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arXiv:1804.10358 · Physical Review A

Measuring Hopf links and Hopf invariants in a quenched topological Raman lattice

Jinlong Yu

Published Apr 17, 2019• 24 citations
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Abstract

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 ${S}^{2}$. The Hopf links between the preimages of any two Bloch vectors on ${S}^{2}$ 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.

DOI
10.1103/physreva.99.043619

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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 (kx,ky,t)\ensuremath∈T3({k}_{x},{k}_{y},t)\ensuremath{\in}{T}^{3}(kx​,ky​,t)\ensuremath∈T3 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 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.
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.
We discuss the topological invariant in the (2+12+12+1)-dimensional quench dynamics of a two-dimensional two-band Chern insulator starting from a topological initial state (i.e., with a nonzero Chern number ci{c}_{i}ci​), evolved by a postquench Hamiltonian (with Chern number cf{c}_{f}cf​). This process is classified by the torus homotopy group ${\ensuremath{\tau}}_{3}({S}^{2})$. In contrast to the process with ci=0{c}_{i}=0ci​=0 studied in previous works, this process cannot be characterized by the Hopf invariant that is described by the sphere homotopy group ${\ensuremath{\pi}}_{3}({S}^{2})=\mathbb{Z}$. It is possible, however, to calculate a variant of the Chern-Simons integral with a complementary part to cancel the Chern number of the initial spin configuration, which at the same time does not affect the (2+12+12+1)-dimensional topology. We show that the modified Chern-Simons integral gives rise to a topological invariant of this quench process, i.e., the linking invariant in the Z2ci{\mathbb{Z}}_{2{c}_{i}}Z2ci​​ class: $\ensuremath{\nu}=({c}_{f}\ensuremath{-}{c}_{i})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.28em}{0ex}}(2{c}_{i})$. We give concrete examples to illustrate this result and also show the detailed deduction to get this linking invariant.
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.
Hopf insulators are exotic topological states of matter outside the standard tenfold-way classification based on discrete symmetries. Its topology is captured by an integer invariant that describes the linking structures of the Hamiltonian in the three-dimensional momentum space. In this paper, we investigate the quantum dynamics of Hopf insulators across a sudden quench and show that the quench dynamics is characterized by a Z2{\mathbb{Z}}_{2}Z2​ invariant $\ensuremath{\nu}$ which reveals a rich interplay between quantum quench and static band topology. We construct the Z2{\mathbb{Z}}_{2}Z2​ topological invariant using the loop unitary operator and prove that $\ensuremath{\nu}$ relates the pre- and postquench Hopf invariants through $\ensuremath{\nu}=(\mathcal{L}\ensuremath{-}{\mathcal{L}}_{0})\phantom{\rule{0.28em}{0ex}}mod\phantom{\rule{0.28em}{0ex}}2$. The Z2{\mathbb{Z}}_{2}Z2​ nature of the dynamical invariant is in sharp contrast to the Z\mathbb{Z}Z invariant for the quench dynamics of Chern insulators in two dimensions. The nontrivial dynamical topology is further attributed to the emergence of $\ensuremath{\pi}$ defects in the phase band of the loop unitary. These $\ensuremath{\pi}$ defects are generally closed curves in the momentum-time space, for example, as nodal rings carrying Hopf charge.