Topological Continuum Research Articles

Topological continuum charges of acoustic phonons in two dimensions and the Nambu-Goldstone theorem

We analyze the band topology of acoustic phonons in 2D materials by considering the interplay of spatial and internal symmetries with additional constraints that arise from the physical context. These supplemental constraints trace back to the Nambu-Goldstone theorem and the requirements of structural stability. We show that this interplay can give rise to previously unaddressed non-trivial nodal charges that are associated with the crossing of the acoustic phonon branches at the center ($\Gamma$-point) of the phononic Brillouin zone. We moreover apply our perspective to the concrete context of graphene, where we demonstrate that the phonon spectrum harbors these kinds of non-trivial nodal charges. Apart from its fundamental appeal, this analysis is physically consequential and dictates how the phonon dispersion is affected when graphene is grown on a substrate. Given the generality of our framework, we anticipate that our strategy that thrives on combining physical context with insights from topology should be widely applicable in characterizing systems beyond electronic band theory.

Quantum decay in a topological continuum

The quantum mechanical decay of two or more overlapped resonances in a common continuum is largely influenced by Fano interference, leading to important phenomena such as the existence of bound states in the continuum, fractional decay and quiescent dynamics for single particle decay, and signature of particle statistics in the many-body quantum decay. An overlooked yet essential requirement to observe Fano interference is time reversal symmetry of the bath. Here we consider multilevel quantum decay in a bath sustaining unidirectional (chiral) propagating states, such as in quantum Hall or in Floquet topological insulators, and show that the chiral nature of scattering states fully suppresses Fano interference among overlapping resonances. As a result, there are not bound states in the continuum, quantum decay is complete, and there is not any signature of particle statistics in the decay process. Nonetheless, some interesting features are disclosed in the multilevel decay dynamics in a topological bath, such as the appearance of high-order exceptional points, long quiescent dynamics followed by a fast decay, and the possibility to observe damped non-Hermitian Bloch oscillations.

Airway pathological heterogeneity in asthma: Visualization of disease microclusters using topological data analysis

Asthma is a complex chronic disease underpinned by pathological changes within the airway wall. How variations in structural airway pathology and cellular inflammation contribute to the expression and severity of asthma are poorly understood. Therefore we evaluated pathological heterogeneity using topological data analysis (TDA) with the aim of visualizing disease clusters and microclusters. A discovery population of 202 adult patients (142 asthmatic patients and 60 healthy subjects) and an external replication population (59 patients with severe asthma) were evaluated. Pathology and gene expression were examined in bronchial biopsy samples. TDA was applied by using pathological variables alone to create pathology-driven visual networks. In the discovery cohort TDA identified 4 groups/networks with multiple microclusters/regions of interest that were masked by group-level statistics. Specifically, TDA group 1 consisted of a high proportion of healthy subjects, with a microcluster representing a topological continuum connecting healthy subjects to patients with mild-to-moderate asthma. Three additional TDA groups with moderate-to-severe asthma (Airway Smooth MuscleHigh, Reticular Basement MembraneHigh, and RemodelingLow groups) were identified and contained numerous microclusters with varying pathological and clinical features. Mutually exclusive TH2 and TH17 tissue gene expression signatures were identified in all pathological groups. Discovery and external replication applied to the severe asthma subgroup identified only highly similar "pathological data shapes" through analyses of persistent homology. We have identified and replicated novel pathological phenotypes of asthma using TDA. Our methodology is applicable to other complex chronic diseases.

Non-trivial ω-limit sets and oscillating solutions in a chemotaxis model in [formula omitted] with critical mass

This paper studies the Cauchy problem for a parabolic–elliptic system in R2 modeling chemotaxis as well as self-attracting particles. In the critical mass case the fine dynamics of the model is ascertained in terms of the structure of the underlying ω-limit sets. According to the results of this paper, any nonnegative radially symmetric bounded solution either stabilizes to a steady-state as t↑∞, or oscillates between two steady-states. Moreover, a rather general class of nonnegative initial data, not necessarily radially symmetric, for which the associated solutions exhibit a complex oscillatory behavior is constructed; their ω-limit sets consist of a nontrivial topological continuum of steady-states. Besides the technical difficulties inherent to the lack of compactness of the resolvent operators, one has to add the challenge that the problem is utterly non-local. Consequently, thought the basic ideas on the foundations of this paper might be considered classical, most of the proofs throughout are extremely sophisticated and absolutely new in their full generality.

Topological description of the Stone-Wales defect formation energy in carbon nanotubes and graphene

We develop a topological continuum framework to compute the formation energies of Stone-Wales defects in graphene and carbon nanotubes. Our approach makes no a priori assumptions about the analytical form of the dislocation strain fields while explicitly accounting for boundary conditions and defect-defect interactions. The continuum formalism reproduces trends observed in the atomistic simulations remarkably well and demonstrates the necessity of considering long-ranged effects to accurately describe defect energetics in graphene-based systems.

Fixed Points of Two-Sided Fractional Matrix Transformations

AbstractLet "Equation missing" and "Equation missing" be "Equation missing" complex matrices, and consider the densely defined map "Equation missing" on "Equation missing" matrices. Its fixed points form a graph, which is generically (in terms of "Equation missing") nonempty, and is generically the Johnson graph "Equation missing"; in the nongeneric case, either it is a retract of the Johnson graph, or there is a topological continuum of fixed points. Criteria for the presence of attractive or repulsive fixed points are obtained. If "Equation missing" and "Equation missing" are entrywise nonnegative and "Equation missing" is irreducible, then there are at most two nonnegative fixed points; if there are two, one is attractive, the other has a limited version of repulsiveness; if there is only one, this fixed point has a flow-through property. This leads to a numerical invariant for nonnegative matrices. Commuting pairs of these maps are classified by representations of a naturally appearing (discrete) group. Special cases (e.g., "Equation missing" is in the radical of the algebra generated by "Equation missing" and "Equation missing") are discussed in detail. For invertible size two matrices, a fixed point exists for all choices of "Equation missing" if and only if "Equation missing" has distinct eigenvalues, but this fails for larger sizes. Many of the problems derived from the determination of harmonic functions on a class of Markov chains.

A topological lattice model

Using random triangulations to define a lattice “metric”, a two-dimensional lattice model of Ising spins which has properties depending only on the topology of the base manifold is described. It is a rather trivial analogue of the much more complicated topological continuum field theories which have recently been introduced. Some general remarks about topological lattice models are made.

Totally nonconnected im kleinen continua

If a connected topological space T' is not connected im kleinen at the point p of T', then there is some open set D' containing p such that p is a boundary point of the p-component of D'. This paper shows that much stronger conditions than those above hold at some points of a Baire topological continuum' which is not connected im kleinen at any point of a certain domain intersection subset. Limits to results of this kind are shown by an example of a bounded plane continuum that is connected im kleinen at each point of a dense inner limiting (i.e., Ga) subset but is not locally connected at any point. DEFINITION. A topological continuum T' is totally nonconnected im kleinen on a subset A of T' if T' is not connected im kleinen at any point of A (i.e., if each point p of A is contained in some open subset U of T' such that p is a boundary point, relative to T', of the pcomponent of U). DEFINITION. If (1) T' is a topological continuum, (2) Z' is the least cardinal number of any topological basis for T', (3) the set U is an open subset of T', and (4) the subset A of U is the nonvacuous common part of not more than Z' open subsets of T', each dense in U, then A is a dense-domain intersection subset of U (relative to T'). If, in addition to (1), (2) and (3), U is not the sum of Z', or fewer than Z', closed nowhere dense subsets of T', then T' is Baire topological on U. If T' is Baire topological on each open subset of U, then T' is locally Baire topological on U.