Searcharxiv⌕ Search

arXiv subjects

Santanu Sarkar

Publications and source records attributed to Santanu Sarkar.

At least 19 recordsLinked to original sources
arXiv2026

Reduced State Stabilizer R\'enyi Entropy as a Probe of Quantum Phase Transitions in Frustrated J_1-J_2 Spin Models

We investigate whether the second-order purity-corrected stabilizer R\'enyi entropy (SRE) of reduced two-qubit density matrices can serve as a reliable local indicator of quantum phase transitions (QPTs) in frustrated quantum spin systems. We consider the one-dimensional isotropic \(J_1-J_2\) Heisenberg model, the one-dimensional XXZ \(J_1-J_2\) model, and the two-dimensional \(J_1-J_2\) Heisenberg model on a \(4\times4\) square lattice. Unlike several previously studied quantum information measures, which fail to detect QPTs in the ground state of these frustrated systems, the reduced ground-state purity-corrected SRE successfully identifies most transitions. For the remaining cases, we consider a low temperature subjacent state, modeled as a statistical mixture of the ground and first excited states with a Maxwell--Boltzmann-type occupation probability. For the 1D isotropic model, the subjacent-state SRE shows a discontinuity at the critical point, yielding \(\alpha_c(\infty)=0.24116\), in excellent agreement with established values; the ground-state SRE shows a point of inflection, yielding \(\alpha_c(\infty)=0.2681\). For the 1D XXZ model, the subjacent-state SRE reproduces the full anisotropy dependent phase diagram, while the ground-state SRE captures transitions only at low anisotropy. For the 2D model, the subjacent-state SRE detects two transitions, at \(\alpha_c(4\times4)=0.40781\) and \(0.6208\), while the ground-state SRE identifies the second at \(0.6230\). Compared with conventional two-qubit entanglement, purity-corrected SRE shows a clear advantage in revealing otherwise-invisible phase transitions, establishing it as a robust, efficient, local probe of frustrated quantum criticality.

quant-ph↗
arXiv2026

Reduced-State Stabilizer R\'enyi Entropy as a Probe of Quantum Criticality in the Transverse ANNNI Model and the Quantum Compass Model

We investigate the effectiveness of the stabilizer R\'enyi entropy (SRE), a quantifier associated with non-stabilizer resources (quantum magic), as an indicator of quantum phase transitions. Specifically, we analyze the behavior of the purity-corrected SRE of reduced density matrices in the ground states of two one-dimensional spin models: the transverse axial next-nearest-neighbor Ising (TANNNI) model and the quantum compass model (QCM). The ground state of the TANNNI model is obtained using exact diagonalization techniques, while the QCM is analyzed using the Jordan--Wigner (JW) transformation followed by Bogoliubov diagonalization of the resulting quadratic fermionic Hamiltonian. For the TANNNI model, the purity-corrected SRE successfully detects the antiphase--floating phase transition in the high-frustration regime, while in the low-frustration regime the raw (purity-uncorrected) SRE reproduces the known ferromagnetic--paramagnetic phase boundaries more accurately. For the QCM, the purity-corrected SRE exhibits a clear signature near the isotropic point \(J_x/J_z=1\), where the system undergoes a first-order quantum phase transition. Our results establish SRE of reduced states as a complementary probe of quantum criticality and provide further insight into the role of non-stabilizer resources in many-body quantum phase transitions.

quant-ph↗
arXiv2026

Vector valued de Branges spaces, CNU contractions and functional models

In this paper, we study vector valued de Branges spaces associated with a de Branges operator, defined as a pair of Fredholm operator valued analytic functions on a domain symmetric with respect to the unit circle. Using a suitable direct sum decomposition of a Hilbert space, we construct a class of vector valued reproducing kernel Hilbert spaces and show that under some assumptions these are vector valued de Branges spaces. We further demonstrate that these spaces provide functional models for certain class of completely non-unitary contraction operators. We also give a Fredholm-type criterion for verifying the hypotheses of the main construction and apply it to several concrete classes of completely non-unitary contractions. Next, we establish connections between the Sz.-Nagy-Foias characteristic function of the contraction operator, the projection operator valued function arising from the Hilbert space decomposition, and the reproducing kernel of the de Branges space. In particular, we show that the characteristic function coincides with the projection operator valued function on the unit disc. Enroute, we also obtain a complete unitary invariance of a certain class of cnu contractions in terms of de Branges quotient operator valued functions. Finally, we discuss certain aspects of the canonical contraction in de Branges model and its $L^2$ realization. These results provide a new perspective on the role of vector valued de Branges spaces in operator model theory.

math.FA↗
arXiv2026

Composition operators on de Branges spaces of entire functions

This paper aims to study the boundedness and compactness of composition operators from model spaces to the Hardy Hilbert spaces in the upper half-plane. Consequently, we investigate the boundedness and compactness of composition operators on de Branges spaces of entire functions. Moreover, we observe that the boundedness of a composition operator on a regular de Branges space forces the inducing symbol to be affine; conversely, affine symbols under appropriate conditions yield bounded composition operators. Furthermore, we show that the behaviour of boundedness and compactness of composition operators on general de Branges spaces is different from that on the Paley-Wiener spaces.

math.FA↗
arXiv2024

de Branges matrices and associated de Branges spaces of vector valued entire functions

This paper extends the concept of de Branges matrices to any finite $m\times m$ order where $m=2n$. We shall discuss these matrices along with the theory of de Branges spaces of $\mathbb{C}^n$-valued entire functions and their associated functions. A parametrization of these matrices is obtained using the Smirnov maximum principle for matrix valued functions. Additionally, a factorization of matrix valued meromorphic functions is discussed.

math.FA↗
arXiv2024

Analytic Kramer sampling and quasi Lagrange-type interpolation in vector valued RKHS

This paper discusses an abstract Kramer sampling theorem for functions within a reproducing kernel Hilbert space (RKHS) of vector valued holomorphic functions. Additionally, we extend the concept of quasi Lagrange-type interpolation for functions within a RKHS of vector valued entire functions. The dependence of having quasi Lagrange-type interpolation on an invariance condition under the generalized backward shift operator has also been discussed. Furthermore, the paper establishes the connection between quasi Lagrange-type interpolation, operator of multiplication by the independent variable, and de Branges spaces of vector valued entire functions.

math.FA↗
arXiv2023

Vector valued de Branges spaces of entire functions based on pairs of Fredholm operator valued functions and functional model

In this paper, we have considered vector valued reproducing kernel Hilbert spaces (RKHS) $\mathcal{H}$ of entire functions associated with operator valued kernel functions. de Branges operators $\mathfrak{E}=(E_- , E_+)$ analogous to de Branges matrices have been constructed with the help of pairs of Fredholm operator valued entire functions on $\mathfrak{X}$, where $\mathfrak{X}$ is a complex seperable Hilbert space. A few explicit examples of these de Branges operators are also discussed. The newly defined RKHS $\mathcal{B}(\mathfrak{E})$ based on the de Branges operator $\mathfrak{E}=(E_-,E_+)$ has been characterized under some special restrictions. The complete parametrizations and canonical descriptions of all selfadjoint extensions of the closed, symmetric multiplication operator by the independent variable have been given in terms of unitary operators between ranges of reproducing kernels. A sampling formula for the de Branges spaces $\mathcal{B}(\mathfrak{E})$ has been discussed. A particular class of entire operators with infinite deficiency indices has been dealt with and shown that they can be considered as the multiplication operator for a specific class of these de Branges spaces. Finally, a brief discussion on the connection between the characteristic function of a completely nonunitary contraction operator and the de Branges spaces $\mathcal{B}(\mathfrak{E})$ has been given.

math.FA↗
arXiv2023

Representing the inverse map as a composition of quadratics in a finite field of characteristic $2$

In 1953, Carlitz~\cite{Car53} showed that all permutation polynomials over $\F_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ ax+b$ (affine functions, where $0\neq a, b\in \F_q$). Recently, Nikova, Nikov and Rijmen~\cite{NNR19} proposed an algorithm (NNR) to find a decomposition of the inverse function in quadratics, and computationally covered all dimensions $n\leq 16$. Petrides~\cite{P23} found a class of integers for which it is easy to decompose the inverse into quadratics, and improved the NNR algorithm, thereby extending the computation up to $n\leq 32$. Here, we extend Petrides' result, as well as we propose a number theoretical approach, which allows us to cover easily all (surely, odd) exponents up to~$250$, at least.

math.NT↗
arXiv2023

Some aspects of vector valued de Branges spaces of entire functions

This paper deals with certain aspects of the vector valued de Branges spaces of entire functions that are based on pairs of Fredholm operator valued functions. Some factorization and isometric embedding results are extended from the scalar valued theory of de Branges spaces. In particular, global factorization of Fredholm operator valued entire functions and analytic equivalence of reproducing kernels of de Branges spaces are discussed. Additionally, the operator valued entire functions associated with these de Branges spaces are studied, and a connection with the operator nodes is established.

math.FA↗
arXiv2022

Spread and asymmetry of typical quantum coherence and their inhibition in response to glassy disorder

We consider the average quantum coherences of typical redits and qudits - vectors of real and complex Hilbert spaces - with the analytical forms stemming from the symmetry of Haar-uniformly distributed random pure states. We subsequently study the response to disorder in spread of the typical quantum coherence in response to glassy disorder. The disorder is inserted in the state parameters. Even in the absence of disorder, the quantum coherence distributions of redits and qudits are not uniform over the range of quantum coherence, and the spreads are lower for higher dimensions. On insertion of disorder, the spreads decrease. This decrease in the spread of quantum coherence distribution in response to disorder is seen to be a generic feature of typical pure states: we observe the feature for different strengths of disorder and for various types of disorder distributions, viz. Gaussian, uniform, and Cauchy-Lorentz. We also find that the quantum coherence distributions become less asymmetric with increase in dimension and with infusion of glassy disorder.

quant-ph↗
arXiv2017

$\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes

In this paper we study $\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes. These codes are identified as $\mathbb{Z}_{2^n}[x]$-submodules of $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}[x]/ \langle x^{α_i}-1\rangle$; $α_i$ and $\rm{i}$ being relatively prime for each $i=1,2,\ldots,n.$ We first define a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then define the distance between two codewords and the minimum distance of such a code. Moreover we relate these to binary codes using the generalized Gray maps. We define the duals of such codes and show that the dual of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code is also cyclic. We then give the polynomial definition of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then determine the structure of such codes and derive a minimal spanning set for that. We also determine the total number of codewords in this code. We finally give an illustrative example of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code.

cs.IT↗
arXiv2015

Properties of singular integral operators $S_{α,β}$

For $α, β\in L^{\infty} (S^1),$ the singular integral operator $S_{α,β}$ on $L^2 (S^1)$ is defined by $S_{α,β}f:= αPf+βQf$, where $P$ denotes the orthogonal projection of $L^2(S^1)$ onto the Hardy space $H^2(S^1),$ and $Q$ denotes the orthogonal projection onto $H^2(S^1)^{\perp}.$ In a recent paper Nakazi and Yamamoto have studied the normality and self-adjointness of $S_{α,β}.$ This work has shown that $S_{α,β}$ may have analogous properties to that of the Toeplitz operator. In this paper we study several other properties of $S_{α,β}.$

math.FA↗
arXiv2015

On Acyclic Edge-Coloring of Complete Bipartite Graphs

An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic ($2$-colored) cycles. The acyclic chromatic index of a graph $G$, denoted by $a'(G)$, is the least integer $k$ such that $G$ admits an acyclic edge-coloring using $k$ colors. Let $Δ= Δ(G)$ denote the maximum degree of a vertex in a graph $G$. A complete bipartite graph with $n$ vertices on each side is denoted by $K_{n,n}$. Basavaraju, Chandran and Kummini proved that $a'(K_{n,n}) \ge n+2 = Δ+ 2$ when $n$ is odd. Basavaraju and Chandran provided an acyclic edge-coloring of $K_{p,p}$ using $p+2$ colors and thus establishing $a'(K_{p,p}) = p+2 = Δ+ 2$ when $p$ is an odd prime. The main tool in their approach is perfect $1$-factorization of $K_{p,p}$. Recently, following their approach, Venkateswarlu and Sarkar have shown that $K_{2p-1,2p-1}$ admits an acyclic edge-coloring using $2p+1$ colors which implies that $a'(K_{2p-1,2p-1}) = 2p+1 = Δ+ 2$, where $p$ is an odd prime. In this paper, we generalize this approach and present a general framework to possibly get an acyclic edge-coloring of $K_{n,n}$ which possess a perfect $1$-factorization using $n+2 = Δ+2$ colors. In this general framework, we show that $K_{p^2,p^2}$ admits an acyclic edge-coloring using $p^2+2$ colors and thus establishing $a'(K_{p^2,p^2}) = p^2+2 = Δ+ 2$ when $p\ge 5$ is an odd prime.

cs.DM↗
arXiv2014

Drastic Minimization in the van der Waals Interaction with the Bottom Epitaxial Graphene Layer by the Diels-Alder Surface Chemistry of the Top Graphene Layer

The Diels-Alder surface modified top epitaxial graphene layer, with newly created pair of sp3 carbon centres, results in abrupt minimization of interlayer van der Waals interactions between two stacked graphene planes, and escapes the wafer during post-reaction manipulation stage, leaving the layer under it almost pristine-like. Above picture shows Diels-Alder functionalized sp2/sp3 graphene adduct leaves the parent wafer. In this communication we systematically address several fundamental questions in graphene surface chemistry, which are of extreme importance for device fabrication, and in successful implementation of covalently modified graphene in electronics industry.

cond-mat.mtrl-sci↗
arXiv2014

A Chemical Route to Graphene for Electronics and Spintronics Device Applications

The development of selective high precision chemical functionalization strategies for device fabrication, in conjunction with associated techniques for patterning graphene wafers with atomic accuracy would provide the necessary basis for a post-CMOS manufacturing technology. This requires a thorough understanding of the principles governing the reactivity and patterning of graphene at the sub-nanometer length scale. This article reviews our quest to delineate the principles of graphene chemistry - that is, the chemistry at the Dirac point and beyond, and the effect of covalent chemistry on the electronic structure, electrical transport and magnetic properties of this low-dimensional material in order to enable the scalable production of graphene-based devices for low- and high-end technology applications.

cond-mat.mtrl-sci↗
arXiv2014

Organometallic Complexes of Graphene and Carbon Nanotubes: Introducing New Perspectives in Atomtronics, Spintronics, High Mobility Graphene Electronics and Energy Conversion Catalysis

Here we present an overview of recent fundamental studies on the nature of the interaction between individual metal atoms and metal clusters and the conjugated surfaces of graphene and carbon nanotube with a particular focus on the electronic structure and chemical bonding at the metal-graphene interface. We discuss the relevance of organometallic complexes of graphitic materials to the development of a fundamental understanding of these interactions and their application in atomtronics as atomic interconnects, high mobility organometallic transistor devices, high-frequency electronic devices, organometallic catalysis (hydrogen fuel generation by photocatalytic water splitting, fuel cells, hydrogenation), spintronics, memory devices and in the next generation energy devices. We touch on CVD graphene grown on metals, the reactivity of its surface, and its use as a template for asymmetric graphene functionalization chemistry (ultrathin Janus discs). We highlight some of the latest advances in understanding the nature of interactions between metals and graphene surfaces from the standpoint of metal overlayers deposited on graphene and SWNT thin films. Finally, we comment on the major challenges facing the field and the opportunities for technological applications.

cond-mat.mtrl-sci↗
arXiv2013

Organometallic Complexes of Graphene

We demonstrate the organometallic hexahapto complexation of chromium with graphene, graphite and carbon nanotubes. All of these extended periodic pi-electron systems exhibit some degree of reactivity toward the reagents CrCO)6 and (eta6-benzene)Cr(CO)3, and we are able to demonstrate the formation of (eta6-rene)Cr(CO)3 or (eta6-arene)2Cr, where arene = single-walled carbon nanotubes (SWNT), exfoliated graphene (XG), epitaxial graphene (EG) and highly-oriented pyrolytic graphite (HOPG). We find that the SWNTs are the least reactive presumably as a result of the effect of curvature on the formation of the hexahapto bond; in the case of HOPG, (eta6-HOPG)Cr(CO)3 was isolated while the exfoliated graphene samples were found to give both (eta6-graphene)2Cr, and (eta6-graphene)Cr(CO)3 structures. We report simple and efficient routes for the mild decomplexation of the graphene-chromium complexes which appears to restore the original pristine graphene state. This study represents the first example of the use of graphene as a ligand and is expected to expand the scope of graphene chemistry in connection with the application of this material in organometallic catalysis.

physics.chem-ph↗