMBER RR EP, one can extract the right-right and left-left DWNs wd According to Eq. (2) in the main text, it seems that the LL = 1 RR + LL and wd , which satisfy wd 2 (wd wd ). One important definition of DWN depends on the initial state, and it is un- φRR φLL thing is that xz (k) and xz (k) defined with the real left-left clear whether such number converges in the long time. Here, and right-right spin textures are accessible in experimental we prove that the initial state can be rather general and the measurements. DWN is convergent. The time average of ψ˜ (k, t )|σ j|ψ(k, t )

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2 is given by with |c−| = 0. Combining Eqs. (A4), (A5), and (A6), we have ψ˜ (k, t )|σ j|ψ(k, t )  ψ˜ (k, t )|σ j|ψ(k, t ) h j T = , 1 ψ˜ (k, t )|σ j|ψ(k, t ) (A7) = lim dt ψ˜ , |σ |ψ , hi →∞ (k t ) i (k t ) T T 0 ψ˜ (k, t )|ψ(k, t )  ∗ On the other hand, the denominator in the left-hand side  ∗ −i(εμ−ε )t T μ χ |σ |ϕ 1 μ,μ cμcμ e μ j μ of Eq. (A7) should be nonzero; that is, the initial states = lim  ∗ dt. →∞ 2 −i(εμ−εμ )t | |2 =| |2 T T 0 μ |cμ| e should satisfy c+(k) c−(k) for Hermitian systems and 2 2 |c+(k)| = 0 ∧|c−(k)| = 0 for non-Hermitian systems. Ac- (A1) cording to Eq. (A7), one can obtain   We assume that the eigenenergy εμ(k) = μ(A + iB), μ =±, χ |σ |ϕ η = μ j μ = φ , where A and B are k-dependent real numbers. Then Eq. (A1) ji(k) arctan ji(k) χμ|σi|ϕμ can be written as   ψ˜ k, t |σ |ψ k, t η = ( ) j ( ) , ψ˜ (k, t )|σ j|ψ(k, t ) ji(k) arctan  ψ˜ (k, t )|σ |ψ(k, t ) T ∗ −2Bt i 1 c−c−e χ−|σ j|ϕ− = lim  ∗ dt φ ji(k) = arctan(h j/hi ), (A8) →∞ 2 −i(εμ−εμ )t T T 0 μ |cμ| e  where η (k) and φ (k) correspond to the dynamical az- T ∗ 2Bt χ |σ |ϕ ji ji 1 c+c+e + j + imuthal angle and equilibrium azimuthal angle, respectively. + lim  ∗ dt →∞ 2 −i(εμ−εμ )t T T 0 μ |cμ| e It means that the DWN also converges in the long-time limit  T ∗ −i2At when the initial state satisfies a few constraints. c+c e χ−|σ |ϕ+ + 1 − j lim − ε −ε∗ dt →∞ | |2 i( μ μ )t T T 0 μ cμ e APPENDIX B: RELATION BETWEEN DYNAMIC WINDING  T ∗ i2At NUMBER AND TIME-AVERAGED SPIN TEXTURES 1 c−c+e χ+|σ j|ϕ− + lim  ∗ dt. (A2) →∞ 2 −i(εμ−εμ )t η T T 0 μ |cμ| e For the non-Hermitian case, the azimuthal angles ji(k) and φ ji(k) are generally complex angles, so that they do not Here, we note that the of the third and fourth terms represent physical observables in the biorthogonal system. in Eq. (A2) are periodically oscillating functions of time t. This problem can be fixed by decomposing the azimuthal Hence, these terms vanish in the long-time average, leaving angle into two parts, φ ji(k) = Re[φ ji(k)] + Im[φ ji(k)], where only the diagonal terms as Re[φ ji(k)] and Im[φ ji(k)] represent the real part and the   φ k T | |2 −2Bt χ |σ |ϕ imaginary part of ji( ). The azimuthal angle satisfies ∼ 1 c− e − j − ψ˜ (k, t )|σ |ψ(k, t ) = lim + φ + j →∞ | |2 −2Bt +| |2 2Bt φ φ − φ 1 i tan( ji) hi ihj T T 0 c− e c+ e ei2 ji = ei2Re( ji)e 2Im( ji) = = ,  − φ − 2 2Bt 1 i tan( ji) hi ihj |c+| e χ+|σ |ϕ+   + j .   − dt (A3) h + ih | |2 2Bt +| |2 2Bt −2Im(φ ji)  i j  c− e c+ e e =  . (B1) hi − ihj For Hermitian systems, the energies are purely real, i.e., φ φ ε = ε∗ = Re[ ji(k)] and Im[ ji(k)] contribute to the argument and μ μ and B 0. The above equation can be simplified as i2φ (k) amplitude of e ji , respectively. Im[φ ji(k)] is a real, con-  ∂ φ = ψ˜ , |σ |ψ , =∼ | |2χ |σ |ϕ tinuous, and periodic function of k, so that S kIm( ji)dk (k t ) j (k t ) cμ μ j μ ∂ η = μ S kIm( ji)dk 0. It means that only the real part of the azimuthal angle contributes to the DWN,   2 2 h j = (|c+| −|c−| ) . (A4) 1 1 ε+ wd = ∂kRe(η ji)dk = ∂kRe(φ ji)dk. (B2) 2π S 2π S 2 2 Here, |c+| =|c−| ; otherwise ψ˜ (k, t )|σ j|ψ(k, t )=0 and Next, we will show that the real part of azimuthal angle is a its relation to the static Hamiltonian is lost. physical observable. According to Eq. (B1), the real part of For non-Hermitian systems, when B > 0, Eq. (A3) is ap- azimuthal angle satisfies proximately given by £hi+i£hj Im £h −i£h 2 φ = i j , |c+| h tan[2Re( ji)] + (B3) ψ˜ , |σ |ψ , =∼ χ |σ |ϕ = j . £hi i£hj (k t ) j (k t ) + j + (A5) Re £h −i£h |c+|2 ε+ i j where £ is a nonzero arbitrary constant. After some algebras, | |2 = Here, c+ 0, because the state will completely decay if one can rewrite the above relation as the initial state is prepared as |ϕ−. Similarly, when B < 0, tan φRR + tan φLL Eq. (A3) is approximately given by tan[2Re(φ )] = ji ji ji − φRR φLL 2 1 tan ji tan ji | −| h ψ˜ , |σ |ψ , ∼ c χ |σ |ϕ = j , (k t ) j (k t ) = − j − (A6) = φRR + φLL , |c−|2 ε− tan ji ji (B4)

023043-6 DYNAMIC WINDING NUMBER FOR EXPLORING BAND … PHYSICAL REVIEW RESEARCH 2, 023043 (2020) where APPENDIX C: MEASUREMENT OF CONVENTIONAL Re(£h ) + Im(£h ) WINDING NUMBER IN THE PRESENCE OR ABSENCE OF φRR = j i , tan ji CHIRAL SYMMETRY Re(£hi ) − Im(£hj ) The winding number has been widely used for characteriz- Re(£hj ) − Im(£hi ) tan φLL = , (B5) ing the topology of Hermitian systems with chiral symmetry. ji Re(£h ) + Im(£h ) i j In one dimension, the winding number can apply to both φRR φLL Hermitian and non-Hermitian systems with or without chiral which define two real angles ji (k) and ji (k), respec- φRR symmetry. Here, we consider a 1D two-band topological tively. It is worth noting that the two real angles ji (k) φLL system governed by the Hamiltonian, and ji (k) will be changed by different parameters £,but φ φ Re[ ji(k)] remains the same. The relation between Re[ ji(k)] H(k) = hxσx + hyσy + hzσz. (C1) and φRR(k),φLL(k) is given by ji ji The conventional winding number for each band is defined as π φ = η = 1 φRR + φLL + , [8,59] Re( ji) Re( ji) ji ji n (B6)  2 2 1 wμ = dkχμ|i∂ |ϕμ where n is an integer. It indicates the DWN w = 1 (wRR + π k  d 2 d S wLL). Here, wτ = 1 ∂ φτ · dk, τ ∈ RR, LL. ∂ − ∂ d d 2π S k jl = 1 hx khy hy khx , φRR φLL dk (C2) Interestingly, the two real angles ji (k) and ji (k) can 2π S εμ(εμ − hz ) be respectively replaced by time-averaged spin textures corre- π μ = sponding to |ψ(k, t ) and |ψ˜ (k, t ), where S is a closed loop with k varying from 0 to 2 ,   ±. Next, we will build a relation between the conventional ψ(k, t )|σ |ψ(k, t ) winding number and the DWN in different situations. φRR(k) = arctan j , ji When hz = 0, the Hamiltonian (C1) has chiral symmetry ψ(k, t )|σi|ψ(k, t )   H(k) =−H(k) with  = iσxσy. The conventional winding ψ˜ (k, t )|σ |ψ˜ (k, t ) numbers for different bands are the same, which we denote as φLL = j ,  ji (k) arctan (B7) ψ˜ , |σ |ψ˜ , 1 hx∂khy − hy∂khx (k t ) i (k t ) w± = dk . (C3) 2 2  2π S h + h • = 1 T • x y where limT →∞ T 0 dt. Equation (B7) indicates that the two real angles φRR(k) and φLL(k) are physical observ- The expression reduces to that in the Hermitian cases ji ji |χ =|ϕ ables. For simplicity, we prove the relation with two real when μ(k) μ(k) . If we define an azimuthal angle as φ = / angles, φRR(k) and φLL(k) in the case of B > 0. The right- yx(k) arctan(hy hx ), the above equation is given by yx yx  right spin textures defined with |ψ(k, t ) satisfy 1 w± = ∂ φ dk, π k yx (C4) ψ k, t |σ |ψ k, t ϕ+|σ |ϕ+ 2 S ( ) y ( ) = y , ϕ |σ |ϕ (B8) ψ(k, t )|σx|ψ(k, t ) + x + According to Eq. (B2), we can immediately conclude that the conventional winding number is equal to the DWN, and the left-left spin textures defined with |ψ˜ (k, t ) satisfy w± = wd , (C5) ψ˜ k, t |σ |ψ˜ k, t χ+|σ |χ+ ( ) y ( ) y 2 2 = . (B9) under a few constraints on the initial state: |c+(k)| =|c−(k)| ψ˜ , |σ |ψ˜ , χ+|σx|χ+ 2 2 (k t ) x (k t ) for Hermitian systems and |c+(k)| = 0 ∧|c−(k)| = 0for According to the Hamiltonian (1) in the main text, the right non-Hermitian systems. To numerically verify our theory, we = + + and left biorthogonal eigenvectors are given by consider the systems with hx J0 J1 cos(k) J2 cos(2k) and hy = J1 sin(k) + J2 sin(2k) − iδ. In the non-Hermitian 1 Tˆ case with δ = 0, the conventional winding number w± can |ϕμ(k)= (hx − ihy,εμ − hz ) , 2εμ(εμ − hz ) appear half of integer in some parameter regions, different 1 from integer value in the Hermitian systems. For simplicity, χμ(k)|= (h + ih ,εμ − h ), (B10) we take J = 1, and J , J , and δ are real. The dispersion of ε ε − x y z 1 0 2 2 μ( μ hz ) this Hamiltonian is  Tˆ −ik −2ik ik 2ik where the superscript is the transposition operation. Com- ε±(k) =± (J0 − δ + e + J2e )(J0 + δ + e + J2e ). bining Eqs. (B8), (B9), and (B10), we can immediately obtain (C6) ϕ |σ |ϕ + + y + = Re(hy£1) Im(hx£1), The energy is symmetric about zero energy, which is en- ϕ+|σx|ϕ+ Re(hx£1) − Im(hy£1) sured by the chiral symmetry. Since the energy gap must close at phase transition points, we can determine the phase χ+|σy|χ+ Re(hy£1) − Im(hx£1) = , (B11) boundaries by the band-crossing condition ε±(k) = 0, which χ+|σx|χ+ Re(hx£1) + Im(hy£1) yields J0 =±δ + 1 − J2 and J0 =±δ − 1 − J2 for arbitrary = ∗ + ε∗ = ∗ − ε∗ = ± δ | | > . = = where £1 hz +. Similarly, one can obtain £1 hz + J2. Particularly, J0 J2 if J2 0 5. Fixing J1 1, J2 for the case of B < 0. Combining Eqs. (B5) and (B11), one 1, and hz = 0 and changing both δ and J0, we calculate can easily obtain the relation in Eq. (B7). topological phase diagram distinguished by the conventional

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FIG. 4. (a) The phase diagram of the 1D chiral-symmetric FIG. 5. (a) The phase diagram of the 1D non-chiral-symmetric topological systems. The white, blue, green, dark-yellow, and topological systems. The white, green, and yellow regions share = bright-yellow regions respectively share winding numbers as w± winding number as wt = 0, 1, and 2, respectively. [(b)–(f)] , / , , / η 0 1 2 1 3 2, and 2. [(b)–(f)] Re[ yx (k)] as a function of k in differ- Re[ηyx (k)] as a function of k in different parameters (J0,δ), which are ent parameters (J0,δ), which are (1.5,0.2), (1.5,1), (0.5,1), (0.2,0.5), (1.7,0.3), (1,0.3), (0.3,0.3), (0.3,1), and (0.3,1.7), corresponding to and (0.5,0.2), corresponding to points b, c, d, e, and f in panel (a), points b, c, d, e, and f in panel (a), respectively. The other parameters = = respectively. The other parameters are chosen as J1 1, J2 1, and are chosen as J1 = 1, J2 = 0, and hz = 0.5. hz = 0. where the integral is also taken along a loop with k varying winding number; see Fig. 4(a). Here, the white, blue, green, from 0 to 2π. According to Eq. (B2), we can relate the dark-yellow, and bright-yellow regions possess conventional topological invariant wt to the DWN winding numbers w± = 0, 1/2, 1, 3/2, and 2, respectively. In w = w+ + w− = 2w , (C9) Figs. 4(b)–4(f), we also give the angle Re[ηyx(k)] as a function t d of the quasimomentum k with different parameters (J0,δ) 2 2 under a few constraints on the initial state: |c+(k)| =|c−(k)| marked as b, c, d, e, f in Fig. 4(a). 2 2 for Hermitian systems and |c+(k)| = 0 ∧|c−(k)| = 0for The DWNs are 0, 1, 1/2, 3/2, and 2, respectively. The non-Hermitian systems. Fixing J = 1, J = 0, and h = 0.5, DWNs extracted from Figs. 4(b)–4(f) are in good agreement 1 2 z we calculate the topological invariant w as functions of δ and with the conventional winding numbers in Fig. 4(a), demon- t J ; see Fig. 5(a). Here, the white, green, and bright-yellow strating the validity for our dynamic approach once again. 0 regions possess topological invariant w = 0, 1, and 2, respec- When h = 0, the Hamiltonian (C1) breaks the chiral sym- t z tively. In Figs. 5(b)–5(f), we also give the angle Re[η (k)] metry. Unlike the systems with chiral symmetry, the con- yx versus the quasimomentum k with different parameters (J ,δ) ventional winding number for each band is not a quantized 0 marked as b, c, d, e, and f in Fig. 5(a). number, which indicates that w± is no longer a topological The DWNs are 0, 1/2, 1, 1/2, and 0, respectively. The invariant. However, the sum of the winding numbers for double of DWNs based on Figs. 5(b)–5(f) are consistent with different bands, the equilibrium winding numbers of the phase diagrams in  Fig. 5(a), demonstrating the validity of our dynamic approach. 1 hx∂khy − hy∂khx w = w+ + w− = dk , (C7) t π 2 + 2 S hx hy APPENDIX D: ALTERNATE CHOICE OF REFERENCE has been demonstrated to be a topological invariant [35,59]. AXIS AND MEASUREMENT OF LARGER CHERN The topological invariant wt is independent of hz, though NUMBER its definition is related to the eigenvector of H(k). The pa- In the main text, we only consider θy(k) = arccos(hy/|h|) rameters hx and hy become very important for the definition φ = / 2 + 2 = and xz(k) arctan(hx hz ). Alternatively, we can also of topological invariant. Except for the EPs hx hy 0, we take θz(k) = arccos(hz/|h|) and φyx(k) = arctan(hy/hx ), or introduce a complex angle φyx(k) satisfying tan[φyx(k)] = θ = /| | φ = / hy/hx.Intermofφyx(k), wt can be represented as x (k) arccos(hx h(k) ) and zy(k) arctan(hz hy ). These  two choices lead to distinct observations, but give the same = 1 ∂ φ , Chern number. In Figs. 6(a) and 6(b), based on the different wt k yxdk (C8) φ π S choices of reference axis, we give the azimuthal angle yx(k)

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FIG. 6. Topologically nontrivial phase with Chern number C = 1. Hermitian case (δ = 0): (a) Azimuthal angle φyx (k)andφzy(k) in the parameter space k → (kx, ky ), where blue and red points FIG. 7. Topologically nontrivial phase with Chern number C = represent north and south poles of the Bloch spherical surface. 3. Hermitian case at top (δ = 0): panels (a) and (b) correspond to δ = . φ Non-Hermitian case ( 0 5): (b) Azimuthal angle Re[ yx (k)] and the azimuthal angle φxz(k)andηxz(k) in the parameter space k → φ , Re[ zy(k)] in the parameter space (kx ky). Here, blue and red points (kx, ky ). (c) Blue and red points represent north and south poles of represent north and south EPs of the virtual Bloch spherical surface. the Bloch spherical surface. Non-Hermitian case at bottom (δ = 0.1): The other parameters are chosen as Jx(y,z) = 1, mz = 1. panels (d) and (e) respectively correspond to the azimuthal angle Re[φxz(k)] and Re[ηxz(k)] in the parameter space (kx, ky). (f) Blue and red points represent north and south EPs of the virtual Bloch = = φ k δ = δ = . spherical surface. The other parameters are chosen as Jx Jy and zy( ) in Hermitian( 0) and non-Hermitian( 0 5) . , = = . cases, where the other parameters are the same as those in 0 2 Jz 1, and mz 0 5. Fig. 2(c) of the main text. Around the singularity points, one can also easily obtain the DWN, and the Chern number C = 1 in both the Hermitian and non-Hermitian cases, consistent APPENDIX E: EXPERIMENTAL CONSIDERATION with the ideal Chern number. This is because these references One can immediately apply the dynamical approach in only differ from a gauge transformation and the Chern number topological Hermitian systems. A cold-atom system is an does not depend on the choice of reference. excellent platform to realize topological band models and The dynamic approach is clearly applicable to topological detect topological invariants. One- and two-dimensional phases with larger Chern numbers. To show this, we consider spin-orbit couplings have been realized in a highly another two-band model which supports band structure with controllable Raman lattice [27,73–75]. Initial states are larger Chern numbers, this is, quite easily prepared by loading the atoms into the lattices. 2 2 Here, the initial constraint |c+(k)| =|c−(k)| may be not satisfied for some specific momentum k,butthe hx = Jx sin(2kx ); hy = Jy sin(2ky ); occurrence probability is so small that the global dynamical = − − − δ. hz mz Jz cos(kx ) Jz cos(ky ) i (D1) azimuthal angle is not affected due to the topological nature. The spin population N↑(↓)(k, t ) with different For the Hermitian case δ = 0, the trivial phase is lying in momenta can be measured by spin-resolved time-of-flight |m | > 2J , while the topological phases are distinguished (TOF) absorption imaging [27]. Thus, one can obtain z z ψ , |σ |ψ , = as (i) Jz < mz < 2Jz with the Chern number C =−1; (ii) the spin population difference (k t ) z (k t ) 0 < m < J with C = 3; (iii) −J < m < 0 with C =−3; [N↑(k, t ) − N↑(k, t )]/[N↑(k, t ) + N↑(k, t )]. The spin z z z z ψ , σ |ψ , and (iv) −2Jz < mz < −Jz with C = 1. Here we only verify textures (k t ) x(y) (k t ) can be transferred to the = spin population difference by applying π/2 pulse, that is, topological phase with Chern number C 3, where the other σ σ −i π y(x) i π y(x) parameters are chosen as Jx = Jy = 0.2, Jz = 1, and mz = ψ(k, t )|σx(y)|ψ(k, t )=ψ(k, t )|e 2 2 σze 2 2 |ψ(k, t ). 0.5. In Figs. 7(a) and 7(b), we give the azimuthal angle φxz(k) Because the cold-atom systems have long coherent time, and ηxz(k) in the parameter space (kx, ky ), respectively. One there is no obstacle to extract the DWN via long-time average can find that more north and south poles appear in Fig. 7(c), of the spin textures. compared with Fig. 2.FromEq.(14), one can also easily To apply the dynamical approach in topological non- obtain the Chern number C = 3 via DWN. For the non- Hermitian systems, we should first consider how to realize Hermitian case, we consider δ = 0.1 and the other parameters the topological non-Hermitian models in experiments. Since are the same as those in the Hermitian case. In Figs. 7(d) two-level non-Hermitian models have been widely realized in and 7(e), we also give the azimuthal angles Re[φxz(k)] and optical systems, such as two coupled optical cavities [76,77], Re[ηxz(k)] in the parameter space (kx, ky ), respectively. The optical waveguides [67,78,79], and optomechanical cavity DWNs around EPs become half, while the EPs become double [52,80,81], we mainly discuss how to extract DWN with two as the Hermitian counterpart; see Fig. 7(f). Eventually, the optical waveguides with tunable parameters. A two-level non- Chern number remains the same as that in the Hermitian case. Hermitian system can be realized by introducing gain and loss

023043-9 ZHU, KE, ZHONG, AND LEE PHYSICAL REVIEW RESEARCH 2, 023043 (2020) in the two waveguides. The coupling strength can be tuned the eigenstates in the long distance. Thus, the output intensity by the waveguide separation. We regard the two different difference of the waveguides is sufficient, and long-distance waveguides as two spin components. The initial states can average of the intensity difference is not necessary. One can be prepared by randomly splitting the light injecting into also obtain ψ(k, l)|σx(y)|ψ(k, l) by insetting a beam splitter the two waveguides. One can obtain ψ(k, l)|σz|ψ(k, l) by before intensity measurement. By designing the waveguide measuring the intensity difference between two waveguides separation and the gain (loss) rates, one can simulate the at propagating distance l. Here, the distance l plays the role two-band model. Repeating the above operations, one can of time. Actually, the final states will collapse onto one of finally construct the DWN.

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