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arXiv:1904.12552 · Physical Review A

Linking invariant for the quench dynamics of a two-dimensional two-band Chern insulator

Xin Chen, Ce Wang, Jinlong Yu

Published Mar 10, 2020• 17 citations
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Abstract

We discuss the topological invariant in the ($2+1$)-dimensional quench dynamics of a two-dimensional two-band Chern insulator starting from a topological initial state (i.e., with a nonzero Chern number ${c}_{i}$), evolved by a postquench Hamiltonian (with Chern number ${c}_{f}$). This process is classified by the torus homotopy group ${\ensuremath{\tau}}_{3}({S}^{2})$. In contrast to the process with ${c}_{i}=0$ 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 ($2+1$)-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 ${\mathbb{Z}}_{2{c}_{i}}$ 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.

DOI
10.1103/physreva.101.032104

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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 \ensuremathτ3(S2){\ensuremath{\tau}}_{3}({S}^{2})\ensuremathτ3​(S2). 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 \ensuremathπ3(S2)=Z{\ensuremath{\pi}}_{3}({S}^{2})=\mathbb{Z}\ensuremathπ3​(S2)=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: \ensuremathν=(cf\ensuremath−ci)mod(2ci)\ensuremath{\nu}=({c}_{f}\ensuremath{-}{c}_{i})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.28em}{0ex}}(2{c}_{i})\ensuremathν=(cf​\ensuremath−ci​)mod(2ci​). 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.
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 $N\ensuremath{\in}{\ensuremath{\pi}}_{3}({S}^{3})=\mathbb{Z}$.
Recent experiments began to explore the topological properties of quench dynamics, i.e., the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well-established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. However, it only works for a topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench, regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.
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.
We classify the topology of the quench dynamics by homotopy groups. A relation between the topological invariant of a postquench order parameter and the topological invariant of a static Hamiltonian is shown in d+1d+1d+1 dimensions (d=1,2,3d=1,2,3d=1,2,3). The midgap states in the entanglement spectrum of the postquench states reveal their topological nature. When a trivial quantum state is under a sudden quench to a Chern insulator, the midgap states in the entanglement spectrum form rings. These rings are analogous to the boundary Fermi rings in the Hopf insulators. Finally, we show a postquench order parameter in 3+1 dimensions can be characterized by the second Chern number. The number of Dirac cones in the entanglement spectrum is equal to the second Chern number.