Topological Insulators in Amorphous Systems

Abstract

Our understanding of topological insulators is based on an underlying crystalline lattice where the local electronic degrees of freedom at different sites hybridize with each other in ways that produce nontrivial band topology, and the search for material systems to realize such phases have been strongly influenced by this. Here we theoretically demonstrate topological insulators in systems with a random distribution of sites in space, i. e., a random lattice. This is achieved by constructing hopping models on random lattices whose ground states possess nontrivial topological nature (characterized e. g., by Bott indices) that manifests as quantized conductances in systems with a boundary. By tuning parameters such as the density of sites (for a given range of fermion hopping), we can achieve transitions from trivial to topological phases. We discuss interesting features of these transitions. In two spatial dimensions, we show this for all five symmetry classes (A, AII, D, DIII and C) that are known to host nontrivial topology in crystalline systems. We expect similar physics to be realizable in any dimension and provide an explicit example of a Z2 topological insulator on a random lattice in three spatial dimensions. Our study not only provides a deeper understanding of the topological phases of non-interacting fermions, but also suggests new directions in the pursuit of the laboratory realization of topological quantum matter. Introduction:The band insulating state of many fermions has received renewed attention in recent times [1][2][3]. Clues that the band insulating state may support additional nontrivial physics -attributed to topology -was provided by the discovery of the integer quantum Hall effect[4] and the theoretical work that followed [5][6][7]. These ideas saw a resurgence with the discovery of the two dimensional spin Hall insulator[8-13], soon followed [14][15][16][17] by the three dimensional topological insulator (see [1][2][3]).A complete classification of gapped phases of noninteracting fermions soon appeared [18][19][20][21]. This classification hinges on the ten symmetry classes of Altland and Zirnbauer [22]. In any given spatial dimension d only five symmetry classes of the ten host topologically nontrivial phases (see e.g Kitaev [21]). Two gapped systems are considered to be topologically equivalent if the ground state of one can be reached starting from the ground state of the other by a symmetry preserving adiabatic deformation of the Hamiltonian which does not close the gap during the deformation process. The ground state of a gapped crystalline system is characterized by a set of filled bands. For each point in the Brillouin zone (BZ), this amounts to a Slater determinant state made of Bloch wavefunctions (whose character is determined by the symmetry class) of the filled bands. In d-dimensions this turns out to be a map from the BZ (d-dimensional torus) to points on a symmetric space whose character is determined by the total number of bands and the symmetry of the system [21]. Whethe...

Topological phase transitions in glassy quantum matter

“…Previously it has been shown that various hopping models with randomly distributed dopants will undergo a topological phase transition at sufficiently high density [2,11]. The purpose of this work is to establish a quantitative description of this phase transition in representative topological models with randomly-generated geometries in two dimensions.…”

“…The Z 2 invariant can be evaluated through a configuration-averaged spin Bott indexC s [23], for which we assume a similar scaling form. It should be stressed that while the scaling hypotheses for topological invariants are superficially similar to the scaling of percolation probability [17], they characterize the topology of quantum ground states of models (1), (2), which is conceptually completely independent on the classical percolation problem.…”

Topological and conventional phases of a three-dimensional electronic glass

“…9,10 A particularly interesting and recent setting to explore such physics occur in the three dimensional Z 2 free fermion SPTs (e.g., the topological band insulators (TBI)), protected by timereversal symmetry (TRS) and particle number conservation, in simple hopping models on a structurally amorphous network. 9 The role of potential disorder in three dimensional Z 2 TBI and associated disorder driven quantum have transitions have so far almost exclusively investigated using a variety of methods [11][12][13][14][15][16][17][18] starting with an underlying crystal. Interestingly in this context Ref.…”

“…Model for amorphous Z 2 insulator in three spatial dimensions : Our starting point is the the hopping Hamiltonian introduced in Ref. 9 :…”

Topological states on fractal lattices

“…This approach, while efficacious, fails when no such structure exists i.e., for aperiodic systems, including amorphous, quasiperiodic, and fractal systems. Nonetheless, topological phenomena have been shown to exist in both amorphous [39][40][41][42][43][44] and quasiperiodic [45][46][47][48][49][50] systems.…”