Quenching a quantum system involves three basic ingredients: the initial phase, the postquench target phase, and quantum dynamics, which may carry the information of the former two. Here we propose a dynamical theory, based on an interaction quench, to characterize both the equilibrium symmetry-breaking order and topological phases by nonequilibrium correlated quantum dynamics. We illustrate the theory with the Haldane-Hubbard model, which is quenched from an initial correlated magnetic phase to a topologically nontrivial regime. We show that the quench dynamics exhibit profound universal behaviors on the so-called band-inversion surfaces (BISs), from which both the topological phase in the weakly interacting regime and the correlated magnetic phase in the strongly interacting regime can be extracted. In particular, the topology is characterized by dynamical topological patterns emerging on BISs, which are robust against interaction-induced dephasing and heating; the symmetry-breaking order can be read out from a universal dynamical scaling behavior, which is valid beyond the mean-field theory. This work uncovers the first paradigm of nonequilibrium characterization of equilibrium symmetry-breaking and topological phases.
Quenching a quantum system involves three basic ingredients: the initial phase, the postquench target phase, and quantum dynamics, which may carry the information of the former two. Here we propose a dynamical theory, based on an interaction quench, to characterize both the equilibrium symmetry-breaking order and topological phases by nonequilibrium correlated quantum dynamics. We illustrate the theory with the Haldane-Hubbard model, which is quenched from an initial correlated magnetic phase to a topologically nontrivial regime. We show that the quench dynamics exhibit profound universal behaviors on the so-called band-inversion surfaces (BISs), from which both the topological phase in the weakly interacting regime and the correlated magnetic phase in the strongly interacting regime can be extracted. In particular, the topology is characterized by dynamical topological patterns emerging on BISs, which are robust against interaction-induced dephasing and heating; the symmetry-breaking order can be read out from a universal dynamical scaling behavior, which is valid beyond the mean-field theory. This work uncovers the first paradigm of nonequilibrium characterization of equilibrium symmetry-breaking and topological phases.
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.
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.
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.
Equilibrium topological phases are robust against weak static disorder but may break down in the strongdisorder regime.Here we propose to explore how the quench-induced emergent dynamical topology evolves under dynamical noise and uncover novel dynamical topological physics beyond equilibrium counterparts.We develop an analytic theory and show that for weak noise, the quantum dynamics induced by quenching an initial trivial phase to the Chern insulating regime exhibits robust emergent topology on certain momentum subspaces called band-inversion surfaces (BISs).The dynamical topology is protected by the minimal oscillation frequency over BISs, mimicking a bulk gap of the dynamical phase.Two novel types of dynamical transitions classified by distinct exceptional points or rings are predicted if increasing the noise to critical strength, with critical points being exactly obtained.At the exceptional points on the BISs the minimal oscillation frequency vanishes, manifesting the dynamical bulk-gap closing, beyond which the dynamical topology breaks down.Interestingly, we predict a sweet spot region of the transition, in which the dynamical topology survives surprisingly at an arbitrarily strong noise regime.This work unveils novel universal physics of dynamical topology under dynamical noise, which can be probed with control in experiment.
Interactions generically have important effects on topological quantum phases. For a quantum anomalous Hall (QAH) insulator, the presence of interactions can qualitatively change a topological phase diagram, which, however, is typically hard to measure in an experiment. Here we propose a scheme based on quench dynamics to detect the mean-field topological phase diagram of an interacting Chern insulator described by the QAH-Hubbard model, with nontrivial dynamical quantum physics being uncovered. We focus on the dynamical properties of the system at a weak to intermediate Hubbard interaction, which mainly induces a ferromagnetic order under the mean-field level. Remarkably, three characteristic times---ts,tc, and t∗---are found in the quench dynamics. The first two (ts,tc) capture the emergences of dynamical self-consistent particle density and the dynamical topological phase transition, respectively, while the third one (t∗) gives a linear scaling time on the topological phase boundaries. We show generically that the characteristic times obey ts>t∗>tc (t∗<ts<tc) in the repulsive (attractive) interacting regime, when the system is quenched from an initial nearly fully polarized state to the topologically nontrivial regimes. Moreover, the Chern number of the equilibrium phase of postquench interacting Hamiltonian can be determined by any two of the three timescales, providing a dynamical way to determine the equilibrium mean-field topological phases. Experimentally, the measurement of ts is challenging, while tc and t∗ can be directly read out by measuring the spin polarizations of four Dirac points and the time-dependent particle density, respectively, showing the feasibility of the present dynamical scheme. Our work opens a way to detect by quench dynamics the mean-field phase diagram of Chern insulators with interactions.