High-entropy alloys (HEAs) are a relatively new class of materials that have gained considerable attention from the metallurgical research community over recent years. They are characterised by their unconventional compositions, in that they are not based around a single major component, but rather comprise multiple principal alloying elements. Four core effects have been proposed in HEAs: (1) the entropic stabilisation of solid solutions, (2) the severe distortion of their lattices, (3) sluggish diffusion kinetics and (4) that properties are derived from a cocktail effect. By assessing these claims on the basis of existing experimental evidence in the literature, as well as classical metallurgical understanding, it is concluded that the significance of these effects may not be as great as initially believed. The effect of entropic stabilisation does not appear to be overarching, insufficient evidence exists to establish the strain in the lattices of HEAs, and rapid precipitation observed in some HEAs suggests their diffusion kinetics are not necessarily anomalously slow in comparison to conventional alloys. The meaning and influence of the cocktail effect is also a matter for debate. Nevertheless, it is clear that HEAs represent a stimulating opportunity for the metallurgical research community. The complex nature of their compositions means that the discovery of alloys with unusual and attractive properties is inevitable. It is suggested that future activity regarding these alloys seeks to establish the nature of their physical metallurgy, and develop them for practical applications. Their use as structural materials is one of the most promising and exciting opportunities. To realise this ambition, methods to rapidly predict phase equilibria and select suitable HEA compositions are needed, and this constitutes a significant challenge. However, while this obstacle might be considerable, the rewards associated with its conquest are even more substantial. Similarly, the challenges associated with comprehending the behaviour of alloys with complex compositions are great, but the potential to enhance our fundamental metallurgical understanding is more remarkable. Consequently, HEAs represent one of the most stimulating and promising research fields in materials science at present.

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms We show that topology can protect exponentially localized, zero energy edge modes at critical points between one-dimensional symmetry-protected topological phases. This is possible even without gapped degrees of freedom in the bulk-in contrast to recent work on edge modes in gapless chains. We present an intuitive picture for the existence of these edge modes in the case of noninteracting spinless fermions with time-reversal symmetry (BDI class of the tenfold way). The stability of this phenomenon relies on a topological invariant defined in terms of a complex function, counting its zeros and poles inside the unit circle. This invariant can prevent two models described by the same conformal field theory (CFT) from being smoothly connected. A full classification of critical phases in the noninteracting BDI class is obtained: Each phase is labeled by the central charge of the CFT, c ∈ 1 2 N, and the topological invariant, ω ∈ Z. Moreover, c is determined by the difference in the number of edge modes between the phases neighboring the transition. Numerical simulations show that the topological edge modes of critical chains can be stable in the presence of interactions and disorder. Introduction.-Topology is fundamental to characterizing quantum phases of matter in the absence of local order parameters [1]. In one spatial dimension, such zerotemperature phases have topological invariants generically protecting edge modes, i.e., zero energy excitations localized near boundaries. These phases are usually referred to as topological insulators or superconductors when noninteracting [2] and as symmetry-protected topological phases when interacting [3,4]. The topological invariants of one-dimensional systems require the presence of a symmetry and have been classified for phases with a gap above the ground state [5][6][7][8][9][10][11]. Topology and Edge Modes in Quantum Critical ChainsConventional wisdom says topological edge modes require a bulk gap. Recently, there has been work on gapless phases hosting edge modes [12][13][14][15][16][17][18][19][20][21][22][23][24], but, when their localization is exponential, it is attributed to gapped degrees of freedom (meaning there are exponentially decaying correlation functions).We indicate that this picture is at odds with the critical points between topological superconductors in the BDI class (noninteracting spinless fermions with time-reversal symmetry) [25]. Curiously, a 2001 work by Motrunich, Damle, and Huse with different aims implied that some of these transitions host topological edge modes [26]. This phenomenon and its consequences have not been explored since. It is of particular importance given the recent interest in the interplay between topology and criticality, since-as we will argue-the bulk has no gapped degrees of freedom.After revi...