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In recent years, synthesis and experimental research of fractalized materials has evolved in a paradigmatic crossover with topological phases of matter. We present here a theoretical investigation of the helical edge transport in Sierpinski carpets (SCs), combining the Bernevig-Hughes-Zhang (BHZ) model and the Landauer approach. Starting from a pristine two-dimensional topological insulator (2DTI), according to the BHZ model, our results reveal resonant transport modes when the SC fractal generation reaches the same scale as the space discretization; these modes are analyzed within a contour plot mapping of the local spin-polarized currents, shown spanned and assisted by inner-edge channels. From such a deeply fractalized SC building block, we introduce a rich tapestry formed by superior SC hierarchies, enlightening intricate patterns and unique fingerprints that offer valuable insights into how helical edge transport occurs in these fractal dimensions.

Given a (possibly singular) Riemannian foliation $\mathcal{F}$ with closed leaves on a compact manifold $M$ with an adapted metric, we investigate the wave trace invariants for the basic Laplacian about a non-zero period. We compare them to the wave invariants of the underlying Riemannian orbifold that exists when the leaves in the regular region are identified to points, equipped with the metric that is transverse to the leaves of the foliation. Recalling that the basic Laplacian differs from the underlying orbifold Laplacian by a term that is the mean curvature vector field associated to the foliation, we show that the first wave invariant about any non-zero period $T$ corresponding to geodesics perpendicular to the leaves that all lie entirely in the regular region of $M$ is independent of the mean curvature vector field and depends only on the underlying orbifold structure of the leaf space quotient. Similar results hold on the singular strata whenever the transverse geodesic flow remains confined to the strata defined by the leaf dimension of the foliation. Conversely, closed geodesics that pass through the exceptional leaves depend on the full laplacian, including the leaf-wise metric on the ambient space. We also discuss families of representations that yield the same leaf space and basic spectrum. We use this to give conditions under which non-trivial isotropy for orbifold quotients can be detected., Comment: 30 pages

We report here an experimental and theoretical study on the magnetoresistance properties of heavily phosphorous doped germanium on the metallic side of the metal-nonmetal transition. An anomalous regime, formed by negative values of the magnetoresistance, was observed by performing low-temperature measurements and explained within the generalized Drude model, due to the many-body effects. It reveals a key mechanism behind the magnetoresistance properties at low temperatures and, therefore, constitutes a path to its manipulation in such materials of great interest in fundamental physics and technological applications, Comment: 5 pages, 7 figures

In a recent paper by M. Mantoiu and M. Ruzhansky, a global pseudo-differential calculus has been developed for unimodular groups of type I. In the present article we generalize the main results to arbitrary locally compact groups of type I. Our methods involve the use of Plancherel's theorem for non-unimodular groups. We also make connections with a $C^*$-algebraic formalism, involving dynamical systems, and give explicit constructions for the group of affine transformations of the real line., Comment: 21 pages

Mainstream among topological insulators, GaSb/InAs quantum wells present a broken gap alignment for the energy bands which supports the quantum spin Hall insulator phase and forms an important building block in the search of exotic states of matter. Such structures allow the band-gap inversion with electrons and holes confined in adjacent layers, providing a fertile ground to tune the corresponding topological properties. Using a full 3D 8-band ${\bf k}\cdot{\bf p}$ method we investigate the inverted band structure of GaSb/InAs/GaSb and InAs/GaSb/InAs multilayers and the behavior of the helical edge states, under the influence of an electric field applied along the growth direction. By tuning the electric field modulus, we induce the change of the energy levels of both conduction and valence bands, resulting in a quantum spin Hall insulator phase where the helical edge states are predominantly confined in the GaSb layer. In particular, we found that InAs/GaSb/InAs has a large hybridization gap of about $12\,\textrm{meV}$ and, therefore, are promising to observe massless Dirac fermions with a large Fermi velocity. Our comprehensive characterization of GaSb/InAs multilayers creates a basis platform upon which further optimization of III-V heterostructures can be contrasted., Comment: 17 pages, 10 figures

In recent papers and books, a global quantization has been developed for unimodular groups of type I. It involves operator-valued symbols defined on the product between the group $\mathsf{G}$ and its unitary dual $\widehat{\mathsf{G}}$, composed of equivalence classes of irreducible representations. For compact or for graded Lie groups, this has already been developed into a powerful pseudo-differential calculus. In the present article we extend the formalism to arbitrary locally compact groups of type I, making use of the Fourier theory of non-unimodular second countable groups. The unitary dual and its Plancherel measure being quite abstract in general, we put into evidence situations in which concrete forms are available. Kirillov theory and parametrizations of large parts of $\widehat{\mathsf{G}}$ allow rewriting the basic formulae in a manageable form. Some examples of completely solvable groups are worked out., Comment: 25 pages

In this paper, the authors consider leaf spaces of singular Riemannian foliations $\mathcal{F}$ on compact manifolds $M$ and the associated $\mathcal{F}$-basic spectrum on $M$, $spec_B(M, \mathcal{F}),$ counted with multiplicities. Recently, a notion of smooth isometry $\varphi: M_1/\mathcal{F}_1\rightarrow M_2/\mathcal{F}_2$ between the leaf spaces of such singular Riemannian foliations $(M_1,\mathcal{F}_1)$ and $(M_2,\mathcal{F}_2)$ has appeared in the literature. In this paper, the authors provide an example to show that the existence a smooth isometry of leaf spaces as above is not sufficient to guarantee the equality of $spec_B(M_1,\mathcal{F}_1)$ and $spec_B(M_2,\mathcal{F}_2).$ The authors then prove that if some additional conditions involving the geometry of the leaves are satisfied, then the equality of $spec_B(M_1,\mathcal{F}_1)$ and $spec_B(M_2,\mathcal{F}_2)$ is guaranteed. Consequences and applications to orbifold spectral theory, isometric group actions, and their reductions are also explored., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1607.05593 This version corrects a missing hypothesis and includes a cleaner presentation with a shorter proof of the main result. An appendix has also been added with the original proof of one of the main results

Given a compact boundaryless Riemannian manifold $Y$ on which a compact Lie group $G$ acts, there is always a metric on $Y$ such that the action is by isometries. Assuming $Y$ is equipped with such a metric, recall that the $G$-invariant Laplacian is the restriction of the ordinary Laplacian to the space of functions which are constant along the orbits of $G$. In this paper, the author analyzes the wave trace of the $G$-invariant Laplacian and shows that the singularities of this wave trace occur at the lengths of certain geodesic arcs--those that are orthogonal to the orbits and whose endpoints are related by the action of $G$. This defines a notion of the length spectrum for the group action. Further, using the deep connection between foliated manifolds and isometric group actions on manifolds, the asymptotics of these singularities are calculated for an arbitrary non-zero number in the length spectrum.

We apply techniques of microlocal analysis to the study of the transverse geometry of Riemannian foliations in order to analyze spectral invariants of the basic Laplacian acting on functions on a Riemannian foliation with a bundle-like metric. In particular, we consider the trace of the basic wave operator when the mean curvature form is basic. We extend the concept of basic functions to distributions and demonstrate the existence of the basic wave kernel. The singularities of the trace of this basic wave kernel occur at the lengths of certain geodesic arcs which are orthogonal to the closures of the leaves of the foliation. In cases when the foliation has regular closure, a complete representation of the trace of the basic wave kernel can be computed for $t\not=0$. Otherwise, a partial trace formula over a certain set of lengths of well-behaved geodesic arcs is obtained.