Reactivity to sustainability metrics: A configurational study of motivation and capacity

Previous research on reactivity – defined as changing organisational behaviour to better conform to the criteria of measurement in response to being measured – has found significant variation in company responses towards sustainability metrics. We propose that reactivity is driven by dialogue, motivation and capacity in a configurational way. Empirically, we use fuzzy set Qualitative Comparative Analysis (fsQCA) to analyse company responses to the sustainability index FTSE4Good. We find evidence of complimentary and substitute effects between motivation and capacity. Based on these effects we develop a typology of reactivity to sustainability metrics, which also theorises the use of metrics as tools for performance feedback and the building of calculative capacity. We show that when reactivity is studied configurationally, we can identify previously underacknowledged types of responses. We discuss the theoretical and practical implications for studying and using sustainability metrics as governance tools for responsible behaviour

Interplay between electronic topology and crystal symmetry: Dislocation-line modes in topological band-insulators

We elucidate the general rule governing the response of dislocation lines in three-dimensional topological band insulators. According to this ${\bf K}\text{-}{\bf b}\text{-}{\bf t}$ rule, the lattice topology, represented by dislocation lines oriented in direction ${\bf t}$ with Burgers vector ${\bf b}$, combines with the electronic-band topology, characterized by the band-inversion momentum ${\bf K}_{\rm inv}$, to produce gapless propagating modes when the plane orthogonal to the dislocation line features a band inversion with a nontrivial ensuing flux $\Phi={\bf K}_{\rm inv}\cdot {\bf b}\,\, ({\rm mod\,\,2\pi})$. Although it has already been discovered by Y. Ran {\it et al.}, Nature Phys. {\bf 5}, 298 (2009), that dislocation lines host propagating modes, the exact mechanism of their appearance in conjunction with the crystal symmetries of a topological state is provided by the ${\bf K}\text{-}{\bf b}\text{-}{\bf t}$ rule . Finally, we discuss possible experimentally consequential examples in which the modes are oblivious for the direction of propagation, such as the recently proposed topologically-insulating state in electron-doped BaBiO$_3$.Comment: Main text + supplementary material, published versio

The translational side of topological band insulators

Spin-orbit coupled materials have attracted revived prominent research interest as of late, especially due their connection with topological materials. A hallmark of this pursuit is arguably formed by the advent of topological band insulators (TBIs). Whereas such topological systems are often characterized by the presence of metallic edge states, we here provide a review on the equally rich, although far less explored, aspects of the bulk. Starting from the fact that pi fluxes unambiguously probe the nontrivial nature of time reversal invariant TBIs upon binding special excitations, we summarize the criteria for the formation of similar localized modes in the presence of crystal dislocations. These criteria under which dislocations act as effective fluxes directly relate to their status as unique probes of translational symmetry breaking and as such pinpoint an indexing of TBIs beyond the standard tenfold way. Apart from elucidating refined classification schemes that include crystal symmetries, we also review why the predicted dislocation physics may be perceived compelling in its own right. For example, the study of these defects could result in new routes to explore spin-charge separated excitations. Similarly, arrays of dislocations, as realizable in ordinary grain boundaries, can be shown to host self organized semimetals that have distinct transport properties. Most importantly, however, research efforts on the material side as well as recent experimental signatures further indicate that the outlined perspective can very well provide for an attractive, timely and experimentally viable research agenda beyond the edge state focussed activities

Impurity Bound States and Greens Function Zeroes as Local Signatures of Topology

We show that the local in-gap Greens function of a band insulator $\mathbf{G}_0 (\epsilon,\mathbf{k}_{\parallel},\mathbf{r}_{\perp}=0)$, with $\mathbf{r}_\perp$ the position perpendicular to a codimension-1 or -2 impurity, reveals the topological nature of the phase. For a topological insulator, the eigenvalues of this Greens function attain zeros in the gap, whereas for a trivial insulator the eigenvalues remain nonzero. This topological classification is related to the existence of in-gap bound states along codimension-1 and -2 impurities. Whereas codimension-1 impurities can be viewed as 'soft edges', the result for codimension-2 impurities is nontrivial and allows for a direct experimental measurement of the topological nature of 2d insulators.Comment: 11 pages, 8 figure

Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry

We present a general methodology towards the systematic characterization of crystalline topological insulating phases with time reversal symmetry (TRS).~In particular, taking the two-dimensional spinful hexagonal lattice as a proof of principle we study windings of Wilson loop spectra over cuts in the Brillouin zone that are dictated by the underlying lattice symmetries.~Our approach finds a prominent use in elucidating and quantifying the recently proposed ``topological quantum chemistry" (TQC) concept.~Namely, we prove that the split of an elementary band representation (EBR) by a band gap must lead to a topological phase.~For this we first show that in addition to the Fu-Kane-Mele $\mathbb{Z}_2$ classification, there is $C_2\mathcal{T}$-symmetry protected $\mathbb{Z}$ classification of two-band subspaces that is obstructed by the other crystalline symmetries, i.e.~forbidding the trivial phase. This accounts for all nontrivial Wilson loop windings of split EBRs \textit{that are independent of the parameterization of the flow of Wilson loops}.~Then, we show that while Wilson loop winding of split EBRs can unwind when embedded in higher-dimensional band space, two-band subspaces that remain separated by a band gap from the other bands conserve their Wilson loop winding, hence revealing that split EBRs are at least "stably trivial", i.e. necessarily non-trivial in the non-stable (few-band) limit but possibly trivial in the stable (many-band) limit.~This clarifies the nature of \textit{fragile} topology that has appeared very recently.~We then argue that in the many-band limit the stable Wilson loop winding is only determined by the Fu-Kane-Mele $\mathbb{Z}_2$ invariant implying that further stable topological phases must belong to the class of higher-order topological insulators.Comment: 27 pages, 13 figures, v2: minor corrections, new references included, v3: metastable topology of split EBRs emphasized, v4: prepared for publicatio

Self-organized pseudo-graphene on grain boundaries in topological band insulators

Semi-metals are characterized by nodal band structures that give rise to exotic electronic properties. The stability of Dirac semi-metals, such as graphene in two spatial dimensions (2D), requires the presence of lattice symmetries, while akin to the surface states of topological band insulators, Weyl semi-metals in three spatial dimensions (3D) are protected by band topology. Here we show that in the bulk of topological band insulators, self-organized topologically protected semi-metals can emerge along a grain boundary, a ubiquitous extended lattice defect in any crystalline material. In addition to experimentally accessible electronic transport measurements, these states exhibit valley anomaly in 2D influencing edge spin transport, whereas in 3D they appear as graphene-like states that may exhibit an odd-integer quantum Hall effect. The general mechanism underlying these novel semi-metals -- the hybridization of spinon modes bound to the grain boundary -- suggests that topological semi-metals can emerge in any topological material where lattice dislocations bind localized topological modes.Comment: 14 pages, 6 figures. Improved discussion compared to the earlier versio