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.
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 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.
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.
Quantum simulation, as a state-of-the-art technique, provides a powerful way to explore topological quantum phases beyond natural limits. Nevertheless, it is usually hard to simulate both the bulk and surface topological physics at the same time to reveal their correspondence. Here we build up a quantum simulator using nitrogen-vacancy center to investigate a three-dimensional (3D) chiral topological insulator, and demonstrate the study of both the bulk and surface topological physics by quantum quenches. First, a dynamical bulk-surface correspondence in momentum space is observed, showing that the bulk topology of the 3D phase uniquely corresponds to the nontrivial quench dynamics emerging on 2D momentum hypersurfaces called band inversion surfaces (BISs). This is the momentum-space counterpart of the bulk-boundary correspondence in real space. Further, the symmetry protection of the 3D chiral phase is uncovered by measuring dynamical spin textures on BISs, which exhibit perfect (broken) topology when the chiral symmetry is preserved (broken). Finally, we measure the topological charges to characterize directly the bulk topology and identify an emergent dynamical topological transition when varying the quenches from deep to shallow regimes. This work demonstrates how a full study of topological phases can be achieved in quantum simulators.
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.
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.
The detection of topological phases of matter has become a central issue in recent years. Conventionally, the realization of a specific topological phase in condensed matter physics relies on probing the underlying surface band dispersion or quantum transport signature of a real material, which may be imperfect or even absent. On the other hand, quantum simulation offers an alternative approach to directly measure the topological invariant on a universal quantum computer. However, experimentally demonstrating high-dimensional topological phases remains a challenge due to the technical limitations of current experimental platforms. Here, we investigate the three-dimensional topological insulators in the AIII (chiral unitary) symmetry class, which yet lack experimental realization. Using the nuclear magnetic resonance system, we experimentally demonstrate their topological properties, where a dynamical quenching approach is adopted and the dynamical bulk-boundary correspondence in the momentum space is observed. As a result, the topological invariants are measured with high precision on the band-inversion surface, exhibiting robustness to the decoherence effect. Our Letter paves the way toward the quantum simulation of topological phases of matter in higher dimensions and more complex systems through controllable quantum phases transitions.