Dynamical detection of topological charges
Long Zhang Lin Zhang Xiong-Jun Liu



𝕀1
i
d




International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China.
Collaborative Innovation Center of Quantum Matter, Beijing 100871, China.


International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China.
Collaborative Innovation Center of Quantum Matter, Beijing 100871, China.


Corresponding author: xiongjunliu@pku.edu.cn
International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China.
Collaborative Innovation Center of Quantum Matter, Beijing 100871, China.


We propose a generic scheme to characterize topological phases via detecting topological charges by quench dynamics.
A topological charge is defined as the chirality of a monopole at Dirac or Weyl point of spin-orbit field, and topological phases can
be classified by total charges in the region enclosed by the so-called band-inversion surfaces (BISs).
We show that both the topological monopoles and BISs can be identified by non-equilibrium spin dynamics caused by a sequence of quenches.
From an emergent dynamical field given by time-averaged spin textures, the topological charges, as well as the topological invariant, can be readily obtained. An explicit advantage in this scheme is that only a single spin component needs to be measured to detect all the information of the topological phase.
We numerically examine two realistic models, and propose a feasible experimental setup for the measurement. This work opens a new
way to dynamically classify topological phases.

Dynamical detection of topological charges
    Xiong-Jun Liu
    April 14, 2020
==========================================




Introduction.—
Topological quantum matter <cit.> has ignited extensive research in recent years.
In particular, experimental advances in cold atoms have promoted the realization of
novel topological states including the one-dimensional (1D) Su-Schrieffer-Heeger (SSH) chain <cit.>,
1D chiral topological phase <cit.> and 2D Chern insulators <cit.>.
Compared with solid state materials,
ultracold atoms are ideal platforms with high controllability, which makes it accessible to
experimentally investigate non-equilibrium dynamics caused by quenches <cit.>.

Very recently, several works <cit.> have focused on characterizing
topology of Hamiltonian by quench dynamics. In particular, a non-equilibrium classification of topological states,
which in equilibrium are characterized by integer invariants,
is established and shows experimental feasibility to detect bulk topology with high precision <cit.>.
It is shown that the bulk topology of the post-quench phase can be classified by
a dynamical topological invariant defined on the so-called band-inversion surfaces (BISs) <cit.>.
This classification theory has been applied in a latest experiment based on 2D Chern insulator <cit.>,
which shows that the dynamical measurement of topological states has a much higher precision over the equilibrium measurement strategies <cit.>.
Applications to dynamical topological phase transition <cit.> and non-Hermitian topological phases <cit.> are also considered.

In this letter, we propose a new scheme of dynamical classification to characterize topology by detecting topological charges.
The topological charge is defined as the chirality of a monopole (vanishing point) of the spin-orbit (SO) field, and the topology can
be determined by the total charges in the region enclosed by BISs. The key idea is that through a sequence of quenches
with respect to all (pseudo)spin quantization axes, both the BISs and topological charges of monopoles can be directly identified by
measuring the time-averaged spin polarization. 

Unlike our previous dynamical scheme <cit.> which quenches along a certain (pseudo)spin axis but needs to measure all the (pseudo)spin components, the present new detection method necessitates measuring only a single spin component.
On the other hand, compared with the method of measuring the linking number of trajectories in the momentum-time space <cit.>, which is valid for 2D phases, the present scheme can be applied to characterize topological states of all dimensions.
In the last part of this work, we propose an experimental setup to simulate the dynamical detection on 2D quantum anomalous Hall model, which can be well achieved in experiment.



Generic model.—
We start with the model of a generic d-dimensional (dD) gapped topological phase, which can be insulator or superconductor, and is described by the basic Hamiltonian

    H( k)= h( k)·γ=h_0( k)γ_0+∑_i=1^d h_i( k)γ_i.

Here the γ matrices define a (pseudo)spin and obey the Clifford algebra {γ_i,γ_j} =2δ_ij1
for i,j=0,1…,d, and the (d+1)D vector field h(𝐤) describes an effective Zeeman field depending on the momentum 𝐤 in the Brillouin zone (BZ) <cit.>.
The Hilbert space at each k are of dimensionality n_d=2^d/2 (or 2^(d+1)/2) if d is even (or odd), involving
the minimal bands to open a topological gap <cit.>.
In the 1D and 2D regimes, the γ matrices simply reduce to the Pauli matrices, and ℋ(𝐤) describes a two-band model for the topological states, e.g., the well-known SSH model <cit.> for 1D and the quantum anomalous Hall model <cit.> for 2D. Similarly, for 3D and 4D phases, the γ matrices take the Dirac forms, and a fully gapped topological phase has at least four bands <cit.>.
Similar to the convention used in Ref. <cit.>, we choose h_0(𝐤) to characterize the `dispersion' of the n_d decoupled bands, and define the remaining components as the SO field h_so(𝐤)≡(h_1,…,h_d), which depict the coupling between different bands.

In the previous work <cit.>, a bulk-surface duality was shown that the dD bulk topology
can be characterized by a (d-1)D invariant defined on
BISs. A BIS refers to
the (d-1)D band-crossing surface with h_0(𝐤)=0.
The SO field h_so( k) opens the gap, and brings out nontrivial topology.
The defined (d-1)D topological invariant counts the winding number of the SO field on the BISs.
Analogous to the magnetic field, one can also consider the monopoles of SO field, and
the topological invariant is thus viewed as the flux of the monopoles through the BIS [see Fig. <ref>(a)].
It is easily seen that the monopoles are located at k=ϱ where h_so(ϱ)=0, such that
the gap is closed and then reopened as a monopole passes through a BIS, indicating a topological phase transition.
As detailed in the Supplementary Information <cit.>, we show that the topological invariant reads

    W   =∑_i∈ V_ BIS C_i,

with

    C_i   = sgn[J_ h_ so(ϱ_i)]

being the topological charge of the ith monopole
in the region V_ BIS enclosed by BISs. Here J_ h_ so( k)≡[(∂ h_ so,i/∂ k_j)] is the Jacobian determinant
and V_ BIS denotes the region h_0( k)<0.

The invariant in Eq. (<ref>), as a summation of topological charges, is directly related to the Brouwer degree <cit.> of
the mapping ĥ( k)≡ h( k)/| h( k)| from the BZ torus T^d to dD spherical surface S^d. This formula can be intuitively interpreted
as the effective number of times that the parametric surface D traced by the vector h( k) passes through the negative γ_0 axis <cit.>.
The intersection points are locations (h_ so=0) of monopoles, with the charges ±1 indicating the orientation (or chirality) of the manifold at these intersections. By summing up all the orientation numbers,
we obtain the winding number or Chern number of the dD topological phase.

Dynamical detection of the topological charges.—
We shall show that BISs and topological charges of monopoles can be identified via
quantum spin dynamics induced by, respectively, quenching h_0 and a sequence of quenches of h_i>0.
The basic idea is as follows.
We focus on the time evolution of spin polarization triggered by quenches.
For quenching h_i (i=0,1,2,⋯,d), we initialize a polarized state along this axis,
which is achieved by tuning a large constant magnetization for this component h_i(𝐤)≈ m_i.
After the quench, the momentum-linked spin ⟨γ(t)⟩
processes around the vector field h(𝐤).
In the unitary evolution, the time-averaged spin polarization ⟨γ(t)⟩ directly reflects the h_i component [see Fig. <ref>(b)].
On the surfaces with h_i( k)=0, the spin orientation is always perpendicular to the procession axis,
leading to vanishing polarization. In the region with h_i( k)<0 (or h_i( k)>0), the vector ⟨γ(t)⟩ is in the (or opposite)
direction of the field h. Taking ⟨γ_0⟩ as the measurement,
the observations fall into two categories—the first corresponds to quenching h_0
and the other corresponds to quenching h_ so.
For quenching h_0,  one can identify the BISs as the momentum points with vanishing polarization ⟨γ_0⟩_0=0 [see Fig. <ref>(c)].
On the other hand, for the second category of quenching h_i>0, the time-averaged spin polarization ⟨γ_0⟩_i also vanishes on surfaces with h_i( k)=0.
The topological monopoles, located at the intersections of all the h_i>0( k)=0 surfaces, are then dynamically identified by the points with ⟨γ_0⟩_i>0=0 for all h_i>0 quenches. The topological charges can further be characterized from a dynamical field constructed by time-averaged spin textures [Fig. <ref>(c)], as detailed below.

To be more specific, the system described by density matrix ρ_i(0) is initialized in the ground state of the pre-quench Hamiltonian, with the spins being
fully polarized in the opposite direction of the γ_i axis.
The quantum dynamics after the quench is governed by the unitary evolution operator U(t)=exp(-ℋt).
The time-dependent density matrix ρ_i(t)=U(t)ρ_i(0)U^†(t) and the spin polarization is given by ⟨γ_0⟩_i= Tr[ρ_i(t)γ_0]. After some calculations <cit.>, the time-averaged spin texture reads

    ⟨γ_0( k)⟩_i=-h_0( k)h_i( k)/E^2( k),

where E( k)=[∑_i=0^dh_i^2]^1/2.
It is interesting that no matter which axis is quenched, the spin texture ⟨γ_0( k)⟩_i always vanishes on the BISs (with h_0=0).
Hence, the BISs are the surfaces with vanishing time-averaged spin polarization independent of the quench way, i.e.
BISs={𝐤|∀ i,⟨γ_0(𝐤)⟩_i=0}.
Besides, the Eq. (<ref>) also gives ⟨γ_0( k)⟩_i=0 on the surface
with h_i( k)=0. Accordingly, the location k=ϱ of a monopole can be found by ⟨γ_0(ϱ)⟩_i=0 for all i≠0 but ⟨γ_0(ϱ)⟩_0≠0. Finally, to characterize the charge,
one notices that near the point k=ϱ,
the time-averaged spin texture in quenching h_i can measure the i-th component of the SO field as
⟨γ_0( k)⟩_i|_ k→ϱ=-h_i( k)/h_0(ϱ).
Enlightened by this result, we define a dynamical field Θ( k) to characterize the topological charge, with the components being given by

    Θ_i( k)≡- sgn[h_0( k)]/ N_ k⟨γ_0( k)⟩_i,

where N_ k is the normalization factor. It can be shown directly that near the location ϱ of the topological monopole, the dynamical field
Θ_i( k)|_ k→ϱ=h_ so,i( k).
With this result, we reach finally that the topological charge is dynamically determined by

    C_i= sgn[J_Θ(ϱ_i)].





Application to two models.—
We illustrate our scheme by numerally examining two different models.
First, we consider the 2D quantum anomalous Hall (QAH) model H( k)= h( k)·σ <cit.>, where the vector field reads
h( k)=(m_x+t_ sosin k_x,m_y+t_ sosin k_y,m_z-t_0cos k_x-t_0cos k_y). This model has been realized in cold atom experiments <cit.>.
The bulk topology is determined by m_z (m_x=m_y=0) that the Chern number Ch_1=- sgn(m_z) for 0<|m_z|<2t_0 and Ch_1=0 for |m_z|≥2t_0. We take h_0≡ h_z,  h_ so≡(h_y,h_x) <cit.>, and the quench is performed by suddenly varying m_α (α=x,y,z). Note that only the time evolution of spin polarization of the σ_z-component needs to be measured after each quench process to obtain all the information of topology [see Fig. <ref>(a-c)]. The spin textures ⟨σ_z( k)⟩_α in all three quenches (α=x,y,z) show clearly a ring-shape structure, which characterizes the BIS. Besides, spin textures in (b) and (c), respectively, exhibits two lines with vanishing polarization, which indicate the surfaces with h_y( k)=0 [for (b)] and h_x( k)=0 [for (c)]. These four lines have four intersection points marking the monopoles at Γ, M and X_1,2 points, and the dynamical field Θ( k) obtained by spin textures in (b-c) determines the charge ±1 at each point via
sgn[J_Θ( k)] [see Fig.  <ref>(d)].
For m_z=t_0, the ring encloses the monopole with C=-1 at Γ point, giving the Chern number Ch_1=-1.



We further consider the application to a 3D topological phase, whose Hamiltonian reads ℋ(𝐤)= h(𝐤)·γ,
with h_0(𝐤)=m_0-t_0∑_icos k_r_i and
h_i=m_i+t_sosin k_r_i [(r_1,r_2,r_3)=(x,y,z) for i=1,2,3]. Here we take
γ_0=σ_z⊗τ_x, γ_1=σ_x⊗𝕀,
γ_2=σ_y⊗𝕀 and γ_3=σ_z⊗τ_z, where
σ_x,y,z and τ_x,y,z are both Pauli matrices.
The topological phases are classified by 3D winding numbers.
The trivial phase corresponds
to |m_0|>3t_0, while the topological phases include three regions (m_i>0=0):
(I) t_0<m_0<3t_0 with winding number ν_3=1; (II) -t_0<m_0<t_0
with ν_3=-2; and (III) -3t_0<m_0<-t_0 with ν_3=1.
We observe the time-averaged spin textures ⟨γ_0( k)⟩_i after quenching different components <cit.>.
In all cases, the post-quench state takes the parameters m_0=1.3, m_i>0=0 and t_ so=t_0. The pre-quench state is with m_i=25t_0, corresponding to quenching m_i (i=0,1,2,3). The surface with ⟨γ_0( k)⟩_0=0 characterizes the BIS, and ⟨γ_0( k)⟩_i>0=0 locate the topological monopoles as their intersections [see Fig. <ref>(a)].
The dynamical field Θ( k), constructed from ⟨γ_0( k)⟩_i by following Eq. (<ref>),
reflects the charges [Fig. <ref>(b)]. One can see that the BIS only surrounds a single monopole with C=+1,
which reveals that the post-quench state lies in the topological phase with ν_3=1.






Experimental realization.—
Now we propose an experimental setup (Fig. <ref>) to detect topological charges in a 2D QAH model <cit.>, which has been realized in Refs. <cit.>.
This setup exploits the so-called optical Raman lattice scheme <cit.>: two beams E_x,y to generate both 2D lattice V_ latt(x,y) and Raman potentials M_x,y(x,y) simultaneously.
The quench process is realized by manipulating the relative symmetries between Raman and lattice potentials [Fig. <ref>(a)].



First, the 2D QAH model can be realized as described in Ref. <cit.>.
Two electro-optic modulators (EOMs) are set to induce an additional π/2-phase shift for the ẑ-component field by manipulating the voltage [Fig. <ref>(b)].
This gives the light fields
E_x=ŷE_xycos k_0 x+ẑE_xzsin k_0x
and E_y=x̂E_yxcos k_0 y+ẑE_yzsin k_0y,
where E_μν (μ,ν=x,y,z) is the amplitude of the field in the μ direction and with the ν polarization.
For alkali metal atoms, the optical lattice potential is spin independent V_ latt(x,y)=V_0xcos^2 k_0x+V_0ycos^2 k_0y (see Ref. <cit.> for optical transitions of ^40K atoms), with the amplitudes V_0x∝(E_xy^2-E_xz^2)/Δ and V_0y∝(E_yx^2-E_yz^2)/Δ.
One Raman potential M_x(x,y)=M_0xcos k_0ysin k_0x, with
M_0x∝ E_xzE_yx/Δ, is generated by the E_xz and E_yx components, and the other M_y(x,y)=M_0ycos k_0xsin k_0y
with M_0y∝ E_xyE_yz/Δ by the E_xy and E_yz components <cit.>.
The Raman M_x(y) and lattice potentials satisfy a relative antisymmetric configuration along the x (y) direction,
which ensures that M_x(y) leads to spin-flipped hopping only along the x (y) direction [see Fig. <ref>(a)].
Hence, for the s-band, the Bloch Hamiltonian is <cit.>
H( q)= h( q)·σ=[m_z-t_0(cos q_xa+cos q_ya)]σ_z+t_ sosin q_xaσ_y+t_ sosin q_yaσ_x,
which is the QAH model.
Here a is the lattice constant, q_x,y are Bloch momenta, m_z measures the two-photon detuning,
and the spin-conserved (t_0) and spin-flipped (t_ so) hopping coefficients are respectively determined by the lattice
and Raman potentials.





The quench for all the components  h_x,y,z are proposed as follow.
Quenching h_z can be easily achieved by varying the two-photon detuning via the bias magnetic field.
Quenching  h_x,y requires the generation of constant magnetization m_xσ_x or m_yσ_y, which
can be realized by turning off one of the EOMs.
Take quenching h_y as an example. If turning off the EOM in the x direction,
the beams form the fields E'_x=ŷE_xycos k_0 x+ẑE_xzcos k_0x
and E_y=x̂E_yxcos k_0 y+ẑE_yzsin k_0y, which produce the lattice potential V_ latt(x,y)=V'_0xcos^2 k_0x+V'_0ycos^2 k_0y,
with V'_0x∝(E_xy^2+E_xz^2)/Δ and V'_0y=V_0y given as before, and Raman potentials
M_x(x,y)=0 and M_y(x,y)=M_0xcos k_0ycos k_0x+M_0ycos k_0xsin k_0y.
The M_0y-term of the Raman potential induces spin-flipped hopping along y direction, corresponding to
t_ sosin q_yaσ_x in the model, while
the former term (M_0x) is symmetric with respect to each lattice site,
leads to on-site spin-flipping [see Fig. <ref>(a)] and generates a constant
magnetization m_yσ_y <cit.>.
Similarly, turning off the EOM in the y direction can generate the term m_xσ_x. Thus, the quenches regarding h_x,y can be readily implemented by suddenly varying the relative phase of lights
via the EOMs. An initial magnetization m_y(x)∼20t_0 is achieved in typical parameter conditions with
|V'_0x|=|V'_0y|=6.3E_ r, M_0x(0y)=2.5E_ r, M_0y(0x)=0 (E_ r≡ħ^2k_0^2/2m) <cit.>.


Conclusions.— We have proposed a generic scheme to characterize the phases by
a dynamical topological invariant defined at the monopoles of the SO field. The scheme is applicable to topological phases of integer ℤ invariants and of any dimension, with 2D and 3D models being numerically examined. Besides, this method
exhibits explicit advantages for experimental investigation due to the simplification of measurement.
In the future, we expect to generalize our theory to broader classes of topological phases, including those
classified by ℤ_2.

This work was supported by the National Key R&D Program of China (2016YFA0301604), National Nature Science Foundation of China
(under grants No. 11674301 and No. 11761161003), and the Thousand-Young-Talent Program of China.

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*definitionDefinition











§ SUPPLEMENTARY INFORMATION




 §.§ I. Characterize Topological Phases by Charges


We consider the generic d-dimensional (dD) Hamiltonian

    H( k)= h( k)·γ=h_0( k)γ_0+∑_i=1^d h_i( k)γ_i,

where the matrices γ  obey the Clifford algebra {γ_i,γ_j} =2δ_ij1 (i,j=0,1…,d), and are of dimensionality n_d=2^d/2 (or 2^(d+1)/2) if d is even (or odd).

This model involve the minimal bands to open a topological gap for the dD topological phase.
Without loss of the generality, we choose h_0(𝐤) to characterize the band structure and the remaining d-components
h_i≠0(𝐤) to depict the coupling between bands.
We denote these components by the spin-orbit (SO) field h_so(𝐤)≡(h_1,…,h_d).
We emphasize that the γ matrices are constructed to satisfy the trace property Tr[∏_j=0^dγ_j]=(-2)^n for even d=2n
or Tr[γ∏_j=0^dγ_j]=(-2)^n for odd d=2n-1, with γ=^n∏_j=0^dγ_j being the chiral matrix.
For example, in 2D we should have Tr[γ_0γ_1γ_2]=-2; if γ_0=σ_z, one should set γ_1=σ_y and γ_2=σ_x
(it is just the case in our numerical calculations of the 2D model).

In Ref. <cit.>, we have proved that the dD bulk topology can be characterized by a (d-1)D invariant defined on the band-inversion surfaces (BISs):

    W =∑_jΓ(d/2)/2π^d/21/(d-1)!∫_ BIS_jĥ_so(ĥ_so)^d-1,

where Γ(x) is the Gamma function, ĥ_so(𝐤)≡ h_so(𝐤)/| h_so(𝐤)| denotes the unit SO field, ĥ_so(ĥ_so)^d-1≡ϵ^i_1i_2⋯ i_dĥ_so,i_1ĥ_so,i_2∧⋯∧ĥ_so,i_d, with ϵ^i_1i_2⋯ i_d being the fully anti-symmetric tensor and i_1,2,…,d∈{1,2,…,d}, and `' denotes the exterior derivative.
The above formula can also be written as

    W   =∑_i∈ V_ BIS C_i,

with

    C_i   =Γ(d/2)/2π^d/2∫_ S_i1/| h_so|^d∑_i=1^d(-1)^i-1h_so,ih_so,1∧⋯∧h_so,i∧⋯∧h_so,d.

Here S_i denotes a (d-1)D surface enclosing the ith monopole k=ϱ_i with h_ so(ϱ_i)=0, and the term with a big hat is omitted.
We further have

    C_i   =Γ(d/2)/2π^d/2∫_ S_iJ_ h_ so(ϱ_i)/| h_so|^d∑_i=1^d(-1)^i-1k_ik_1∧⋯∧k_i∧⋯∧k_d
       = sgn[J_ h_ so(ϱ_i)]Γ(d/2)/2π^d/2∫_ S_i|J_ h_ so(ϱ_i)|/| h_so|^d∑_i=1^d(-1)^i-1k_ik_1∧⋯∧k_i∧⋯∧k_d
       = sgn[J_ h_ so(ϱ_i)]Γ(d/2)/2π^d/2∫_ h_ so( S_i)[∑_i=1^d(-1)^i-1h_so,i/| h_so|^dh_so,1∧⋯∧h_so,i∧⋯∧h_so,d].

Here J_ h_ so(ϱ_i)≡[(∂ h_ so,i/∂ k_j)|_ k=ϱ_i] denotes the Jacobian determinant.
The integral in the last line gives the the area of the (d-1)D sphere h_ so( S_i), i.e. Vol[S^d-1]=2π^d/2/Γ(d/2). Thus, we have

    C_i   = sgn[J_ h_ so(ϱ_i)].






The repression (<ref>) using a summation of topological charges is directly related to the Brouwer degree <cit.>
of the mapping
ĥ( k)≡ h( k)/| h( k)|.
We first introduce the definition of mapping degree here.
Let M and N be oriented d-dimensional manifolds without boundary and f: M→ N be a smooth map. If M is compact and
N is connected, then the degree of f is defined as follows <cit.>:

Let x∈ M be a regular point of f, so that df_x:T M_x→ T N_f(x) is a linear isomorphism between oriented tangent spaces. Define the sign of
df_x to be +1 or -1 according as df_x preserves or reverses orientation. For any regular value y∈ N define the mapping degree

    deg(f;y)=∑_x∈ f^-1(y) sgn(df_x).



Since h( k) is nonzero for all k in BZ, h: T^d→ℝ^d+1∖{ 0} is a mapping from the Brillouin
zone torus T^d to a (d+1)D closed surface D. Thus ĥ is a composition
ĥ=π∘ h, where π:ℝ^d+1∖{ 0}→ S^d is the central projection to the unit sphere.
According to the definition, for any regular point z∈ S^d, the Brouwer degree is

    deg(ĥ; z)=∑_ k∈ĥ^-1(z) sgn[dĥ( k)].

The degree is the same for all regular points, so one can choose z as a point in an axis of the unit sphere S^d, such as in the negative γ_0 axis. Denote
O as the set of all the k points with h_0( k)<0 and h_ so( k)=0.  We then have

    deg(ĥ)=∑_ϱ_i∈ O sgn[J_ h_ so(ϱ_i)].

We reobtain the expression of the topological invariant [Eqs. (<ref>) and (<ref>)].
Therefore, as shown in Fig. <ref>, the topological monopoles are the intersection points of the closed surface D and the negative γ_0 axis, and the charges
denote the orientation (or chirality) of the surface at each intersection.




 §.§ II.  Dynamical Detection of Topological Charges



  §.§.§ A. General ideas


We consider the spin dynamics of ⟨γ_0⟩ following different quenches. Each quench process is
realized by tuning h_i(𝐤)=m_i+h̃_i( k) from a deep trivial regime with m_i≫ max[|h̃_i( k)|] (i>0)  to
0 or tuning m_0 from
m_0≫ max[|h̃_0( k)|] to a topological regime m_0≲ max[|h̃_0( k)|] (m_i≠0=0).
Here h̃_i( k) represents momentum-dependent field and m_i demotes a constant magnetization.
The system with density matrix ρ_i(0) is prepared in the ground state of the pre-quench Hamiltonian
in the deep trivial regime, where the spins are
fully polarized in the opposite direction of the γ_i axis.
After a sudden quench, the quantum dynamics is governed by the unitary evolution operator U(t)=exp(-ℋt).
The time-dependent density matrix is then ρ_i(t)=U(t)ρ_i(0)U^†(t) and the spin texture is given by ⟨γ_0⟩_i= Tr[ρ_i(t)γ_0].

The time-averaged spin polarization after quenching h_i is measured as

    ⟨γ_0⟩_i=lim_T→∞1/T∫_0^T t Tr[ρ_i(0)e^ℋtγ_0e^-ℋt].

With the identity e^ℋt=cos(Et)+sin(Et)ℋ/E, where E( k)=√(∑_i=0^dh_i^2), we have

    ⟨γ_0⟩_i=h_0 Tr[ρ_i(0) H]/E^2.

Since the initial state ρ_i(0) is chosen such that Tr[ρ_i(0) H]=-h_i,
the time averaged spin texture in the BZ is

    ⟨γ_0( k)⟩_i=-h_0( k)h_i( k)/E^2( k).

It is worthwhile to note that if we set the quench from m_i≪0, the spin texture should be ⟨γ_i⟩=+h_0h_i/E^2.

From the result, one can see that no matter which axis is quenched,
the time-averaged spin polarization ⟨γ_0( k)⟩_i always vanishes on BISs.
Hence, a BIS can be dynamically determined by the surface with vanishing time-averaged spin polarization independent of the quench axis:
BIS={𝐤|∀ i∈{0,1,2,⋯,d},⟨γ_0(𝐤)⟩_i=0}.
In particular, ⟨γ_0⟩_0=0 occurs only on BISs, so one can find out these surfaces simply by quenching h_0.
Besides, the time-averaged spin texture ⟨γ_0( k)⟩_i≠0 also vanishes on the surface
with h_i( k)=0 (see Eq. <ref>). Accordingly, the monopoles of the SO field can be found out by ⟨γ_0(ϱ)⟩_i=0 for all i>0 but ⟨γ_0(ϱ)⟩_0≠0. To characterize the charge, we find that near the point k=ϱ, E( k)≈|h_0(ϱ)| and the time-averaged spin texture directly reflects the SO field:

    ⟨γ_0( k)⟩_i|_ k→ϱ≃-h_i( k)/h_0(ϱ).

Thus, we define a dynamical field Θ( k), whose components are

    Θ_i( k)≡- sgn[h_0( k)]/ N_ k⟨γ_0( k)⟩_i,

with N_ k being the normalization factor. It can be shown that near a topological monopole ϱ, we have

    Θ_i( k)|_ k→ϱ=h_ so,i( k).

With this result, we reach immediately that the topological charge can be dynamically determined by

    C_i= sgn[J_Θ(ϱ_i)].






  §.§.§ B. Application to 3D topological phases


Here we present the details of numerical calculations on the 3D model. The Hamiltonian reads ℋ(𝐤)= h(𝐤)·γ,
with γ_0=σ_z⊗τ_x, γ_1=σ_x⊗𝕀,
γ_2=σ_y⊗𝕀, γ_3=σ_z⊗τ_z, h_0(𝐤)=m_0-t_0∑_icos k_r_i and
h_i=m_i+t_sosin k_r_i [i=1,2,3 and (r_1,r_2,r_3)=(x,y,z)].
This model has a chiral symmetry γ=^2γ_0γ_1γ_2γ_3=-σ_z⊗τ_y.
One can check that the constructed matrices satisfy the trace property
Tr[γγ_0γ_1γ_2γ_3]=(-2)^2.

In our calculations, the topological phase is set with the parameters {m_0,m_1,m_2,m_3}={1.3t_0,0,0,0}, and one quench process is realized by tuning m_i from 25t_0 (others are all zero) to the set value (m_0 should be tuned to 1.3t_0).
After the quench, the spin polarization will oscillate with time at each k. The observed time-averaged spin textures can determine both the BIS
and the topological monopoles with ⟨γ_0( k)⟩_i=0 [see Fig. <ref>(a)]: The surface with
⟨γ_0( k)⟩_0=0 corresponds to the BIS (the green surface) and the
interaction points of ⟨γ_0( k)⟩_i>0=0 mark the locations of monopoles (colored dots).
Furthermore, based on the spin textures ⟨γ_0( k)⟩_i>0, we construct the dynamical field Θ( k) via the formula (<ref>).
The results are shown in Fig. <ref>(b-c), from which one can easily read out the charge of each monopole. As an example, we illustrate how to construct
the dynamical field on the planes S_0^z (k_z=0) and S_f^z (k_z=-π) in Fig. <ref>(d-e).






 §.§ III. Experimental Realization for ^40K Atoms

Our experimental setup exploits the 2D optical Raman lattice, which has been realized in ^87Rb bosons <cit.>.
Here, we take ^40K atoms as an example while all our results are applicable to other alkali atoms. For ^40K, the spin-1/2 system can be constructed
by |g_↑⟩=|F=9/2, m_F=+9/2⟩ and |g_↓⟩=|9/2, +7/2⟩. As shown in Fig. <ref>,
the lattice and Raman coupling potentials are contributed from both the D_2 (4^2S_1/2→4^2P_3/2) and D_1 (4^2S_1/2→4^2P_1/2) lines.
The total Hamiltonian reads (ħ=1)

    H=[ k^2/2m+V_ latt(x,y)]⊗ 1+ M_x(x,y)σ_x+ M_y(x,y)σ_y+m_zσ_z,

where V_ latt(x,y) denotes the square lattice potential, M_x,y(x,y) are the Raman coupling potentials, and m_z=δ/2 measures the two-photon detuning of Raman coupling.





  §.§.§ A. 2D QAH model

Details of the realization of a QAH model can be also found in Supplementary Material of Ref. <cit.>.
To generate 2D spin-orbit coupling, we use electro-optic modulators (EOMs) to introduce an additional π/2 phase for the ẑ-component field (see Fig. 4 of the main text).
Passing through a EOM twice leads to a π-phase shift.
Hence, the laser beams form the standing-wave fields for atoms (see Fig. 4 of the main text):

    E_x   =ŷE_xye^(α+α_L/2)cos(k_0 x-α_L/2)+ẑE_xze^(α+α_L/2)sin(k_0 x-α_L/2)
     E_y   =x̂E_yxe^(β+β_L/2)cos (k_0 y-β_L/2)+ẑE_yze^(β+β_L/2)sin(k_0 y-β_L/2),

where α and β denote the initial phases, and α_L (β_L) is the phase acquired by E_x (E_y) for an additional optical path
 back to the atom cloud (excluding the EOM effect).

 As shown in Fig. <ref>, two independent Raman transitions are driven by the components E_xz, E_yx and E_xy, E_yz, respectively,
 which leads to
 
    M_1   =∑_FΩ^(3/2)*_↑ F,xz·Ω^(3/2)_↓ F,yx/Δ_D_2+∑_FΩ^(1/2)*_↑ F,xz·Ω^(1/2)_↓ F,yx/Δ_D_1
    
     M_2   =∑_FΩ^(3/2)*_↑ F,xy·Ω^(3/2)_↓ F,yz/Δ_D_2+∑_FΩ^(1/2)*_↑ F,xy·Ω^(1/2)_↓ F,yz/Δ_D_1,

 where Δ_D_1>0, Δ_D_2<0, Ω^(J)_σ F,μ z=⟨σ|ez|F,m_Fσ,J⟩ê_z· E_μ z and
 Ω^(J)_σ F,μν=⟨σ|ex|F,m_Fσ+1,J⟩ê_+· E_μν+⟨σ|ex|F,m_Fσ-1,J⟩ê_-· E_μν (μ,ν=x,y). Here ê_+· E_yx=E_yx/√(2) and ê_-· E_xy= E_xy/√(2) represent the right- and left-handed components, respectively. From the dipole matrix elements of ^40K <cit.>, we obtain
 
    M_x   =M_0xsin(k_0 x-α_L/2)cos (k_0 y-β_L/2)
     M_y   =M_0ycos(k_0 x-α_L/2)sin(k_0 y-β_L/2),

 where

    M_0x/0y=t_D_1^2/9(1/|Δ_D_1|+1/|Δ_D_2|)E_xz/xyE_yx/yz,

with the transition matrix elements t_D_1≡⟨ J=1/2||e r||J'=1/2⟩, t_D_2≡⟨ J=1/2||e r||J'=3/2⟩ and t_D_2≈√(2)t_D_1.
The optical lattice is given by

    V_σ=↑,↓   =∑_F1/Δ_D_2(|Ω^(3/2)_σ F,xz|^2+|Ω^(3/2)_σ F,xy|^2
    +|Ω^(3/2)_σ F,yz|^2+|Ω^(3/2)_σ F,yx|^2) 
       +∑_F1/Δ_D_1(|Ω^(1/2)_σ F,xz|^2+|Ω^(1/2)_σ F,xy|^2
    +|Ω^(1/2)_σ F,yz|^2+|Ω^(1/2)_σ F,yx|^2)
       =V_0xcos^2(k_0 x-α_L/2)+V_0ycos^2(k_0 y-β_L/2),

where

    V_0x/0y=t_D_1^2/3(1/|Δ_D_1|-2/|Δ_D_2|)(E_xy/yx^2-E_xz/yz^2).


In the tight-binding limit and only considering s-bands, the Hamiltonian (<ref>) takes the form in the momentum space <cit.>
H=∑_ qΨ^†_ q H_ fΨ_ q,
where Ψ_ q=(c_ q↑,c_ q↓) and

    H_ f=[m_z-t_0(cos q_xa+cos q_y a)]σ_z+t_ sosin q_xaσ_y+t_ sosin q_yaσ_x.

Here  a is the lattice constant, q is the Bloch momentum,
and  t_0 and t_ so are, respectively, the spin-conserved and spin-flipped hopping coefficients

    t_0   =-2∫ d rϕ_s(x,y)[ k^2/2m+V_ latt( r)]ϕ_s(x-a,y),
    
    t_ so   =2M_0∫ d rϕ_s(x,y)cos(k_0y)sin(k_0x)ϕ_s(x-a,y),

where ϕ_s(x,y) denotes the Wannier function and  M_0x=M_0y=M_0 is assumed.



  §.§.§ B. Hamiltonian for quenching h_x,y


In our dynamical detection scheme, quenching h_z can be easily achieved by varying the two-photon detuning δ.
Quenching the other two components require producing the constant magnetization m_xσ_x or m_yσ_y, which means that
on-site spin-flipping needs to be generated.  It can be verified that the transverse magnetization m_xσ_x (or m_yσ_y) can be induced by
turning off the EOM in the y (or x) direction [see Fig. 4(b) in the main text]. Here we take m_yσ_y as an example.

When turning off the EOM in the x direction, we have the light fields

    E'_x   =ŷE_xye^(α+α_L/2)cos(k_0 x-α_L/2)+ẑE_xze^(α+α_L/2)cos(k_0 x-α_L/2),
     E_y   =x̂E_yxe^(β+β_L/2)cos (k_0 y-β_L/2)+ẑE_yze^(β+β_L/2)sin(k_0 y-β_L/2),

where E_y is unchanged and the phases α, β, α_L and β_L are defined as before.
The lattice and Raman couplings are the same as in Fig. <ref>. The Raman transitions M_1,2 [Eq. (<ref>)] both contribute to
the σ_y coupling.
Thus the Raman potentials read

    M_x=0,   M_y=M_0xcos(k_0 x-α_L/2)cos(k_0 y-β_L/2)+M_0ycos(k_0 x-α_L/2)sin(k_0 y-β_L/2),

with M_0x/0y given by Eq. (<ref>). The lattice potential is

    V_σ=↑,↓=V'_0xcos^2(k_0 x-α_L/2)+V'_0ycos^2(k_0 y-β_L/2),

with

    V'_0x=t_D_1^2/3(1/|Δ_D_1|-2/|Δ_D_2|)(E_xy^2+E_xz^2)m,

and V'_0y=V_0y.
One can see that in the Raman potential M_y, the M_0x term is symmetric with respect to each lattice site along x or y direction,
thus leading to on-site spin-flipping if only s-bands are considered. The other term (M_0y) keeps spin-flipped hopping in the y direction.

Since

    ∫ d rϕ_s↑^i⃗( r)cos(k_0x)sin(k_0y)ϕ_s↓^j⃗( r)   =(-1)^i_x+i_y∫ d rϕ_s↑( r)cos(k_0x)sin(k_0y)ϕ_s↓( r- r_j⃗+ r_i⃗).,
    ∫ d rϕ_s↑^i⃗( r)cos(k_0x)cos(k_0y)ϕ_s↓^i⃗( r)   =(-1)^i_x+i_y∫ d rϕ_s↑( r)cos(k_0x)cos(k_0y)ϕ_s↓( r),

the tight-binding Hamiltonian should take the form

    H   =   -t_0^x/2∑_⟨ i_xj_x⟩,σc^†_i_xσc_j_xσ-t_0^y/2∑_⟨ i_yj_y⟩,σc^†_i_yσc_j_yσ-[∑_i_y(-1)^i_x+i_yt^x_ so/2(c^†_i_x↑c_i_x+1↓-c^†_i_x↑c_i_x-1↓)+ h. c.] 
       -    m_y∑_i⃗(-1)^i_x+i_y(c^†_i⃗↑c_i⃗↓-c^†_i⃗↓c_i⃗↑)+∑_i⃗m_z(n_i⃗↑-n_i⃗↓),

with 

    t^x_0   =-2∫ d rϕ_s(x,y)[ k^2/2m+V_ latt( r)]ϕ_s(x-a,y), 
    
    t^y_0   =-2∫ d rϕ_s(x,y)[ k^2/2m+V_ latt( r)]ϕ_s(x,y-a),
    
    t^x_ so   =2M_0y∫ d rϕ_s(x,y)cos(k_0x)sin(k_0y)ϕ_s(x-a,y),
    
    m_y   =M_0x∫ d rϕ_s( r)cos(k_0x)cos(k_0y)ϕ_s( r).

After the transformation c_j⃗↓→ e^iπ r_j⃗/ac_j⃗↓, the Hamiltonian becomes

    H   =   -t_0^x/2∑_⟨ i_xj_x⟩(c^†_i_x↑c_j_x↑-c^†_i_x↓c_j_x↓)
    -t_0^y/2∑_⟨ i_yj_y⟩(c^†_i_y↑c_j_y↑-c^†_i_y↓c_j_y↓)
    -[∑_i_xt^x_ so/2(c^†_i_x↑c_i_x+1↓-c^†_i_x↑c_i_x-1↓)+ h. c.] 
       -    m_y∑_i⃗(c^†_i⃗↑c_i⃗↓-c^†_i⃗↓c_i⃗↑)+∑_i⃗m_z(n_i⃗↑-n_i⃗↓),

The Fourier transform 

    c_j⃗σ=1/√(N)∑_ q e^i q· r_j⃗c_ qσ,

with N being the number of lattice sites,
yields the Bloch Hamiltonian

    H_ i=[m_z-(t^x_0cos q_xa+t_0^ycos q_y a)]σ_z+t^x_ sosin q_xaσ_x+m_yσ_y.


Similarly, when turning off the EOM in the y direction, one can generate the constant magnetization m_xσ_x.



  §.§.§ C. Quench steps and parameter estimation


In summary, one can implement the quenches as follows:
(1) The EOMs are both tuned to introduce a π/2 phase shift and the system is governed by the Hamiltonian H_ f all the time.
Quench h_z by suddenly changing the two-photon detuning m_z.
(2) Turn on only the EOM in the y direction, and set E_xz,E_yx,E_xy,E_yz at desired values;
we have H_ i≈ m_yσ_y. Turn on the other one such that both EOMs take effect; the system is described by H_ f.
The process of quenching h_x is thus accomplished.
(3) Similar to (2), the only difference is tuning on the EOM in the x direction before both EOMs take effect.
This is equivalent to quenching h_y.
After each step of the three, one can observe the time evolution of spin polarization and obtain the time-averaged spin texture ⟨σ_z( k)⟩_α (α=x,y,z).

Here we take quenching h_y as an example for parameter estimation. The laser beams can be set at the wavelength of 768nm (between the D_1 and D_2 lines <cit.>). The detunings are then Δ_D_1=(2π)1.07THz, and  Δ_D_2=-(2π)0.66THz, respectively.
Before the quench, one can set E_xz=E_yx=E_0 and E_xy=E_yz≃0, and have the results V'_0x=V_0y=-V_0,
M_0x≈0.39V_0, and M_0y≃0. Here V_0≡ t_D_1^2E_0^2(2/|Δ_D_2|-1/|Δ_D_1|)/3.
When V_0=6.3E_ r with E_ r≡ k_0^2/(2m), we have m_y≈ 20t_0, which is large enough for magnetization.
After the quench, one should ensure E_xy>E_xz and E_yx>E_yz such that the lattice is kept red-detuned.
With proper parameters, such as V_0x/0y=4E_ r and M_0x/0y=1E_ r, the lifetime of ^40K degenerate gas can be ∼150ms <cit.>, long enough for experimental observation.


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