Supersymmetry In Quantum Mechanics Research Articles

Study of quarkonium properties using SUSYQM method with baryonic chemical potential

In this paper, we employed the quasi-particle (QP) Debye mass at finite baryonic chemical potential which can be used in the medium modified heavy quark potential to solve the [Formula: see text]-dimensional Schrödinger equation. The bound state solution of the Schrödinger equation using Cornell potential been obtained by Super-Symmetry Quantum Mechanics (SUSYQM) method. The thermodynamical properties of quark matter are calculated by using baryonic chemical potential ([Formula: see text]). We found that the binding energy of quarkonia dissociates more with QP Debye mass in comparison to nonperturbative and leading order Debye mass. The medium modified form of potential (real part) has been used to study the thermodynamical properties of quark matter with different equation of states (EoSs) (i.e. pressure, energy density and speed of sound) with [Formula: see text]. The mass spectra of quarkonia have been also calculated in the [Formula: see text]-dimensional space and compared with the experimental data at [Formula: see text]. We have also calculated the dissociation temperature ([Formula: see text]) for the ground states of quarkonium using the dissociation criteria of thermal width.

Dispersive one-dimensional Majorana modes with emergent supersymmetry in one-dimensional proximitized superconductors via spatially modulated potentials and magnetic fields

In condensed matter systems, zero-dimensional or one-dimensional Majorana modes can be realized respectively as the end and edge states of one-dimensional and two-dimensional topological superconductors. In this $\textit{top-down}$ approach, $(d-1)$-dimensional Majorana modes are obtained as the boundary states of a topologically nontrivial $d$-dimensional bulk. In a $\textit{bottom-up}$ approach instead, $d$-dimensional Majorana modes in a $d$-dimensional system can be realized as the continuous limit of a periodic lattice of coupled $(d-1)$-dimensional Majorana modes. We illustrate this idea by considering one-dimensional proximitized superconductors with spatially-modulated potential or magnetic fields. The ensuing inhomogeneous topological state exhibits one-dimensional counterpropagating Majorana modes with finite dispersion, and with a Majorana gap that can be controlled by external fields. In the massless case, the Majorana modes have opposite Majorana polarizations and pseudospins, are conformally invariant, and realize centrally extended quantum mechanical supersymmetry. The supersymmetry exhibits spontaneous partial breaking. Consequently, the massless Majorana fermion can be identified as a Goldstino, i.e., the Nambu-Goldstone fermion associated with the spontaneously broken supersymmetry.

1D Majorana Goldstinos and partial supersymmetry breaking in quantum wires

Realizing Majorana modes in topological superconductors, i.e., the condensed-matter counterpart of Majorana fermions in particle physics, may lead to a major advance in the field of topologically-protected quantum computation. Here, we introduce one-dimensional, counterpropagating, and dispersive Majorana modes as bulk excitations of a periodic chain of partially-overlapping, zero-dimensional Majorana modes in proximitized nanowires via periodically-modulated fields. This system realizes centrally-extended quantum-mechanical supersymmetry with spontaneous partial supersymmetry breaking. The massless Majorana modes are the Nambu-Goldstone fermions (Goldstinos) associated with the spontaneously broken supersymmetry. Their experimental fingerprint is a dip-to-peak transition in the zero-bias conductance, which is generally not expected for Majorana modes overlapping at a finite distance. Moreover, the Majorana modes can slide along the wire by applying a rotating magnetic field, realizing a “Majorana pump”. This may suggest new braiding protocols and implementations of topological qubits.

One parameter family of rationally extended isospectral potentials

We start from a given one dimensional rationally extended shape invariant potential associated with Xm exceptional orthogonal polynomials and using the idea of supersymmetry in quantum mechanics, we obtain one continuous parameter (λ) family of rationally extended strictly isospectral potentials. We illustrate this construction by considering three well known rationally extended potentials, two with pure discrete spectrum (the extended radial oscillator and the extended Scarf-I) and one with both the discrete and the continuous spectrum (the extended generalized Pöschl–Teller) and explicitly construct the corresponding one continuous parameter family of rationally extended strictly isospectral potentials. Further, in the special case of λ=0 and −1, we obtain two new exactly solvable rationally extended potentials, namely the rationally extended Pursey and the rationally extended Abraham–Moses potentials respectively. We illustrate the whole procedure by discussing in detail the particular case of the X1 rationally extended one parameter family of potentials including the corresponding Pursey and the Abraham Moses potentials.

Solitons in lattice field theories via tight-binding supersymmetry

Reflectionless potentials play an important role in constructing exact solutions to classical dynamical systems (such as the Korteweg-de Vries equation), non-perturbative solutions of various large-N field theories (such as the Gross-Neveu model), and closely related solitonic solutions to the Bogoliubov-de Gennes equations in the theory of superconductivity. These solutions rely on the inverse scattering method, which reduces these seemingly unrelated problems to identifying reflectionless potentials of an auxiliary one-dimensional quantum scattering problem. There are several ways of constructing these potentials, one of which is quantum mechanical supersymmetry (SUSY). In this paper, motivated by recent experimental platforms, we generalize this framework to develop a theory of lattice solitons. We first briefly review the classical inverse scattering method in the continuum limit, focusing on the Korteweg-de Vries (KdV) equation and SU(N) Gross-Neveu model in the large N limit. We then generalize this methodology to lattice versions of interacting field theories. Our analysis hinges on the use of trace identities, which are relations connecting the potential of an equation of motion to the scattering data. For a discrete Schrödinger operator, such trace identities had been known as far back as Toda; however, we derive a new set of identities for the discrete Dirac operator. We then use these identities in a lattice Gross-Neveu and chiral Gross-Neveu (Nambu-Jona-Lasinio) model to show that lattice solitons correspond to reflectionless potentials associated with the discrete scattering problem. These models are of significance as they are equivalent to a mean-field theory of a lattice superconductor. To explicitly construct these solitons, we generalize supersymmetric quantum mechanics to tight-binding models. We show that a matrix transformation exists that maps a tight-binding model to an isospectral one which shares the same structure and scattering properties. The corresponding soliton solutions have both modulated hopping and onsite potential, the former of which has no analogue in the continuum limit. We explicitly compute both topological and non-topological soliton solutions as well as bound state spectra in the aforementioned models.

Non-Hermiticity-Governed Active Photonic Resonances.

Photonic resonances play an essential role in the generation and propagation of light in optical and photonic devices, as well as in light-matter interaction, including nonlinear optical responses. Previous studies in lasers and other open systems have shown exotic roles played by non-Hermiticity on modifying passive resonances, defined in the absence of optical gain and loss. Here we report a new type of resonances in non-Hermitian photonic systems that does not originate from a passive resonance, identified by analyzing a unique quantization condition in the non-Hermitian extension of the Wentzel-Kramers-Brillouin method. Termed active photonic resonances, these unique resonances are found in non-Hermitian systems with a spatially correlated complex dielectric function, which is related to supersymmetry quantum mechanics after a Wick rotation. Remarkably, such an active photonic resonance shifts continuously on the real frequency axis as optical gain increases, suggesting the possibility of a tunable on-chip laser that can span a wavelength range over 100nm.

Bound state solution of the Schrodinger equation for the Woods–Saxon potential plus coulomb interaction by Nikiforov–Uvarov and supersymmetric quantum mechanics methods

The generalized Woods–Saxon potential plus repulsive Coulomb interaction is considered in this work. The supersymmetry quantum mechanics method is used to get the energy spectrum of Schrodinger equation and also the Nikiforov–Uvarov approach is employed to solve analytically the Schrodinger equation in the framework of quantum mechanics. The potentials with centrifugal term include both exponential and radial terms, hence, the Pekeris approximation is considered to approximate the radial terms. By using the step-by-step Nikiforov–Uvarov method, the energy eigenvalue and wave function are obtained analytically. After that, the spectrum of energy is obtained by the supersymmetry quantum mechanics method. The energy eigenvalues obtained from each method are the same. Then in special cases, the results are compared with former result and a full agreement is observed. In the [Formula: see text]-state, the standard Woods–Saxon potential has no bound state, but with Coulomb repulsive interaction, it may have bound state for zero angular momentum.

Solution of d-dimension time independent cosmic string using supersymmetry quantum mechanics method for rosen morse, scarf II and scarf I non-central potentials

Solution of d-dimension time independent cosmic string for rosen morse, scarf II and scarf I non-central potentials was solved using supersymmetry quantum mechanics method. Variable separation method was applied to the d-dimension time independent cosmic string equation, and then the d-dimension time independent cosmic string equation was reduced into d one dimensional schrodinger equation. Supersymmetry quantum mechanics method was used to analyze radial wave function, angular wave functions, and energy level. Energy levels were higher by the presence of rosen morse, scarf II and scarf I potentials parameters. The increased value of nr , n θ 2 and n θ 4 caused the increase in energy levels, the decrease of cosmic string α parameter, caused the decrease of energy levels and the increasing of number n θ 1 and n θ 3 caused the decrease of energy levels.

Generalized Uncertainty Principle in Cosmology with Supersymmetry Quantum Mechanics

In this paper, we exactly solve the Wheeler-DeWitt equation in the presence of a conformally coupled scalar field and in the context of the generalized uncertainty principle (GUP). The GUP-corrected Wheeler—De Witt equation in momentum space leads us to introduce factorization method. This method helps us to obtain the exact solution for the corresponding system. So, here we factorize the second-order equation in terms of first-order operators. These first-order operators help us to arrange partner potential and superpotential. Also, we achieve the general quantum stats and energy spectrum for GUP with conformally coupled scalar field system. Also, we show that the stability of a system with the energy spectrum with some conditions.

Symmetry classes, many-body zero modes, and supersymmetry in the complex Sachdev-Ye-Kitaev model

The complex Sachdev-Ye-Kitaev (cSYK) model is a charge-conserving model of randomly interacting fermions. The interaction term can be chosen such that the model exhibits chiral symmetry. Then, depending on the charge sector and the number of interacting fermions, level spacing statistics suggests a fourfold categorization of the model into the three Wigner-Dyson symmetry classes. In this work, inspired by previous findings for the Majorana Sachdev-Ye-Kitaev model, we embed the symmetry classes of the cSYK model in the Altland-Zirnbauer framework and identify consequences of chiral symmetry originating from correlations across different charge sectors. In particular, we show that for an odd number of fermions, the model hosts exact many-body zero modes that can be combined into a generalized fermion that does not affect the system's energy. This fermion directly leads to quantum-mechanical supersymmetry that, unlike explicitly supersymmetric cSYK constructions, does not require fine-tuned couplings, but only chiral symmetry. Signatures of the generalized fermion, and thus supersymmetry, include the long-time plateau in time-dependent correlation functions of fermion-parity-odd observables: The plateau may take nonzero value only for certain combinations of the fermion structure of the observable and the system's symmetry class. We illustrate our findings through exact diagonalization simulations of the system's dynamics.

Supersymmetry in the time domain and its applications in optics

Supersymmetry is a conjectured symmetry between bosons and fermions aiming at solving fundamental questions in string and quantum field theory. Its subsequent application to quantum mechanics led to a ground-breaking analysis and design machinery, later fruitfully extrapolated to photonics. In all cases, the algebraic transformations of quantum-mechanical supersymmetry were conceived in the space realm. Here, we demonstrate that Maxwell’s equations, as well as the acoustic and elastic wave equations, also possess an underlying supersymmetry in the time domain. We explore the consequences of this property in the field of optics, obtaining a simple analytic relation between the scattering coefficients of numerous time-varying systems, and uncovering a wide class of reflectionless, three dimensional, all-dielectric, isotropic, omnidirectional, polarisation-independent, non-complex media. Temporal supersymmetry is also shown to arise in dispersive media supporting temporal bound states, which allows engineering their momentum spectra and dispersive properties. These unprecedented features may enable the creation of novel reconfigurable devices, including invisible materials, frequency shifters, isolators, and pulse-shape transformers.

Fractional Supersymmetric Hermite Polynomials

We provide a realization of fractional supersymmetry quantum mechanics of order r, where the Hamiltonian and the supercharges involve the fractional Dunkl transform as a Klein type operator. We construct several classes of functions satisfying certain orthogonality relations. These functions can be expressed in terms of the associated Laguerre orthogonal polynomials and have shown that their zeros are the eigenvalues of the Hermitian supercharge. We call them the supersymmetric generalized Hermite polynomials.

Extended supersymmetry with central charges in Dirac action with curved extra dimensions

We discuss a new realization of $\mathcal{N}$-extended quantum-mechanical supersymmetry (QM SUSY) with central charges hidden in the four-dimensional (4D) mass spectrum of higher dimensional Dirac action with curved extra dimensions. We show that this $\mathcal{N}$-extended QM SUSY results from symmetries in extra dimensions, and the supermultiplets in this supersymmetry algebra correspond to the Bogomol'nyi--Prasad--Sommerfield states. Furthermore, we examine the model of the $S^2$ extra dimension with a magnetic monopole background and confirm that the $\mathcal{N}$-extended QM SUSY explains the degeneracy of the 4D mass spectrum.

Solution of time independent cosmic string for harmonic oscillator plus poschl teller non-central potentials using supersymmetry quantum mechanics methods

Time independent cosmic string equation for Harmonic Oscillator and Poschl Teller non-central potential have been solved using supersymmetry quantum mechanics method. The time independent cosmic string with non-central potential was reduced into three one dimensional Schrodinger equations by using variable separation method. The energy levels and radial wave function were analyzed using supersymmetry quantum mechanics method from radial part of Schrodinger like equation. The angular wave function was obtained from angular part of Schrodinger equation. The effect of the non-central potential caused energy levels increasing. Energy spectra for λ = 0 and κ = 0 have smaller value than energy levels for λ ≠ 0 and κ ≠ 0. The increased value of parameter λ and κ caused increased in energy levels. However, the decreased of cosmic string parameter α, caused the increased of energy levels.

Extended supersymmetry with central charges in higher dimensional Dirac action

A new realization of extended quantum-mechanical supersymmetry (QM SUSY) with central extension is investigated. We first show that two different sets of $d+2$ ($d+1$) supercharges for $d=$ even (odd), each of which satisfies an $\mathcal{N}=d+2$ ($d+1$) extended QM SUSY algebra without central extension, are hidden in the four-dimensional (4D) mass spectrum of the $(4+d)$-dimensional Dirac action. We then find that the whole set of the supercharges forms an $\mathcal{N}=2d+4$ ($2d+2$) extended QM SUSY algebra with central charges for $d=$ even (odd). The representation of the supersymmetry algebra is shown to be $1/2$-Bogomol'nyi--Prasad--Sommerfield states that correspond to a short representation for the supersymmetry algebra with central extension. We explicitly examine the 4D mass spectrum of the models with the hyperrectangle and the torus extra dimensions, and discuss their supersymmetric structures.

Extended supersymmetry in Dirac action with extra dimensions

We investigate a new realization of extended quantum-mechanical supersymmetry. We first show that an quantum-mechanical supersymmetry is hidden in the four-dimensional (4D) spectrum of the Kaluza–Klein decomposition for the higher dimensional Dirac field, that is, Kaluza–Klein mode functions of 4D right-handed spinors and 4D left-handed ones form supermultiplets. In addition to supersymmetry, we discover that an -extended supersymmetry ( for d = even (odd) extra dimensions) is further hidden in the 4D spectrum. The extended symmetry can explain additional degeneracy of the spectrum. Furthermore, we show that a superpotential can be introduced into the -extended supercharges and clarify the condition to preserve the supersymmetry. The partial breaking of the supersymmetry is also demonstrated.

Higher-order supersymmetric partners of generalized quantum nonlinear oscillators

We apply an extended version of the quantum-mechanical supersymmetry (SUSY) formalism to a generalized quantum nonlinear oscillator model. Considering both the standard and the inverted version of quantum nonlinear oscillator, we use SUSY transformations of various orders to generate partner systems, focusing on regular potentials and spectral modifications.

Bound and Scattering State of Position Dependent Mass Klein–Gordon Equation with Hulthen Plus Deformed-Type Hyperbolic Potential

In this article we use supersymmetry quantum mechanics and factorization methods to study the bound and scattering state of Klein–Gordon equation with deformed Hulthen plus deformed hyperbolical potential for arbitrary state in D-dimensions. The analytic relativistic bound state eigenvalues and the scattering phase factor are found in closed form. We report on the numerical results for the bound state energy in D-dimensions.

Extended supersymmetry and hidden symmetries in one-dimensional matrix quantum mechanics

We study properties of nonlinear supersymmetry algebras realized in the one-dimensional quantum mechanics of matrix systems. Supercharges of these algebras are differential operators of a finite order in derivatives. In special cases, there exist independent supercharges realizing an (extended) supersymmetry of the same super-Hamiltonian. The extended supersymmetry generates hidden symmetries of the super-Hamiltonian. Such symmetries have been found in models with (2×2)-matrix potentials.

Commutative vs. Noncommutative Space Statistical Properties of Two-Dimensional Harmonic Oscillator in Magnetic Field

The Schrodinger equation and its solutions are still a challenging subject in physics. Just as many other areas of science, where we have to solve a differential equation to obtain the required information, we have to first solve the building block of nonrelativistic quantum mechanics. In doing so, we have to use numerical and analytical techniques depending on the structure of the equation. In particular, the analytical approaches are attractive as they provide a deeper and more touchable insight into the physics of the problem [1–6]. Cooper et al. reviewed the theoretical formulation of supersymmetry quantum mechanics and discussed its applications in dealing with both relativistic and nonrelativistic equations of quantum mechanics [7]. Ciftci et al. used the asymptotic iteration method for finding solutions of the Schrodinger equation [8]. Slater considered a simplification of the Hartree–Fock method to analyze the related problems [9]. Stevenson applied the optimized perturbation theory to the field [10]. Dong et al. proposed the quasi-exact solutions of the Schrodinger equation via the ansatz technique which is a quasi-exact approach [11]. Another frequently used tool is the Nikiforov–Uvarov (NU) technique which transforms classes of equations of mathematical physics into hypergeometric form. A reason of recent renewed interests in the wave equations of quantum mechanics is the implications of fundamental theories such as string theory. To be more precise, the noncommutative (NC) formulations of quantum mechanics and the proposition of the so-called minimal length, has motivated many recent studies on the quantum equations. The NC formulation, which is the focus of the present work, originates from fundamental theories