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arXiv:1611.03304 · Physical Review Letters

Scheme to Measure the Topological Number of a Chern Insulator from Quench Dynamics

Ce Wang, Pengfei Zhang, Xin Chen, Jinlong Yu, Hui Zhai

Published May 5, 2017• 159 citations
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Abstract

We show how the topological number of a static Hamiltonian can be measured from a dynamical quench process. We focus on a two-band Chern insulator in two dimension, for instance, the Haldane model, whose dynamical process can be described by a mapping from the [k_{x},k_{y},t] space to the Bloch sphere, characterized by the Hopf invariant. Such a mapping has been constructed experimentally by measurements in cold atom systems. We show that, taking any two constant vectors on the Bloch sphere, their inverse images of this mapping are two trajectories in the [k_{x},k_{y},t] space, and the linking number of these two trajectories exactly equals the Chern number of the static Hamiltonian. Applying this result to a recent experiment from the Hamburg group, we show that the linking number of the trajectories of the phase vortices determines the phase boundary of the static Hamiltonian.

DOI
10.1103/physrevlett.118.185701

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We show how the topological number of a static Hamiltonian can be measured from a dynamical quench process. We focus on a two-band Chern insulator in two dimension, for instance, the Haldane model, whose dynamical process can be described by a mapping from the [k_{x},k_{y},t] space to the Bloch sphere, characterized by the Hopf invariant. Such a mapping has been constructed experimentally by measurements in cold atom systems. We show that, taking any two constant vectors on the Bloch sphere, their inverse images of this mapping are two trajectories in the [k_{x},k_{y},t] space, and the linking number of these two trajectories exactly equals the Chern number of the static Hamiltonian. Applying this result to a recent experiment from the Hamburg group, we show that the linking number of the trajectories of the phase vortices determines the phase boundary of the static Hamiltonian.
In a recent experimental work [Z. Wu et al., Science 354, 83 (2016)], the PKU-USTC group realized a two-dimensional two-band quantum anomalous Hall model on a square Raman lattice. By quenching the atom-laser detuning of such a Raman lattice, the time-dependent Bloch vectors for each quasimomentum points define a Hopf mapping from quasimomentum-time space $({k}_{x},{k}_{y},t)\ensuremath{\in}{T}^{3}$ to the Bloch sphere S2{S}^{2}S2. The Hopf links between the preimages of any two Bloch vectors on S2{S}^{2}S2 can be measured experimentally through Bloch-state tomography using spin-resolved time-of-flight measurement together with suitable radio-frequency manipulations. The dynamical Hopf invariants, which are quasimomentum-time integrations of Chern-Simons densities, can also be extracted experimentally through certain quench processes. As the Hopf invariant equals the Chern number of the postquench Hamiltonian, the measurements of Hopf links and Hopf invariants provide an alternative way of understanding the topological phase diagram of the equilibrium system.
We show evidence which suggests that the quench dynamics of general N -level Chern insulators can be topologically characterised by a linking number that is related to the equilibrium topology of the insulator. Recently, it was demonstrated that the intuitive picture of the Chern number in terms of the wrapping of a sphere, that famously exists in two-level systems, can be generalised to N-level systems, where it corresponds to the wrapping of N − 1 spheres that are nested in one another. The exterior sphere in this nested structure plays a dominant role in determining the Chern number. Inspired by a previous work which focuses on the quench dynamics of two-level systems, we make use of the dominant role played by the exterior sphere and use its surface vector as a Hopf map. For a topologically trivial initial Hamiltonian, the homotopy classes of such maps give rise to the homotopy group π3(S2) = Z, and is used in the characterisation of the quenched topological phase. The topological index is the Hopf invariant and is equal to the linking number of the preimages of the Hopf map.
In the creation of Hopf topological matters, the old paradigm is to conceive the Hopf invariant first, and then display its intuitive topology through links. Here we brush aside this effort and put forward a recipe for unraveling the quenched two-dimensional (2D) two-band Chern insulators under a parallel quench protocol, which implies that the quench quantities with different momentum $\mathbit{k}$ are parallel or antiparallel to each other. We find that whether the dynamical Hopf invariant exists or not, the links in (2+1)D(2+1)\mathrm{D}(2+1)D space always keep their standard shape even for topological initial states, and trace out the trajectories of phase vortices. The linking number is exactly equal to the difference between pre- and postquench Chern numbers regardless of the construction of homotopy groups. We employ two concrete examples to illustrate these results, highlighting the polarity reversal at fixed points.
Using the recently developed Bloch-state tomography technique, the quasimomentum k\mathbf{k}k-dependent Bloch states ${[sin\left({\ensuremath{\theta}}_{\mathbf{k}}/2\right),\phantom{\rule{0.28em}{0ex}}\ensuremath{-}cos\left({\ensuremath{\theta}}_{\mathbf{k}}/2\right){e}^{i{\ensuremath{\phi}}_{\mathbf{k}}}]}^{T}$ of a two-band tight-binding model with two sublattices can be mapped out. We show that if we prepare the initial Bloch state as the lower-band eigenstate of a topologically trivial Haldane Hamiltonian Hi{H}_{i}Hi​ and then quench the Haldane Hamiltonian to Hf{H}_{f}Hf​, the time-dependent azimuthal phase ${\ensuremath{\phi}}_{\mathbf{k}}(t)$ supports two types of vortices. The first type of vortices is static, with the corresponding Bloch vectors pointing to the north pole (${\ensuremath{\theta}}_{\mathbf{k}}=0$). The second type of vortices is dynamical, with the corresponding Bloch vectors pointing to the south pole (${\ensuremath{\theta}}_{\mathbf{k}}=\ensuremath{\pi}$). In the (kx,ky,t)({k}_{x},{k}_{y},t)(kx​,ky​,t) space, the linking number between the trajectories of these two types of vortices exactly equals the Chern number of the lower band of Hf{H}_{f}Hf​, which provides an alternative method to directly map out the topological phase boundaries of the Haldane model.
Integer-valued topological indices, characterizing nonlocal properties of quantum states of matter, are known to directly predict robust physical properties of equilibrium systems. The Chern number, e.g., determines the quantized Hall conductivity of an insulator. Using non-interacting fermionic atoms in a periodically driven optical lattice, here we demonstrate experimentally that the Chern number determines also the far-from-equilibrium dynamics of a quantum system. Extending a respective proposal to Floquet systems, we measure the linking number that characterizes the trajectories of momentum-space vortices emerging after a strong quench. We observe that it directly corresponds to the ground-state Chern number. This one-to-one relation between a dynamical and a static topological index allows us to experimentally map out the phase diagram of our system. Furthermore, we measure the instantaneous Chern number and show that it remains zero under the unitary dynamics.
A quantum quench is a nonequilibrium dynamics governed by the unitary evolution. We propose a two-band model whose quench dynamics is characterized by an arbitrary Hopf number belonging to the homotopy group ${\ensuremath{\pi}}_{3}({S}^{2})=\mathbb{Z}$. When we quench a system from an insulator with the Chern number ${C}_{i}\ensuremath{\in}{\ensuremath{\pi}}_{2}({S}^{2})=\mathbb{Z}$ to another insulator with the Chern number Cf{C}_{f}Cf​, the preimage of the Hamiltonian vector forms links having the Hopf number ${C}_{f}\ensuremath{-}{C}_{i}$. We also investigate a quantum-quench dynamics for a four-band model carrying an arbitrary second Chern number $N\ensuremath{\in}{\ensuremath{\pi}}_{4}({S}^{4})=\mathbb{Z}$, which can be realized by quenching a three-dimensional topological insulator having the three-dimensional winding number $N\ensuremath{\in}{\ensuremath{\pi}}_{3}({S}^{3})=\mathbb{Z}$.