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arXiv:2309.01344 · Physical review. B./Physical review. B

From orthogonal link to phase vortex in generalized dynamical Hopf insulators

Yuxuan Ma, Xin Li, Yu Wang, Shun‐Cai Zhao, Guangqin Xiong, Tongxin Sun

Published Aug 11, 2023• 0 citations
arXivHTMLPDF

Abstract

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 $(2+1)\mathrm{D}$ 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.

DOI
10.1103/physrevb.108.075128

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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 \mathbitk\mathbit{k}\mathbitk 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.
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.
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.
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.
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 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.
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 $N\ensuremath{\in}{\ensuremath{\pi}}_{3}({S}^{3})=\mathbb{Z}$.