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Bhabani Prasad Mandal

Publications and source records attributed to Bhabani Prasad Mandal.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2024

Polyadic Cantor potential of minimum lacunarity: Special case of super periodic generalized unified Cantor potential

To bridge the fractal and non-fractal potentials we introduce the concept of generalized unified Cantor potential (GUCP) with the key parameter $N$ which represents the potential count at the stage $S=1$. This system is characterized by total span $L$, stages $S$, scaling parameter $ρ$ and two real numbers $μ$ and $ν$. Notably, the polyadic Cantor potential (PCP) system with minimal lacunarity is a specific instance within the GUCP paradigm. Employing the super periodic potential (SPP) formalism, we formulated a closed-form expression for transmission probability $T_{S}(k, N)$ using the $q$-Pochhammer symbol and investigated the features of non-relativistic quantum tunneling through this potential configuration. We show that GUCP system exhibits sharp transmission resonances, differing from traditional quantum systems. Our analysis reveals saturation in the transmission profile with evolving stages $S$ and establishes a significant scaling relationship between reflection probability and wave vector $k$ through analytical derivations.

quant-ph↗
arXiv2024

Exploring topological entanglement through Dehn surgery

We compute the $\text{PSL}(2,\mathbb{C})$ Chern-Simons partition function of a closed 3-manifold obtained from Dehn fillings of the link complement $\mathbf S^3\backslash {\mathcal{L}}$, where $\mathcal{L}=\mathcal{K}# H$ is the connected sum of the knot $\mathcal {K}$ with the Hopf link $H$. Motivated by our earlier work on topological entanglement and the reduced density matrix $σ$ for such link complements, we wanted to determine a choice of Dehn filling so that the trace of the matrix $σ$ becomes equal to the $\text{PSL}(2,\mathbb{C})$ partition function of the closed 3-manifold. We use the SnapPy program and numerical techniques to show this equivalence up to the leading order. We have given explicit results for all hyperbolic knots $\mathcal{K}$ up to six crossings.

hep-th↗
arXiv2024

New solutions of Isochronous potentials in terms of exceptional orthogonal polynomials in heterostructures

Point canonical transformation (PCT) has been used to find out new exactly solvable potentials in the position-dependent mass (PDM) framework. We solve $1$-D Schrödinger equation in the PDM framework by considering two different fairly generic position-dependent masses $ (i) M(x)=λg'(x)$ and $(ii) M(x) = c \left( {g'(x)} \right)^ν$, $ν=\frac{2η}{2η+1},$ with $η= 0,1,2\cdots $. In the first case, we find new exactly solvable potentials that depend on an integer parameter $m$, and the corresponding solutions are written in terms of $X_m$-Laguerre polynomials. In the latter case, we obtain a new one parameter $(ν)$ family of isochronous solvable potentials whose bound states are written in terms of $X_m$-Laguerre polynomials. Further, we show that the new potentials are shape invariant by using the supersymmetric approach in the framework of PDM.

quant-ph↗
arXiv2023

Quantum tunneling from a new type of generalized Smith-Volterra-Cantor potential

In this paper, we introduce and analyze the Smith-Volterra-Cantor potential of power \( n \), denoted as SVC\(\left(ρ, n\right)\). Bridging the gap between the general Cantor and SVC systems, this novel potential offers a fresh perspective on Cantor-like potential systems within quantum mechanics that unify fractal and non-fractal potentials. Utilizing the Super Periodic Potential (SPP) formalism, we derive the close form expression of the transmission probability \( T_{G}(k) \). Notably, the system exhibits exceptionally sharp transmission resonances, a characteristic that distinguishes it from other quantum systems. Furthermore, the multifaceted transmission attributes of the SVC\(\left(ρ, n\right)\) are found to be critically dependent on both parameters, \( ρ\) and \( n \), offering an intricate interplay that warrants deeper exploration. Our findings highlight a pronounced scaling behavior of reflection probability with \( k \), which is underpinned by analytical derivations.

quant-ph↗
arXiv2023

Renormalization Group Approach for Modified vdP Oscillator with $\mathcal{PT}$ Symmetric Non-Hermitian Interaction

We consider a modified version of the well-known 2d vdP oscillator with a new non-Hermitian interaction. The usual perturbative approach fails to provide the classical dynamics of the system as the classical solutions become divergent in the long time limit. These kinds of divergences are similar to what occurs in quantum field theory and critical phenomena. The Renormalization Group (RG) technique for the dynamical system has been used to eliminate the divergences in the perturbative solution of the 2d vdP oscillator and to provide a physically acceptable solution which is shown to be consistent with numerical study. We further investigate the model in the framework of non-Hermitian quantum mechanics to show the $\mathcal{PT}$ phase transition in the system.

quant-ph↗
arXiv2023

Hellmann Feynman Theorem in Non-Hermitian system

We revisit the celebrated Hellmann-Feynman theorem (HFT) in the PT invariant non-Hermitian quantum physics framework. We derive a modified version of HFT by changing the definition of inner product and explicitly show that it holds good for both PT broken, unbroken phases and even at the exceptional point of the theory. The derivation is extremely general and works for even PT non-invariant Hamiltonian. We consider several examples of discrete and continuum systems to test our results. We find that if the eigenvalue goes through a real to complex transition as a function of the Hermiticity breaking parameter, both sides of the modified HFT expression diverge at that point. If that point turns out to be an EP of the PT invariant quantum theory, then one also sees the divergence at EP. Finally, we also derive a generalized Virial theorem for non-Hermitian systems using the modified HFT, which potentially can be tested in experiments.

quant-ph↗
arXiv2023

Quantum tunneling from a new type of Unified Cantor Potential

We introduce a new type of potential system that combines the families of general Cantor (fractal system) and general Smith-Volterra-Cantor (non-fractal system) potentials. We call this system as Unified Cantor Potential (UCP) system. The UCP system of total span $L$ is characterized by scaling parameter $ρ>1$, stage $G$ and two real numbers $α$ and $β$. For $α=1$, $β=0$, the UCP system represents general Cantor potential while for $α=0$, $β=1$, this system represent general Smith-Volterra-Cantor (SVC) potential. We provide close-form expression of transmission probability from UCP system for arbitrary $α$ and $β$ by using $q$-Pochhammer symbol. Several new features of scattering are reported for this system. The transmission probability $T_{G}(k)$ shows a scaling behavior with $k$ which is derived analytically for this potential. The proposed system also opens up the possibility for further generalization of new potential systems that encompass a large class of fractal and non-fractal systems. The analytical formulation of tunneling from this system would help to study the transmission feature at breaking threshold when a system transit from fractal to non-fractal domain.

quant-ph↗
arXiv2023

Effect of Quantum Gravity on Specific Heat of Solid

All possible theories of quantum gravity suggest the existence of a minimal length. As a consequence, the usual Heisenberg Uncertainty Principle (HUP) is replaced by a more general uncertainty principle known as the Generalised Uncertainty Principle(GUP). The dynamics of all quantum mechanical system gets modified due to GUP. In this work, we consider both Einstein's and Debye's models to find the quantum gravity effect on the specific heat of solids. GUP modified specific heat in Einstein's model shows usual exponential dominance at low temperatures. Further, the modification to Debye's specific heat is calculated by considering the GUP modified dispersion relation, which becomes time dependent for elastic waves.

quant-ph↗
arXiv2023

Effect of Minimal Length on Landau Diamagnetism and de Haas-van Alphen Effect

We study Landau diamagnetism in the framework of generalised uncertainty principle(GUP). We calculate the correction to magnetisation and susceptibility by constructing the grand partition function of diamagnetic material in this framework. We explicitly show that Curie's law gets a temperature independent correction which vanishes when quantum gravity effects are neglected. We further consider the low temperature limit to find how GUP affects the de Haas-van Alphen effect.

hep-th↗
arXiv2023

Uncertainty Relation for Non-Hermitian Systems

We construct uncertainty relation for arbitrary finite dimensional PT invariant non-Hermitian quantum systems within a special inner product framework. This construction is led by good observables which are a more general class of operators. We show that the cumulative gain in the quantum Fisher information when measuring two good observables for such non-Hermitian systems is way better than their Hermitian counterpart. Minimum uncertainty states being the best candidates for this gain near the exceptional point supports the intelligent or simultaneous non-Hermitian quantum sensors.

quant-ph↗
arXiv2023

Tunneling from general Smith-Volterra-Cantor potential

We study the tunneling problem from general Smith-Volterra-Cantor (SVC) potential of finite length $L$ characterized by the scaling parameter $ρ$ and stage $G$. We show that the SVC($ρ$) potential of stage $G$ is the special case of super periodic potential (SPP) of order $G$. By using SPP formalism developed by us earlier, we provide the close form expression of tunneling probability $T_{G}(k)$ with the help of $q$-Pochhammer symbol. The profile of $T_{G}(k)$ with wave vector $k$ is found to saturate with increasing stage $G$. Very sharp transmission resonances are found to occur from this system which may find applications in the design of sharp transmission filters.

quant-ph↗
arXiv2022

Quantum tunneling from family of Cantor potentials in fractional quantum mechanics

We explore the features of non-relativistic quantum tunneling in space fractional quantum mechanics through a family of Cantor potentials. We consider two types of potentials: general Cantor and general Smith-Volterra-Cantor potential. The Cantor potential is an example of fractal potential while the Smith-Volterra-Cantor potential doesn't belong to the category of a fractal system. The present study brings for the first time, the study of quantum tunneling through fractal potential in fractional quantum mechanics. We report several new features of scattering in the domain of space fractional quantum mechanics including the emergence of energy-band like features from these systems and extremely sharp transmission features. Further the scaling relation of the scattering amplitude with wave vector $k$ is presented analytically for both types of potentials.

quant-ph↗
arXiv2022

Solutions of (1+1)-dimensional Dirac equation associated with exceptional orthogonal polynomials and the parametric symmetry

We consider $1+1$-dimensional Dirac equation with rationally extended scalar potentials corresponding to the radial oscillator, the trigonometric Scarf and the hyperbolic Poschl-Teller potentials and obtain their solution in terms of exceptional orthogonal polynomials. Further, in the case of the trigonometric Scarf and the hyperbolic Poschl-Teller cases, new family of Dirac scalar potentials are generated using the idea of parametric symmetry and their solutions are obtained in terms of conventional as well as exceptional orthogonal polynomials.

math-ph↗
arXiv2022

New Forms of BRST Symmetry on a Prototypical First-Class System

We scrutinize the many known forms of BRST symmetries, as well as some new ones, realized within a prototypical first-class system. Similarities and differences among ordinary BRST, anti-BRST, dual-BRST and anti-dual-BRST symmetries are highlighted and discussed. We identify a precise $\mathbb{Z}_4\times\mathbb{Z}_2$ discrete group of symmetries of the ghost sector, responsible for connecting the various forms of BRST transformations. Considering a Hamiltonian approach, those symmetries can be interrelated by canonical transformations among ghost variables. However, the distinguished characteristic role of the dual BRST symmetries can be fully appreciated within a gauge-fixed Lagrangian viewpoint. New forms of BRST symmetries are given, a set generalizing particular ones previously reported in the literature as well as a brand new unprecedented set. The featured gauge invariant prototypical first-class system encompasses an extensive class of physical models and sheds light on previous controversies in the current quantum field theory literature.

hep-th↗
arXiv2022

BRST symmetry as a mechanism for confinement in the inter gauge framework

The finite field dependent BRST (FFBRST) technique generally interrelates two different gauge fixed theories. Here we propose a new and unique FFBRST transformation that transforms the effective theory in Lorenz gauge, which represents perturbative or deconfined phase of QCD to massive gauge theory. The exercise serves two purposes. First is of course that it is a distinct connection where one of the theories is not gauge fixed and BRST invariant therefore, it is expected to give newer insights into FFBRST technique itself. Second is that a particular case of this transformation holds a novel physical significance of acting as general mechanism for confinement across two different gauge theories. Thus, the present task identifies a profound non-perturbative implication of the BRST symmetry in a general setting.

hep-th↗