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arXiv:2012.13494 · Physical Review A

Dynamically characterizing topological phases by high-order topological charges

Wei Jia, Lin Zhang, Long Zhang, Xiong-Jun Liu

Published May 21, 2021• 15 citations
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Abstract

We propose a theory to characterize equilibrium topological phase with nonequilibrium quantum dynamics by introducing the concept of high-order topological charges, with novel phenomena being predicted. Through a dimension reduction approach, we can characterize a $d$-dimensional ($d\mathrm{D}$) integer-invariant topological phase with lower dimensional topological number quantified by high-order topological charges, of which the $s\mathrm{th}$-order topological charges denote the monopoles confined on the $(s\ensuremath{-}1)\mathrm{th}$-order band inversion surfaces (BISs) that are $(d\ensuremath{-}s+1)\mathrm{D}$ momentum subspaces. The bulk topology is determined by the $s\mathrm{th}$-order topological charges enclosed by the $s\mathrm{th}$-order BISs. By quenching the system from trivial phase to topological regime, we show that the bulk topology of postquench 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. This characterization theory has essential advantages in two aspects. First, the highest ($d\mathrm{th}$) order topological charges are characterized by only discrete signs of spin polarization in zero dimension (i.e., the $0\mathrm{th}$ Chern numbers), whose measurement is much easier than the first-order topological charges that are characterized by the continuous charge-related spin texture in higher dimensional space. Second, a more striking result is that a first-order high integer valued topological charge always reduces to multiple highest order topological charges with unit charge value, and the latter can be readily detected in experiments. The two fundamental features greatly simplify the characterization and detection of the topological charges and also topological phases, which shall advance the experimental studies in the near future.

DOI
10.1103/physreva.103.052213

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We propose a theory to characterize equilibrium topological phase with nonequilibrium quantum dynamics by introducing the concept of high-order topological charges, with novel phenomena being predicted. Through a dimension reduction approach, we can characterize a ddd-dimensional (dDd\mathrm{D}dD) integer-invariant topological phase with lower dimensional topological number quantified by high-order topological charges, of which the sths\mathrm{th}sth-order topological charges denote the monopoles confined on the (s\ensuremath−1)th(s\ensuremath{-}1)\mathrm{th}(s\ensuremath−1)th-order band inversion surfaces (BISs) that are (d\ensuremath−s+1)D(d\ensuremath{-}s+1)\mathrm{D}(d\ensuremath−s+1)D momentum subspaces. The bulk topology is determined by the sths\mathrm{th}sth-order topological charges enclosed by the sths\mathrm{th}sth-order BISs. By quenching the system from trivial phase to topological regime, we show that the bulk topology of postquench 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. This characterization theory has essential advantages in two aspects. First, the highest (dthd\mathrm{th}dth) order topological charges are characterized by only discrete signs of spin polarization in zero dimension (i.e., the 0th0\mathrm{th}0th Chern numbers), whose measurement is much easier than the first-order topological charges that are characterized by the continuous charge-related spin texture in higher dimensional space. Second, a more striking result is that a first-order high integer valued topological charge always reduces to multiple highest order topological charges with unit charge value, and the latter can be readily detected in experiments. The two fundamental features greatly simplify the characterization and detection of the topological charges and also topological phases, which shall advance the experimental studies in the near future.
How to characterize topological quantum phases is a fundamental issue in the broad field of topological matter. Based on dimension reduction approach, here we propose a new theory to characterize equilibrium topological phases by far-from-equilibrium quantum dynamics, and further report the experimental observation. For a d-dimensional (dD) topological phase with integer invariants, we show that its characterization can be reduced to lower-dimensional topological invariants in the high-order band inversion surfaces (BIS-s), of which the nth-order BIS is a (d-n)D interface in momentum space. In quenching the system from trivial phase to topological regime, we show that the quantum dynamics exhibits nontrivial topological pattern in the arbitrary nth-order BISs, which universally corresponds to and so characterizes the equilibrium topological phase of the post-quench Hamiltonian. This prediction unveils a high-order dynamical bulk-surface correspondence in the momentum space, and provides new dynamical schemes with experimental advantages to detect topological states. In particular, based on the highest-order BISs which are of zero dimension, the detection of topological phase is achieved with minimal measurements. We experimentally build up a quantum simulator using nitrogen-vacancy center to measure the high-order dynamical bulk-surface correspondence in 3D chiral topological insulator, and show the advantages of characterizing the topological phase with high-order BISs in experiment.
We investigate a generic dynamical theory to characterize by quantum quenches the topological quantum phases defined for the equilibrium ground state and study the emergent topology of quantum dynamics when the quenches start from a deep trivial (far from the phase boundary) or shallow trivial (close to the phase boundary) phase to topological regimes. Two dynamical schemes are examined: One is to characterize topological phases via quantum dynamics induced by a single quench along an arbitrary axis of (pseudo)spin polarization, and the other applies a set of quenches with respect to all (pseudo)spin axes. These two schemes are both built on the so-called dynamical bulk-surface correspondence, which shows that the ddd-dimensional (dDd\mathrm{D}dD) topological phases with integer invariants can be characterized by the dynamical topological pattern emerging on particular $(d\ensuremath{-}1)\mathrm{D}$ momentum subspaces called band inversion surfaces (BISs). We show that the first dynamical scheme works for both deep and shallow quenches, while for the second scheme, an emergent topological transition, associated with topological charges crossing BISs, is predicted in the joint quench dynamics. A generic criterion for the emergent transition is precisely obtained. Above the criterion (deep quench regime), quantum dynamics on BISs can directly characterize the topology of the postquench Hamiltonian. Below the criterion (shallow quench regime), the quench dynamics may depict a new dynamical topology; the postquench topology can be characterized by the emergent topological invariant plus the total charges moving outside the region enclosed by BISs. We illustrate our results by numerically calculating the 2D quantum anomalous Hall model, which has been realized in ultracold atoms. This work broadens the way to classify topological phases by nonequilibrium quantum dynamics and is feasible for experimental realization.
Quantum dynamics induced in quenching a d-dimensional topological phase across a phase transition may exhibit a nontrivial dynamical topological pattern on the (d-1)D momentum subspace, called band inversion surfaces (BISs), which have a one-to-one correspondence to the bulk topology of the postquench phase. Here we report the experimental observation of such dynamical bulk-surface correspondence through measuring the topological charges in a 2D quantum anomalous Hall model realized in an optical Raman lattice. The system can be quenched with respect to every spin axis by suddenly varying the two-photon detuning or phases of the Raman couplings, in which the topological charges and BISs are measured dynamically by the time-averaged spin textures. We observe that the total charges in the region enclosed by BISs define a dynamical topological invariant, which equals the Chern number of the postquench band and also characterizes the topological pattern of a dynamical field emerging on the BISs, rendering the dynamical bulk-surface correspondence. This study opens a new avenue to explore topological phases dynamically.
We propose a scheme to characterize topological phases with integer invariants through detecting topological charges via quench dynamics. The essential advantage of the current scheme is that only a single-(pseudo)spin component needs to be measured to extract the complete information of topological charges which determine the bulk topology of the phase. A topological charge depicts the chirality of a monopole at Dirac or Weyl point captured by the node of the (pseudo)spin-orbit coupling, and the topological phases are determined by total charges enclosed by the so-called band-inversion surfaces (BISs). We show a duality that measuring the ithi\mathrm{th}ith spin component in quenching jthj\mathrm{th}jth spin axis precisely equals to measuring the jthj\mathrm{th}jth spin component in quenching the ithi\mathrm{th}ith spin axis. With this we find that characterizing the complete information of topological charges is precisely mapped to measuring the quench (pseudo)spin dynamics of a single spin component caused by quenches along all spin axes. Therefore, the detection of topological physics involves measuring only a single spin component and has great experimental advantages. We numerically examine two concrete models, and propose a cold-atom experimental setup with high feasibility for measurement. This work paves the way to dynamically classify topological phases.
Dynamical characterization of equilibrium topological phases has attracted considerable attention in recent years. In this paper, we make a thorough exploration of the nonadiabatic characterization of topological phases under slow quench protocol. We first propose an exactly solvable multistate Landau-Zener model that can be directly applied to the nonadiabatic slow quench dynamics of topological systems. Then we present two different schemes to characterize the bulk topology of the system based on the so-called spin inversion surface. The first one needs the least number of quenching processes, but requires measuring the gradients of time-averaged spin-polarization on the spin inversion surface. The second one only needs to measure the value of time-averaged spin-polarization on the spin inversion surface, thus making it possible to directly characterize the topological phases by introducing an extra quenching process. Moreover, high-order spin inversion surface or band inversion surface relying on the dimension reduction approach, is also generalized to the above two different characterization schemes. One can extract the topological invariant from pairs of points with opposite signs both on the zero-dimensional (0D) highest order band inversion surface and on the 0D highest order spin inversion surface, which greatly simplifies the measurement strategy and characterization process. In a word, superior to the sudden quench protocol, we demonstrate that the topological invariant can be captured not only by the topological information on band inversion surface, but also on the spin inversion surface. In particular, direct characterization of topological phases based on band inversion surface and spin inversion surface can be realized.
The complete characterization of a generic ddd-dimensional Floquet topological phase is usually hard for the requirement of information about the micromotion throughout the entire driving period. In a recent work [L. Zhang et al., Phys. Rev. Lett. 125, 183001 (2020)], an experimentally feasible dynamical detection scheme was proposed to characterize the integer Floquet topological phases using quantum quenches. However, this theory is still far away from completion, especially for free-fermion Floquet topological phases, where the states can also be characterized by Z2{\mathcal{Z}}_{2}Z2​ invariants. Here we develop a full and unified dynamical characterization theory for the Z2{\mathcal{Z}}_{2}Z2​ Floquet topological phases of different dimensionality and tenfold-way symmetry classes by quenching the system from a trivial and static initial state to the Floquet topological regime through suddenly changing the parameters and turning on the periodic driving. By measuring the minimal information of Floquet bands via the stroboscopic time-averaged spin polarizations, we show that the topological spin texture patterns emerging on certain discrete momenta of Brillouin zone called the 0 or $\ensuremath{\pi}$ gap highest-order band-inversion surfaces provide a measurable dynamical Z2{\mathcal{Z}}_{2}Z2​ Floquet invariant, which uniquely determines the Floquet boundary modes in the corresponding quasienergy gap and characterizes the Z2{\mathcal{Z}}_{2}Z2​ Floquet topology. The applications of our theory are illustrated via one- and two-dimensional models that are accessible in current quantum simulation experiments. Our work provides a highly feasible way to detect the Z2{\mathcal{Z}}_{2}Z2​ Floquet topology and completes the dynamical characterization for the full tenfold classes of Floquet topological phases, which shall advance the research in theory and experiments.