Concrete experimental realization of the Lin–Yau doubled topological band structures: two nearly degenerate double bands with second Chern numbers $(m+n,-n)$, interpenetrating insulator lattices, 3D photonic crystals, and Weyl/Bogoliubov–Weyl heterostructure signatures

The 2013 paper by Lin and Yau proposed mapping exotic 7-sphere fibrations onto condensed-matter band topology: an 8x8 'doubled' Hamiltonian, built from two coupled topological-insulator models, whose two nearly degenerate double bands carry second Chern numbers $(m+n,-n)$ over the combined crystal-momentum and tuning-parameter space. Proposed hosts were interpenetrating lattices of topological and normal insulators (e.g., Bi2Se3-based), engineered 3D photonic crystals with two second Chern numbers, TI/NI/TI/NI' multilayer Weyl semimetals, TI/SC/TI/SC' superconducting multilayers with Bogoliubov-Weyl nodes, semiconductor light-hole/heavy-hole multilayers, and cold atoms with F=3/2; the authors left concrete realization explicitly open. The open problem is to identify concrete compositions, geometries, or protocols achieving the required doubled band structure and to experimentally verify its predicted signatures, including robustness to disorder and finite temperature and the practical tunability of Chern-number transfer between the two double bands.

Background

A topological insulator (TI) is a material that is insulating in its interior but hosts protected conducting surface states; the canonical three-dimensional (3D) examples are Bi2Se3, Bi2Te3, and Sb2Te3, described near the band-inversion point by a 4x4 effective Hamiltonian with two orbitals and two spin states (H. Zhang et al., 2009). The topology of a set of bands is captured by Chern numbers. The first Chern number of an isolated band is the integral of its Berry curvature - the geometric curvature of the Bloch eigenstates, which accumulates as a Berry phase over closed loops in parameter space - over a closed two-dimensional parameter surface. The second Chern number generalizes this to a degenerate multiplet of bands: it is the integral of the non-Abelian Berry curvature (a matrix-valued 2-form built from the multiplet's eigenvectors) over a four-dimensional (4D) parameter space, such as the three crystal momenta $(k_1,k_2,k_3)$ together with a tuning parameter $\xi$. Zhang and Hu (2001) showed that a 4D generalization of the quantum Hall effect exists in which the quantized response is governed by the second Chern number, whose source is a unit 'Yang monopole' - the SU(2) analogue of a magnetic monopole - located at a degeneracy point in 4D parameter space. In 3D TIs the same invariant underlies the quantized magnetoelectric ('axion') angle $\theta = 2\pi c_2$ (Qi, Hughes, and Zhang, 2008).

In differential topology, Milnor's exotic 7-spheres are smooth manifolds that are homeomorphic, but not diffeomorphic, to the standard 7-sphere; they arise as total spaces of $S^3$-bundles over $S^4$ (the quaternionic Hopf fibration being the prototype), classified by pairs of second Chern numbers of two SU(2) bundles, writable as $(m+n,-n)$ with conserved total $m$. Lin and Yau (2013) proposed mapping this fibration data onto band topology: they doubled the 4x4 Bi2Se3-type Hamiltonian to 8x8 by adding an su(2) 'pseudospin' label - physically, two interleaved copies of the system, such as an interpenetrating lattice of two insulating materials (one TI plus one normal insulator for $m=1$, two TIs for $m=2$) - so that the two resulting nearly degenerate double bands carry second Chern numbers $c_2^{(1)} = m+n$ and $c_2^{(2)} = -n$ over the $(k,\xi)$ space. The boundary of such a system is predicted to carry $m+n$ right-moving and $n$ left-moving chiral edge modes, and interfaces with a medium of different magnetoelectric angle carry currents proportional to the second Chern numbers. The same paper proposed photonic analogs - photonic crystals, periodic dielectric structures whose photon bands can carry topological Chern numbers in the spirit of Raghu and Haldane (2008), here with two second Chern numbers from two almost-degenerate band doublets - as well as periodic TI/NI/TI/NI' multilayer heterostructures (NI = normal insulator) that realize Weyl semimetals, i.e., 3D crystals whose bands touch at isolated 'Weyl nodes' acting as monopoles of Berry curvature, generalizing the TI-multilayer proposal of Burkov and Balents (2011); and TI/SC/TI/SC' superconducting multilayers hosting 'Bogoliubov-Weyl nodes' - the analogous touching points in the Bogoliubov quasiparticle spectrum - following Meng and Balents (2012). Semiconductor multilayers exploiting the light-hole and heavy-hole bands (the two doubly degenerate valence bands of zincblende semiconductors) and cold atoms with total angular momentum $F=3/2$ (whose four internal states form the required multiplet) were suggested as further hosts. All of this was explicitly theoretical: the paper states that interpenetrating lattices are discussed 'only ... theoretically' and closes by asking 'whether these fibrations can be realized concretely in these experimental settings.' A related experimental program has meanwhile developed around 4D quantum Hall physics: Kraus, Ringel, and Zilberberg (2013) showed that a 2D quasicrystal can host the 4D quantum Hall response, and analog realizations followed in cold-atom topological charge pumps (Lohse et al., 2018), photonic waveguide arrays (Zilberberg et al., 2018), electrical circuits (Wang et al., 2020), and acoustics (Chen et al., 2021) - but all of these realize a single second Chern number through dimensional reduction, not the doubled $(m+n,-n)$ structure. Weyl nodes themselves were discovered experimentally in the TaAs family (Xu et al., 2015), in a different material class from the TI/NI multilayers.

Problem Statement

The question posed by the source is whether the fibration-motivated doubled band topology it proposed can be realized concretely and verified experimentally:

1. Which specific material compositions - e.g., interpenetrating lattices of a topological insulator such as Bi2Se3 with a normal insulator ($m=1$), or of two topological insulators ($m=2$) - or which engineered 3D photonic crystal geometries (including interpenetrating lattices of two photonic crystals), semiconductor light-hole/heavy-hole multilayer stacks, or cold-atom ($F=3/2$) protocols can achieve the required band structure: two nearly degenerate double bands carrying second Chern numbers $(m+n,-n)$ over the combined crystal-momentum and tuning-parameter space, and in what parameter windows?

2. Can the predicted observable consequences be verified experimentally: the chiral edge-mode counting ($m+n$ right-moving versus $n$ left-moving modes, or the one-way photonic analog); the Weyl nodes predicted for TI/NI/TI/NI' heterostructures at the momenta given by the source's band model; the Bogoliubov-Weyl nodes predicted for TI/SC/TI/SC' superconducting multilayers; and the interface responses proportional to the two second Chern numbers?

3. Do these topological properties survive disorder, lattice imperfections, and finite temperature (unstudied in the source), and is the discussed Chern-number transfer between the two double bands - retuning $(m+n,-n)$ to $(m+n',-n')$ at fixed total $m$ through band touchings - practically achievable in real systems?

These facets share one scientific objective: a concrete realization of the doubled fibration band topology - a concrete design or protocol (named compositions, or a specified geometry, stack, or preparation scheme) with a quantitative band model and the computed pair of second Chern numbers - and, ultimately, its experimental verification. The platform classes above, together with any platform engineering the same doubled structure by equivalent means, are the candidate host systems for this realization. Measured verification of the predicted signatures, their robustness to disorder and finite temperature, and the achievability of Chern-number transfer at fixed total m are part of this same objective.

References

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