Emil Prodan

Quantifying the configuration space and the Gibbs measure of thermally disordered condensed matter systems has been a long-standing problem. The challenge is to avoid the Gibbs paradox, which forbids any ordering or labeling of the atoms. Our key observation is that the lattice of a thermally disordered condensed matter system, in either solid, liquid, or gas phase, can be fully reconstructed from the Voronoi cells of the atoms alone, even if these Voronoi cells are disassembled and randomly scrambled. In the example of the crystalline phase of silicon, the statistics of the Voronoi cells reveals the existence of four, and only four, large facets that are present with probability one for all temperatures up to the solid–liquid melting line. These four largest facets, which separate nearest-neighboring atoms, can be also be used to reconstruct the lattice of the crystal. Hence, their collection supplies the optimal representation of the configuration of the crystal. We conjecture that the existence of Voronoi facets that, despite their large thermal fluctuations, survive with probability one up to the melting temperature, is the fundamental signature of the crystalline solid phase and therefore key to quantifying the Gibbs measure over the entire solid phase.

Given a triangulation of a closed orientable surface, we place single-mode resonators or single-orbital artificial atoms at its vertices, edges, and facets, and we devise near-neighbor hopping terms derived from the boundary and Poincaré duality maps of the simplicial complex of the triangulation. Regardless of the surface or its triangulation, these terms always lead to tight-binding Hamiltonians with large and clean topological spectral gaps, carrying non-trivial Chern numbers in the limit of infinite refinement of the triangulation. We confirm this via numerical simulations, and demonstrate how these models enable topological edge modes at the surfaces of real-world objects. Furthermore, we describe a metamaterial whose dynamics reproduces that of the proposed model, thus bringing the topological metamaterials closer to real-world applications.

Let \mathcal{L}_{\mathbf{v}}\subset{\mathbb{Z}}^{D} be a suitable cone semigroup and \mathfrak{A}_{\mathbf{v}} its reduced semigroup C^{*} -algebra. In this paper, we compute the \mathcal{L}_{\mathbf{v}} -invariant measures in the transversal hull of the semigroup \mathcal{L}_{\mathbf{v}} that exhibit regularity in the boundaries of \mathcal{L}_{\mathbf{v}} . These measures enable the construction of a trace per-unit hypersurface for observables in \mathfrak{A}_{\mathbf{v}} supported near the boundaries of \mathcal{L}_{\mathbf{v}} , leading to the construction of appropriate Chern cocycles in the “boundary” ideals of \mathfrak{A}_{\mathbf{v}} . Our approach applies to both finitely and non-finitely generated cone semigroups. Applications for the bulk-defect correspondence of lattice models of topological insulators are also provided.

We study the dynamics of synthetic molecules whose architectures are generated by space transformations from a point group acting on seed resonators. We show that the dynamical matrix of any such molecule can be reproduced as the left regular representation of a self-adjoint element from the stabilized group's algebra. Furthermore, we use elements of representation theory and K-theory to rationalize the dynamical features supported by such synthetic molecules up to topological equivalences. These tools enable us to identify a set of fundamental models which generate by superposition all possible dynamical matrices up to homotopy equivalences. Interpolations between these fundamental models give rise to topological spectral flows.

In space-adiabatic approaches one can approximate Hamiltonians that are modulated slowly in space by phase-space functions that depend on position and momentum. In this paper, we establish a rigorous relation between this approach and the operator-theoretic approach for topological insulators with defects, which employs $C^*$-algebras and operator K-theory. Using such tools, we show that by quantizing phase-space functions one can construct lattice Hamiltonians which are gapped at certain spatial limits and carry protected states at defects such as boundaries, hinges, and corners. Moreover, we show that the topological invariants that protect the latter can be computed in terms of the symbol functions. This enables us to compute boundary maps in K-theory that are relevant for bulk-defect correspondences.