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arXiv:2609.30260

Projected amorphous topological insulators

Archisman Panigrahi, Bitan Roy

Published Sep 24, 2026• 0 citations
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Abstract

We introduce projected amorphous topological insulators (PATIs) that despite containing only a fraction ($x$) of otherwise randomly selected {\it disconnected} sites of a parent crystal, feature a quantized global topological invariant ({\it strong} PATIs) when its effective Hamiltonian is constructed by integrating out the residual sites of the original lattice. Within the topological regime of the parent model, additionally, there exists a critical $x$ below which such systems foster a {\it fragile} PATI that only supports quantized local topological marker on a small fraction of sites therein. Both strong and fragile PATIs accommodate in-gap modes near the boundary. The system also hosts a normal insulator in the entire trivial parameter regime of the lattice-based model for any $x$, devoid of gapless boundary modes, for which both global topological invariant and local topological marker vanish. We demonstrate these {\it possibly} generic outcomes by starting with a parent square lattice-based model for time-reversal symmetry breaking insulators. Then the strong-to-fragile PATI quantum phase transition, tunable via $x$, is characterized by the {\it mean} correlation length exponent $ν\in (1.00,1.46)$.

DOI
10.48550/arxiv.2609.30260

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We introduce projected amorphous topological insulators (PATIs) that despite containing only a fraction (xxx) of otherwise randomly selected {\it disconnected} sites of a parent crystal, feature a quantized global topological invariant ({\it strong} PATIs) when its effective Hamiltonian is constructed by integrating out the residual sites of the original lattice. Within the topological regime of the parent model, additionally, there exists a critical xxx below which such systems foster a {\it fragile} PATI that only supports quantized local topological marker on a small fraction of sites therein. Both strong and fragile PATIs accommodate in-gap modes near the boundary. The system also hosts a normal insulator in the entire trivial parameter regime of the lattice-based model for any xxx, devoid of gapless boundary modes, for which both global topological invariant and local topological marker vanish. We demonstrate these {\it possibly} generic outcomes by starting with a parent square lattice-based model for time-reversal symmetry breaking insulators. Then the strong-to-fragile PATI quantum phase transition, tunable via xxx, is characterized by the {\it mean} correlation length exponent ν∈(1.00,1.46)ν\in (1.00,1.46)ν∈(1.00,1.46).
Nontrivial geometry of electronic Bloch states gives rise to topological insulators which are robust against sufficiently weak randomness inevitably present in any quantum material. However, increasing disorder triggers a quantum phase transition into a featureless normal insulator. As the underlying quantum critical point is approached from the topological side, small scattered droplets of normal insulators start to develop in the system and their coherent nucleation causes ultimate condensation into a trivial insulator. Unless disorder is too strong, the normal insulator accommodates disjoint tiny topological puddles. Furthermore, in the close vicinity of such a transition the emergent islands of topological and trivial insulators display spatial fractal structures, a feature that is revealed only by local topological markers. Here, we showcase this (possibly) generic phenomenon that should be apposite to dirty topological crystals of any symmetry class in any dimension from the Bott index and local Chern marker for a square-lattice-based disordered Chern insulator model.
When time-reversal symmetry is present there is a sharp distinction between topological and trivial band insulators which ensures that, as parameters are varied, these phases are separated by a phase transition at which the bulk gap closes. Surprisingly we find that even in the absence of time-reversal symmetry, gapless regions originating from the phase boundaries persist. Moreover, the critical line generically opens up to enclose Chern insulating phases that are thin but of finite extent in the phase diagram. We explain the topological origin of this effect in terms of quantized charge pumping, showing in particular that it is robust to the effect of disorder and interactions.
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In addition to topological lattice defects such as dislocations and disclinations, crystals are also accompanied by unavoidable ordinary defects, devoid of any non-trivial geometry or topology, among which vacancies, Schottky defects, substitutions, interstitials, and Frenkel pairs are the most common. In this work, we demonstrate that these ubiquitous ordinary lattice defects, although geometrically trivial, can nonetheless serve as universal probes of the non-trivial topology of electronic Bloch bands, and any change in the local topological environment in an otherwise normal insulator in terms of mid-gap bound states in their vicinity. We theoretically establish these generic findings by implementing a minimal model Hamiltonian describing time-reversal symmetry breaking topological and normal insulators on a square lattice, fostering such point defects. The defect-bound mid-gap modes are also shown to be robust against sufficiently weak random point-like charge impurities. Furthermore, we showcase experimental observation of such bound states by embedding ordinary crystal defects in two-dimensional acoustic Chern lattices, where precision-controlled hopping amplitudes are implemented via active meta-atoms and Green's-function-based spectroscopy is used to reconstruct spectra and eigenstates. Our combined theory-experiment study establishes ordinary lattice defects as probes of topology that should be germane in crystals of any symmetry and dimension, raising the possibility of arresting localized Majorana modes near such defects in the bulk of topological superconductors and to emulate ordinary-defect-engineered topological devices.
Abstract The topology of insulators is usually revealed through the presence of gapless boundary modes: this is the so-called bulk-boundary correspondence. However, the many-body wavefunction of a crystalline insulator is endowed with additional topological properties that do not yield surface spectral features, but manifest themselves as (fractional) quantized electronic charges localized at the crystal boundaries. Here, we formulate such bulk-corner correspondence for the physical relevant case of materials with time-reversal symmetry and spin-orbit coupling. To do so we develop partial real-space invariants that can be neither expressed in terms of Berry phases nor using symmetry-based indicators. These previously unknown crystalline invariants govern the (fractional) quantized corner charges both of isolated material structures and of heterostructures without gapless interface modes. We also show that the partial real-space invariants are able to detect all time-reversal symmetric topological phases of the recently discovered fragile type.
Topological crystalline phases (TCPs) are topological states protected by spatial symmetries. A broad range of TCPs have been conventionally studied by formulating topological invariants (symmetry indicators) at invariant momenta in the Brillouin zone, which leaves open the question of their stability in the absence of translational invariance. In this work, we show that robust basis-independent topological invariants can be generically constructed for TCPs using projected symmetry operators. Remarkably, we show that the real-space topological markers of these invariants are exponentially localized to the fixed points of the spatial symmetry. As a result, this real-space structure protects them against the presence of impurities that are located away from the fixed points and also sufficiently slowly varying disorder potentials. By considering all possible symmetry centers in a crystalline system we can generate a mesh of real-space topological markers that can provide a local topological distinction for TCPs. We illustrate the robustness of this mesh of invariants with one- and two-dimensional TCPs protected by inversion, rotational, and mirror symmetries. Finally, we find that the boundary modes of these TCPs can also exhibit robust topological invariants with localized markers on the edges. We illustrate this with the gapless Majorana boundary modes of mirror-symmetric topological superconductors, and relate their integer topological edge invariant with a quantized effective edge polarization.