arXiv subjects

Sudhanshu Shekhar

Publications and source records attributed to Sudhanshu Shekhar.

15 recordsLinked to original sources

arXiv2026

Quantum transport and unified scaling law in graphene with polyadic Cantor electrostatic barriers

We study the quantum transport of Dirac electrons in graphene subjected to a polyadic Cantor-structured electrostatic potential. Using the superperiodic potential formalism, we obtain a closed-form expression for the transmission probability. As the Cantor stage increases, the transmission spectrum evolves from a sparse set of superlattice resonances to a near-transparent regime, with the polyadic order setting the rate of this evolution. The angular response depends on the doping configuration, showing distinct behavior in the $n$--$n$--$n$, $n$--$p$--$n$, and Dirac-point cases. In the near-transparent regime, the transmission follows double-logarithmic scaling laws with respect to four independent control parameters: the Cantor stage, the potential height, the initiator length, and the angle of incidence. By combining these individual scaling relations, we establish a unified scaling law governing quantum transport. These results show that the hierarchical self-similarity of the potential governs the transport properties of such systems.

cond-mat.mes-hall↗

arXiv2026

A Gauge-Covariant Geometric Framework for Non-Hermitian Quantum Systems

We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence $\sim |ε_μ|^{-2}$, while the Berry curvature shows a weaker, subleading divergence $\sim |ε_μ|^{-3/2}$, with $ε_μ$ denoting the parameter displacement from the EP along an individual parameter axis $μ$. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.

quant-ph↗

arXiv2025

Iwasawa theory of fine Selmer groups over global fields

The $p^\infty$-fine Selmer group of an elliptic curve $E$ over a number field $F$ is a subgroup of the classical $p^\infty$-Selmer group of $E$ over $F$. Fine Selmer group is closely related to the 1st and 2nd Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group of $E$ over a $p$-adic Lie extension of a number field is intricately related to some deep questions in classical Iwasawa theory; for example, Iwasawa's classical $μ$-invariant vanishing conjecture. In this article, we study the properties of the $p^\infty$-fine Selmer group of an elliptic curve over certain $p$-adic Lie extensions of a number field. We also define and discuss $p^\infty$-fine Selmer group of an elliptic curve over function fields of characteristic $p$ and also of characteristic $\ell \neq p.$ We relate our study with a conjecture of Jannsen.

math.NT↗

arXiv2025

2-Selmer companion modular forms

Let $N$ be a positive integer and $K$ be a number field. Suppose that $f_1,f_2 \in S_k(Γ_0(N))$ are two newforms such that their residual Galois representations at $2$ are isomorphic. Let $ω_2: G_{\mathbb Q} \rightarrow {\mathbb Z}^*_2$ be the $2$-adic cyclotomic character. Then, under suitable hypotheses, we have shown that for every quadratic character $χ$ of $K$ and each critical twist $j$, the residual Greenberg $2$-Selmer groups of $f_1χω_2^{-j}$ and $f_2χω_2^{-j}$ over $K$ are isomorphic. This generalizes the corresponding result of Mazur-Rubin on $2$-Selmer companion elliptic curves. Conversely, if the difference of the residual Greenberg (respectively Bloch-Kato) $2$-Selmer ranks of $f_1χ$ and $f_2χ$ is bounded independent of every quadratic character $χ$ of $K$, then under suitable hypotheses we have shown that the residual Galois representations at $2$ of $f_1$ and $f_2$ are isomorphic as $G_K$-modules. The corresponding result for elliptic curves was a conjecture of Mazur-Rubin, which was proved by M. Yu.

math.NT↗

arXiv2025

Klein Tunneling in Uniaxial Strained Graphene under Super-Periodic Potential

In this article, we employ the transfer matrix method (TMM) to analytically explore the impact of uniaxial strain on electron scattering in graphene under locally periodic and super-periodic electrostatic potential. Our study reveals that strain significantly influences electron transmission through the merging parameter $(δ)$, which modulates the Dirac cone positions. a positive merging parameter ($δ> 0$) reduces the transmission probability by opening an energy gap at the merging point, while a negative merging parameter ($δ< 0$) enhances transmission by bringing the Dirac cones closer, facilitating electron transport at certain angles. However, the resonance peaks in super-periodic potential (SPP) are sharper for $δ< 0$, making them more pronounced but increasingly difficult to resolve as the number of barriers increases.

cond-mat.mes-hall↗

arXiv2024

Relativistic particles in super-periodic potentials: exploring graphene and fractal systems

In this article, we employ the transfer matrix method to investigate relativistic particles in super-periodic potentials (SPPs) of arbitrary order $n \in I^{+}$. We calculate the reflection and transmission probabilities for spinless Klein particles encountering rectangular potential barriers with super-periodic repetition. It is found that spinless relativistic particles exhibit Klein tunneling and a significantly higher degree of reflection compared to their non-relativistic counterparts. Additionally, we analytically explore the behavior of experimentally realizable massless Dirac electrons as they encounter rectangular potential barriers with a super-periodic pattern in a monolayer of graphene. In this system, the transmission probability, conductance, and Fano factor are evaluated as functions of the number of barriers, the order of super-periodicity, and the angle of incidence. Our findings reveal that the transmission probability shows a series of resonances that depend on the number of barriers and the order of super-periodicity. We extend our analysis to specific cases within the Unified Cantor Potentials (UCPs)-$γ$ system ($γ$ is a scaling parameter greater than $1$), focusing on the General Cantor fractal system and the General Smith-Volterra-Cantor (GSVC) system. For the General Cantor fractal system, we calculate the tunneling probability, which reveals sharp transmission peaks and progressively thinner unit cell potentials as $G$ increases. In the GSVC system, we analyze the potential segment length and tunneling probability, observing nearly unity tunneling coefficients when $γ\approx 1$, as well as saturation behavior in transmission coefficients at higher stages $G$.

cond-mat.mes-hall↗

arXiv2024

Generating QES potentials supporting zero energy normalizable states for an extended class of truncated Calogero Sutherland model

Motivated by recent interest in the search for generating potentials for which the underlying Schrödinger equation is solvable, we report in the recent work several situations when a zero-energy state becomes bound depending on certain restrictions on the coupling constants that define the potential. In this regard, we present evidence of the existence of regular zero-energy normalizable solutions for a system of quasi-exactly solvable (QES) potentials that correspond to the rationally extended many-body truncated Calogero-Sutherland (TCS) model. Our procedure is based upon the use of the standard potential group approach with an underlying $so(2, 1)$ structure that utilizes a point canonical transformation with three distinct types of potentials emerging having the same eigenvalues while their common properties are subjected to the evaluation of the relevant wave functions. These cases are treated individually by suitably restricting the coupling parameters.

quant-ph↗

arXiv2023

Iwasawa theory for Rankin-Selberg convolution at an Eisenstein prime

Let $p$ be an odd prime, $ f$ be a $ p $-ordinary newform of weight $ k $ and $ h $ be a normalized cuspidal $ p $-ordinary Hecke eigenform of weight $ l < k$. In this article, we study the $p$-adic $ L $-function and $ p^{\infty} $-Selmer group of the Rankin-Selberg product of $f$ and $h$ under the assumption that $ p $ is an Eisenstein prime for $ h $ i.e. the residual Galois representation of $ h $ at $ p $ is reducible. We show that the $ p $-adic $ L $-function and the characteristic ideal of the $p^\infty$-Selmer group of the Rankin-Selberg product of $f, h$ generate the same ideal modulo $ p $ in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for $f \otimes h$ holds mod $p$. As an application to our results, we explicitly describe a few examples where the above congruence holds.

math.NT↗

arXiv2021

Twisting lemma for $Λ$-adic modules

A classical twisting lemma says that given a finitely generated torsion module $M$ over the Iwasawa algebra $\mathbb{Z}_p[[Γ]]$ with $Γ\cong \mathbb{Z}_p, \ \exists$ a continuous character $θ: Γ\rightarrow \mathbb{Z}_p^\times$ such that, the $ Γ^{n}$-Euler characteristic of the twist $M(θ)$ is finite for every $n$. This twisting lemma has been generalized for the Iwasawa algebra of a general compact $p$-adic Lie group $G$. In this article, we consider a further generalization of the twisting lemma to $\mathcal{T}[[G]]$ modules, where $G$ is a compact $p$-adic Lie group and $\mathcal{T}$ is a finite extension of $\mathbb{Z}_p[[X]]$. Such modules naturally occur in Hida theory. We also indicate arithmetic application by considering the twisted Euler Characteristic of the big Selmer (respectively fine Selmer) group of a $Λ$-adic form over a $p$-adic Lie extension.

math.NT↗

arXiv2020

Multiplicities in Selmer groups and root numbers for Artin twists

Let $K/F$ be a finite Galois extension of number fields and $σ$ be an absolutely irreducible, self-dual representation of $\mathrm{Gal}(K/F)$. Let $p$ be an odd prime and consider two elliptic curves $E_1, E_2$ with good, ordinary reduction at primes above $p$ and equivalent mod-$p$ Galois representations. In this article, we study the variation of the parity of the multiplicities of $σ$ in the representation space associated to the $p^\infty$-Selmer group of $E_i$ over $K$. We also compare the root numbers for the twist of $E_i/F$ by $σ$ and show that the $p$-parity conjecture holds for the twist of $E_1/F$ by $σ$ if and only if it holds for the twist of $E_2/F$ by $σ$. We also express Mazur-Rubin-Nekovář's arithmetic local constants in terms of certain local Iwasawa invariants.

math.NT↗

arXiv2019

Relating Tate-Shafarevich group of an elliptic curve with class group

The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb Z}/p)$ operates by its standard 2 dimensional representation over ${\mathbb Z}/p$. We establish such a relationship in most cases.

math.NT↗

arXiv2019

$p^r$-Selmer companion modular forms

The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $χ$ of $K$, the $n$-Selmer groups of $E_1^χ$ and $E_2^χ$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogue of this for Bloch-Kato $p^r$-Selmer group of modular forms. We compare the Bloch-Kato Selmer groups of a modular form respectively with the Greenberg Selmer group when the modular form is $p$-ordinary and with the signed Selmer group of Lei-Loeffler-Zerbes when the modular form is non-ordinary at $p$. We also indicate the corresponding results over $\Q_\cyc$ and its relation with the well known congruence results of the special values of the corresponding $L$-functions due to Vatsal.

math.NT↗

arXiv2017

Non-commutative twisted Euler characteristic

It is well known that given a finitely generated torsion module $M$ over the Iwasawa algebra $\mathbb Z_p[[Γ]]$, where $Γ\cong \mathbb Z_p$, there exists a continuous $p$-adic character $ρ$ of $Γ$ such that, for the twist $M(ρ)$ of $M$, the $Γ_n := Γ^{p^n}$ Euler characteristic, i.e. $χ(Γ_n, M(ρ))$, is finite for every $n$. We prove a generalization of this result by considering modules over the Iwasawa algebra of a general $p$-adic Lie group $G$, instead of $Γ$. We relate this twisted Euler characteristic to the evaluation of the {\it Akashi series} at the twist and in turn use it to indicate some application to the Iwasawa theory of elliptic curves. This article is a natural generalization of the result established in [JOZ].

math.NT↗

arXiv2017

Root numbers and parity of local Iwasawa invariants

Given two elliptic curves $E_1$ and $E_2$ defined over the field of rational numbers, $\mathbb{Q}$, with good reduction at an odd prime $p$ and equivalent mod $p$ Galois representation, we compare the $p$-Selmer rank, global and local root numbers of $E_1$ and $E_2$ over number fields.

math.NT↗

arXiv2016

Visibility of Shafarevich-Tate group of abelian varieties over number field extensions

Given an abelian variety J and an abelian subvariety A of J over a number field K, we study the visible elements of the Shafarevich-Tate group of A with respect to J over certain number field extension M of K. The notion of visible elements in Shafarevich-Tate group of an abelian variety was introduced by Mazur. In this article, we study the image of Visible elements of A with respect to J under the natural restriction map of the Galois cohomology of A over K to the Galois cohomology of A over M. In particular, for a fixed odd prime p, we investigate the conditions under which visible elements of order p can be produced over a quadratic extension or a degree p extension M of K.

math.NT↗