Dynamic winding number for exploring band topology
Bo Zhu Yongguan Ke Honghua Zhong Chaohong Lee
quant-ph cond-mat.quant-gas




Laboratory of Quantum Engineering and Quantum Metrology, School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China
State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University (Guangzhou Campus), Guangzhou 510275, China



Laboratory of Quantum Engineering and Quantum Metrology, School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China
Nonlinear Physics Centre, Research School of Physics, The Australian National University, Canberra ACT 2601, Australia


Institute of Mathematics and Physics, Central South University of Forestry and Technology, Changsha 410004, China



lichaoh2@mail.sysu.edu.cn.
Laboratory of Quantum Engineering and Quantum Metrology, School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China
State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University (Guangzhou Campus), Guangzhou 510275, China
Nonlinear Physics Centre, Research School of Physics, The Australian National University, Canberra ACT 2601, Australia



Topological invariants play a key role in the characterization of topological states.

Due to 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 dynamical winding number directly gives the conventional winding number.

In two-dimension, the Chern number relates to the weighted sum of dynamic winding numbers of all phase singularity points.

This work opens a new avenue to measure topological invariants not requesting any prior knowledge of system topology via time-averaged spin textures.



Dynamic winding number for exploring band topology
    Chaohong Lee
    December 30, 2023
==================================================



Introduction. Topological invariant, a global quantity defined with static Bloch functions, has been widely used for classifying and characterizing topological states of matters, including insulators, superconductors, semimetals and waveguides etc <cit.>.

Nontrivial topological invariants attribute to novel topological effects, such as winding number for quantized geometric phase <cit.>, and Chern number for integer Hall effect <cit.> and Thouless pumping <cit.>.

Measuring topological invariants provide undoubted evidence of topological states, beneficial for precision measurement <cit.>, error-resistant spintronics <cit.> and quantum computing <cit.>.

Most existed methods to measure topological invariants are based on adiabatic band sweeping <cit.>. 

However, these methods do not work well for imperfect initial states and small energy gaps and become invalid for non-Hermitian systems.

In recent, topological invariants have been measured via linking numbers and band-inversion surfaces in quench dynamics <cit.>.

However, these quench schemes request prior knowledge of topology before and after quench.

As non-Hermitian systems may exhibit complex spectra and exceptional points (EPs) <cit.>, their topological states have stimulated extensive interests <cit.>.

Conventional topological invariants such as winding number and Chern number have been generalized to non-Hermitian systems <cit.>, and new topological invariants such as vorticity have been introduced <cit.>.

Due to the EPs,  the winding number in non-Hermitian systems may take half-integers <cit.>.

Besides, non-Bloch definition of Chern number strictly gives the numbers of chiral edge modes <cit.>.

How to measure these topological invariants is more challenging than that in Hermitian systems.

For an example, the Hall conductivity is no longer quantized despite being classified as a Chern insulator based on non-Hermitian topological band theory <cit.>.

In one-dimension, the winding number in a non-Hermitian system has been determined via the mean displacement in long-time quantum walk <cit.>, but it does not works for measuring Chern numbers and half-integer winding numbers.

One may ask, is there a unified dynamic approach for measuring topological invariants in both Hermitian and non-Hermitian systems?

In this Letter, we study a generic two-band model which supports nontrivial topological invariants in both Hermitian and non-Hermitian regions.

We define a dynamic winding number (DWN) for the time-averaged spin textures, which is robust against various initial states.

In one-dimension, we prove that the DWNs directly gives the conventional winding numbers in both chiral- and non-chiral-symmetric systems.

In two-dimension, the Chern number relates to the weighted sum of DWNs around all singularity points (SPs), where the weight is +1 for the north pole and -1 for the south pole.

When the system change from Hermitian to non-Hermitian, each singularity point will split into two EPs (which are also SPs), the Chern number can still be extracted via the DWNs of all EPs.

Without requesting any prior knowledge of their topology, our approach provides a general guidance for measuring topological invariants in both Hermitian and non-Hermitian systems.


 Dynamic winding number. We consider a general two-band model in d-dimension.

The Hamiltonian in momentum space is composed of three Pauli matrices,

    H(k)=h_x(k)σ_x+h_y(k)σ_y+h_z(k)σ_z.

Here, k is the quasi-momentum, h_x(y,z) are periodic functions of k.

The Hamiltonian could be Hermitian H^†=H or non-Hermitian H^† H.

Then, the right and left eigenvectors are respectively given by H(k)|φ_μ⟩=ε_μ|φ_μ⟩ ( H^†(k)|χ_μ⟩=ε^*_μ|χ_μ⟩), where μ=±, and ε_±=±(h_x^2+h_y^2+h_z^2)^1/2 are the eigenvalues.

For Hermitian systems, |φ_μ⟩=|χ_μ⟩ and ε_μ=ε_μ^*.

For non-Hermitian systems, neither the eigenstates |φ_μ⟩ nor ⟨χ_μ| are orthogonal.

We adopt biorthogonal vectors which fulfil ⟨χ_ν| φ_μ⟩=δ_ν,μ and ∑_μ|φ_μ⟩⟨χ_μ |=1 by normalizing  |φ_μ⟩=|φ_μ⟩/N_μ and ⟨χ_μ|=⟨χ_μ|/N_μ with N_μ=√(⟨χ_μ| φ_μ⟩).


For an arbitrary initial state |ψ_k(0)⟩=∑_μ c_μ|φ_μ⟩ and its associated state
⟨ψ̃_k(0)|=∑_μ c_μ^*⟨χ_μ|,

the time-evolution of |ψ_k(t)⟩ and ⟨ψ̃_k(t)| respectively satisfy |ψ_k(t)⟩=∑_μ c_μ e^-iε_μ t|φ_μ⟩ and ⟨ψ̃_k(t)|=∑_μ c_μ^* e^iε_μ^* t⟨χ_μ|.

According to the biorthogonal quantum mechanics <cit.>, the spin
textures are given by the expectation values of Pauli matrices, ⟨σ_j (k,t)⟩ =⟨ψ̃_k(t) | σ_j |ψ_k(t)⟩/⟨ψ̃_k(t) | ψ_k(t)⟩,
where j∈x,y,z.

We are interesting in its long-time average,

σ_j( k)=lim_T→∞1/T∫_0^T⟨σ_j (k,t)⟩ dt.


As the quasimomentum continuously varies, (σ_i, σ_j) will form a trajectory in the polarization plane.

The DWN of the spin vector (σ_i, σ_j) is defined as

    w_d=1/2π∮_S∂_ kη_ji( k) dk,

where S is a close loop in parameter space k, and dynamical azimuthal angle η_ji( k)= arctan [σ_j( k)/σ_i( k)].

It is easy to prove that the DWN is convergent to the equilibrium azimuthal angle,
 
    η_ji=ϕ_ji≡arctan[h_j(k)/h_i(k)],


if |c_+|^2|c_-|^2 for Hermitian systems and |c_+|^2 0⋂ |c_-|^2 0 for non-Hermitian systems <cit.>.

For Hermitian systems, the DWN can be directly probed by the long-time average of spin textures.

For non-Hermitian systems, η_ji is a complex angle which cannot be directly observed.

This problem can be fixed by decomposing the azimuthal angle into real and imaginary parts.

We find that only the real part of η_ji contributes to the DWN and it satisfies,

    (η_ji)=1/2(ϕ_ji^RR+ ϕ_ji^LL)+n π/2,


 where ϕ_ji^RR=arctan(⟨ψ_k |σ_j|ψ_k⟩/⟨ψ_k |σ_i|ψ_k⟩) and ϕ_ji^LL=arctan(⟨ψ̃_k |σ_j|ψ̃_k⟩/⟨ψ̃_k |σ_i|ψ̃_k⟩) are both real <cit.>, (η_ji) represents the real part of η_ji.

Thus we have

w_d=1/2(w_d^RR+w_d^LL),

where w_d^τ=1/2π∮_S ∂_kϕ_jl^τ dk, τ∈ RR, LL.

This means that the DWN can also be observed by the time-evolution of left-left and right-right spin textures whose dynamics are respectively governed by Ĥ and Ĥ^†.

In the following, we show how to utilize the DWN to uncover the topology in both Hermitian and non-Hermitian systems.


 Connection between conventional winding number and dynamic winding number. In one-dimension, if h_z=0, the Hamiltonian (<ref>) has chiral symmetry Γ H(k) Γ =-H(k) with Γ=i σ_x σ_y and k→ k.
 
The conventional winding number w_± for the Hamiltonian (<ref>) reads as,

    w_±=1/2π∮_S dk h_x∂_k h_y-h_y∂_k h_x/(ε_±)^2,

which associates with the Zak phase.

According to Eqs. (<ref>) and (<ref>), one can find that w_± = w_d.









If h_z 0, the Hamiltonian (<ref>) breaks the chiral symmetry.

The winding numbers w_± can be given as,

    w_±=1/2π∮_S dk h_x∂_k h_y-h_y∂_k h_x/ε_±(ε_±-h_z).


Unlike the systems with chiral symmetry, the conventional winding number for each band is not a quantized number, which indicates that w_± is no longer a topological invariant.

However, the sum of two conventional winding numbers,

    w_t=w_++w_-=1/π∮_S dk h_x∂_k h_y-h_y∂_k h_x/h_x^2+h_y^2,

relates to the dynamic winding number via w_t=2w_d and thus it can be used as a topological invariant.







As an example, we consider a system with h_x=J_0+J_1cos(k), h_y=J_1sin(k)-iδ and h_z=0.

In the Hermitian case, the parameters are chosen as δ=0 and J_1=1.


The conventional winding number w_±=1 for |J_0|<J_1, and w_±=0 for |J_0|>J_1.

We first calculate the time-evolution of ⟨σ_x(y)(k,t)⟩ and their long-time average with J_0=0.5J_1, see  Figs. <ref>(a)-(c).

The spin textures ⟨σ_x(y)(k,t)⟩ oscillate with a momentum-dependent period t_k=π/|ε_μ|, and their long-time averages σ_x(y) depend on quasi-momentum, see the black and red lines in Fig. <ref> (c).

With σ_x and σ_y, we calculate η_yx as a function of k in Fig. <ref>(d), where two discontinuity points k_1 and k_2 appear.

The DWN can be obtained via the integral of piecewise function,

    w_d   =   1/2π(∫_-π^k_1∂_kη_yx dk+∫_k_1^k_2∂_kη_yx dk+∫_k_2^π∂_kη_yx dk).

We find that the DWN is equal to 1, the same as the conventional winding number w_±.


When δ 0, the system becomes non-Hermitian and one always needs to measure both ϕ_ji^RR and ϕ_ji^LL to extract the DWN.

For a chiral-symmetric system (whose parameters are given as J_0=J_1=1, δ=0.3, h_z=0 and w_±=1/2), the two dynamic azimuthal angles ϕ_ji^RR=ϕ_ji^LL, and w_d^RR=w_d^LL=1/2, see in Fig. <ref>(e).

It means that we only need to measure ϕ_ji^RR or ϕ_ji^LL in experiments.

For a non-chiral-symmetric system (whose parameters are given as J_0=J_1=1, δ=0.3, h_z=0.5, and w_t=1), we find ϕ_ji^RR≠ϕ_ji^LL, and w_d^RR=1, w_d^LL=0, see Fig. <ref>(f).



Nevertheless, the conventional winding number can be obtain by measuring the w_d=(w_d^RR+w_d^LL)/2 in both chiral and non-chiral symmetric systems.


Connection between Chern number and dynamic winding number.
By generalizing the concept of gapped band structures from Hermitian to non-Hermitian systems,

the Chern number for an energy separable band can be constructed in a similar way <cit.>.

In contrast to Hermitian systems, there are left-right, right-right, left-left, right-left Chern numbers in non-Hermitian systems, dependent on the definitions of Berry connection, A_k^LR= i ⟨χ_μ| ∂_k |φ_μ⟩, A_k^RR= i ⟨φ_μ| ∂_k |φ_μ⟩, A_k^LL= i ⟨χ_μ| ∂_k |χ_μ⟩, and A_k^RL= i ⟨φ_μ| ∂_k |χ_μ⟩.

Although the corresponding Berry curvatures are locally different quantities, but the four kinds of Chern numbers are the same <cit.>.

Here, we only focus on analyzing the Chern number defined with left-right Berry connection A_k^LR=i ⟨χ_μ| ∂_k|φ_μ⟩, which naturely reduces the Chern number in Hermitian systems as |φ_μ⟩=|χ_μ⟩.



We map the Hamiltonian to a normalized vector,
n⃗(k)=(sin(θ_i)cos(ϕ_jl), sin(θ_i)sin(ϕ_jl), cos(θ_i)), which reduces a Bloch vector in Hermitian systems.

Here, θ_i denotes the angle between the vector and the axis-i, and ϕ_jl denotes the equilibrium azimuthal angle in the j-l plane.

The reference axis is free to choose without affecting the validity of the dynamic approach.



Then, the left and right eigenstates for the low-energy band are given as

    ⟨χ_-(θ_i,ϕ_jl)|=[ -e^iϕ_jl/2cos(θ_i/2),
    e^-iϕ_jl/2sin(θ_i/2) ], 
          |φ_-(θ_i,ϕ_jl)⟩=[ -e^-iϕ_jl/2cos(θ_i/2);   e^iϕ_jl/2sin(θ_i/2) ].




The right and left eigenstates have a phase singularity at n⃗(k)=(0, 0,± 1), in which + and - respectively correspond to north and south poles.

In the parameter space (k_x,k_y), the location k_0 of the poles satisfy h_j(k_0)^2+h_l(k_0)^2=0.



The left-right Berry connection A_k_x(y)^LR  of the low-energy band are given by

    A_k_x(y)^LR=i⟨χ_-|∂_k_x(y)|φ_-⟩=cos(θ_i)/2∂ϕ_jl/∂_k_x(y).


We discreterize the parameter space (k_x,k_y) by N× M mesh grids in the first Briliouin zone <cit.>.

For each grid, a direct application of the two-dimensional Stokes theorem implies <cit.>

    C^LR=1/2π∑_l_x=1^N∑_l_y=1^M∮_S_l_x,l_y(A_k_xdk_x+A_k_ydk_y),

where S_l_x,l_y represents the clockwise path integration of the (l_x,l_y) grid.

We find that the Chern number is determined by the winding numbers for all SPs, where cos(θ_i)=h_i(k_0)/|h⃗(k_0)|=sgn([h_i(k_0)])=1 for the north SPs and cos(θ_i)=-1 for the south SPs,  [h_i(k_0)] represents the real part of h_i(k_0), and h⃗=(h_x, h_y, h_z).


At last, we can deduce the left-right Chern number as

    C^LR=1/2∑_k_0 ∈SPs sgn([h_i(k_0)])w_d(k_0),


where w_d(k_0) is the DWN for the SP at k_0 <cit.>.
In non-Hermitian case, w_d(k_0) is relevant to two real angles ϕ_jl^RR and ϕ_jl^LL, which can be respectively extracted via right-right spin textures ⟨ψ_k(t)| σ_j(l)| ψ_k(t)⟩ and left-left spin textures ⟨ψ̃_k(t)| σ_j(l)| ψ̃_k(t)⟩.

As an example, we consider

h_x=J_x sin(k_x), h_y=J_y sin(k_y) and h_z=m_z-J_z cos(k_x)-J_z cos(k_y)-i δ <cit.>.

Here, J_x(y,z) denote spin-orbit coupling parameters, m_z is the effective magnetization, and δ is a gain or loss strength.

When δ=0, the system is a quantum anomalous Hall model <cit.>, which has  been realized in recent experiments <cit.>.


In the Hermitian case (J_x(y,z)=1, m_z=1, δ=0), the north and south poles in the parameter space (k_x,k_y) can be determined as following.

Since the poles are related to the chosen axis, we select θ=θ_y=arccos(h_y/|h⃗(k)|) and ϕ=ϕ_xz=arctan(h_x/h_z).

However, the validity of our dynamic approach is independent on the choice of reference axis <cit.>.

In the parameter space (k_x,k_y), by solving h_x^2+h_z^2=0, we find that the north and south poles locate at k_0=(k_x,k_y)=(0,±π/2).



Then, we need to extract the DWN around the two poles.

We randomly choose an initial state |ψ_k(0)⟩=∑_μ c_μ|φ_μ⟩ with |c_+|^2>|c_-|^2 and calculate the spin textures ⟨σ_x( k, t)⟩ and ⟨σ_z( k, t)⟩.




The dynamical azimuthal angle η_xz( k) can be extracted via long-time average values σ_x(z)( k), see Figs. <ref>(a) and <ref>(b).

We also calculate the equilibrium azimuthal angle ϕ_xz( k) via the eigenstates, see Fig. <ref>(c).

The difference between η_xz( k) and ϕ_xz( k) gradually disappears with the increase of total time T.



Thus we can obtain the DWNs for the north and south poles via integrating the gradient of η_xz( k) in Fig. <ref>(d) and <ref>(e), respectively.

The DWNs for the north and south poles are respectively given as w_d=± 1.

Applying Eq. (<ref>), one can obtain the Chern number as 1, which is consistent with the one calculated via integrating the static Berry curvature over the whole in the parameter space (k_x,k_y).




In the more generalized case, we first show the topological phase diagram in the parameter plane (m_z,δ) by setting J_x(y,z)=1, see Fig. <ref>(a).


The Chern numbers of the first band are C=0 and 1 in the green and grey regions, and not well defined in the white region.

The boundaries satisfy (m_z-1)^2+δ^2<1 for the gray region and m_z>2 for the green region.

Varying δ along the dashed red arrow, in Figs. <ref>(b)–(d) we explore the correspondence between the Chern number and energy modes under open boundary condition, corresponding to the parameter points b-d.

For the topological nontrivial phase, one can see that the complex bulk bands are still gapped and the edge-state modes are still preserved in the real energy axis, see Fig. <ref>(b).

Now we consider δ=0.5 and keep other parameters the same as those in Fig. <ref>(b).

We find that the two poles in Hermitian case are split into four EPs, see Fig. <ref>(e).





The blue and red points represent north and  south EPs, respectively.



In Figs. <ref>(f), we give (η_xz) in parameter space (k_x,k_y), and the DWNs can be obtain by
integral of the dynamic azimuthal-angle gradient along the trajectory enclosing the north and south EPs.

Here, the DWNs for the north and south EPs are w_d=1/2 and -1/2, respectively.

Although the SPs are doubled, as each DWN is reduced by half, the Chern number keep unchanged.

For completeness, we also show the dynamic azimuthal angles defined with the right-right and left-left spin textures, see Fig. <ref>(g) and <ref>(h) .

The dynamic azimuthal angles are quite different from each other, corresponding to ϕ_xz^RR and ϕ_xz^LL, respectively.

Nevertheless, the north and south EPs are the same as those in the Fig. <ref>(e).

Around an EP, one can extract the right-right and left-left DWN w_d^RR and w_d^LL, which satisfy w_d=1/2(w_d^RR+w_d^LL).



One important thing is that the ϕ_xz^RR and ϕ_xz^LL defined with the real left-left and right-right spin textures are accessible in experimental measurements.

In addition, our approach can also extract larger Chern numbers without extra efforts <cit.>, which is very hard to access by adiabatic band sweeping.


Conclusions and Discussions. We put forward a new concept of dynamic winding number (DWN) and uncover its connection to conventional topological 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 dynamic azimuthal-angle gradient enclosing a single singularity point.

We find that, (i) the conventional winding numbers in one-dimensional systems can be directly given by the corresponding DWNs, and (ii) the Chern numbers in two-dimensional systems relates to the weighted sum of all corresponding DWNs.

Our scheme has two main advantages.

Firstly, in contrast to the quench schemes via measuring linking numbers <cit.> and band-inversion surfaces <cit.>, which request prior knowledge of topology before and after quench, our scheme does not request any prior knowledge.

Secondly, our scheme can be used to measure half-integer winding numbers in non-Hermitian one-dimensional systems, which can not be measured via previous methods.

Our scheme is readily realized in various systems, ranging from cold atoms in optical lattices, optical waveguide arrays, to optomechanical devices.


In Supplementary Material <cit.>, we provide more details about the experimental realization of our scheme via cold atoms and optical waveguide arrays.

In future, it would be interesting to extend our scheme to measure topological invariants in high-dimensional systems, multi-band systems, periodically driven systems and disordered systems.





B.Z. and Y.K. made equal contributions. The authors thank Yuri S. Kivshar, Andrey A. Sukhorukov, Zhihuang Luo, Yuangang Deng, Shi Hu, Ling Lin, and Zhoutao Lei for discussions. This work is supported by the National Natural Science Foundation of China (NNSFC) under Grants No. 11374375, No. 11574405, No. 11805283 and No. 11904419, the Hunan Provincial Natural Science Foundation under Grants No. 2019JJ30044, and the International Postdoctoral Exchange Fellowship Program No. 20180052.










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  (S1) Convergence of dynamic winding number; (S2) Relation between dynamic
  winding number and time-averaged spin textures; (S3) Dynamic winding number
  in the present/absent of chiral symmetry; (S4) Chern number in 2D systems;
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Supplemental Material




sfigure




 
 









§ S1. CONVERGENCE OF DYNAMIC WINDING NUMBER

According to Eq.(2) in the main text, it seems that the definition of dynamic winding number depends on the initial state, and it is unclear whether such number is convergent in the long time.

Here, we prove that the initial state can be rather general and the dynamic winding number is convergent.

The time-average of ⟨σ_j⟩ is given as


    σ_j=lim_T→∞1/T∫_0^T∑_μ,μ'c_μc_μ'^*e^-i(ε_μ-ε_μ'^*)t⟨χ_μ'|σ_j|φ_μ⟩/∑_μ|c_μ|^2e^-i(ε_μ-ε_μ^*)tdt.


For Hermitian systems, ε_μ=ε_μ^*, and the periodic terms vanish in the long time average and only the diagonal terms preserve.

The above equation can be simplified as

    σ_j= ∑_μ|c_μ|^2
      ⟨χ_μ|σ_j|φ_μ⟩=(|c_+|^2-|c_-|^2)h_j/ε_+


For the non-Hermitian systems, we assume that the eigenenergy ε_μ=μ(A+iB), where A and B are real numbers.

When B>0, Eq. (<ref>) is approximately given as

    σ_j   =   ⟨χ_+|σ_j|φ_+⟩=h_j/ε_+.


Similarly, when  B<0, Eq. (<ref>) is approximately given as

    σ_j   =   ⟨χ_-|σ_j|φ_-⟩=-h_j/ε_+.


Combining with Eqs. (<ref>) and  (<ref>), we can also obtain

    σ_j/σ_i=h_j/h_i,

in the conditions |c_+|^2|c_-|^2 for Hermitian systems and  c_+ 0 for B>0 or c_- 0 for B<0 in non-Hermitian systems.

It means that the dynamic winding number also converges in the long time limits when the initial state satisfies a few constraint.

According to the Eq. (<ref>), one can also obtain

    η_ji=ϕ_ji=arctan(⟨χ_μ|σ_j|φ_μ⟩/⟨χ_μ|σ_i|φ_μ⟩),

where the azimuthal angle η_ji=arctan(σ_j/σ_i) and ϕ_ji=arctan(h_j/h_i).



§ S2. RELATION BETWEEN DYNAMIC WINDING NUMBER AND TIME-AVERAGED SPIN TEXTURES 

For the non-Hermitian case, the azimuthal angle η_ji and ϕ_ji is generally a complex angle, so that they do not represent physical observables in the biorthogonal system.
 
This problem can be fixed by  decomposing the azimuthal angle into two parts, ϕ_ji=(ϕ_ji)+(ϕ_ji), where (ϕ_ji) and (ϕ_ji) represents the real part and image part of ϕ_ji.

The azimuthal angle satisfies

    e^i2ϕ_ji=e^i2(ϕ_ji)e^-2(ϕ_ji)=1+itan(ϕ_ji)/1-itan(ϕ_ji)=h_i+ih_j/h_i-ih_j,
    
          e^-2(ϕ_ji)=|h_i+ih_j/h_i-ih_j|,

(ϕ_ji) and (ϕ_ji) contribute to the argument and amplitude, respectively.

(ϕ_ji) is a real continuous periodic function of k, so that ∮_S∂_ k[ϕ_ji( k)]dk=∮_S∂_ k[η_ji( k)]dk=0. It means that only the real part of azimuthal angle contributes to the dynamic winding number,

    w_d=1/2π∮_S∂_ k(η_ji) dk=1/2π∮_S∂_ k(ϕ_ji) dk,

Next, we will show that the real part of azimuthal angle is a physical observable.

According to the Eq. (<ref>), the real part of azimuthal angle satisfies


    tan(2(ϕ_ji))=( h_i+i h_j/ h_i-i h_j)/( h_i+i h_j/ h_i-i h_j),

where  is a nonzero arbitrary constant.

After some algebras, one can rewrite the above relation as

    tan(2(ϕ_ji))=tan(ϕ_ji^RR)+tan(ϕ_ji^LL)/1-tan(ϕ_ji^RR)tan(ϕ_ji^LL)=tan(ϕ_ji^RR+ϕ_ji^LL),

where

    tan(ϕ_ji^RR)=( h_j)+( h_i)/( h_i)-( h_j), 
    tan(ϕ_ji^LL)=( h_j)-( h_i)/( h_i)+( h_j),

which define two real angles ϕ_ji^RR and ϕ_ji^LL, respectively.

It is worth noting that the two real angles ϕ_ji^RR and ϕ_ji^LL will be changed by different parameters , but (ϕ_ji) still keeps the same.

The relation between (ϕ_ji) and ϕ_ji^RR, ϕ_ji^LL satisfies

    (ϕ_ji)=(η_ji)=1/2(ϕ_ji^RR+ ϕ_ji^LL)+n π/2,

where n is an integer. It means the dynamic winding number w_d=1/2(w_d^RR+w_d^LL).

Here, w_d^τ=1/2π∮_S ∂_kϕ_jl^τ dk, τ∈ RR, LL.

Interestingly, the two real angles ϕ_ji^RR and ϕ_ji^LL can be respectively replaced by time-averaged spin textures corresponding to |ψ_k(t)⟩ and |ψ̃_k(t)⟩,

    ϕ_ji^RR=arctan(⟨ψ_k(t) |σ_j|ψ_k(t)⟩/⟨ψ_k(t) |σ_i|ψ_k(t)⟩),
    
          ϕ_ji^LL=arctan(⟨ψ̃_k(t) |σ_j|ψ̃_k(t)⟩/⟨ψ̃_k(t) |σ_i|ψ̃_k(t)⟩),

where ⟨∙⟩= lim_T →∞1/T∫_0^T⟨∙⟩ dt.

The Eqs. (<ref>) indicates that the two real angles ϕ_ji^RR and ϕ_ji^LL are physical observables.

For simplicity, we prove the relation with two real angles, ϕ_yx^RR and ϕ_yx^LL in the case of B>0.



The right-right spin textures defined with |ψ_k(t)⟩ satisfies

    ⟨ψ_k(t) |σ_y|ψ_k(t)⟩/⟨ψ_k(t) |σ_x|ψ_k(t)⟩=⟨φ_+|σ_y|φ_+⟩/⟨φ_+|σ_x|φ_+⟩,

and the left-left spin textures defined with |ψ̃_k(t)⟩ satisfies

    ⟨ψ̃_k(t)|σ_y|ψ̃_k(t)⟩/⟨ψ̃_k(t) |σ_x|ψ̃_k(t)⟩=⟨χ_+|σ_y|χ_+⟩/⟨χ_+|σ_x|χ_+⟩.


According to the Hamiltonian (1) in main text, neither the eigenstates |φ_μ⟩ nor ⟨χ_μ| are orthogonal in the non-Hermitian system.

We adopt biorthogonal vectors which fulfill ⟨χ_ν| φ_μ⟩=δ_ν,μ and ∑_μ|φ_μ⟩⟨χ_μ |=1 by normalizing  |φ_μ⟩=|φ_μ⟩/N_μ and ⟨χ_μ|=⟨χ_μ|/N_μ with N_μ=√(⟨χ_μ| φ_μ⟩), this is,


    |φ_μ⟩   =   1/√(2ε_μ(ε_μ-h_z))(h_x-ih_y,ε_μ-h_z)^T̂, 
    ⟨χ_μ|   =   1/√(2ε_μ(ε_μ-h_z))(h_x+ih_y,ε_μ-h_z),

where the superscript T̂ is the transpose operation.

Combining with Eq. (<ref>), (<ref>) and (<ref>), we can immediately obtain,

    ⟨φ_+|σ_y|φ_+⟩/⟨φ_+|σ_x|φ_+⟩= (h_y_1)+(h_x_1)/(h_x_1)-(h_y_1), 
          ⟨χ_+|σ_y|χ_+⟩/⟨χ_+|σ_x|χ_+⟩= (h_y_1)-(h_x_1)/(h_x_1)+(h_y_1),

where _1=h_z^*+ε_+^*. Similarly, one can obtain _1=h_z^*-ε_+^* for the case of B<0.

Combining with Eq. (<ref>) and (<ref>), one can easily obtain the relations of Eq. (<ref>).



§ S3. DYNAMIC WINDING NUMBER IN THE PRESENCE/ABSENCE OF CHIRAL SYMMETRY 

Winding number has been widely used for characterizing the topology of Hermitian systems with chiral symmetry.

In one dimension, winding number can be applied to both Hermitian and non-Hermitian systems with or without chiral symmetry.

Here, we consider a 1D two-band topological system governed by the Hamiltonian,

    H(k)=h_x(k)σ_x+h_y(k)σ_y+h_z(k)σ_z.

The conventional winding number for each band is defined as <cit.>,

    w_μ=1/π∮_S dk ⟨χ_μ|i∂_k|φ_μ⟩
    =1/2π∮_S dk h_x∂_k h_y-h_y∂_k h_x/ε_μ(ε_μ-h_z).


where S is a closed loop with k varying from 0 to 2π.
Next, we will build relation between conventional winding number to the dynamic winding number in different situations.



 §.§ A. Chiral symmetric systems
 
When h_z=0, the Hamiltonian (<ref>) has chiral symmetry Γ H(k) Γ =-H(k) with Γ=i σ_x σ_y.
The conventional winding numbers for different bands are the same, and we denote as

    w_±=1/2π∮_S dk h_x∂_k h_y-h_y∂_k h_x/h_x^2+h_y^2.


The expression reduces to the Hermitian cases when ⟨χ_μ|=⟨φ_μ|.

If we define an azimuthal angle as ϕ_yx=arctan(h_y/h_x), the above equation is given as

    w_±=1/2π∮_S ∂_k ϕ_yx dk,


According to Eq. (<ref>), we can immediately conclude that the conventional winding number is equal to the dynamic winding number,

    w_±=w_d,

under  a few constraints of initial state:  |c_+|^2|c_-|^2 for Hermitian systems and |c_+|^2 0⋂ |c_-|^2 0 for non-Hermitian systems.





To numerically verify our theory, we consider the systems with h_x=J_0+J_1cos(k)+J_2cos(2k) and h_y=J_1sin(k)+J_2sin(2k)-iδ.

In the non-Hermitian case with δ≠0, the conventional winding numbers w_± can appear half integer values in some parameter ranges, different from the integer values in the Hermitian systems.

For simplicity, we take J_1=1, and J_0, J_2, δ are real.
The dispersion of this Hamiltonian is

    ε_±(k)=±√((J_0-δ+e^-ik+J_2e^-2ik)(J_0+δ+e^ik+J_2e^2ik)).


The energy is symmetric about zero energy, which is ensured by the chiral symmetry.
Since the energy gap must close at phase transition points, we can determine the phase boundaries by the band-crossing condition ε_±(k)=0, which yields J_0=±δ+1-J_2 and J_0=±δ-1-J_2 for arbitrary J_2. Particularly, J_0=J_2±δ if |J_2|>0.5.

Fixing J_1=1, J_2=1 and h_z=0 and changing both δ and J_0, we calculate topological phase diagram distinguished by their winding numbers, see Fig. <ref> (a).
Here, the white, blue, green, dark-yellow and bright-yellow regions possess conventional winding number w_±=0, 1/2, 1, 3/2 and 2, respectively.

In Figs. <ref> (b)-(f), we also give the angle (η_yx) as a function of quasi-momentum k with different parameters (J_0,δ) marked as b, c, d, e, f in the Figs. <ref> (a).
 
 The dynamic winding number are 0, 1, 1/2, 3/2 and 2, respectively.

The numerical results are in well agreement with the theoretical prediction, which prove the validity for our dynamic approach once again.



 §.§ B. Non-chiral symmetric systems

When h_z 0, the Hamiltonian (<ref>) breaks the chiral symmetry.

Unlike the systems with chiral symmetry, the conventional winding number for each band is not a quantized number, which indicates that w_± is no longer a topological invariant.

However, the sum of the winding numbers for different bands,

    w_t=w_++w_-=1/π∮_S dk h_x∂_k h_y-h_y∂_k h_x/h_x^2+h_y^2,

has been demonstrated to be a topological invariant <cit.>.








The topological invariant w_t is independent of h_z, although its definition is related to the eigenvector of H(k).

The parameters h_x and h_y become very important for the definition of topological invariant.

Except for the exceptional point h_x^2+h_y^2=0, we introduce a complex angle ϕ_yx satisfying tan(ϕ_yx)=h_y/h_x.

In terms of ϕ_yx, w_t can be represented as

    w_t=1/π∮_S ∂_kϕ_yx dk,

where the integral is also taken along a loop with k from 0 to 2π.

According to Eq. (<ref>), we can relate the topological invariant w_t to the dynamic winding number

    w_t=w_++w_-= 2 w_d,

under a few constraints of initial state:  |c_+|^2|c_-|^2 for Hermitian systems and |c_+|^2 0⋂ |c_-|^2 0 for non-Hermitian systems.

Fixing J_1=1, J_2=0 and h_z=0.5 in the same model as that in Subsec. <ref>, we calculate the topological invariant w_t as a function of δ and J_0, see Fig. <ref> (a).
Here, the white, green, and bright-yellow regions possess topological invariant w_t=0, 1 and 2, respectively.

In Figs. <ref> (b)-(f), we also give the angle (η_yx) versus the quasi-momentum k with different parameters (J_0,δ) marked as b, c, d, e, f in the Fig. <ref> (a).
 
The dynamic winding number are 0, 1/2, 1, 1/2 and 0, respectively.

The numerical results are also in well agreement with the theoretical prediction, which demonstrate the validity of our dynamic approach.





§ S4. CHERN NUMBER IN 2D SYSTEMS  



 §.§ A. Alternate choice of reference axis


In our main text, we only consider θ=θ_y=arccos(h_y/|h⃗(k)|) and ϕ=ϕ_xz=arctan(h_x/h_z).

Alternatively, we can also take θ=θ_z=arccos(h_z/|h⃗(k)|)  and ϕ=ϕ_yx=arctan(h_y/h_x), or θ=θ_x=arccos(h_x/|h⃗(k)|)  and ϕ=ϕ_zy=arctan(h_z/h_y).

These two choices lead to distinct observations, but give the same Chern number.

In Fig. <ref>(a) and (b), based on different choices of reference axis, we give the azimuthal angle ϕ_yx and ϕ_zy in Hermitian(δ=0) and non-Hermitain(δ=0.5) cases, where the other parameters are the same as the Fig. 2(c) of the main text.

Around the  singularity points, one can also easily obtain the dynamic winding number, and the Chern number C=1 in both the Hermitian and non-Hermitian cases, consistent with the ideal Chern number.

This is because the different references only differ from a gauge transformation and the Chern number do not depend on the choice of reference.



 §.§ B. Larger Chern number  



The dynamic approach is clearly applicable to topological phases with larger Chern numbers.
To show this, we consider another two-band model which supports band structure with larger Chern number, this is,

    h_x   =   J_x sin(2k_x); h_y=J_y sin(2k_y); h_z=m_z-J_z cos(k_x)-J_z cos(k_y)-i δ.


For the Hermitian case δ=0, the trivial phase is lying in |m_z|>2J_z, while the topological phases are distinguished as: (i) J_z<m_z<2J_z with the Chern number C=-1; (ii) 0<m_z<J_z with C=3; (iii) -J_z<m_z<0 with C=-3; (iv) -2J_z<m_z<-J_z with C=1.
Here we only verify topological phase with Chern number C=3, where the other parameters are chosen as J_x=J_y=0.2, J_z=1 and m_z=0.5.

In Figs. <ref>(a) and <ref>(b), we give the azimuthal angle ϕ_xz and η_xz in parameter space (k_x,k_y), respectively.

One can find that more north and south poles appear in Fig. <ref>(c), compared with Fig. 2 in the main text.

From the Eq. (10) in main text, one can also easily obtain the Chern number C=3 via dynamic winding number.

For the non-Hermitian case, we consider δ=0.1 and the other parameters are the same as those in the Hermitian case.


In Figs. <ref>(d) and <ref>(e), we also give the azimuthal angle (ϕ_xz) and (η_xz) in parameter space (k_x,k_y), respectively.

The dynamic winding numbers around EPs become half, while the EPs become double as the Hermitian counterpart, see Fig. <ref>(f).

Eventually, the Chern number keeps the same as that in the Hermitian case.





§ S5. EXPERIMENTAL CONSIDERATION 

One can immediately apply the dynamical approach for topological Hermitian systems.

Cold atom systems is an excellent platform to realize topological band models and detect topological invariants.

One and two dimensional spin-orbit couplings have been realized in a highly controllable Raman lattice <cit.>.

Initial states are quite easily prepared by loading the atoms into the lattices.

Here, the initial constraint |c_k+|^2 |c_k-|^2 may be not satisfied for some specific momentum k, but the occurred probability is so small that the global dynamical azimuthal angle is not affected due to the topological nature.

The spin population N_↑(↓)(k) with different momentum can be measured by spin-resolved time-of-flight (TOF)  absorption imaging<cit.>.

Thus, one can obtain the spin population difference ⟨ψ_k(t)| σ_z | ψ_k(t)⟩=(N_↑(k)-N_↑(k))/(N_↑(k)+N_↑(k)).

The spin textures ⟨ψ_k(t)| σ_x(y) | ψ_k(t)⟩ can be transferred to the spin population difference by applying π/2 pulse,

that is, ⟨ψ_k(t)| σ_x(y) | ψ_k(t)⟩=⟨ψ_k(t)|e^-iπ/2σ_y(x)/2σ_ze^iπ/2σ_y(x)/2| ψ_k(t)⟩.

Because the cold atom systems have long coherent time, there is no obstacle to extract the dynamic winding number via long time average of the spin textures.

To apply the dynamical approach in topological non-Hermitian systems, we should first consider how to realize the topological non-Hermitian models in experiments.

Since two-level non-Hermitian models have been widely realized in optical systems, such as two coupled optical cavities <cit.>, optical waveguides <cit.>, optomechanical cavity<cit.> etc.

We mainly discuss how to extract dynamic winding number with two optical waveguides with tunable parameters.

A two-level non-Hermitian system can be realized by introducing gain and loss in the two waveguides.

The coupling strength can be tuned by the waveguide separation.

We regard the two different waveguides as two spin components.

The initial states can be prepared by randomly split the light injecting into the two waveguides.

One can obtain ⟨ψ(l)|σ_z|ψ(l)⟩ by measuring the intensity difference between two waveguides at propagating distance l. Here, the distance l plays the role of time.

Actually, the final states will collapse into one of the eigenstate in the long distance.

Thus, the output intensity difference of the waveguides is sufficient and long distance average of the intensity difference is not necessary.

One can also obtain  ⟨ψ(l)|σ_x(y)|ψ(l)⟩ by insetting a beam splitter before intensity measurement.

By designing the waveguide separation and gain and loss rates, one can simulate the two-band model.

Repeating the above operations, one can finally construct the dynamic winding number.









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