Dynamically characterizing topological phases by high-order topological charges

Topological quantum phases have become a mainstream of research in various areas, including condensed matter physics Klitzing et al. (1980); Tsui et al. (1982); Hasan and Kane (2010); Qi and Zhang (2011); Elliott and Franz (2015); Wen (2017); Kosterlitz (2017), ultracold atoms Greiner et al. (2002); Bloch et al. (2008); Miyake et al. (2013); Atala et al. (2013); Aidelsburger et al. (2013); Liu et al. (2014); Jotzu et al. (2014); Aidelsburger et al. (2015); Wu et al. (2016); Goldman et al. (2016); Lohse et al. (2018); Song et al. (2018), and photonic systems Verbin et al. (2013); Lu et al. (2016); Mittal et al. (2019). In equilibrium theory a topological phase is characterized by the bulk topological invariant defined in the ground state of the system and host protected boundary modes through the bulk-boundary correspondence. Based upon the equilibrium characterization the topological phases can be detected in experiment from the bulk-boundary correspondence, e.g. by resolving the boundary modes with angle-resolved photoelectron spectroscopy (ARPES) or transport measurements, which identify the equilibrium topological features König et al. (2007); Hsieh et al. (2008); Xia et al. (2009). The great success has been achieved in discovering the topological matter, such as topological insulators Fu et al. (2007); Fu and Kane (2007); Fu (2011); Chang et al. (2013), topological semimetals Burkov and Balents (2011); Young et al. (2012); Xu et al. (2015); Lv et al. (2015), and topological superconductors Qi et al. (2009); Bernevig and Hughes (2013); Ando and Fu (2015); Sato and Ando (2017); Zhang et al. (2018a).

Recently, as a momentum-space counterpart of the bulk-boundary correspondence, a dynamical bulk-surface correspondence was proposed for generic $d$-dimensional ($d$D) topological phases with integer invariants, and connects the bulk topology of such equilibrium topological phase and nontrivial dynamical pattern of quench-induced quantum dynamics emerging on the so-called band inversion surfaces (BISs) Zhang et al. (2018b, 2019a). The BISs are $(d-1)$D interfaces in Brillouin zone (BZ) where the band inversion occurs, and are characterized by that the coupling between momentum and one (pseudo)spin component in the Hamiltonian vanishes Zhang et al. (2018b). By suddenly tuning the system from initially trivial phase to topological regime, the induced quench dynamics exhibit novel dynamical topological pattern on the $(d-1)$D BISs, which is uniquely related to the bulk topology of the $d$D equilibrium phase of the post-quench Hamiltonian. The dynamical bulk-surface correspondence establishes a universal correspondence between the equilibrium topological phases and far-from-equilibrium quantum dynamics. It provides conceptually new schemes to characterize and detect with high precision the equilibrium topological phases via non-equilibrium quench dynamics, which have been widely studied in the recent experiments with ultracold atoms Sun et al. (2018a, b); Yi et al. (2019); Song et al. (2019); Wang et al. , solid state spin systems Wang et al. (2019a); Ji et al. (2020); Xin et al. (2020), and superconducting circuits Niu et al. . Many novel issues have been further investigated in theory, such as the dynamical characterization of both symmetry-breaking order and topological phases in correlated systems Zhang et al. , the topological phases in non-Hermitian systems Zhou and Gong (2018); Qiu et al. (2019); Wang et al. (2019b); Zhu et al. (2020) and Floquet bands Zhang et al. (2020); Hu et al. (2020), generalization to generic quenches from a trivial or nontrivial phase via loop unitary construction Hu and Zhao (2020), and to the regime with slow nonadiabatic quantum quenches Ye and Li (2020). These studies also benefit from the high controllability of the synthetic quantum systems, which facilitates the exploration of non-equilibrium quantum dynamics Caio et al. (2015); Hu et al. (2016); Wang et al. (2017); Fläschner et al. (2018); Qiu et al. (2018); McGinley and Cooper (2019); Yang and Chen (2019); Tian et al. (2019); Xiong et al. (2020); Chen et al. (2020); Lu et al. (2020); Su et al. (2020).

The topology emerging on BISs can also be characterized by the topological charges enclosed in the BISs Zhang et al. (2019b), as an analogy to the Gaussian theorem, and such topological charges are dual to BISs and denote the monopole charges located at the nodes of the (pseudo)spin-orbit (SO) couplings Wang et al. (2019a); Ji et al. (2020); Yi et al. (2019). In this picture the topological invariant is viewed as the quantized flux of the monopoles through the BISs, which provides an intuitive perspective for the nontrivial bulk topology. More recently, the high-order BISs are proposed based on a dimension reduction approach Yu et al. , and the $s$th-order BIS correspond to $(d-s)$D momentum subspace which is reduced from $(s-1)$-order BIS by further taking the coupling between momentum and the $s$th (pseudo)spin component to be zero. In the quench dynamics the equilibrium topological phase can be characterized by the dynamical topology emerging on arbitrary high-order BISs. Since the higher-order BISs can be determined with less information, the dynamical theory based on high-order BISs can simplify the characterization of topological phases Yu et al. . The concept of high-order BIS is novelly extended by Li etal Li et al. to characterize the high-order topological phases Benalcazar et al. (2017a, b); Schindler et al. (2018). An interesting consideration is to extend the topological charges to the high-order regime based on the dimension reduction approach, which are dual to the high-order BISs and may have exceptional features and advantages in the dynamical characterization of topological phases, but have yet to be studied.

In this article, we introduce the concept of high-order topological charges, with which we propose a new dynamical characterization theory of topological phases. The equilibrium bulk topology is generically determined by the total $s$th-order topological charges confined on the $(s-1)$th-order BISs and enclosed in the $s$th-order BISs. By quenching the system from trivial phase to topological regime, we further show that the topological phase of post-quench Hamiltonian can be detected through a high-order dynamical bulk-surface correspondence, in which both the high-order topological charges and high-order BISs are identified from quench dynamics. The proposed new characterization theory has two essential advantages: (i) Unlike the $1$st-order topological charge whose characterization necessitates to measure the continuous charge-related (pseudo)spin texture in $d$D space, which could be tedious, the highest ($d$th) order topological charges are characterized by only discrete signs of spin-polarization in the zero dimension. (ii) A high integer-valued topological charge of the first order always reduces to multiple highest-order topological charges with unit charge value. Then the high integer-valued topological invariant can be read by the summation of the highest-order topological charges enclosed by the highest-order BISs. The two fundamental features greatly simplify the characterization and detection the equilibrium topological phases. Finally, these advantages of the dynamical characterization are illustrated with concrete examples.

The remaining part of this paper is organized as follows. In Sec. II, we introduce the generic theory of the new dynamical characterization. In Sec. III, our dynamical scheme is applied to two realistic models. In Sec. IV, we show the decomposition of high integer-valued topological charges. Finally, we summarize the main results and provide the brief discussion in Sec. V.

We consider a simple $d$D model with

which can be realized with recent advances. Here $\mathbf{k}=(k_{r_{1}},k_{r_{2}},...,k_{r_{d}})$ is the $d$D momentum, $m_{0}$ and $m_{i}$ are the effective Zeeman coupling, and $t_{0}$, $t_{\text{so}}$ are the nearest-neighbor spin-conserved and spin-flipped hopping coefficients, respectively.

The 2D quantum anomalous Hall (QAH) model with $(r_{1},r_{2})=(y,x)$ is considered first, which has been realized in cold atoms experiments Liu et al. (2014); Wu et al. (2016); Wang et al. (2018) and widely studied Zhang and Yi (2013); Pan et al. (2016); Liu et al. (2016); Poon and Liu (2018); Jia et al. (2019). The $\boldsymbol{\gamma}$ matrices are Pauli matrices $\gamma_{0,1,2}=\sigma_{z,y,x}$. For $m_{1}=m_{2}=0$, the bulk topology is characterized by the 1st Chern number ($\text{Ch}_{1}$), where the topological phase corresponds to $0<|m_{0}|<2t_{0}$ with $\text{Ch}_{1}=-\text{sgn}(m_{0})$, but the trivial phases are for $|m_{0}|\geqslant 2t_{0}$ with $\text{Ch}_{1}=0$. We perform the quench by suddenly varying $(m_{0},m_{1},m_{2})$ from $(30t_{0},0,0)$ to $(t_{0},0,0)$ for $h_{0}$, from $(0,30t_{0},0)$ to $(t_{0},0,0)$ for $h_{1}$, and from $(0,0,30t_{0})$ to $(t_{0},0,0)$ for $h_{2}$. The time evolution of spin polarization for $\sigma_{z}$-component only needs to be measured, which can present the second-order BISs and second-order topological charges and then gives the information of bulk topology.

A ring structure characterizes the first-order BIS $\mathcal{B}_{1}$ with $h_{0}=0$ is observed from the vanishing TASP $\overline{\langle\sigma_{z}(\mathbf{k})\rangle}_{z}=0$ in Fig. 2(a). The vanishing polarization $\overline{\langle\sigma_{z}(\mathbf{k})\rangle}_{y}=0$ in Fig. 2(b) shows the surfaces of $h_{0}(\mathbf{k})=0$ and $h_{1}(\mathbf{k})=0$, where the second-order BISs $\mathcal{B}_{2}$ are given by $h_{1}(\mathbf{k})=0$ on the first-order BIS $\mathcal{B}_{1}$ and present two points when taking $\mathbf{h}^{(1)}_{\text{eff-so}}(\tilde{\mathbf{k}})=h_{2}$. Moreover, the second-order topological charges determined by $h_{2}(\tilde{\mathbf{k}})=0$ sit on the first-order BIS $\mathcal{B}_{1}$ and are obtained by the vanishing polarization $\overline{\langle\sigma_{z}(\mathbf{k})\rangle}_{x}=0$, as shown in Fig. 2(c). Because the effective BZ is reduced as $\{\tilde{\mathbf{k}}|h_{0}(\mathbf{k})=0\}$ (or say $\tilde{\mathbf{k}}\in\mathcal{B}_{1}$) and the bottom half-ring of first-order BIS holds $h_{1}(\tilde{\mathbf{k}})<0$, the second-order topological charge $\mathcal{C}_{1}^{(2)}=1$ is enclosed into by the second-order BISs, which gives the 1st Chern number $\text{Ch}_{1}=\mathcal{C}_{1}^{(2)}=-1$ and is shown in Fig. 2(e). Compared with the dynamical characterization by using the first-order topological charges in Fig. 2(d), the spin textures around second-order topological charges are determined by the sign of $h_{2}(\tilde{\mathbf{k}})$ at two sides of monopoles, which is more convenient for the experimental measurement.

We further consider the application to a 3D chiral topological insulator with $(r_{1},r_{2},r_{3})=(x,y,z)$, which has been simulated by using nitrogen-vacancy center Ji et al. (2020). The $\boldsymbol{\gamma}$ matrices are taken as $\gamma_{0}=\sigma_{z}\otimes\tau_{x}$, $\gamma_{1}=\sigma_{x}\otimes\mathbbm{1}$, $\gamma_{2}=\sigma_{y}\otimes\mathbbm{1}$, and $\gamma_{3}=\sigma_{z}\otimes\tau_{z}$, where $\sigma_{x,y,z}$ and $\tau_{x,y,z}$ are both Pauli matrices. For $m_{1}=m_{2}=m_{3}=0$, the topological phases are classified by the 3D winding number and are distinguished as: (i) $t_{0}<m_{0}<3t_{0}$ with winding number $\nu_{3}=1$; (ii) $-t_{0}<m_{0}<t_{0}$ with $\nu_{3}=-2$; and (iii) $-3t_{0}<m_{0}<-t_{0}$ with $\nu_{3}=1$. Beyond these regions the phase is trivial. We perform the quench by suddenly varying $(m_{0},m_{1},m_{2},m_{3})$ from $(30t_{0},0,0,0)$ to $(1.5t_{0},0,0,0)$ for $h_{0}$, from $m_{i}=30t_{0}$ to $0$ for $h_{i}$ (tuning $m_{0}$ to $1.5t_{0}$ and keeping $m_{j\neq i}=0$), then the bulk topology in region (i) can be read out by measuring the time evolution of pseudospin polarization of the $\gamma_{0}$-component.

The vanishing $\overline{\langle\gamma_{0}(\mathbf{k})\rangle}_{0,1,2,3}$ in six 2D planes of BZ implies $h_{1,2,3}(\mathbf{k})=0$, but the spherical-like surface is for $h_{0}(\mathbf{k})=0$, which identifies the first-order BIS $\mathcal{B}_{1}$ [see Fig. 3(a)]. When taking the effective SO vector field as $\mathbf{h}^{(1)}_{\text{eff-so}}(\tilde{\mathbf{k}})=(h_{2},h_{3})$, the second-order BIS $\mathcal{B}_{2}$ presents the ring-shape structure produced by $h_{0}=h_{1}=0$ in vanishing polarization of $\overline{\langle\gamma_{0}(\mathbf{k})\rangle}_{1}$, which are confined on the first-order BIS $\mathcal{B}_{1}$ [see Fig. 3(b)]. The corresponding second-order topological charges $\mathcal{C}^{(2)}_{n=1,2}$ with $h_{2}=h_{3}=0$ are determined by the vanishing polarization of $\overline{\langle\gamma_{0}(\mathbf{k})\rangle}_{2,3}$ on the first-order BIS $\mathcal{B}_{1}$. The second-order topological charge $\mathcal{C}_{1}^{2}=1$ is enclosed by the second-order BIS $\mathcal{B}_{2}$, giving the 3D winding number $\nu_{3}=\mathcal{C}_{1}^{(2)}=1$. Moreover, when the effective SO vector field $\mathbf{h}^{(3)}_{\text{eff-so}}(\tilde{\mathbf{k}})=h_{3}$ is taken, the third-order BISs $\mathcal{B}_{3}$ are produced by $h_{0}=h_{1}=h_{2}=0$ in vanishing polarization of $\overline{\langle\gamma_{0}(\mathbf{k})\rangle}_{0,1,2}$, which are confined on the second-order BIS $\mathcal{B}_{2}$ and present two points [see Fig. 3(c)]. The effective BZ is reduced as $\{\tilde{\mathbf{k}}|h_{0}(\mathbf{k})=h_{1}(\mathbf{k})=0\}$ (or say $\tilde{\mathbf{k}}\in\mathcal{B}_{2}$) and the front half-ring structure of the second-order BIS $\mathcal{B}_{2}$ holds $h_{2}(\tilde{\mathbf{k}})<0$. Therefore, the 3D winding number is given by $\nu_{3}=\mathcal{C}_{1}^{(3)}=1$. Similarly, the observation of third-order topological charges are simpler to determine the bulk topology.

Particularly, here we can also identify the third-order topological charges by a minimum measurement scheme (see Appendix B) with advantage in future experiments. By measuring the TASP on some 2D planes of BZ, $\overline{\langle\gamma_{0}(\mathbf{k})\rangle}_{1}=0$ is for all momentum $\mathbf{k}$ on the 2D plane of $k_{x}=0$ (The 2D plane of $k_{x}=-\pi$ is failed to identify the third-order BISs $\mathcal{B}_{3}$). Thus $\overline{\langle\gamma_{0}(\mathbf{k})\rangle}_{0,2}$ on this plane reflects the locations of the second-order BISs $\mathcal{B}_{2}$ and third-order BISs $\mathcal{B}_{3}$ [see Figs. 3(d1) and 3(d2)]. Therefore, the TASP $\overline{\langle\gamma_{0}(\mathbf{k})\rangle}_{3}$ and the normalized dynamic field $\Theta(\mathbf{k})$ on 2D plane of $k_{x}=0$ give the properties of the third-order topological charges and the topological number of the system [see Figs. 3(d3) and 3(d4)].

For a monopole charge without linear dispersion, the charge value is larger than one and the system has a high-valued winding or Chern number. If detecting the topological charges with high charge value to identify the topological phases, it is cumbersome for the measurements of the continuous charge-related (pseudo)spin texture. Nevertheless, we can avoid these redundant measurements by reducing the high integer-valued topological charges to multiple highest-order topological charges with unit charge value. This essential advantage of the highest-order topological charge greatly simplifies topological characterization, especially for high-dimensional systems. Next we use two extended models to illustrate this point.

We first extend the 2D QAH model as follows:

with positive integer $p$. For $m_{1}=m_{2}=0$, the topological phase corresponds to $0<|m_{0}|<2t_{0}$ with $\text{Ch}_{1}=-p\times\text{sgn}(m_{0})$, but the trivial phase is still for $|m_{0}|\geqslant 2t_{0}$. By quenching the system with $p=2$ in the same parameters as 2D QAH model, a ring-shape structure is identified as the first-order BIS $\mathcal{B}_{1}$ from the vanishing polarization $\overline{\langle\sigma_{z}(\mathbf{k})\rangle}_{z}=0$ [see Fig. 4(a)]. Further, four first-order topological charges $\mathcal{C}^{(1)}_{n=1,2,3,4}$ with high charge value $|\mathcal{C}^{(1)}_{n}|=2$ are given by $\overline{\langle\sigma_{z}(\mathbf{k})\rangle}_{y}=\overline{\langle\sigma_{z}(\mathbf{k})\rangle}_{x}=0$ [see Fig. 4(b)]. Thus the bulk topology is determined by the summation of the first-order topological charges enclosed by the first-order BIS $\mathcal{B}_{1}$, i.e. $\text{Ch}_{1}=\mathcal{C}^{(1)}_{2}=-2$ [see Fig. 4(d)].

After taking $\mathbf{h}^{(1)}_{\text{so}}(\tilde{\mathbf{k}})=h_{2}$ for the dimension reduction, we observe the second-order BISs $\mathcal{B}_{2}$ from the vanishing polarization $\overline{\langle\sigma_{z}(\mathbf{k})\rangle}_{y}=0$, which is confined on the first-order BIS $\mathcal{B}_{1}$ [see Fig. 4(b)]. Correspondingly, four second-order topological charges $\mathcal{C}^{(2)}_{n=1,2,3,4}$ are obtained by the vanishing polarization $\overline{\langle\sigma_{z}(\mathbf{k})\rangle}_{x}=0$ [see Fig. 4(c)]. We emphasize that now each second-order topological charge has unit charge value $|\mathcal{C}^{(2)}_{n}|=1$, and then the bulk topology is calculated by the summation of second-order topological charges enclosed by the second-order BISs $\mathcal{B}_{2}$, i.e. $\text{Ch}_{1}=\mathcal{C}^{(2)}_{1}+\mathcal{C}^{(2)}_{3}=-1-1=-2$ [see Fig. 4(e)]. One can find that a first-order topological charge $\mathcal{C}^{(1)}_{2}$ is separated into two second-order topological charges $\mathcal{C}^{(2)}_{n=1,3}$ with unit negative charge by dimension reduction, and the total contribution of the remaining first-order topological charges $\mathcal{C}^{(1)}_{n=1,3,4}$ is equivalent to two second-order topological charges $\mathcal{C}^{(2)}_{n=2,4}$ with unit positive charge [see Fig. 4(f)]. The 2D topology with high integer-valued $\text{Ch}_{1}=2$ is transformed to 0D topology given by the summation of two 0th Chern numbers with $\text{Ch}_{0}=1$, i.e. $\text{Ch}_{1}\rightarrowtail 2\text{Ch}_{0}$.

Similarly, we extend the 3D chiral topological insulator model as the following case,

For $m_{1}=m_{2}=m_{3}=0$, the topological phases are classified by: (i) $t_{0}<m_{0}<3t_{0}$ with $\nu_{3}=3$; (ii) $-t_{0}<m_{0}<t_{0}$ with $\nu_{3}=-6$; and (iii) $-3t_{0}<m_{0}<-t_{0}$ with $\nu_{3}=3$. By taking the quenched parameters as the same as 3D chiral topological insulator model, eight first-order topological charges with high charge value $|\mathcal{C}^{(1)}_{n=1,2,\cdots,8}|=3$ and a spherical-like first-order BIS $\mathcal{B}_{1}$ [see Figs. 5(a) and 5(b)] can be observed by vanishing polarization of $\langle\gamma_{0}(\mathbf{k})\rangle_{0,1,2,3}$ when taking $\mathbf{h}^{(0)}_{\text{so}}(\mathbf{k})=(h_{1},h_{2},h_{3})$. Thus the bulk topology is determined by the summation of first-order topological charges enclosed by the first-order BISs $\mathcal{B}_{1}$, i.e. $\nu_{3}=\mathcal{C}^{(1)}_{1}=3$.

After taking $\mathbf{h}^{(2)}_{\text{so}}(\tilde{\mathbf{k}})=h_{3}$ for the dimension reduction, six third-order topological charges $\mathcal{C}^{(3)}_{n=1,2,\cdots,6}$ sit on the second-order BISs $\mathcal{B}_{2}$ [see Fig. 5(c)]. Each third-order topological charge has unit charge value $|\mathcal{C}^{(3)}_{n=1,2,\cdots,6}|=1$, and then the bulk topology is given by the summation of third-order topological charges enclosed by the third-order BISs $\mathcal{B}_{3}$, i.e. $\nu_{3}=\mathcal{C}^{(3)}_{1}+\mathcal{C}^{(3)}_{3}+\mathcal{C}^{(3)}_{5}=3$ [see Fig. 5(d)]. Similarly, a first-order topological charge $\mathcal{C}^{(1)}_{1}$ is separated into three third-order topological charges $\mathcal{C}^{(3)}_{n=1,3,5}$ with unit positive charge, and the total contribution of the remaining first-order topological charges $\mathcal{C}^{(1)}_{n=2,\cdots,8}$ is equivalent to the summation of three third-order topological charges $\mathcal{C}^{(3)}_{n=2,4,6}$ with unit negative charge [see Fig. 5(e)]. Thus a high integer-valued $3$D winding number with $\nu_{3}=3$ is transformed to the sum of three $0$th Chern numbers, i.e. $\nu_{3}\rightarrowtail 3\text{Ch}_{0}$.

The above results strongly demonstrate the advantages of the highest-order topological charge in characterization of topological phases. Although the definition of topological charge depends on the selection of the $h$-components, choosing a different $h$-component to define the highest-order topological charge will not change the essence of its unit charge value, which is different from the first-order topological charge. Therefore, for a more general system, we only need to measure the properties of the highest-order topological charge to determine the bulk topology.

In conclusion, we have proposed a new dynamical scheme to characterize the equilibrium topological phases based on the high-order topological charges, which correspond to monopoles confined in low dimensional subspaces. Through a dimensional reduction approach for a $d$D bulk Hamiltonian, the topology of the $d$D system can be determined by the arbitrary $s$th order topological charges enclosed by the $s$th-order BISs. In quenching the system from a trivial phase to a topologically nontrivial regime, both the high-order BISs and the high-order topological charges are directly observed by the quench induced (pseudo)spin dynamics, for which the topological phases of post-quench Hamiltonian can be detected dynamically.

The high-order topological charges have essential advantages in characterizing topological phases due to their intrinsic features. We compare the first-order and highest-order topological charges. For the first-order topological charge with unit or high charge value, as defined in $d$D momentum space, its characterization generically necessitates to measure the continuous charge-related (pseudo)spin texture in $d$D space. In comparison, the highest-order topological charges are defined in the zero dimension, and are characterized by the discrete signs of spin-polarization in zero dimension. This intrinsic feature determines that the charge value of a highest-order topological charge only takes $\mathcal{C}^{(d)}=\pm 1$. Accordingly, a high integer-valued lower-order topological charge can always reduce to multiple highest-order topological charges with unit charge value, which can be easily measured in experiment, hence simplifying the characterization and detection of topological phases.

This work was supported by National Natural Science Foundation of China (Grants No. 11761161003, No. 11825401, and No. 11921005), the National Key R& D Program of China (Project No. 2016YFA0301604), Strategic Priority Research Program of the Chinese Academy of Science (Grant No. XDB28000000), and by the Open Project of Shenzhen Institute of Quantum Science and Engineering (Grant No.SIQSE202003).