arXiv subjects

Abhishek Banerjee

Publications and source records attributed to Abhishek Banerjee.

80 records · Page 5Linked to original sources

arXiv2016

Emergence of a weak topological insulator from the Bi$_x$Se$_y$ family and the observation of weak anti-localization

The discovery of strong topological insulators led to enormous activity in condensed matter physics and the discovery of new types of topological materials. Bisumth based chalcogenides are exemplary strong three dimensional topological insulators that host an odd number of massless Dirac fermionic states on all surfaces. A departure from this notion is the idea of a weak topological insulator, wherein only certain surface terminations host surface states characterized by an even number of Dirac cones leading to exciting new physics. Experimentally however, weak topological insulators have proven to be elusive. Here, we report a discovery of a weak topological insulator (WTI), BiSe, of the Bi-chalcogenide family with an indirect band gap of 42 meV. Its structural unit consists of bismuth bilayer (Bi$_2$), a known quantum spin hall insulator sandwiched between two units of Bi$_2$Se$_3$ which are three dimensional strong topological insulators. Angle resolved photo-emission spectroscopy (ARPES) measurements on cleaved single crystal flakes along with density fucntional theory (DFT) calculations confirm the existence of weak topological insulating state of BiSe. Additionally, we have carried out magneto-transport measurements on single crystal flakes as well as thin films of BiSe, which exhibit clear signatures of weak anti-localization at low temperatures, consistent with the properties of topological insulators.

cond-mat.mtrl-sci↗

arXiv2016

Granular topological insulators

Granular conductors form an artificially engineered class of solid state materials wherein the microstructure can be tuned to mimic a wide range of otherwise inaccessible physical systems. At the same time, topological insulators (TIs) have become a cornerstone of modern condensed matter physics as materials hosting metallic states on the surface and insulating in the bulk. However it remains to be understood how granularity affects this new and exotic phase of matter. We perform electrical transport experiments on highly granular topological insulator thin films of Bi$_2$Se$_3$ and reveal remarkable properties. We observe clear signatures of topological surface states despite granularity with distinctly different properties from conventional bulk TI systems including sharp surface state coupling-decoupling transitions, large surface state penetration depths and exotic Berry phase effects. We present a model which explains these results. Our findings illustrate that granularity can be used to engineer designer TIs, at the same time allowing easy access to the Dirac-fermion physics that is inaccessible in single crystal systems.

cond-mat.mtrl-sci↗

arXiv2016

Noetherian Schemes over abelian symmetric monoidal categories

In this paper, we develop basic results of algebraic geometry over abelian symmetric monoidal categories. Let $A$ be a commutative monoid object in an abelian symmetric monoidal category $(\mathbf C,\otimes,1)$ satisfying certain conditions and let $\mathcal E(A)=Hom_{A-Mod}(A,A)$. If the subobjects of $A$ satisfy a certain compactness property, we say that $A$ is Noetherian. We study the localisation of $A$ with respect to any $s\in \mathcal E(A)$ and define the quotient $A/\mathscr I$ of $A$ with respect to any ideal $\mathscr I\subseteq \mathcal E(A)$. We use this to develop appropriate analogues of the basic notions from usual algebraic geometry (such as Noetherian schemes, irreducible, integral and reduced schemes, function field, the local ring at the generic point of a closed subscheme, etc) for schemes over $(\mathbf C,\otimes,1)$ . Our notion of a scheme over a symmetric monoidal category $(\mathbf C,\otimes,1)$ is that of Toën and Vaquié.

math.AG↗

arXiv2016

On Noetherian schemes over $(\mathcal C,\otimes,1)$ and the category of quasi-coherent sheaves

Let $(\mathcal C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let $X$ be a scheme over $(\mathcal C,\otimes,1)$ in the sense of Toën and Vaquié. In this paper we show that when $X$ is quasi-compact and semi-separated, any quasi-coherent sheaf on $X$ may be expressed as a directed colimit of its finitely generated quasi-coherent submodules. Thereafter, we introduce a notion of "field objects" in $(\mathcal C,\otimes,1)$ that satisfy several properties similar to those of fields in usual commutative algebra. Finally we show that the points of a Noetherian, quasi-compact and semi-separated scheme $X$ over such a field object $K$ in $(\mathcal C,\otimes,1)$ can be recovered from certain kinds of functors between categories of quasi-coherent sheaves. The latter is a partial generalization of some recent results of Brandenburg and Chirvasitu.

math.AG↗

arXiv2015

Quasimodular Hecke algebras and Hopf actions

Let $Γ=Γ(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$. In this paper, we extend the theory of modular Hecke algebras due to Connes and Moscovici to define the algebra $\mathcal Q(Γ)$ of quasimodular Hecke operators of level $Γ$. Then, $\mathcal Q(Γ)$ carries an action of "the Hopf algebra $\mathcal H_1$ of codimension $1$ foliations" that also acts on the modular Hecke algebra $\mathcal A(Γ)$ of Connes and Moscovici. However, in the case of quasimodular forms, we have several new operators acting on the quasimodular Hecke algebra $\mathcal Q(Γ)$. Further, for each $σ\in SL_2(\mathbb Z)$, we introduce the collection $\mathcal Q_σ(Γ)$ of quasimodular Hecke operators of level $Γ$ twisted by $σ$. Then, $\mathcal Q_σ(Γ)$ is a right $\mathcal Q(Γ)$-module and is endowed with a pairing $(\_\_,\_\_):\mathcal Q_σ(Γ)\otimes \mathcal Q_σ(Γ)\longrightarrow \mathcal Q_σ(Γ)$. We show that there is a "Hopf action" of a certain Hopf algebra $\mathfrak{h}_1$ on the pairing on $\mathcal Q_σ(Γ)$. Finally, for any $σ\in SL_2(\mathbb Z)$, we consider operators acting between the levels of the graded module $\mathbb Q_σ(Γ)=\underset{m\in \mathbb Z}{\oplus}\mathcal Q_{σ(m)}(Γ)$, where $σ(m)=\begin{pmatrix} 1 & m \\ 0 & 1 \\ \end{pmatrix}\cdot σ$ for any $m\in \mathbb Z$. The pairing on $\mathcal Q_σ(Γ)$ can be extended to a graded pairing on $\mathbb Q_σ(Γ)$ and we show that there is a Hopf action of a larger Hopf algebra $\mathfrak{h}_{\mathbb Z}\supseteq \mathfrak{h}_1$ on the pairing on $\mathbb Q_σ(Γ)$.

math.NT↗

arXiv2014

Dennis trace map for certain $K$-groups of categories with cofibrations

Let $\mathcal C$ be a small category with cofibrations. In this paper, we define the $K$-theory and Hochschild homology groups of $\mathcal C$ of order $Y$, where $Y$ is an ordered finite simplicial set with basepoint. Further, we construct the Dennis trace map between these groups.

math.KT↗

arXiv2014

The $q$-analog of higher order Hochschild homology and the Lie derivative

Let $A$ be a commutative algebra over $\mathbb C$. Given a pointed simplicial finite set $Y$ and $q\in \mathbb C$ a primitive $N$-th root of unity, we define the $q$-Hochschild homology groups of $A$ of order $Y$. When $D$ is a derivation on $A$, we construct the corresponding Lie derivative on these groups. We also define the Lie derivative for a higher derivation $\{D_n\}_{n\geq 0}$ on $A$. Finally, we describe the morphisms induced on the bivariant $q$-Hochschild cohomology groups of order $Y$ by a derivation $D$ on $A$.

math.RA↗

arXiv2014

Sur la catégorie dérivée des faisceaux tordus

Soit $X$ un schéma quasi-compact et séparé et soit $α\in \check{C}^2(X, \mathcal O_X^*)$ un cocycle de Cěch. Nous considérons la catégorie dérivée $D(QCoh(X,α))$ des faisceaux quasi-cohérents sur $X$ tordu par $α$. Soit $Δ(X,α)$ la plus petite sous-catégorie triangulée de $D(QCoh(X,α))$ contenant tous les objets $u_*\mathcal F^\bullet$, où $u:U\longrightarrow X$ est une immersion ouverte avec $U$ affine et $\mathcal F^\bullet\in D(QCoh(U,u^*(α)))$. Alors, le but de cet article est de montrer que $Δ(X,α)=D(QCoh(X,α))$. -- Let $X$ be a quasi-compact and separated scheme and let $α\in \check{C}^2(X, \mathcal O_X^*)$ be a Cěch cocycle. We consider the derived category $D(QCoh(X,α))$ of quasi-coherent sheaves on $X$ twisted by $α$. Let $Δ(X,α)$ be the smallest triangulated subcategory of $D(QCoh(X,α))$ containing all the objects $u_*\mathcal F^\bullet$, where $u:U\longrightarrow X$ is an open immersion with $U$ affine and $\mathcal F^\bullet\in D(QCoh(U,u^*(α)))$. Then, the purpose of this article is to show that $Δ(X,α)=D(QCoh(X,α))$.

math.AG↗