Uncover Topology by Quantum Quench Dynamics

Abstract

Topological quantum states are characterized by nonlocal invariants, and their detection is intrinsically challenging. Various strategies have been developed to study topological Hamiltonians through their equilibrium states. We present a fundamentally new, high-precision dynamical approach, revealing topology through the unitary evolution after a quench from a topological trivial initial state with a two-dimensional Chern band realized in an ultracold 87 Rb atom gas. The emerging ring structure in the spin dynamics uniquely determines the Chern number for the post-quench band and enables probing the full phase diagram of the band topology with high precision. Besides, we also measure the topological band gap and the domain wall structure dynamically formed in the momentum space during the long-term evolution. Our dynamical approach provides a way towards observing a universal bulk-ring correspondence, and has broad applications in exploring topological quantum matter.The discovery of the quantum Hall effect introduced a new fundamental concept, topological quantum phase, to condensed-matter physics [1,2]. In contrast to symmetry-breaking quantum phases that are characterized by local order parameters in the Landau paradigm, topological quantum matter is classified by nonlocal topological invariants [3], which usually do not directly correspond to the local physical observables. In consequence the detection of topological states is intrinsically challenging. In solid-state experiments, various strategies have been developed and great success has been achieved in the discovery of topological quantum matter like topological insulators [4][5][6][7] and semimetals [8,9]. For instance, transport measurements and angle-resolved photoemission spectroscopy are used to detect gapless boundary modes that reflect the bulk topology [10,11]. In some circumstances, these strategies do not provide fully unambiguous evidences for topological quantum phases, as they do not directly measure topological numbers. An important example is topological superconductivity, which supports a kind of exotic non-Abelian quasiparticle called Majorana modes [12][13][14][15] and remains to be rigorously confirmed by experiment.To precisely detect the topology for an ultracold-atom system can be more challenging, whereas the extensive tool box of manipulating and probing ultracold atoms may offer distinct new strategies for measurement. For a one-dimensional (1D) Su-Schrieffer-Heeger model simulated with a 1D double well lattice, the band topology can be determined by measuring the Zak phase [16]. Furthermore, the bulk topology of a 2D Chern insulator, characterized by Chern invariants, can be observed by Hall transport studies [17,18], by Berry curvature mapping [19], and by a minimal measurement strategy [20] of imaging the spin texture at symmetric Bloch momenta [21]. The Chern invariants are precisely detectable only in a few special cases, e.g. when the bulk band is flat [17] or the system is of inversion symmetry [20,21]. Nevertheless, the ...

Characterizing topological phases by quantum quenches: A general theory

“…The central idea of our nonequilibrium theories is to characterize topological phases by dynamical topological invariants emerging from unitary dynamics of a topologically trivial state. This idea is embodied in the dynamical bulk-surface correspondence, which is not only of theoretical significance, but also has practical benefits for experimental realization [40]. Former studies [33,34] all focused on a simple case that the initial trivial state is fully polarized.…”

“…Here we take the 2D QAH model [21,58,59] as an example, whose Hamiltonian reads H QAH (k) = h(k)·σ, where h(k) = (t so sin k x , t so sin k y , m z − t 0 cos k x − t 0 cos k y ). This model has been realized in cold atoms [19,22,40], and the dynamical characterization via deep quenches has been analyzed in Ref. [33].…”

“…In particular, a novel dynamical bulk-surface correspondence was proven recently and applies to the generic d-dimensional (dD) topological phase [33], showing that the equilibrium bulk topology for a generic dD topological phase universally corresponds to the dynamical topology emerging in (d − 1)D momentum subspaces, dubbed band inversion surfaces (BISs). Similar to the bulk-boundary correspondence for equilibrium topological phases in the real space, the dynamical bulk-surface correspondence is of broad applicability and opens up generic non-equilibrium characterization or classification for topological phases [34][35][36][37][38][39], with novel experimental progresses having been made recently [40][41][42][43][44].Building on the dynamical bulk-surface correspon-arXiv:1907.08840v2 [cond-mat.mes-hall]…”

Imaging topology of Hofstadter ribbons

“…Experimental signatures of Chern numbers generally leverage one of two physical effects: in condensed matter systems the edge-bulk correspondence allows the Chern number to be inferred from the quantized Hall conductivity s = C R H K , and in cold-atom experiments direct probes of the underlying band structure at every value of crystal momentum give access to the Chern number through either static [10,11] or dynamic [12][13][14][15] signatures. Both of these connections derive from the pioneering work of Thouless, Kohmoto, Nightingale, and den Nijs [16], in the now famous TKNN paper.…”

Interacting symmetry-protected topological phases out of equilibrium

“…Recently, theoretical progress has been made in generalising notions of topology to this setting. This includes studying periodically driven Hamiltonians, whose Floquet eigenstates can be topologically characterised [15][16][17][18][19][20], as well as identifying fingerprints of static topological phases in quench dynamics [21][22][23][24][25][26]. In addition to these works, which describe features of the system's trajectory over time, the instantaneous topological properties of wavefunctions undergoing time evolution have also been (a) -Equilibrium [27,28].…”