Previous studies have shown that the bulk topology of single-particle systems can be captured by the band inversion surface or by the spin inversion surface emerged on the time-averaged spin polarization. Most of the studies, however, are based on the single-particle picture even though the systems are fermionic and of multi-bands. Here, we study the many-body quench dynamics of topological systems with all the valence bands fully occupied, and show that the concepts of band inversion surface and spin inversion surface are still valid. More importantly, the many-body quench dynamics is shown to be reduced to a nontrivial three-level Landau-Zener model, which can be solved exactly. Based on the analytical results, the topological spin texture revealed by the time-averaged spin polarization can be applied to characterize the bulk topology and thus provides a direct comparison for future experiments.
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.
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
d-dimensional (
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.
We investigate the dynamical characterization theory for periodically driven systems in which Floquet topology can be fully detected by emergent topological patterns of quench dynamics in momentum subspaces called band-inversion surfaces. We improve the results of a recent work [Zhang et al., Phys. Rev. Lett. 125, 183001 (2020)] and propose a more flexible scheme to characterize a generic class of
d-dimensional Floquet topological phases classified by
Z-valued invariants by applying a quench along an arbitrary spin-polarization axis. Our basic idea is that by disassembling the Floquet system into multiple static subsystems that are periodic in quasienergy, a full characterization of Floquet topological phases reduces to identifying a series of bulk topological invariants for time-independent Hamiltonians, which greatly enhances the convenience and flexibility of the measurement. We illustrate the scheme by numerically analyzing two experimentally realizable models in two and three dimensions, respectively, and adopting two different but equivalent viewpoints to examine the dynamical characterization. Finally, considering the imperfection of experiment, we demonstrate that the present scheme can also be applied to a general situation where the initial state is not completely polarized. This study provides an immediately implementable approach for dynamically classifying Floquet topological phases in ultracold atoms or other quantum simulators.
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 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 (
dD) integer-invariant topological phase with lower dimensional topological number quantified by high-order topological charges, of which the
sth-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
sth-order topological charges enclosed by the
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 (
dth) order topological charges are characterized by only discrete signs of spin polarization in zero dimension (i.e., the
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.